four lectures on generalized nash equilibrium problems in ...€¦ · nash equilibrium problems in...

Four Lectures on GeneralizedNash Equilibrium Problems in Finite Dimensions

Jong-Shi Pang

Daniel J Epstein Department of Industrial and Systems Engineering

University of Southern California

presented at

Autumn School on Quasi-Variational Inequalities

Theory Algorithms and Applications

Universitat Wurzburg Germany

Monday September 23 through Wednesday September 25 2019

Foreword Brief history of generalized Nash games

Lecture I Formulations and a recent model

Lecture II Existence results with and without convexity

Lecture III Exact penalization theory

Lecture IV Best-response algorithms

2

A brief historical account

Gerard Debreu A social equilibrium existence theorem Proceedings of theNational Academy of Sciences 38(10) 886ndash893 1952

Kenneth J Arrow and Gerard Debreu Existence of an equilibrium fora competitive economy Econometrica Journal of the Econometric Society22(3) 265ndash290 1954

Hukukane Nikaido and Kazuo Isoda Note on noncooperative convexgames Pacific Journal of Mathematics 5(Suppl 1) 807-815 1955

J Ben Rosen Existence and uniqueness of equilibrium points for concaven-person games Econometrica Journal of the Econometric Society 33(3)520ndash534 1965

Alain Bensoussan Points de Nash dans le cas de fontionnelles quadratiques

et jeux differentiels lineaires a N personnes SIAM Journal on Control 12(3)

460ndash499 1974

3

A brief historical account (cont)Patrick T Harker Generalized Nash games and quasi-variationalinequalities European Journal of Operational Research 54(1) 81ndash94 1991

Jong-Shi Pang and Masao Fukushima Quasi-variational inequalitiesgeneralized Nash equilibria and multi-leader-follower games ComputationalManagement Science 2 21-56 2005 [Erratum same journal 6 373-3752009]

Francisco Facchinei and Christian Kanzow Generalized Nashequilibrium problems Annals of Operations Research 175(1)177ndash211 2007

Francisco Facchinei and Jong-Shi Pang Nash equilibria the variationalapproach Chapter 12 in Daniel P Palomar and Yonica C Eldar editorsConvex optimization in signal processing and communications CambridgeUniversity Press pages 443ndash493 2010

Francisco Facchinei Computation of generalized Nash equilibria Recent

advances Part II in Roberto Cominetti Francisco Facchinei and Jean-Bernard

Lasserre Editors and authors Modern Optimization Modeling Techniques

Birkhauser Springer Basel pp 133ndash184 2012

4

Some recent references

Extensive papers by our host Christian Kanzowhttpswwwmathematikuni-wuerzburgdeoptimizationteam

kanzow-christian

Q Ba and JS Pang Exact penalization of generalized Nashequilibrium problems Operations Research (accepted August 2019) inprint

DA Schiro JS Pang and UV Shanbhag On the solution ofaffine generalized Nash equilibrium problems with shared constraints byLemkersquos method Mathematical Programming Series A 142(1ndash2) 1ndash46(2013)

5

Modelling

From single decision maker minusrarr multiple agents

From optimization minusrarr equilibrium

From cooperation minusrarr non-cooperation

From centralized decision making minusrarr distributed responses

Mathematics

From WeierstrassCauchy minusrarr BrouwerBanach

From smooth calculus minusrarr variational analysis

From coordinated descent minusrarr asynchronous computation

From potential function minusrarr non-expansion of mappings

6


930 ndash 1030 AM Monday September 23 2019

7

The Abstract Generalized Nash Equilibrium Problem (GNEP)

N decision makers each (labeled ν = 1 middot middot middot N) with

bull a moving strategy set Ξν(xminusν) sube Rnν and

bull a cost function θν(bull xminusν) Rnν rarr R

both dependent on the rivalsrsquo strategy tuple

xminusν ≜983059xν

prime983060

ν prime ∕=νisin Rminusν ≜

983132

ν prime ∕=ν

Rnν prime

Anticipating rivalsrsquo strategy xminusν player ν solves

minimizexνisinΞν(xminusν)

θν(xν xminusν)

A Nash equilibrium (NE) is a strategy tuple xlowast ≜ (xlowast ν)Nν=1 such that

xlowast ν isin argminxνisinΞν(xlowastminusν)

θν(xν xlowastminusν) forall ν = 1 middot middot middot N

In words no player can improve individual objective by unilaterallydeviating from an equilibrium strategy

8

Notation

bull x ≜ (xν)Nν=1

bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)

bull FIXΞ ≜ x x isin Ξ(x)

bull the GNEP (Ξ θ)

Necessarily a NE xlowast must belong to FIXΞ the ldquofeasible setrdquo of theGNEP

9

The GNEP in actionbull basic case Ξν(xminusν) ≜ Xν is independent of xminusν for all νbull finitely representable with convexity and constraint qualifications

Ξν(xminusν) ≜

983099983103

983101xν isin Rnν hν(xν) le 0

private constraints

gν(xν xminusν) le 0

coupled constraints

983100983104

983102

where hν Rnν rarr Rℓν is C1 and for each i = 1 middot middot middot ℓν the componentfunction hν

i is convex and gν Rn rarr Rmν is C1 and for each i = 1 middot middot middot mν

and every xminusν the component function gνi (bull xminusν) is convex

bull joint convexity graph of Ξν ≜ Ctimes 983141X where 983141X ≜N983132

ν=1

Xν and C sub Rn

where n ≜N983131

ν=1

nν are closed and convex

bull roles of multipliers these can be different for same constraints in differentplayersrsquo problems

bull common multipliers for C ≜ x isin Rn g(x) le 0 where g Rn rarr Rm is C1

each component function gi is convex and Lagrange multipliers for the

constraint g(x) le 0 is the same for all players leading to variational equilibria10

Quasi variational inequality (QVI) formulation

bull If θν(bull xminusν) is convex and Ξν is convex-valued then xlowast ≜ (xlowast ν)Nν=1 isa NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x

lowast) such that

(xν minus xlowast ν )⊤alowast ν ge 0 forallxν isin Ξ ν(xlowastminusν)

or equivalently existalowast isin Θ(xlowast) ≜N983132

ν=1

partxνθν(xlowast) such that

(xminus xlowast )⊤alowast =

N983131

ν=1

(xν minus xlowast ν )Talowast ν ge 0 forallx ≜ (xlowast ν)Nν=1 isin Ξ(xlowast)

bull If in addition θν(bull xminusν) is differentiable then Θ(x) = (nablaxνθν(x) )Nν=1

is a single-valued map

bull If further Ξ ν(xlowastminusν) = Xν then get the VI (Θ 983141X)

(xminus xlowast )⊤Θ(x) ge 0 forallx isin 983141X

11

Complementarity formulationbull Each θν(bull xminusν) is differentiable and

Ξν(xminusν) = xν isin Rnν+ gν(xν xminusν) le 0

Karush-Kuhn-Tucker conditions (KKT) of the GNEP983099983105983103

983105983101

0 le xν perp nablaxνθν(x) +

mν983131

i=1

nablaxνgνi (x)λνi ge 0

0 le λν perp gν(x) le 0

983100983105983104

983105983102ν = 1 middot middot middot N

yielding the nonlinear complementarity problem (NCP) formulationunder suitable constraint qualifications

0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107

983075nablaxνθν(x) +

mν983131

i=1

nablaxνgνi (x)λνi

983076N

ν=1

minus ( gν(x) )Nν=1

983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12

One recent model in transportation e-hailing

Contributions

bull realistic modeling of a topical research problem

bull novel approach for proof of solution existence

bull awaiting design of convergent algorithm

bull opportunity in model refinements and abstraction

J Ban M Dessouky and JS Pang A general equilibrium modelfor transportation systems with e-hailing services and flow congestionTransportation Research Series B (2019) in print

13

Model background Passengers are increasingly using e-hailing as ameans to request transportation services Goal is to obtain insights intohow these emergent services impact traffic congestion and travelersrsquomode choices

Formulation as a GNEMOPECMulti-VI

bull e-HSP companiesrsquo choices in making decisions such as where to pickup the next customer to maximize the profits under given fare structures

bull travelersrsquo choices in making driving decisions to be either a solo driveror an e-HSP customer to minimize individual disutility

bull network equilibrium to capture traffic congestion as a result ofeveryonersquos travel behavior that is dictated by Wardroprsquos route choiceprinciple

bull market clearance conditions to define customersrsquo waiting cost andconstraints ensuring that the e-HSP OD demands are served

14

Network set-up

N set of nodes in the network

A set of links in the network subset of N timesNK set of OD pairs subset of N timesNO set of origins subset of N O = Ok k isin KD set of destinations subset of N D = Dk k isin K

besides being the destinations of the OD pairs where

customers are dropped off these are also the locations

where the e-HSP drivers initiate their next trip to pick up

other customers

Ok Dk origin and destination (sink) respectively of OD pair k isin KM labels of the e-HSPs M ≜ 1 middot middot middot MM+ union of the solo drivercustomer label (0) and the

e-HSP labels

15

Overall model is to determine

bull e-HSP vehicle allocations983153zmjk m isin M j isin D k isin K

983154for each type

m destination j and OD pair k

bull travel demands Qmk m isin M k isin K for each e-HSP type m and

OD pair k

bull travel times tij (i j) isin N timesN of shortest path from node i to j

bull vehicular flows hp p isin P on paths in network

16

Interaction of modules in model

17

The e-HSP module

The per-customer (or per-pickup) profit of an e-HSPm trip for m isin Mat location j who plans to serve OD pair k can be modeled as

Rmjk ≜ 983141Rm

jk minus βm3 983141wm

k983167983166983165983168e-HSPm waiting timeincurring loss of revenue

where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk

)983167 983166983165 983168travel time based cost

minus βm2 ( djOk

+ dOkDk)

983167 983166983165 983168travel distance based cost

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168time based revenue

+ αm2 dOkDk983167 983166983165 983168

distance based revenue

Anticipating Rmjk as a parameter e-HSPm company decides on the

allocation zmjk of vehicles from location j to serve OD pair k by solving

18

e-HSPrsquos choice module of profit maximization for m isin M

maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk

983167 983166983165 983168average trip profit

minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096

983167 983166983165 983168cost due to waiting

subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime

983167 983166983165 983168available e-HSP vehicles for service

for all j isin D

983131

jisinDzmjk ge Qm

k

983167 983166983165 983168OD demands served by e-HSPm

for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk

983167 983166983165 983168trip hours of vehiclesen route to service calls

+983131

kisinKQm

k tOkDk

983167 983166983165 983168trip hours of vehiclesserving travel demands

le Nm983167983166983165983168

e-HSPm vehicleavailability

19

The passenger module

An e-HSPm customerrsquos disutility V mk for m isin M

FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare

+ γm1 tOkDk983167 983166983165 983168in-vehicle baseddisutility

+wmk (disutility due to waiting for e-HSPsrsquo pick up)

For a solo driver the disutility can be expressed as

V 0k = γ01 tOkDk983167 983166983165 983168

travel timebased disutility

+ β02 dOkDk983167 983166983165 983168

distancebased disutility

Anticipating V mk and e-HSP vehicle allocations zmkj passengers decide on

travel modes (solo or e-hailing) by solving

20

Passenger mode choice of disutility minimization

minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints

for all ( km ) isin K timesM

983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K

where M+ = M cup solo mode

21

Network congestion module

Wardroprsquos principle of route choice for the determination of travel timetij from node i to j and traffic flow hp on path p joining these two nodes

0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131

(kℓ)isinKtimesKδeminusHSPijkℓ

983131

misinMzmiℓ

983094

983096 ge 0

for all (i j) isin N timesN

0 le hp perp Cp(h)minus tij ge 0 for all p isin Pij where

δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070

δeminusHSPi primej primekℓ ≜

9830691 if i prime = Dk and j prime = Oℓ

0 otherwise

983070

22

Customer waiting disutility

wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk

983167 983166983165 983168average travel time to origin ofOD-pair k from all locations

for all m isin M

Remark need a complementarity maneuver to rigorously handle thedenominator to avoid division by zero

End model is a highly complex generalized Nash equilibrium problem withside conditions for which state-of-the-art theory and algorithms are notdirectly applicable

A penalty approach is employed for proving solution existence23



24

Recall QVI formulation convex player problems

If θν(bull xminusν) is convex and Ξν is convex-valued thenxlowast ≜ (xlowast ν)Nν=1 isin FIXΞ is a NE if and only if for every ν = 1 middot middot middot N exist alowast ν isin partxνθν(x

lowast) such that



ν=1



N983131

ν=1


where Ξ(x) ≜N983132

ν=1

Ξν(xminusν) and FIXΞ ≜ x x isin Ξ(x)

25

Existence results for convex player problemsIcchiishi 1983 with compactness A NE exists ifbull each Ξν Rminusν rarr Rnν is a continuous convex-valued multifunction ondom(Ξν)bull each θν Rn rarr R is continuous and θν(bull xminusν) is convexforallxminusν isin dom(Ξν)bull exist compact convex set empty ∕= 983141K sube Rn such that Ξ(y) sube 983141K for all y isin 983141K

Proof Apply Kakutani fixed-point theorem to the self-map

y isin 983141K 983041rarrN983132

ν=1

argminxνisinΞ ν(yminusν)

θν(xν yminusν) sube 983141K

Assumptions ensure applicability of this theorem

Applicable to the jointly convex compact case with

graph Ξ ν = C983167983166983165983168shared constraints

times 983141X983167983166983165983168private constraints

for all ν

26

Facchinei and Pang 2009 no compactness Instead of the lastassumptionbull exist a bounded open set Ω with Ω sube dom(Ξ) a vector

xref ≜983059xrefν

983060N

ν=1isin Ω and a continuous function s Ω rarr Ω such that

ndash (continuous selection) s(y) ≜ (sν(y))Nν=1 isin Ξ(y) for all y isin Ωndash the open line segment joining xref and s(y) is contained in Ω for ally isin Ω

ndash (an abstract form of weak coercivity) Llt cap partΩ = empty where

Llt ≜983083

y ≜ ( yν )Nν=1 isin FIXΞ for each ν such that yν ∕= sν(y)

( yν minus sν(y) )Tuν lt 0 for some uν isin partxνθν(y)

983084

bull Facchinei and Pang 2003 A NE exists if the solutions (if they exist) ofthe NCP 0 le z perp F(z) + τz ge 0 over all scalars τ gt 0 are bounded

27



983060N




Llt ≜983083



983084


27

Nonconvex games with shared constraintsbull θν(bull xminusν) nonconvex and

bull Ξν(xminusν) ≜

983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101

xν isin Xν hν(xν) le 0983167 983166983165 983168private constraintsdenoted X ν

G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102

where G Rn rarr Rp and 983141G Rn rarr R983141p Denote H(x) ≜ minus (hν(xν) )Nν=1

Given prices (π 983141π) and anticipating xminusν player νrsquos problem

minimizexν isinX ν

983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1

πj Gj(xν xminusν) +

983141p983131

j=1

983141πj 983141Gj(xν xminusν)

983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28

The game G(Θ HG 983141G)

A Nash (variational) equilibrium (NE) is a tuple (xlowastπlowast 983141π lowast) withxlowast ≜ (xlowastν )Nν=1 such that

xlowastν isin argminxν isinX ν

Lν(xν xlowastminusν πlowast 983141π lowast) (1)

for every ν = 1 middot middot middot N and

0 le πlowast perp G(xlowast) le 0 and 0 le 983141πlowast perp 983141G(xlowast) le 0

Multipliers (πlowast 983141π lowast) of the common shared constraints are the same forall players they are optimal solutions of the market playerrsquos problemwho anticipating xlowast solves

(πlowast 983141π lowast) isin minimize(π983141π)ge 0

π⊤G(xlowast) + 983141π⊤ 983141G(xlowast)

A local Nash equilibrium (LNE) is a tuple (xlowastπlowast 983141π lowast ) for which anopen neighborhood N ν of xlowastν exists such that (1) is replaced by

xlowastν isin argminxν isinX ν capN ν

Lν(xν xlowastminusν πlowast 983141π lowast)

29













29

A quasi-Nash equilibrium (QNE) is a a tuple (xlowastπlowast λlowast ) solving thegamersquos variational formulation ie the (linearly constrained) VI (KΦ)

where K ≜ Ξ times Rp+ times

N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1

Xν | 983141G(x) le 0983167 983166983165 983168coupled constraints

983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)

983076nonconvexconstraints

983092

983110983110983110983110983110983110983108

30

Roadmap of the Analysis

For a QNE

bull Formulate VI as a system of (nonsmooth) equations and use degreetheory to show existence and uniqueness of a QNE

mdash need a Slater condition plus a copositivity assumption

bull Invoke second-order sufficiency condition in NLP to yield a LNE from aQNE

For a NE (model remains nonconvex)

bull Derive sufficient conditions for best-response map to be single-valuedyielding existence of a NE

mdash rely on bounds of multipliers and positive definiteness of the Hessianmatrices of the Lagrangian

bull Use a distributed Jacobi or Gauss-Seidel iterative scheme to compute aNE whose convergence is based on a contraction argument

31

A Review of VI TheoryLet K be a closed convex subset of Rn and F K rarr Rn be a continuousmapbull The solution set of the VI defined by this pair (KF ) is denotedSOL(KF )bull The critical cone at xlowast isin SOL(KF ) is

C(KF xlowast) ≜ T (Kxlowast) cap F (xlowast)perp = T (Kxlowast) cap (minusF (xlowast) )lowast

where T (Kxlowast) is the tangent conebull The normal map of the VI (KF ) is

F norK (z) ≜ F (ΠK(z)) + z minusΠK(z) z isin Rn

where ΠK is the Euclidean projector onto Kbull F nor

K (z) = 0 implies x ≜ ΠK(z) isin SOL(KF ) converselyx isin SOL(KF ) implies F nor

K (z) = 0 where z ≜ xminus F (x)bull A matrix M isin Rntimesn is copositive on a cone C sube Rn if xTMx ge 0 forall x isin C strictly copositive if xTMx ge 0 for all x isin C 0

32

Solution Existence and Uniqueness of VIs(a) SOL(KF ) ∕= empty if exist a vector xref isin K such that the set

Llt ≜ x isin K | (xminus xref )TF (x) lt 0 is bounded

(b) SOL(KF ) ∕= empty and bounded if exist a vector xref isin K such that the set

Lle ≜ x isin K | (xminus xref )TF (x) le 0 is bounded

(c) SOL(KF ) is a singleton for a polyhedral K if(i) the set Lle is bounded and

(ii) for every xlowast isin SOL(KF ) JF (xlowast) is copositive on C(KF xlowast)

and

C(KF xlowast) ni v perp JF (xlowast)v isin C(KF xlowast)lowast rArr v = 0

ie if (C(KF xlowast) JF (xlowast)) is a R0 pair

Proof by applying degree theory to the normal map F norK (z)

33

Existence of QNE

The game G(Θ HG 983141G) has a QNE if exist a tuple xref ≜983059xrefν

983060N

ν=1isin Ξ

such that

(Slater condition of nonconvex constraints) Ψ(xref) ≜983061

H(xref)G(xref)

983062lt 0

(copositivity of private constraints) the Hessian matrix nabla2hνi (xν) is

copositive on T (Xν xrefν) for every xν isin Xν and all i = 1 middot middot middot ℓν andevery ν = 1 middot middot middot N

(copositivity of coupled constraints) so are nabla2Gj(x) on T (Ξxref) for allj = 1 middot middot middot p

(weak coercivity of playersrsquo objectives) the set983153x isin Ξ | (xminus xref)TΘ(x) lt 0

983154is bounded (possibly empty)

34

Uniqueness of QNE

For uniqueness examine the pair (JΦ(xπλ) C(KΦ (xπλ)) First

JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0

983078 where χ ≜ (πλ ) and

A(xχ) ≜ JΘ(x) +

p983131

j=1

πj nabla2Gj(x) + blkdiag

983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N

ν=1983167 983166983165 983168Jacobian of VI Lagrangian

The critical cone of the VI (KΦ) at y ≜ (xπλ) isin SOL(KΦ)

C(KΦ y) =

983141C(xχ) times983045T (Rp

+π) cap G(x)perp983046times

N983132

ν=1

983147T (Rℓν

+ λν) cap hν(x)perp983148

where 983141C(xχ) ≜ T (Ξx) cap (Θ(x) + JΨ(x)χ )perp35

Thus JΦ(xχ) is copositive on C(KΦ y) if and only if A(xχ) iscopositive on 983141C(xχ)

The game G(Θ HG 983141G) has a unique QNE if in addition to theconditions for existence

bull the set983153x isin Ξ | (xminus xref)TΘ(x) le 0

983154is bounded

bull for every QNE (xlowastπlowastλlowast) the matrix A(xlowastπlowastλlowast) is strictlycopositive on 983141C(xlowastχlowast)

bull a strict Mangasarian-Fromovitz constraint qualification holds thatensures the uniqueness of (πlowastλlowast)

36




983154is bounded



36

When is a QNE a LNE

Answer Under a second-order sufficiency condition as in NLP

Let L(2)ν (xπλν) ≜ nabla2

xνθν(x) +

p983131

j=1

πj nabla2xνGj(x) +

ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that

A(xχ) = Diag983059L(2)ν (xπλν)

983060N

ν=1983167 983166983165 983168diagonal blocks

+ Off-diagonal blocks of JΘ(x)

The critical cone of player qrsquos optimization problem

C ν(xlowastπlowast 983141π lowast) ≜

T (X ν xlowastν) cap

983093

983095nablaxνθν(xlowast) +

p983131

j=1

πlowastj nablaxνGj(x

lowast) +

983141p983131

j=1

983141π lowastj nablaxν 983141Gj(x

lowast)

983094

983096perp

bull Let (xlowastπlowastλlowast) isin SOL(KΦ) If exist 983141π lowast such that for each

ν = 1 middot middot middot N L(2)ν (xlowastπlowastλlowastν) is strictly copositive on C ν(xlowastπlowast 983141π lowast)

then (xlowastπlowast 983141π lowast) is a LNE

37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp



then (xlowastπlowast 983141π lowast) is a LNE37

Existence and Uniqueness of NERecall 2 challengesbull nonconvexity of playersrsquo optimization problems

minimizexν isinXν and h ν(xν)le 0

Lν(xπ 983141π) (2)

bull no explicit bounds on prices π and 983141πminimize

983141πge 0983141π T 983141G(x) and minimize

πge 0πTG(x)

Remediesbull impose uniqueness (guaranteeing continuity) of playersrsquo best responsesfor given rivalsrsquo strategies xminusν and prices (π 983141π) how to justify suchuniquenessbull for t gt 0 consider a price-truncated game Gt(Θ HG 983141G ) with

minimize983141π isin 983141St

983141π T 983141G(x) and minimizeπ isinSt

πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104

983102 and similarly for 983141St

38

The price-truncated game Gt(Θ HG 983141G ) for fixed t gt 0

Write 983141X ≜N983132

ν=1

Xν and let Λνt ≜

983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜

max(micro983141micro)isinSttimes983141St

983099983103

983101maxxisin 983141X

983055983055983055983055983055983055

Q983131

q=1

(xrefν minus xν )TnablaxνLν(x micro 983141micro)

983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min

1leileℓν(minushνi (x

refν) )

983062

Suppose 983141X is bounded and Slater and copositivity hold Let t gt 0bull For each pair (π 983141π) isin St times 983141St every KKT multiplier λq belongs to Λtbull The game Gt(Θ HG 983141G ) has a NE if in addition(a) a CQ holds at every optimal solution of playersrsquo optimization problem(2)

(b) the matrix L(2)ν (x microλν) is positive definite for all

(x microλν) isin 983141X times St times Λνt

Proof by Brouwer fixed-point theorem applied to best-response map andprice regularization

39

Bounding the prices and removing truncation

There exists a scalar ξlowast such that every KKT tuple (xt 983141π tπtλt) of thegame Gt(Θ HG 983141G) satisfies

p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast

If t gt ξlowast then the price-truncation constraints983141p983131

j=1

983141πj le t and

p983131

j=1

πj le t are not binding thus recovering the

price complementarity condition

40

Existence and Uniqueness of NE

(A) Assume boundedness of 983141X Slater and copositivity and that exist ascalar t gt ξlowast satisfying for all ν = 1 middot middot middot N

bull a CQ holds at every optimal solution of (2) for every(xminusν π 983141π) isin Xminusν times St times 983141St

bull the matrix L(2)ν (xπλq) is positive definite for all

(xπλν) isin 983141X times S t times Λνt

Then the game G(Θ HG 983141G ) has a NE (xlowastπlowast 983141π lowast) with πlowast isin St and983141π lowast isin 983141St

(B) If the matrix A(xπλ) is positive definite for all

(xπλ) isin 983141X times St timesN983132

ν=1

Λνt then the x-component of a NE of the game

G(Θ HG 983141G ) is unique

41

A special multi-leader-follower gameSetting There are Q dominant players Associated with dominant player q is agroup Fq of Nash followers who react non-cooperatively to hisher strategy zqAssume that the Nash equilibria of the followers in Fq denoted yq isin Rℓq arecharacterized by the linear complementarity problem parameterized by zq

0 le yq perp wq ≜ rq +Nqzq +Mqyq ge 0

for some constant vector rq and matrices Nq and Mq with the latter being asquare matrix of order ℓq

Dominant player qrsquos optimization problem is

minimizezq yq wq

θq(zq yq zminusq)

subject to ( zq yq ) isin Z q ≜ (zq yq) | Aqzq +Bqyq le bq

0 le yq wq perp rq +Nqzq +Mqyq ge 0

The overall non-cooperative game among the selfish dominant players is anequilibrium program with equilibrium constraints

Let Xq ≜ ( zq yq ) isin Z q | yq ge 0 and rq +Nqzq +Mqyq ge 0 42

Existence of ε-QNEFor every ε gt 0 consider the relaxed problem

minimizezq yq wq

θq(zq yq zminusq)


0 le yq wq ≜ rq +Nqzq +Mqyq ge 0

and yqi wqi le ε i = 1 middot middot middot ℓq

Suppose that for every q = 1 middot middot middot Q θq(bull bull xminusq) is continuously differentiableon an open set containing Xq and zrefq exists such that Aqzrefq le bq andrq +Nqzrefq = 0 Assume further that the set983099983105983105983105983105983105983103

983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1

983045( zq minus zrefq )Tnablazqθq(z

q yq zminusq) + ( yq )Tnablayqθq(zq yq zminusq)

983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102

is bounded (possibly empty) Then for every ε gt 0 an ε-QNE to the

multi-leader-follower game exists43

Lecture III Exact penalization via error boundsand constraint qualifications

1100 ndash noon Tuesday September 24 2019

44

Recall the GNEP which we denote G(CX θ) where player ν solves

minimizexν isinC ν capXν(xminusν)

θν(xν xminusν)

where C ν and Xν(xminusν) are both closed and convex in Rnν Letrν(bull xminusν) be a C ν-residual function of the latter set ie rν(bull xminusν) isnonnegative on C ν and

983045xν isin C ν and rν(x

ν xminusν) = 0983046

hArr xν isin C ν cap Xν(xminusν)

Let θνρX(xν xminusν) = θν(xν xminusν) + ρ rν(x

ν xminusν) Consider the Nashequilibrium problem which we denote NEP (C θρX) where player νsolves

minimizexν isinC ν

θνρX(xν xminusν)

(Partial) exact penalization is concerned with the question Does exist afinite ρ gt 0 such that for all ρ ge ρ

G(CX θ) hArr NEP(C θρX)

45

Review (partial) exact penalization of optimization problems

ConsiderminimizexisinW capS

f(x)

where W and S are two closed sets of constraints with S consideredmore complex than W Assume that S cap W ∕= empty

The penalized optimization problem for a ρ gt 0

minimizexisinW

f(x) + ρ rS(x)

where rS(x) is a W -residual function of the set S

Classical Let f be Lipschitz on W with constant Lipf gt 0 If rS(x) is aW -residual function of the set S satisfying a W -Lipschitz error bound forthe set S with constant γ gt 0 ie rS(x) ge γdist(xS capW ) for allx isin W then for all ρ gt Lipfγ

argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46

Exact penalization of games Preliminary result

bull If xlowast is an equilibrium solution of penalized NEP(C θρX) then xlowast is aNE of G(CX θ) if and only if xlowast isin FIXΞ

bull Suppose exist positive constants Lipν and γν such that for everyxminusν isin C minusν

ndash C ν cap Xν(xminusν) ∕= empty larrminus main weakness

ndash θν(bull xminusν) is Lipschitz continuous with constant Lipν on the set C ν and

ndash rν(bull xminusν) provides a C ν-Lipschitz error bound with constant γν forthe set Xν(xminusν)

Then for all ρ gt maxνisin[N ]

Lipνγν every equilibrium solution of the

penalized NEP (C θρX) is an equilibrium solution of the GNEP(CX θ) and vice versa

47

Example Consider the shared-constrained GNEP with 2 players

Player 1 minimizex1isinR

x2 (x1 minus 1 )2 | Player 2 minimizex2isinR+

(x2 minus 12 )2

subject to x1 + x2 le 1 | subject to x1 + x2 le 1

The Karush-Kuhn-Tucker (KKT) conditions of this GNEP are

0 = 2x2 (x1 minus 1 ) + λ1

0 le x2 perp 2 (x2 minus 12 ) + λ2 ge 0

0 le λ1 perp 1minus x1 minus x2 ge 0


This GNEP has no variational equilibrium ie solutions with λ1 = λ2Partial exact penalization is valid for this GNEP In particular the NE98304312

12

983044is not a normalized equilibrium in the sense of Rosen

Thus penalization can yield solutions that are otherwise not obtainableby Rosenrsquos normalization approach

48


C1 == C2 = [ 0 4 ] private constraints

commonconstraints

983070X1(x2) = x1 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2 X2(x1) = x2 | x1 + x2 le 2 2x1 minus x2 le 2 minusx1 + 2x2 le 2

(A) θ1(x1 x2) = minusx1 and θ2(x1 x2) = minusx2

(B) θ1(x1 x2) = minus 12 x1 x2 and θ2(x1 x2) = minus1

4x1 x2

x2

0 x1

g1

g2

g32

249

bull ℓ1 penalization is not exact

r1(x1 x2) = (x1 + x2 minus 2 )+ + ( 2x1 minus x2 minus 2 )+ + (minusx1 + 2x2 minus 2 )+

bull Squared ℓ2 penalization is not exact

r2(x1 x2)2 =

[(x1 + x2 minus 2 )+]2 + [( 2x1 minus x2 minus 2 )+]

2 + [(minusx1 + 2x2 minus 2 )+]2

bull ℓ2 penalization is exact

bull ℓ1 penalization is exact for variational equilibria (VE)

bull ℓ1 penalization is exact when C is restricted by redefining983144C = 983144C1 times 983144C2 with

983144C1 ≜ x1 isin C1 | existx2 isin C2 such that x1 isin X1(x2)

and similarly for 983144C2 Further research is needed

bull ℓ2-penalized NE is not VE50

The jointly convex GNEP (CD θ)

bull C ≜N983132

ν=1

C ν is the product of the private constraint sets

bull for some closed convex subset D of Rn

graph X ν = D for all ν

bull Let rD be a directionally differentiable C-residual function of the setD yielding for ρ gt 0 the penalized NEP (C θ + ρ rD )

Notation

Let T (xS) denote the tangent cone of the set S at x isin S

Let ϕ prime(x dx) ≜ limτdarr0

ϕ(x+ τ dx)minus ϕ(x)

τbe the directional derivative of

ϕ at x along the direction dx

51

Theorem Suppose

bull each θν(bull xminusν) is directionally differentiable and locally Lipschitzcontinuous on C ν for all xminusν isin C minusν

bull for every x = (xν)Nν=1 isin C D either one of the following holds

ndash for some ν and some nonzero d ν isin T (xν C ν) sube Rnν

rD(bull xminusν) prime(xν d ν) le minusα prime 983042 d ν 9830422

ndash for some nonzero tangent vector d isin T (xC)

983131

νisin[N ]

rD(bull xminusν) prime(xν d ν) le r primeD(x d)

983167 983166983165 983168sum property of directional derivative

le minusα 983042 d 9830422

then exist a finite number ρ such that for every ρ gt ρ every equilibriumsolution of the NEP (C θ + ρrD) is an equilibrium solution of the GNEP(CD θ)

52

Linear metric regularity

Two closed convex subsets C and D of Rn with a nonempty intersectionare linearly metrically regular if there exists a constant γ prime gt 0 such that

dist(xC capD) le γ prime max ( dist(xC) dist(xD) ) forallx isin Rn

This holds if either

bull C capD is bounded and rint(C) cap rint(D) ∕= empty or

bull C and D are polyhedra

Corollary Exact penalization holds for shared constrained GNEP if

bull C and D are linear metrically regular

bull rD is convex on C satisfies a C-Lipschitz error bound for D anddifferentiable on C D

53

Shared finitely representable setsLet C be convex and compact and

D = x isin Rn | g(x) le 0 and h(x) = 0

where h is affine and each gj is convex and differentiable Let

rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn

for a given q isin [1infin) which is differentiable at x isin D

Theorem Suppose each θν(bull xminusν) is convex and Lipschitz continuouson C ν with the same Lipschitz constant for all xminusν isin C minusν Then

bull for every ρ gt 0 the NEP (C θ + ρ rq) has an equilibrium solution

Assume further that a vector x isin rint(C) exists such that h(x) = 0 andg(x) lt 0 Then ρ gt 0 exists such that for every ρ gt ρ

NEP (C θ + ρ rq) rArr GNEP (CD θ)

54

Non-shared finitely representable sets

Similar setting but with

X ν(xminusν) ≜

983099983105983105983103

983105983105983101

xν isin Rnν | gν(xν xminusν) le 0 and

| hν(x) = 0983167 983166983165 983168linear equalities allowed

983100983105983105983104

983105983105983102

where each hν Rn rarr Rpν is an affine function and each gνj Rn rarr Rfor j = 1 middot middot middot mν is such that gνj (bull xminusν) is convex and differentiable forevery xminusν isin C minusν Let

rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn

for a given q isin [1infin) be the ℓq-residual function for the set X ν(xminusν)

Let rXq(x) ≜ ( rνq(x) )Nν=1

55

Theorem Suppose there exists α gt 0 such that for every

x isin C FIXΞ there exists 983141x isin C satisfying for some 983141ν with

x983141ν ∕isin X983141ν(xminus983141ν)

bull for all i isin 1 middot middot middot mν

g983141νi (x) gt 0 rArr nablax983141νg983141νi (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα983167 983166983165 983168

uniform descent condition at infeasible x

bull for all j isin 1 middot middot middot pν

h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν

j (x983141ν xminus983141ν)T ( 983141x 983141ν minus x983141ν ) le minusα

983167 983166983165 983168uniform descent condition at infeasible x

Then ρ gt 0 exists such that for every ρ gt ρ every equilibrium solution ofthe NEP (C θ+ ρ rXq) is an equilibrium solution of the GNEP (CX θ)

56

The Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ)for equalities only

X ν(xminusν) ≜983051xν isin Rnν | gν(xν xminusν) le 0

983052no equalities

For every x isin C and every ν isin 1 middot middot middot N there exists 983141x isin C suchthat

gνi (x) ge 0 rArr nablaxνgνi (xν xminusν)T ( 983141x ν minus xν ) lt 0

Remark Imposed condition applies to all x isin C cap FIXΞ which is toensure the validity of the KKT conditions at a solution

EMFCQ is more restrictive than required for the validity of exactpenalization

57


930 ndash 1030 AM Wednesday September 25 2019

58

A fixed-point (best-response) schemeReturning to the GNEP (Ξ θ) we define the proximal response mapderived from the regularized Nikaido-Isoda function

y ≜ ( yν )Nν=1 983041rarr 983141x(y) ≜ argminxisinΞ(y)

N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046

Fact y is a NE if and only if y is a fixed point of the proximal responsemap 983141x

A fixed-point iteration yk+1 ≜ 983141x(yk) k = 0 1 2 middot middot middot

Theoretically when is this a contraction a non-expansion

Computationally allow distributed player optimization

minimizexνisinΞν(ykminusν)

θν(xν ykminusν) + 1

2 983042xν minus ykν 9830422 for ν = 1 middot middot middot N

each of which is a strongly convex optimization problem

59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59

Convergence theoryBasic version requires

bull Ξν(xminusν) = Xν for all ν and

bull the second derivatives nabla2xν primexνθν(x) ≜

983077part2θν(x)

partxνprime

j xνi

983078(nν nν prime )

(ij)=1

exist and are

bounded on 983141X ≜N983132

ν=1

Xν Weakening to a 4-point condition of first

derivatives is possible (Hao 2018 PhD thesis)

A matrix-theoretic criterion Let

ζ νmin ≜ inf

xisin983141Xsmallest eigenvalue of nabla2

xνθν(x)

ξ νν primemax ≜ sup

xisin983141X

983056983056983056nabla2xνxν primeθν(x)

983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60

Define the N timesN Z-matrix (all off-diagonal entries are non-positive)

Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095

ζ1min minusξ12max minusξ13max middot middot middot minusξ1Nmax

minusξ21max ζ2min minusξ23max middot middot middot minusξ2Nmax

minusξ(Nminus1) 1max minusξ

(Nminus1) 2max middot middot middot ζNminus1

min minusξ(Nminus1)Nmax

minusξN 1max minusξN 2

max middot middot middot minusξN Nminus1max ζNmin

983094

983098983098983098983098983098983098983098983098983098983098983098983098983096

bull If Υ is a P-matrix (all principal minors are positive) then the proximalresponse map is a contraction moreover the NE is unique and can beobtained by the fixed-point iteration

bull If |983042Υ983042| le 1 for a matrix norm induced by some monotonic vectornorm then the proximal response map is nonexpansive in this case anaveraging scheme can compute a NE if one exists

61

Foreword Brief history of generalized Nash games



Lecture III Exact penalization theory


2








460ndash499 1974

3









4



kanzow-christian



5

Modelling





Mathematics





6



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61








460ndash499 1974

3









4



kanzow-christian



5

Modelling





Mathematics





6



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61









4



kanzow-christian



5

Modelling





Mathematics





6



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



kanzow-christian



5

Modelling





Mathematics





6



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Modelling





Mathematics





6



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



7







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61







prime983060


983132

ν prime ∕=ν

Rnν prime



θν(xν xminusν)





8

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Notation


bull Ξ(x) ≜N983132

ν=1

Ξν(xminusν)




9


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


Ξν(xminusν) ≜

983099983103


private constraints


coupled constraints

983100983104

983102





ν=1

Xν and C sub Rn

where n ≜N983131

ν=1








lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



lowast) such that



ν=1



N983131

ν=1






11




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




983105983101


mν983131

i=1



983100983105983104



0 le983061

xλ

983062

983167 983166983165 983168denoted z

perp

983091

983109983109983107


mν983131

i=1


983076N

ν=1


983092

983110983110983108

983167 983166983165 983168denoted F(z)

ge 0

12


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


Contributions






13







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61







14

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Network set-up






other customers


e-HSP labels

15



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983154for each type



OD pair k



16


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


17

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

The e-HSP module


Rmjk ≜ 983141Rm



where

983141Rmjk ≜ Fm

Okminus βm

1 ( tjOk+ tOkDk


minus βm2 ( djOk

+ dOkDk)


+ αm1 ( tOkDk

minus f 0OkDk


+ αm2 dOkDk983167 983166983165 983168




18


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


maximizezmjkge0

983131

kisinK

983131

jisinD

983141Rmjk z

mjk


minusβm3

983093

983095Nm minus983131

kisinK

983131

jisinDzmjktjOk

minus983131

kisinKQm

k tOkDk

983094

983096


subject to983131

kisinKzmjk =

983131

k prime j=Dk prime

Qmk prime


for all j isin D

983131

jisinDzmjk ge Qm

k


for all k isin K

and983131

kisinK

983131

jisinDzmjk tjOk


+983131

kisinKQm

k tOkDk


le Nm983167983166983165983168


19



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



FmOk

+ αm1 ( tOkDk

minus f 0OkDk

)983167 983166983165 983168

time based fare

+ αm2 dOkDk983167 983166983165 983168

distance based fare




V 0k = γ01 tOkDk983167 983166983165 983168


+ β02 dOkDk983167 983166983165 983168




20


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


minimizeQm

k ge0

983131

misinM+

983131

kisinKV mk Qm

k

subject to

983131

jisinDzmjk ge Qm

k

shared constraints


983131

misinM+

Qmk = Qk983167983166983165983168

given

for all k isin K


21



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



0 le tij perp983131

pisinPij

hp minus

983093

983095983131

kisinKδODijk Qk +

983131


983131

misinMzmiℓ

983094

983096 ge 0



δODijk ≜

9830691 if i = Ok and j = Dk

0 otherwise

983070



0 otherwise

983070

22


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


wmk = γm2

983131

jisinDzmjk tjOk

983131

jisinDzmjk


for all m isin M






24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



24



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



lowast) such that



ν=1



N983131

ν=1



ν=1


25



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



y isin 983141K 983041rarrN983132

ν=1







for all ν

26



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983060N




Llt ≜983083



983084


27



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983060N




Llt ≜983083



983084


27



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983099983105983105983105983105983105983105983105983103

983105983105983105983105983105983105983105983101


G(x) le 0983167 983166983165 983168nonconvex

983141G(x) le 0983167 983166983165 983168polyhedral983167 983166983165 983168

shared

983100983105983105983105983105983105983105983105983104

983105983105983105983105983105983105983105983102




983093

983097983097983097983097983097983097983095θν(x

ν xminusν) +

p983131

j=1


983141p983131

j=1


983167 983166983165 983168denoted Lν(xπ 983141π)

983094

983098983098983098983098983098983098983096

28













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61













29













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61













29



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



N983132

ν=1

Rℓν+ with

Ξ ≜

983099983105983105983103

983105983105983101x = (xν )

Nν=1 isin

N983132

ν=1


983100983105983105983104

983105983105983102 polyhedral

Φ(xπλ) ≜983091

983109983109983109983109983109983109983107

983091


p983131

j=1

πj nablaxνGj(x) +

ℓν983131

i=1

λνi nablahν

i (xν)

983092

983108N

ν=1

VI Lagrangian

Ψ(x) ≜983075

minusG(x)

minusH(x)


983092

983110983110983110983110983110983110983108

30


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


For a QNE








31








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61








32







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61







and




33

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Existence of QNE


983060N

ν=1isin Ξ

such that


H(xref)G(xref)

983062lt 0






34

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Uniqueness of QNE


JΦ(xχ) =

983077A(xχ) JΨ(x)T

minusJΨ(x) 0


A(xχ) ≜ JΘ(x) +

p983131

j=1


983077ℓν983131

i=1

λνi nabla2hνi (x

ν)

983078N



C(KΦ y) =

983141C(xχ) times983045T (Rp


N983132

ν=1

983147T (Rℓν






983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




983154is bounded



36




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




983154is bounded



36

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp




37

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

When is a QNE a LNE



xνθν(x) +

p983131

j=1


ℓν983131

i=1

λνi nabla2hνi (x

ν)

note that


983060N






983093


p983131

j=1


lowast) +

983141p983131

j=1


lowast)

983094

983096perp






Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



Lν(xπ 983141π) (2)



πge 0πTG(x)




πTG(x)

where St ≜

983099983103

983101π ge 0 |p983131

j=1

πj le t

983100983104


38


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


Write 983141X ≜N983132

ν=1


983083λν ge 0 |

ℓν983131

i=1

λνi le ξt

983084 where

ξt ≜


983099983103


983055983055983055983055983055983055

Q983131

q=1


983055983055983055983055983055983055

983100983104

983102

min1leνleN

983061min


refν) )

983062





39



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



p983131

j=1

πtj +

983141p983131

j=1

983141π tj +

N983131

ν=1

ℓν983131

i=1

λtνi le ξlowast


j=1

983141πj le t and

p983131

j=1



40









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61









ν=1



41





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61





minimizezq yq wq

θq(zq yq zminusq)






minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


minimizezq yq wq

θq(zq yq zminusq)





983105983105983105983105983105983101

( zq yq )Qq=1 isin

Q983132

q=1

Xq |

Q983131

q=1



983046lt 0

983100983105983105983105983105983105983104

983105983105983105983105983105983102





44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



44



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



θν(xν xminusν)











45



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



f(x)



minimizexisinW

f(x) + ρ rS(x)



argminxisinS capW

f(x) = argminxisinW

f(x) + ρ rS(x)

46










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61










47




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




(x2 minus 12 )2



0 = 2x2 (x1 minus 1 ) + λ1





12



48



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



commonconstraints




4x1 x2

x2

0 x1

g1

g2

g32

249




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




r2(x1 x2)2 =


2 + [(minusx1 + 2x2 minus 2 )+]2








bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


bull C ≜N983132

ν=1





Notation






51

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

Theorem Suppose






983131

νisin[N ]



le minusα 983042 d 9830422


52










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61










53




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




rq(x) ≜983056983056983056983056983056

983075max( g(x) 0 )

h(x)

983076983056983056983056983056983056q

x isin Rn






54



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



X ν(xminusν) ≜

983099983105983105983103

983105983105983101



983100983105983105983104

983105983105983102


rνq(x) ≜983056983056983056983056983056

983075max( gν(x) 0 )

hν(x)

983076983056983056983056983056983056q

x isin Rn



55








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61








h983141νj (x) ∕= 0 rArr

983147sgn h983141ν

j (x)983148nablax983141νh983141ν




56



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



983052no equalities





57



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



58



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61



N983131

ν=1

983045θν(x

ν yminusν) + 12 983042x

ν minus yν 9830422983046









59




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61




983077part2θν(x)

partxνprime

j xνi


(ij)=1

exist and are


ν=1




ζ νmin ≜ inf


xνθν(x)


xisin983141X


983056983056983056

60


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61


Υ ≜

983093

983097983097983097983097983097983097983097983097983097983097983097983097983095








983094

983098983098983098983098983098983098983098983098983098983098983098983098983096



61

four lectures on generalized nash equilibrium problems in ...€¦ · nash equilibrium problems in...

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