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Data Driven Decisions @ The Ohio State UniversityIndustrial and Systems Engineering

Advances in Stochastic Mixed Integer Programming

Lecture at the INFORMS Optimization Section Conference in Miami, February 26, 2012

Suvrajeet SenData Driven Decisions Lab

Integrated Systems EngineeringOhio State University

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Some Historical Remarks Classification of SMIP SMIP Models: Risk, Recourse, Resilience Structural Properties Decomposition: Benders’ and Beyond Illustrative Computational Results

Overview of this Lecture

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Historical Remarks: IOS

Age of the INFORMS Optimization Section isa) 0 < age 10

b) 10 < age 15

c) 15 < age 20

d) 20 < age 25

e) age > 25 INFORMS OS was founded at the Spring

ORSA/TIMS Meeting in Los Angeles, April 1995.

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Historical Remarks:My assessment of SIP/SMIP

StochasticInteger

Programming

IntegerProgramming

LinearProgramming

StochasticLinear

Programming

Discrete Choice

Discrete Choice1950s-

Present

1960s - Present

Uncertainty Uncertainty

Major Hurdles Still Remain!!

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Why model uncertainty? For most people, that’s just reality In the real world:

“Risk is everywhere” Certainty:

“Risk is merely a 4-letter word” In the real world:

“There is a market for information” Certainty:

“Information has no value” In the real world:

“Hind sight is 20/20” Certainty:

Foresight is 20/20

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Some data: [1980 – 2000) … prior to 2000 annotated bibliography (Stougie and van der Vlerk) Theory: (7+) papers

Simple Integer Recourse: 2 Structural Properties of Expected Recourse Function: 4 Complexity: (1+)

General Purpose Algorithms: 17 papers Benders’-type methods: 5 Grobner-basis methods: 2 Convex Approximations for Simple Integer Recourse: 2 Other: 8 (Sampling with first-stage integer, disjunctive cuts)

More Historical Remarks:“Walk Before You Can Run”

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Books/Surveys: 6 altogether Dissertations: 3-4 (1 prior to 2000 in North America) Habilitation: 1 Published surveys: 3 (includes hierarchical planning)

Special Purpose Models/Algorithms: 25 papers Production Planning/Scheduling: 3 Network and Routing: 11 Location: 7 Other: 4

In the 12 years: more than 350 articles listed in http://mally.eco.rug.nl/index.html?BIBLIO/SIP.HTML

More Historical Remarks:“Walk Before You Can Run” [1980-2000)

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Schultz, R (2003) “Stochastic Programming with Integer Variables,” Mathematical Programming-B, 285-309

Stougie, L. and M.H. van der Vlerk (2005) “Approximation in Stochastic Integer Programming”

Sen, S. (2005) “Stochastic Mixed-Integer Programming Algorithms,” Handbook of Discrete Optimization, (Aardal, Nemhauser, Weismantel, eds.)

…. Some newer surveys are also available ….

“Now we are running” (Survey Articles 2000-)

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… And you know we’re serious because of applications with realistic data

Manufacturing Supply Chain: Two-stage Design (IBM, Intel)

Biofuel Supply Chain: Multi-stage Design (Fan et al) Homeland Security – Defender/Attacker/Defender

(Wood et al – NPS, Ordonez/Tambe, Smith) Electric Power – Unit Commitment (Birge/Takriti,

Philpott, Guan/Zhang), Fuel Price Hedging (Sen et al) Military – Prioritizing Choices (Morton), UAV/MAV

(Evers et al) Fighting Forest Fire (Ntaimo)

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SMIP Classification: A (B-C-D-E) Notation for SMIP

Two Stage Stochastic Linear Programming

Min cTx + E[f(x, ω)]

Ax = b, x ≥ 0

where,f(x, ω) = Min gTy

Wy ≥ r(ω) – T(ω)x

y ≥ 0

Variations depend on where the randomness appears

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Stochastic MIP with First Stage Integers

Min cTx + E[f(x, ω)]

Ax ≥ b, x ∈ R n

1 × Z n

2

where,f(x, ω) = Min gTy

Wy ≥ r(ω) – T(ω)x

y ∈ Rn

3

Z n

denotes integer vectors of length n. With second-stage integers, extremely

difficult!

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Stochastic Combinatorial Optimization

Min cTx + E[f(x, ω)]

Ax ≥ b, x ∈ Bn

1

where,(0l18f(x, ω) = Min gTy

Wy ≥ r(ω) – T(ω) x

y ∈ Rn

2 × Bn

3

Here Bn denotes binary vectors of length n. Many different structures for SMIP!

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Describing SMIP Problems B = Set of stages with Binary Vars. C = Set of stages with Continuous Vars. D = Set of stages with Discrete Vars.

(arbitrary integers, not just binary) E = Endogenous Uncertainty (Y/N)

Louveaux has proposed a notation that covers all SP problems (e.g. notation includes whether random variables are cont/discrete)

Above notation helps clarify domain of applicability of results/algorithms etc.

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Traditional Benders’ Decomposition SLP: B = {∅}, C={1,2}, D ={∅} Wollmer, Norkin et al, Poojari/Mitra:

B = {1}, C={1,2}, D ={1} Special Structure: Simple Integer

Recourse: B = {2}, C={1}, D ={2} + structure of

second stage Global Optimization and IP

Ahmed, Tawarmalani, Sahinidis: B = {2}, C={1,2}, D ={2} ; + Fixed Tenders

Grossman & Co. (E = Y)

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Disjunctive Programming for Two-Stage Caroe/Tind, Sherali/Fraticelli,

Sen/Higle, Sen/Sherali:B = {1,2}, C={2}, D ={ }∅

Ntaimo/Sen: B = {1,2}, C={1,2}, D ={ }∅

Lagrangian-based Methods for Multi-stage Multi-stage SMIPs: Caroe/Schultz, Roemisch et al,

Alonso-Ayuso et al, Lulli/Sen, Guan et al B = {1,2, … N}, C={1,2 … N}, D ={1,2 … N}b

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SMIP Models Modeling Risk Modeling Recourse Modeling Resilience Multi-stage Models

Models not Covered (Chance Constraints with Discrete Distributions) Special Structured IP (Knapsack, Mixing etc.) See

Prékopa, Dentcheva, Ruszczynski … Leudtke et al (2010), Küçükyavuz (2010), Saxena et al

(2009), Shen et al (2010)

Stochastic MIP Models: Risk, Recourse, and Resilience

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Risk in SMIP

We have only stated models via “Expected Values”

Is the reliance on “Expectation” a handicap?

Of course! But many risk measures (e.g. down-side risk, mean absolute deviation, CVaR, etc.) can be re-formulated using expectation of a slightly modified, though mathematically similar function.

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SCO for Modeling Risk

We have only stated models via “Expected Values”

Is the reliance on “Expectation” a handicap?

Of course! But many risk measures (e.g. down-side risk, mean absolute deviation, CVaR, etc.) can be re-formulated using expectation of a slightly modified, though mathematically similar function

Important: Inequalities are indispensible for risk modeling

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SCO for Modeling RiskExample: Kahneman/Tversky “S” curve for risk-aversion can be linearized using 0-1 variables.

Similar to non-convex piecewise linear programming.

Each piece requires a binary (switch variable)

r

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SCO for Modeling Recourse:Stochastic Server Location Problem

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SCO for Modeling Recourse: SSLP

This SCO has two sets of decisions:1. Choose server locations (e.g. bases)2. Once demand nodes (e.g threats) appear, then assign servers to demand nodes

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SCO for Modeling Recourse: SSLP

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Our Stochastic Server Location Problem (SSLP) also includes some policy constraints:

Policy that each customer will receive service from only one site has been established. Moreover, service site must be located within a prescribed zone (z).

Max number to be located is v, with each zone having no more than wz servers.

SCO for Modeling Recourse: SSLP

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The SSLP model objective: minimize Cost – Expected Revenue Potential (last term denotes Penalty for lost demand)

Min Σj cjxj – E[Σijqijyij(ω) + Σj QjYj(ω)]

subject to: constraints on supply-side,Σj xj ≤ v, Σj ∈ J(z) xj ≥ wz, ∀z

demand-side,Σjyij(ω) + Yj(ω) = ωi, ∀i

supply/demand:Σi yij(ω) – Yj(ω) ≤ ujxj, ∀j

Plus: All variables are binary

SCO for Modeling Recourse: SSLP

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Modeling Resilience

Logical conditions are as follows:

y0

jk ≤ 1 – xj

y1jk ≤ xj

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Multi-Stage SMIP ModelsNon-anticipativity in the Two Stage Model

(*) is the non-anticipativity constraint …

all scenarios must agree on first-stage

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Two-stage: NA only on Root Node

Multi-stage: Difficult, unless Ocotillo-type Trees

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Recursive Formulation using State Variables

Challenge of Coniferous Trees

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SMIP with Recursive Formulation

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Most structural properties and algorithms for SMIP assume relatively complete and sufficiently expensive recourse. -∞ < f(x,ω) < +∞ with probability 1.

Under the above assumption, the expected recourse function is real-valued and lower semi-continuous.

Structural Properties

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Complexity of Two-Stage SMIP

Two-stage stochastic programs with recourse having finitely many scenarios is #P-hard (The class #P asks for the count (i.e. “how many”, rather than “are there any”?) The proof reduces any graph reliability problem to

a two-stage stochastic combinatorial optimization problem

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What Do we Need?

Two Issues in Algorithm Design:- Cuts for Second Stage IP - Approximation of f (also convexification)

A Potent Brew! Decomposition (SP) and Convexification (IP)

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Gomory Cuts for SIP Decomposition Hot off the Printer! First stage 0-1, Second-stage General Integer,

Disjunctive Decomposition (D2) First stage: 0-1 Second-stage: mixed 0-1

Disjunctive Decomposition with Branch-and-Cut (D2-BAC) First stage 0-1 Second-stage: mixed-integer

Beyond Benders’ Decomposition: Second-stage IP

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Recall --- SLP

Two Stage Stochastic Linear Programming

Min cTx + E[f(x, ω)]

Ax = b, x ≥ 0

where,f(x, ω) = Min gTy

Wy ≥ r(ω) – T(ω)x

y ≥ 0

Variations depend on where the randomness appears

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Recall --- Benders’ Decomposition or L-shaped Method

Standard Benders’ Master (OR – 501)

Where denote Non-negative Second-stage Dual Multipliers

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Caroe-Tind extension of Benders’ orL-shaped Method for Second-stage SIP

Gomory Cuts to represent Subadditive Value Functions

Where denote Non-decreasing Second-stage value function approximations

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Structure Similar to Benders’ And a More General Framework But … Need to overcome bottlenecks

Subproblems are Integer Programs Master Problems are required to Optimize Non-

convex functions

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Our Recommendation: maintain Benders’ piecewise linear approximations Notice the change below!

Where denote Non-negative Second-stage Dual Multipliers

Notice that RHS r has changed to and T to

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Our Suggestion: Solve Second stage using Updated LP approximations

Each iteration will involve only LP solutions in the second-stage

Solve LP relaxation TWICE Once solve with an Old Convexification Derive a Cut to Update the Convexification

We will have First-stage is same as Benders’ original proposal Second-stage are LPs.

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But can this be achieved? Yes under certain assumptions!

First stage pure binary (B = {1,2}) C = , D={2} Use Gomory Cuts (Gade,

Küçükyavuz, Sen) If C= {2}, B = {2} Use Disjunctive Set Convexification

(Sen and Higle) If C= {2}, D = {2} Use Disjunctive Value

Approximations for Branch-and-Cut (Sen and Sherali) First stage general MILP (Global Optimization)

{∅}

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Second Stage Set Convexification

Original ConstraintsValid Inequalities as Functions of x

Parametric Gomory Cuts: Affine

Parametric Disjunctive Cuts: Piecewise Linear Concave

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Parametric Gomory Cuts

Finiteness with Lexicographic Dual Simplex (Gade, Küçükyavuz, Sen)

Scen Obj Vars Cons GDD-S GDD-R B&B Nodes

B&B + Gom Nodes

4 -63.50 22 24 7 (13) 7 (32) 54 2 (6)

9 -66.17 47 54 7 (39) 6 (76) 306 8 (13)

36 -67.33 182 216 10 (183) 6 (384) 1.55E7 52 (50)

121 -67.67 607 726 9 (526) 6 (1032) 7.60E6 13224 (167)

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Convexify π0(x,ω) by viewing its epigraph as a disjunctive set such as the one shown below.

π0(x,ω)

0 1First stage binary variable

Parametric Disjunctive Cuts

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Convergence for Disjunctive Decomposition (Set Convexification)Assumptions

•Complete recourse•All integer variables are 0-1•Maintain all cuts in Wk •Certain rules of order hold (a la lexicographic dual simplex in

Gomory’s proof)Under these assumptions, the D2 methodresults in a convergent algorithm (Sen and Higle).

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π0(x,ω)

Value Approximations for Branch-and-Cut in Second Stage (Sen and Sherali)

—There will be one piece per node of a truncated BAC tree in the second-stage—Disjunctive Programming lets us convexify the function (for each outcome )

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Illustrative Computational Results

with D2 and D2-BACComputational Results for Problem Instance SSLP_10_50 CPU Time (secs.)

Scenarios Binaries Constraints % ZIP Gap Iterations D2 Cuts D2 L2 DEP DEP % Gap

5 2,510 301 10.49 209 189 78.25 997.74 80.5310 5,010 601 11.38 264 257 171.49 1284.47 Failed 0.1925 12,510 1,501 10.81 286 281 248.81 1339.24 Failed 0.3450 25,010 3,001 10.89 252 250 295.95 1982.60 Failed 0.44

100 50,010 6,001 11.07 300 299 480.46 2782.88 Failed 9.02500 250,010 30,001 10.75 309 307 1902.20 Failed Failed 38.17

1,000 500,010 60,001 11.07 322 321 5410.10 Failed Failed 99.602,000 1,000,010 120,001 11.01 308 307 9055.29 Failed Failed 46.24

SCALABILITY: D2 scales well with increase in number of scenarios (linear) D2 does not scale well with increase in size of master program (x)

Computational Results for Problem Instance SSLP_15_45 CPU Time (secs.)

Scenarios Binaries Constraints % ZIP Gap Iterations D2 Cuts D2 L2 DEP DEP % Gap

5 3,390 301 6.88 146 145 110.34 Failed Failed 1.1910 6,765 601 6.53 454 453 1,494.89 Failed Failed 0.2715 10,140 901 5.62 814 813 7,210.63 Failed Failed 0.72

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T = 4.6631S

R2 = 0.9888

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 500 1000 1500 2000 2500

Number of Scenarios S

CP

U T

ime

(s

)

T

Computational Results Cont…

The D2 Algorithm Solves some of the largest (0-1) instances Scalability - Linear in the number of scenarios

D2 CPU time for SSLP_10_50 with 100 scenarios

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Treating Cut Generation as a Specialized Two-Stage LP

Computational Results (with Y. Yuan)

Instance D2 D2-BAC D2-BAC++ 5.25.50 1.64 0.70 0.365.25.100 2.15 1.73 0.895.50.100 7.10 3.70 1.565.50.500 34.50 23.05 12.365.50.1000 140.47 64.17 22.775.50.2000 603.37 274.40 42.7410.50.50 295.95 373.98 262.1310.50.100 396.76 452.31 486.9910.50.500 1902.2 2772.22 1313.3810.50.1000 5410.1 5677.80 2139.4710.50.2000 9055.29 > 10800

3916.4715.45.5 110.34 232.30 211.7915.45.10 1494.89 222.41 153.4115.45.15 7210.63 1988.26 803.56

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Conclusions

Decomposition (SP) + Valid Inequalities (IP) provide a potent potion!

But … Stochastic MIP still needs a lot of work

Specially structured cuts (already at play in Chance Constrained SP)

Multi-stage extensions (very rich area) Real-world Applications ….