invex&formulations&in&& integerprogramming& · 2013. 1. 20. · "modeling&disjunchve&constraints&...
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Hassan Hijazi AUSSOIS 2012
INVEX FORMULATIONS IN INTEGER PROGRAMMING
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09/01/2012
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Convex functions 1
q Convex opHmizaHon: " Any staHonary point is opHmal
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Invex functions 2
-1 -0,75 -0,5 -0,25 0 0,25 0,5 0,75 1 1,25
-0,25
0,25
0,5
0,75
1
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Definition (Hanson 1981) 4
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Simple characterization 5
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Constrained Optimization 6
We need to look at the Lagrangian funcHon:
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¨ Modeling disjuncHve constraints featuring unbounded variables
¨ Invex formulaHons for a facility locaHon problem
Invex formulations in Integer Programs 7
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Unbounded disjunction 8
¨ How to formulate the constraint:
¨ x must remain unbounded!
Ø Now, we only need to model:
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Unbounded disjunction 9
z
y
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The big-M formulation 10
y
z
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Convex hull formulation 11
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Lifting 12
z
y
ϒ
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A second order cone constraint 13
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Cplex 14
CPLEX 12.2.0.0: best soluGon found, primal-‐dual infeasible; objecGve 3.749997256 50 barrier iteraGons No basis. x = 3.75
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A hidden hypothesis: constraint qualification
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If a point x* saHsfies a constraint qualificaHon condiHon, it is opHmal if and only if it saHsfies the KKT condiHons. q LICQ: the gradients of the acHve inequality constraints and the gradients of the equality constraints are linearly independent at x*. q Slater condiGons: there exists a point x’ such that gi(x’) < 0 for all gi acHve in x*.
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A hidden hypothesis: constraint qualification
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Invex formulation 17
z
y
ϒ
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Invex formulation 18
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It works! 19
Using IPOPT open source solver, Interior point method
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With Ipopt 20
Total CPU secs in IPOPT (w/o funcGon evaluaGons) = 0.003 Total CPU secs in NLP funcGon evaluaGons = 0.000 EXIT: OpGmal SoluGon Found. Ipopt 3.8.3: OpGmal SoluGon Found x = 4 gamma = 0 y = 0 z = 0
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Concentrator placement in Smart Energy Grids
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Concentrator
Smart meter
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Mathematical modeling 22
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Mathematical modeling 23
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Mathematical modeling 24
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Mathematical modeling 25
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Mathematical modeling 26
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Mathematical modeling 27
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Example 28
minimize cost: 100*z1 + 100*z2 + 25*x11 + 35*x12 + 50*x21 + 35*x22; subject to demand1: 1 -‐ x11 -‐ x12
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LP relaxation 29
CPLEX 12.2.0.0: opGmal soluGon; objecGve 172.5 4 dual simplex iteraGons (0 in phase I) z1 = 0.5 z2 = 0.5 x11 = 0.5 x12 = 0.5 x21 = 0.5 x22 = 0.5
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Example 30
minimize cost: 100*z1^2 + 100*z2^2 + 25*y11 + 35*y12 + 50*y21 + 35*y22; subject to demand1: 1 -‐ x11 -‐ x12
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Invex relaxation 31
ObjecGve...............: 168.859 Ipopt 3.8.3: OpGmal SoluGon Found z1 = 0.5 z2 = 0.559344 x11 = 1 x12 = 4.55569e-‐09 x21 = 5.02689e-‐09 x22 = 1
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Finding a feasible solution
|I| |J| Bonmin’s best Invex
rdata1 15 250 2300 39
rdata2 20 250 >3000 43
rdata3 30 250 >3000 120
rdata4 40 250 >3000 150
rdata5 100 250 >3000 300
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Bonmin 1.5 using CBC-‐IPOPT, Hme limit = 3000 sec
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