Optimizing over the Split Closure Optimizing over the Split Closure Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. (Joint Work with Egon Balas)
MIP Model MIP Model min cx Contains x j ¸ 0 j 2 N x j · u j j 2 N 1 Ax ¸ b x j 2 Z 8 j 2 N 1 N 1 : set of integer variables Incumbent Fractional Solution Anureet Saxena, TSoB 2
Split Disjunctions Split Disjunctions π 2 Z N , π 0 2 Z • π j = 0, j 2 N 2 • π 0 < π < π 0 + 1 • π x · π 0 π x ¸ π 0 + 1 Split Disjunction Anureet Saxena, TSoB 3
Split Cuts Split Cuts v u Ax ¸ b Ax ¸ b π x · π 0 π x ¸ π 0 +1 u 0 v 0 α L x ¸ β L α R x ¸ β R α x ¸ β Split Cut Anureet Saxena, TSoB 4
Split Closure Split Closure Elementary Split Closure of P = { x | Ax ¸ b } is the polyhedral set defined by intersecting P with the valid rank - 1 split cuts. How much duality gap can be closed by optimizing over the split closure? Rank - 1 Chvatal Closure Elementary Disjunctive Closure M. Fischetti & A.Lodi P. Bonami & M. Minoux Anureet Saxena, TSoB 5
Algorithmic Framework Algorithmic Framework min cx Add Cuts Solve Ax ¸ b Master LP α t x ¸ β t t 2 Π Integral Sol? Yes MIP Solved Unbounded? Infeasible? No No Split Cuts Optimum over Split Cuts Generated Rank - 1 Split Cut Generated Split Closure Separation attained Anureet Saxena, TSoB 6
Algorithmic Framework min cx Add Cuts Solve Ax ¸ b Master LP α t x ¸ β t t 2 Π Integral Sol? Yes MIP Solved Unbounded? Infeasible? No No Split Cuts Optimum over Split Cuts Rank - 1 Split Cut Generated Rank - 1 Split Cut Generated Split Closure Separation Separation attained Anureet Saxena, TSoB 7
SC Separation Theorem SC Separation Theorem Theorem: lies in the split closure of P if and only if the o ptimal value of the following parametric mixed integer linear program is non - negative. Parameter Parametric Mixed Integer Linear Program i Anureet Saxena, TSoB 8
SC Separation Theorem SC Separation Theorem Theorem: lies in the split closure of P if and only if the o ptimal value of the following parametric mixed integer linear program is non - negative. (u,v, π , π 0 , θ ): α = uA - θ π β = ub - θ π 0 α x ¸ β Split Cut Anureet Saxena, TSoB 9
Deparametrization Deparametrization Parameteric Mixed Integer Linear Program Anureet Saxena, TSoB 10
Deparametrization Deparametrization Parameteric Mixed Integer Linear Program If θ is fixed, then PMILP reduces to a MILP Anureet Saxena, TSoB 11
Deparametrization Deparametrization MILP( ) Deparametrized Mixed Integer Linear Program Maintain a dynamically updated grid of parameters Anureet Saxena, TSoB 12
Separation Algorithm Separation Algorithm Initialize Parameter Grid ( Σ ) For θ 2 Σ , Diversification • Solve MILP( θ ) using CPLEX 9.0 • Enumerate τ branch and bound nodes • Store all the separating split disjunctions which Strengthening are discovered At least one no yes STOP Grid Enrichment split disjunction discovered? Bifurcation Anureet Saxena, TSoB 13
Implementation Details Implementation Details Processor Details • Pentium IV • 2Ghz, 2GB RAM COIN - OR CPLEX 9.0 Solving MILP( θ ) Core Implementation • Solving Master LP • Setting up MILP • Disjunctions/Cuts Management • L&P cut generation+strengthening Anureet Saxena, TSoB 14
Computational Results Computational Results • MIPLIB 3.0 instances • OR - Lib (Beasley) Capacitated Warehouse Location Problems Anureet Saxena, TSoB 15
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Instance # Int Var # Cont Var % Gap Closed 10teams 1800 225 100.00% dcmulti 75 473 100.00% egout 55 86 100.00% flugpl 11 7 100.00% gen 150 720 100.00% gesa2_o 720 504 100.00% khb05250 24 1326 100.00% misc06 112 1696 100.00% qnet1 1417 124 100.00% qnet1_o 1417 124 100.00% rgn 100 80 100.00% vpm1 168 210 100.00% fixnet6 378 500 99.76% gesa2 408 816 98.66% 14 Instances 98 - 100% Gap Closed Anureet Saxena, TSoB 16
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Instance # Int Var # Cont Var % Gap Closed 10teams 1800 225 100.00% dcmulti 75 473 100.00% egout 55 86 100.00% flugpl 11 7 100.00% gen 150 720 100.00% gesa2_o 720 504 100.00% khb05250 24 1326 100.00% misc06 112 1696 100.00% qnet1 1417 124 100.00% qnet1_o 1417 124 100.00% rgn 100 80 100.00% vpm1 168 210 100.00% fixnet6 378 500 99.76% gesa2 408 816 98.66% 14 Instances 98 - 100% Gap Closed Anureet Saxena, TSoB 17
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Instance # Int Var # Cont Var % Gap Closed pp08aCUTS 64 176 97.01% modglob 98 324 96.48% pp08a 64 176 95.81% gesa3 384 768 95.78% gesa3_o 672 480 95.31% set1ch 240 472 89.41% bell5 58 46 87.44% arki001 538 850 83.05% vpm2 168 210 81.22% mod011 96 10862 80.72% qiu 48 792 77.51% 11 Instances Unsolved MIP Instance In MIPLIB 3.0 75 - 98% Gap Closed Anureet Saxena, TSoB 18
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Instance # Int Var # Cont Var % Gap Closed rout 315 241 70.73% bell3a 71 62 55.19% blend2 264 89 46.77% 3 Instances 25 - 75% Gap Closed Anureet Saxena, TSoB 19
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Instance # Int Var # Cont Var % Gap Closed danoint 56 465 7.44% dano3mip 552 13321 0.12% pk1 55 31 0.00% 3 Instances 0- 25% Gap Closed Anureet Saxena, TSoB 20
MIPLIB 3.0 MIP Instances MIPLIB 3.0 MIP Instances Summary of MIP Instances (MIPLIB 3.0) Total Number of Instances: 34 Number of Instances included: 33 No duality gap: noswot , dsbmip Instance not included: rentacar Results 98 - 100% Gap closed in 14 instances 75 - 98% Gap closed in 11 instances 25 - 75% Gap closed in 3 instances 0 - 25% Gap closed in 3 instances Average Gap Closed: 82.53% Anureet Saxena, TSoB 21
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed air03 10757 100.00% gt2 188 100.00% mitre 10724 100.00% mod008 319 100.00% mod010 2655 100.00% nw04 87482 100.00% p0548 548 100.00% p0282 282 99.90% fiber 1254 98.50% 9 Instances 98 - 100% Gap Closed Anureet Saxena, TSoB 22
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed lseu 89 93.75% p2756 2756 92.32% l152lav 1989 87.56% p0033 33 87.42% 4 Instances 75 - 98% Gap Closed Anureet Saxena, TSoB 23
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed p0201 201 74.93% air04 8904 62.42% air05 7195 62.05% seymour 1372 61.94% misc03 159 51.47% cap6000 6000 37.63% 6 Instances Ceria, Pataki et al closed around 50% of the gap using 10 rounds of L&P cuts 25 - 75% Gap Closed Anureet Saxena, TSoB 24
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed misc07 259 19.48% fast0507 63009 18.08% stein27 27 0.00% stein45 45 0.00% 4 Instances 0- 25% Gap Closed Anureet Saxena, TSoB 25
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Summary of Pure IP Instances (MIPLIB 3.0) Total Number of Instances: 25 Number of Instances included: 24 No duality gap: enigma Instance not included: harp2 Results 98 - 100% Gap closed in 9 instances 75 - 98% Gap closed in 4 instances 25 - 75% Gap closed in 6 instances 0 - 25% Gap closed in 4 instances Average Gap Closed: 71.63% Anureet Saxena, TSoB 26
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed (SC) %Gap Closed (CG) air03 10757 100.00% 100.00% 8904 62.42% 27.60% air04 air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% 100 - - enigma 63009 18.08% 4.70% fast0507 1254 98.50% 98.50% fiber 188 100.00% 100.00% gt2 1989 87.56% 69.20% l152lav % Gap Closed by 89 93.75% 91.30% lseu 159 51.47% 51.20% misc03 First Chvatal Closure 259 19.48% 16.10% misc07 ( Fischetti - Lodi Bound) 10724 100.00% 100.00% mitre 319 100.00% 100.00% mod008 2655 100.00% 100.00% mod010 87482 100.00% 100.00% nw04 33 87.42% 85.40% p0033 201 74.93% 60.50% p0201 282 99.90% 99.90% p0282 p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% 1372 61.94% 23.50% seymour 27 0.00% 0.00% stein27 stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59% Anureet Saxena, TSoB 27
MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances Instance # Variables %Gap Closed (SC) %Gap Closed (CG) air03 10757 100.00% 100.00% 8904 62.42% 27.60% air04 air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% 100 - - enigma 63009 18.08% 4.70% fast0507 1254 98.50% 98.50% fiber 188 100.00% 100.00% gt2 1989 87.56% 69.20% l152lav 89 93.75% 91.30% lseu 159 51.47% 51.20% misc03 259 19.48% 16.10% misc07 10724 100.00% 100.00% mitre 319 100.00% 100.00% mod008 2655 100.00% 100.00% mod010 87482 100.00% 100.00% nw04 33 87.42% 85.40% p0033 201 74.93% 60.50% p0201 282 99.90% 99.90% p0282 p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% 1372 61.94% 23.50% seymour 27 0.00% 0.00% stein27 stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59% Anureet Saxena, TSoB 28
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