Optimizing over the Split Closure Optimizing over the Split Closure - - PowerPoint PPT Presentation

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)

Anureet Saxena, TSoB 2

MIP Model MIP Model

min cx Ax ¸ b xj 2 Z 8 j2N1

N1: set of integer variables

Contains xj ¸ 0 j2 N xj · uj j2 N1

Incumbent Fractional Solution

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Split Disjunctions Split Disjunctions

• π 2 Z N, π0 2 Z
• πj = 0, j2 N2
• π0 < π

< π0 + 1

Split Disjunction

π x · π0 π x ¸ π0 + 1

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Split Cuts Split Cuts

Ax ¸ b π x · π0 Ax ¸ b π x ¸ π0+1 u u0 v0 v αL x ¸ βL αR x ¸ βR α x ¸ β Split Cut

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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

• ptimizing over the split closure?

Rank-1 Chvatal Closure Elementary Disjunctive Closure

• M. Fischetti & A.Lodi
• P. Bonami & M. Minoux

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Algorithmic Framework Algorithmic Framework

Solve Master LP Integral Sol? Unbounded? Infeasible? Rank-1 Split Cut Separation MIP Solved Optimum over Split Closure attained

Split Cuts Generated No Split Cuts Generated min cx Ax ¸ b αt x¸ βt t2Π Yes No Add Cuts

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Algorithmic Framework

Solve Master LP Integral Sol? Unbounded? Infeasible? Rank-1 Split Cut Separation Rank-1 Split Cut Separation MIP Solved Optimum over Split Closure attained

Split Cuts Generated No Split Cuts Generated min cx Ax ¸ b αt x¸ βt t2Π Yes No Add Cuts

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SC Separation Theorem SC Separation Theorem

Theorem: lies in the split closure of P if and only if the optimal value of the following parametric mixed integer linear program is non-negative.

i

Parameter

Parametric Mixed Integer Linear Program

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SC Separation Theorem SC Separation Theorem

Theorem: lies in the split closure of P if and only if the optimal value of the following parametric mixed integer linear program is non-negative.

(u,v,π,π0,θ): α = uA - θ π β = ub - θ π0 α x ¸ β

Split Cut

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Deparametrization Deparametrization

Parameteric Mixed Integer Linear Program

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Deparametrization Deparametrization

Parameteric Mixed Integer Linear Program

If θ is fixed, then PMILP reduces to a MILP

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Deparametrization Deparametrization

MILP( ) Deparametrized Mixed Integer Linear Program

Maintain a dynamically updated grid of parameters

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Separation Algorithm Separation Algorithm

Initialize Parameter Grid ( Σ ) For θ 2 Σ,

• Solve MILP(θ) using CPLEX 9.0
• Enumerate τ branch and bound nodes
• Store all the separating split disjunctions which

are discovered At least one split disjunction discovered? Grid Enrichment Diversification Strengthening

STOP Bifurcation

yes no

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Implementation Details Implementation Details

Processor Details

• Pentium IV
• 2Ghz, 2GB RAM

COIN-OR CPLEX 9.0

Core Implementation

• Solving Master LP
• Setting up MILP
• Disjunctions/Cuts Management
• L&P cut generation+strengthening

Solving MILP( θ )

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Computational Results Computational Results

• MIPLIB 3.0 instances
• OR-Lib (Beasley) Capacitated Warehouse Location

Problems

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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

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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

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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

75-98% Gap Closed Unsolved MIP Instance In MIPLIB 3.0

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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

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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

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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%

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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

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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

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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

25-75% Gap Closed Ceria, Pataki et al closed around 50%

• f the gap using

10 rounds of L&P cuts

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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

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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%

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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% air04 8904 62.42% 27.60% air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% enigma 100

• fast0507

63009 18.08% 4.70% fiber 1254 98.50% 98.50% gt2 188 100.00% 100.00% l152lav 1989 87.56% 69.20% lseu 89 93.75% 91.30% misc03 159 51.47% 51.20% misc07 259 19.48% 16.10% mitre 10724 100.00% 100.00% mod008 319 100.00% 100.00% mod010 2655 100.00% 100.00% nw04 87482 100.00% 100.00% p0033 33 87.42% 85.40% p0201 201 74.93% 60.50% p0282 282 99.90% 99.90% p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% seymour 1372 61.94% 23.50% stein27 27 0.00% 0.00% stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59%

% Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound)

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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% air04 8904 62.42% 27.60% air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% enigma 100

• fast0507

63009 18.08% 4.70% fiber 1254 98.50% 98.50% gt2 188 100.00% 100.00% l152lav 1989 87.56% 69.20% lseu 89 93.75% 91.30% misc03 159 51.47% 51.20% misc07 259 19.48% 16.10% mitre 10724 100.00% 100.00% mod008 319 100.00% 100.00% mod010 2655 100.00% 100.00% nw04 87482 100.00% 100.00% p0033 33 87.42% 85.40% p0201 201 74.93% 60.50% p0282 282 99.90% 99.90% p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% seymour 1372 61.94% 23.50% stein27 27 0.00% 0.00% stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59%

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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% air04 8904 62.42% 27.60% air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% enigma 100

• fast0507

63009 18.08% 4.70% fiber 1254 98.50% 98.50% gt2 188 100.00% 100.00% l152lav 1989 87.56% 69.20% lseu 89 93.75% 91.30% misc03 159 51.47% 51.20% misc07 259 19.48% 16.10% mitre 10724 100.00% 100.00% mod008 319 100.00% 100.00% mod010 2655 100.00% 100.00% nw04 87482 100.00% 100.00% p0033 33 87.42% 85.40% p0201 201 74.93% 60.50% p0282 282 99.90% 99.90% p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% seymour 1372 61.94% 23.50% stein27 27 0.00% 0.00% stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59%

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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% air04 8904 62.42% 27.60% air05 7195 62.05% 15.50% cap6000 6000 37.63% 26.90% enigma 100

• fast0507

63009 18.08% 4.70% fiber 1254 98.50% 98.50% gt2 188 100.00% 100.00% l152lav 1989 87.56% 69.20% lseu 89 93.75% 91.30% misc03 159 51.47% 51.20% misc07 259 19.48% 16.10% mitre 10724 100.00% 100.00% mod008 319 100.00% 100.00% mod010 2655 100.00% 100.00% nw04 87482 100.00% 100.00% p0033 33 87.42% 85.40% p0201 201 74.93% 60.50% p0282 282 99.90% 99.90% p0548 548 100.00% 100.00% p2756 2756 92.32% 69.20% seymour 1372 61.94% 23.50% stein27 27 0.00% 0.00% stein45 45 0.00% 0.00% 24 Instances 71.63% 62.59%

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MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances

Instance # Variables %Gap Closed (SC) %Gap Closed (CG) Ratio air04 8904 62.42% 27.60% 2.261 air05 7195 62.05% 15.50% 4.003 cap6000 6000 37.63% 26.90% 1.347 fast0507 63009 18.08% 4.70% 3.847 l152lav 1989 87.56% 69.20% 1.265 lseu 89 93.75% 91.30% 1.027 misc03 159 51.47% 51.20% 1.005 misc07 259 19.48% 16.10% 1.210 p0033 33 87.42% 85.40% 1.024 p0201 201 74.93% 60.50% 1.239 p2756 2756 92.32% 69.20% 1.334 seymour 1372 61.94% 23.50% 2.636 24 Instances 62.42% 45.09% 1.850

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MIPLIB 3.0 Pure IP Instances MIPLIB 3.0 Pure IP Instances

Instance # Variables %Gap Closed (SC) %Gap Closed (CG) Ratio air04 8904 62.42% 27.60% 2.261 air05 7195 62.05% 15.50% 4.003 cap6000 6000 37.63% 26.90% 1.347 fast0507 63009 18.08% 4.70% 3.847 l152lav 1989 87.56% 69.20% 1.265 lseu 89 93.75% 91.30% 1.027 misc03 159 51.47% 51.20% 1.005 misc07 259 19.48% 16.10% 1.210 p0033 33 87.42% 85.40% 1.024 p0201 201 74.93% 60.50% 1.239 p2756 2756 92.32% 69.20% 1.334 seymour 1372 61.94% 23.50% 2.636 24 Instances 62.42% 45.09% 1.850

Comparison of Split Closure vs CG Closure Total Number of Instances: 24 CG closure closes >98% Gap: 9 Results (Remaining 15 Instances) Split Closure closes significantly more gap in 9 instances Both Closures close almost same gap in 6 instances

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OrLib CWLP OrLib CWLP

• Set 1

– 37 Real-World Instances – 50 Customers, 16-25-50 Warehouses

• Set 2

– 12 Real-World Instances – 1000 Customers, 100 Warehouses

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OrLib CWLP Set 1 OrLib CWLP Set 1

Instance % Gap Closed Instance % Gap Closed cap41 100.000% cap93 100.000% cap42 100.000% cap94 100.000% cap43 100.000% cap101 100.000% cap44 100.000% cap102 100.000% cap51 100.000% cap103 100.000% cap61 100.000% cap104 100.000% cap62 100.000% cap111 100.000% cap63 100.000% cap112 100.000% cap64 100.000% cap113 100.000% cap71 100.000% cap114 100.000% cap72 100.000% cap121 100.000% cap73 100.000% cap122 100.000% cap74 100.000% cap123 100.000% cap81 100.000% cap124 100.000% cap82 100.000% cap131 100.000% cap83 100.000% cap132 100.000% cap84 100.000% cap133 100.000% cap91 100.000% cap134 100.000% cap92 100.000%

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OrLib CWLP Set 1 OrLib CWLP Set 1

Instance % Gap Closed Instance % Gap Closed cap41 100.000% cap93 100.000% cap42 100.000% cap94 100.000% cap43 100.000% cap101 100.000% cap44 100.000% cap102 100.000% cap51 100.000% cap103 100.000% cap61 100.000% cap104 100.000% cap62 100.000% cap111 100.000% cap63 100.000% cap112 100.000% cap64 100.000% cap113 100.000% cap71 100.000% cap114 100.000% cap72 100.000% cap121 100.000% cap73 100.000% cap122 100.000% cap74 100.000% cap123 100.000% cap81 100.000% cap124 100.000% cap82 100.000% cap131 100.000% cap83 100.000% cap132 100.000% cap84 100.000% cap133 100.000% cap91 100.000% cap134 100.000% cap92 100.000%

Summary of OrLib CWLP Instances (Set 1) Number of Instances: 37 Number of Instances included: 37 Results 100% Gap closed in 37 instances

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OrLib CWLP Set 2 OrLib CWLP Set 2

Instance % Gap Closed capa_8000 87.00% capa_10000 87.57% capa_12000 91.53% capa_14000 97.53% capb_5000 94.18% capb_6000 92.80% capb_7000 92.42% capb_8000 93.74% capc_5000 93.42% capc_5750 93.90% capc_6500 94.69% capc_7250 95.06%

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OrLib CWLP Set 2 OrLib CWLP Set 2

Instance % Gap Closed capa_8000 87.00% capa_10000 87.57% capa_12000 91.53% capa_14000 97.53% capb_5000 94.18% capb_6000 92.80% capb_7000 92.42% capb_8000 93.74% capc_5000 93.42% capc_5750 93.90% capc_6500 94.69% capc_7250 95.06%

Summary of OrLib CWFL Instances (Set 2) Number of Instances: 12 Number of Instances included: 12 Results >90% Gap closed in 10 instances 85-90% Gap closed in 2 instances Average Gap Closed: 92.82%

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Support Size & Support Size & Sparsity Sparsity

The support of a split disjunction D(π, π0) is the set of non-zero components of π

π x · π0 π x ¸ π0 + 1

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Support Size & Support Size & Sparsity Sparsity

The support of a split disjunction D(π, π0) is the set of non-zero components of π

Sparse Split Disjunctions Sparse Split Cuts

• Computationally Faster
• Avoid fill-in

Disjunctive argument Non-negative row combinations Basis Factorization Sparse Matrix Op

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Support Size & Support Size & Sparsity Sparsity

air05 [ MIPLIB 3.0 Pure IP instance, 7195 Int Var, 62.05% Gap closed by Split Closure ]

5 10 15 20 25 30 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 Iteration Support Size

Mean Support Size Standard Deviation

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Support Size & Support Size & Sparsity Sparsity

arki001** [ MIPLIB 3.0 Mixed IP instance, 538 Int Var, 83.35% Gap closed by Split Closure ]

1 2 3 4 5 6 7 8 9 10 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 Iteration Support Size

Mean Support Size Standard Deviation

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Support Size & Support Size & Sparsity Sparsity

Pure IP Instance # Int Variables Mean Support Size nw04 87482 2.084 air05 7195 8.210 seymour 1372 5.263 misc03 159 3.771 p0033 33 4.847 Mixed IP Instance # Int Variables Mean Support Size qnet1_o 1417 6.690 gesa2_o 720 4.937 arki001 538 3.146 vpm1 168 4.503 pp08aCUTS 64 3.850

Anureet Saxena, TSoB 43

Support Size & Support Size & Sparsity Sparsity

Pure IP Instance # Int Variables Mean Support Size nw04 87482 2.084 air05 7195 8.210 seymour 1372 5.263 misc03 159 3.771 p0033 33 4.847 Mixed IP Instance # Int Variables Mean Support Size qnet1_o 1417 6.690 gesa2_o 720 4.937 arki001 538 3.146 vpm1 168 4.503 pp08aCUTS 64 3.850

Empirical Observation

Substantial Duality gap can be closed by using split cuts generated from sparse split disjunctions

Anureet Saxena, TSoB 44

Support Coefficients Support Coefficients

Practice

• Elementary 0/1 disjunctions
• Mixed Integer Gomory Cuts
• Lift-and-project cuts

Theory

• Determinants of sub-matrices
• Andersen, Cornuejols & Li (’05)
• Cook, Kannan & Scrhijver (’90)

1

det (B) Huge Gap

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Support Coefficients Support Coefficients

air05 [ MIPLIB 3.0 Pure IP instance, 7195 Int Var, 62.05% Gap closed by Split Closure ]

1 2 3 4 5 6 7 8 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 Iteration Coefficient Size

Mean Coef Size Standard Deviation

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Support Coefficients Support Coefficients

arki001** [ MIPLIB 3.0 Mixed IP instance, 538 Int Var, 83.35% Gap closed by Split Closure ]

5 10 15 20 25 30 35 40 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 Iteration Coefficient Size

Mean Coef Size Standard Deviation

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Pure IP Instance #Int Variables Mean Coef Size nw04 87482 1.228 air05 7195 1.156 seymour 1372 1.099 misc03 159 1.227 p0033 33 2.099 Mixed IP Instance #Int Variables Mean Coef Size qnet1_o 1417 2.381 gesa2_o 720 1.767 arki001 538 3.044 vpm1 168 1.833 pp08aCUTS 64 1.418

Support Coefficients Support Coefficients

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Pure IP Instance #Int Variables Mean Coef Size nw04 87482 1.228 air05 7195 1.156 seymour 1372 1.099 misc03 159 1.227 p0033 33 2.099 Mixed IP Instance #Int Variables Mean Coef Size qnet1_o 1417 2.381 gesa2_o 720 1.767 arki001 538 3.044 vpm1 168 1.833 pp08aCUTS 64 1.418

Support Coefficients Support Coefficients

Empirical Observation

Substantial Duality gap can be closed by using split cuts generated from split disjunctions containing small support coefficients.

Anureet Saxena, TSoB 49

arki001 arki001

• MIPLIB 3.0 & 2003 instance
• Metallurgical Industry
• Unsolved for the past 10 years [1996-2000-2005]

Anureet Saxena, TSoB 50

arki001 arki001

• MIPLIB 3.0 & 2003 instance
• Metallurgical Industry
• Unsolved for the past 10 years [1996-2000-2005]

Problem Stats 1048 Rows 1388 Columns 123 Gen Integer Vars 415 Binary Vars 850 Continuous Vars

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Strengthening + CPLEX 9.0 Strengthening + CPLEX 9.0

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Strengthening + CPLEX 9.0 Strengthening + CPLEX 9.0

Crossover Point (227 rank-1 cuts)

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Strengthening + CPLEX 9.0 Strengthening + CPLEX 9.0

Solved to optimality

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CPLEX 9.0 CPLEX 9.0

43 million B&B nodes 22 million active nodes 12GB B&B Tree

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Comparison Comparison

Crossover Point

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Thank You