Optimizing over the Split Closure Anureet Saxena ACO

Optimizing over the Split Closure Anureet Saxena ACO Ph. D Student, Tepper School of Business, Carnegie Mellon University. (Joint Work with Egon Balas) Anureet Saxena, TSo. B

MIP Model min cx Ax ¸ b xj 2 Z 8 j 2 N 1 Contains xj ¸ 0 j 2 N xj · uj j 2 N 1 N 1: set of integer variables Incumbent Fractional Solution Anureet Saxena, TSo. B 1

Split Disjunctions • • • 2 Z N, 0 2 Z j = 0, j 2 N 2 0 < < 0 + 1 x · 0 x ¸ 0 + 1 Split Disjunction Anureet Saxena, TSo. B 2

Split Cuts u u 0 Ax ¸ b x · 0 Ax ¸ b x ¸ 0+1 L x ¸ L R x ¸ R x¸ Anureet Saxena, TSo. B v v 0 Split Cut 3

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, TSo. B 4

Algorithmic Framework Add Cuts min cx Ax ¸ b t x¸ t t 2 Solve Master LP Integral Sol? Unbounded? Infeasible? Yes MIP Solved No Split Cuts Generated Rank-1 Split Cut Separation No Split Cuts Generated Anureet Saxena, TSo. B Optimum over Split Closure attained 5

Algorithmic Framework Add Cuts min cx Ax ¸ b t x¸ t t 2 Solve Master LP Integral Sol? Unbounded? Infeasible? Yes MIP Solved No Split Cuts Generated Rank-1 Split Cut Separation No Split Cuts Generated Anureet Saxena, TSo. B Optimum over Split Closure attained 6

SC Separation 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. Parameter (u, v, , 0, ): = u. A - Parametric = ub - 0 Mixed Integer x¸ Linear Program Split Cut Anureet Saxena, TSo. B 7

Deparametrization Parameteric Mixed Integer Linear Program Anureet Saxena, TSo. B 8

Deparametrization Parameteric Mixed Integer Linear Program If is fixed, then PMILP reduces to a MILP Anureet Saxena, TSo. B 9

Deparametrization MILP( ) Deparametrized Mixed Integer Linear Program Maintain a dynamically updated grid of parameters Anureet Saxena, TSo. B 10

Separation Algorithm Initialize Parameter Grid ( ) For 2 , Diversification • Solve MILP( ) using CPLEX 9. 0 • Enumerate branch and bound nodes • Store all the separating split disjunctions which are discovered Grid Enrichment no Strengthening At least one split disjunction yes STOP discovered? Bifurcation Anureet Saxena, TSo. B 11

Implementation Details Processor Details • Pentium IV • 2 Ghz, 2 GB RAM COIN-OR CPLEX 9. 0 Core Implementation • Solving Master LP • Setting up MILP • Disjunctions/Cuts Management • L&P cut generation+strengthening Anureet Saxena, TSo. B Solving MILP( ) 12

Computational Results • MIPLIB 3. 0 instances • OR-Lib (Beasley) Capacitated Warehouse Location Problems Anureet Saxena, TSo. B 13

MIPLIB 3. 0 MIP Instances 98 -100% Gap Closed Anureet Saxena, TSo. B 14

MIPLIB 3. 0 MIP Instances 98 -100% Gap Closed Anureet Saxena, TSo. B 15

MIPLIB 3. 0 MIP Instances Unsolved MIP Instance In MIPLIB 3. 0 75 -98% Gap Closed Anureet Saxena, TSo. B 16

MIPLIB 3. 0 MIP Instances 25 -75% Gap Closed Anureet Saxena, TSo. B 17

MIPLIB 3. 0 MIP Instances 0 -25% Gap Closed Anureet Saxena, TSo. B 18

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, TSo. B 19

MIPLIB 3. 0 Pure IP Instances 98 -100% Gap Closed Anureet Saxena, TSo. B 20

MIPLIB 3. 0 Pure IP Instances 75 -98% Gap Closed Anureet Saxena, TSo. B 21

MIPLIB 3. 0 Pure IP Instances Ceria, Pataki et al closed around 50% of the gap using 10 rounds of L&P cuts 25 -75% Gap Closed Anureet Saxena, TSo. B 22

MIPLIB 3. 0 Pure IP Instances 0 -25% Gap Closed Anureet Saxena, TSo. B 23

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: harp 2 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, TSo. B 24

MIPLIB 3. 0 Pure IP Instances % Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound) Anureet Saxena, TSo. B 25

MIPLIB 3. 0 Pure IP Instances Anureet Saxena, TSo. B 26

MIPLIB 3. 0 Pure IP Instances Anureet Saxena, TSo. B 27

MIPLIB 3. 0 Pure IP Instances 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 Anureet Saxena, TSo. B 28

Or. Lib CWLP • Set 1 – 37 Real-World Instances – 50 Customers, 16 -25 -50 Warehouses • Set 2 – 12 Real-World Instances – 1000 Customers, 100 Warehouses Anureet Saxena, TSo. B 29

Or. Lib CWLP Set 1 Summary of Or. Lib CWLP Instances (Set 1) Number of Instances: 37 Number of Instances included: 37 Results 100% Gap closed in 37 instances Anureet Saxena, TSo. B 30

Or. Lib CWLP Set 2 Summary of Or. Lib 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% Anureet Saxena, TSo. B 31

Support Size & Sparsity The support of a split disjunction D( , 0) is the set of non-zero components of x · 0 x ¸ 0 + 1 Anureet Saxena, TSo. B 32

Support Size & Sparsity The support of a split disjunction D( , 0) is the set of non-zero components of • Computationally Faster • Avoid fill-in Sparse Split Disjunctions Disjunctive argument Non-negative row combinations Sparse Split Cuts Anureet Saxena, TSo. B Basis Factorization Sparse Matrix Op 33

Support Size & Sparsity Anureet Saxena, TSo. B 34

Support Size & Sparsity Anureet Saxena, TSo. B 35

Support Size & Sparsity Empirical Observation Substantial Duality gap can be closed by using split cuts generated from sparse split disjunctions Anureet Saxena, TSo. B 36

Support Coefficients Practice Theory • Determinants of sub-matrices • Andersen, Cornuejols & Li (’ 05) • Cook, Kannan & Scrhijver (’ 90) • Elementary 0/1 disjunctions • Mixed Integer Gomory Cuts • Lift-and-project cuts Huge Gap det (B) 1 Anureet Saxena, TSo. B 37

Support Coefficients Anureet Saxena, TSo. B 38

Support Coefficients Anureet Saxena, TSo. B 39

Support Coefficients Empirical Observation Substantial Duality gap can be closed by using split cuts generated from split disjunctions containing small support coefficients. Anureet Saxena, TSo. B 40

arki 001 • MIPLIB 3. 0 & 2003 instance • Metallurgical Industry Problem Stats • Unsolved for the past 10 years [1996 -2000 -2005] 1048 Rows 1388 Columns 123 Gen Integer Vars 415 Binary Vars 850 Continuous Vars Anureet Saxena, TSo. B 41

Strengthening + CPLEX 9. 0 Solved to optimality Crossover Point (227 rank-1 cuts) Anureet Saxena, TSo. B 42

CPLEX 9. 0 43 million B&B nodes 22 million active nodes 12 GB B&B Tree Anureet Saxena, TSo. B 43

Comparison Crossover Point Anureet Saxena, TSo. B 44

Thank You Anureet Saxena, TSo. B 45