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Optimizing over the Split Closure. Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. (Joint Work with Egon Balas). Talk Outiline. Cutting Planes Commercial Split Closure Separation Problem PMILP & Deparametrization Computational Results.
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Optimizing over the Split Closure Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. (Joint Work with Egon Balas)
Talk Outiline • Cutting Planes Commercial • Split Closure Separation Problem • PMILP & Deparametrization • Computational Results • Support Size & Sparsity • Support Coefficients • Cuts Statistics • arki001solved Anureet Saxena, TSoB
MIP Model Contains xj¸ 0 j2N xj· uj j2N1 min cx Ax ¸ b xj2Z8 j2N1 N1: set of integer variables Incumbent Fractional Solution Anureet Saxena, TSoB
Taxonomy of Cutting Planes MIR Fractional Gomory Fractional Basic Split Cuts Fractional Basic Feas Chvatal MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Taxonomy of Cutting Planes MIR Fractional Gomory Fractional Basic Split Cuts Fractional Basic Feas Chvatal MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Taxonomy of Cutting Planes MIR Fractional Gomory Fractional Basic Split Cuts Fractional Basic Feas Chvatal MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Taxonomy of Cutting Planes MIR Fractional Gomory Fractional Basic Split Cuts Fractional Basic Feas Chvatal MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Taxonomy of Cutting Planes MIR Fractional Gomory Fractional Basic Split Cuts Elementary Closure Elementary closure of P w.r.t a family of cutting planes is defined by intersecting P with all rank-1 cuts in . Eg: CG Closure, Split Closure Fractional Basic Feas Chvatal MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Elementary Closures MIR Fractional Gomory Fractional Basic Split Cuts CG Closure Fractional Basic Feas Chvatal Split Closure MIG Mixed Integer Basic Feas Intersection Basic + Strengthening Mixed Integer Basic L&P + Strengthening Intersection Basic Feas + Strengthening L&P Closure Intersection Basic Intersection Basic Feas Simple Disjunctive L&P Anureet Saxena, TSoB
Elementary Closures Operations Research Inference Dual Rank-1 cuts have short polynomial length proofs max v x2PI )P cx¸v P2 Elementary Closures Proof Family Constraint Programming Complexity Theory Anureet Saxena, TSoB
Elementary Closures How much duality gap can be closed by optimizing over elementary closures? Split Closure ? CG Closure Fischetti and Lodi L&P Closure Bonami and Minoux Anureet Saxena, TSoB
Elementary Closures How much duality gap can be closed by optimizing over elementary closures? Split Closure Balas and Saxena CG Closure Fischetti and Lodi L&P Closure Bonami and Minoux Anureet Saxena, TSoB
Split Disjunctions • 2ZN, 02Z • j = 0, j2 N2 • 0 < < 0 + 1 x ·0 x ¸0 + 1 Split Disjunction Anureet Saxena, TSoB
Split Cuts Ax ¸ b x ·0 Ax ¸ b x ¸0+1 u v u0 v0 L x ¸L R x ¸R x ¸ Split Cut Anureet Saxena, TSoB
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. C = { x2 P | x ¸8 rank-1 split cuts x¸} Without Recursion Anureet Saxena, TSoB
Algorithmic Framework min cx Ax ¸ b t x¸t t2 Solve Master LP Add Cuts Integral Sol? Unbounded? Infeasible? Yes MIP Solved No No Split Cuts Generated Split Cuts Generated Optimum over Split Closure attained Rank-1 Split Cut Separation Anureet Saxena, TSoB
Algorithmic Framework min cx Ax ¸ b t x¸t t2 Solve Master LP Add Cuts Integral Sol? Unbounded? Infeasible? Yes MIP Solved No No Split Cuts Generated Split Cuts Generated Optimum over Split Closure attained Rank-1 Split Cut Separation Anureet Saxena, TSoB
Split Closure Separation Problem Theorem: lies in the split closure of P if and only if the optimal value of the following program is non-negative. Cut Violation • = 1 • u.e + v.e + u0 + v0 = 1 • u0 + v0 = 1 • y = 1 • ||2=1 Disjunctive Cut Split Disjunction Normalization Set Anureet Saxena, TSoB
Split Closure Separation Problem Theorem: lies in the split closure of P if and only if the optimal value of the following program is non-negative. Mixed Integer Non-Convex Quadratic Program u0 + v0 = 1 Anureet Saxena, TSoB
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 programis non-negative. Parameter Parametric Mixed Integer Linear Program Anureet Saxena, TSoB
Deparametrization Parameteric Mixed Integer Linear Program Anureet Saxena, TSoB
Deparametrization Parameteric Mixed Integer Linear Program If is fixed, then PMILP reduces to a MILP Anureet Saxena, TSoB
Deparametrization MILP( ) Deparametrized Mixed Integer Linear Program Maintain a dynamically updated grid of parameters Anureet Saxena, TSoB
Separation Algorithm Initialize Parameter Grid ( ) • For 2, • Solve MILP() using CPLEX 9.0 • Enumerate branch and bound nodes • Store all the separating split disjunctions which are discovered Diversification Strengthening At least one split disjunction discovered? Grid Enrichment no yes STOP Bifurcation Anureet Saxena, TSoB
Implementation Details • Processor Details • Pentium IV • 2Ghz, 2GB RAM COIN-OR CPLEX 9.0 Solving MILP( ) • Core Implementation • Solving Master LP • Setting up MILP • Disjunctions/Cuts Management • L&P cut generation+strengthening Anureet Saxena, TSoB
Computational Results • MIPLIB 3.0 instances • OR-Lib (Beasley) Capacitated Warehouse Location Problems Anureet Saxena, TSoB
MIPLIB 3.0 MIP Instances 98-100% Gap Closed Anureet Saxena, TSoB
MIPLIB 3.0 MIP Instances 98-100% Gap Closed Anureet Saxena, TSoB
MIPLIB 3.0 MIP Instances Unsolved MIP Instance In MIPLIB 3.0 75-98% Gap Closed Anureet Saxena, TSoB
MIPLIB 3.0 MIP Instances 25-75% Gap Closed Anureet Saxena, TSoB
MIPLIB 3.0 MIP Instances 0-25% Gap Closed Anureet Saxena, TSoB
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
MIPLIB 3.0 Pure IP Instances 98-100% Gap Closed Anureet Saxena, TSoB
MIPLIB 3.0 Pure IP Instances 75-98% Gap Closed Anureet Saxena, TSoB
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, TSoB
MIPLIB 3.0 Pure IP Instances 0-25% Gap Closed Anureet Saxena, TSoB
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
MIPLIB 3.0 Pure IP Instances % Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound) Anureet Saxena, TSoB
MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB
MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB
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 6instances Anureet Saxena, TSoB
OrLib 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, TSoB
OrLib CWLP Set 1 Summary of OrLib CWLP Instances (Set 1) Number of Instances: 37 Number of Instances included: 37 Results 100% Gap closed in 37 instances Anureet Saxena, TSoB
OrLib CWLP Set 2 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% Anureet Saxena, TSoB
Algorithmic Framework min cx Ax ¸ b t x¸t t2 Solve Master LP Add Cuts Integral Sol? Unbounded? Infeasible? Yes MIP Solved No No Split Cuts Generated Split Cuts Generated Optimum over Split Closure attained Rank-1 Split Cut Separation Anureet Saxena, TSoB
Algorithmic Framework min cx Ax ¸ b t x¸t t2 Solve Master LP Add Cuts What are the characteristics of the cuts which are binding at the final optimal solution? Integral Sol? Unbounded? Infeasible? Yes MIP Solved What can one say about the split disjunctions which were used to generate cuts? No No Split Cuts Generated Split Cuts Generated Optimum over Split Closure attained Rank-1 Split Cut Separation Anureet Saxena, TSoB
Support Size & Sparsity The support of a split disjunction D(, 0) is the set of non-zero components of x ·0 x ¸0 + 1 (2x1 + 3x3 – x5· 1) Ç (2x1 + 3x3 – x5¸ 2) Support Size = 3 Anureet Saxena, TSoB
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 Basis Factorization Sparse Matrix Op Disjunctive argument Non-negative row combinations Sparse Split Cuts Anureet Saxena, TSoB
Support Size & Sparsity Anureet Saxena, TSoB
Support Size & Sparsity Anureet Saxena, TSoB