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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). Talk Outiline. Cutting Planes Commercial Split Closure Separation Problem PMILP & Deparametrization Computational Results.

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Optimizing over the Split Closure

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

  2. Talk Outiline • Cutting Planes Commercial • Split Closure Separation Problem • PMILP & Deparametrization • Computational Results • Support Size & Sparsity • Support Coefficients • Cuts Statistics • arki001solved Anureet Saxena, TSoB

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. Split Disjunctions • 2ZN, 02Z • j = 0, j2 N2 • 0 <  < 0 + 1 x ·0 x ¸0 + 1 Split Disjunction Anureet Saxena, TSoB

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. Deparametrization Parameteric Mixed Integer Linear Program Anureet Saxena, TSoB

  22. Deparametrization Parameteric Mixed Integer Linear Program If  is fixed, then PMILP reduces to a MILP Anureet Saxena, TSoB

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

  24. 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

  25. 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

  26. Computational Results • MIPLIB 3.0 instances • OR-Lib (Beasley) Capacitated Warehouse Location Problems Anureet Saxena, TSoB

  27. MIPLIB 3.0 MIP Instances 98-100% Gap Closed Anureet Saxena, TSoB

  28. MIPLIB 3.0 MIP Instances 98-100% Gap Closed Anureet Saxena, TSoB

  29. MIPLIB 3.0 MIP Instances Unsolved MIP Instance In MIPLIB 3.0 75-98% Gap Closed Anureet Saxena, TSoB

  30. MIPLIB 3.0 MIP Instances 25-75% Gap Closed Anureet Saxena, TSoB

  31. MIPLIB 3.0 MIP Instances 0-25% Gap Closed Anureet Saxena, TSoB

  32. 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

  33. MIPLIB 3.0 Pure IP Instances 98-100% Gap Closed Anureet Saxena, TSoB

  34. MIPLIB 3.0 Pure IP Instances 75-98% Gap Closed Anureet Saxena, TSoB

  35. 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

  36. MIPLIB 3.0 Pure IP Instances 0-25% Gap Closed Anureet Saxena, TSoB

  37. 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

  38. MIPLIB 3.0 Pure IP Instances % Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound) Anureet Saxena, TSoB

  39. MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB

  40. MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

  49. Support Size & Sparsity Anureet Saxena, TSoB

  50. Support Size & Sparsity Anureet Saxena, TSoB

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