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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). MIP Model. Contains x j ¸ 0 j 2 N x j · u j j 2 N 1. min cx Ax ¸ b x j 2 Z 8 j 2 N 1.

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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. 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

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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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 (u,v,,0,): = uA -  = ub - 0 x ¸ Parametric Mixed Integer Linear Program Split Cut Anureet Saxena, TSoB

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

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

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

  12. 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

  13. 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

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

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

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

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

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

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

  20. 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 98-100% Gap Closed Anureet Saxena, TSoB

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

  23. 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

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

  25. 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 % Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound) Anureet Saxena, TSoB

  27. MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB

  28. MIPLIB 3.0 Pure IP Instances Anureet Saxena, TSoB

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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, TSoB

  34. 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

  35. Support Size & Sparsity Anureet Saxena, TSoB

  36. Support Size & Sparsity Anureet Saxena, TSoB

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

  38. 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) Huge Gap det (B) 1 Anureet Saxena, TSoB

  39. Support Coefficients Anureet Saxena, TSoB

  40. Support Coefficients Anureet Saxena, TSoB

  41. 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

  42. 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 Anureet Saxena, TSoB

  43. Strengthening + CPLEX 9.0 Solved to optimality Crossover Point (227 rank-1 cuts) Anureet Saxena, TSoB

  44. CPLEX 9.0 43 million B&B nodes 22 million active nodes 12GB B&B Tree Anureet Saxena, TSoB

  45. Comparison Crossover Point Anureet Saxena, TSoB

  46. Thank You Anureet Saxena, TSoB

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