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

Anureet Saxena, TSoB1 Talk Outiline Cutting Planes Commercial Split Closure Separation Problem PMILP & Deparametrization Computational Results Support Size & Sparsity Support Coefficients Cuts Statistics arki001 solved

Anureet Saxena, TSoB2 MIP Model min cx Ax ¸ b x j 2 Z 8 j2N 1 N 1 : set of integer variables Contains x j ¸ 0 j2N x j · u j j2N 1 Incumbent Fractional Solution

Anureet Saxena, TSoB3 Taxonomy of Cutting Planes Fractional Basic Feas Fractional Basic Mixed Integer Basic Feas Intersection Basic Feas Intersection Basic L&P Simple Disjunctive Chvatal Fractional Gomory Mixed Integer Basic Intersection Basic + Strengthening L&P + Strengthening MIG Split Cuts Intersection Basic Feas + Strengthening MIR

Anureet Saxena, TSoB4 Taxonomy of Cutting Planes Fractional Basic Feas Fractional Basic Mixed Integer Basic Feas Intersection Basic Feas Intersection Basic L&P Simple Disjunctive Chvatal Fractional Gomory Mixed Integer Basic Intersection Basic + Strengthening L&P + Strengthening MIG Split Cuts Intersection Basic Feas + Strengthening MIR

Anureet Saxena, TSoB5 Taxonomy of Cutting Planes Fractional Basic Feas Fractional Basic Mixed Integer Basic Feas Intersection Basic Feas Intersection Basic L&P Simple Disjunctive Chvatal Fractional Gomory Mixed Integer Basic Intersection Basic + Strengthening L&P + Strengthening MIG Split Cuts Intersection Basic Feas + Strengthening MIR

Anureet Saxena, TSoB6 Taxonomy of Cutting Planes Fractional Basic Feas Fractional Basic Mixed Integer Basic Feas Intersection Basic Feas Intersection Basic L&P Simple Disjunctive Chvatal Fractional Gomory Mixed Integer Basic Intersection Basic + Strengthening L&P + Strengthening MIG Split Cuts Intersection Basic Feas + Strengthening MIR

Anureet Saxena, TSoB7 Taxonomy of Cutting Planes 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

Anureet Saxena, TSoB8 Elementary Closures Intersection Basic L&P Simple Disjunctive Chvatal Fractional Gomory MIG Split Cuts MIR Split Closure CG Closure L&P Closure

Anureet Saxena, TSoB9 Elementary Closures Operations Research Constraint Programming Complexity Theory max v x2P I ) P cx¸v P2  Inference Dual Proof Family Rank-1 cuts have short polynomial length proofs

Anureet Saxena, TSoB10 Elementary Closures How much duality gap can be closed by optimizing over elementary closures? L&P Closure Bonami and Minoux CG Closure Fischetti and Lodi Split Closure ?

Anureet Saxena, TSoB11 Elementary Closures How much duality gap can be closed by optimizing over elementary closures? L&P Closure Bonami and Minoux CG Closure Fischetti and Lodi Split Closure Balas and Saxena

Anureet Saxena, TSoB12 Split Disjunctions  2 Z N,  0 2 Z  j = 0, j2 N 2  0 <  <  Split Disjunction  x ·  0  x ¸  0 + 1

Anureet Saxena, TSoB13 Split Cuts Ax ¸ b  x ·  0 Ax ¸ b  x ¸  0 +1 u u0u0 v0v0 v  L x ¸  L  R x ¸  R  x ¸  Split Cut

Anureet Saxena, TSoB14 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, TSoB15 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  t x¸  t t2  Yes No Add Cuts

Anureet Saxena, TSoB16 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  t x¸  t t2  Yes No Add Cuts

Anureet Saxena, TSoB17 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. Disjunctive Cut Cut Violation Split Disjunction Normalization Set  = 1 u.e + v.e + u 0 + v 0 = 1 u 0 + v 0 = 1  y = 1 |  | 2 =1

Anureet Saxena, TSoB18 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 u 0 + v 0 = 1

Anureet Saxena, TSoB19 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. Parameter Parametric Mixed Integer Linear Program

Anureet Saxena, TSoB20 Deparametrization Parameteric Mixed Integer Linear Program

Anureet Saxena, TSoB21 Deparametrization Parameteric Mixed Integer Linear Program If  is fixed, then PMILP reduces to a MILP

Anureet Saxena, TSoB22 Deparametrization MILP( ) Deparametrized Mixed Integer Linear Program Maintain a dynamically updated grid of parameters

Anureet Saxena, TSoB23 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 At least one split disjunction discovered? Grid Enrichment Diversification Strengthening STOP Bifurcation yes no

Anureet Saxena, TSoB24 Implementation Details Processor Details Pentium IV 2Ghz, 2GB RAM COIN-ORCPLEX 9.0 Core Implementation Solving Master LP Setting up MILP Disjunctions/Cuts Management L&P cut generation+strengthening Solving MILP(  )

Anureet Saxena, TSoB25 Computational Results MIPLIB 3.0 instances OR-Lib (Beasley) Capacitated Warehouse Location Problems

Anureet Saxena, TSoB26 MIPLIB 3.0 MIP Instances % Gap Closed

Anureet Saxena, TSoB27 MIPLIB 3.0 MIP Instances % Gap Closed

Anureet Saxena, TSoB28 MIPLIB 3.0 MIP Instances 75-98% Gap Closed Unsolved MIP Instance In MIPLIB 3.0

Anureet Saxena, TSoB29 MIPLIB 3.0 MIP Instances 25-75% Gap Closed

Anureet Saxena, TSoB30 MIPLIB 3.0 MIP Instances 0-25% Gap Closed

Anureet Saxena, TSoB31 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 % 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, TSoB32 MIPLIB 3.0 Pure IP Instances % Gap Closed

Anureet Saxena, TSoB33 MIPLIB 3.0 Pure IP Instances 75-98% Gap Closed

Anureet Saxena, TSoB34 MIPLIB 3.0 Pure IP Instances 25-75% Gap Closed Ceria, Pataki et al closed around 50% of the gap using 10 rounds of L&P cuts

Anureet Saxena, TSoB35 MIPLIB 3.0 Pure IP Instances 0-25% Gap Closed

Anureet Saxena, TSoB36 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 % 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, TSoB37 MIPLIB 3.0 Pure IP Instances % Gap Closed by First Chvatal Closure (Fischetti-Lodi Bound)

Anureet Saxena, TSoB38 MIPLIB 3.0 Pure IP Instances

Anureet Saxena, TSoB39 MIPLIB 3.0 Pure IP Instances

Anureet Saxena, TSoB40 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, TSoB41 OrLib CWLP Set 1 –37 Real-World Instances –50 Customers, Warehouses Set 2 –12 Real-World Instances –1000 Customers, 100 Warehouses

Anureet Saxena, TSoB42 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, TSoB43 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, TSoB44 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  t x¸  t t2  Yes No Add Cuts

Anureet Saxena, TSoB45 Algorithmic Framework What can one say about the split disjunctions which were used to generate cuts? What are the characteristics of the cuts which are binding at the final optimal solution?

Anureet Saxena, TSoB46 Support Size & Sparsity The support of a split disjunction D( ,  0 ) is the set of non-zero components of   x ·  0  x ¸  (2x 1 + 3x 3 – x 5 · 1) Ç (2x 1 + 3x 3 – x 5 ¸ 2) Support Size = 3

Anureet Saxena, TSoB47 Support Size & Sparsity The support of a split disjunction D( ,  0 ) is the set of non-zero components of  Sparse Split Disjunctions Sparse Split Cuts Computationally Faster Avoid fill-in Disjunctive argument Non-negative row combinations Basis Factorization Sparse Matrix Op

Anureet Saxena, TSoB48 Support Size & Sparsity

Anureet Saxena, TSoB49 Support Size & Sparsity

Anureet Saxena, TSoB50 Support Size & Sparsity Empirical Observation Substantial Duality gap can be closed by using split cuts generated from sparse split disjunctions

Anureet Saxena, TSoB51 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

Anureet Saxena, TSoB52 Support Coefficients

Anureet Saxena, TSoB53 Support Coefficients

Anureet Saxena, TSoB54 Support Coefficients Empirical Observation Substantial Duality gap can be closed by using split cuts generated from split disjunctions containing small support coefficients.

Anureet Saxena, TSoB55 Cuts Statistics

Anureet Saxena, TSoB56 Number of Cuts Average:

Anureet Saxena, TSoB57 #Cuts/m vs log(n) Average: 45.76%

Anureet Saxena, TSoB58 Average Cut Density vs log(n) Average: 20.82%

Anureet Saxena, TSoB59 Cuts Statistics Internet Checkable Proofs Strengthened formulations for MIPLIB 3.0 instances available at Google Query: anureet saxena (I’m feeling lucky)

Anureet Saxena, TSoB60 arki001 MIPLIB 3.0 & 2003 instance Metallurgical Industry Unsolved for the past 10 years [ ] Problem Stats 1048 Rows 1388 Columns 123 Gen Integer Vars 415 Binary Vars 850 Continuous Vars

Anureet Saxena, TSoB61 Solution Strategy Original Problem Strengthened Formulation Preprocessed Problem CPLEX 9.0 Presolver Rank-1 Split Cut Generation Emphasis on optimality Strong Branching

Anureet Saxena, TSoB62 Strengthening + CPLEX 9.0 Crossover Point (227 rank-1 cuts) Solved to optimality

Anureet Saxena, TSoB63 Strengthening + CPLEX 9.0 arki001 Solution Statistics % Gap closed by rank-1 split cuts: 83.05% Time spent in generating rank-1 split cuts: hrs Time taken by CPLEX 9.0 after strengthening: hrs No. of branch-and-bound nodes enumerated by CPLEX: Total time taken to solve the instance to optimality: hrs

Anureet Saxena, TSoB64 CPLEX 9.0 After 100 hours: 43 million B&B nodes 22 million active nodes 12GB B&B Tree

Anureet Saxena, TSoB65 Comparison Crossover Point

Anureet Saxena, TSoB66 Thank You