IBM CPLEX Global Non-Convex MIQP

1. IBM CPLEXGlobal Non-Convex MIQP Christian Bliek & Pierre Bonami

2. Global Non-Convex MIQP Quadratic Program (QP) • Standard form • Convex or Positive Semi-Definite • Indefinite

3. Non-Convex QP Local optimum Available since IBM CPLEX 12.3 Interior Point Algorithm Solution target Parameter FIRSTORDER Global Non-Convex MIQP

4. Local Non-Convex QP Benchmark Performance Cplex versus Ipopt with Wsmp Global Non-Convex MIQP

5. Non-Convex MIQP Global optimum NEW in CPLEX 12.6 Branch and Bound Global Non-Convex MIQP

6. Global Non-Convex MIQP Example Local Optimum Global Optimum 6

7. Global Non-Convex QP Even if Q has only 1 negative eigenvalue, Non-Convex QP is NP-hard Checking if a feasible solution is not a local minimum is NP-complete Checking if a Non-Convex QP is unbounded is NP-complete Global Non-Convex MIQP

8. We consider 2 formulations Original Factorized Eigenvalue Overview Global Non-Convex MIQP

9. Factorized Eigenvalue Formulation Global Non-Convex MIQP

10. Factorized Eigenvalue Formulation Global Non-Convex MIQP

11. Factorized Eigenvalue Formulation Global Non-Convex MIQP

12. Factorized Eigenvalue Formulation Global Non-Convex MIQP

13. Factorized Eigenvalue Formulation Global Non-Convex MIQP

14. Factorized Eigenvalue Formulation Global Non-Convex MIQP Advantage • Sparse • Efficient • Proper identification of negative eigenvalues

15. Original Formulation Factorized Eigenvalue Formulation Global Non-Convex MIQP Example

16. We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview

17. We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Global Non-Convex MIQP Overview

18. We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation Global Non-Convex MIQP Overview

19. Relaxation of Non-Convex MIQP Global Non-Convex MIQP

20. Global Non-Convex MIQP Relaxation of Non-Convex MIQP • Relaxation of individual Non-Convex quadratic terms using McCormick envelopes 20

21. We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation Global Non-Convex MIQP Overview

22. We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation Branch and Bound Global Non-Convex MIQP Overview

23. Global Non-Convex MIQP Branching for Non-Convex MIQP • Branch on continuous variables and update envelopes 23

24. Other Ingredients QP simplex for convex QP relaxation Pseudocost branching Local interior point solver for incumbents Bound strengthening Detection of unboundedness Linearize quadratic terms involving binaries Global Non-Convex MIQP

25. Global Non-Convex MIQP Global Non-Convex QP Benchmark • internal non-convex miqp testset • globallib GAMS • minlp.org • boxqp • From miqp testset generated 50% mixed miqp set • Comparison with SCIP and Couenne on 1 thread

26. CPLEX versus SCIP on individual testsets Global Non-Convex MIQP Global Non-Convex QP Benchmark

27. CPLEX versus SCIP and Couenne on combined testset Global Non-Convex MIQP Global Non-Convex QP Benchmark

28. CPLEX versus SCIP and Couenne on combined testset Global Non-Convex MIQP Global Non-Convex QP Benchmark

29. CPLEX 1 versus 4 threads on combined testset Global Non-Convex MIQP Global Non-Convex QP Benchmark

30. Available in CPLEX 12.6 By default Non-Convex MIQP are not accepted Set Solution Target Parameter to OPTIMALGLOBAL Global Non-Convex MIQP How to use it

31. CPLEX versus SCIP and Couenne on combined testset Global Non-Convex MIQP Global Non-Convex QP Benchmark