IBM CPLEX Global Non-Convex MIQP Christian Bliek & Pierre Bonami
Quadratic Program (QP) Global Non-Convex MIQP Quadratic Program (QP) Standard form Convex or Positive Semi-Definite Indefinite
Non-Convex QP Local optimum Available since IBM CPLEX 12.3 Global Non-Convex MIQP Non-Convex QP Local optimum Available since IBM CPLEX 12.3 Interior Point Algorithm Solution target Parameter FIRSTORDER
Local Non-Convex QP Benchmark Global Non-Convex MIQP Local Non-Convex QP Benchmark Performance Cplex versus Ipopt with Wsmp
Non-Convex MIQP Global optimum NEW in CPLEX 12.6 Branch and Bound Global Non-Convex MIQP Non-Convex MIQP Global optimum NEW in CPLEX 12.6 Branch and Bound
Global Non-Convex MIQP Example Local Optimum Global Optimum 6
Global Non-Convex MIQP 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
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation
Factorized Eigenvalue Formulation Global Non-Convex MIQP Factorized Eigenvalue Formulation Advantage Sparse Efficient Proper identification of negative eigenvalues
Example Original Formulation Factorized Eigenvalue Formulation Global Non-Convex MIQP Example Original Formulation Factorized Eigenvalue Formulation
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation
Relaxation of Non-Convex MIQP Global Non-Convex MIQP Relaxation of Non-Convex MIQP
Relaxation of Non-Convex MIQP Global Non-Convex MIQP Relaxation of Non-Convex MIQP Relaxation of individual Non-Convex quadratic terms using McCormick envelopes 20
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation
Overview We consider 2 formulations Original Factorized Eigenvalue Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue Automatically select most promising one Do Term by Term McCormick Relaxation Branch and Bound
Branching for Non-Convex MIQP Global Non-Convex MIQP Branching for Non-Convex MIQP Branch on continuous variables and update envelopes 23
Other Ingredients QP simplex for convex QP relaxation Global Non-Convex MIQP 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 QP Benchmark 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
Global Non-Convex QP Benchmark Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP on individual testsets
Global Non-Convex QP Benchmark Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset
Global Non-Convex QP Benchmark Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset
Global Non-Convex QP Benchmark Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX 1 versus 4 threads on combined testset
How to use it Available in CPLEX 12.6 Global Non-Convex MIQP How to use it Available in CPLEX 12.6 By default Non-Convex MIQP are not accepted Set Solution Target Parameter to OPTIMALGLOBAL
Global Non-Convex QP Benchmark Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset