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IBM CPLEX Global Non-Convex MIQP
Christian Bliek & Pierre Bonami
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Quadratic Program (QP)
Global Non-Convex MIQP Quadratic Program (QP) Standard form Convex or Positive Semi-Definite Indefinite
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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
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Local Non-Convex QP Benchmark
Global Non-Convex MIQP Local Non-Convex QP Benchmark Performance Cplex versus Ipopt with Wsmp
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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
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Global Non-Convex MIQP
Example Local Optimum Global Optimum 6
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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
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Overview We consider 2 formulations Original Factorized Eigenvalue
Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation
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Factorized Eigenvalue Formulation
Global Non-Convex MIQP Factorized Eigenvalue Formulation Advantage Sparse Efficient Proper identification of negative eigenvalues
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Example Original Formulation Factorized Eigenvalue Formulation
Global Non-Convex MIQP Example Original Formulation Factorized Eigenvalue Formulation
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Overview We consider 2 formulations Original Factorized Eigenvalue
Global Non-Convex MIQP Overview We consider 2 formulations Original Factorized Eigenvalue
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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
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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
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Relaxation of Non-Convex MIQP
Global Non-Convex MIQP Relaxation of Non-Convex MIQP
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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
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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
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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
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Branching for Non-Convex MIQP
Global Non-Convex MIQP Branching for Non-Convex MIQP Branch on continuous variables and update envelopes 23
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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
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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
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Global Non-Convex QP Benchmark
Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP on individual testsets
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Global Non-Convex QP Benchmark
Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset
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Global Non-Convex QP Benchmark
Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset
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Global Non-Convex QP Benchmark
Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX 1 versus 4 threads on combined testset
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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
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Global Non-Convex QP Benchmark
Global Non-Convex MIQP Global Non-Convex QP Benchmark CPLEX versus SCIP and Couenne on combined testset
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