Outer approximation with conic certificates for mixed-integer convex problems

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K ∗ cuts...

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Veröffentlicht in:	Mathematical programming computation Jg. 12; H. 2; S. 249 - 293
Hauptverfasser:	Coey, Chris, Lubin, Miles, Vielma, Juan Pablo
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2020 Springer Nature B.V
Schlagworte:	Algorithms Approximation Certificates Cones Full Length Paper Heuristic methods Integers Mathematical analysis Mathematics Mathematics and Statistics Mathematics of Computing Operations Research/Decision Theory Optimization Theory of Computation Well posed problems 90C25 Convex programming 90C26 Nonconvex programming, global optimization 90C11 Mixed integer programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
ISSN:	1867-2949, 1867-2957
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Abstract	A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K ∗ cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed , the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K ∗ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate K ∗ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K ∗ cuts. Our new open source MI-conic solver Pajarito ( github.com/JuliaOpt/Pajarito.jl ) uses an external mixed-integer linear solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX’s specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K ∗ cuts.
AbstractList	A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K∗cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K∗ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregateK∗ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K∗ cuts. Our new open source MI-conic solver Pajarito (github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX’s specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K∗ cuts. A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K ∗ cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed , the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K ∗ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate K ∗ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K ∗ cuts. Our new open source MI-conic solver Pajarito ( github.com/JuliaOpt/Pajarito.jl ) uses an external mixed-integer linear solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX’s specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K ∗ cuts.
Author	Coey, Chris Vielma, Juan Pablo Lubin, Miles
Author_xml	– sequence: 1 givenname: Chris surname: Coey fullname: Coey, Chris email: coey@mit.edu organization: Operations Research Center, Massachusetts Institute of Technology – sequence: 2 givenname: Miles surname: Lubin fullname: Lubin, Miles organization: Operations Research Center, Massachusetts Institute of Technology – sequence: 3 givenname: Juan Pablo surname: Vielma fullname: Vielma, Juan Pablo organization: Sloan School of Management, Massachusetts Institute of Technology
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Copyright	Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020 Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020.
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Snippet	A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound...
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SubjectTerms	Algorithms Approximation Certificates Cones Full Length Paper Heuristic methods Integers Mathematical analysis Mathematics Mathematics and Statistics Mathematics of Computing Operations Research/Decision Theory Optimization Theory of Computation Well posed problems
Title	Outer approximation with conic certificates for mixed-integer convex problems
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