Undercover: a primal MINLP heuristic exploring a largest sub-MIP

We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover , such that each constrai...

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Published in:	Mathematical programming Vol. 144; no. 1-2; pp. 315 - 346
Main Authors:	Berthold, Timo, Gleixner, Ambros M.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2014 Springer Nature B.V
Subjects:	Algorithms Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Complement Domain propagation Full Length Paper Heuristic Integer programming Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Neighborhoods Numerical Analysis Optimization Solvers State of the art Studies Theoretical Variables 90C20 Primal heuristic Nonconvex optimization 90C30 Mixed-integer nonlinear programming Large neighborhood search 90C11 90C26 90C59 Mixed-integer quadratically constrained programming
ISSN:	0025-5610, 1436-4646
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Abstract	We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover , such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP .
AbstractList	We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover, such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP.[PUBLICATION ABSTRACT] We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover , such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP . We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover, such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP.
Author	Berthold, Timo Gleixner, Ambros M.
Author_xml	– sequence: 1 givenname: Timo surname: Berthold fullname: Berthold, Timo email: berthold@zib.de organization: Department of Optimization, Zuse Institute Berlin – sequence: 2 givenname: Ambros M. surname: Gleixner fullname: Gleixner, Ambros M. organization: Department of Optimization, Zuse Institute Berlin
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Snippet	We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of... We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of...
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SubjectTerms	Algorithms Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Complement Domain propagation Full Length Paper Heuristic Integer programming Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Neighborhoods Numerical Analysis Optimization Solvers State of the art Studies Theoretical Variables
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Title	Undercover: a primal MINLP heuristic exploring a largest sub-MIP
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