Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce s...

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Vydané v:	Mathematical programming Ročník 124; číslo 1-2; s. 383 - 411
Hlavní autori:	Saxena, Anureet, Bonami, Pierre, Lee, Jon
Médium:	Journal Article Konferenčný príspevok..
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer-Verlag 01.07.2010 Springer Springer Nature B.V
Predmet:	Applied sciences Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Constraints Cutting Eigenvalues Eigenvectors Exact sciences and technology Full Length Paper Inequality Linear equations Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Programming Theoretical global optimization 90C26 Nonconvex programming Non convex programming Eigenvector Constraint satisfaction Global optimum Mixed integer programming Quadratic programming Modeling Cutting plane method Convex programming Cut generation Integer programming Disjunction Disjunctive programming Eigenvalue problem Convexity Non convex analysis
ISSN:	0025-5610, 1436-4646
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Abstract	This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.
AbstractList	This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x super( )T We use the non-convex constraint Y - x x T <= 0 to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y - x x super( )Twith positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint Y - x x T [sccue] 0 to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings. Issue Title: Series B - Special Issue: Combinatorial Optimization and Integer Programming (ProQuest: Abstract omitted; see image)[PUBLICATION ABSTRACT] This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.
Author	Lee, Jon Bonami, Pierre Saxena, Anureet
Author_xml	– sequence: 1 givenname: Anureet surname: Saxena fullname: Saxena, Anureet email: asaxena@axiomainc.com organization: Axioma Inc – sequence: 2 givenname: Pierre surname: Bonami fullname: Bonami, Pierre organization: Laboratoire d’Informatique Fondamentale de Marseille, CNRS-Aix Marseille Universités – sequence: 3 givenname: Jon surname: Lee fullname: Lee, Jon organization: IBM T.J. Watson Research Center
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Keywords	global optimization 90C26 Nonconvex programming Non convex programming Eigenvector Constraint satisfaction Global optimum Mixed integer programming Quadratic programming Modeling Cutting plane method Convex programming Cut generation Integer programming Disjunction Disjunctive programming Eigenvalue problem Convexity Non convex analysis
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Snippet	This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP... Issue Title: Series B - Special Issue: Combinatorial Optimization and Integer Programming (ProQuest: Abstract omitted; see image)[PUBLICATION ABSTRACT]
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SubjectTerms	Applied sciences Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Constraints Cutting Eigenvalues Eigenvectors Exact sciences and technology Full Length Paper Inequality Linear equations Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Programming Theoretical
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Title	Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations
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