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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Veröffentlicht in:Mathematical programming Jg. 124; H. 1-2; S. 383 - 411
Hauptverfasser: Saxena, Anureet, Bonami, Pierre, Lee, Jon
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer-Verlag 01.07.2010
Springer
Springer Nature B.V
Schlagworte:
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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Zusammenfassung: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0371-9