Semidefinite relaxations for non-convex quadratic mixed-integer programming

We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem co...

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Bibliographic Details
Published in:Mathematical programming Vol. 141; no. 1-2; pp. 435 - 452
Main Authors: Buchheim, Christoph, Wiegele, Angelika
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2013
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computation
Computer programming
Full Length Paper
Integer programming
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Optimization techniques
Programming
Routines
Studies
Theoretical
Variables
United States > US
90C20
90C10
90C26
90C11
90C22
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0534-y