An effective branch-and-bound algorithm for convex quadratic integer programming

We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of th...

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Vydáno v:	Mathematical programming Ročník 135; číslo 1-2; s. 369 - 395
Hlavní autoři:	Buchheim, Christoph, Caprara, Alberto, Lodi, Andrea
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer-Verlag 01.10.2012 Springer Springer Nature B.V
Témata:	Algorithms Applied sciences Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Ellipsoids Enumeration Exact sciences and technology 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 Operational research and scientific management Operational research. Management science Optimization Preprocessing Theoretical Trees Variables 90C10 90C57 Computational results 90C25 90C90 Convex quadratic minimization Branch-and-bound algorithm Closest vector problem Lower bound Tree(graph) Enumeration Branch and bound method Linear time Quadratic programming Convex programming Lattice Integer programming Linear process Implicit enumeration method Convex function Objective function Ellipsoid Quadratic function Mathematical programming
ISSN:	0025-5610, 1436-4646
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Popis
Shrnutí:	We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids. Experiments show that our approach is very fast on both unconstrained problems and problems with box constraints. The main reason is that all expensive calculations can be done in a preprocessing phase, while a single node in the enumeration tree can be processed in linear time in the problem dimension.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23
ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-011-0475-x