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Quadratic 0–1 programming: Tightening linear or quadratic convex reformulation by use of relaxations

Published online by Cambridge University Press:  17 May 2008

Alain Billionnet
Affiliation:
Laboratoire CEDRIC, ENSIIE, 18 allée Jean Rostand, 91025 Evry, France; billionnet@ensiie.fr
Sourour Elloumi
Affiliation:
Laboratoire CEDRIC, Conservatoire National des Arts et Métiers, 292 rue Saint Martin, 75141 Paris, France; e-mail:
Marie-Christine Plateau
Affiliation:
Laboratoire CEDRIC, Conservatoire National des Arts et Métiers, 292 rue Saint Martin, 75141 Paris, France; e-mail:
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Abstract

Many combinatorial optimization problems can be formulated as the minimization of a 0–1 quadratic function subject to linear constraints. In this paper, we are interested in the exact solution of this problem through a two-phase general scheme. The first phase consists in reformulating the initial problem either into a compact mixed integer linear program or into a 0–1 quadratic convex program. The second phase simply consists in submitting the reformulated problem to a standard solver. The efficiency of this scheme strongly depends on the quality of the reformulation obtained in phase 1. We show that a good compact linear reformulation can be obtained by solving a continuous linear relaxation of the initial problem. We also show that a good quadratic convex reformulation can be obtained by solving a semidefinite relaxation. In both cases, the obtained reformulation profits from the quality of the underlying relaxation. Hence, the proposed scheme gets around, in a sense, the difficulty to incorporate these costly relaxations in a branch-and-bound algorithm.

Type
Research Article
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

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