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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 ,
Sourour Elloumi  and
Marie-Christine Plateau
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 asthe minimization of a 0–1 quadratic function subject to linear constraints. Inthis paper, we are interested in the exact solution of this problem through atwo-phase general scheme. The first phase consists in reformulating theinitial problem either into a compact mixed integer linear program or into a0–1 quadratic convex program. The second phase simply consists insubmitting the reformulated problem to a standard solver. The efficiency ofthis scheme strongly depends on the quality of the reformulation obtained inphase 1. We show that a good compact linear reformulation can be obtained bysolving a continuous linear relaxation of the initial problem. We also showthat a good quadratic convex reformulation can be obtained by solving asemidefinite relaxation. In both cases, the obtained reformulation profitsfrom the quality of the underlying relaxation. Hence, the proposed scheme getsaround, in a sense, the difficulty to incorporate these costly relaxations ina branch-and-bound algorithm.

Keywords

Combinatorial optimizationquadratic 0–1 programminglinear reformulationquadratic convex reformulation.
Type
Research Article
Information
RAIRO - Operations Research , Volume 42 , Issue 2: CIRO 05 , April 2008 , pp. 103 - 121
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

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