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Quadratic 0-1 programming : tightening linear or quadratic convex reformulation by use of relaxations

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.
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Alain Billionnet, Sourour Elloumi, Marie-Christine Plateau. Quadratic 0-1 programming : tightening linear or quadratic convex reformulation by use of relaxations. RAIRO - Operations Research, EDP Sciences, 2008, 42 (2), pp.103-121. ⟨10.1051/ro:2008011⟩. ⟨hal-01125219⟩

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