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

Alain Billionnet; Sourour Elloumi; Marie-Christine Plateau

RAIRO - Operations Research (2008)

  • Volume: 42, Issue: 2, page 103-121
  • ISSN: 0399-0559

Abstract

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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.

How to cite

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Billionnet, Alain, Elloumi, Sourour, and Plateau, Marie-Christine. "Quadratic 0–1 programming: Tightening linear or quadratic convex reformulation by use of relaxations." RAIRO - Operations Research 42.2 (2008): 103-121. <http://eudml.org/doc/250384>.

@article{Billionnet2008,
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. },
author = {Billionnet, Alain, Elloumi, Sourour, Plateau, Marie-Christine},
journal = {RAIRO - Operations Research},
keywords = {Combinatorial optimization; quadratic 0–1 programming; linear reformulation; quadratic convex reformulation.; combinatorial optimization; quadratic 0-1 programming; quadratic convex reformulation},
language = {eng},
month = {5},
number = {2},
pages = {103-121},
publisher = {EDP Sciences},
title = {Quadratic 0–1 programming: Tightening linear or quadratic convex reformulation by use of relaxations},
url = {http://eudml.org/doc/250384},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Billionnet, Alain
AU - Elloumi, Sourour
AU - Plateau, Marie-Christine
TI - Quadratic 0–1 programming: Tightening linear or quadratic convex reformulation by use of relaxations
JO - RAIRO - Operations Research
DA - 2008/5//
PB - EDP Sciences
VL - 42
IS - 2
SP - 103
EP - 121
AB - 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.
LA - eng
KW - Combinatorial optimization; quadratic 0–1 programming; linear reformulation; quadratic convex reformulation.; combinatorial optimization; quadratic 0-1 programming; quadratic convex reformulation
UR - http://eudml.org/doc/250384
ER -

References

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