# 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

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topBillionnet, 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 -

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