Le problème de la minimisation d'une fonction quadratique en variables 0-1 sous contraintes linéaires permet de modéliser de nombreux problèmes d'Optimisation Combinatoire. Nous nous intéressons à sa résolution exacte par un schéma général en deux phases. La première phase permet de reformuler le problème de départ soit en un programme linéaire compact en variables mixtes soit en un programme quadratique convexe en variables 0-1. La deuxième phase consiste simplement à soumettre le problème reformulé à un solveur standard. L'efficacité de ce schéma est étroitement liée à la qualité de la reformulation obtenue à la fin de la phase 1. Nous montrons qu'une bonne reformulation linéaire compacte peut être obtenue par la résolution d'une relaxation linéaire. De même, une bonne reformulation quadratique convexe peut être obtenue par une relaxation semi-définie positive. Dans les deux cas, la reformulation obtenue tire profit de la qualité de la relaxation sur laquelle elle se base. Ainsi, le schéma proposé contourne, d'une certaine façon, la difficulté d'intégrer des relaxations, coûteuses en temps de calcul, dans un algorithme de branch-and-bound.
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.
Mots clés : combinatorial optimization, quadratic 0-1 programming, linear reformulation, quadratic convex reformulation
@article{RO_2008__42_2_103_0,
author = {Billionnet, Alain and Elloumi, Sourour and Plateau, Marie-Christine},
title = {Quadratic 0-1 programming : tightening linear or quadratic convex reformulation by use of relaxations},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {103--121},
publisher = {EDP-Sciences},
volume = {42},
number = {2},
year = {2008},
doi = {10.1051/ro:2008011},
mrnumber = {2431395},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2008011/}
}
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 - Recherche Opérationnelle PY - 2008 SP - 103 EP - 121 VL - 42 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro:2008011/ DO - 10.1051/ro:2008011 LA - en ID - RO_2008__42_2_103_0 ER -
%0 Journal Article %A Billionnet, Alain %A Elloumi, Sourour %A Plateau, Marie-Christine %T Quadratic 0-1 programming : tightening linear or quadratic convex reformulation by use of relaxations %J RAIRO - Operations Research - Recherche Opérationnelle %D 2008 %P 103-121 %V 42 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro:2008011/ %R 10.1051/ro:2008011 %G en %F RO_2008__42_2_103_0
Billionnet, Alain; Elloumi, Sourour; Plateau, Marie-Christine. Quadratic 0-1 programming : tightening linear or quadratic convex reformulation by use of relaxations. RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 2, pp. 103-121. doi : 10.1051/ro:2008011. http://www.numdam.org/articles/10.1051/ro:2008011/
[1] , and , Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optim. 1(2) (2004) 99-120. | MR | Zbl
[2] and , A tight linearization and an algorithm for 0-1 quadratic programming problems. Manage. Sci. 32 (1986) 1274-1290. | MR | Zbl
[3] and , Mixed-integer bilinear programming problems. Math. Program. 59 (1993) 279-305. | MR | Zbl
[4] , Heuristic algorithms for the unconstrained binary quadratic programming problem. Technical report, Department of Mathematics, Imperial College of Science and Technology, London, England (1998).
[5] and , Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 problem. Math. Program. 109 (2007) 55-68. | MR
[6] , and , Improving the performance of standard solvers for quadratic 0-1 programs by a toight convex reformulation: the QCR method. Discrete Appl. Math., http://dx.doi.org/10.1016/j.dam.2007.007 (to appear). | MR
[7] , and , Quadratic convex reformulation: a computational study of the graph bisection problem. Technical Report CEDRIC, http://cedric.cnam.fr/PUBLIS/RC1003.pdf (2005).
[8] and , Using a mixed integer programming tool for solving the 0-1 quadratic knapsack problem. INFORMS J. Comput. 16 (2004) 188-197. | MR
[9] , The indefinite zero-one quadratic problem. Discrete Appl. Math. 7 (1984) 23-44. | MR | Zbl
[10] , Linear programming versus convex quadratic programming for the module allocation problem. Technical Report CEDRIC 1100, http://cedric.cnam.fr/PUBLIS/RC1100.pdf (2005).
[11] , Applications de l'algèbre de boole en recherche opérationnelle. Rev. Fr. d'Automatique d'Informatique et de Recherche Opérationnelle 4 (1959) 5-36. | MR | Zbl
[12] , L'algèbre de boole et ses applications en recherche opérationnelle. Cahiers du Centre d'Etudes de Recherche Opérationnelle 4 (1960) 17-26. | Zbl
[13] and , Computers and intractibility: a guide to the theroy of np-completeness. W.H. freeman & Co. (1979). | MR | Zbl
[14] , Improved linear integer programming formulation of non linear integer problems. Manage. Sci. 22 (1975) 445-460. | MR | Zbl
[15] , and , Adaptative memory tabu search for binary quadratic programs. Manage. Sci. 44 (1998) 336-345. | Zbl
[16] and , Miniaturized linearizations for quadratic 0/1 problems. Ann. Oper. Res. 140 (2005) 235-261. | MR | Zbl
[17] and , Some remarks on quadratic programming with 0-1 variables. RAIRO 3 (1970) 67-79. | Numdam | MR | Zbl
[18] , and , Roof duality, complementation and persistency in quadratic 0-1 optimization. Math. Program. 28 (1984) 121-155. | MR | Zbl
[19] , , and , Optimization by simulated annealing: an experimental evaluation; part1, graph partitioning. Oper. Res. 37 (1989) 865-892. | Zbl
[20] and , An efficient heuristic procedure for partitioning graphs. The Bell System Technical Journal 49 (1970) 291-307. | Zbl
[21] , and , An evolutionary heuristic for quadratic 0-1 programming. Eur. J. Oper. Res. 119 (1999) 662-670. | Zbl
[22] and , Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (1991) 166-190. | MR | Zbl
[23] and , Greedy and local search heuristics for unconstrained quadratic programming. J. Heuristics 8 (2002) 197-213. | Zbl
[24] , and , Eigenvalue methods for linearly constrained quadratic 0-1 problems with application to the densest k-subgraph problem. In 6e congrès ROADEF, Tours, 14-16 février, Presses Universitaires Francois Rabelais, http://cedric.cnam.fr/PUBLIS/RC723.pdf (2005) 55-66.
[25] and , A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publ., Norwell, MA (1999). | MR | Zbl
[26] and , A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7 (1995) 1-31. | MR | Zbl
Cité par Sources :
