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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; [email protected]
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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References

W.P. Adams, R. Forrester and F. Glover, Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discrete Optim. 1(2) (2004) 99–120.
Adams, W.P. and Sherali, H.D., A tight linearization and an algorithm for 0–1 quadratic programming problems. Manage. Sci. 32 (1986) 1274–1290. CrossRef
Adams, W.P. and Sherali, H.D., Mixed-integer bilinear programming problems. Math. Program. 59 (1993) 279–305. CrossRef
J.E. Beasley, Heuristic algorithms for the unconstrained binary quadratic programming problem. Technical report, Department of Mathematics, Imperial College of Science and Technology, London, England (1998).
Billionnet, A. and Elloumi, S., Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 problem. Math. Program. 109 (2007) 55–68. CrossRef
A. Billionnet, S. Elloumi and M.C. Plateau, 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). CrossRef
A. Billionnet, S. Elloumi and M.C. Plateau, Quadratic convex reformulation: a computational study of the graph bisection problem. Technical Report CEDRIC, http://cedric.cnam.fr/PUBLIS/RC1003.pdf (2005).
Billionnet, A. and Soutif, E., Using a mixed integer programming tool for solving the 0-1 quadratic knapsack problem. INFORMS J. Comput. 16 (2004) 188–197. CrossRef
Carter, M.W., The indefinite zero-one quadratic problem. Discrete Appl. Math. 7 (1984) 23–44. CrossRef
S. Elloumi, Linear programming versus convex quadratic programming for the module allocation problem. Technical Report CEDRIC 1100, http://cedric.cnam.fr/PUBLIS/RC1100.pdf (2005).
Fortet, R., Applications de l'algèbre de boole en recherche opérationnelle. Rev. Fr. d'Automatique d'Informatique et de Recherche Opérationnelle 4 (1959) 5–36.
Fortet, R., L'algèbre de boole et ses applications en recherche opérationnelle. Cahiers du Centre d'Etudes de Recherche Opérationnelle 4 (1960) 17–26.
M. Garey and D. Johnson, Computers and intractibility: a guide to the theroy of np-completeness. W.H. freeman & Co. (1979).
Glover, F., Improved linear integer programming formulation of non linear integer problems. Manage. Sci. 22 (1975) 445–460. CrossRef
Glover, F., Kochenberger, G.A. and Alidaee, B., Adaptative memory tabu search for binary quadratic programs. Manage. Sci. 44 (1998) 336–345. CrossRef
Gueye, S. and Michelon, P., Miniaturized linearizations for quadratic 0/1 problems. Ann. Oper. Res. 140 (2005) 235–261. CrossRef
Hammer, P.L. and Rubin, A.A., Some remarks on quadratic programming with 0-1 variables. RAIRO 3 (1970) 67–79. CrossRef
Hammer, P.L., Hansen, P. and Simeone, B., Roof duality, complementation and persistency in quadratic 0-1 optimization. Math. Program. 28 (1984) 121–155. CrossRef
Johnson, D.S., Aragon, C.R., McGeoch, L.A. and Schevon, C., Optimization by simulated annealing: an experimental evaluation; part1, graph partitioning. Oper. Res. 37 (1989) 865–892. CrossRef
Kernighan, B.W. and Lin, S., An efficient heuristic procedure for partitioning graphs. The Bell System Technical Journal 49 (1970) 291–307. CrossRef
Lodi, A., Allemand, K. and Liebling, T.M., An evolutionary heuristic for quadratic 0-1 programming. Eur. J. Oper. Res. 119 (1999) 662–670. CrossRef
Lovász, L. and Schrijver, S., Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (1991) 166–190. CrossRef
Merz, P. and Freisleben, B., Greedy and local search heuristics for unconstrained quadratic programming. J. Heuristics 8 (2002) 197–213. CrossRef
M.C. Plateau, A. Billionnet and S. Elloumi, 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.
H.D. Sherali and W.P. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publ., Norwell, MA (1999).
Sherali, H.D. and Tuncbilek, H., A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7 (1995) 1–31. CrossRef