Skip to main content
SpringerLink
Log in

“Miniaturized” Linearizations for Quadratic 0/1 Problems

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In order to solve a quadratic 0/1 problem, some techniques, consisting in deriving a linear integer formulation, are used. Those techniques, called “linearization”, usually involve a huge number of additional variables. As a consequence, the exact resolution of the linear model is, in general, very difficult.

Our aim, in this paper, is to propose “economical” linear models. Starting from an existing linearization (typically the so-called “classical linearization”), we find a new linearization with fewer variables. The resulting model is called “Miniaturized” linearization. Based on this approach, we propose a new linearization scheme for which numerical tests have been performed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Adams, W., A. Billionnet, and A. Sutter. (1990). “Unconstrained 0-1 Optimization and Lagrangean Relaxation.” Discrete Applied Mathematics 29, 131–142.

    Article  Google Scholar 

  • Alidaee, B., G. Kochenberger, and A. Ahmadian. (1994). “0-1 Quadratic Programming Approach for the Optimal Solution of Two Scheduling Problems.” International Journal of System Science 25, 401–408.

    Article  Google Scholar 

  • Beasley, J.E. (1998). “Heuristic Algorithms for the Unconstrained Binary Quadratic Programming Problem.” Technical report, Department of Mathematics, Imperial College of Science and Technology, London, England.

  • Balas, E. and J.B. Mazzola. (1984). “Non-Linear 0-1 Programming i : Linearization Techniques.” Mathematical Programming 30, 1–21.

    Google Scholar 

  • Billionnet, A. and A. Sutter. (1994). “Minimization of a Quadratic Pseudo-Boolean Function.” European Journal of Operational Research 78, 106–115.

    Article  Google Scholar 

  • Chang, C.T. and C.C. Chang. (2000). “A Linearization Method for Mixed 0-1 Polynomial Programs.” Computers and Operations Research 27, 1005–1016.

    Article  Google Scholar 

  • Chardaire, P. and A. Sutter. (1995). “A Decomposition method for Quadratic Zero-One Programming.” Management Science 41(4), 704–712.

    Google Scholar 

  • Djabali, R. (1998). Optimisation non linéaire en variables bivalentes et applications. Thèse de doctorat, CNAM.

  • Deza, M. and M. Laurent. (1992a). “Facets for the Cut Cone I.” Mathematical Programming 56(2), 121–160.

    Article  Google Scholar 

  • Deza, M. and M. Laurent. (1992b). “Facets for the Cut Cone II.” Mathematical Programming 56(2), 161–188.

    Article  Google Scholar 

  • Fiduccia, C. and R. Mattheyses. (1982). “A Linear-Time Heuristic for Improving Network Partitions.” In ACM/IEEE 19th Design Autom. Conference, Las Vegas, 175–181.

  • Fortet, R. (1959). “L'algèbre De Boole et ses Applications En Recherche Opérationnelle.” Cahier du Centre d'Etudes de Recherche Opérationnelle 1, 5–36.

    Google Scholar 

  • Fortet, R. (1960). “Application De L'algèbre De Boole En Recherche Opérationnelle.” Revue Française de Recherche Opérationnelle 4, 17–26.

    Google Scholar 

  • Garey, M. and D. Johnson. (1979). Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman & Company.

  • Glover, F., G.A. Kochenberger, and B. Alidaee. (1998). “Adaptative Memory Tabu Search for Binary Quadratic Programs.” Management Science 44, 336–345.

    Google Scholar 

  • Glover, F. (1975). “Improved Linear Integer Programming Formulations of Nonlinear Integer Problems.” Management Science 22(4), 455–460.

    Google Scholar 

  • Gueye, S. (2002). Linéarisation et relaxation lagrangienne pour problèmes quadratiques en variables binaires. Thèse de doctorat, Université d'Avignon.

  • Helmberg, C. and F. Rendl. (1998). “Solving Quadratic (0,1)-Problems by Semidefinite Programs and Cutting Planes.” Mathematical Programming 82(3), 291–315.

    Article  Google Scholar 

  • Johnson, D., C. Aragon, L. McGeoch, and C. Schevon. (1989). “Optimization by Simulated Annealing: An Experimental Evaluation: Part i, Graph Partitioning.” Operations Research 37, 865–892.

    Google Scholar 

  • Kumar, V., A. Grama, A. Gupta, and G. Karypis. (1994). Introduction to parallel computing: design and analysis of algorithms. Benjamin/Cummings Publishing Company, Redwood City, CA.

    Google Scholar 

  • Kernighan, B.W. and S. Lin. (1970). “An Efficient Heuristic Procedure for Partitioning Graphs.” Bell Systems Technical Journal 49, 291–307.

    Google Scholar 

  • Kuznar, R. (1996). Partitioning and optimization of digital integrated circuits. Doctoral thesis, University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia.

  • Lodi, A., K. Allemand, and T.M. Liebling. (1999). “An Evolutionary Heuristic for Quadratic 0-1 Programming.” European Journal of Operational Research 119, 662–670.

    Article  Google Scholar 

  • Laguna, M., T.A. Feo, and H.C. Herold. (1994). “A Greedy Randomized Adaptive Search Procedure for the Two-Partition Problem.” Operations Research 42(4), 677–687.

    Article  Google Scholar 

  • Merz, P. and B. Freisleben (2002). “Greedy and Local Search Heuristics for Unconstrained Quadratic Programming.” Journal of Heuristics 8(2), 197–213.

    Article  Google Scholar 

  • Michelon, P. (1992). “Unconstrained Non-Linear 0-1 Programming: A Non differentiable Approach.” The Journal of Global Optimization 2, 155–165.

    Article  Google Scholar 

  • Michelon, P., S. Ripeau, and N. Maculan. (1995). “Un Algorithme pour La Bipartition D'un Graphe En Sous Graphes De Cardinalité fixée.” In FRANCORO I.

  • Padberg, M. (1989). “The Boolean Quadric Polytope: Some Characteristics, Facets and Relatives.” Mathematical Programming 45, 139–172.

    Article  Google Scholar 

  • Poljak, S., F. Rendl, and H. Wolkowicz. (1995). “A Recipe for Semidefinite relaxation for (0,1)-Quadratic Programming.” Journal of Global Optimization 7(1), 51–73.

    Article  Google Scholar 

  • Rolland E. and H. Pirkul. (1992). “Heuristic Solution Procedures for the Graph Partitioning Problem.” In S. Zenios, O. Balci, and R. Sharda (eds.), Computer Science and Operations Research: New Developments in their Interfaces. Pergamon Press, Oxford.

    Google Scholar 

  • Sherali, H.D. and W.P. Adams. (1999). A Reformulation Linearization Technique for Solving Discrete and Continous Nonconvex Problems. Kluwer Academic Publishers.

  • De Simone, C. (1989). “The Cut Polytope and Boolean Quadric Polytope.” Discrete Mathematics 79, 71–75.

    Article  Google Scholar 

  • Soutif, E. (2000). Résolution du problème quadratique en variables bivalentes. Thèse de doctorat, CNAM.

Download references

Author information

Authors and Affiliations

  1. Université du Havre, Laboratoire de Mathématiques Appliquées du Havre, 25 rue Philippe Lebon, BP 540, 76058, Le Havre cedex

    Serigne Gueye

  2. Université d'Avignon et des Pays de Vaucluse, Laboratoire d'Informatique, d'Avignon – CNRS FRE n∘ 2487, 339 chemin des Meinajariès, Agroparc BP 1228, 84911, Avignon Cedex 9, France

    Philippe Michelon

Authors
  1. Serigne Gueye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Michelon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serigne Gueye.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gueye, S., Michelon, P. “Miniaturized” Linearizations for Quadratic 0/1 Problems. Ann Oper Res 140, 235–261 (2005). https://doi.org/10.1007/s10479-005-3973-5

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-005-3973-5

Keywords

Search

Navigation