A linearization framework for unconstrained quadratic (0-1) problems

In this paper, we are interested in linearization techniques for the exact solution of the Unconstrained Quadratic (0-1) Problem. Our purpose is to propose ''economical'' linear formulations. We first extend current techniques in a general linearization framework containing many other schemes and propose a new linear formulation. Numerical results comparing classical, Glover's and the new linearization are reported.

[1]  Lawrence J. Watters Letter to the Editor - Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems , 1967, Oper. Res..

[2]  Warren P. Adams,et al.  A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems , 1998 .

[3]  Alain Billionnet,et al.  Persistency in quadratic 0–1 optimization , 1992, Math. Program..

[4]  P. Hammer,et al.  Pseudo-Boolean Functions and Their Graphs , 1984 .

[5]  Endre Boros,et al.  Cut-Polytopes, Boolean Quadric Polytopes and Nonnegative Quadratic Pseudo-Boolean Functions , 1993, Math. Oper. Res..

[6]  Recherche Opérationnelle,et al.  REVUE FRANÇAISE D'AUTOMATIQUE, D'INFORMATIQUE ET DE , 1985 .

[7]  Serigne Abdoulaye Gueye Linéarisation et relaxation lagrangienne pour problèmes quadratiques en variables binaires , 2002 .

[8]  Laurence A. Wolsey,et al.  Cutting planes in integer and mixed integer programming , 2002, Discret. Appl. Math..

[9]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[10]  Ching-Ter Chang,et al.  An efficient linearization approach for mixed-integer problems , 2000, Eur. J. Oper. Res..

[11]  David S. Johnson,et al.  Computers and Inrracrobiliry: A Guide ro the Theory of NP-Completeness , 1979 .

[12]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[13]  Andrea Lodi,et al.  An evolutionary heuristic for quadratic 0-1 programming , 1999, Eur. J. Oper. Res..

[14]  Muhittin Oral,et al.  A Linearization Procedure for Quadratic and Cubic Mixed-Integer Problems , 1992, Oper. Res..

[15]  Caterina De Simone A note on the Boolean quadric polytope , 1996, Oper. Res. Lett..

[16]  W. Art Chaovalitwongse,et al.  A new linearization technique for multi-quadratic 0-1 programming problems , 2004, Oper. Res. Lett..

[17]  Alain Billionnet,et al.  Minimization of a quadratic pseudo-Boolean function , 1994 .

[18]  Y. Crama,et al.  Upper-bounds for quadratic 0-1 maximization , 1990 .

[19]  R. Fortet L’algebre de Boole et ses applications en recherche operationnelle , 1960 .

[20]  Fred W. Glover,et al.  Technical Note - Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program , 1974, Oper. Res..

[21]  F. Glover IMPROVED LINEAR INTEGER PROGRAMMING FORMULATIONS OF NONLINEAR INTEGER PROBLEMS , 1975 .

[22]  Éric Soutif,et al.  Decomposition and Linearization for 0-1 Quadratic Programming , 2000, Ann. Oper. Res..

[23]  John E. Beasley,et al.  Heuristic algorithms for the unconstrained binary quadratic programming problem , 1998 .

[24]  Philippe Michelon,et al.  “Miniaturized” Linearizations for Quadratic 0/1 Problems , 2005, Ann. Oper. Res..

[25]  Willard I. Zangwill,et al.  Media Selection by Decision Programming , 1976 .

[26]  Martin Grötschel,et al.  An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design , 1988, Oper. Res..

[27]  Fred W. Glover,et al.  Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs , 2004, Discret. Optim..

[28]  Franz Rendl,et al.  A recipe for semidefinite relaxation for (0,1)-quadratic programming , 1995, J. Glob. Optim..

[29]  Talal M. Alkhamis,et al.  Simulated annealing for the unconstrained quadratic pseudo-Boolean function , 1998, Eur. J. Oper. Res..

[30]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[31]  Hanif D. Sherali,et al.  CONVEX ENVELOPES OF MULTILINEAR FUNCTIONS OVER A UNIT HYPERCUBE AND OVER SPECIAL DISCRETE SETS , 1997 .

[32]  Anatoliy D. Rikun,et al.  A Convex Envelope Formula for Multilinear Functions , 1997, J. Glob. Optim..

[33]  Ching-Ter Chang,et al.  A linearization method for mixed 0-1 polynomial programs , 2000, Comput. Oper. Res..

[34]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[35]  M. Rao Cluster Analysis and Mathematical Programming , 1971 .

[36]  F. Glover,et al.  Adaptive Memory Tabu Search for Binary Quadratic Programs , 1998 .

[37]  J. Ben Rosen,et al.  A quadratic assignment formulation of the molecular conformation problem , 1994, J. Glob. Optim..

[38]  Hanif D. Sherali,et al.  Evolution and state-of-the-art in integer programming , 2000 .

[39]  Manfred W. Padberg,et al.  The boolean quadric polytope: Some characteristics, facets and relatives , 1989, Math. Program..

[40]  Bernd Freisleben,et al.  Greedy and Local Search Heuristics for Unconstrained Binary Quadratic Programming , 2002, J. Heuristics.

[41]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[42]  H. Weingartner Capital Budgeting of Interrelated Projects: Survey and Synthesis , 1966 .

[43]  P. Chardaire,et al.  A Decomposition Method for Quadratic Zero-One Programming , 1995 .

[44]  C. Helmberg,et al.  Solving quadratic (0,1)-problems by semidefinite programs and cutting planes , 1998 .

[45]  Alain Billionnet,et al.  Unconstrained 0-1 optimization and Lagrangean relaxation , 1990, Discret. Appl. Math..

[46]  Muhittin Oral,et al.  Reformulating nonlinear combinatorial optimization problems for higher computational efficiency , 1992 .