A polyhedral approach for a constrained quadratic 0-1 problem

In this paper we consider the problem of optimizing a quadratic pseudo-Boolean function subject to the cardinality constraint @?"1"=<"i"=<"nx"i=k with a polyhedral method. More precisely we propose a study of the convex hull of feasible points included in the Padberg's Boolean quadric polytope and satisfying the cardinality constraint. Specifically, we investigate the connection with the Boolean quadric polytope and study a facet family. The relationship with two other polytopes of the literature is also explored.

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

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

[3]  Antonio Sassano,et al.  The equipartition polytope. I: Formulations, dimension and basic facets , 1990, Math. Program..

[4]  George L. Nemhauser,et al.  Min-cut clustering , 1993, Math. Program..

[5]  Antonio Sassano,et al.  The equipartition polytope. II: Valid inequalities and facets , 1990, Math. Program..

[6]  Helmuth Späth,et al.  Heuristically determining cliques of given cardinality and with minimal cost within weighted complete graphs , 1985, Z. Oper. Research.

[7]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[8]  Ulrich Faigle,et al.  A cutting-plane approach to the edge-weighted maximal clique problem , 1993 .

[9]  Nicolas Boissin.  Optimisation des fonctions quadratiques en variables bivalentes , 1994 .

[10]  Sungsoo Park,et al.  An extended formulation approach to the edge-weighted maximal clique problem , 1996 .

[11]  Alain Billionnet,et al.  A Lower Bound for a Constrained Quadratic 0-1 Minimization Problem , 1997, Discret. Appl. Math..

[12]  Yves Crama,et al.  Valid inequalities and facets for a hypergraph model of the nonlinear knapsack and the FMS part selection problems , 1995, Ann. Oper. Res..

[13]  Endre Boros,et al.  Chvátal Cuts and ODD Cycle Inequalities in Quadratic 0 - 1 Optimization , 1992, SIAM J. Discret. Math..

[14]  Egon Balas,et al.  On the Dimension of Projected Polyhedra , 1998, Discret. Appl. Math..

[15]  A. Brøndsted.  An Introduction to Convex Polytopes , 1982 .

[16]  Anuj Mehrotra.  Cardinality Constrained Boolean Quadratic Polytope , 1997, Discret. Appl. Math..