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ABSTRACT. The 0-1 exact k-item quadratic knapsack problem (E − k QK P) consists of maximizing a quadratic function subject to two linear constraints: the first one is the classical linear capacity constraint; the second one is an equality cardinality constraint on the number of items in the knapsack. Most instances of this NP-hard problem with more than forty variables cannot be solved within one hour by a commercial software such as CPLEX 12.1. We propose therefore a fast and efficient heuristic method which produces both good lower and upper bounds on the value of the problem in reasonable time. Specifically, it integrates a primal heuristic and a semidefinite programming reduction phase within a surrogate dual heuristic. A large computational experiments over randomly generated instances with up to 200 variables validates the relevance of the bounds produced by our hybrid dual heuristic, which yields known optima (and prove optimality) in 90% (resp. 76%) within 100 seconds on the average.

Article published by EDP Sciences c EDP Sciences, ROADEF, SMAI 2008 Résumé. 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 a 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 simplementà soumettre le problème reformuléà un solveur standard. L'efficacité de ce schéma estétroitement liéeà la qualité de la reformulation obtenueà la fin de la phase 1. Nous montrons qu'une bonne reformulation linéaire compacte peutêtre obtenue par la résolution d'une relaxation linéaire. De même, une bonne reformulation quadratique convexe peutêtre 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.

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This is a summary of the author's PhD thesis supervised by A. Billionnet and S. Elloumi and defended on November 2006 at the CNAM, Paris (Conservatoire National des Arts et Métiers). The thesis is written in French and is available from http://www.cedric.cnam.fr/PUBLIS/RC1115. This work deals with exact solution methods based on reformulations for quadratic 0-1 programs under linear constraints. These problems are generally not convex; more precisely, the associated continuous relaxation is not a convex problem. We developed approaches with the aim of making the initial problem convex and of obtaining a good lower bound by continuous relaxation. The main contribution is a general method (called QCR) that we implemented and applied to classical combinatorial optimization problems.

We consider the problem of covering the edge set of an unweighted, undirected graph with the minimum number of connected bipartite subgraphs (where the subgraphs are not necessarily bicliques). We show that this is an NP-hard problem, provide lower bounds through an integer programming formulation, propose several constructive heuristics and a local search, and discuss computational results. Finally, we consider a constrained variant of the problem which we show to be NP-hard, and provide an integer programming formulation for the variant.