ALGORITHM RELAX-AND-CUT FOR THE 0-1 QUADRATIC KNAPSACK PROBLEM / UM ALGORITMO RELAX-AND-CUT PARA O PROBLEMA QUADRÁTICO DA MOCHILA 0-1

AUTOR(ES)
DATA DE PUBLICAÇÃO

1999

RESUMO

We consider the 0-1 Quadratic Knapsack Problem (QKP), which consists of maximizing a quadratic Boolean function subject to a linear capacity constraint. The problem has applications in several areas such as telecommunications, financial engineering, location problems, graph theory (Max Clique). We propose a Branch-and-Bound algorithm to solve the QKP to optimality based on lagrangian Relaxation. Initially, we linearize the formulation of the problem given above and then we relax-and-cut dinamicaly its continous relaxation using a few classes of valid inequalities. In the process the Subgradient Method is applied. We also propose a new primal heuristic for the QKP that has improved upon previous approaches, and finds an optimal solution for all of the instances we considered. The good quality of our upper and lower bounds is translated into small gaps at the root node of the enumeration tree (usually below 1%, even for difficult instances). That, coupled with tests for fixing variables, allowed optimality to be proven within only a few nodes of the enumeration tree. We provide a way to randomly generate instances of the QKP harder than those in the literature. We report computational results for randomly generated instances (the ones in the literature and the new harder ones) of QKP with different densities and sizes; and also for Known instances of Max Clique problems.

ASSUNTO(S)

integer programming math programming combinatory optimization programacao matematica programacao inteira otimizacao combinatoria

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