Linear programming for the 0–1 quadratic knapsack problem

In this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We propose a new method for computing an upper bound. This method is based on the solution of a continuous linear program constr...

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Bibliographic Details
Published in:European journal of operational research Vol. 92; no. 2; pp. 310 - 325
Main Authors: Billionnet, Alain, Calmels, Frédéric
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 19.07.1996
Elsevier
Elsevier Sequoia S.A
Series:European Journal of Operational Research
Subjects:
Algorithms
Branch & bound algorithms
Branch-and-bound
Computational analysis
Computational Complexity
Computer Science
Knapsack problem
Linear programming
Linearization
Mathematical models
Optimization algorithms
Studies
Zero-one quadratic programming
Zero-one quadratic programming
Branch-and-bound
Computational analysis
Knapsack problem
Linearization
ISSN:0377-2217, 1872-6860
Online Access:Get full text
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Summary:In this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We propose a new method for computing an upper bound. This method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0–1 variables but nonredundant in continuous variables. The obtained upper bound is better than the bounds given by other known methods. We also propose an algorithm for computing a good feasible solution. This algorithm is an elaboration of the heuristic methods proposed by Chaillou, Hansen and Mahieu and by Gallo, Hammer and Simeone. The relative error between this feasible solution and the optimum solution is generally less than 1%. We show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum. Finally we propose a branch-and-bound algorithm for solving the quadratic knapsack problem and report extensive computational tests.
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ISSN:0377-2217
1872-6860
DOI:10.1016/0377-2217(94)00229-0