Iterated responsive threshold search for the quadratic multiple knapsack problem
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The quadratic multiple knapsack problem (QMKP) consists in assigning objects with both individual and pairwise profits to a set of limited knapsacks in order to maximize the total profit. QMKP is a NP-hard combinatorial optimization problem with a number of applications. In this paper, we present an iterated responsive threshold search (IRTS) approach for solving the QMKP. Based on a combined use of three neighborhoods, the algorithm alternates between a threshold-based exploration phase where solution transitions are allowed among those satisfying a responsive threshold and a descent-based improvement phase where only improving solutions are accepted. A dedicated perturbation strategy is utilized to ensure a global diversification of the search procedure. Extensive experiments performed on a set of 60 benchmark instances in the literature show that the proposed approach competes very favorably with the current state-of-the-art methods for the QMKP. In particular, it discovers 41 improved lower bounds and attains all the best known results for the remaining instances. The key components of IRTS are analyzed to shed light on their impact on the performance of the algorithm.
KeywordsQuadratic multiple knapsack problem Constrained quadratic optimization Responsive threshold search Multiple neighborhood Heuristics
We are grateful to the reviewers for their insightful comments which helped us improve the paper. We would like to thank Dr. García-Martínez for answering our questions and making the codes of García-Martínez et al. (2014), García-Martínez et al. (2014) available to us. This work is partially supported by the RaDaPop (2009–2013) and LigeRo projects (2009–2013) from the Region of Pays de la Loire (France). Support for Yuning Chen from the China Scholarship Council is also acknowledged.
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