Polynomial-size formulations and relaxations for the quadratic multiple knapsack problem

•Polynomial-size formulations of the quadratic multiple knapsack problem from classical 0-1 quadratic programming reformulations.•New level-1 decomposable reformulation linearization techniques for the problem.•Comparison of LP, surrogate, and Lagrangian relaxations.•Theoretical properties and domin...

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Bibliographic Details
Published in:European journal of operational research Vol. 291; no. 3; pp. 871 - 882
Main Authors: Galli, Laura, Martello, Silvano, Rey, Carlos, Toth, Paolo
Format: Journal Article
Language:English
Published: Elsevier B.V 16.06.2021
Subjects:
Binary quadratic programming
Combinatorial optimization
Lagrangian relaxation
Quadratic multiple knapsack
Reformulation linearization technique
Binary quadratic programming
Combinatorial optimization
Quadratic multiple knapsack
Lagrangian relaxation
Reformulation linearization technique
ISSN:0377-2217, 1872-6860
Online Access:Get full text
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Summary:•Polynomial-size formulations of the quadratic multiple knapsack problem from classical 0-1 quadratic programming reformulations.•New level-1 decomposable reformulation linearization techniques for the problem.•Comparison of LP, surrogate, and Lagrangian relaxations.•Theoretical properties and dominances.•Computational experiments on a large set of benchmark instances. The Quadratic Multiple Knapsack Problem generalizes, simultaneously, two well-known combinatorial optimization problems that have been intensively studied in the literature: the (single) Quadratic Knapsack Problem and the Multiple Knapsack Problem. The only exact algorithm for its solution uses a formulation based on an exponential-size number of variables, that is solved via a Branch-and-Price algorithm. This work studies polynomial-size formulations and upper bounds. We derive linear models from classical reformulations of 0-1 quadratic programs and analyze theoretical properties and dominances among them. We define surrogate and Lagrangian relaxations, and we compare the effectiveness of the Lagrangian relaxation when applied to a quadratic formulation and to a Level 1 reformulation linearization that leads to a decomposable structure. The proposed methods are evaluated through extensive computational experiments.
ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2020.10.047