The Quadratic Assignment Problem

Lawler, Edward E.

doi:10.1287/mnsc.9.4.586

Cited by 771 publications

(335 citation statements)

References 5 publications

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“…Roughly speaking, these bounds can be categorized into two groups. The first group includes several bounds that are not very strong but can be computed efficiently such as the well-known Gilmore-Lawler bound (GLB) [13,23], the bound based on projection [31] (denoted by PB) and the bound based on convex quadratic programming (denoted by QPB) [3]. The second group contains strong bounds that require expensive computation such as the bounds derived from lifted integer linear programming [1,2,15] and bounds based on SDRs [29,35].…”

Section: Introductionmentioning

confidence: 99%

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

et al. 2014

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Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25,28]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the so-called redundant and non-redundant matrix splitting and show that the relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a non-redundant matrix splitting via solving an auxiliary semi-definite programming problem (SDP). We show that applying the minimal trace principle directly leads to the so-called orthogonal matrix splitting introduced in [28]. To find other non-redundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the so-called one-matrix and the sum-matrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can provide a non-redundant matrix splitting. The lower bounds from these two splitting schemes are compared theoretically. Promising numerical results on some large QAP instances are reported, which further validate our theoretical conclusions.

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Section: Introductionmentioning

confidence: 99%

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

et al. 2014

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“…In [3] a more general expression of (1.1) was introduced by using a four-dimensional arraŷ q ijkl instead of the flow-distance products f ik d jl : Without loss of generality, we can assume that q ijkl are nonnegative. If they are negative, we add a sufficiently large constant to all q ijkl , which does not change the optimal permutation and increase the objective function by n 2 times the added constant.…”

Section: Introductionmentioning

confidence: 99%

“…Lawler [3] replaced the quadratic terms x ij x kl in the objective function by n 4 variables y ijkl = x ij x kl . The main drawback of this approach is the huge number of variables.…”

Section: Introductionmentioning

confidence: 99%

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

2012

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The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearization are tight, but hardly exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation [1] is the smallest of these linearizations, but very weak. In this paper, we analyze how Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As we show in our numerical experiments, these tightened formulations remain small but computationally effective in order to solve the QAP by means of general purpose MILP solvers.

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“…Nesta linha, diversos limites inferiores do tipo linear foram determinados, inclusive um dos mais antigos deles, conhecido por limite de Gilmore e Lawler e suas variações, [Gi62] e [La63];…”

Section: Introductionunclassified

Ordenações parciais nos conjuntos das soluções dos problemas de alocação linear e quadrático

Rangel

Abreu

2003

Pesqui. Oper.

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ResumoO Problema Quadrático de Alocação, PQA, pode ser abordado através de uma relaxação na forma do Problema de Alocação Linear, PAL. Introduzimos um poset (conjunto parcialmente ordenado) no conjunto das soluções lineares que nos permite comparar também os custos das soluções do problema quadrático, sem o conhecimento prévio das matrizes que definem seus exemplares. Construímos um algoritmo polinomial capaz de determinar pares de permutações livremente comparáveis, conceito apresentado neste trabalho. Provamos um teorema que garante que os custos associados a tais permutações preservam a ordem dada pelo número de inversões das mesmas. Associando soluções do problema a permutações, testes empíricos são apresentados, visando a validação do número de inversões como um parâmetro de referência para a qualidade das soluções. Este trabalho é uma extensão do artigo [RA01] publicado nos anais do XXXIII SBPO, em CD-ROM, 1277-1287, Sobrapo, ILTC, 2001.Palavras-chave: ordenação parcial; problema quadrático de alocação; problema de alocação linear. AbstractWe present an approach to Quadratic Assignment Problem, QAP, via a linear relaxation, through the use of the Linear Assignment Problem, LAP. We define one ordering an LAP-solution-set that makes us compare QAP costs. From here, this poset, partial ordering set, considers coordinates of instances as free variables. We have managed to build a capable algorithm to determine freely comparable permutation pairs. A theorem shows that the order mentioned above is compatible to the one that involves inversion numbers of permutations. We can apply linear and quadratic solutions to permutations and empiric tests are presented to show that the inversion numbers can be used to measure the quality of QAP-solutions. This work is an extension of the article [RA01] published in

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The Quadratic Assignment Problem

Cited by 771 publications

References 5 publications

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

Ordenações parciais nos conjuntos das soluções dos problemas de alocação linear e quadrático

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