A Dual Framework for Lower Bounds of the Quadratic Assignment Problem Based on Linearization

Karisch, Stefan E.; Çela, Eranda; Clausen, Jens; Espersen, T.

doi:10.1007/s006070050040

Cited by 31 publications

(28 citation statements)

References 16 publications

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“…An obvious approach for obtaining lower bounds is to make use of the linear programming relaxation of the mixed integer linear program and its dual linear program (see for example, Assad and Xu (1985), Adams and Johnson (1994), Ramachandran and Pekny (1998), and Karisch et al (1999). Using ideas from , Resende et al (1995) implemented an interior point algorithm to solve a relaxation of the mixed integer program.…”

Section: Alternative Branch-and-bound Approaches To the Quadratic Assmentioning

confidence: 99%

“…However, researchers have realized that the simplicity of computing the Gilmore and Lawler bound comes at a cost, as the bound is often not very tight for large instances of the quadratic assignment problem. Since the publication of the GilmoreLawler bound, research efforts have been directed toward finding improved bounds.An obvious approach for obtaining lower bounds is to make use of the linear programming relaxation of the mixed integer linear program and its dual linear program (see for example, Assad and Xu (1985), Adams and Johnson (1994), Ramachandran andPekny (1998), andKarisch et al (1999). Using ideas from Drezner (1995), Resende et al (1995) implemented an interior point algorithm to solve a relaxation of the mixed integer program.…”

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Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. PrefaceThis book is the final result of a number of incidents that occurred to us while discussing location issues with colleagues at conferences. Frequently we attended presentations, in which the authors quoted the well-known references that helped to make the discipline what it is today. Upon further inquiry, though, it turned out that some of these colleagues had never actually read the original papers. We then discussed among ourselves which contributions could be credited with shaping the field. And, lo and behold, we found that we, too, had neglected to read some of the papers that form the foundation of our science. Whether it was laziness or other things that got in the way, it had become clear that something had to be done. Our first thought was to collect the original contributions (once we could agree on what they were) and reprint them. When discussing this possibility with a publisher, we immediately ran into a roadblock in the form of copyright. While this appeared to have stopped our enthusiastic effort dead in its tracks, we kept on collecting and reading what we considered original contributions.This went on until we met Camille Price, who suggested that, rather than reprinting the original contributions, we should invite some of the leaders in our field and ask them to describe the original contribution, explain and interpret it, and comment on the impact that it had to the field. This was, of course, an excellent idea, and the response by our colleagues to our pertinent requests was equally enthusiastic. What you hold in your hands is the result of this effort.In other words, the purpose of this book is to provide easy access to the main contributions to location theory. The book is organized as follows. The introductory chapter provides an overview of some of the many facets of location analysis. This is followed by contributions in the main three fields of inquiry: minisum, minimax, and covering problems. The next chapters are part of an ever-growing list of nonstandard location models: models including competitive components, those that locate undesirable facilities, those with probabilistic features, and those that allow interactions between facilities. The following chapters discuss solution techniques: after a discussion of exact and heuristic techniques, we devote an entire chapter to Weiszfeld's method, and another to Lagrangean techniques. The last chapters of this book deal with the spheres of influence that the facilities generate and that attract viii customers to them, an...

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Section: Alternative Branch-and-bound Approaches To the Quadratic Assmentioning

confidence: 99%

mentioning

confidence: 99%

Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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“…Recent developments have produced improved, that is, tighter, bounds. The new methodologies include the interior point bound by Resende et al (1995), the level-1 RLT-based dual-ascent bound by Hahn and Grant (1998), the dual-based bound by Karisch et al (1999), the convex quadratic programming bound by Anstreicher and Brixius (2001), the level-2 RLT interior point bound by Ramakrishnan et al (2002), the SDP bound by Roupin (2004), the lift-and-project SDP bound by Burer and Vandenbussche (2006), the bundle method bound by Rendl and Sotirov (2007), and the HahnHightower level-2 RLT-based dual-ascent bound by Adams et al (2007). The tightest bounds are the lift-and-project SDP bound and the two level-2 RLT-based bounds.…”

Section: Introductionmentioning

confidence: 99%

A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

Hahn

Zhu²,

Guignard

et al. 2012

INFORMS Journal on Computing

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Abstract:We apply the level-3 Reformulation Linearization Technique (RLT3) to the Quadratic Assignment Problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm, based on this RLT3 formulation. This algorithm is not guaranteed to calculate the RLT3 lower bound exactly, but approximates it very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3-based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problems sizes larger than size 25 still presents a challenge due to the large memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP. NotationEntries of a matrix E of size mxnx…xp, indexed by i,j,…,k, are denoted e ij…k . Conversely, given numbers , one can form a corresponding matrix of appropriate size. Z(P) will denote the optimal value of optimization problem (P).

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“…We refer to [Ma00] for the discussion of QAP origins and for its definitions, from which we retain only that a QAP instance is a pair of matrices (M F In what concerns the theoretical work on QAP the majority of the studies concerned the definition of better lower bounds for the optimal solution to be used in exact algorithms, such as in [KÇCE00]. A characterization of some polynomial cases has also been presented [Çe98].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial instruments in the design of a heuristic for the quadratic assignment problem

Boaventura-Netto

2003

Pesqui. Oper.

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This work discusses the use of a neighbouring structure in the design of specific heuristics for the Quadratic Assignment Problem (QAP). This structure is formed by the 4-and 6-cycles adjacent to a vertex in the Hasse diagram of the permutation lattice and it can be adequately partitioned in subsets of linear and quadratic cardinalities, a characteristics which frequently allows an economy in the processing time. We propose also a restart strategy and a mechanism for generating initial solutions which constitute, together with the neighbouring structure, a possible QAP-specific heuristic proposal. For the construction of these instruments we used the relaxed ordered set of QAP solutions.Keywords: quadratic assignment problem; combinatory; heuristics. ResumoEste trabalho discute o uso de uma estrutura de vizinhança em heurísticas específicas para o Problema Quadrático de Alocação (PQA). Esta estrutura envolve os ciclos de comprimento 4 e 6 adjacentes a um vértice do diagrama de Hasse do reticulado das permutações e pode ser particionada em subconjuntos de cardinalidade linear e quadrática em relação à ordem da instância, o que permite frequentemente uma economia de tempo de processamento. Propõem-se ainda uma estratégia de repartida e um mecanismo de geração de soluções iniciais, que constituem, ao lado da estrutura de vizinhança, uma proposta de heurística específica para o PQA. Na construção desses instrumentos foi utilizada a noção de conjunto relaxado ordenado das soluções do PQA.Palavras-chave: problema quadrático de alocação; combinatória; heurísticas.Boaventura-Netto -Combinatorial instruments in the design of a heuristic for the quadratic assignment problem

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A Dual Framework for Lower Bounds of the Quadratic Assignment Problem Based on Linearization

Cited by 31 publications

References 16 publications

Hub Location Problems: The Location of Interacting Facilities

Hub Location Problems: The Location of Interacting Facilities

A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

Combinatorial instruments in the design of a heuristic for the quadratic assignment problem

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