A New Lower Bound Via Projection for the Quadratic Assignment Problem

Hadley, Scott; Rendl, Franz; Wolkowicz, Henry

doi:10.1287/moor.17.3.727

Cited by 99 publications

(75 citation statements)

References 12 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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“…The difficulty of a QAP instance for the branch and bound algorithm measures how well the lower bounds techniques approximate the optimum solution. In Table 3, we compare our cQAP instances and the random QAP instances, considered difficult for both exact and heuris- (Gilmore 1962) is given in third column, and the gap with the elimination lower bound (Hadley et al 1992) is given in the forth column. Note that the gap of the cQAP instances is higher than for the random instances indicating that these instances are difficult for branch and bound methods.…”

Section: Comparing Qap Instances With Heuristic and Exact Methodsmentioning

confidence: 99%

Generating QAP instances with known optimum solution and additively decomposable cost function

Drugan

2013

J Comb Optim

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Link to publication Citation for published version (APA):Drugan, M. M. (2015). Generating QAP instances with known optimum solution and additively decomposable cost function. Journal of Combinatorial Optimization, 30(4), 1138-1172. DOI: 10.1007 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract Quadratic assignment problems (QAPs) is a NP-hard combinatorial optimization problem. QAPs are often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator that can be used for benchmarking for heuristic algorithms. Our QAP generator combines small size QAPs with known optimum solution into a larger size QAP instance. We call these instances composite QAPs (cQAPs), and we show that the cost function of cQAPs is additively decomposable. We give mild conditions for which a cQAP instance has known optimum solution. We generate cQAP instances using uniform distributions with different bounds for the component QAPs and for the rest of the cQAP elements. Numerical and analytical techniques that measure the difficulty of the cQAP instances in comparison with other QAPs from the literature are introduced. These methods point out that some cQAP instances are difficult for local search with many local optimum of various values, low epistasis and non-trivial asymptotic behaviour.

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“…Unfortunately, the resulting eigenvalue bound has proven to be somewhat weak, but has been improved by enforcing additional constraints. For example, Hadley et al (1992) enforce constraints on row and column sums resulting in a projected eigenvalue bound. Sometimes their bound was better than that by Gilmore and Lawler, and sometimes not.…”

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

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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), 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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A New Lower Bound Via Projection for the Quadratic Assignment Problem

Cited by 99 publications

References 12 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

Generating QAP instances with known optimum solution and additively decomposable cost function

Hub Location Problems: The Location of Interacting Facilities

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