A comparison of two dual-based procedures for solving the p-median problem

Hanjoul, Pierre; Peeters, Dominique

doi:10.1016/0377-2217(85)90012-8

Cited by 91 publications

(36 citation statements)

References 33 publications

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“…Kariv and Hakimi [21] proved that the PMP is N P-hard. Optimization algorithms based on Lagrangean relaxation have been proposed by Narula et al [33], Cornuéjols et al [11], Christofides and Beasley [9] and Beasley [4]; approaches based on dual formulations are illustrated in Galvao [15] and Hanjoul and Peeters [18]. Among the most recent contributions we cite the branch-and-cut-and-price algorithm by Avella et al [2], the branch-and-price algorithm by Senne et al [41] and the semi-Lagrangean relaxation algorithm by Beltran et al [5].…”

Section: Cardinality Constraintsmentioning

confidence: 99%

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A computational evaluation of a general branch-and-price framework for capacitated network location problems

2008

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The purpose of this paper is to illustrate a general framework for network location problems, based on column generation and branch-andprice. In particular we consider capacitated network location problems with single-source constraints. We consider several different network location models, by combining cardinality constraints, fixed set-up costs, concentrator restrictions and regional constraints. Our general branchand-price-based approach can be seen as a natural counterpart of the branch-and-cut-based commercial ILP solvers, with the advantage of exploiting the tightness of the lower bound provided by the set partitioning reformulation of network location problems. Branch-and-price and branch-and-cut are compared through an extensive set of experimental tests.

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Section: Cardinality Constraintsmentioning

confidence: 99%

“…On the contrary the computing time of branch-and-price is worse, even if the N constraints (15) can be removed from formulation (13) - (18).…”

Section: Regional Constraints and Concentratorsmentioning

confidence: 99%

A computational evaluation of a general branch-and-price framework for capacitated network location problems

2008

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“…For minimum sum-of-stars partitioning (the M -median problem) the well-known DUALOC algorithm [42] combined with Lagrangian relaxation of the cardinality constraint [57] is very efficient. Problems with N ≤ 900 are solved exactly and the dimension of the space considered does not appear to be an obstacle.…”

Section: Branch-and-boundmentioning

confidence: 99%

Cluster analysis and mathematical programming

Hansen

Jaumard

1997

Mathematical Programming

413

251

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Les textes publiés dans la série des rapports de recherche HEC n'engagent que la responsabilité de leurs auteurs. La publication de ces rapports de recherche bénéficie d'une subvention du Fonds F.C.A.R. Cluster Analysis and Mathematical ProgrammingPierre Hansen GERAD andÉcole des HautesÉtudes Commerciales Montréal, CanadaBrigitte Jaumard GERAD andÉcole Polytechnique de Montréal Canada February, 1997Les Cahiers du GERAD G-97-10Abstract Given a set of entities, Cluster Analysis aims at finding subsets, called clusters, which are homogeneous and/or well separated. As many types of clustering and criteria for homogeneity or separation are of interest, this is a vast field. A survey is given from a mathematical programming viewpoint.Steps of a clustering study, types of clustering and criteria are discussed. Then algorithms for hierarchical, partitioning, sequential, and additive clustering are studied. Emphasis is on solution methods, i.e., dynamic programming, graph theoretical algorithms, branch-and-bound, cutting planes, column generation and heuristics.

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“…Lagrangian relaxation methods enable the exact solution of PMM instances with n ≤ 500 objects (3,4). For larger problems, a vertex substitution heuristic (VSH) developed in (5) has been the standard for comparison for nearly four decades.…”

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confidence: 99%

Comment on "Clustering by Passing Messages Between Data Points"

Brusco

Köhn

2008

Science

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Frey and Dueck (Reports, 16 February 2007, p. 972) described an algorithm termed "affinity propagation" (AP) as a promising alternative to traditional data clustering procedures. We demonstrate that a well-established heuristic for the p-median problem often obtains clustering solutions with lower error than AP and produces these solutions in comparable computation time.F rey and Dueck (1) described an algorithm for analyzing complex data sets termed "affinity propagation" (AP). The algorithm extracts a subset of representative objects or "exemplars" from the complete object set by exchanging real-valued messages between data points. Clusters are formed by assigning each data point to its most similar exemplar. The authors reported that "[a]ffinity propagation found clusters with much lower error than other methods, and it did so in less than one-hundredth the amount of time" (1). We demonstrate that an efficient implementation of a 40-year-old heuristic for the well-known p-median model (PMM) often provides lower-error solutions than AP in comparable central processing unit (CPU) time.For consistency with AP in (1), we present the PMM as a sum of similarities maximization problem, while recognizing that this is equivalent to the more common form of minimizing the sum of dissimilarities (e.g., distances or costs). The PMM is a general mathematical problem that can be concisely stated as follows: Given an m × n similarity matrix, S, select p columns from S such that the sum of the maximum values within each row of the selected columns is maximized (2). Thus, each row is effectively assigned to its most similar selected column (exemplar) with the goal of maximizing overall similarity. One classic example of the PMM occurs in facility location planning: Locate p plants such that the total distance (or cost) required to serve m demand points is minimized. In data analysis applications where S is an n × n matrix of negative squared Euclidean distances between objects, clustering the n objects using the PMM corresponds to the selection of p exemplars to minimize error, which is defined as the sum of the squared Euclidean distances of each object to its nearest exemplar.Lagrangian relaxation methods enable the exact solution of PMM instances with n ≤ 500 objects (3, 4). For larger problems, a vertex substitution heuristic (VSH) developed in (5) has been the standard for comparison for nearly four decades. The VSH begins with the random selection of a subset of p exemplars, which is iteratively refined by evaluating the effects of substituting an unselected point for one of the selected exemplars. Frey and Dueck assert that this type of strategy "works well only when the number of clusters is small and chances are good that at least one random initialization is close to a good solution" (1). To the contrary, the VSH is remarkably effective and is often the engine for metaheuristics such as tabu search (6) and variable neighborhood search (7).We compared AP to an efficient implementation of VSH (7) across eight data set...

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A comparison of two dual-based procedures for solving the p-median problem

Cited by 91 publications

References 33 publications

A computational evaluation of a general branch-and-price framework for capacitated network location problems

A computational evaluation of a general branch-and-price framework for capacitated network location problems

Cluster analysis and mathematical programming

Comment on "Clustering by Passing Messages Between Data Points"

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