Optimal clustering: A model and method

Klein, Gary; Aronson, Jay E.

doi:10.1002/1520-6750(199106)38:3<447::aid-nav3220380312>3.0.co;2-0

Cited by 61 publications

(42 citation statements)

References 10 publications

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“…It solves problems with N ≤ 120 and a few well-separated clusters of points of R 2 , but its performance deteriorates in higher dimensional spaces. Another algorithm [84], for minimum sum-of-cliques partitioning, uses bounds based on ranking dissimilarities, which are not very sharp. Problems with N ≤ 50, M ≤ 5 can be solved.…”

Section: Branch-and-boundmentioning

confidence: 99%

Cluster analysis and mathematical programming

Hansen

Jaumard

1997

Mathematical Programming

402

246

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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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Section: Branch-and-boundmentioning

confidence: 99%

Cluster analysis and mathematical programming

Hansen

Jaumard

1997

Mathematical Programming

402

246

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“…We use the computation of a lower bound proposed in [49], which takes into account the unassigned variables. The sum defining V can be split into three parts V = V 1 + V 2 + V 3 , where:…”

Section: Within-cluster Sum Of Dissimilarities Criterionmentioning

confidence: 99%

Constrained clustering by constraint programming

Dao

Duong

Vrain

2017

Artificial Intelligence

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Constrained Clustering allows to make the clustering task more accurate by integrating user constraints, which can be instance-level or cluster-level constraints. Few works consider the integration of different kinds of constraints, they are usually based on declarative frameworks and they are often exact methods, which either enumerate all the solutions satisfying the user constraints, or find a global optimum when an optimization criterion is specified. In a previous work, we have proposed a model for Constrained Clustering based on a Constraint Programming framework. It is declarative, allowing a user to integrate user constraints and to choose an optimization criterion among several ones. In this article we present a new and substantially improved model for Constrained Clustering, still based on a Constraint Programming framework. It differs from our earlier model in the way partitions are represented by means of variables and constraints. It is also more flexible since the number of clusters does not need to be set beforehand; only a lower and an upper bound on the number of clusters have to be provided. In order to make the model-based approach more efficient, we propose new global optimization constraints with dedicated filtering algorithms. We show that such a framework can easily be embedded in a more general process and we illustrate this on the problem of finding the optimal Pareto front of a bi-criterion constrained clustering task. We compare our approach with existing exact approaches, based either on a branch-and-bound approach or on graph coloring on twelve datasets. Experiments show that the model outperforms exact approaches in most cases.

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“…We denote these distances by e ij for stands i and j. We use an auxiliary binary variable y ij , which obtains a value 1 if the distance e ij from the stand i to stand j is selected (see Rosing and ReVelle, 1986;Klein and Aronson, 1991). Given this notation, the goal of the society is to maximize the sum of ecological distances in the selected conservation network over all stands,…”

Section: Site Selection Modelsmentioning

confidence: 99%