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A Global Optimization RLT-based Approach for Solving the Fuzzy Clustering Problem

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Abstract

The field of cluster analysis is primarily concerned with the partitioning of data points into different clusters so as to optimize a certain criterion. Rapid advances in technology have made it possible to address clustering problems via optimization theory. In this paper, we present a global optimization algorithm to solve the fuzzy clustering problem, where each data point is to be assigned to (possibly) several clusters, with a membership grade assigned to each data point that reflects the likelihood of the data point belonging to that cluster. The fuzzy clustering problem is formulated as a nonlinear program, for which a tight linear programming relaxation is constructed via the Reformulation-Linearization Technique (RLT) in concert with additional valid inequalities. This construct is embedded within a specialized branch-and-bound (B&B) algorithm to solve the problem to global optimality. Computational experience is reported using several standard data sets from the literature as well as using synthetically generated larger problem instances. The results validate the robustness of the proposed algorithmic procedure and exhibit its dominance over the popular fuzzy c-means algorithmic technique and the commercial global optimizer BARON.

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References

  • E. Balas (1988) ArticleTitleOn the convex hull of the union of certain polyhedra Operations Research Letters 7 IssueID6 279–283 Occurrence Handle10.1016/0167-6377(88)90058-2

    Article  Google Scholar 

  • A. Baraldi P. Blonda (1999a) ArticleTitleSurvey of fuzzy clustering analysis for pattern recognition – Part I IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 29 IssueID6 778–785

    Google Scholar 

  • A. Baraldi P. Blonda (1999b) ArticleTitleSurvey of fuzzy clustering analysis for pattern recognition – Part II IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 29 IssueID6 786–801

    Google Scholar 

  • J.C. Bezdek (1981) Pattern recognition with fuzzy objective function algorithms Plenum Press New York

    Google Scholar 

  • Chazelle, B. (1991), An optimal convex hull algorithm and new results on cuttings. In: proceedings of the Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Silver Spring, MD, pp.29–38.

  • J.C. Dunn (1973) ArticleTitleA fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters Journal of Cybernetics 3 IssueID3 32–57 Occurrence Handle10.1080/01969727308546046

    Article  Google Scholar 

  • I. Gath A.B. Geva (1989) ArticleTitleUnsupervised optimal fuzzy clustering IEEE Transactions on Pattern Analysis and Machine Intelligence 11 IssueID7 773–781 Occurrence Handle10.1109/34.192473

    Article  Google Scholar 

  • F. Guoyao (1998) ArticleTitleOptimization methods for fuzzy clustering Fuzzy Sets and Systems 93 301–309

    Google Scholar 

  • Gustafson, D.E. and Kessel, W.C. (1979), Fuzzy clustering with a fuzzy covariance matrix. In: proceedings of the International Proceedings of the IEEE Conference on Decision and Control, 78, New York, pp. 761–766.

  • J.A. Hartigan (1975) Clustering Algorithms Wiley New York

    Google Scholar 

  • F. Höppner F. Klawonn R. Kruse T. Runkler (1999) Fuzzy Cluster Analysis Wiley Chichester, UK

    Google Scholar 

  • M.A. Ismail S.Z. Selim (1986) ArticleTitleFuzzy c-means: optimality of solutions and effective termination of the algorithm Pattern Recognition 19 481–485 Occurrence Handle10.1016/0031-3203(86)90048-8

    Article  Google Scholar 

  • M.S. Kamel S.Z. Selim (1994) ArticleTitleNew algorithms for solving the fuzzy clustering problem Pattern Recognition 27 IssueID3 421–428 Occurrence Handle10.1016/0031-3203(94)90118-X

    Article  Google Scholar 

  • A. Klapper (1987) ArticleTitleLower bound on the complexity of the convex hull problem for simply polyhedra Information Processing Letters 25 IssueID3 159–161 Occurrence Handle10.1016/0020-0190(87)90126-8

    Article  Google Scholar 

  • Klawonn, F. and Keller, A. (1998), Fuzzy clustering based on modified distance measures. In: proceedings of the Advances in Intelligent Data Analysis, Proceedings of the 3rd International Symposium, eds., Hand, D.J., Kok, J.N. and Berthold, K.R., 291–301, Amsterdam, The Netherlands.

  • Y. Leung J. Zhang Z. Xu (1997) ArticleTitleNeural networks for convex hull computation IEEE Transactions on Neural Networks 8 IssueID3 601–611

    Google Scholar 

  • U. Manber (1989) Introduction to Algorithms: A Creative Approach Addison-Wesley Publishing Company Reading, MA

    Google Scholar 

  • P. Mangiameli K.S. Chen D. West (1996) ArticleTitleA comparison of SOM neural network and hierarchical clustering methods European Journal of Operations Research 93 402–417 Occurrence Handle10.1016/0377-2217(96)00038-0

    Article  Google Scholar 

  • B. Mirkin (1996) Mathematical Classification and Clustering Kluwer Academic Publishers Dordrecht, The Netherlands

    Google Scholar 

  • M. Roubens (1982) ArticleTitleFuzzy clustering algorithms and their cluster validity European Journal of Operational Research 10 IssueID3 294–301 Occurrence Handle10.1016/0377-2217(82)90228-4

    Article  Google Scholar 

  • E.H. Ruspini (1973) ArticleTitleNew experimental results in fuzzy clustering Information Science 6 273–284 Occurrence Handle10.1016/0020-0255(73)90043-1

    Article  Google Scholar 

  • N.V. Sahinidis (1996) ArticleTitleBARON: A general purpose global optimization software package Journal of Global Optimization 8 IssueID2 201–205 Occurrence Handle10.1007/BF00138693

    Article  Google Scholar 

  • H.D. Sherali W.P. Adams (1990) ArticleTitleA hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems SIAM Journal on Discrete Mathematics 3 IssueID3 411–430 Occurrence Handle10.1137/0403036

    Article  Google Scholar 

  • H.D. Sherali W.P. Adams (1994) ArticleTitleA hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems Discrete Applied Mathematics 52 83–106 Occurrence Handle10.1016/0166-218X(92)00190-W

    Article  Google Scholar 

  • H.D. Sherali W.P. Adams (1999) Reformulation-linearization techniques for discrete optimization problems D.Z. Du P.M. Pardalos (Eds) Handbook of Combinatorial Optimization 1 Kluwer Academic Publishers Dordrecht 479–532

    Google Scholar 

  • H.D. Sherali J. Desai (2005) ArticleTitleAn RLT-based global optimization algorithm for solving the hard clustering problem Journal of Global Optimization 32 IssueID2 281–306 Occurrence Handle10.1007/s10898-004-2706-7

    Article  Google Scholar 

  • H.D. Sherali V. Ganesan (2003) ArticleTitleA pseudo-global optimization approach with application to the design of containerships Journal of Global Optimization 26 335–360 Occurrence Handle10.1023/A:1024792717467

    Article  Google Scholar 

  • H.D. Sherali C. Tuncbilek (1992) ArticleTitleA global optimization algorithm for polynomial programming problems using a reformulation-linearization technique Journal of Global Optimization 2 101–112

    Google Scholar 

  • H.D. Sherali C.H. Tuncbilek (1997) ArticleTitleComparison of two Reformulation-Linearization Technique based linear programming relaxations for polynomial programming problems Journal of Global Optimization 10 381–390 Occurrence Handle10.1023/A:1008237515535

    Article  Google Scholar 

  • H.D. Sherali H. Wang (2001) ArticleTitleGlobal optimization of nonconvex factorable programming problems Mathematical Programming 89 IssueID3 459–478

    Google Scholar 

  • H. Späth (1980) Cluster Analysis Algorithms for Data Reduction and Classification of Objects Wiley New York

    Google Scholar 

  • M. Sultan D.A. Wigle C.A. Cumbaa M. Maziarz J. Glasgow M.S. Tsao I. Jurisica (2002) ArticleTitleBinary tree-structured vector quantization approach to clustering and visualizing microarray data Bioinformatics 18 IssueID1 111–119

    Google Scholar 

  • M.P. Windham (1982) ArticleTitleCluster validity for the fuzzy c-means clustering algorithm IEEE Transactions on Pattern Analysis and Machine Intelligence 4 357–363 Occurrence Handle10.1109/TPAMI.1982.4767266

    Article  Google Scholar 

  • M.P. Windham (1983) ArticleTitleGeometric fuzzy clustering algorithms Fuzzy Sets and Systems 10 271–279 Occurrence Handle10.1016/S0165-0114(83)80120-1

    Article  Google Scholar 

  • N. Zahid M. Limouri A. Essaid (1999) ArticleTitleNew cluster-validity for fuzzy clustering Pattern Recognition 32 IssueID7 1089–1097 Occurrence Handle10.1016/S0031-3203(98)00157-5

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Grado Department of Industrial and Systems Engineering (0118), Virginia Polytechnic Institute and State University, 250 Durham Hall, Blacksburg, VA, 24061, USA

    Hanif D. Sherali & Jitamitra Desai

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  1. Hanif D. Sherali
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  2. Jitamitra Desai
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Correspondence to Hanif D. Sherali.

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Sherali, H.D., Desai, J. A Global Optimization RLT-based Approach for Solving the Fuzzy Clustering Problem. J Glob Optim 33, 597–615 (2005). https://doi.org/10.1007/s10898-004-7390-0

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  • DOI: https://doi.org/10.1007/s10898-004-7390-0

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