Applying the Gradient Projection Method to a Model of Proportional Membership for Fuzzy Cluster Analysis

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2 Citations (Scopus)


This paper presents a fuzzy proportional membership model for clustering (FCPM). Unlike the other clustering models, FCPM requires that each entity may express an extent of each prototype, which makes its criterion to loose the conventional prototype-additive structure. The methods for fitting the model at different fuzziness parameter values are presented. Because of the complexity of the clustering criterion, minimization of the errors requires the gradient projection method (GPM).We discuss how to find the projection of a vector on the simplex of the fuzzy membership vectors and how the stepsize length of the GPM had been fixed. The properties of the clusters found with the FCPM are discussed. Especially appealing seems the property to keep the extremal cluster prototypes stable even after addition of many entities around the grand mean.
Original languageEnglish
Title of host publicationOptimization and its Applications in Control and Data Sciences
Subtitle of host publicationIn Honor of Boris T. Polyak’s 80th Birthday
EditorsBoris Goldengorin
Place of PublicationCham
PublisherSpringer International Publishing
ISBN (Electronic)978-3-319-42056-1
ISBN (Print)978-3-319-42054-7
Publication statusPublished - 1 Sept 2016

Publication series

NameSpringer Optimization and Its Applications
PublisherSpringer International Publishing
ISSN (Print)1931-6828


  • Fuzzy Proportional membership
  • Gradient projection method
  • Extremal cluster prototype


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