A lower bound on the complexity of the convex hull problem for simple polyhedra
Information Processing Letters
Unsupervised Optimal Fuzzy Clustering
IEEE Transactions on Pattern Analysis and Machine Intelligence
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
An optimal convex hull algorithm and new results on cuttings (extended abstract)
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Discrete Applied Mathematics
Optimization methods for fuzzy clustering
Fuzzy Sets and Systems
Introduction to Algorithms: A Creative Approach
Introduction to Algorithms: A Creative Approach
Clustering Algorithms
A Pseudo-Global Optimization Approach with Application to the Design of Containerships
Journal of Global Optimization
A Global Optimization RLT-based Approach for Solving the Hard Clustering Problem
Journal of Global Optimization
A survey of fuzzy clustering algorithms for pattern recognition. I
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
A survey of fuzzy clustering algorithms for pattern recognition. II
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
Neural networks for convex hull computation
IEEE Transactions on Neural Networks
Journal of Global Optimization
Clustering high dimensional data: A graph-based relaxed optimization approach
Information Sciences: an International Journal
A review of recent advances in global optimization
Journal of Global Optimization
Calibrating Steady-State Traffic Stream and Car-Following Models Using Loop Detector Data
Transportation Science
Journal of Global Optimization
An efficient algorithm for maximal margin clustering
Journal of Global Optimization
Journal of Global Optimization
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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.