A global optimization RLT-based approach for solving the fuzzy clustering problem

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.