Clustering by Hyperbolic Smoothing

This paper addresses the minimum sum-of-squares clustering problem. The mathematical modeling of this problem leads to a min-sum-min formulation which, in addition to its intrinsic bi-level nature, has the significant characteristic of being strongly nondifferentiable. To overcome these difficulties, the solution method proposed adopts a smoothing strategy using a special C ∞ differentiable class function. The final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essentials of the method is presented. For the purpose of illustrating both the reliability and the efficiency of the method, a set of computational experiments was performed making use of traditional test problems described in the literature. New best known solutions were produced by the method for two of these test problems.