On the Solution of Min-sum-min Problems

Abstract

One class of min-sum-min problems is discussed in the paper. Min-sum-min problems appear in a natural way in many applications (e.g., in cluster analysis, pattern recognition, classification theory etc.). Like min-max-min problems, min-sum-min problems represent a very important family of nonsmooth problems. Problems of this type can be treated by means of the existing tools of Nonsmooth Analysis. However, most of algorithms available provide a local minimizer only, since they are based on necessary conditions which are of local nature. In the paper it is proved that the original problem can be reduced to the problem of minimizing a finite number of sum-functions. A necessary condition for a global minimum and a sufficient condition for a local minimum are stated. The necessary condition is of nonlocal nature. An algorithm (so-called Exchange algorithm) for finding points, satisfying necessary conditions, is described. An ɛ-Exchange algorithm is formulated, allowing, in principle, to escape from a ‘shallow’ local minimizer. An example is presented to illustrate the results and algorithms. An application of the proposed algorithms to solving one clustering problem is also given. Numerical results are provided.