How to Divide a Territory? A New Simple Differential Formalism for Optimization of Set Functions

Authors

Hung T. Nguyen
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-1998

Comments

Technical Report: UTEP-CS-97-25a

Published in International Journal of Intelligent Systems, 1999, Vol. 14, No. 3, pp. 223--251.

Abstract

In many practical problems, we must optimize a set function, i.e., find a set A for which f(A) is maximum, where f is a function defined on the class of sets. Such problems appear in design, in image processing, in game theory, etc.

Most optimization problems can be solved (or at least simplified) by using the fact that small deviations from an optimal solution can only decrease the value of the objective function; as a result, some derivative must be equal to 0. This approach has been successfully used, e.g., for set functions in which the desired set A is a shape, i.e., a smooth (or piece-wise smooth) surface. In some real-life problems, in particular, in the territorial division problem, the existing methods are not directly applicable. For such problems, we design a new simple differential formalism for optimizing set functions.