Necessary conditions for min-max problems and algorithms by a relaxation procedure

Abstract

For decision making under uncertainty, a rational optimality criterion is min-max. Min-max problems such that the minimizer makes an optimal decision against the worst case that might be chosen by the maximizer are studied. This paper presents necessary conditions and computational methods for a min-max solution (not a saddle point solution). Those conditions are stated in a form like Kuhn-Tucker theorem. The computational methods are based on the relaxation procedure. A min-max problem such that the minimizer and the maximizer are subject to separate constraints is primarily studied. But it is shown that the obtained results can be applied for the unseparate constraint case by use of duality theory.