Necessary Conditions for Min-Max Problems and Algorithms by a Relaxation Procedure

For decision making under uncertainty, a rational opthnality criterion is min-max. Min-max problems such that the minhnizer makes an optimal decision against the worst case that might be chosen by the maximlzer 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 Knhn-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.