Abstract
This paper is concerned with a min-max problem for decision-making under uncertainty that excludes even stochastic information. A min-max solution is sought for the problem that a function to be maximized with respect to the maximizer's variables is to be minimized with respect to the minimizer's variables. The solution is rational in the sense that the minimizer makes an optimal decision against the worst case that might be chosen by the opponent (the maximizer).
First we present a necessary condition and a computational method for the min-max problem in which the minimizer and the maximizer are constrained separately. This condition is stated in a form like that of Kuhn-Tucker's Theorem and is closely related to the subgradients of a Lagrangian. The proposed computational method is based on a relaxation procedure. We next study the min-max problem of the minimizer and the maximizer subject to unseparated constraint. It is shown that the results obtained for the separated case can be applied for the unseparat ed one by the use of duality theory.
