Optimality Conditions and Generalized Gradient Methods for Mathematical Programming Problems Including Extremal-Value Functions

Abstract

We study a mathematical programming problem whose objective and constraint functions include the extremal-value functions. This problem can be regarded as an extension of well-known “min-max problem” and/or infinitely constrained problem”, and it becomes a nondifferentiable optimization problem. In this paper, we present some optimality conditions for such problem by use of a new theorem of the alternative and Gauvin's results concerned with the extremal-value functions. Moreover, as a computational method for our problem, we propose applying the Mifflin's nondifferentiable optimization algorithm by means of the generalized gradients. A numerical example is also presented.