Optimality Conditions For Max-Type Nonsmooth Minimization Problems

The study of max-type nonsmooth minimization problems was initiated by Ben-Tal and Zowe ([2], 1982). In this paper we develop similar optimality conditions by using conditions developed in Di’s recent paper ([7], 1996). Along with the general max-type problems, four specific kinds of max-type problems are considered: the discrete minimax problem, the discrete l 1-approximation, the minimization of an exact penalty function, and the minimization of the exterior penalty function. The assumptions in this paper are weaker than the conditions imposed by Ben-Tal and Zowe, and an abstract constraint set can be incorporated. The functions involved are only assumed to be Frechet differentiable (first order and second order) at the point of interest instead of being smooth, as in [2]. However a constraint qualification needs to be assumed (most results in [2] do not assume a constraint qualification)