Differentiability of a maximum function. I

Abstract

A topic discussed in [1] was the differentiability with respect to a direction of a function of the form ϑ(y) = max x∈Ω x ƒ (x, y) . A variety of problems can be solved by means of θ( y), including the problem of best approximation of functions, the problem of finding a minimax, and time-optimal problems. Such functions have been discussed e.g. in [2]. It is sometimes of interest to investigate the differentiability with respect to a direction of the function ϑ(z) = max x min y ƒ(x,y,z) . Here, finite-dimensional sets are considered. The result can also be extended to the case of infinite-dimensional spaces.