A Modified Algorithm of Finding an Element of Clarke Generalized Gradient for a Smooth Composition of Max-Type Functions

Abstract

This paper refers to Clarke generalized gradient for a smooth composition of max-type functions of the form: $f(x)=g(x,\ {\rm max}_{j \in J_1} \ f_{1j}(x),\cdots, \ {\rm max}_{j \in J_m} \ f_{mj}(x))$, where $x\in R^n, \ J_i, \ i=1, ..., \ m$ are finite index sets, $g$ and $f_{ij}, \ j \in J_i, \ i=1,...,m$, are continuously differentiable on $R^{m+n}$ and $R^n$ respectively. In a previous paper, we proposed an algorithm of finding an element of Clarke generalized gradient for $f$, at a point. In that paper, finding an element of Clarke generalized gradient for $f$ , at a point, is implemented by determining the compatibilities of systems of linear inequalities many times. So its computational amount is very expensive. In this paper, we will modify the algorithm to reduce the times that the compatibilities of systems of linear inequalities have to be determined.