On directions of ∊-steepest descent for real-valued lipschitz continuous functions

Abstract

Suppose that f is a real-valued locally Lipschitz function having as domain n-dimensional real Euclidean space E_x. If f is continuously differentiable, the classical method of steepset descent can be used to generate a sequence whose limit points are likely to be local minimizers or stationary points. If f is not Continuously differentiable or not even differentiable, then a member of the generalized gradient may replace the gradient in the method of steepest descent.Examination of the resulting generalized algorithm has led to the study of a multifunction h, defined as follows: gievn (x,ε) with x in E _n and ε>0, h(x,ε) is the set of all points in the closed ball B(x,ε) of center x and radius e which minimize f on B(x,ε). in this paper, several properties of h are discussed. The Principal result gives a relationship concerning the angles between members of the set h(x,ε) and the set - .