Abstract

We consider the gradient method x t+1 = x t +# t (s t +w t ), where s t is a descent direction of a function f : # n # # and w t is a deterministic or stochastic error. We assume that #f is Lipschitz continuous, that the stepsize # t diminishes to 0, and that s t and w t satisfy standard conditions. We show that either f(x t ) #-# or else f(x t ) converges to a finite value and #f(x t ) # 0 (with probability 1 in the stochastic case), and in doing so, we remove various boundedness conditions that are assumed in existing results, such as boundedness from below of f , or boundedness of #f(x t ), or boundedness of x t . 1 Research supported by NSF under Grant DMI-9625489 2 Dept. of Electrical Engineering and Computer Science, M.I.T., Cambridge, Mass., 02139. 1 1. Introduction 1. INTRODUCTION We consider the problem minimize f(x) subject to x # # n , (1.1) where # n denotes the n-dimensional Euclidean space and f : # n ## # is a continuously di#erentiable fu...

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