Special Classes of Nondifferentiable Functions and Generalizations of the Concept of the Gradient

Abstract

When solving many mathematical programming problems one has to consider a broader class of functions than that of continuously differentiable functions. In such cases We, as a rule, deal with continuous and almost everywhere differentiable functions, whose gradient is discontinuous on a set of null measure. Although in principle functions may arise with even greater pathology (for example, functions that are nowhere differentiable on a set of positive measure), such functions are not encountered in practice; moreover, construction of any nontrivial algorithm for minimizing such functions does not appear possible.