Favorable Classes of Lipschitz Continuous Functions in Subgradient Optimization

Abstract

Clarke has given a robust definition of subgradients of arbitrary Lipschitz continuous functions f on R^n, but for purposes of minimization algorithms it seems essential that the subgradient multifunction partial f have additional properties, such as certain special kinds of semicontinuity, which are not automatic consequences of f being Lipschitz continuous. This paper explores properties of partial f that correspond to f being subdifferentially regular, another concept of Clarke's, and to f being a pointwise supremum of functions that are k times continuously differentiable.