A comparison of two approaches to second-order subdifferentiability concepts with application to optimality conditions

Abstract

The graphical derivative and the coderivative when applied to the proximal subdifferential are in general not generated by a set of linear operators Nevertheless we find that in directions at which the subject (or subhessian) is supported, in a rank-1 sense, we have these supported operators interpolating the contingent cone. Thus under a prox-regularity assumption we are able to make a selection from the contingent graphical derivative in certain directions, using the exposed facets of a convex set of symmetric matrices. This allows us to make a comparison between some optimality conditions. A nonsmooth formulation of a standard smooth mathematical programming problem is used to derive a novel set of sufficient optimality conditions.