Invexity: an updated survey and new results for nonsmooth functions

The purpose of this paper is to introduce two recent developments in optimization theory which have the potential for significant impact when applied to the many optimization problems in mathematical statistics. The needs of optimization theory have served as the catalyst for the introduction and recent development of two important concepts: invexity and generalized differentiability. The former is a significant generalization of the simple yet powerful concept of convexity, while the latter extends the differentiabe calculus to functions that are not differentiabe in any traditional two-sided sense. For the most part these concepts have been developed independently in the literature. In this paper we provide an updated review of many of the results concerning smooth invex functions and merge the two concepts by introducing invexity for nonsmooth functions. For both real-valued and vector-valued nonsmooth functions we present recent results on optimality conditions, duality, and converse duality. No previous knowledge of invexity or nonsmooth analysis is assumed; a self-contained section on generalized differentiability for Lipschitz functions is included.