Regular Subgradients of Marginal Functions with Applications to Calculus and Bilevel Programming

The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which frequently appear in variational analysis, parametric optimization, and a variety of applications. Functions of this type are intrinsically nonsmooth and require the usage of tools of generalized differentiation. The main results of this paper provide novel evaluations and exact calculations of regular/Fréchet subgradients and their singular counterparts for general classes of marginal functions via their given data. The obtained results are applied to establishing new calculus rules for such subgradients and necessary optimality conditions in bilevel programming.