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

TL;DR

This paper develops new methods for calculating subgradients of marginal functions, which are crucial in variational analysis and bilevel programming, enhancing the understanding and solution of complex optimization problems.

Contribution

It provides novel formulas for regular subgradients of marginal functions and applies these to derive calculus rules and optimality conditions in bilevel programming.

Findings

Exact calculations of subgradients for broad classes of marginal functions.

New calculus rules for subgradients in nonsmooth analysis.

Necessary optimality conditions in bilevel programming derived from these results.

Abstract

The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared 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/Fr\'echet 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