Regular Subgradients of Marginal Functions with Applications to Calculus and Bilevel Programming
Le Phuoc Hai, Felipe Lara, Boris S. Mordukhovich
TL;DR
This work develops a comprehensive regular and singular subdifferential calculus for marginal functions $\mu(x)=\inf_{y\in G(x)}\varphi(x,y)$ in normed spaces, using tools from generalized differentiation. It delivers new lower and upper estimates and exact formulas for $\widehat{\partial}\mu(\bar{x})$ and $\widehat{\partial}^{\infty}\mu(\bar{x})$, under calmness-type and metric-qualification conditions, including when costs are nondifferentiable. The authors then derive calculus rules (chain, product, quotient, and sums) for these subgradients and apply the results to variational calculus and to optimistic bilevel programming in Asplund spaces, obtaining necessary optimality conditions that broaden prior finite-dimensional results. The framework supports sensitivity analysis and optimization in nonsmooth parametric settings, with potential impact on infinite-dimensional bilevel problems and related applications.
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/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