A viscosity solution approach to regularity properties of the optimal value function

TL;DR

This paper uses viscosity solution theory to analyze the regularity of the optimal value function in parametric optimization, establishing new conditions for Lipschitz continuity and differentiability.

Contribution

It introduces a viscosity solution framework to derive regularity properties of the optimal value function, including Lipschitz and differentiability conditions in various spaces.

Findings

Established Lipschitz conditions in Banach spaces.

Derived optimality conditions via comparison principles.

Connected viscosity and Clarke solutions for differentiability.

Abstract

In this paper we analyze the optimal value function $v$ associated to a general parametric optimization problems via the theory of viscosity solutions. The novelty is that we obtain regularity properties of $v$ by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of $v$. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get further differentiability properties of $v$ in Euclidean spaces.