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

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.