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

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