# Differentiability of value of convex optimization problem

Setup: Consider the problem $$V(y) \quad = \quad \min_{x \in \mathbb R^N} f(x) \quad \text{s.t.} \quad g(x+y) \leq 0$$ where $$f$$ and $$g$$ are convex functions and $$y \in \mathbb R^N$$ is a parameter taken as given. Assume that this problem admits a (potentially non-unique) solution for all $$y \in \mathbb R^N$$.

Question: Under what conditions is $$V(y)$$ twice continuously differentiable in $$y$$?

This is related to the envelope theorem (e.g. Milgrom Segal [2002]) but deals with a higher derivative.

Cross posted from Math Stack Exchange.

(1) If either $$f$$ or $$g$$ is strictly convex, then (as I have already assumed a solution exists) there exists a unique solution $$x(y)$$.
(2) If $$f$$ and $$g$$ are continuously differentiable and one imposes assumptions so that the constraint binds, then this solution and its associated Lagrange multiplier (which exists by convexity) are $$N+1$$ unknowns solving a system of $$N+1$$ equations (first order condition plus constraint). If $$f$$ and $$g$$ are twice continuously differentiable and some rank conditions hold, one can apply the implicit function theorem to argue that $$x(y)$$ is continuously differentiable.
(3) Theorem 3 of Milgrom & Segal (2002) is an envelope theorem implying that---under some conditions implied by those assumed in (2)---$$V(y)$$ is differentiable and $$V_y(y) = -f_x(x(y))$$. (Here it helps to rewrite the problem as suggested by Ilya's comment on Math Stack Exchange.) Since $$f(x)$$ and $$x(y)$$ are both continuously differentiable, so is $$V_y(y)$$. So $$V(y)$$ is twice continuously differentiable.