Convexity of the Marginal (Optimal Value) Function

Convexity of the Marginal Function: Let $X$ and $Y$ be vector spaces . Assume that $\varphi:X\times Y\to\overline{\mathbb{R}}$ is a convex function and that $F:X\rightrightarrows Y$ is a convex set-valued mapping (i.e., its graph is convex). Define the marginal function $\mu$ by

Then $\mu$ is convex on $X$.

Context: This result explains why optimal value functions in convex optimization are convex in parameters: convexity of $\varphi$ and convexity of the feasible graph propagate through the infimum operation.

Proof sketch (idea): Fix $x_1,x_2$ and choose near-minimizers $y_1\in F(x_1)$, $y_2\in F(x_2)$. By graph convexity, $\lambda y_1+(1-\lambda)y_2\in F(\lambda x_1+(1-\lambda)x_2)$. Use convexity of $\varphi$ to bound $\mu(\lambda x_1+(1-\lambda)x_2)$ by $\lambda\mu(x_1)+(1-\lambda)\mu(x_2)$ (letting the approximation error go to $0$).