Marginal (Optimal Value) Function

Given a set-valued mapping $F:X\rightrightarrows Y$ and a function $\varphi:X\times Y\to\overline{\mathbb{R}}$ , the optimal value (marginal) function $\mu:X\to\overline{\mathbb{R}}$ is defined by

\mu(x):=\inf\{\varphi(x,y)\mid y\in F(x)\}, \qquad x\in X.

We use the convention $\inf(\emptyset):=\infty$ , and in the notes it is assumed that $\mu(x)>-\infty$ for all $x\in X$ .

The marginal function captures "minimize over $y$ given $x$ " and is central in parametric optimization and convex analysis; its convexity is addressed in the convexity theorem for marginal functions.

Examples

If $F(x)\equiv C$ is a fixed nonempty set and $\varphi(x,y)=g(x,y)$ , then $\mu(x)=\inf_{y\in C}g(x,y)$ .
If $F(x)$ is the feasible set of a constraint system depending on $x$ , then $\mu(x)$ is the optimal value of that parameterized problem.