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

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.