Value Function¶

Definition¶

For a parametric optimization problem, the value function assigns to each parameter value the optimized objective value. For example,

when $\boldsymbol{x}^\ast(s)$ solves the problem at parameter $s$.

Interpretation¶

The value function records the best achievable outcome after the decision variables have been chosen optimally.

$V(s)=f(x^\ast(s),s)=\max_{x} f(x,s)$ is the best value attainable at $s$: its graph is never below any single curve $s\mapsto f(x,s)$ with $x$ fixed (grey) and touches the curve of $x^\ast(s)$; here $V(s)=s^2$$s^2$, for instance $V(2)=f(x^\ast(2),2)$ with $x^\ast(2)=2$$2$.

A global maximum or minimum determines the best feasible objective at each parameter value. When several optimizers attain it, they give the same value; recording only a local optimum would not in general define the global value function.

Key Results¶

• The value function depends on parameters, not directly on arbitrary feasible choices.
• Envelope theorems provide formulas for derivatives of value functions under regularity conditions.
• A value function can be nonsmooth if the optimizer changes abruptly or constraints switch status.

Examples¶

• If $V(I)$ is the maximum utility attainable with income $I$, then $V$ is a value function.
• If $V(K)$ is maximum profit when available capital is $K$, then changing $K$ changes the optimization problem and hence the value.

Prerequisites¶

• Parametric Constrained Optimization Problem
• Global Maximum and Minimum on $\mathbb{R}^n$

Related Concepts¶

• Envelope Theorem for Constrained Optimization
• Shadow Price in Constrained Optimization
• Method of Lagrange for Equality Constraints

Sources¶

• Math IV: Multivariate Calculus · Week 6, Lecture 1

Aliases: optimal value function, indirect objective function, indirect objective, maximum value function Symbols: real numbers

Last updated: 17 October 2026 21:53