Binding and Nonbinding Constraints¶

Definition¶

For an inequality constraint written as $g_i(\boldsymbol{x})\le b_i$$b_i$, the constraint is binding at a feasible point $\boldsymbol{x}^\ast$ if

It is nonbinding or slack at $\boldsymbol{x}^\ast$ if

The same idea applies to constraints written in other equivalent sign conventions.

Interpretation¶

A binding constraint is tight at the chosen point. A nonbinding constraint still has room left and does not directly limit the current choice.

At the illustrated optimum, $g_1(\boldsymbol{x}^\ast)=3.9$ binds and has multiplier $\lambda_1>0$$0$, while $g_2(\boldsymbol{x}^\ast)=2.2<3$ is slack and has $\lambda_2=0$$0$. In this example, relaxing the first constraint increases the maximum, whereas relaxing the second does not change it.

The function $g_i$ evaluates the constrained quantity at the chosen point. Comparing that value with $b_i$ through the defining inequality distinguishes equality at the boundary from strict slack inside the allowed region.

Key Results¶

• Only binding inequality constraints can carry nonzero multipliers in standard Kuhn-Tucker conditions.
• The set of binding constraints is often called the active set.
• Changing a nonbinding constraint slightly usually does not change the optimum locally.

Examples¶

• If $x\le 10$ and the optimum has $x=10$$10$, the constraint is binding.
• If $x+y\le 5$ and the chosen point is $(1,2)$, the constraint is nonbinding because $1+2<5$$5$.

Prerequisites¶

• Inequality
• Function

Related Concepts¶

• Complementary Slackness
• Method of Lagrange for Inequality-Constrained Maximization
• Lagrange Multiplier
• Constraint

Sources¶

• Math IV: Multivariate Calculus · Week 5, Lecture 1

Aliases: active and inactive constraints, inactive constraint, tight constraint Keywords: binding constraint, nonbinding constraint, slack constraint, active constraint Symbols: lambda

Last updated: 17 October 2026 21:53