Complementary Slackness¶

Definition¶

For an inequality constraint written as $c_i(\boldsymbol{x})\le 0$ (for example $c_i=g_i-b_i$ for a constraint $g_i(\boldsymbol{x})\le b_i$ $b_{i}$ ) with multiplier $\lambda_i\ge 0$ $0$ , complementary slackness is the condition

\lambda_i c_i(\boldsymbol{x}^\ast)=0.

Equivalently, either the constraint is binding or its multiplier is zero.

Interpretation¶

Complementary slackness says that a constraint can matter through its multiplier only when it is actually tight.

Two panels. Left: the optimum lies on the boundary of the feasible region, the constraint is active and its multiplier is positive. Right: the optimum lies inside, the gradient of f is zero and the multiplier is zero.

On the left, the binding constraint holds back the maximum, with $\boldsymbol{\nabla} f=\lambda\boldsymbol{\nabla} c$ $λ \nabla c$ , $\lambda>0$ $0$ , and $c=0$ $0$ . On the right, the optimum lies in the interior and the multiplier is zero. The grey curves are objective level sets and the blue region is feasible.

Move $b$ and the peak: either the constraint binds, or its multiplier is $0$ , so the product $\lambda\,(b-g(\boldsymbol{x}^*))$ is always $0$ .

The inequality fixes the sign of the slack. A strictly negative constraint value forces its multiplier to zero because their product must vanish; equality permits a nonzero multiplier.

Key Results¶

If $c_i(\boldsymbol{x}^\ast)<0$ $0$ , then $\lambda_i=0$ $0$ .
If $\lambda_i>0$ $0$ , then $c_i(\boldsymbol{x}^\ast)=0$ $0$ .
The condition is part of the Kuhn-Tucker system for inequality-constrained optimization.

Examples¶

If $x\le 4$ is written as $x-4\le 0$ and the optimum has $x=3$ $3$ , then the multiplier must be $0$ .
If a multiplier for $x\le 4$ is positive, then the optimum must satisfy $x=4$ $4$ .

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 5, Lecture 1

Aliases: complementary slackness condition, kt slackness, complementarity condition Keywords: slackness condition Symbols: lambda, nabla gradient

Last updated: 17 October 2026 21:53