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.

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