Kuhn-Tucker Lagrangian¶

Definition¶

For an optimization problem with equality constraints $h_i(\boldsymbol{x})=0$ and inequality constraints $g_j(\boldsymbol{x})\le 0$ $0$ , a Kuhn-Tucker Lagrangian is an expression of the form

L(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\mu}) = f(\boldsymbol{x}) +\sum_i \lambda_i h_i(\boldsymbol{x}) +\sum_j \mu_j g_j(\boldsymbol{x}),

with sign restrictions such as $\mu_j\ge 0$ or $\mu_j\le 0$ depending on the maximization or minimization convention.

Interpretation¶

The Kuhn-Tucker Lagrangian brings the objective and all active constraint pressures into one stationarity expression.

Feasible triangle x1 + x2 at most 3.8, with the unconstrained maximum of f outside it. At x star on the boundary the gradients of f and g are parallel and point outwards.

At the illustrated constrained maximum, the gradient points outwards, towards the infeasible unconstrained maximum. Writing the constraints as $g_i(\boldsymbol{x})\le b_i$ and using the maximization convention $\displaystyle L=f-\sum_i\lambda_i(g_i-b_i)$ with $\lambda_i\ge0$ $0$ , the multipliers satisfy $\displaystyle \boldsymbol{\nabla} f(\boldsymbol{x}^\ast)=\sum_i\lambda_i\boldsymbol{\nabla} g_i(\boldsymbol{x}^\ast)$ $i \sum λ_{i} \nabla g_{i} (x^{*})$ , $\lambda_i\ge0$ $0$ , and $\lambda_i(g_i(\boldsymbol{x}^\ast)-b_i)=0$ $0$ . Any slack inequality has multiplier zero.

At the optimum $\boldsymbol{\nabla} f$ is a nonnegative multiple of the gradients of the binding constraints.

This extends the Lagrangian function by including inequality residuals. Their multiplier signs depend on how the residual and optimization direction are written; complementary slackness prevents a strictly slack constraint from contributing a nonzero multiplier.

Key Results¶

Equality multipliers are usually unrestricted in sign.
Inequality multipliers have sign restrictions tied to the chosen convention.
Complementary slackness connects each inequality multiplier to whether its constraint is binding.

Examples¶

For maximizing $f(x)$ subject to $g(x)\le 0$ $0$ , one common convention uses $L=f-\mu g$ with $\mu\ge 0$ $0$ .
For a nonnegativity condition $x_i\ge 0$ $0$ , the Kuhn-Tucker system may include a multiplier attached to that boundary condition.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 7, Lecture 1

Aliases: KKT Lagrangian, Karush-Kuhn-Tucker Lagrangian Keywords: Lagrangian with inequality constraints Symbols: lambda, mu, nabla gradient, sum

Last updated: 22 October 2026 19:04