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Kuhn-Tucker Conditions with Nonnegativity Constraints

Theorem

Let A⊆RnA\subseteq\mathbb{R}^n be open and let ff and the finitely many constraint functions gig_i be C1C^1 on AA. For the maximization problem

max⁡x∈Af(x)subject togi(x)≤bi,xh≥0,\begin{aligned} &\max_{\boldsymbol{x}\in A} f(\boldsymbol{x})\\ &\text{subject to}\quad g_i(\boldsymbol{x})\le b_i,\qquad x_h\ge0, \end{aligned}

define the Kuhn-Tucker Lagrangian

L~(x,λ)=f(x)+∑iλi(bi−gi(x)).\widetilde L(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})+\sum_i\lambda_i(b_i-g_i(\boldsymbol{x})).

Assume the nondegenerate constraint qualification holds for all active inequalities, including the active bounds −xh≤0-x_h\le0: their gradients at the candidate point are linearly independent. At such a regular local maximum, there exists λ∗≥0\boldsymbol{\lambda}^\ast\ge0 such that

∂L~∂xh(x∗,λ∗)≤0,xh∗∂L~∂xh(x∗,λ∗)=0,xh∗≥0,\frac{\partial \widetilde L}{\partial x_h}(\boldsymbol{x}^\ast,\boldsymbol{\lambda}^\ast)\le0,\qquad x_h^\ast\frac{\partial \widetilde L}{\partial x_h}(\boldsymbol{x}^\ast,\boldsymbol{\lambda}^\ast)=0,\qquad x_h^\ast\ge0,

and

∂L~∂λi(x∗,λ∗)≥0,λi∗∂L~∂λi(x∗,λ∗)=0,λi∗≥0.\frac{\partial \widetilde L}{\partial \lambda_i}(\boldsymbol{x}^\ast,\boldsymbol{\lambda}^\ast)\ge0,\qquad \lambda_i^\ast\frac{\partial \widetilde L}{\partial \lambda_i}(\boldsymbol{x}^\ast,\boldsymbol{\lambda}^\ast)=0,\qquad \lambda_i^\ast\ge0.

Interpretation

Nonnegative variables can be either interior or stuck at the boundary. The Kuhn-Tucker conditions encode that boundary logic together with ordinary inequality slackness.

Triangular feasible region x1 + x2 at most 3.6 with x1, x2 nonnegative. The unconstrained maximum lies below the x1 axis, so the optimum x star = (2.6, 0) sits on the axis where only the constraint x2 at least 0 binds.

In this maximization example, x1∗>0x_1^\ast>0 gives ∂L/∂x1=0\partial L/\partial x_1=0, while x2∗=0x_2^\ast=0 gives ∂L/∂x2≤0\partial L/\partial x_2\le0 and x2∗(∂L/∂x2)=0x_2^\ast(\partial L/\partial x_2)=0. The inequality g(x)≤bg(\boldsymbol{x})\le b is slack, so its multiplier is λ=0\lambda=0; only the nonnegativity of x2x_2 binds. The objective would increase by moving below the x1x_1-axis, but that move is infeasible.

At a corner two constraints bind at once, each with its own nonnegative multiplier.

These conditions specialize the inequality-constrained maximization method to coordinate restrictions. Treating xh≥0x_h\ge0 as inequalities gives a zero derivative for a positive coordinate and a nonpositive derivative at a coordinate fixed at zero, with the stated maximization convention.

Key Results

  • If xh∗>0x_h^\ast>0, the derivative condition for that variable becomes equality.
  • If a multiplier is positive, the corresponding ordinary inequality constraint is binding.
  • The conditions are necessary at regular local maxima.

Examples

  • If xh∗>0x_h^\ast>0, then xh∗∂L~/∂xh=0x_h^\ast\partial\widetilde L/\partial x_h=0 forces ∂L~/∂xh=0\partial\widetilde L/\partial x_h=0.
  • If λi∗>0\lambda_i^\ast>0, then ∂L~/∂λi=0\partial\widetilde L/\partial\lambda_i=0, so the constraint is binding.

Prerequisites

Sources

Aliases: KKT conditions with nonnegativity Keywords: KT conditions, KKT conditions, Kuhn-Tucker Lagrangian, nonnegativity constraint Symbols: element of, lambda, partial, real numbers, subset, sum

Last updated: 22 October 2026 19:04