Method of Lagrange for Mixed Constraints¶

Theorem¶

Let $A\subseteq\mathbb{R}^n$ be open, and let the objective $f$ and all constraint functions $g_i,h_j$ be continuously differentiable on $A$ . Consider

\begin{aligned} &\max_{\boldsymbol{x}\in A} f(\boldsymbol{x})\\ &\text{subject to}\quad g_i(\boldsymbol{x})\le b_i,\quad h_j(\boldsymbol{x})=c_j. \end{aligned}

The course convention defines

L(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\mu}) =f(\boldsymbol{x})+\sum_i\lambda_i(b_i-g_i(\boldsymbol{x})) +\sum_j\mu_j(c_j-h_j(\boldsymbol{x})).

If $\boldsymbol{x}^\ast$ is a local constrained maximizer and the binding inequalities are indexed by $i_1,\ldots,i_\ell$ , then either

\operatorname{rank}\boldsymbol{D}(g_{i_1},\ldots,g_{i_\ell},h_1,\ldots,h_m)(\boldsymbol{x}^\ast) <\ell+m,

or there exist multipliers satisfying stationarity, complementary slackness, inequality feasibility, equality feasibility, and $\lambda_i^\ast\ge0$ $0$ .

Interpretation¶

Mixed problems combine exact equations with one-sided restrictions. The first-order candidate conditions combine equality stationarity with inequality sign and slackness rules.

An arc of the equality constraint h(x) = c cut off by the vertical inequality constraint x1 at most 3.1. At x star, where both constraints are active, the gradient of f lies between the gradients of h and g.

The illustrated feasible arc satisfies $h(\boldsymbol{x})=c$ and $g(\boldsymbol{x})\le3.1$ $3.1$ . Both constraints bind at $\boldsymbol{x}^\ast$ , where the objective gradient is a combination of the active constraint gradients with the appropriate multiplier signs. The theorem above specifies the normalization and regularity requirements for the multipliers.

The equality-constraint method contributes unrestricted multipliers for the equalities. The inequality-constrained maximization method contributes nonnegative multipliers and complementary slackness for the inequalities. Both kinds enter the same stationarity equation.

Key Results¶

Only inequalities require nonnegative multipliers.
Only binding inequalities are included in the active NDCQ matrix.
All equality constraints must hold exactly.

Examples¶

For $\max xy$ subject to $x+y\le10$ and $x-y=0$ $0$ , the method uses one nonnegative multiplier and one unrestricted multiplier.
If an inequality is nonbinding at the solution, its multiplier is zero and it is not part of the active rank test.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 5, Lecture 2

Aliases: Lagrange method with mixed constraints Keywords: mixed constraints, mixed-constraint FONC, KKT conditions with equalities Symbols: element of, lambda, mu, real numbers, subset, sum

Last updated: 22 October 2026 19:04