General Method of Lagrange¶

Theorem¶

Let $A\subseteq\mathbb{R}^n$ be open, and let $f,g_i,h_j$ be continuously differentiable on $A$ . For minimization, write the inequalities as $g_i(\boldsymbol{x})\le0$ $0$ ; for maximization, write them as $g_i(\boldsymbol{x})\ge0$ $0$ . With equalities $h_j(\boldsymbol{x})=0$ $0$ , define the generalized Lagrangian

\Lambda(\boldsymbol{x},\lambda_0,\boldsymbol{\lambda},\boldsymbol{\mu}) = \lambda_0 f(\boldsymbol{x})+\sum_i\lambda_i g_i(\boldsymbol{x}) +\sum_j\mu_j h_j(\boldsymbol{x}).

If $\boldsymbol{x}^\ast$ is a local constrained optimum, then there exist multipliers with $\lambda_0\in\{0,1\}$ ${0, 1}$ , $\lambda_i\ge0$ $0$ , and $(\lambda_0,\boldsymbol{\lambda},\boldsymbol{\mu})\neq0$ such that stationarity, feasibility, and complementary slackness hold.

Interpretation¶

This method avoids a separate regularity check by adding a multiplier on the objective itself. The extra case captures points where the active constraints are degenerate.

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 mixed-constraint method is recovered when the objective multiplier can be normalized to one. The nondegenerate constraint qualification rules out an abnormal candidate with zero objective multiplier: independence of the active constraint gradients would then force every multiplier to vanish.

Key Results¶

If $\lambda_0=1$ $1$ , the regular first-order necessary conditions are recovered.
If $\lambda_0=0$ $0$ , the condition signals linear dependence among active constraint gradients.
The method keeps nonregular feasible points inside the theorem.

Examples¶

For a regular optimum, choosing $\lambda_0=1$ gives the usual Lagrangian conditions.
If active constraint gradients are linearly dependent, a solution may appear with $\lambda_0=0$ and nonzero constraint multipliers.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 7, Lecture 1

Aliases: Fritz John conditions, Fritz John theorem Keywords: generalized Lagrange method, objective multiplier method, Lagrange method without checking NDCQ Symbols: element of, lambda, mu, real numbers, subset, sum

Last updated: 22 October 2026 19:04