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Method of Lagrange for Constrained Minimization

Theorem

Let A⊆RnA\subseteq\mathbb{R}^n be open, and let the objective ff and all constraint functions gi,hjg_i,h_j be continuously differentiable on AA. For

min⁡x∈Af(x)subject togi(x)≥bi,hj(x)=cj,\begin{aligned} &\min_{\boldsymbol{x}\in A} f(\boldsymbol{x})\\ &\text{subject to}\quad g_i(\boldsymbol{x})\ge b_i,\quad h_j(\boldsymbol{x})=c_j, \end{aligned}

the course convention writes the inequality part as bi−gi(x)≤0b_i-g_i(\boldsymbol{x})\le0 and defines

L(x,λ,μ)=f(x)+∑iλi(bi−gi(x))+∑jμj(cj−hj(x)).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 x∗\boldsymbol{x}^\ast is a local constrained minimizer, then either nondegenerate constraint qualification (NDCQ) fails or there are multipliers satisfying

∇xL=0,λi∗(gi(x∗)−bi)=0,gi(x∗)≥bi,hj(x∗)=cj,λi∗≥0.\begin{gathered} \boldsymbol{\nabla}_xL=0,\quad \lambda_i^\ast(g_i(\boldsymbol{x}^\ast)-b_i)=0,\\ g_i(\boldsymbol{x}^\ast)\ge b_i,\quad h_j(\boldsymbol{x}^\ast)=c_j,\quad \lambda_i^\ast\ge0. \end{gathered}

Interpretation

For minimization, the inequality is written so the same nonnegative multiplier convention can be used. The resulting conditions are still only a candidate filter.

Feasible half-plane x1 + x2 at least 3.2, with the unconstrained minimum of f outside it. At the constrained minimum x star on the boundary line, the gradients of f and g point the same way, into the feasible side.

In the illustrated minimization problem, gi(x)≥big_i(\boldsymbol{x})\ge b_i and the nonnegative multipliers satisfy ∇f(x∗)=∑iλi∇gi(x∗)+∑jμj∇hj(x∗)\displaystyle \boldsymbol{\nabla} f(\boldsymbol{x}^\ast)=\sum_i\lambda_i\boldsymbol{\nabla} g_i(\boldsymbol{x}^\ast)+\sum_j\mu_j\boldsymbol{\nabla} h_j(\boldsymbol{x}^\ast) and λi(gi(x∗)−bi)=0\lambda_i(g_i(\boldsymbol{x}^\ast)-b_i)=0. At the boundary optimum, the gradient points into the feasible side. Equivalently, one can maximize −f-f under the same constraints.

The mixed-constraint method separates equality multipliers from signed inequality multipliers. Passing from maximization to minimization requires the residual convention written above; multiplier signs cannot be copied without checking that convention.

Key Results

  • The sign of the inequality term in the Lagrangian depends on the feasible side of the constraint.
  • Nonbinding inequalities have zero multipliers.
  • Equality multipliers remain unrestricted.

Examples

  • For min⁡ex−y\min \mathrm{e}^{x-y} subject to x2+y2≥4x^2+y^2\ge4, use L=ex−y+λ(4−x2−y2)L=\mathrm{e}^{x-y}+\lambda(4-x^2-y^2).
  • If x+y>1x+y>1 at a local minimizer subject to x+y≥1x+y\ge1, complementary slackness forces the multiplier on that constraint to be zero.

Prerequisites

Sources

Aliases: inequality minimization Lagrangian Keywords: minimization FONC, Lagrange method for minimization Symbols: element of, lambda, mu, nabla gradient, real numbers, subset, sum

Last updated: 22 October 2026 19:04