Lagrangemultiplier

A Lagrange multiplier is a variable used in constrained optimization to find extrema of a function subject to equality constraints. The method introduces multipliers, collectively called λ, and forms the Lagrangian L(x, λ) = f(x) + λ^T g(x), where f is the objective function and g(x) = (g1(x), ..., gm(x)) represents the constraints gi(x) = 0.

To locate candidate extrema, one solves the stationary equations obtained from the Lagrangian together with the constraints: ∇x L(x, λ) = ∇f(x) + ∑i λi ∇gi(x) = 0 and gi(x) = 0 for all i. For a single constraint g(x) = 0, this reduces to ∇f(x) + λ ∇g(x) = 0 and g(x) = 0. The condition ∇f(x) = −λ ∇g(x) expresses that at an optimum the gradient of f lies in the span of the gradients of the active constraints.

Interpretation often given is that the multipliers measure the sensitivity of the optimum value to small changes in the constraints: increasing the constraint boundary by a small amount changes the optimal objective by approximately λ times that amount.

Existence of multipliers generally requires a constraint qualification, such as the gradients ∇gi(x) being linearly independent at the solution. After finding candidate points, one may apply second-order conditions to classify them as minima or maxima.

Extensions of the basic method address inequality constraints through Karush-Kuhn-Tucker conditions and related augmented Lagrangian methods. Lagrange multipliers are widely used in physics, economics, engineering, and various areas of applied mathematics.