LagrangeMultiplierTests

Lagrange multipliers are a method for finding extrema of a function f(x) subject to equality constraints g_i(x) = 0. The idea is to introduce auxiliary variables, called Lagrange multipliers, one for each constraint, and to form the Lagrangian L(x, λ) = f(x) + sum_i λ_i g_i(x). Under regularity conditions, a constrained extremum occurs at points where the gradient of the Lagrangian with respect to x vanishes while the constraints hold.

The standard procedure is to form the Lagrangian and then solve the system given by the stationarity condition ∇_x L(x, λ) = 0 together with the constraint equations g_i(x) = 0. The multipliers λ_i encode how much the objective would change with a small relaxation of each constraint, and the constraints themselves are enforced by ∇_λ L(x, λ) = g(x) = 0. In practice, one also checks second-order conditions or evaluates the objective at each candidate point to determine which are maxima or minima.

A simple example is maximizing f(x, y) = xy subject to the constraint x^2 + y^2 = 1. The Lagrangian is L = xy + λ(1 − x^2 − y^2). Setting the partial derivatives to zero gives y − 2λx = 0, x − 2λy = 0, and 1 − x^2 − y^2 = 0. Solving yields λ = ±1/2 with x = y for the plus case and x = −y for the minus case. The extrema on the circle occur at (x, y) = (±1/√2, ±1/√2), giving f = ±1/2.

Extensions include inequality constraints via the Karush–Kuhn–Tacker (KKT) conditions, which add nonnegativity and complementary slackness. Lagrange multipliers are widely used in economics, engineering, and physics, and connect to duality theory. They require differentiability and a suitable constraint qualification; without these, solutions may not be found or may be local rather than global.