Lagrange multiplier method

The Lagrange multiplier method is used when one needs to find the extreme or stationary points of a function on a set which is a subset of the domain.


Suppose that f(𝐱) and gi(𝐱),i=1,,m (𝐱n) are differentiable functions that map n, and we want to solve

minf(𝐱),maxf(𝐱)such thatgi(𝐱)=0,i=1,,m

By a calculus theoremMathworldPlanetmath, if the constaints are independent, the gradientMathworldPlanetmath of f, f, must satisfy the following equation at the stationary points:


The constraints are said to be independent iff all the gradients of each constraint are linearly independentMathworldPlanetmath, that is:

{g1(𝐱),,gm(𝐱)} is a set of linearly independent vectors on all points where the constraints are verified.

Note that this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to finding the stationary points of:


for 𝐱 in the domain and the Lagrange multipliers λi without restrictionsPlanetmathPlanetmath.

After finding those points, one applies the gi constraints to get the actual stationary points subject to the constraints.

Title Lagrange multiplier method
Canonical name LagrangeMultiplierMethod
Date of creation 2013-03-22 12:25:10
Last modified on 2013-03-22 12:25:10
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 10
Author cvalente (11260)
Entry type Definition
Classification msc 49K30
Related topic ExampleOfCalculusOfVariations
Related topic IsoperimetricProblem
Defines Lagrange multiplier