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.

Method

Suppose that $f(\mathbf{x})$ and $g_{i}(\mathbf{x}),i=1,...,m$ ( $\mathbf{x}\in\mathbbmss{R}^{n}$ ) are differentiable functions that map $\mathbbmss{R}^{n}\mapsto\mathbbmss{R}$ , and we want to solve

\min f(\mathbf{x}),\max f(\mathbf{x})\quad\mbox{such that}\quad g_{i}(\mathbf{% x})=0,\quad i=1,\ldots,m

By a calculus theorem, if the constaints are independent, the gradient of $f$ , $\nabla f$ , must satisfy the following equation at the stationary points:

\nabla f=\sum_{i=1}^{m}\lambda_{i}\nabla g_{i}

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

$\left\{\nabla g_{1}(\mathbf{x}),\ldots,\nabla g_{m}(\mathbf{x})\right\}$ is a set of linearly independent vectors on all points where the constraints are verified.

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

f(\mathbf{x})-\sum_{i=1}^{m}\lambda_{i}(g_{i}(\mathbf{x}))

for $\mathbf{x}$ in the domain and the Lagrange multipliers $\lambda_{i}$ without restrictions.

After finding those points, one applies the $g_{i}$ 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