PlanetMath: Lagrange multiplier method

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Lagrange multiplier method

(Definition)

The Lagrange multiplier method is used when one needs to find the extreme values of a function whose domain is constrained to lie within a particular 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

$\begin{displaymath}\min f(\mathbf{x})\quad\mbox{such that}\quad g_{i}(\mathbf{x})=0,\quad i=1,\ldots,m\end{displaymath}$

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

$\begin{displaymath}\nabla f = \sum_{i=1}^{m} \lambda_{i} \nabla g_{i}\end{displaymath}$

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 solving the following problem:

$\begin{displaymath}\min f(\mathbf{x})-\sum_{i=1}^{m} \lambda_{i}( g_{i}(\mathbf{x}))\end{displaymath}$

for $\mathbf{x}, \lambda_{i}$ , without restrictions.

"Lagrange multiplier method" is owned by cvalente. [ full author list (3) | owner history (3) ]

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Keywords:

constraint, extrema

Attachments:: proof of calculus theorem used in the Lagrange method (Proof) by mathcam
proof of Lagrange multiplier method (Proof) by aplant
Lagrange multipliers on manifolds (Topic) by stevecheng
proof of arithmetic-geometric means inequality using Lagrange multipliers (Example) by stevecheng
Lagrange multipliers on Banach spaces (Theorem) by stevecheng
tests for local extrema in Lagrange multiplier method (Result) by stevecheng

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Cross-references: restrictions, equivalent, points, vectors, linearly independent, iff, equation, gradient, independent, Calculus, map, differentiable functions, subset, domain, function
There are 3 references to this entry.

This is version 5 of Lagrange multiplier method, born on 2002-02-21, modified 2005-12-03.
Object id is 2352, canonical name is LagrangeMultiplierMethod.
Accessed 19252 times total.

Classification:

AMS MSC:

49K30 (Calculus of variations and optimal control; optimization :: Necessary conditions and sufficient conditions for optimality :: Optimal solutions belonging to restricted classes)

Pending Errata and Addenda

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