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# 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.

## Mathematics Subject Classification

49K30*no label found*

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## Attached Articles

Lagrange multiplier method, proof of by aplant

Lagrange multipliers on manifolds by stevecheng

proof of arithmetic-geometric means inequality using Lagrange multipliers by stevecheng

Lagrange multipliers on Banach spaces by stevecheng

tests for local extrema in Lagrange multiplier method by stevecheng

example of using Lagrange multipliers by pahio

example needing two Lagrange multipliers by pahio