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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

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