Lagrange multiplier

Description

The method of Lagrange multipliers is a way of finding optimal solution which also need to fulfill certain equations. Apparently, this is often used in physics.

Definition

Let $U\subseteq {\mathbb{R}}^n$ and $f:U\to \mathbb{R}$ be a function where we want to find a maximal point. We also want the solution $a$ to fulfill a set of $d\le n$ equations $g_1(x),...,g_d(x)=0$ written as functions $g_1,...,g_d:U\to \mathbb{R}$. Let $M:=\{x\in U:g_1(x)=...=g_d(x)=0\}$. If the gradients of all conditions $\nabla g_1(x),...,\nabla g_d(x)$ are linearly independent, then the following theorem holds:

Let $a\in M$ be a locally maximal value. Then there are numbers ${\lambda }_1,...,{\lambda }_d$, called Lagrange multipliers such that $(\nabla f)(a)=\sum _{i=1}^d{\lambda }_i(\nabla g_i)(a)$

The idea is extremely well explained in the German Wikipedia. It can be shown that $M$ is a Untermannigfaltigkeit von R of dimension $n-d$. $M$ is the manifold which is perpendicular to all gradients of $g_i$. A solution on $M$ is maximal if and only if $M$ is tangent to the level sets of $f$. This is the case if the gradient of $f$ is perpendicular to $M$ or if it can be written as a linear combination of the gradients of $g_i$.

Properties

Tip

Examples

Example