Method of Lagrange Multipliers

Lagrange Multipliers

Lagrange multipliers allow us to determine if a function subject to some constraints has local extrema.

Theorem: Lagrange Multipliers

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a continuously partially differentiable real scalar field and let

$$\begin{array}{rl}{g}_1(\mathbf{x})& =c_1\\ & ⋮\\ g_k(\mathbf{x})& =c_k\end{array}$$

be some constraints, where $g_1,\dots ,g_k:\mathcal{D}\to \mathbb{R}$ are also continuously partially differentiable, and let $S\{\mathbf{x}\mid g_1(\mathbf{x})=c_1\text{ and }\cdots \text{ and }{g}_k(\mathbf{x})=c_k\}$.

If $f{|}_S$ has a local extremum at $\mathbf{a}$ and the gradients $\nabla g_1(\mathbf{a}),\dots ,\nabla g_k(\mathbf{a})$ are linearly independent, then $\nabla f(\mathbf{a})$ is a linear combination of $\nabla g_1(\mathbf{a}),\dots ,\nabla g_k(\mathbf{a})$.

PROOF

TODO

Definition: Lagrange Multipliers

The coefficients ${\lambda }_1,\dots ,{\lambda }_k$ in the representation

$$\nabla f(\mathbf{a})={\lambda }_1\nabla g_1(\mathbf{a})+\cdots +{\lambda }_k\nabla g_k(\mathbf{a})$$

of $\nabla f(\mathbf{a})$ as a linear combination of $\nabla g_1(\mathbf{a}),\dots ,\nabla g_k(\mathbf{a})$ are known as Lagrange multipliers.

EXAMPLE

TODO