Local Extrema

Definition: Local Minimum

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field.

We say that $f({\vec{x}}_0)$ is a local minimum of $f$ if there is an open ball $B_{\epsilon }({\vec{x}}_0)$ around ${\vec{x}}_0\in D$ where $f({\vec{x}}_0)$ is the smallest funtional value.

$f({\vec{x}}_0)\le f(\vec{x})\forall \vec{x}\in B_{\epsilon }({\vec{x}}_0)$

We say that ${\vec{x}}_0$ is a place of a local minimum for $f$.

Definition: Global Maximum

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field.

We say that $f({\vec{x}}_0)$ is a local maximum of $f$ if there is an open ball $B_{\epsilon }({\vec{x}}_0)$ around ${\vec{x}}_0\in D$ where $f({\vec{x}}_0)$ is the greatest funtional value.

$f({\vec{x}}_0)\ge f(\vec{x})\forall \vec{x}\in B_{\epsilon }({\vec{x}}_0)$

We say that ${\vec{x}}_0$ is a place of a local maximum for $f$.

Definition: Local Extremum

The local minima and local maxima of a real scalar field are collectively known as its local extrema.

Theorem: Finding Local Extrema

Let $f:\mathcal{D}\to \mathbb{R}$ be a real scalar field.

If $f$ has a local extremum at $\mathbf{a}\in \mathcal{D}$, then $\mathbf{a}$ is a critical point of $f$.

PROOF

TODO

Theorem: Hessian Matrix Criteria for Local Extrema

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field which is twice continuously partially differentiable in Cartesian coordinates on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^n$.

A critical point $\mathbf{p}\in \mathcal{D}$ is:

• a place of a local maximum if the Hessian matrix $H_f(\mathbf{p})$ is negative-definite;

• a place of a local minimum if the Hessian matrix $H_f(\mathbf{p})$ is positive-definite;

• a place of a saddle point if the Hessian matrix $H_f(\mathbf{p})$ is indefinite;

If the Hessian matrix $H_f(\mathbf{p})$ is semi-definite, then it cannot be used to make any predictions.

PROOF

TODO