Global Extrema of Real Scalar Fields

Definition: Global Minimum

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field and let ${\mathbf{x}}_0\in \mathcal{D}$.

We call ${\mathbf{x}}_0$ a place of a global minimum of $f$ iff

$$f({\mathbf{x}}_0)\le f(\mathbf{x})\forall \mathbf{x}\in \mathcal{D}$$

The value $f({\mathbf{x}}_0)$ is known as the global minimum of $f$.

Theorem: Uniqueness of the Global Minimum

If $f$ has a global minimum, then its value is unique.

PROOF

TODO

NOTE

The global minimum is unique but may occur at multiple places.

Definition: Global Maximum

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real scalar field and let ${\mathbf{x}}_0\in \mathcal{D}$.

We say that $f({\mathbf{x}}_0)$ is the global maximum of $f$ if it is the greatest functional value:

We call ${\mathbf{x}}_0$ a place of a global maximum of $f$ iff.

$$f({\mathbf{x}}_0)\ge f(\mathbf{x})\forall \mathbf{x}\in \mathcal{D}$$

The value $f({\mathbf{x}}_0)$ is known as the global maximum of $f$.

Theorem: Uniqueness of the Global Maximum

If $f$ has a global maximum, then its value is unique.

PROOF

TODO

NOTE

The global maximum is unique but may occur at multiple places.

Definition: Global Extremum

The global minimum and the global maximum of a real scalar field are collectively known as its global extrema.