# 3.1 Local optimization

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Describes conditions for local optimization in Hilbert Spaces

We also must define the notion of an extremum in an arbitrary normed space.

Definition 1 Let $f$ be a real-valued functional defined on $\Omega \subseteq X$ where $X$ is a normed space. A point ${x}_{0}\in \Omega$ is a local/relative minimum of $f$ on $\Omega$ if $f\left({x}_{0}\right)\le f\left(x\right)$ for all $x\in \Omega$ such that $\parallel x-{x}_{0}\parallel <ϵ$ for some $ϵ>0$ .

Definition 2 Let $f$ be a real-valued functional defined on $\Omega \subseteq X$ where $X$ is a normed space. A point ${x}_{0}\in \Omega$ is a local maximum of $f$ on $\Omega$ if $f\left({x}_{0}\right)\ge f\left(x\right)$ for all $x\in \Omega$ such that $\parallel x-{x}_{0}\parallel <ϵ$ for some $ϵ>0$ .

Definition 3 Let $f$ be a real-valued functional defined on $\Omega \subseteq X$ where $X$ is a normed space. A point ${x}_{0}\in \Omega$ is a local strict minimum of $f$ on $\Omega$ if $f\left({x}_{0}\right) for all $x\in \Omega$ such that $\parallel x-{x}_{0}\parallel <ϵ$ for some $ϵ>0$ .

Definition 4 Let $f$ be a real-valued functional defined on $\Omega \subseteq X$ where $X$ is a normed space. A point ${x}_{0}\in \Omega$ is a local strict maximum of $f$ on $\Omega$ if $f\left({x}_{0}\right)>f\left(x\right)$ for all $x\in \Omega$ such that $\parallel x-{x}_{0}\parallel <ϵ$ for some $ϵ>0$ .

It turns out the notion of a gradient is intrinsically linked to the directional derivatives we have introduced.

Definition 5 Let $X$ be a Hilbert space and $f:X\to R$ . If $f$ is a Fréchet differentiable functional, then for each $x\in X$ there exists a vector in $X$ such that $\delta f\left(x;h\right)=⟨h,\nabla f\left(x\right)⟩$ for all $h\in X$ ; the vector $\nabla f\left(x\right)$ is called the gradient of $f$ at $x$ , and can be written as a functional $\nabla f:X\to X$ .

This definition can be seen to correspond to an application of the Riesz representation theorem to the Fréchet derivative $\delta f\left(x;h\right)$ , which is a linear bounded functional on $h$ .

Example 1 We know now that:

$\delta f\left(x;h\right)=⟨h,\nabla f\left(x\right)⟩.$

By the Cauchy-Schwarz Inequality, we have:

$|\delta f\left(x;h\right)|=|⟨h,\nabla f\left(x\right)⟩|\le \parallel h\parallel \parallel \nabla f\left(x\right)\parallel .$

If $h=\nabla f\left(x\right)$ then $\delta f\left(x;h\right)$ is maximized.

Example 2 Recall that if $f:{\mathbb{R}}^{n}\to \mathbb{R}$ , then

$\begin{array}{c}\hfill \nabla f\left(x\right)=\left[\begin{array}{c}\frac{\partial f}{\partial {x}_{1}}\\ \frac{\partial f}{\partial {x}_{2}}\\ ⋮\\ \frac{\partial f}{\partial {x}_{n}}\end{array}\right].\end{array}$

Theorem 1 Let $f:X\to \mathbb{R}$ have a Gâteaux differential on $X$ . A necessary condition for $f$ to have an extremum at ${x}_{0}\in X$ is that $\delta f\left({x}_{0};h\right)=0$ for all $h\in X$ . Alternatively, if $X$ is a Hilbert space, we can write $⟨h,\nabla f\left({x}_{0}\right)⟩=0$ for all $h\in X$ , which implies $\nabla f\left({x}_{0}\right)=0$ .

Suppose ${x}_{0}$ is a local minimum. Then there exists $ϵ>0$ such that if $\parallel x-{x}_{0}\parallel <ϵ$ then $f\left({x}_{0}\right)\le f\left(x\right)$ . Fix $h\ne 0$ and let $\theta =\frac{ϵ}{\parallel h\parallel }$ . Next, consider $x={x}_{0}+\alpha h$ . For $\alpha \in \left(-\theta ,\theta \right)$ :

• If $\alpha >0$ then $\frac{f\left({x}_{0}+\alpha h\right)-f\left({x}_{0}\right)}{\alpha }\ge 0$ , and therefore $\delta f\left({x}_{0};h\right)\ge 0$ .
• If $\alpha <0$ then $\frac{f\left({x}_{0}+\alpha h\right)-f\left({x}_{0}\right)}{\alpha }\le 0$ , and therefore $\delta f\left({x}_{0};h\right)\le 0$ .

Therefore, $\delta f\left({x}_{0};h\right)=0$ for arbitrary nonzero $h$ . Now since $\delta f\left(x;h\right)$ is linear on $h$ we must have $\delta f\left(x;h\right)=0$ for $h=0$ . Therefore, the equality is true for all $h\in X$ .

Definition 6 A point at which $\delta f\left(x;h\right)=0$ for all $h\in X$ is called a stationary point of $f$ .

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