# 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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Joseph
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Lohitha
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
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Damian
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Professor
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Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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Rafiq
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Anam
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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what king of growth are you checking .?
Renato
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