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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 Ω X where X is a normed space. A point x 0 Ω is a local/relative minimum of f on Ω if f ( x 0 ) f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 2 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local maximum of f on Ω if f ( x 0 ) f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 3 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local strict minimum of f on Ω if f ( x 0 ) < f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 4 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local strict maximum of f on Ω if f ( x 0 ) > f ( x ) for all x Ω such that x - x 0 < 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 R . If f is a Fréchet differentiable functional, then for each x X there exists a vector in X such that δ f ( x ; h ) = h , f ( x ) for all h X ; the vector f ( x ) is called the gradient of f at x , and can be written as a functional f : X X .

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

Example 1 We know now that:

δ f ( x ; h ) = h , f ( x ) .

By the Cauchy-Schwarz Inequality, we have:

| δ f ( x ; h ) | = | h , f ( x ) | h f ( x ) .

If h = f ( x ) then δ f ( x ; h ) is maximized.

Example 2 Recall that if f : R n R , then

f ( x ) = f x 1 f x 2 f x n .

Theorem 1 Let f : X R have a Gâteaux differential on X . A necessary condition for f to have an extremum at x 0 X is that δ f ( x 0 ; h ) = 0 for all h X . Alternatively, if X is a Hilbert space, we can write h , f ( x 0 ) = 0 for all h X , which implies f ( x 0 ) = 0 .

Suppose x 0 is a local minimum. Then there exists ϵ > 0 such that if x - x 0 < ϵ then f ( x 0 ) f ( x ) . Fix h 0 and let θ = ϵ h . Next, consider x = x 0 + α h . For α ( - θ , θ ) :

  • If α > 0 then f ( x 0 + α h ) - f ( x 0 ) α 0 , and therefore δ f ( x 0 ; h ) 0 .
  • If α < 0 then f ( x 0 + α h ) - f ( x 0 ) α 0 , and therefore δ f ( x 0 ; h ) 0 .

Therefore, δ f ( x 0 ; h ) = 0 for arbitrary nonzero h . Now since δ f ( x ; h ) is linear on h we must have δ f ( x ; h ) = 0 for h = 0 . Therefore, the equality is true for all h X .

Definition 6 A point at which δ f ( x ; h ) = 0 for all h X is called a stationary point of f .

Questions & Answers

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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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