# Optimization over function spaces

There are many methods to solve optimization over standard spaces such as real and complex numbers or vector spaces of these. In my case I have a vector space of functions of a real value to find.

e.g., I need to find the vector $\vec{y}(t)$ s.t.

$$\lVert s(t) - \vec{y}(t)\cdot\vec{y}(t)\rVert\cdot\lVert\vec{y}(t)\rVert$$

is minimal for some given function $s(t)$.

(each component of $\vec{y}(t)$ is a standard function(or, actually, whatever $s(t)$ is).

This is just an example of a more general case. I'm looking for methods that can solve these types of problems rather quickly and not by brute force as it obviously may take an extremely long time.

Obviously we can convert this to a non-function based optimization by approximating $\vec{y}(t)$ by, say, it's finite Taylor series. (for example, if we wanted to brute force it)

What I'm kinda after is something where I can plug in the functional form and "click a button", wait a few minutes, and investigate the solution.

My problems are all of the time $F(s(t), \vec{y}(t)$ where F sort of looks like the original example and I'm trying to find the $\vec{y}(t)$'s that minimize it.

It is analogous to using numbers instead of functions: $F(s, \vec{y})$, which is easy to solve using calculus. e.g., $F(s, \vec{y}) = |s - \vec{y}\cdot\vec{y}| \cdot \lVert\vec{y}\rVert$.

My $F$ will be more complex and likely have a unique solution. The examples I give above are just specific cases of F to get the point across. I'm not looking for a specific solution to the above examples but a general method for the minimization problems and possible software(matlab) based solutions. (F and s(t) will real valued and generally well behaved)

(BTW, if it is not clear I'm looking for the argument that gives the minimum and not the value itself)

• You can get proper norm bars using \lVert and \rVert. – joriki Sep 27 '12 at 1:46
• The name is en.wikipedia.org/wiki/Calculus_of_variations – user31373 Sep 27 '12 at 3:17