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Subject: Re: Minimization of integral problem
Date: Sun, 20 Mar 2011 17:43:04 +0000 (UTC)
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"toms Seidel" wrote in message <im4us8$7kc$1@fred.mathworks.com>...
> Hello!
> 
> I am just about to solve the following minimization problem with equality and inequality constrains:
> 
> min -int_{-\inf}^{\inf} log(1+ a * f(x)) dx
> 
> s.t. f(x) >= 0, int_{-\inf}^{\inf} f(x) dx = C, int_{-\inf}^{\inf} f(x) * b(x) >= d *C
> 
> f(x) and b(x), respectively, are scalar function of x, a, C and d can be considered as constants. The optimization parameter is f(x)  What's the best way to tackle such a problem? I would have started using fmincon, but I am not sure whether this can be used because of the infinite integrals. 
> 
> What integral solver should be employed for this purpose? According to the Matlab reference quadgk seems to be the right choice ...
> 
> Any hints appreciated! 
- - - - - - - - -
  The infinite limits of integration are the least of your problems.  You will not be able to use routines like 'fmincon' to solve your problems, because these are limited to cases involving a finite number of variable parameters.  What you have is, in effect, an infinite number of variable parameters in the form of the unknown function f(x).  That makes it a problem in the calculus of variations which leads to differential equations of the Euler Lagrange type.  See the Wikipedia site:

 http://en.wikipedia.org/wiki/Calculus_of_variations

  I do not know offhand how to combine this with your constraints, but I would suggest taking time to read up about the general field of calculus of variations before undertaking your task.

Roger Stafford