Optimization Problem - Need Stochastic Method

*To*: mathgroup at smc.vnet.net*Subject*: [mg50377] Optimization Problem - Need Stochastic Method*From*: "David Mebane" <david.mebane at mse.gatech.edu>*Date*: Tue, 31 Aug 2004 06:29:00 -0400 (EDT)*Organization*: Georgia Institute of Technology*Sender*: owner-wri-mathgroup at wolfram.com

Hi All, I just started using Mathematica - and I'm hoping that there's a nice trick I can use to get my optimization done. The problem is in stereology - I'm an engineering grad student and I have developed a bivariate size-orientation unfolding equation for cylinders. The equation is extremely complex - a double-integral equation of about 14 terms. In addition, there is a stability problem, meaning that for practical purposes, the problem is ill-posed. Because I need to graduate (this is basically a side project gone mad) I am getting around the problems of finding a solution and dealing with the ill-posedness by simply fitting a distribution that takes a form similar to a bivariate normal distribution to the equation. This means that I have to fit 5 distribution parameters to the experimental data, of which there are about 25 points. This makes for a well-overdetermined system, and I have constructed a square-error function in Mathematica that is a function of the parameters I need to optimize. Here's the problem: I tried to use constrained NMinimize, but it did not converge well - when I plotted the solution graphically against the data points, there was severe and systematic error. I have no idea how the fitting function behaves since it's impossible to visualize, but as described earlier it's extremely complicated when written out, and it involves trig functions and exponentials, so I wouldn't assume that there are no local minima. I think that the best thing to do is to use a stochastic method as the search algorithm instead of the default Nelder-Mead, but I am not sure if one exists. Can anyone help? Thanks