Re: Gradient in FindMinimum

*To*: mathgroup at smc.vnet.net*Subject*: [mg24144] Re: Gradient in FindMinimum*From*: Robert Knapp <rknapp at wolfram.com>*Date*: Wed, 28 Jun 2000 02:11:59 -0400 (EDT)*Organization*: Wolfram Research, Inc.*References*: <8icaf9$99c@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Johannes Ludsteck wrote: > > Dear MathGroup Members, > I want to minimize a complicated function which contains > numerical integrals. Since the function is really complicated, I give > a simple example which captures the structure of the problem: > > f[x_] := NIntegrate[g[t], {t, -Infinity, x}] > > (g is a known function; however symbolical integration is > impossible). > > When I request numerical minimization of this function by typing > > FindMinimum[f[x],{x,1}] > > Mathematica gives me the following error message: > > FindMinimum::fmgl: Gradient {Indeterminate} is not a length 1 > list of real numbers at {x} = {1.}. > > Appearently, Mathematica is not able to find the gradient > symbolically. A simple solution would be to define f using Integrate > (without prefix N) and to wrap it with N[ ]: > > f[x_]:= N[ Integrate[g[t], {t,-Infinity, x}] ] > > However, since the function contains some hundred terms, > evaluation of the function takes several minutes. (Mathematica then > tries to find the integral symbolically before applying the numerical > integration procedure.) This makes optimization impracticable. > (the function I want to optimize has about 40 variables!). > > Are there any suggestions how to avoid computation of the gradient > manually? (minimization algorithms which don't use the gradient > are impracticable.) > I.e. how can I tell Mathematica to use the first definition > > f[x_] := NIntegrate[g[t], {t, -Infinity, x}] > > for evaluation of the function and the second > > f[x_]:= N[ Integrate[g[t], {t,-Infinity, x}] ] > > for the computation of the gradient. > You can specify a function for the Gradient using the Gradient option to FindMinimum. Assuming your function g in f[x_] := NIntegrate[g[t], {t, -Infinity, x}] is not too special, and you can differentiate inside the integral, you could use Gradient->{g[x]} Another way to do it is using finite differences. Right now this requires that you define a function to give to the Gradient option which approximates is using finite differences. We are working on code which will make this automatic in a future version of Mathematica. Rob Knapp Wolfram Research, Inc.