Re: Gradient in FindMininum

*To*: mathgroup at smc.vnet.net*Subject*: [mg24039] Re: Gradient in FindMininum*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>*Date*: Wed, 21 Jun 2000 02:20:01 -0400 (EDT)*Organization*: Universitaet Leipzig*References*: <8in56l$44a@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi, a) with f[x_] := x^2 + NIntegrate[Exp[-t^2], {t, -Infinity, x}] FindMinimum[f[x], {x, -2, 2}] I got {0.666069, {x -> -0.419364}} with out an error message. Just use two values. b) Is it so hard to compute D[Integrate[g[t],{t,-Infinity,x}],x] and put it into the Gradient option of FindMinimum[] ? (Abramowitz, Stegun, formula 3.3.7 ?) D[#,c]& @ Integrate[f,{x,a,b}]== f[b,c]*D[b,c]-f[a,c]*D[a,c]+Integrate[D[f,c],{x,a,b}] Regards Jens Johannes Ludsteck wrote: > > Dear MathGroup Members, > Unfortunately I got no answer when I sent the question below last > week to the mathgroup mailing list. Since I think that the problem > is not a very special one but a general problem of the way how > Mathematica treats numerical integrals in the computation of > gradients, I retry to get an answer. > I want to minimize a complicated function which contains > numerical integrals. Since the function is too complicated for a > direct demonstration, I give a simple example which makes the > structure of the problem clear: > > The (example) function to be minimized is: > 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. > > Thank you > > P.S If you want to reproduce the error message, you can use a > simple definition: > > f[x_]:= x^2 + NIntegrate[ Exp[-t^2], {t, -Infinity, x} ]. > > Johannes Ludsteck > Centre for European Economic Research (ZEW) > Department of Labour Economics, > Human Resources and Social Policy > Phone (+49)(0)621/1235-157 > Fax (+49)(0)621/1235-225 > > P.O.Box 103443 > D-68034 Mannheim > GERMANY > > Email: ludsteck at zew.de