Re: Keeping global parameters in an evaluated function

*To*: mathgroup at smc.vnet.net*Subject*: [mg113264] Re: Keeping global parameters in an evaluated function*From*: Davide Mancusi <d.mancusi at ulg.ac.be>*Date*: Thu, 21 Oct 2010 07:02:05 -0400 (EDT)

On Wed, 20 Oct 2010 10:47:37 -0500, Daniel Lichtblau <danl at wolfram.com> wrote: > Davide Mancusi wrote: > > Hello everyone, > > > > I have a function that depends on some variables and a parameter > > (say c). My goal is to find the global minimum of the function for > > a given value of the parameter, say c==0. However, the function has > > a lot of local minima for c==0, and is thus very difficult to hit > > the global one. For c==1, however, the function is much smoother, > > and the minimization algorithms work pretty well. > > > > What I'm trying to do is to change the value of the parameter > > continuously while Mathematica minimizes the function. By doing so I > > hope to trap the minimization algorithms close to the global minimum > > using the smooth version of the function, and then slowly let c go > > to 0. > > > > I have come up with the following toy model (I use NMinimize in the > > real problem, but that shouldn't be important): > > f[x_] :== (x - c)^4 > > c==1; > > iterations==1; > > FindMinimum[ f[x], {x, 1001}, StepMonitor :> {c == c/2; iterations+ > > +} ] > > Print[{c // N,iterations}] > > If I run this I get > > {4.48416*10^-32, {x -> 1.}} > > {1.45519*10^-11,37} > > The algorithm finds the minimum at x====1 while I would like it to > > find it close to 1.45519*10^-11. > > > > I think the problem lies in the fact that f[x] is evaluated only > > once, at the beginning of the minimization algorithm, and the > > global value of c gets substituted. Thus, even if I change c later, > > that doesn't affect the function being minimized. > > > > Can anyone suggest a way to update c from within the minimization > > routine? > > > > Thanks in advance, > > Davide > > Could instead run a loop of FindMinimum or NMinimize, using the > result from the previous run to initialize the start point for the > next. Below is your example done in this manner. > > In[70]:== f[x_] :== (x - c)^4 > c == 1; > prev == 1001; > Last[Table[min == FindMinimum[f[x], {x, prev}]; > prev == x /. min[[2]]; c /== 2; min, {i, 40}]] > > Out[73]== {7.15727*10^-32, {x -> 1.63582*10^-8}} (Posting back to the list...) Daniel, thanks for your reply. I had thought about this solution, but it is not so straightforward to do that with NMinimize, since most (all?) of the Methods use many-point internal states. I can't see how to use the solution of one step as input for the next one in a simple way. -- --------------------------- Davide Mancusi, Ph.D. Fundamental Interactions in Physics and Astrophysics University of Li=C3=A8ge all=C3=A9e du 6 ao=C3=BBt, 17 B-4000 Li=C3=A8ge 1 Belgium tel.: +32 (0)4 366 3642 ---------------------------