Re: Keeping global parameters in an evaluated function

*To*: mathgroup at smc.vnet.net*Subject*: [mg113267] Re: Keeping global parameters in an evaluated function*From*: "Carl K. Woll" <carlw at wolfram.com>*Date*: Thu, 21 Oct 2010 07:02:39 -0400 (EDT)

On 10/20/2010 3:10 AM, 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 > > You might try using calculus and homotopic continuation. For example, suppose you're trying to minimize f[x, c] for a given value of c. Let x[c] be the value of x that minimizes f for a given value of c. Then, x[c] should be a stationary point of f. That is, let: g[x_, c_] = D[f[x, c], x] Then x[c] should satisfy: g[x[c], c]==0 If we differentiate this equation with respect to c, we obtain an ODE that x[c] should satisfy, which can be solved by NDSolve. As a concrete example, suppose: f[x_,c_]:=(x^4+c x^2-c^2 x) Then the derivative with respect to x is: In[208]:= g[x_,c_]=D[f[x,c],x] Out[208]= -c^2+2 c x+4 x^3 The ODE that needs to be solved is: In[209]:= eqn=D[g[x[c],c],c]==0 Out[209]= -2 c+2 x[c]+2 c (x^\[Prime])[c]+12 x[c]^2 (x^\[Prime])[c]==0 The minimizer for c=10 is: In[210]:= min10=ArgMin[f[x,10],x] Out[210]= Root[-25+5 #1+#1^3&,1] Use NDSolve to find the minimizer for values of c from 1 to 10: In[211]:= min=x/.NDSolve[{eqn,x[10]==min10},x,{c,1,10}][[1]] Out[211]= InterpolatingFunction[{{1.,10.}},<>] Check: In[212]:= ArgMin[f[x, 5.], x] min[5] Out[212]= 1.40072 Out[213]= 1.40072 In[214]:= ArgMin[f[x,2.],x] min[2] Out[214]= 0.682328 Out[215]= 0.682327 So, min is an interpolating function that gives the minimizer x[c] as a function of c. To get the minimum value, one can do: f[min[c], c] for a particular value of c. Carl Woll Wolfram Research