Re: Two successive FindMinimum -> Bug?

*To*: mathgroup at smc.vnet.net*Subject*: [mg74335] Re: Two successive FindMinimum -> Bug?*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Mon, 19 Mar 2007 01:59:57 -0500 (EST)*Organization*: The Open University, Milton Keynes, UK*References*: <etik3j$j32$1@smc.vnet.net>

Dmitry Garanin wrote: > Hi All, > > I am trying to solve a problem that requires application of FindMinimum (or similar rutines two times and it does not work. Here is a simplified example: > > > fxt[x_,t_]=(x^2-t)^2 (* Searching for a minimal t for which fxt turns to zero *) > tminx[x_]:=t/.FindMinimum[fxt[x,t],{t,1}][[2]] (* Finding the minimizing t for a given x *) > tminabs=FindMinimum[tminx[x],{x,1}] (* Minimizing this t with respect to x. The result should be tminabs=0 at x=0 *) > > The last command here does not work since, evidently tminx[x] (that is x^2 for us humans) is not treated as a regular function. > > Is there a bug here or I am doing something wrong?? As written, your code cannot work. In[1]:= Remove[fxt, tminx, tminabs]; fxt[(x_)?NumericQ, (t_)?NumericQ] := (x^2 - t)^2; tminx[(x_)?NumericQ] := t /. FindMinimum[fxt[x, t], {t, 1}][[2]]; tminabs = FindMinimum[tminx[x], {x, 1}] FindMinimum::"lstol" : "The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than (MachinePrecision) digits of working precision to meet these tolerances. More... Out[4]= {0.00309089, {x -> 0.0555958}} So you can try to switch to arbitrary precision, In[5]:= Remove[fxt, tminx, tminabs]; fxt[(x_)?NumericQ, (t_)?NumericQ] := (x^2 - t)^2; tminx[(x_)?NumericQ] := t /. FindMinimum[fxt[x, t], {t, 1}][[2]]; tminabs = FindMinimum[tminx[x], {x, 1}, WorkingPrecision -> 20] FindMinimum::"lstol" : "The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than (MachinePrecision) digits of working precision to meet these tolerances. More... FindMinimum::"fmgz" : "Encountered a gradient which is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point. More... Out[8]= {1.0000000000000000000, {x -> 1.0000000000000000000}} The last message should ring a bell. Moreover, why use two calls to FindMinimum when one is enough? In[9]:= Chop[FindMinimum[(x^2 - t)^2, {x, 1}, {t, 1}]] FindMinimum::"fmgz" : "Encountered a gradient which is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point. More... Out[9]= {0, {x -> 1., t -> 1.}} The graph of the function may shed some light on the issue: In[10]:= Plot3D[(x^2 - t)^2, {x, 0, 1}, {t, 0, 1}, AxesLabel -> {"x", "t", None}]; [...graphic deleted...] Regards, Jean-Marc