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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
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