       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}][] (* 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:=
Remove[fxt, tminx, tminabs];
fxt[(x_)?NumericQ, (t_)?NumericQ] := (x^2 - t)^2;
tminx[(x_)?NumericQ] := t /. FindMinimum[fxt[x, t], {t, 1}][];
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=
{0.00309089, {x -> 0.0555958}}

So you can try to switch to arbitrary precision,

In:=
Remove[fxt, tminx, tminabs];
fxt[(x_)?NumericQ, (t_)?NumericQ] := (x^2 - t)^2;
tminx[(x_)?NumericQ] := t /. FindMinimum[fxt[x, t], {t, 1}][];
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

Out=
{1.0000000000000000000, {x -> 1.0000000000000000000}}

The last message should ring a bell. Moreover, why use two calls to
FindMinimum when one is enough?

In:=
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

Out=
{0, {x -> 1., t -> 1.}}

The graph of the function may shed some light on the issue:

In:=
Plot3D[(x^2 - t)^2, {x, 0, 1}, {t, 0, 1}, AxesLabel -> {"x", "t", None}];

[...graphic deleted...]

Regards,
Jean-Marc

```

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