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Re: NMinimize problem: fct minimized uses FindRoot

• To: mathgroup at smc.vnet.net
• Subject: [mg123552] Re: NMinimize problem: fct minimized uses FindRoot
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 11 Dec 2011 03:45:56 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201112101227.HAA19219@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

1) Never, ever, EVER use Return. In this case, there wasn't even a flimsy  
excuse for it.

2) To prevent computing a function with symbolic arguments, use a pattern  
on the LHS such as _?NumericQ.

3) You used invalid syntax in the second argument of NMinimize.

4) Use Set, not SetDelayed, whenever possible.

Clear[numFct]
numFct[optvar_?NumericQ] :=
  Module[{inteq, n, x}, inteq[x_] = (Sin[x] + 1/2*Cos[x])/optvar;
   Sin[optvar] + n /. FindRoot[inteq[n], {n, 0.1}]]

NMinimize[numFct[var], {var, 0, 6}]

{-1.46365, {var -> 4.71239}}

5) Finding a root for (Sin[x] + 1/2*Cos[x])/optvar is the same as finding  
a root for Sin[x] + 1/2*Cos[x]:

Clear[numFct]
numFct[optvar_?NumericQ] :=
  Module[{inteq, n, x}, inteq[x_] = Sin[x] + 1/2*Cos[x];
   Sin[optvar] + n /. FindRoot[inteq[n], {n, 0.1}]]

NMinimize[numFct[var], {var, 0, 6}]

{-1.46365, {var -> 4.71239}}

6) Don't define functions you don't need:

Clear[numFct]
numFct[optvar_?NumericQ] := Module[{x},
   Sin[optvar] + x /. FindRoot[Sin[x] + Cos[x]/2, {x, 0.1}]]
NMinimize[numFct[var], {var, 0, 6}]

{-1.46365, {var -> 4.71239}}

7) Optimization and root-finding are uncoupled in this case, so:

Clear[numFct]
numFct[optvar_?NumericQ] := Sin[optvar] + Module[{x},
    x /. FindRoot[Sin[x] + Cos[x]/2, {x, 0.1}]]
NMinimize[numFct[var], {var, 0, 6}]

{-1.46365, {var -> 4.71239}}

or

Clear[numFct, x, y]
numFct[x_?NumericQ] = Sin[x] +
     y /. FindRoot[Sin[y] + Cos[y]/2, {y, 0.1}];
NMinimize[numFct[x], {x, 0, 6}]

{-1.46365, {x -> 4.71239}}

or even simpler:

Clear[x]
NMinimize[Sin[x], {x, 0, 6}]
First@% + x /. FindRoot[Sin[x] + Cos[x]/2, {x, 0.1}]

{-1., {x -> 4.71239}}

-1.46365

Bobby

On Sat, 10 Dec 2011 06:27:05 -0600, Doug Tinkham <dtinkham at live.ca> wrote:

> Hello
>
> I'm using NMinimize and FindMinimum to minimize a function that uses
> FindRoot when calculating it's value. The problem is that the equation
> that FindRoot is used on uses the variable that is being optimized, and
> Mathematica appears to be forcing the variable that is being optimized
> to remain symbolic in the FindRoot call, and this leads to recursion and
> a recursion limit error.
>
> Rather than post my actual functions that are quite long, I've reduced
> my problem to the code below that shows my issue. As you will see,
> FindRoot keeps optvar in symbolic form when executing FindRoot. Is there
> a way to force Mathematica to use all numerical calculations using
> NMinimize or FindMinimum?  Is the issue with calculation of the
> gradient, which Mathematica wants to do symbolically?
>
> Many thanks.
>
>
>
> MyNumFct[optvar_] := Module[{inteq, n},
>   inteq[x_] := (Sin[x] + 1/2*Cos[x])/optvar;
>   n = n /. FindRoot[inteq[n], {n, 0.1}];
>   Return[n + Sin[optvar]];
> ]
> NMinimize[{MyNumFct[var], 0 <= var <= 6}, {var, 4.1}]
>
>


-- 
DrMajorBob at yahoo.com

• References:
  • NMinimize problem: fct minimized uses FindRoot
    • From: "Doug Tinkham" <dtinkham@live.ca>