NMinimize fails to find a minimum value

• To: mathgroup at smc.vnet.net
• Subject: [mg120706] NMinimize fails to find a minimum value
• From: Hovhannes Babayan <bhovhannes at gmail.com>
• Date: Sat, 6 Aug 2011 02:12:01 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hello,

I am solving the following task. I have set of points (list of pair of
X, Y coordinates) and list of weights for the each point. If for some
point weight=1, that point is very important of weight=0, that point
is not important. Weight ranges from 0 to 1.
The task is to construct a curve, which will fit given list of points
taking weights into account. The analytic form of function is known,
but I have to find parameters z,u,a,b, which will give the best
fitting curve.
To do that, I am using NMinimize to minimize the following expression:
Sum[  ((f[x[[i]],z,u,a,b]-y[[i]])^2)*weights[[i]],  {i, n}  ]
where "n" is a number of points, "x" and "y" are point coordinates,
and "f(r, z,u,a,b)" is my function, where z,u,a,b are parameters which
should be found and "r" (x) is the only argument.

NMinimize finds something, but does stupid things:
1) when I declare function "f" as f[r_, z_,u_,a_,b_] := z*(r^(-a/
110)*(1+r/u)^(b/100));  I got normal results
2) when I declare function "f" as f[r_, z_,u_,a_,b_] := z*(r^(-a/
10)*(1+r/u)^(b/10));  I got worse results
3) when I declare function "f" as f[r_, z_,u_,a_,b_] := z*(r^(-a)*(1+r/
u)^(b));  I got the worst results

Why I have to divide "a" to get better results?? Is it possible to
avoid using such tricks and get NMinimize work with the 3rd case?
Calculated values of parameters are not very small or very large,
normal numbers....
I'll be grateful for any remarks/suggestions/ideas. To try, just copy/
paste the code below into Mathematica and run it.

points = {
{ 4.68, 0.156},
{ 7.63, 0.100},
{10.56, 0.074},
{13.48, 0.061},
{16.55, 0.048},
{20.33, 0.029},
{25.39, 0.023},
{30.28, 0.018},
{36.05, 0.016},
{42.60, 0.016},
{55.49, 0.028}
} ;
weights = {0.2523, 0.4007, 0.5801,0.6193, 0.6798, 1, 0.7957, 0.5144,
0.37030, 0.1079, 0.0189};

f[r_, z_,u_,a_,b_] := z*(r^(-a/110)*(1+r/u)^(b/100)); (* Here z,u,a,b
are parameters *)

x = points[[All,1]];
y = points[[All,2]];
n = Length[points];
bestFitVals = NMinimize[
Sum[
((f[x[[i]],z,u,a,b]-y[[i]])^2)*weights[[i]],
{i, n}
],
{z,u,a,b}
]
Show[
Plot[
Tooltip[f[r,z,u,a,b] /. bestFitVals[[2]]],
{r, Max[xdata], Max[ydata]}
],
ListPlot[points],
PlotRange->{{0,Automatic},{0,0.2}},
AxesLabel->{"r", "f[r]"}
]

```

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