Lsqcurvefit - different optimal values for different starting values ??

Hi, I have a problem with the optimization by lsqcurvefit function. Briefly, I have a symbolic function which it is a vector contains symbolic elements. Each element contains the desired parameter I would like to optimize. In addition I have an experimental data vector of the same length as the function vector. I tried to optimize the function vector with the experimental data, but when I define different starting values in lsqcurvefit function I get different optimal values to the desired parameters. Someone knows what could be the problem ? Thanks!!

[Avihai Goldfarb]

Below I attached the code:

P and T are symbolic variables.

The symbolic Function Vector:

AnalyticalConc = [3/8 - (3*exp(P*(1/(2*T) - 1)))/8 ;
exp(P*(1/(2*T) - 1))/32 - (3*exp(2*P*(1/(2*T) - 1)))/8 + 11/32 ;
(11*exp(P*(1/(2*T) - 1)))/384 + exp(2*P*(1/(2*T) - 1))/32 - (3*exp(3*P*(1/(2*T) - 1)))/8 + 121/384 ; 
...
]'

(note: I didn't write all the elements! there are three symbolic element more - total six elements).

time = [1 2 3 4 5 6];

The experimental data = [0.039 0.059 0.062 0.072 0.069 0.068]; Now I would like to optimizae the parameters P and T, hence:

x0 = [0.1, 6]; %%[P T] initial values
[solution,resnorm,residual,exitflag] = lsqcurvefit(@fun,x0,time,ExperimentalConc);

function F = fun(x,xdata)
syms P T
load('AnalyticalConc.mat');
time=1:6;
ind=find(time==xdata);
F = double(subs(AnalyticalConc(ind),[P,T],[x(1),x(2)]));
end

In the case that I change the initial values to [0.02 9], for example, the optimal parameters also change!

Answer 1

[Matt J]

I would guess that you have infinite solutions and that it is because your function depends on P and T entirely through the expression

     P*(1/(2*T) - 1)

Therefore, if P0 and T0 are a fixed pair of optimal parameters, then any additional P,T satisfying

    P*(1/(2*T) - 1)   =   P0*(1/(2*T0) - 1)

will also be optimal. The equation above is satisfied by an infinite locus of (P,T).

[Greig]

Since it is only a 2D problem, it might be worth doing some number crunching over a small, but reasonable range of P and T values and contouring the function to see the pattern.