Which solver can be used for a non linear curve fitting with some variables being constrained to integers?

I have a non linear curve fitting problem, say e^(x1+x2)-y=0, where x1,x2 are variables and y is the data against which fitting is done. Here x1 should be an integer and x2 can be a non integer. Which solver can handle such problems?

Answer 1

[Alan Weiss]

If you really have the equation you gave as an example, then I would use

 x = x1 + x2

meaning just fit a continuous x variable, and afterwards take

 x1 = floor(x);
 x2 = x - x1;

If, however, you have a general nonlinear fit with a variable constrained to be integer-valued, then the first thing that I would try is to fit it using any method you like, such as lsqcurvefit, but with no integer constraint. Then I would solve a set of problems with the integer variable set to various values close to the continuous solution.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Thanigaivel Raja T]

Thanks Alan. A small modification in my question...

Can an integer variable defined in a solver, be an index of an vector?

Example : If above question was - fn=@(x) e^(x(1)+z(x(2)))-y, where y is the data against which fitting is done, z is a known vector, x(1) is a noninteger/integer, x(2) should be an integer(because it is an index of z).

I used genetic algorithm with integer constraints(mixed integer non-linear programming). But I get one of the these errors:- "Index exceeds matrix dimensions", "Subscripted assignment dimension mismatch"

I got the reason behind these errors.

But how to solve such problems?

Thanks in advance!

[Alan Weiss]

I think you can do this. But you have to set bounds on your variable x(2) to keep from getting an indexing error.

Alan Weiss

MATLAB mathematical toolbox documentation

[Walter Roberson]

Yes, if you have a specific list of (typically) non-integer values that a parameter can assume but no other values, then a common approach is to convert that to an integer constraint over indices into the vector, and inside the objective function use the indices to access the values. You need to put range constraints to get this to work.

[Thanigaivel Raja T]

Thanks you very much Alan and Walter. I set the bounds for the variables but I get a Fval which is high, even though there is enough possibility of getting a Fval close to zero. What should be the options to get Fval near to zero? The code and results I get are as follows:

z=[1,2,3,4,5]

fn=@(x) [exp(x(1)+ z(x(2)))-25]

options= gaoptimset('TolFun',1e-6,'TolCon',1e-3,'Generations',200,'StallGenLimit',50)

[x,Fval,exitFlag,Output]=ga(fn,2,[],[],[],[],[0,1],[Inf,5],[],[2],options)

Results:

Optimization terminated: average change in the penalty fitness value less than options.TolFun and constraint violation is less than options.TolCon.

x =

    0.0000    1.0000

Fval =

-22.2817

exitFlag =

     1

Output =

      problemtype: 'integerconstraints'
         rngstate: [1x1 struct]
      generations: 51
        funccount: 2081
          message: [1x147 char]
    maxconstraint: 0

While the result should have been: x(1)=.21887,x(2)=3

[Walter Roberson]

You are not getting a high Fval, you are getting the most negative Fval you can get with that particular equation and constraints, exp(1)-25 . Most negative is most minimized, which is what ga is intended to do.

If you want the combination closest to 0, square the expression.

fn=@(x) (exp(x(1)+ z(x(2)))-25).^2;

[Walter Roberson]

Why is x(2)=3 a better solution than x(2)=1 or x(2)=2 ? All three of those lead to the same basic x(1) = 2 * log(5) - x(2) which meets the constraint on x(1) provided x(2) is no more than 3. There is no way for ga to tell the solutions apart.

If you are wanting to minimize x(1) such that exp(x(1)+ z(x(2)))-25=0 then you should be using gamultiobj and using x(1) as one of the objectives and (exp(x(1)+ z(x(2)))-25).^2 as your other objective.

[Thanigaivel Raja T]

Thanks you very much Walter. I just gave one of the possible solutions x(1)=.21887,x(2)=3, it was not a necessary solution for me. Sorry for being unclear on my part. As you suggested, upon squaring the objective function I get Fval in order e^-21.

I actually want to do a non linear curve fitting, as I mentioned in my first question. So how to use ga, to fit an array of values i.e, from previous example, if:

z=[1,2,3,4,5,6,7,8,9];
y=[25,50,75,100,125,150,175,200,225];
fn=@(x)[(exp(x(1)+z(1:x(2)))-y(1:x(2)))^2;(exp(z(x(2):9))-y(x(2):9))^2]
[P1,Fval,exitFlag,Output]=ga(fn,2,[],[],[],[],[0,1],[Inf,9],[],2)

How to use ga for such a problem? I get errors when I solve this. Thanks in advance.