Nonlinear least square minimization

I want to perform a nonlinear least square minimization of the form

Minimize(Sum((Y - F (X ; a, b, c))^2))

Where Y is a vector of response variable; X is a 5 element vector of parameter and a, b and c are vectors of input data. F has the following form:

F = X(5)* Scalar1.* Scalar2.* c;

Scalar 1 is defined as follows.

if X(2)<a<X(1)
     Scalar1=((a-X(2))/((X(1)-X(2));
elseif a<=X(2)
    Scalar1=0;
elseif a>=X(1)
    Scalar1=1;  
else
end;

The Scalar2 has a similar definition

if X(4)<b<X(3)
     Scalar2 = ((-1)/(X(3)-X(4))).*(b-X(4))+1;
elseif b>=X(3)
    Scalar2=0;
elseif b<=X(4)
    Scalar2=1;
else
end;

As you can see, I have a function F whose very definition depends on the optimized parameters.

Can someone provide me any lead about where to look for any guidance to solve a problem like this. Any specific keyword/phrase that will help me Google search relevant books, articles will be of great help.

Answer 1

[Steve Grikschat]

Your answer looks ok. lsqnonlin is a good choice. You could also look at lsqcurvefit:

http://www.mathworks.com/help/toolbox/optim/ug/lsqcurvefit.html

which handles the the responses (Y) and fixed parameters (a,b,c - lsqcurvefit calls this X) automatically. The results would be the same (assuming you've set up lsqnonlin properly) since lsqcurvefit uses the same algorithms as lsqnonlin.

I can't be sure, but there could be problems with the non-smoothness of your function (the piecewise definition). If it works though, that's good.

[Manish]

Thanks a lot for your response. I have nonoverlapping bounds on different parameters, so i guess non-smoothness is not a problem.

Answer 2

[Manish]

This is how I am defining my objective function

function F = myObjective(X, a, b, c, Y)

index1=a>=X(1);     
index2=a<=X(2);    
Scalar1=(a-X(2))./(X(1)-X(2));  
Scalar1(index1)=1;       
Scalar1(index2)=0;  

index1=b<=X(4);   
index2=b>=X(3);   
Scalar2=(X(4)-b)./(X(3)-X(4));  
Scalar2(index1)=1;        
Scalar2(index2)=0;        

F = Y-X(5).* Scalar1.*Scalar2.*c;

I then call 'F' with lsqnonlin.

It gives reasonable answers.

Can someone comment if this is correct.