Please help!! Need to constrain/force LSQNONLIN to pass through a given point e.g. the origin

Hello

I am struggling to implement constraints on nonlinear regression using lsqnonlin (or fmincon), where I need the fitted function to pass through a given point.

e.g. I have the simple function

function diff = fit_simp(x,X,Y)

A=x(1);
B=x(2);

diff = A*log(X+B) + A*log(1+B) - Y;

end

and the simple X and Y data attached (X and Y in columns 1 and 2 respectively of XY.txt). Then use lsqnonlin as follows:

 X0=[-1,0];%Initial conditions
 x=lsqnonlin(@fit_simp,X0,[],[],[],X,Y);
 Y_new = x(1)*log(X+x(2))+x(1)*log(1+x(2));
 plot(X,Y,'+r',X,Y_new,'b');

However the problem I have is that the function needs to pass through the (X,Y) point (1,0).

I have tried reading many help pages and tutorials to solve this issue however I haven't found any solution that applies to this situation. I have tried something similar with fmincon but cannot figure out how to implement the constraints.

So now I am devoid of ideas. I bought the Curve Fitting, Statistics and Optimization Toolboxes to specifically help me with this problem. Please help!!

Many thanks in advance, Chris

[Yu Jiang]

Hi Chris

Can you elaborate more on "the function needs to pass through the (X,Y) point (1,0)."

-Yu

[Chris]

Hi Yu

It is important that the least-squares curve for the model I am adopting [Y = A*log(X+B) + A*log(1+B)] is constrained such that when X=1, Y=0.

I am trying to calculate the values of coefficients A and B that minimize the fit, subject to this constraint.

Hopefully this helps explains the problem better?

Cheers, Chris

[Chris]

Sorry the X=1,Y=0 is a bad example, as the function can't go through this point. I'll change the example to when X=0.3, Y=1.

Answer 1

[Yu Jiang]

Hi Chris

From what I understand, you would like to fit the nonlinear model with your data set, and you would like to know how to impose the nonlinear equality constraint

Fit_simp(x, 0.3, 1) = 0.

In order to achieve data fitting with constraints, I suggest you use the function fmincon (See Documentation) , which allows you to specify nonlinear constraints.

You may consider the following steps:

1. Revise the objective function as follows

function diff = fit_simp(x,X,Y)
A=x(1);
B=x(2);
diff = norm(A*log(X+B) + A*log(1+B) - Y);
end

The new function returns a scalar as the data fitting error. You may save the function as a MATLAB file under the same name as the function. See the documentation for the function norm.

2. Create the nonlinear constraint function

function [c,ceq] = nlcon(x, X, Y)
A=x(1);
B=x(2);
c = [];
ceq = A*log(X+B) + A*log(1+B) - Y;
end

This function has two outputs, c is the output of inequality nonlinear constraints and ceq is the output of the equality constraints. Since we only have equality constraints here, I set c=[]. Again, save this function in a MATLAB file.

3. In the main MATLAB script or in the command window, try the following code (Assuming X and Y are column vectors which contain your data set)

>> X0=[1,0];
>> x=fmincon(@(x) fit_simp(x,X,Y),X0,[],[],[],[],[],[],@(x) nlcon(x,0.3,1));

Then, you should have the expected answer. If you would like to have better data fitting accuracy, you can always specify the options (See Documentation) for fmincon.

For more information regrading nonlinear constraints, you can refer to the link below:

http://www.mathworks.com/help/optim/ug/writing-constraints.html#brhkghv-16

-Yu

Answer 2

[Chris]

Perfect! Thank you so much Yu, your help is greatly appreciated. Chris

[Yu Jiang]

My pleasure :)