Non linear fitting with constraints

I have a function that uses 9 inputs (cts(1) to cts (9)) and provides 1 value as result. The function needs 4 parameters (c(1) to c(4)). The code of the function is:

fun = @(c,cts) ((c(1) .* (4-c(4))) ./ (c(2)+c(3) .* 4)) *cts(1,:) +...
                ((c(1) .* (5-c(4))) ./ (c(2)+c(3) .* 5)) *cts(2,:)+ ...
                ((c(1) .* (6-c(4))) ./ (c(2)+c(3) .* 6)) *cts(3,:) + ...
                ((c(1) .* (7-c(4))) ./ (c(2)+c(3) .* 7)) *cts(4,:) + ...
                ((c(1) .* (8-c(4))) ./ (c(2)+c(3) .* 8)) *cts(5,:) + ... 
                ((c(1) .* (9-c(4))) ./ (c(2)+c(3) .* 9)) *cts(6,:) + ...
                ((c(1) .* (10-c(4))) ./ (c(2)+c(3) .* 10)) *cts(7,:) + ...
                ((c(1) .* (11-c(4))) ./ (c(2)+c(3) .* 11)) *cts(8,:) + ...
                ((c(1) .* (12-c(4))) ./ (c(2)+c(3) .* 12)) *cts(9,:)

I have 11 measured values with each 9 inputs. From this data set I want to fit the parameters c. A condition that applies for only the first row of the input is that

((c(1) .* (4-c(4))) ./ (c(2)+c(3) .* 4)) *cts(1,1) must be greater than ((c(1) .* (5-c(4))) ./ (c(2)+c(3) .* 5)) *cts(2,1). Is there an algorithm which can do that? I already tried lsqcurvefit without this constraint, but the fitted parameters do not follow the condition.

Thank you for your time.

Answer 1

[Matt J]

You would have to use fmincon, which will let you specify non-linear constraints.

[Marco Knobloch]

I tried to use fmincon, but I realized that I have to pass parameters from the optimization to my non linear constraint and it is not obvious to me how I can pass them. In detail, I have my function and inputs from above. The objective to minimize and the constraints function:

fun = @(c) ((c(1) .* (4-c(4))) ./ (c(2)+c(3) .* 4)) *cts(1,:) +...
                ((c(1) .* (5-c(4))) ./ (c(2)+c(3) .* 5)) *cts(2,:)+ ...
                ((c(1) .* (6-c(4))) ./ (c(2)+c(3) .* 6)) *cts(3,:) + ...
                ((c(1) .* (7-c(4))) ./ (c(2)+c(3) .* 7)) *cts(4,:) + ...
                ((c(1) .* (8-c(4))) ./ (c(2)+c(3) .* 8)) *cts(5,:) + ... 
                ((c(1) .* (9-c(4))) ./ (c(2)+c(3) .* 9)) *cts(6,:) + ...
                ((c(1) .* (10-c(4))) ./ (c(2)+c(3) .* 10)) *cts(7,:) + ...
                ((c(1) .* (11-c(4))) ./ (c(2)+c(3) .* 11)) *cts(8,:) + ...
                ((c(1) .* (12-c(4))) ./ (c(2)+c(3) .* 12)) *cts(9,:)

objective = @(c) sum(((fun(c)-y) ./ y).^2); % To minimize  

The constraint is that the first term (c(1) .* (4-c(4))) ./ (c(2)+c(3) .* 4)) *cts(1,1) must be greater than the second term (c(1) .* (5-c(4))) ./ (c(2)+c(3) .* 5)) *cts(2,1), but only in the first row.

function [a, ceq] = nonlcon(m,c,cts)
a = (c(1) .* (5-c(4))) ./ (c(2)+c(3) .* 5)) *cts(2,1) - ...
     (c(1) .* (4-c(4))) ./ (c(2)+c(3) .* 4)) *cts(1,1)  ;
ceq= [];
end

The fmincon line looks like this

A_opt = fmincon(objective,[1 1 1 1],[],[],[],[],[],[],@(c,cts)nonlcon(m,c,cts)

Obviously there is an error because I gave fmincon not enough input arguments. But the additional input arguments c are the parameters which fmincon should fit. Is there a way to pass nonlcon the current values of the optimization for the parameters?

[Matt J]

Obviously there is an error because I gave fmincon not enough input arguments.

It would be more obvious if you copy/pasted the exact error messages, but my guess is that this will fix it,

c_opt = fmincon(objective,[1 1 1 1],[],[],[],[],[],[],@(c)nonlcon(m,c,cts) )

[Marco Knobloch]

That worked. Many thanks