Bad results of lsqnonlin

Hi everybody, I have to solve a 2 non-linear equations system with bounds(upper and lower) I tried lsqnonlin and sometimes it didn't converge to the right answer and i don't understand why!

***A simplified version of the problem (linear and deterministic):

options=optimset('Display','off','MaxIter',1000,'TolFun',1e-009,'TolX',1e-009); 
lb = [0,0];  ub = [13.33,6.66];  x0 = [0;0]; 

xRes  = lsqnonlin(@(x)OptimalStrategy2(x,30,1,10,8),q0,lb,ub,options);

***The defined function is the following:

function y = OptimalStrategy2(x,a,b,c1,c2)

y = [a-b*x(2)-c1-2*b*x(1);
      a-b*x(1)-c2-2*b*x(2)];        
end

*****by running the code I get xRes = 7.0720 6.6600 which is incorrect I'm supposed to get xRes = 6.6600 6.6600 to have both equations satisfied (i.e a-b*x(2)-c1-2*b*x(1)=0 and a-b*x(1)-c2-2*b*x(2)=0)

I need your help to undertand this point please!

Answer 1

[Matt J]

I'm supposed to get xRes = 6.6600 6.6600 to have both equations satisfied

By direct inspection, I find that xRes = [6.6600 6.6600] does not satisfy your equations. Neither does xRes = [7.0720 6.6600], but it is better in a least squares sense:

      K>> f=@(x) OptimalStrategy2(x,30,1,10,8);

      K>> norm(f([6.66,6.66]))

      ans =

          2.0201

      K>> norm(f([  7.0720 6.6600]))

      ans =

          1.7978

Incidentally, your problem is linear, so you should probably be using LSQLIN.

[Matt J]

Because your system is linear and nonsingular, it has a unique unconstrained solution

>> xunc = [-2*b -b; -b -2*b]\[a-c1;a-c2]

   xunc =

     -6
     -8

There is therefore no way you can satisfy both equations exactly with lower bounds lb=[0,0] in place.