Why is lsqnonlin returning initial input values ?

Hello,

I'm trying to do some king of optimization stuff. I have a LS problem to solve which is that form:

C=A*x-B;
Z=Y*x-W;
min_x(C'*C + theta.* Z'*Z)

To solve this, I'm using the lsqnonlin Matlab function this way.

f=@(x)((A*x-B)'*(A*x-B)+ theta.*(Y*x-W)'(Y*x-W));
options=optimset('TolX', 1E-12, 'TolFun', 1E-12, 'ScaleProblem', 'Jacobian','MaxFunEvals', 1000, 'Display', 'off', 'Algorithm', 'levenberg-marquardt');
[Teopt] = lsqnonlin(f,Te,[],[], options);

But the problem is that despite I tried to modify the TolX and TolFun parameters, the Levenberg-Marquardt algorithm seem to stop after no or only on iteration, returning Teopt=Te.

When I look at the lsqnonlin display in the command window, I have both of theses messages:

                                          First-Order                    Norm of 
   Iteration  Func-count    Residual       optimality      Lambda           step
       0          10        0.143131            1.13         0.01
       1          36        0.143131            1.13       1e+014   2.07051e-008
  Optimization terminated: the relative change in the sum-of-squares of the functions is less than options.TolFun
                                          First-Order                    Norm of 
   Iteration  Func-count    Residual       optimality      Lambda           step
       0          10        0.593098            9.95         0.01
  Optimization terminated: no further progress can be made.
  Cannot generate point that reduces the sum of squares.
  Problem may be ill-conditioned.

I'm not satisfied by the range of the residuals after this (non)optimization. I'm far from being an expert in the field of matlab optimization functions: would someone have a hint in order to make this piece of code work better? I'm pretty sure my LS problem is not ill-conditioned as I just implemented an algorithm from a IEEE paper, but I can probably misuse the lsqnonlin function.

Thank you very much,

Guillaume

Answer 1

[John D'Errico]

lsqnonlin is not a function to optimize a scalar nonlinear function. Even though you could actually have written your problem in the form of a linear least squares, you have formulated a scalar objective for lsqnonlin to solve.

So first, you MIGHT have used fmincon. But instead, it would have been simpler and far more efficient to use just backslash. That is:

x = [A;theta*Y]\[B,theta*W];

So ONE line of code is required, with no iterative tool needed.