MATLAB: How to use lsqnonlin together with MultiStart

Question

Hello,

Please, I would like to see how I can use lsqnonlin together with the MultiStart solver to search over many initial points automatically and find the global solution of the portfolio optimization problem below.

I am using lsqnonlin to solve the following system for the n-order vector w and the scalar lambda. There is only a lower bound constraint at 0 for each element of w (short-selling constraint).

      x0=[1/n*ones(n,1);1];
      lb=[zeros(n,1);0];
      [w phi]=lsqnonlin(@sub,x0,lb);
      function fcns=sub(x)
      w=x(1:n,1);
      lambda=x(n+1,1);
      fcns(1:n,1)=lambda^(-1)*covmat^(-1)*(meanbl-(vec1'*covmat^(-1)*vec1)^(-1)*(vec1'*covmat^(-1)*meanbl-lambda)*vec1)-w;
      fcns(n+1,1)=w'*meanbl+phi*sqrt(w'*covmat*wb)-H;

Where the scalars H and phi, the n-order vectors meanbl and vec1, and the nxn matrix covmat are known.

I did not really understand the example at http://www.mathworks.com/help/gads/multistart-using-lsqcurvefit-or-lsqnonlin.html where the local solver used with MultiStart is lsqcurvefit instead of lsqnonlin.

Thank you very much.

Best wishes.

Answer 1

First you need to figure out bounds on all the parameters w and lambda. In general, it is not enough to say that lb = zeros(n+1,1), you also need an upper bound vector.

Once you have that, you can set up your problem for MultiStart easily. You just need to put your problem into a problem structure using the createOptimProblem function:

 problem = createOptimProblem('lsqnonlin','x0',x0,'lb',lb,'ub',ub,'objective',@sub);
 ms = MultiStart;
 [x,fval,eflag,output,manymins] = run(ms,problem,50);

Alan Weiss

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