How To Perform Global Optimization (Least Square Bounded) with Objective Function in Matrix From?

Hello,

I have the following system A x = b with A(1000,1000), x and b both (1000,1). I would like to use a global optimization method such as particle swarm or global search to find the global minimum of the function (as defined in the lsqlin doc page):

function y = parameterfun(x,A,b)

y = zeros(size(b));
for i =1:size(A,1)
    for j =1:size(A,2)

           y(i) = 0.5*(norm(A(i,j)*x(i)-b(i)))^2;

      end
  end

However I get the following error:

Error using particleswarm>makeState 
Objective function must return scalar values.

I can solve that problem just fine with lsqlin, but I find a local minima that doesn't correspond to what the solution should be. I want to find other minimas. Is there a way to go around that limitation of the objective function having to return a scalar (when I want a vector) and apply a global optimization method for my problem?

Thank you.

Answer 1

[Matt J]

Your function is convex, so if it has more than one solution, then they are all global solutions and the problem is numerically unstable. Trying other solvers may produce a different solution, but the difference will be meaningless and unpredictable.

In any case, the function implementation you are seeking is,

function y = parameterfun(x,A,b)

     y=0.5*norm(A*x-b).^2; 

end