unconstrained nonlinear optimization problem

Hello,

Kindly I have a question concerning nonlinear optimization problem. I have the following problem:

How I can define this objective function in matlab?

Thank you

[Matt J]

Why fmincon? You have no constraints. You probably should add some constraints, however, because the solution is currently not unique. Since the objective depends only on the differences , adding any constant to any optimal will produce another optimal solution.

[Bjorn Gustavsson]

Why isn't for all i a solution? Shouldn't that make every term in the sum zero, and that ought to be the smallest possible value...

[John D'Errico]

And why would you bother to use fmincon to minimize a function that can be solved using multiple linear regression in the first place?

Not only are you wanting to use a Mack truck to carry a pea to Boston, but you never even needed any truck at all. A bycycle would suffice.

And as Bjorn points out, a trivial solution exists anyway.

[Matt J]

I think Bjorn mean's . The solution set is for any scalar constant c.

[Heborvi]

Thank you for your replies and I am sorry because there is an error in writing the optimization problem. the problem is the diference between d_{ij} and the euclidean distance between vectors. I have changed it in the question.

Answer 1

[Matt J]

Still, there are no constraints, so I think you should be using fminunc. Also, your revised objective function is not differentiable near theta=X, so you'll have to hope that the optimizer doesn't search there.

N=nchoosek(n,2);
I=1:N;
J=nchoosek(1:n,2);

C=(sparse(I,J(:,1),1,N,n)+sparse(I,J(:,2),-1,N,n)).';

thetaOptimal=fminunc(@(theta) P(theta,X,C,d), theta0);

function fval=P(theta,X,C,d)
%theta - pxn matrix of unknowns
% X    - known pxn matrix
% d    - column vector of length nchoosek(n,2)

   V=vecnorm((X-theta)*C,2,1);

   fval=norm(d-V(:)).^2;
   
end