fmincon for matrix optimization

I am trying to solve the following minimization problem:

• where the is a positive number, Wis a 11-by-3 orthogonal (eigenvector) matrix, and Ω is the 3-by-3 diagonal (eigenvalue) matrix
• the free parameters to be optimized are the last column vector (11-by-1) of W, and the last eigenvalue (1-by-1 schaler) in Ω.

Subject to the constraints:

• 
• .

I am looking at fmincon, but the problem seems to be not as straight forward as I initially thought. Below is my wroking code.

In particular, I want to know:

1. whether the following structure is correct

2. how to make "myObjectivefunction" to optimise the vector and scaler values at the same time

optimized = fmincon(@(x) myObjectivefunction, initialV,[],[],w',beq,[],[],@nonLinconVec,options);

• optimized is a 12-by-1 vector of outputs. I put the eigenvalue in the first element, and the eigenvector in to (2:end) of the vector (not sure if it's correct or not)
• [w', beq] satisfies the first constraint (verified)
• @nonLinconVec: a separate file for the second constraint (verified)

I truly appreciate your help and suggestions!

[David Wilson]

Here's my take on your problem.

I've made up my own numbers, and for that reason it doesn't solve very well, and there are a few possible reasons for this.

I've followed your partitioning suggestion with x defined as [lamda3; w3].

The objective function is straight forward, but the nonlinear constraint less so. Below I've used equality constraints, but I suspect that they are too difficult to exactly solve, so a better approach would be to minimise their residuals.

function rc = foptprob()
A = randn(11); 
[W,~] = qr(A); % W is orthogonal 
omega12 = [1,2]'; 
lam0 = 6; 
target = 105; 

W12 = W(:,[1,2]); 
x0 = [lam0; W(:,end)]; % start guess  
% j = fobj(x0, target, W12, omega12) 


% Now try 
A = []; B = []; Aeq = []; Beq = []; LB = []; UB = []; 
xstar = fmincon(@(x) fobj(x, target, W12, omega12), ... 
                x0,A,B,Aeq,Beq,LB,UB, @(x) nonlcon(x, W12))
rc = xstar; 

end 

function j = fobj(x, target, W12, omega12) 
lam = x(1); 
w3 = x(2:end); 

W = [W12, w3]; 
Omega = diag([omega12(:); lam]); 

j = target - trace(W*Omega*W'); 


end 

function [C, Ceq] = nonlcon(x,W12) 
% Satisfy the 3 equality constraints 
w3 = x(2:end,1); 
w1 = W12(:,1); w2 = W12(:,2); 

Ceq = [w1'*w3; w2'*w3; w3'*w3 - 1];
C = []; 

end 

[Matt J]

As you've stated the problem, the solution is trivially independently of W. This is because, since W is orthogonal,

Thus since you mentioned no bounds on Ω.

[yp78]

Hi David, thank you so much for the nice example. I implimented your code with my data set, yes it does the work which I was trying to do. For further improvement, could I ask what exactly do you mean by minimising the residuals? Which residuals are you talking about?

[yp78]

Hi Matt, thanks to your comments, I become aware of the problem. I started to reform the optimization problem from different perspective now.