how to find the optimal value of a matrix that minimize a function?

I have a matrix W which is a block diagonal matrix with dimensions , and each of its two block diagonal is vector. I want to find the values of its enteries that minimze the difference between the following function:

Where: W is the required block diagonal matrix to be optimized, B is a matrix, H is a given matrix, and A is a matrix. A and B are calculated using the functions used in the attached code.

It tried this attached code, but I think it is now in infinte loop and I don't know what should I do?

while ((B*H)-(A*W)~=zeros(2,4))
            w1=randn(1,2); % generate the first block diagonal vector with dimensions 1*2. The values of each entry of the block 
            % diagonal vector may be not the same.
            w2=randn(1,2); % generate the second block diagonal vector with dimensions 1*2.
            W=blkdiag(w1,w2); % bulid the block diagonal matrix that I want to optimize with dimmensions 2*4.
            
            R=sqrtm(W*inv(inv(P)+(H'*inv(eye(2)+D)*H))*W'); % R is 2*2 matrix that will be used to calculate matrix A using LLL lattice 
            % reduction algorithm. The values of P (4*4 matrix), H (2*4 matrix) and D (2*2 matrix) are given. It's clear here that 
            % matrix R is a function of W.
            
            A= LLL(R,3/4); % I use here LLL lattice reduction algorithm to obtain 2*2 matrix A which is function of R.
            
            B=A'*W*P*H'*inv(eye(2)+D+H*P*H'); % B is 2*2 matrix which is function of A and W. The values of P (4*4 matrix), 
            % H (2*4 matrix) and D (2*2 matrix) are given.
        end

[Stephen Cobeldick]

Do you have the Optimization Toolbox or Global Optimization Toolbox?

[Adi Nor]

Yes, but I don't know how to use them. So, I tried to optmize the matrix using the attached code.

Answer 1

[Fabio Freschi]

If I understand correctly the problem can be recast to the following

A*W = B*H = F;
[a11 a12; a21 a22]*[w1 w2 0 0; 0 0 w3 w4] = [f11 f12 f13 f14; f21 f22 f23 f24];

If so, it is possible to write it in terms of another linear system

A0*w0 = f0;
[A 0 0 0; 0 A 0 0; 0 0 A 0; 0 0 0 A]*[w1 0 w2 0 0 w3 0 w4]' = [f11 f21 f12 f22 f13 f23 f14 f24];

Where A0 is 8x8, w0 is 8x1 and f is 8x1.

This is actually a over determined system, because some of the unknowns (at positions 2 4 5 and 7) are actually zeros, so we can delete the corresponding cols of A0, having a 8x4 matrix. This can be solved with backslash, which gives the linear least squares solution. I don't have the actual matrices, so I play with dummy values

% this is your matrix A
A = eye(2)+rand(2);

% this is the result of the product of your matrices B*H
F = rand(2,4);

% create the 8x8 system
A0 = blkdiag(A,A,A,A);

% create the rhs
f = F(:);

% remove the 2 4 5 7 cols from A0
A0(:,[2 4 5 7]) = [];

% now get the least squares solution
w = A0\f

% result with your notation w1 and w2
w1 = [W(1) W(2)];
w2 = [W(3) W(4)];

hope it helps

[Adi Nor]

Should this code be placed between the while loop?

[Fabio Freschi]

No, you have to pick your A, B, H matrices, create F and run the lines from A0 = ... on

[Ahmed Nasr]

If I understand his code well. How can he pick A, B matrices without determining the value of matrix W? as matrices A and B are function of W?

[Adi Nor]

Thank You. It works well now with me.

I have another question related to the same problem. Using the same method in your answer, How can I find the optimal entries of W that minimze the function F? Where:

F = sqrtm(W*A*W'), F is a 2*2 matrix is the function to be minimized.

A is a "given" 4*4 matrix not random.

W is 2*4 block diagonal matrix in which each block diagonal is 1*2 vector. W' is the transpose of matrix W that we want to minimize.

[Fabio Freschi]

what do you mean with "minimize a matrix", given that F is a 2x2 matrix?

[Adi Nor]

F is 2*2 matrix and equal to F = sqrtm(W*A*W'), where W and A are also matrices. A is 4*4 full rank matrix given with me. I don't to know the values of matrix F and W. So, I want to find the optimal entries of matrix W that minimize for example the L-2 norm of F. Subject to:

F = sqrtm(W*A*W'), F is a 2*2 matrix is the function to be minimized.

A is a "given" 4*4 matrix not random.

W is 2*4 block diagonal matrix in which each block diagonal is 1*2 vector. W' is the transpose of matrix W that we want to minimize.

The l-2 norm value of W and A must be greater than zero.