Minimising a matrix function subject to an equality constraint

Hi,

I have a problem which I am trying to optimise for, where I have two matricies, d and x, of which the values are knows, and I want to find a matrix y, such that f(y) is minimised and d*y = x.

I have d, x and f defined, and y is defined as a sym matrix. I can't find a function that can optimise on the matrix.

Update: The matrix y is not a square matrix as assumed in the answers given. Is there still a way to make this work?

[Matt J]

Update: The matrix y is not a square matrix as assumed in the answers given. Is there still a way to make this work?

But you said previously that y is a symmetric matrix. Therefore, it must be square.

[John D'Errico]

I wonder if y is defined as a symbolic matrix, not a symmetric one? Thus the statement about a sym matrix?

Answer 1

[Alan Weiss]

This looks like a job for fmincon.

Your unknown values are the upper (say) triangular elements in y.

Create f to handle the upper-triangular y, say by using the tril function along with transpose. I mean, if y is upper triangular, then

ysym = tril(y',-1) + y;

is a symmetric matrix with the same upper triangular part as y. Then you can define your objective and constraint for fmincon easily.

Or to make it even easier on you (but harder on fmincon), let y be a general matrix, and add a nonlinear equality constraint y' = y.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Matt J]

[N,N]=size(y0);        %get matrix dimensions from initial guess, y0

        J=(1:N^2).';
        I=reshape(J,N,N).';
        
        T=sparse(I,J,1,N^2,N^2); %transposition operator

Aeq=[speye(N^2)-T;...         %symmetry constraint
    kron(speye(N),d)]  ;      %other constraint
beq=[zeros(N^2,1); 
    x(:)];

y=fmincon(@f,y0, [],[],Aeq,beq)

[Matt J]

If you have no symmetry/squareness assumptions on y, then the above simplifies to,

[M,N]=size(y0); 

Aeq=kron(speye(N),d);
beq=x(:);

y=fmincon(@f,y0, [],[],Aeq,beq,[],[],[],options);

Answer 3

[John D'Errico]

It appears that y is merely a SYMBOLIC matrix, not a symmetric matrix, as others have assumed. But that just means that y is unknown in your eyes.

The constraint that d*y == x is merely a set of linear equality constraints. You can still set them up like this:

[N,N] = size(y0);        %get matrix dimensions from initial guess, y0
Aeq = kron(eye(N),d);
beq = x(:);