MATLAB: How to find the matrix that minimizes a certain objective function, with complex constraints

complex matrix optimizationfminconfmincon complexfmincon with complex variables and complex constraintsmatrix optimisationoptimisation of constrained matrix objective

Hi all.
Given 3 known m by m matrices M1 and M2 and B, I want to find the matrix X that minimize the following objective:
Obj = -trace(M1*inv(M2+X)*M1*inv(M2+X))
subject to X-B being semidefenite, i.e.,
X – B >=0.
The matrix B is purely complex and anti-symmetric (i.e., transpose(B) = -B) while M1, M2 and the solution X are real symmetric matrices.
I tried to use fmincon, but I ran into some issues with fmincon and complex numbers. An example code in which dimension m=2 is provided below.
[M1,M2] = cov_mat(1);
B = 1i*[0 1;-1 0];
fun =@(X) -trace((M1/(M2+X))^2);
X0 = eye(2*m);
%   If I understand correctly, my constraint should be expressed in a vectorised manner, 
%  i.e., X-B>=0 should be replaced with kron(eye(2*m),eye(2*m))*X(:) >= B(:);
A = -eye(2*m);A = kron(A,eye(2*m));
B = B(:);
X = fmincon(fun,X0,A,B);
function [sigma,dsigma] = cov_mat(T)
sigma = coth(1/T) * eye(2);
dsigma = (coth(1/T)^2 - 1)/T^2 * eye(2);
end
Any hints on how to fix this, or any alternative methods to solve this problem are appreciated.

Best Answer

Convert this to an unconstrained problem with the change of variables X=B+Z'*Z and then use fminunc,
M3=M2+B;
fun = @(Z) -norm(  M1/(M3+Z'*Z),'fro')^2;
Z = fminunc(fun,Z0);
X=B+Z'*Z;
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