How to realize the following problem using cvx.

variable x(Ng,Ng) hermitian semidefinite;

minimize(sum(sum(abs(x),1),2)+lambda*sum(sum_square_abs(Ryu-Au*diag(x)*(Au’)),2));

subject to

for k=1:Ng

for n=k:Ng

det_rootn([x(k,k),x(k,n);x(n,k),x(n,n)])==0;

end

end

where x is the optimization variable;Ryu,Au are deterministic matrixs respectively; lambda is a regularization parameters.

Maybe the most troublesome problem is to realize the equality constraint .

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Equalities must be linear. End of story.

That’s a whole lot of singularity going on.

You are attempting to constrain every 2 by 2 submatrix of a hermitian semidefinite matrix to be singular. The only possibility I can think of to reformulate this as a convex problem is if there is a reparameterization of `x`

in a lower dimensional space such that singularity constraints are not needed. I have no idea whether that can be done.

Do you really want all the non-principal 2 by 2 submatrices to be singular, or just the 2 by 2 principal submatricss? And of course, the n = k constraints can be eliminated as vacuous.