How to write this objective function that can be recognized in CVX

How to write this objective function in CVX, see figure below. Thank you very much.

I will presume w >= 0. Of course, if no constraints are added, H = matrix of zeros will be optimal with optimal objective value = 0.

variable H(n,n)
V = sqrt(diag(w))*H'*X;
% place constraints here

Dear Mark,

How to write the following obective function in CVX? Thank you so much.

I believe that is convex for C = ones(n)., i.e., for sum(vec(H*H'))

However, I believe it is not convex for general elementwise nonnegative C. For instance, let C=[1 2;1 1]. Then the Hessian with respect to the 4 elements of H of sum(vec(C.*(H*H'))) has two eigenvalues equal to 5 and two eigenvalues equal to -1, and hence is indefinite, so sum(vec(C.*(H*H'))) is not convex.

You are right. Thanks.