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.
cvx_begin
variable H(n,n)
V = sqrt(diag(w))*H'*X;
minimize(vec(V)'*vec(V))
% place constraints here
cvx_end
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.