I am trying to simulate the following optimization problem in CVX, where F is the optimization variable. It is a SDP problem. However, I ran into Disciplined convex programming error: Illegal operation: log( {convex} ) for the objective function. All the constraints are OK.

As the author claim that the corresponding problem satisfies DCP, so, there should be a way to implement it in the CVX. I would really appreciate any help. Thank you so much.

Provided w(k) >= 0. that objective should be sum(log(concave)), which CVX should accept. The matrix variable F appears in the affine argument of sqrt, which is thus concave (and multiplied by 2*w(k)*abs(g_k'*h_k)), and as an affine term. So altogether, that is log(concave + affine), which is log(concave), which is then summed.

If w(k) < 0, that term would become log(convex), which CVX does not allow.