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w*w' is non-convex… But even if it were real, taking the norm of a non-affine argument would not be allowed. I’m not saying for sure that the overall objective function is non-convex, but will assume it is not unless you prove otherwise.
Note: If z_theta is Hermitian semidefinite, trace(z_theta*w*w') can be reformulated as square_pos(norm(sqrtm(z_theta)*w,'fro')), but then you couldn’t take the norm of that (plus plhi_theta_1) because it is not affine.