I have a question regarding the effect of an objective function on getting rank-1 solution for semidefinite optimization. I'm trying to minimize the following objective function
Minimize $ c_0\ (\Re[Tr(A_1W)] + \Re[ \ Tr(A_2W)]) + [1 - Tr(EW)]^2 $
When I choose $c_0 \leq 1$. I could not get rank-1 solution (have more than 1 positive eigenvalue of $W$, which is positive semi-definite matrix). Could anyone explain to me what is the importance of choosing proper $c_0$ in my objective function?
Thanks,