Suppose $S\in\mathbb{R}^{n\times n}$ is a symmetric positive semidefinite matrix. I wonder how to solve the following optimization problem
\begin{equation*} \begin{aligned} & \underset{\Omega=\Omega^T}{\text{minimize}} & & \mbox{tr} \left( S \Omega^2 \right) \\ & \text{subject to} & & \Omega_{ii}=1, \quad i=1,\dots,n \\ && & \Omega_{ij}=0, \quad (i,j) \in \mathcal{O} \\ &&& \Omega \succeq 0. \end{aligned} \end{equation*}
Here, $\mathcal{O}$ is a set of indexes so that $\Omega_{ij}=0, \forall (i,j)\in\mathcal{O}$.
I think this is a convex optimization problem because the function $\mbox{tr} \left( S \Omega^2 \right)$ is a convex function of $\Omega$ and the constraint set is also convex. This is because
$$\mbox{tr} \left( S \Omega^2 \right) = \sum_{i=1}^n \omega_{i.} S \omega_{.i}$$
which is a sum of convex quadratic form, and for any $\Omega_1$ and $\Omega_2$ satisfying the constraints,
$$\forall \lambda \in (0,1), \lambda \Omega_1 + (1 - \lambda) \Omega_2$$
also satisfies the constraints. Is there a (fast) way to solve this optimization problem?
I understand that if we remove the zero, we can use MATLAB CVX, etc. But what if there is zero constraints?
