I need to know about what is the solution to this convex optimization problem. The feasible set is all $6\times 6$ doubly-stochastic symmetric matrices with all non-first column and non-first row, off-diagonal entries being zero. And the objective function is the 2-norm (largest singular value). Where $1\in \mathbb{R}^6$ means an all-one vector.
\begin{aligned} \min_{W} \quad & { \| W - \frac{1}{6}11^T \|_2 } \\ \textrm{subject to} \quad & W\in \mathbb{S}^6 \\ & W1 = 1\\ & \forall i\neq j , (i\neq1 \land j\neq 1) \implies W_{ij}=0 \\ \end{aligned}
It is easy to see that the problem is convex: objective is convex since it is a norm function composed with an affine function, and norm is convex. The feasible set is also convex because any constraint lead to a convex set, and intersection of convex sets are still convex.
However, I do not know how to put into a vector form to do optimization in CVX, can someone help me? Thank you very much.