I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian conjugate to $\vec v_i$) and with $9\times9$ identity matrix $I$,
minimize $Tr(D) - \max_i(\vec v_i\!' * D * \vec v_i)$
(1) $\vec v_i\!' * D *\vec v_j = 0$ for all $i\ne j$
(2) $||I - X|| = R$ (using any matrix norm here that will work)
(5) $X + D - \sum_i (\vec v_i\!' * X * \vec v_i) \vec v_i*\vec v_i\!'\ge0$
(6) $X$ is a tensor product of some (unknown) pair of $3\times3$ matrices.
Constraint (5) essentially just subtracts off the diagonal elements (in the basis of the $\vec v_i$) of $X$, replacing them with those of $D$. For this particular case, the $\vec v_i$ are real, if that matters. But $X$ can be complex.
$X$ and $D$ are the ($9\times9$ matrix) variables, and without the last constraint involving the tensor product, it appears to almost be a semidefinite, or at least a convex, program. The constraint involving $R$, confining things to a spherical shell centered on the identity, may also be problematic. I've tried using cvx to solve this, but without luck, apparently because of these two issues.
Is there a way to deal with these issues to obtain a solution that we know is a global, and not just a local, minimum? Any help would be greatly appreciated.