  The conic bundle implementation of the spectral bundle method for large scale
semidefinite programming solves in each iteration a semidefinite quadratic
subproblem by an interior point approach. For larger cutting model sizes the
limiting operation is collecting and factorizing a Schur complement of the
primal-dual KKT system. We explore possibilities to improve on this by an
iterative approach that exploits structural low rank properties. Two
preconditioning approaches are proposed and analyzed. Both might be of interest
for rank structured positive definite systems in general. The first employs
projections onto random subspaces, the second projects onto a subspace that is
chosen deterministically based on structural interior point properties. For
both approaches theoretic bounds are derived for the associated condition
number. In the instances tested the deterministic preconditioner provides
surprisingly efficient control on the actual condition number. The results
suggest that for large scale instances the iterative solver is usually the
better choice if precision requirements are moderate or if the size of the
Schur complemented system clearly exceeds the active dimension within the
subspace giving rise to the cutting model of the bundle method.
