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.

翻译：针对大规模半定规划的谱束方法中，锥束实现每步迭代均通过内点法求解一个半定二次子问题。当切割模型规模较大时，性能瓶颈在于对原始-对偶KKT系统的舒尔补进行收集与因式分解。我们探索利用结构低秩特性的迭代方法以改善此过程，提出并分析了两种预条件策略（这两种策略对一般秩结构正定系统亦具参考价值）。第一种采用随机子空间投影，第二种则基于结构内点性质选择确定性子空间进行投影。针对两种方法均推导了条件数的理论界。在测试实例中，确定性预条件子对实际条件数展现出惊人的高效控制能力。结果表明：在大规模问题中，若精度要求适中，或舒尔补系统规模显著超过束方法切割模型对应的子空间活跃维度时，迭代求解器通常是更优选择。