The concept of the optimal partition was originally introduced for linear optimization and linear complementarity problems and subsequently extended to semidefinite optimization. For linear optimization and sufficient linear complementarity problems, from a central solution sufficiently close to the optimal set, the optimal partition and a maximally complementary optimal solution can be identified in strongly polynomial time. In this paper, we consider the identification of the optimal partition of semidefinite optimization, for which we provide an approximation from a bounded sequence of solutions on, or in a neighborhood of the central path. Using bounds on the magnitude of the eigenvalues, we identify the subsets of eigenvectors of the interior solutions whose accumulation points are orthonormal bases for the subspaces of the optimal partition. The magnitude of the eigenvalues of an interior solution is quantified using a condition number and an upper bound on the distance of an interior solution to the optimal set. We provide a measure of proximity of the approximation obtained from the central solutions to the true optimal partition of the problem.

最优分区的概念最初是针对线性优化和线性互补问题引入的，随后扩展到半定优化。对于线性优化和足够的线性互补性问题，从足够接近最优集合的中心解中，可以在强多项式时间内确定最优划分和最大互补最优解。在本文中，我们考虑确定半定优化的最佳分区，为此我们从中心路径上或附近的解的有界序列提供了一个近似值。利用特征值幅值的界限，我们确定了内部解的特征向量的子集，这些解的累积点是最优分区子空间的正交基。内部条件特征值的大小使用条件数和内部解决方案与最佳集合之间距离的上限来量化。我们提供了从中心解决方案到问题的真正最佳划分的近似值的度量。