On the identification of the optimal partition for semidefinite optimization

Abstract 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.