Properties of the Central Path

In this chapter, we take a careful look at the behavior of the central path, when it approaches the optimal solution set of a semidefinite program. We will demonstrate that the primal-dual central path converges to the analytic center of the optimal solution set. Moreover, the distance to this analytic center from any point on the central path is shown to converge at the same R-rate as the duality gap. This result can be interpreted as an error-bound for solutions on the central path, with respect to the optimal solution set. Underlying the analysis is an assumption that a strictly complementary solution pair for the semidefinite program exists.