On the Convergence of the Central Path in Semidefinite Optimization

Abstract

The central path in linear optimization always converges to the analytic center of the optimal set. This result was extended to semidefinite optimization in [D. Goldfarb and K. Scheinberg, SIAM J. Optim., 8 (1998), pp. 871--886]. In this paper we show that this latter result is not correct in the absence of strict complementarity. We provide a counterexample, where the central path converges to a different optimal solution. This unexpected result raises many questions. We also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry.