The central path in linear optimization always converges to the analytic center of the optimal set. This result was extended to semidefinite programming by Goldfarb and Scheinberg (SIAM J. Optim. 8: 871-886, 1998). 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 rigorous proof that the central path always converges in semidefinite optimization, by using ideas from algebraic geometry.
Technical report Faculty ITS, Delft University of Technology Mekelweg 4, 2628 CD Delft, The Netherlands