Limiting behavior of the Alizadeh-Haeberly-Overton weighted paths in semidefinite programming
Optimization Methods & Software
Strong duality and minimal representations for cone optimization
Computational Optimization and Applications
Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials
Computational Optimization and Applications
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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.