Counter Example to A Conjecture on Infeasible Interior-Point Methods

Based on extensive computational evidence (hundreds of thousands of randomly generated problems) the second author conjectured that $\bar{\kappa}(\zeta)=1$, which is a factor of $\sqrt{2n}$ better than that has been proved, and which would yield an $O(\sqrt{n})$ iteration full-Newton step infeasible interior-point algorithm. In this paper we present an example showing that $\bar{\kappa}(\zeta)$ is in the … Read more

Full Nesterov-Todd Step Interior-Point Methods for Symmetric Optimization

Some Jordan algebras were proved more than a decade ago to be an indispensable tool in the unified study of interior-point methods. By using it, we generalize the infeasible interior-point method for linear optimization of Roos [SIAM J. Optim., 16(4):1110–1136 (electronic), 2006] to symmetric optimization. This unifies the analysis for linear, second-order cone and semidefinite … Read more

Full Nesterov-Todd Step Primal-Dual Interior-Point Methods for Second-Order Cone Optimization

After a brief introduction to Jordan algebras, we present a primal-dual interior-point algorithm for second-order conic optimization that uses full Nesterov-Todd-steps; no line searches are required. The number of iterations of the algorithm is $O(\sqrt{N}\log ({N}/{\varepsilon})$, where $N$ stands for the number of second-order cones in the problem formulation and $\varepsilon$ is the desired accuracy. … Read more