Polynomial Convergence of Primal-Dual Path-Following Algorithms for Symmetric Cone Programming Based on Wide Neighborhoods and a New Class of Directions

Abstract

This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming (SCP) based on wide neighborhoods and new directions with a parameter $$\theta $$ . When the parameter $$\theta =1$$ , the direction is exactly the classical Newton direction. When the parameter $$\theta $$ is independent of the rank of the associated Euclidean Jordan algebra, the algorithm terminates in at most $${\mathcal {O}}\left( \kappa r\log \varepsilon ^{-1}\right) $$ iterations, which coincides with the best known iteration bound for the classical wide neighborhood algorithms. When the parameter $$\theta =\sqrt{n/\beta \tau }$$ and Nesterov–Todd search direction is used, the algorithm has $${\mathcal {O}}\left( \sqrt{r}\log \varepsilon ^{-1}\right) $$ iteration complexity, the best iteration complexity obtained so far by any interior-point method for solving SCP. To our knowledge, this is the first time that a class of interior-point algorithms including the classical wide neighborhood path-following algorithm is proposed and analyzed over symmetric cone.