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.
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This paper is dedicated to Professor Lian-Sheng Zhang in celebration of his 80th birthday.
This work was partially supported by the National Natural Science Foundation of China (No. 11471102) and the Key Basic Research Foundation of the Higher Education Institutions of Henan Province (No. 16A110012).
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Liu, CH., Huang, YY. & Shang, YL. Polynomial Convergence of Primal-Dual Path-Following Algorithms for Symmetric Cone Programming Based on Wide Neighborhoods and a New Class of Directions. J. Oper. Res. Soc. China 5, 333–346 (2017). https://doi.org/10.1007/s40305-017-0172-4
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DOI: https://doi.org/10.1007/s40305-017-0172-4
Keywords
- Path-following interior-point algorithm
- Wide neighborhood
- Symmetric cone programming
- Euclidean Jordan algebra
- Polynomial complexity