In this paper, we propose a predictor–corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on a new one-norm neighbourhood, which is an even wider neighbourhood than a given negative infinity neighbourhood. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We show that the algorithm has $\text{O}\left(\sqrt{r}\right(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left(G\right){)}^{1/4}\log {ϵ}^{-1})$ iteration complexity bound which is better than that of the usual wide neighbourhood algorithm $\text{O}(r\sqrt{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left(G\right)}\log {ϵ}^{-1})$. To our knowledge, these are the best complexity results obtained so far for the solution of symmetric cone optimization. We prove that beside the predictor steps, each corrector step also reduces the duality gap by a rate of $1-\frac{1}{\text{O}\left(\sqrt{r}\right)}$. Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

在本文中，我们提出了一种用于对称锥优化的预测器-校正器内点方法。所提出的算法基于一个新的单范数邻域，该邻域比给定的负无穷大邻域更宽。收敛性显示为可交换的搜索方向类别，包括 Nesterov-Todd 方向以及xs和sx方向。我们证明该算法有$\text{○}\left(\sqrt{r}\right(\mathrm{C}\mathrm{○}\mathrm{n}\mathrm{d}\left(G\right){)}^{1/4}\mathrm{日志}{\epsilon }^{-1})$迭代复杂度界限优于通常的宽邻域算法$\text{○}(r\sqrt{\mathrm{C}\mathrm{○}\mathrm{n}\mathrm{d}\left(G\right)}\mathrm{日志}{\epsilon }^{-1})$. 据我们所知，这些是迄今为止解决对称锥优化的最佳复杂性结果。我们证明，除了预测器步骤之外，每个校正器步骤还可以将对偶差距缩小$1-\frac{1}{\text{○}\left(\sqrt{r}\right)}$. 最后，数值实验表明所提算法是高效可靠的。