In this paper, we propose an arc-search infeasible interior-point algorithm with infeasible central path wide neighborhood for semidefinite optimization. In every iteration, the algorithm uses an analytic expression of an ellipse and searches an \(\varepsilon \)-approximation solution of the problem along an ellipsoidal approximation of the infeasible central path. Based on the commutative class of scaling matrices at an iterate \((X, S)\succ (0, 0)\), we show that the algorithm has the complexity order \({\mathcal {O}}(n^{\frac{3}{2}}L)\) to Nesterov-Todd (NT) search directions, which coincides with the results for the corresponding algorithm for linear optimization. Then, we present a simplified version of the algorithm and show that the iteration complexity bound of the algorithm is the same as the best iteration bound for feasible interior-point algorithms.

在本文中，我们提出了一种具有不可行中心路径宽邻域的弧搜索不可行内点算法，用于半定优化。在每次迭代中，该算法使用椭圆的解析表达式并沿着不可行中心路径的椭圆近似搜索问题的\(\varepsilon \) - 近似解。基于迭代\((X, S)\succ (0, 0)\)的缩放矩阵的交换类，我们证明了该算法具有复杂性阶\({\mathcal {O}}(n^{ \frac{3}{2}}L)\)到 Nesterov-Todd (NT) 搜索方向，这与相应的线性优化算法的结果一致。然后，我们提出了该算法的简化版本，并表明该算法的迭代复杂度界限与可行内点算法的最佳迭代界限相同。