Title: Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone Show Chinese title

Title: 交替投影方法在仿射子空间与二阶锥交集上的收敛速率

Abstract: We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example. Show Chinese abstract

Abstract: 我们研究交替投影方法（APM）应用于仿射子空间与二阶锥的交集时的收敛速率。 我们证明当它们非横截相交时，收敛速率为$O(k^{-1/2})$，其中$k$是 APM 的迭代次数。 特别是，当交集不在原点或以原点为端点形成半直线时，得到的收敛速率可以是精确的，因为收敛速率的下界被评估出来了。 这些结果与 [Borwein 等人，SIOPT，2014] 和 [Drusvyatskiy 等人，Math. Prog., 2017] 中讨论的误差界所得出的最坏情况收敛速率一致。 此外，我们考虑了 APM 在仿射子空间与两个二阶锥的乘积的交集时的收敛速率。 我们提供了一个例子，说明 APM 的最坏情况收敛速率优于从该例子的误差界所预期的速率。