Finite convergence of locally proper circumcentered methods

Circumcentered methods induced by isometries

Motivated by the circumcentered Douglas--Rachford method recently introduced by Behling, Bello Cruz and Santos to accelerate the Douglas--Rachford method, we study the properness of the circumcenter mapping and the circumcenter method induced by isometries. Applying the demiclosedness principle for circumcenter mappings, we present weak convergence results for circumcentered isometry methods, which include the Douglas--Rachford method (DRM) and circumcentered reflection methods as special instances. We provide sufficient conditions for the linear convergence of circumcentered isometry/reflecti…

关于外心反射方法的中心化

On the centralization of the circumcentered-reflection method

本文提出首个不依赖乘积空间重构的外心反射方法（CRM）中心化版本（cCRM），证明了其在特定区域内收敛，并在误差界条件下实现线性或超线性收敛，数值实验验证了其有效性。

On the linear convergence of the circumcentered-reflection method

In order to accelerate the Douglas--Rachford method we recently developed the circumcentered--reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.

欧几里得空间中闭多面锥上的圆心反射法有限收敛性

Finite Convergence of Circumcentered-Reflection Method on Closed Polyhedral Cones in Euclidean Spaces

本研究证明圆心反射法（CRM）在二维欧几里得空间中两闭凸锥、两多面体集及n维中两楔状集交集的有限收敛性，提出球心反射法扩展至三维真多面锥交集，并举例说明初始点位置对收敛性的影响。

Best approximation mappings in Hilbert spaces

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^…