标题： 欧氏空间中闭多面锥上的外心反射法的有限收敛性 显示英文标题

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

摘要： 圆心反射法（CRM）是一种最近开发的投影方法，用于解决凸可行性问题。 与经典的Douglas-Rachford方法和交替投影方法相比，它具有更优的收敛性质。 在本研究中，我们的第一个主要定理证明了CRM可以从欧几里得平面上的任何起始点识别出两个闭合凸锥在\(\mathbb{R}^2\)中的可行点。 然后我们将该定理应用于\(\mathbb{R}^2\)中的两个多面体集的交集以及\(\mathbb{R}^n\)中的两个楔形集的交集，证明了从任何初始位置出发，CRM可以有限次迭代收敛到交集中的一个点。 此外，我们引入了一种基于CRM的改进技术，称为球心反射法。 借助该技术，我们证明了当初始点位于交集的极锥补集的一个子集内时，CRM可以在有限次迭代中定位到\(\mathbb{R}^3\)中两个适当多面体锥的交集中的可行点。 最后，我们提供了一个例子，说明如果初始猜测位于指定集之外，那么在\(\mathbb{R}^3\)中两个适当多面体锥的交集中，有限收敛可能会失败。 显示英文摘要

摘要： The Circumcentered Reflection Method (CRM) is a recently developed projection method for solving convex feasibility problems. It offers preferable convergence properties compared to classic methods such as the Douglas-Rachford and the alternating projections method. In this study, our first main theorem establishes that CRM can identify a feasible point in the intersection of two closed convex cones in \(\mathbb{R}^2\) from any starting point in the Euclidean plane. We then apply this theorem to intersections of two polyhedral sets in \(\mathbb{R}^2\) and two wedge-like sets in \(\mathbb{R}^n\), proving that CRM converges to a point in the intersection from any initial position finitely. Additionally, we introduce a modified technique based on CRM, called the Sphere-Centered Reflection Method. With the help of this technique, we demonstrate that CRM can locate a feasible point in finitely many iterations in the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) when the initial point lies in a subset of the complement of the intersection's polar cone. Lastly, we provide an example illustrating that finite convergence may fail for the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) if the initial guess is outside the designated set.