论文标题
关于凸出可行性问题的边缘反射方法
On the Circumcentered-Reflection Method for the Convex Feasibility Problem
论文作者
论文摘要
圆周中心的古老概念最近诞生了束缚反思方法（CRM）。 CRM首先被用来解决涉及仿射子空间的最佳近似问题。在这种情况下，它显示出胜过最负盛名的基于投影的方案，即道格拉斯·拉赫福德方法（DRM）和交替投影的方法（MAP）。现在，我们证明CRM的收敛性用于在有限数量的封闭凸组的交点中找到一个点。这种引人注目的结果是在合适的产品空间重新印象下得出的，在合适的产品空间重新印象中，寻求封闭凸组与仿射子空间的相交点的点。事实证明，CRM巧妙地解决了完整的问题。我们还表明，对于仿射集中的一个点，CRM迭代总是比MAP和DRM迭代更接近解决方案集。我们的理论结果和数值实验显示出杰出的性能，将CRM确立为解决一般凸的可行性问题的强大工具。
The ancient concept of circumcenter has recently given birth to the Circumcentered-Reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.
