论文标题
道格拉斯·拉赫福德（Douglas-Rachford
The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces
论文作者
论文摘要
道格拉斯 - 拉赫福德算法是解决凸优化问题的最突出的拆分算法之一。最近，该方法已成功地找到了在不一致的情况下（即不存在解决方案时）在不一致的情况下进行优化问题的广义解决方案（只要存在）。不一致的案例的收敛分析取决于与道格拉斯 - 拉赫福德分裂操作员和相应的最小位移载体相关的位移操作员范围的研究。在本文中，我们为（可能）在基础操作员的轻度假设下（可能是）无限二维的希尔伯特空间中的道格拉斯 - 拉赫福德分裂操作员范围提供了一个公式。我们的新结果补充了有限维的希尔伯特空间中的已知结果。几个例子说明了我们的结论并加强了我们的结论。
The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case, i.e., when a solution does not exist. The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas-Rachford splitting operator and the corresponding minimal displacement vector. In this paper, we provide a formula for the range of the Douglas-Rachford splitting operator in (possibly) infinite-dimensional Hilbert space under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.
