论文标题
Ryu，Malitsky-Tam和Campoy的分裂算法应用于线性子空间的正常锥，强烈收敛到投影到交叉口
The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection
论文作者
论文摘要
在现代优化和非平滑分析中，找到最大单调运算符的总和是零。假设可以使用操作员的分解，则可以使用道格拉斯·拉赫福德算法解决此问题。但是，在与三个或三个或更多操作员打交道时，必须在产品空间中工作，并且具有与操作员一样多的因素。在Ryu和Malitsky和Tam的最新作品中，这表明可以减少一个因素的数量。克鲁格（Kruger）最初提出的巧妙重新制定，坎波伊（Campoy）最近也实现了类似的减少。所有三种分裂方法都确保弱收敛到基础总和问题的某些解决方案；强烈的收敛性在均匀的单调性存在下。 在本文中，当涉及的操作员是子空间的正常圆锥操作员时，我们提供了一个案例研究，因此解决方案集是子空间的相交。即使这些操作员缺乏严格的凸度，我们表明在这种情况下可以得出明显的结论：强（而不是弱）收敛性，而所获得的解决方案是（不是任意的，但）投射到交叉点上。还提供了用于说明我们结果的数值实验。
Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. A similar reduction was achieved recently by Campoy through a clever reformulation originally proposed by Kruger. All three splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity. In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. Numerical experiments to illustrate our results are also provided.
