论文标题
在统一的单调性和没有最小化器的情况下，在道格拉斯·拉赫福德和和平脑 - 拉赫福德算法上
On the Douglas-Rachford and Peaceman-Rachford algorithms in the presence of uniform monotonicity and the absence of minimizers
论文作者
论文摘要
Douglas-Rachford和Peaceman-Rachford算法已成功地用于解决凸优化问题，或者更普遍地找到了单调包含物的零。最近，在不一致的情况下，这些方法的行为，即在没有解决方案的情况下引发了重大考虑。已经表明，在温和的假设下，当基础操作员是适当的下半连续凸函数的亚差异时，道格拉斯 - rachford算法的阴影序列薄弱地收敛到广义溶液。但是，在Peaceman-Rachford算法的情况下，没有证明融合行为。在本文中，我们证明了与道格拉斯 - 拉赫福德算法和和平人 - 拉赫福德算法相关的阴影序列的融合，当操作员之一均匀单调单调和$ 3^*$单调时，但不一定是细分。几个例子说明并加强了我们的结论。我们使用优化问题实例进行数值实验。
The Douglas-Rachford and Peaceman-Rachford algorithms have been successfully employed to solve convex optimization problems, or more generally find zeros of monotone inclusions. Recently, the behaviour of these methods in the inconsistent case, i.e., in the absence of solutions has triggered significant consideration. It has been shown that under mild assumptions the shadow sequence of the Douglas-Rachford algorithm converges weakly to a generalized solution when the underlying operators are subdifferentials of proper lower semicontinuous convex functions. However, no convergence behaviour has been proved in the case of Peaceman-Rachford algorithm. In this paper, we prove the convergence of the shadow sequences associated with the Douglas-Rachford algorithm and Peaceman-Rachford algorithm when one of the operators is uniformly monotone and $3^*$ monotone but not necessarily a subdifferential. Several examples illustrate and strengthen our conclusion. We carry out numerical experiments using example instances of optimization problems.
