论文标题
在反射巴拉克空间中解决单调包容问题的两种简单方法的收敛性
Convergence of Two Simple Methods for Solving Monotone Inclusion Problems in Reflexive Banach Spaces
论文作者
论文摘要
我们提出了两种非常简单的方法，第一种方法具有恒定的步骤大小，第二种方法具有自适应步长大小，用于在实际反射性Banach空间中找到两个单调操作员的总和中的零。我们的方法在每次迭代时只需要对单值操作员进行一个评估。当设定值的运算符是最大单调的，并且单值操作员是Lipschitz的连续，并且当需要这两个操作员中的任何一个中的任何一个是强烈单调时，就会获得强大的收敛结果，从而获得了弱收敛结果。我们还获得了在实际反射性巴拉赫空间中提出的方法的收敛速率。最后，我们将结果应用于解决天然气市场的广义NASH平衡问题。
We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous, and strong convergence results are obtained when either one of these two operators is required, in addition, to be strongly monotone. We also obtain the rate of convergence of our proposed methods in real reflexive Banach spaces. Finally, we apply our results to solving generalized Nash equilibrium problems for gas markets.
