In this article, we propose and study a stochastic and relaxed preconditioned Douglas--Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas--Rachford splitting methods.

翻译：本文针对具有可分离对偶变量的鞍点问题，提出并研究了一种随机松弛预条件Douglas-Rachford分裂方法。我们在希尔伯特空间中证明了一类凸凹非光滑鞍点问题迭代序列的几乎必然收敛性。同时，我们给出了遍历序列关于受限原始-对偶间隙函数期望值的次线性收敛速率。数值实验表明，所提出的随机松弛预条件Douglas-Rachford分裂方法具有高效性。