Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.

翻译：节俭预解分裂是一类不动点算法，用于寻找有限个集值单调算子之和的零点，其中不动点算子每次迭代仅使用向量加法、标量乘法以及每个单调算子的预解一次。文献中，这些方案的收敛分析是在低效的逐算法基础上进行的。本研究通过开发一个通用的节俭预解分裂框架来解决这一问题，该框架同时涵盖并扩展了文献中的若干重要方案。该框架还衍生出一种新的预解分裂算法，适用于正则网络上的去中心化实现。