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.

节俭解析分裂是一类定点算法，用于在有限多个集值单调算子的总和中找到零，其中定点算子仅使用向量加法、标量乘法以及每个单调算子的解析一次迭代。在文献中，这些方案的收敛分析是在低效的逐个算法的基础上进行的。在这项工作中，我们通过开发一个节俭解析分裂的通用框架来解决这个问题，该框架同时涵盖和扩展了文献中的几个重要方案。该框架还产生了一种新的解析分裂算法，适合在常规网络上分散实施。