Frugal and Decentralised Resolvent Splittings Defined by Nonexpansive Operators

Abstract: 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.