In this work, we study fixed point algorithms for finding a zero in the sum
of nβ₯2 maximally monotone operators by using their resolvents. More
precisely, we consider the class of such algorithms where each resolvent is
evaluated only once per iteration. For any algorithm from this class, we show
that the underlying fixed point operator is necessarily defined on a d-fold
Cartesian product space with dβ₯nβ1. Further, we show that this bound is
unimprovable by providing a family of examples for which d=nβ1 is attained.
This family includes the Douglas-Rachford algorithm as the special case when
n=2. Applications of the new family of algorithms in distributed
decentralised optimisation and multi-block extensions of the alternation
direction method of multipliers (ADMM) are discussed.Comment: 23 page