Resolvent splitting with minimal lifting for composite monotone inclusions

Abstract

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of n 2 maximally monotone operators involving the composition with a linear bounded operator. The resolvent of each monotone operator, the linear operator, and its adjoint are computed exactly once in the proposed algorithm. In the case when the linear operator is the identity, we recover the resolvent splitting with minimal lifting developed in [8]. We also derive a new resolvent splitting for solving the composite monotone inclusion in the case n = 2 with minimal 1−fold lifting.

Distributed Forward-Backward Methods without Central Coordination

“…While at first it may seem unusual that the number of set-valued and single-valued monotone operators in (7) are not the same, we note that this same situation arises in Davis-Yin splitting as described above.…”

“…. , [16] proposed a splitting algorithm with (n − 1)-fold lifting for finding a zero of the sum of n 2 maximally monotone operators; see also [7] for recent extensions. In this section, we adapt the methodology developed in [1] to obtain a splitting method of forward-backward-type for the inclusion (7) by modifying the splitting method in [16] without increasing the dimension of the ambient space.…”

“…. , z 0 n−1 ) ∈ H n−1 , our proposed algorithm for (7) generates two sequences, (z k ) ⊆ H n−1 and (x k ) ⊆ H n , according to…”

“…Although the number of set-valued and single-valued monotone operators in (7) differ by one, it is straightforward to derive a scheme where this is not the case by setting A 1 = 0. In this case, x 1 = J λA 1 (z 1 ) = z 1 can be used to eliminate x 1 so that ( 9) and ( 10) respectively become…”