On the asymptotic behaviour of the Aragón Artacho–Campoy algorithm
Introduction
Throughout this note, with inner product and associated norm . The notation of our paper is standard and mainly follows [6] to which we also refer to basic results on convex analysis and monotone operator theory. A central problem is to find a zero (critical point) of the sum of two maximally monotone operators. The Douglas–Rachford and Peaceman–Rachford algorithms (see Fact 2.1 ) are classical approaches to solve this problem. If the monotone operators are normal cone operators of nonempty closed convex subsets of , then one obtains a feasibility problem, i.e., the problem of finding a point in the intersection of the two sets. Suppose, however, that we are interested in finding the nearest point in the intersection. One may then apply several classical best approximation algorithms (see, e.g., [6, Chapter 30]). In the recently published paper [2], Aragón Artacho and Campoy presented a novel algorithm, which we term the Aragón Artacho–Campoy Algorithm (AACA) to solve this best approximation problem. Even more recently, they extended this algorithm in [1] to deal with general maximally monotone operators.
The aim of this paper is to re-derive the AACA from the view point of the proximal and resolvent average. We also complete their analysis by describing the asymptotic behaviour of the underlying curve.
This note is organized as follows. In Section 2, we collect a few facts and results that will make the subsequent analysis more clear. Section 3 contains a new variant of a convergence result for AACA (Theorem 3.2) as well as the announced asymptotic behaviour of the curve (Theorem 3.4).
Section snippets
Auxiliary results
Fact 2.1 Douglas–Rachford and Peaceman–Rachford Let and be maximally monotone on . Suppose that , let , and set where and . Let and define Then there exists such that and the following hold: If or is strongly monotone, then . If , then and . If and or is strongly monotone, then . If and is strongly monotone, then .
Proof This follows from [6, Theorem 26.11 and Proposition 26.13].
The Aragón Artacho–Campoy algorithm (AACA)
From now on, we suppose that Let and be such that and we also assume that which will make all results more tidy. (See also Remark 3.3 .) Next, as in Remark 2.3, we introduce the resolvent averages between and : and
Proposition 3.1 The following hold true: , , and are maximally monotone. , , and
Acknowledgements
The authors thank two anonymous referees for their careful reading and constructive comments. SA was partially supported by Saudi Arabian Cultural Bureau . HHB and XW were partially supported by NSERC Discovery Grants. WMM was partially supported by NSERC Postdoctoral Fellowship.
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