On the asymptotic behaviour of the Aragón Artacho–Campoy algorithm

Abstract

Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.

Survey: Sixty Years of Douglas–rachford

“…For two closed sets A and B and an initial point x 0 ∈ H, the DR generates a sequence (x n ) ∞ n=1 as follows. x n+1 ∈ T A,B (x n ) where T A,B := 1 2 (I + R B R A ).…”

“…Both DR and AP are special cases of averaged relaxed projection methods. We denote a relaxed projection by R γ C (x) := (2 − γ)(P C − Id) + Id, (1)(2)(3)(4) for a fixed reflection parameter γ ∈ [0, 2). Observe that, when γ = 0, the operator R γ=0 C = 2P C − Id is the standard reflection employed by DR, and, for γ = 1, we obtain the projection R γ C = R 1 C = P C .…”

“…For γ ∈ (1, 2), the operator R γ C can be called an underrelaxed projection following [71]. Here we are using the terminology in (1)(2)(3)(4). However, the reader is cautioned that in some articles R γ C is written as P γ C , while in others the role of γ is reversed so that γ = 2 corresponds to a reflection and γ = 0 is the identity: γ(P C − Id) + Id.…”

“…However, the reader is cautioned that in some articles R γ C is written as P γ C , while in others the role of γ is reversed so that γ = 2 corresponds to a reflection and γ = 0 is the identity: γ(P C − Id) + Id. In addition to using relaxed projections as in (1)(2)(3)(4), the averaging step of the DR iteration can also be relaxed by choosing an arbitrary point on the interval between the second reflection and the initial iterate. This can be parametrized by some λ ∈ (0, 1].…”

Computing the resolvent of the sum of operators with application to best approximation problems

“…By product space reformulation, this problem was then handled in [5] for finitely many operators. Recently, the so-called averaged alternating modified reflections algorithm was used in [2] to study this problem, and was soon after re-derived in [1] from the view point of the proximal and resolvent average. Because computing the resolvent of a finite sum of operators can be transformed into that of the sum of two operators by a standard product space setting, as done in [5,2], we will focus our consideration to the case of two operators for reason of clarity.…”