Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators
Comput Optim Appl
https://doi.org/10.1007/s10589-018-9996-z
Complexity of the relaxed Peaceman–Rachford splitting
method for the sum of two maximal strongly monotone
operators
Renato D. C. Monteiro1 · Chee-Khian Sim2
Received: 3 November 2016
© The Author(s) 2018
Abstract This paper considers the relaxed Peaceman–Rachford (PR) splitting method
for finding an approximate solution of a monotone inclusion whose underlying operator consists of the sum of two maximal strongly monotone operators. Using general
results obtained in the setting of a non-Euclidean hybrid proximal extragradient framework, we extend a previous convergence result on the iterates generated by the relaxed
PR splitting method, as well as establish new pointwise and ergodic convergence rate
results for the method whenever an associated relaxation parameter is within a certain interval. An example is also discussed to demonstrate that the iterates may not
converge when the relaxation parameter is outside this interval.
Keywords Relaxed Peaceman–Rachford splitting method · Strongly monotone
operators · Non-Euclidean hybrid proximal extragradient framework
Renato D. C. Monteiro: The work of this author was partially supported by NSF Grant CMMI-1300221.
Chee-Khian Sim: This research was made possible through support by Centre of Operational Research
and Logistics, University of Portsmouth.
B Chee-Khian Sim
Renato D. C. Monteiro
1
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta,
GA 30332-0205, USA
2
Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace,
Portsmouth PO1 3HF, UK
123
R. D. C. Monteiro, C.-K. Sim
1 Introduction
In this paper, we consider the relaxed Peaceman–Rachford (PR) splitting method for
solving the monotone inclusion
0 ∈ (A + B)(u)
(1)
where A : X ⇒ X and B : X ⇒ X are maximal β-strongly monotone (point-toset) operators for some β ≥ 0 (with the convention that 0-strongly monotone means
simply monotone, and β-strongly monotone with β > 0 means strongly monotone in
the usual sense). Recall that the relaxed PR splitting method is given by
xk = xk−1 + θ (J B (2J A (xk−1 ) − xk−1 ) − J A (xk−1 )),
(2)
where θ > 0 is a fixed relaxation parameter and JT := (I + T )−1 . The special case of
the relaxed PR splitting method in which θ = 2 is known as the Peaceman–Rachford
(PR) splitting method and the one with θ = 1 is the widely-studied Douglas–Rachford
(DR) splitting method. Convergence results for them are studied for example in [1–
4,8,13,14,22].
The analysis of the relaxed PR splitting method for the case in which β = 0
has been undertaken in a number of papers which are discussed in this paragraph.
Convergence of the sequence of iterates generated by the relaxed PR splitting method
is well-known when θ < 2 (see for example [1,7,14]) and, according to [16], its
limiting behavior for the case in which θ ≥ 2 is not known. We actually show in Sect.
√
5.2 that the sequence (2) does not necessarily converge when θ ≥ 2. An O(1/ k)
(strong) pointwise convergence rate result is established in [18] for the relaxed PR
splitting method when θ ∈ (0, 2). Moreover, when A = ∂ f and B = ∂g where f
and g are proper lower semi-continuous convex functions, papers [9–11] derive strong
pointwise (resp., ergodic) convergence rate bounds for the relaxed PR method when
θ ∈ (0, 2) (resp., θ ∈ (0, 2]) under different assumptions on the functions. Assuming
only β-strong monotonicity of A = ∂ f , where β > 0, some smoothness property on f ,
and maximal monotonicity of B, [16] shows that the relaxed PR splitting method has
linear convergence rate for θ ∈ (0, 2 + τ ) for some τ > 0. Linear rate of convergence
of the relaxed PR splitting method and its two special cases, namely, the DR splitting
and PR splitting methods, are established in [2–4,11,15,16,22] under relatively strong
assumptions on A and/or B (see also Table 2).
This paper assumes that β ≥ 0, and hence its analysis applies to the case in which
both A and B are monotone (β = 0) and the case in which both A and B are strongly
monotone (β > 0). This paragraph discusses papers dealing with the latter case. Paper
[12] establishes convergence of the sequence generated by the relaxed PR splitting
method for any θ ∈ (0, 2 + β) and, under some strong assumptions on A and B,
establishes its linear convergence rate. We complement the convergence results in [12]
by showing that for θ = 2 + β, the sequence of iterates generated by the relaxed PR
splitting method also converge, and describe an instance showing its nonconvergence
when θ ≥ min{2 + 2β, 2 + β + 1/β}. Moreover, we establish strong pointwise and
123
Complexity of the relaxed Peaceman–Rachford splitting...
ergodic convergence rate results (Theorems 4.6 and 4.8) for the relaxed PR splitting
method when θ ∈ (0, 2 + β) and θ ∈ (0, 2 + β], respectively.
Finally, by imposing strong assumptions requiring one of the operators to be strong
monotone and one of them to be Lipschitz (and hence point-to-point), [11,15,16]
establish linear convergence rate of the relaxed PR splitting method. As opposed to
these papers, the assumptions in [12] and this paper do not imply the operators A or
B to be point-to-point.
Our analysis of the relaxed PR splitting method for solving (1) is based on viewing
it as an inexact proximal point method, more specifically, as an instance of a nonEuclidean hybrid proximal extragradient (HPE) framework for solving the monotone
inclusion problem. The proximal point method, proposed by Rockafellar [29], is a
classical iterative scheme for solving the latter problem. Paper [30] introduces an
Euclidean version of the HPE framework which is an inexact version of the proximal
point method based on a certain relative error criterion. Iteration-complexities of the
latter framework are established in [25] (see also [26]). Generalizations of the HPE
framework to the non-Euclidean setting are studied in [17,21,31]. Applications of the
HPE framework can be found for example in [19,20,25,26].
This paper is organized as follows. Section 2 describes basic concepts and notation
used in the paper. Section 3 discusses the non-Euclidean HPE framework which is used
to the study the convergence properties of the relaxed PR splitting method in Sects. 4
and 5. Section 4 derives convergence rate bounds for the relaxed Peaceman–Rachford
(PR) splitting method. Section 5, which consists of two subsections, discusses a convergence result of the relaxed PR splitting method in the first subsection and provides an
example showing that its iterates may not converge when θ ≥ min{2+2β, 2+β +1/β}
in the second subsection. Finally, Sect. 6 discusses the numerical performance of the
relaxed PR splitting method for solving the weighted Lasso minimization problem.
Section 7 gives some concluding remarks.
2 Basic concepts and notation
This section presents some definitions, notation and terminology which will be used
in the paper.
We denote the (...truncated)
