Elsevier

Operations Research Letters

Volume 40, Issue 5, September 2012, Pages 332-336
Operations Research Letters

Strong convergence of a splitting proximal projection method for the sum of two maximal monotone operators

https://doi.org/10.1016/j.orl.2012.06.002Get rights and content

Abstract

In this paper, combining the new splitting method proposed by Eckstein and Svaiter and the forcing strong convergence method of Solodov and Svaiter, we propose a splitting proximal projection method for the sum of two maximal monotone operators. We prove that the proposed method is well-defined whether the solution set of the problem is nonempty or not and the sequence generated by the method converges strongly to an extended solution of the problem.

Introduction

Let H be a real Hilbert space endowed with an inner product , and norm . Let A,B:HH be two maximal monotone operators. We consider the following inclusion problem: find xH such that 0A(x)+B(x), which has been extensively studied by many authors (see, for example, [16], [13], [10], [1], [5], [6], [7], [14], [11], [2], [3], [4], [8]). The set of all roots to problem (1) is denoted by (A+B)1(0).

One of the effective methods for problem (1) is the splitting method (see, for example, [5], [6], [7], [14], [11], [2], [3], [4], [8]). Recently, Eckstein and Svaiter [6] developed the basic theory of a new category of splitting methods by using the notion of (overrelaxed) projection onto separating hyperplanes for a certain extended solution set for problem (1). Very recently, Eckstein and Svaiter [7] further extended this new approach to more general problem which consists of finding xH such that 0T1(x)+T2(x)++Tn(x), where every Ti is maximal monotone and n>2. In infinite dimensional Hilbert spaces, the weak convergence results were obtained in [6], [7]. However, we would like to emphasize that strong convergence algorithms are of fundamental importance for solving problems in infinite dimensional spaces.

Forcing strong convergence iteration, as named by Solodov and Svaiter [15], is one of the important techniques with strong convergence property in infinite dimensional spaces. The basic idea of this technique is to construct two hyperplanes per iteration and then project the initial point onto the intersection of the two hyperplanes. The advantages of this technique lie in that not only it can lead to strong convergence of the algorithm in infinite dimensional spaces, but also it possesses the property that the iteration sequence converges to the solution of corresponding problems closest to the initial point in some sense. Some works related to this technique can be found in [15], [12], [17], [9]. As pointed out by Eckstein and Svaiter in the last section of both references [6], [7], applying the techniques of [15] to force strong convergence for problem (1) is an interesting topic to examine.

Motivated and inspired by the research works mentioned above, in this paper, combining the new splitting method proposed by Eckstein and Svaiter [6] and the forcing strong convergence method of Solodov and Svaiter [15], we propose a splitting method for solving problem (1). Under the assumption that (A+B)1(0) is nonempty, we prove that the sequence {pk} generated by the proposed method converges strongly to an extended solution of problem (1), which is closest to the initial point p0. However, quite often, when we try to solve the inclusion problem (1), we do not know if (A+B)1(0) is nonempty. Therefore, it is natural that we hope the algorithm is well defined whether (A+B)1(0) is nonempty or not. To prove the well definedness of the algorithm in the case of (A+B)1(0)=, we employ the technique of Theorem 2 in [15] and require the assumption that A+B is maximal monotone. Therefore, though we observe the work of Bauschke [2] where the author removed the assumption that T1+T2++Tn is maximal monotone appeared in the convergence analysis of Eckstein and Svaiter [7], we would like to adopt the standard assumptions of [6], [7], [10] in the rest of this paper that A+B is also maximal monotone. As a by-product, we obtain another theorem which shows the equivalence between (A+B)1(0) and the behavior of the generated sequence.

Section snippets

Preliminaries

Definition 2.1

Let T:HH be a set-valued mapping. Then the mapping T is said to be

  • (i)

    monotone if, for any x,yH, xT(x) and yT(y), yx,yx0;

  • (ii)

    maximal monotone if, it is monotone and the graph of T, denoted by GphT, is not properly contained in the graph of any other monotone operator.

Lemma 2.1

Theorem 2 of [5]

If T is maximal monotone, then JλT is well-defined on the whole Hilbert space H and is firmly nonexpansive, i.e.,yy2xx,yy,(x,y),(x,y)Gph(JλT),and is in turn necessarily single-valued, where JλT(I+λT)1.

Suppose

A splitting proximal projection method

We are now ready to formally state our algorithm.

Algorithm 3.1

Step 0. Start with an arbitrary p0=(z0,w0)H×H and scalar constants, λ̄λ̃>0 and ρ̃, ρ̄, with 0<ρ̃ρ̄<2. For k=0,1,2,,

(splitting proximal step) Choose some αkR and λk, μk[λ̃,λ̄] satisfying the condition μkλk(αk2)2>0. Find the unique points (xk,bk) and (yk,ak) such that xk+λkbk=zk+λkwk,bkB(xk),yk+μkak=(1αk)zk+αkxkμkwk,akA(yk).

(projection step) Define φk:H×HR to be the decomposable separator corresponding to (xk,bk)Gph(B) and (yk,ak)Gph(

Convergence analysis and verification of solution existence

We start with establishing some properties of the algorithm which hold regardless of whether (A+B)1(0) is nonempty or not.

Lemma 4.1

Suppose that Algorithm 3.1 reaches an iteration k+1 . Thenpk+1p02pkp02+pk+1pk2.

Proof

By the definition of Wk, it is clear that pk=PWk(p0). Applying Lemma 2.2 (ii), we obtain PWk(pk+1)PWk(p0)2pk+1p02PWk(pk+1)pk+1+p0PWk(p0)2. Since pk+1Wk, we know that PWk(pk+1)=pk+1. Furthermore, in view of Lemma 2.2 (i), we have PWk(p0)=pk and so pk+1p02pkp02+pk+1pk

Acknowledgments

This work was supported by the Key Program of NSFC (Grant No. 70831005) and the National Natural Science Foundation of China (11171237). The authors are grateful to the editor and referee for their valuable comments and suggestions.

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