Strong convergence of a splitting proximal projection method for the sum of two maximal monotone operators
Introduction
Let be a real Hilbert space endowed with an inner product and norm . Let be two maximal monotone operators. We consider the following inclusion problem: find such that 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 .
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 such that where every is maximal monotone and . 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 is nonempty, we prove that the sequence generated by the proposed method converges strongly to an extended solution of problem (1), which is closest to the initial point . However, quite often, when we try to solve the inclusion problem (1), we do not know if is nonempty. Therefore, it is natural that we hope the algorithm is well defined whether is nonempty or not. To prove the well definedness of the algorithm in the case of , we employ the technique of Theorem 2 in [15] and require the assumption that is maximal monotone. Therefore, though we observe the work of Bauschke [2] where the author removed the assumption that 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 is also maximal monotone. As a by-product, we obtain another theorem which shows the equivalence between and the behavior of the generated sequence.
Section snippets
Preliminaries
Definition 2.1 Let be a set-valued mapping. Then the mapping is said to be monotone if, for any , and , maximal monotone if, it is monotone and the graph of , denoted by Gph, is not properly contained in the graph of any other monotone operator.
Lemma 2.1 If is maximal monotone, then is well-defined on the whole Hilbert space and is firmly nonexpansive, i.e.,and is in turn necessarily single-valued, where .Theorem 2 of [5]
Suppose
A splitting proximal projection method
We are now ready to formally state our algorithm. Algorithm 3.1 Step 0. Start with an arbitrary and scalar constants, and , , with . For , (splitting proximal step) Choose some and , satisfying the condition Find the unique points and such that (projection step) Define to be the decomposable separator corresponding to and
Convergence analysis and verification of solution existence
We start with establishing some properties of the algorithm which hold regardless of whether is nonempty or not.
Lemma 4.1 Suppose that Algorithm 3.1 reaches an iteration . Then
Proof By the definition of , it is clear that . Applying Lemma 2.2 (ii), we obtain Since , we know that . Furthermore, in view of Lemma 2.2 (i), we have and so
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.
References (17)
- et al.
A family of operator splitting methods revisited
Nonlinear Anal.
(2010) - et al.
Forcing strong convergence of Korpelevich’s method in Banach spaces with its applications in game theory
Nonlinear Anal.
(2010) On the regularization of the sum of two maximal monotone operators
Nonlinear Anal.
(2000)- et al.
Convergence of a splitting inertial proximal method for monotone operators
J. Comput. Appl. Math.
(2003) - et al.
A general duality principle for the sum of two operators
J. Convex Anal.
(1996) A note on the paper by Eckstein and Svaiter on General projective splitting methods for sums of maximal monotone operators
SIAM J. Control Optim.
(2009)Iterative construction of the resolvent of a sum of maximal monotone operators
J. Convex Anal.
(2009)- et al.
On the Douglas-Rachford splitting method and the proximal point algorithm for the maximal monotone operators
Math. Program.
(1992)