Tseng-Type Algorithms for the Split Variational Inclusion

Zhu, Li-Jun; Yin, Tzu-Chien

doi:https://doi.org/10.1155/2022/4819157

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4819157 | https://doi.org/10.1155/2022/4819157

Tseng-Type Algorithms for the Split Variational Inclusion

Li-Jun Zhu¹and Tzu-Chien Yin²

Academic Editor: Yongqiang Fu

Received07 May 2022

Accepted02 Aug 2022

Published29 Aug 2022

Abstract

In this paper, we study iterative methods for solving the split variational inclusion in Hilbert spaces. We consider a Tseng-type algorithm with self-adaptive step sizes for finding a solution of the split variational inclusion. We show the weak convergence of the proposed sequence under some mild assumptions.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let and be two operators. Recall that the variational inclusion is to find such that

Throughout, we use to denote the solution set of the variational inclusion (1).

Let be two proper, lower semicontinuous and convex functions. It is known that the variational inclusion (1) is closely related to the following minimization problem:

Assume that is differentiable and is subdifferentiable. If choose and , then solving (1) is equivalent to solving (2) [1].

Variational inclusion is at the core of the modeling of many problems, such as variational inequalities [2–4], optimization problems [5], split problems [6–9], equilibrium problems [10], and fixed point problems [11]. Variational inclusion (1) has been extended and studied in different ways, see [12–18]. An efficient way for solving (1) is the forward-backward iterate [19–21] defined bywhere is the resolvent of , and is a positive constant.

If is (inverse) strongly monotone and is maximal monotone, then the sequence defined by (3) converges weakly to some point in . In order relaxed the conditions imposed on , Tseng [22] proposed the following algorithm:where is a sequence in . Note that the Lipschitz constant of is usually unknown or difficult to estimate in many problems. Tseng’s method has been applied extensively, and some self-adaptive techniques are used, see [14, 23–26]. Especially, Cholamjiak, Hieu, and Cho [23] suggested an iterative method described as follows:where

Let and be two real Hilbert spaces. Let and be two single-valued operators. Let and be two multivalued operators. Let be a bounded linear operator. In this paper, we investigate the split variational inclusion of finding a point such that

Observe that the prototype of the split variational inclusion is the following split feasibility problem:where and are two nonempty closed convex sets. The split feasibility problem in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [27] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Now, the split feasibility problem has been extended in many different directions by many scholars, see [2, 6, 7, 10, 28]. The split variational inclusion (7) was first studied in [29]. Note that the following equivalent relation holds

By applying (9), Moudafi [29] suggested the following iterate for solving the split variational inclusion (7):where and .

It should be pointed out that the operators and in (10) were assumed to be inverse strongly monotone. The split variational inclusion (7) has been studied extensively, see [24, 28, 30–36].

The main purpose of this paper is to further study the split variational inclusion (7) in which the operators and are plain monotone. We suggest a self-adaptive Tseng-type algorithm for finding a solutions of the split variational inclusion (7). Weak convergence theorem is shown under some additional conditions.

2. Preliminaries

In this section, we assume that is a real Hilbert space. The following equality is well known:for all and for all .

Let be a single-valued operator. Recall that is said to be(i)-Lipschitz if there is a constant fulfilling(ii)monotone if

Let be a multivalued operator. is said to be monotone ifwhere and .

A multivalued monotone operator is said to be maximal monotone if and only if its graph is not strictly contained in the graph of any other monotone operator.

Let be a maximal monotone operator. Let be a constant. Define an operator bywhich is called the resolvent of .

It is well-known that the resolvent is a single-valued firmly nonexpansive operator [1] and .

In what follows, we use the following expressions: ” indicates weak convergence “” indicates strong convergence indicates the set of all weak cluster points of the sequence , that is,

Lemma 1 (see [37]). Let be a nonempty closed convex set. Let be a sequence. Assume that(i), exists(ii)Then, .

3. Main Results

Let and be two real Hilbert spaces. Let and be two multivalued maximal monotone operators. Let be a -Lipschitz monotone operator and be a -Lipschitz monotone operator. Let be a nonzero bounded linear operator and be the adjoint of . Throughout, suppose that .

Let and be two sequences in . Let , , and be three constants. Suppose that the following restrictions are fulfilled:(i) and for all (ii), , and

Next, we present an iterative sequence for solving (7).

Algorithm 1. Let be a fixed point in . Let and be two initial points in . Let . Step 1. Let the iterate be known. Calculate Step 2. Compute Step 3. Compute Step 4. SetLet and return to Step 1.

Remark 1. According to (22), we have for all . Meanwhile, owing to -Lipschitz continuity of , we obtain . It follows that for all . Therefore, exists. Similarly, from (23), we deduce that for all , and hence, exists.

Next, we state our main convergence theorem.

Theorem 1. The sequence generated by Algorithm 1 converges weakly to some point in .

Next, we will divide the proof details into several propositions. In the sequel, assume that . Then, and .

Proposition 1. exists.

Proof. Observe thatwhich together with (24) implies thatBased on (17), we getIt follows thatAs a result of and the monotonicity of , by (28), we receiveUsing (22), we have . With the help of (26) and (29), we haveUtilizing (11), from (18), we obtain
Combining (30) and (31), we haveThe following two equalities hold:Then,Thanks to (19), we getApplying the monotonicity of and , by (36), we gainOwing to (23), we get . In the light of (35) and (37), we acquireApplying (11) to (20), we attainIn view of (38) and (39), we deduceIt follows thatOn account of (21), (32), and (41), we achievewhich implies that exists.

Proposition 2.

Proof. In virtue of (32) and (42), we receivewhich results in thatThen, the sequences , , , , and are all bounded.
By (42), we haveThis together with conditions (i) and (ii) and Remark 1 implies thatDue to (32), we obtainwhich results in thatFrom (18), we getBased on the Lipschitz continuity of and (50), we deriveBy (21), (47), and (52), we deduce

By Proposition 1, the sequence is bounded. Let be any point in and be a subsequence of satisfying as .

Proposition 3. .

Proof. Let . Then, . From (17), we have which yields thatBy the monotonicity of and (54), we attain Thus,Note that because of the monotonicity of . It follows from (55) thatSince , by Proposition 3, we conclude that , , , and . With the help of (50) and (56), we obtain for all which leads to that .

Proposition 4. .

Proof. Let . Thus, . From (19), we get which results in thatApplying the monotonicity of to (57), we derive . It follows thatNotice that . This together with (58) implies thatOwing to and , from (59), we conclude that for all which implies that .

According to Propositions 1–4, we can prove Theorem 1.

Proof. In fact, we have obtained the following results:(i), exists by Propositions 1(ii) due to Propositions 3 and 4.Utilizing Lemma 1, we can conclude that converges weakly to some point in . This completes the proof.

Remark 2. In this paper, we study the split variational inclusion problem (7). We construct an iterative algorithm (Algorithm 1) for finding a solution of (7). It should be pointed out that the involved two operators and are all plain monotone. Our result improves and extends some corresponding results in [2, 23, 29].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

Li-Jun Zhu was supported by the National Natural Science Foundation of China (grant no. 11861003), the Natural Science Foundation of Ningxia Province (grant no. NXYLXK2017B09), Innovation Project of Science and Technology of YinChuan (grant no. 2022RKX03), and Innovation Project of Science and Technology of Xixia District of 2021.

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Copyright © 2022 Li-Jun Zhu and Tzu-Chien Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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