Abstract
We study a splitting algorithm for problems involving the sum of two accretive operators. We prove the strong convergence of the algorithm. Applications to variational inequality, fixed point, equilibrium, and minimization problems are provided.
Similar content being viewed by others
1 Introduction
Many important real world problems have reformulations which require finding zero points of some nonlinear operators, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [1–11] and the references therein. It is well known that minimizing a convex function f can be reduced to finding zero points of the subdifferential mapping ∂f. Forward-backward splitting algorithms were proposed by Lions and Mercier [12], by Passty [13], and, in a dual form for convex programming, by Han and Lou [14]. The algorithms, which provide a range of approaches to solving large-scale optimization problems and variational inequalities, have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation, and this operator is decomposed as the sum of two nonlinear operators. This paper concerns a forward-backward splitting algorithm with computational errors designed to find zeros of the sum of two accretive operators A and B.
The paper is organized in the following way. In Section 2, we present the preliminaries that are needed in our work. In Section 3, we present a splitting algorithm for finding zeros of the sum of two accretive operators A and B. Convergence analysis of the algorithms is investigated. In Section 4, applications of our main results are provided.
2 Preliminaries
Let E be a real Banach space with the dual . Given a continuous strictly increasing function , where denotes the set of nonnegative real numbers, such that and , we associate with it a (possibly multivalued) generalized duality map , defined as , . In this paper, we use the generalized duality map associated with the gauge function for . Let be the modulus of smoothness of E by . A Banach space E is said to be uniformly smooth if as . Let . E is said to be q-uniformly smooth if there exists a fixed constant such that . The modulus of convexity of E is the function defined by . Recall that E is said to be uniformly convex if for any . Let E be a smooth Banach space, and let C be a nonempty subset of E. Let be a retraction and be the normalized duality mapping on E. Then the following are equivalent [15]: (1) is sunny and nonexpansive; (2) , , .
Let I denote the identity operator on E. An operator with domain and range is said to be accretive iff, for and , , , . It follows from Kato [16] that A is accretive iff, for , there exists such that . An accretive operator A is said to be m-accretive iff for all . In this paper, we use to denote the set of zeros of A. For an accretive operator A, we can define a nonexpansive single-valued mapping by for each , which is called the resolvent of A. Recall that a single-valued operator is said to be α-inverse strongly accretive if there exists a constant and some such that , .
Let be a mapping. Recall that T is said to be κ-contractive iff there exists a constant such that , . T is said to be nonexpansive iff . T is said to be κ-strictly pseudocontractive iff there exists a constant such that
for some . T is said to be pseudocontractive iff , for some .
In order to obtain our main results, we also need the following lemmas.
Lemma 2.1 [17]
Let E be a real q-uniformly smooth Banach space. Then the following inequality holds: and , , where is some fixed positive constant.
Lemma 2.2 [18]
Let E be a real Banach space, and let C be a nonempty closed and convex subset of E. Let be a single-valued operator, and let be an m-accretive operator. Then , where is the resolvent of B for .
Lemma 2.3 [19]
Let E be a real Banach space, and A be an m-accretive operator. For , , and , we have , where and .
Lemma 2.4 [20]
Let be a sequence of nonnegative real numbers such that , where is a sequence of nonnegative real numbers, and is a number sequence. Assume that , , and . Then .
Lemma 2.5 [21]
Let . Then the following inequality holds: , for arbitrary positive real numbers a and b.
Lemma 2.6 [22]
Let C be a nonempty closed convex subset of a real uniformly smooth Banach space E. Let be a contractive mapping, and let be a nonexpansive mapping. For each , let be the unique solution of the equation . Then converges strongly to a fixed point .
3 Main results
Theorem 3.1 Let E be a real q-uniformly smooth Banach space with the constant . Let be an m-accretive operator such that is convex. Let be an α-inverse strongly accretive operator. Assume that . Let be a fixed κ-contraction. Let , , , and be positive real number sequences, where , , and are in . Let be a sequence generated in the following iterative process:
where , is a sequence in E, and is a bounded sequence in . Assume that the sequences , , , , and satisfy the following restrictions:
-
(1)
;
-
(2)
, ;
-
(3)
;
-
(4)
, , ;
-
(5)
, .
Then the sequence converges strongly to , where is the unique sunny nonexpansive retraction of C onto .
Proof First, we show that is bounded. In view of Lemma 2.1, we find that
From restriction (4), we find that is nonexpansive. Fixing , we find from Lemma 2.2 that . It follows from restriction (5) that
where . This shows that is bounded. Set . It follows that
Using Lemma 2.3, we find that
On the other hand, we have
Using (3.1) and (3.2), we find
Using restrictions (3), (4), and (5), we obtain from Lemma 2.4 that
On the other hand, we have
Using restrictions (2) and (5), we obtain from (3.3) that
Since
we find from (3.4) and restriction (5) that
Without loss of generality, let us assume that there exists a real number r such that . Since B is accretive, we have
It follows that
This implies from (3.5) that
Since is nonexpansive and f is contractive, we find that the mapping is contractive. Let be the unique fixed point of the mapping , that is, , . Setting , we have , where is the unique sunny nonexpansive retraction from C onto .
Next, we show that
Since
we have
Fix t and let . It follows from (3.6) that
Since the duality map is single-valued and strong-weak∗ uniformly continuous on bounded sets of a Banach space E with a uniformly Gâteaux differentiable norm, one has
Hence, , such that , one has
Using (3.7), we see that
Using Lemma 2.5, one has
It follows that
Using restrictions (2) and (5), we see from (3.8) that converges strongly to x. This completes the proof. □
Remark 3.2 The framework of the space in Theorem 3.1 can be applicable to , where .
Corollary 3.3 Let E be a real q-uniformly smooth Banach space with the constant . Let be an m-accretive operator such that is convex. Assume that . Let be a fixed κ-contraction. Let and be positive real number sequences, where is in . Let be a sequence generated in the following iterative process:
where . Assume that the sequences and satisfy the following restrictions:
-
(1)
, ;
-
(2)
;
-
(3)
, .
Then the sequence converges strongly to , where is the unique sunny nonexpansive retraction of C onto .
Corollary 3.4 Let E be a real q-uniformly smooth Banach space with the constant , and let C be a nonempty closed and convex subset of E. Let be a fixed κ-contraction. Let be an α-strictly pseudocontractive mapping with a nonempty fixed point set. Let , , , and be positive number sequences, where , , and in . Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the sequences , , , and satisfy the following restrictions:
-
(1)
;
-
(2)
, ;
-
(3)
;
-
(4)
, , ;
-
(5)
.
Then the sequence converges strongly to , where is the unique sunny nonexpansive retraction of C onto .
Proof Putting , we find that A is α-inverse strongly accretive and . Notice that
Using Theorem 3.1, we find the desired conclusion immediately. □
4 Applications
In this section, we give some applications of our main results in the framework of Hilbert spaces.
From now on, we always assume that C is a nonempty closed and convex subset of a real Hilbert space H and stands for the metric projection from H onto C. Let be a monotone operator. Recall that the classical variational inequality is to find such that
The solution set of the variational inequality is denoted by .
Let be a function defined by
It is easy to see that is a proper lower and semicontinuous convex function on H, and the subdifferential of is maximal monotone. Define the resolvent of the subdifferential operator . Letting , we find that
where .
Putting in Theorem 3.1, we find that . Hence, the following result can be obtained immediately.
Theorem 4.1 Let be a fixed κ-contraction. Let be an α-inverse strongly monotone operator with . Let be a positive number sequence. Let , , and be real number sequences in . Let be a sequence generated in the following iterative process:
where is a sequence in H and is a bounded sequence in C. Assume that the sequences , , , , and satisfy the following restrictions:
-
(1)
;
-
(2)
, ;
-
(3)
;
-
(4)
, , ;
-
(5)
, .
Then the sequence converges strongly to , where is the unique metric projection from C onto .
Next, we consider the problem of finding a solution of a Ky Fan inequality [23], which is known as an equilibrium problem in the terminology of Blum and Oettli; see [24] and the references therein.
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem:
The solution set of the problem is denoted by in this section.
To study the equilibrium problem (4.2), we may assume that F satisfies the following restrictions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
The following lemma can be found in [24].
Lemma 4.2 Let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that , . Further, define
for all and . Then (1) is single-valued and firmly nonexpansive; (2) is closed and convex.
Lemma 4.3 Let F be a bifunction from to ℝ which satisfies (A1)-(A4), and let be a multivalued mapping of H into itself defined by
Then is a maximal monotone operator with the domain , , and , , , where is defined as in (4.3).
Based on Lemmas 4.2 and 4.3, we find from Theorem 3.1 the following immediately.
Theorem 4.4 Let be a bifunction satisfying (A1)-(A4) such that is not empty. Let be a fixed κ-contraction. Let be a positive number sequence. Let , , and be real number sequences in . Let be a sequence generated in the following iterative process:
where , is a sequence in H and is a bounded sequence in C. Assume that the sequences , , , , and satisfy the following restrictions:
-
(1)
;
-
(2)
, ;
-
(3)
;
-
(4)
, , ;
-
(5)
, .
Then the sequence converges strongly to some point in .
For any matrix , we denote its transpose by and its operator norm by .
Consider the inclusion problem [10], where , and and are maximal monotone mappings on and , respectively, and , , . Then, A and B are maximal monotone and A is Lipschitz continuous on with the constant
The special case where , yields the following convex program:
where and are closed proper convex functions on, respectively, and . The special case where , and yields the inclusion .
Finally, we consider finding minimizers of proper lower semicontinuous convex functions.
For a proper lower semicontinuous convex function , the subdifferential mapping ∂h of h is defined by
Rockafellar [25] proved that ∂h is a maximal monotone operator. It is easy to verify that if and only if .
Theorem 4.5 Let be a fixed κ-contraction. Let be a proper convex lower semicontinuous function such that is not empty. Let be a positive number sequence. Let , , and be real number sequences in . Let be a sequence generated in the following iterative process:
where is a sequence in H and is a bounded sequence in C. Assume that the sequences , , , , and satisfy the following restrictions:
-
(1)
;
-
(2)
, ;
-
(3)
;
-
(4)
, , ;
-
(5)
, .
Then the sequence converges strongly to some minimizer of h.
Proof Since is a proper convex and lower semicontinuous function, we see that the subdifferential ∂h of h is maximal monotone. Putting and , we see that
is equivalent to
It follows that
By using Theorem 3.1, we draw the desired conclusion immediately. □
References
Combettes PL, Wajs VR: Single recovery by proximal forward-backward splitting. Multiscale Model. Simul. 2005, 4: 1168–1200. 10.1137/050626090
Chen GHG, Rockafellar RT: Convergence rates in forward-backward splitting. SIAM J. Optim. 1997, 7: 421–444. 10.1137/S1052623495290179
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47–57.
Peaceman DH, Rachford HH: The numerical solutions of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 1955, 3: 28–41. 10.1137/0103003
Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 94
Moudafi A, Thera M: Finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 1997, 94: 425–448. 10.1023/A:1022643914538
Bauschke HH: 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, 48: 2513–2515. 10.1137/090759690
Tseng P: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 1991, 29: 119–138. 10.1137/0329006
Tseng P: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 2000, 38: 431–446. 10.1137/S0363012998338806
Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.
Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071
Passty GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 1979, 72: 383–390. 10.1016/0022-247X(79)90234-8
Han SP, Lou G: A parallel algorithm for a class of convex programs. SIAM J. Control Optim. 1988, 26: 345–355. 10.1137/0326019
Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3
Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen; 1976.
Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Mitrinovic DS: Analytic Inequalities. Springer, New York; 1970.
Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148
Fan K: A minimax inequality and applications. In Inequalities III. Edited by: Shisha O. Academic Press, New York; 1972:103–113.
Blum E, Oettli W: From optimization and variational inequalities to equilibriums problems. Math. Stud. 1994, 63: 123–145.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
Acknowledgements
The authors are grateful to the anonymous referees for useful suggestions, which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Qin, X., Cho, S.Y. & Wang, L. Convergence of splitting algorithms for the sum of two accretive operators with applications. Fixed Point Theory Appl 2014, 166 (2014). https://doi.org/10.1186/1687-1812-2014-166
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-166