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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 207323, 10 pages
A Perturbed Projection Algorithm with Inertial Technique for Split Feasibility Problem
1School of Management, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China
2Department of Mathematics and Information Science, Henan Polytechnic University, Henan 454001, China
Received 25 February 2012; Revised 29 March 2012; Accepted 2 April 2012
Academic Editor: Giuseppe Marino
Copyright © 2012 Yazheng Dang et al. 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.
This paper deals with the split feasibility problem that requires to find a point closest to a closed convex set in one space such that its image under a linear transformation will be closest to another closed convex set in the image space. By combining perturbed strategy with inertial technique, we construct an inertial perturbed projection algorithm for solving the split feasibility problem. Under some suitable conditions, we show the asymptotic convergence. The results improve and extend the algorithms presented in Byrne (2002) and in Zhao and Yang (2005) and the related convergence theorem.
Let and be nonempty closed convex sets, and let be an real matrix. The split feasibility problem (SFP) is to find a point
This problem was first presented and analyzed by Censor and Elfving  and appeared in signal processing, image reconstruction , and so on. Many well-known iterative algorithms for solving (1.1) were established, see the papers [3–5]. Denoted by , the orthogonal projection operator onto convex set , that is where indicates the 2-norm. The CQ algorithm proposed by Byrne in  has the following iterative process: where , denotes the largest eigenvalue of the matrix , and is the identity operator.
In some cases, it is difficult or even impossible to compute orthogonal projection; to avoid computing projection, Zhao and Yang in  proposed the perturbed projections algorithm for the SFP. This development was based on results of Santos and Scheimberg  who suggested replacing each nonempty closed convex set of the convex feasibility problem by a convergent sequence of supersets. If such supersets can be constructed with reasonable efforts and projecting onto them is simpler than projecting onto the original convex sets, then a perturbed algorithm is favorable. The concrete iterative process of perturbed CQ algorithm  is as follows: where , , , defined as in algorithm (1.3), and and (see the definitions in the Section 2), while the perturbed projections algorithm sometime converges slowly by reason of using only the current point to get the next iterative point.
Many papers have studied the inertial-type extrapolation recently, see [9–12], which uses the term and the two previous iterative points , to get the next iterative point . As an acceleration process, it can considerably improve the speed of convergence for the following causes: one is that the vector acts as an impulsion term, the other is that the parameter acts as a speed regulator.
To the best of our knowledge, no publications deal with perturbed projection algorithm and inertial process simultaneously. In this paper, we apply the inertial technique to the perturbed projection algorithm to get a perturbed inertial projection algorithm for the split feasibility problem. The results improve and extend the algorithms presented in  and in  and the related convergence theorem.
The paper is organized as follows. In Section 2, some preliminaries are given. The inertial perturbed algorithm and the corresponding convergence theorem for the split feasibility problem are presented in Section 3.
Throughout the rest of the paper, denotes the identity operator, denotes the fixed points of an operator , that is, .
An operator is said to be nonexpansive (ne) if It is well known that the projection operator is nonexpansive.
Recall the following notions of the convergence and -distance.
Definition 2.1. Let be an operator on a Hilbert space ; let be a family of operators on a Hilbert space. is said to be convergent to if as for all .
Definition 2.2. The -distance () for operators and on is given by
Now we introduce the Mosco-convergence for sequences of sets in a reflexive Banach space.
Definition 2.3 (see ). Let be a reflexive Banach space and and ( is a set of natural numbers) a sequence of subsets of . The sequence is Mosco-convergent to , denoted by , if where and denote the strong and weak topologies, respectively. In particular, if and are in , then is equivalent to
Using the notation NCCS() for the family of nonempty closed convex subsets of , let and be sets in NCCS(), for . It is easy to verify that if the sequence converges to in the Mosco sense, then the operator sequence converges to .
Definition 2.4. Let and be elements in NCCS(). The -distance is defined by
Let and be sets in NCCS(), then if and only if for all .
The following lemmas will be used in convergence analysis later on.
Lemma 2.5 (see ). Let and be nonnegative sequences satisfying and , . Then, is a convergent sequence.
Lemma 2.6 (see ). Let , , and , satisfying, ,, , where .Then, is a converging sequence and , where for any .
Lemma 2.7 (Opial ). Let be a Hilbert space, and let be a sequence in such that there exists a nonempty set satisfying:for every exists,any weak cluster point of belongs to .Then, there exists such that weakly converges to .
Lemma 2.8 (see ). Let be a Hilbert space, a nonexpansive operator, and a weak cluster point of a sequence , and let . Then .
3. The Inertial Perturbed Algorithm and the Asymptotic Convergence for the SPF
It is well known that the operator is -Lipschitz continuous with . The same is true for the operators for ; it is easy to obtain the following conclusion.
Lemma 3.1 (see ). Let and be sets in , and let and be sets in , for , with and . Then, the operators and defined in (3.1) are nonexpansive operators for . Moreover, the operator sequence converges to .
Now we give the perturbed inertial KM-type algorithm for SFP.
Algorithm 3.2. Given arbitrary elements in for , let where , , for any and with , , .
The following theorem is necessary for the convergence analysis of Algorithm 3.2.
Theorem 3.3. Let and for be nonexpansive operators in finite-dimensional Hilbert space, with , and let be a sequence in satisfying for all . Then, the sequence defined by the iterative step converges to a fixed point of provided that we choose parameter satisfying whenever such fixed points exist.
Proof. We first prove that the sequence is bounded and is convergent for all , where denotes the set of the fixed points of the operator , that is, . Since and are ne operators, we have
where . From the selection of parameter , we have
It is easy to get
By (3.4), (3.8)-(3.10), we obtain from Lemma 2.5 that the sequence is convergent and hence the sequence is bounded.
We next prove that . Let and notice the fact that Then, we have From (3.6), we obtain By (3.12) and observing that (since ), we have Combining with (3.14), we get We have known that the sequence is bounded and is convergent; hence, there exist and such that and for all .
Thus Moreover, one has that Denoting we get Similarly, from the selection of parameter , we have It is easy to get Both (3.10) and (3.21) manifest Then, from (3.4), (3.21), and (3.22), we get According to Lemma 2.6, we obtain .
From (3.17), it follows that Because of (3.3) , we conclude that .
Finally, we prove that converges to a fixed point of . From the above computation, we know that the sequence is also bounded; hence there exist and a subsequence of (denoted ) such that From Lemmas 2.7 and 2.8, we have . It is easy to obtain that , because by (3.10). Since is convergent, it follows that . The proof is completed.
Remark 1. Since the current value of is known when choosing the parameter , then is well defined in Theorem 3.3. In fact, from the process of proof for the Theorem 3.3, we can get the following assert: the convergence result of Theorem 3.3 always holds provided that we select , , with
Now let us return to the convergence analysis of Algorithm 3.2.
Proof. For any , , by reason of the nonexpansive properties of projection, we have
Since for any given , the result of this theorem can be obtained using Theorem 3.3.
This paper is supported by National Science Foundation of China (under Grant no. 11171221), Shanghai Municipal Committee of Science and Technology (under Grant no. 10550500800), Shanghai Municipal Government (under Grant no. S30501), Basic and Frontier Research Program of Science and Technology Department of Henan Province (under Grant nos. 112300410277, 082300440150), and China Coal Industry Association Scientific and Technical Guidance to Project (under Grant no. MTKJ-2011-403).
- Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994.
- Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365, 2006.
- H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
- C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
- Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,” Inverse Problems, vol. 27, no. 1, Article ID 015007, 2011.
- C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002.
- J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1791–1799, 2005.
- P. S. M. Santos and S. Scheimberg, “A projection algorithm for general variational inequalities with perturbed constraint sets,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 649–661, 2006.
- P.-E. Maingé, “Inertial iterative process for fixed points of certain quasi-nonexpansive mappings,” Set-Valued Analysis, vol. 15, no. 1, pp. 67–79, 2007.
- P.-E. Maingé, “Convergence theorems for inertial KM-type algorithms,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 223–236, 2008.
- A. Moudafi and E. Elisabeth, “An approximate inertial proximal method using the enlargement of a maximal monotone operator,” International Journal of Pure and Applied Mathematics, vol. 5, no. 3, pp. 283–299, 2003.
- A. Moudafi and M. Oliny, “Convergence of a splitting inertial proximal method for monotone operators,” Journal of Computational and Applied Mathematics, vol. 155, no. 2, pp. 447–454, 2003.
- H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, Boston, Mass, USA, 1984.
- A. Auslender, M. Teboulle, and S. Ben-Tiba, “A logarithmic-quadratic proximal method for variational inequalities,” Computational Optimization and Applications, vol. 12, no. 1–3, pp. 31–40, 1999.
- Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.