Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
Special Issue

Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 350479 | https://doi.org/10.1155/2014/350479

Chung-Chien Hong, Young-Ye Huang, "A Strong Convergence Algorithm for the Two-Operator Split Common Fixed Point Problem in Hilbert Spaces", Abstract and Applied Analysis, vol. 2014, Article ID 350479, 8 pages, 2014. https://doi.org/10.1155/2014/350479

A Strong Convergence Algorithm for the Two-Operator Split Common Fixed Point Problem in Hilbert Spaces

Academic Editor: Qing-bang Zhang
Received27 Feb 2014
Accepted13 Jun 2014
Published06 Jul 2014

Abstract

The two-operator split common fixed point problem (two-operator SCFP) with firmly nonexpansive mappings is investigated in this paper. This problem covers the problems of split feasibility, convex feasibility, and equilibrium and can especially be used to model significant image recovery problems such as the intensity-modulated radiation therapy, computed tomography, and the sensor network. An iterative scheme is presented to approximate the minimum norm solution of the two-operator SCFP problem. The performance of the presented algorithm is compared with that of the last algorithm for the two-operator SCFP and the advantage of the presented algorithm is shown through the numerical result.

1. Introduction

Throughout this paper, denotes a real Hilbert space with inner product and its induced norm , the identity mapping on , the set of all natural numbers, the set of all real numbers, and the metric projection onto set . is the upper bound of sequence , while is the lower bound. For a self-mapping on , Fix denotes the set of all fixed points of .

It has been an interesting topic of finding zero points of maximal monotone operators. A set-valued map with domain is called monotone if for all , and for any and , where is defined to be is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. For a positive real number , we denote by the resolvent of a monotone operator ; that is, for any . A point is called a zero point of a maximal monotone operator if . In the sequel, we will denote the set of all zero points of by , which is equal to for any . A well-known method to solve this problem is the proximal point algorithm which starts with any initial point and then generates the sequence in by where is a sequence of positive real numbers. This algorithm was first introduced by Martinet [1] and then generally studied by Rockafellar [2], who devised the iterative sequence by where is an error sequence in . Rockafellar showed that the sequence generated by (4) converges weakly to an element of provided that and . Since then, many authors have conducted research on modifying the sequence in (4) so that the strong convergence is guaranteed; compare [312] and the references therein.

On the other hand, let and be nonempty closed convex subsets of two Hilbert spaces and , respectively, and let be a bounded linear mapping. The split feasibility problem (SFP) is the problem of finding a point with the property: The SFP was first introduced by Censor and Elfving [13] for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. The most popular algorithm for the SFP is the algorithm introduced by Byrne [14, 15]. The sequence generated by the algorithm converges weakly to a solution of SFP (5); compare [1416]. Under the assumption that SFP (5) has a solution, there are many algorithms designed to approximate a solution of SFP; compare [1623] and the references therein.

Later, Censor and Segal [24] extended the SFP to the split common fixed point problem (SCFP) which is to find a point with the property: where , , and , , are directed operators in Hilbert spaces. Censor and Segal [24] gave an algorithm for SCFP (6) in spaces. Then, Moudafi [25] named SCFP (6) with the two-operator SCFP and gave an algorithm which generates a sequence weakly converging to the solution of the two-operator SCFP. Till very recently, Cui et al. [26] provided a damped projection algorithm, shown as below, to approach the solution of SCFP (6).

Assume that the solution set of the SCFP is nonempty. Start with any and generate a sequence through the iteration: where , , and satisfying that(i) and ;(ii);(iii).Then, the sequence converges strongly to .

Inspired by the work of [25, 26], this paper presents another algorithm to find the minimum norm solution of two-operator SCFP. We note that the two-operator SCFP contains the SFP and the zero point problem of maximal monotone operators. Let and be metric projections onto and , respectively. Putting and , the two-operator SCFP (6) is reduced to SFP (5). Let and be two maximal monotone operators on and , respectively. Replacing and with and , respectively, in (6), the SFP becomes a two-operator SCFP: Putting , the above two-operator SCFP is reduced to the common zero point problem of two maximal monotone operators and :

Let be in the SCFP (6), and let be . The target of the two-operator SCFP (6) is to find a fixed point of directed operator . Since the definition of a directed operator is based on its fixed point set, it may be difficult to show that is a directed operator before the two-operator SCFP is solved. Therefore, and are only considered as firmly nonexpansive mappings in our presented algorithm. The main result in this paper is as follows.

Let and be two firmly nonexpansive self-mappings on and , respectively. Assume that the solution set of the two-operator SCFP is nonempty. For any , start with any and define the sequence by where and and are sequences in satisfying that(i) and ;(ii).Then the sequence converges strongly to .

The two-operator SCFP covers problems of split feasibility, convex feasibility, and equilibrium as special cases. The presented algorithm can be considered as a unified methodology for solving the aforementioned problems. In Section 4, we use the numerical result to prove that the performance of the presented algorithm is more efficient and more consistent than that of the recent damped projection algorithm [26].

2. Preliminaries

In order to facilitate our investigation in this paper, we recall some basic facts. A mapping is said to be(i)nonexpansive if (ii)firmly nonexpansive if (iii)directed if It is well-known that the fixed point set of a nonexpansive mapping is closed and convex; compare [27].

Let be a nonempty closed convex subset of . The metric projection from onto is the mapping that assigns each the unique point in with the property It is known that is firmly nonexpansive and characterized by the inequality, for any ,

There is a strongly convergent algorithm for a nonexpansive mapping with , which is related to the iteration scheme in our main result; for any , choose arbitrarily a point and define a sequence recursively by where is sequence in satisfying Then, the sequence converges strongly to ; compare [28, 29].

We need some lemmas that will be quoted in the sequel.

Lemma 1. For any and , the following hold:(a);(b).

Lemma 2 (see [27], demiclosedness principle). Suppose that is a nonexpansive self-mapping on and suppose that is a sequence in such that converges weakly to some and . Then, .

Lemma 3. Let be a maximal monotone operator on . Then(a) is single-valued and firmly nonexpansive;(b) and .

Lemma 4 (see [12]). Suppose that is a sequence of nonnegative real numbers satisfying where and verify the following conditions:(i), ;(ii).Then .

Lemma 5 (see [30]). Let be a sequence in that does not decrease at infinity in the sense that there exists a subsequence such that For any , define . Then as and .

3. Main Theorems

Throughout this section, and denote two firmly nonexpansive self-mappings on and , respectively, and denotes a bounded linear operator from to .

Under the assumption that the solution set of two-operator SCFP is nonempty, the following lemma says that the two-operator SCFP is equivalent to the fixed point problem for the operator .

Lemma 6 (see [17]). Let be the solution set of two-operator (6); that is, . For any , let . Suppose that . Then .

Theorem 7. Let and be two firmly nonexpansive self-mappings on and , respectively. Assume that the solution set of the two-operator SCFP is nonempty. For any , start with any and define the sequence by where and and are sequences in satisfying that(i) and ;(ii).Then the sequence converges strongly to .

Proof. Putting , we see that . By Lemmas 1 and 6, we have In addition, Furthermore, since is nonexpansive and , one has from which it follows that Therefore, it follows from (21), (22), and (24) that Hence, by induction, we see that This shows that is bounded. Now, by Lemma 1 and (22), we have We now carry on with the proof by considering the following two cases: (I) is eventually decreasing and (II) is not eventually decreasing.
Case I. Suppose that is eventually decreasing; that is, there is such that is decreasing. In this case, exists in . From inequality (27), we have which together with the boundedness of and conditions (i) and (ii) implies Since is bounded, it has a subsequence such that converges weakly to some and where the last inequality follows from (15) since by Proposition 8 of [17], (29), and Lemmas 2 and 6. Moreover, from (27), we have
Accordingly, applying Lemma 4 to inequality (31), we conclude that
Case II. Suppose that is not eventually decreasing. In this case, by Lemma 5, there exists a nondecreasing sequence in such that and Then it follows from (27) and (33) that Therefore, which implies that and then it follows that From (35), we obtain and thus, letting , we obtain Also, since which together with (36) and conditions (i) and (ii) implies that , by virtue of (39). Consequently, we conclude that via (33) and (41). This completes the proof.

This theorem says that the sequence converges strongly to a point of which is nearest to . In particular, if is taken to be , then the limit point of the sequence is the unique minimum solution of two-operator SCFP (6).

Corollary 8. Let and be nonempty closed convex subsets of two Hilbert spaces and , respectively. Assume that the solution set of the SFP is nonempty. For any , start with any and define a sequence iteratively by where and and are sequences in satisfying that(i) and ;(ii).Then the sequence converges strongly to .

Proof. Putting and in (20), the conclusion follows from Theorem 7.

Corollary 9. Suppose that and are two maximal monotone operators on and , respectively. Assume that the solution set of problem is nonempty. Let . For any , start with any and define a sequence iteratively by where and and are sequences in satisfying that(i) and ;(ii).Then the sequence converges strongly to .

Proof. By Lemma 3, a resolvent of a maximal monotone operator is firmly nonexpansive. Hence, we may put and in (20) to get the conclusion which follows from Theorem 7.

Corollary 10. Let be a maximal monotone operator on with , and let . For any , start with any and define a sequence iteratively by where and and are sequences in satisfying that(i) and ;(ii).Then, the sequence converges strongly to .

Proof. Putting , , , and in Corollary 9, the result follows immediately.

4. Numerical Results

There are four examples in this section provided to demonstrate our presented algorithm. The first three examples are the SFP, while the fourth example is the common zero point problem of two maximal monotone operators. The performance of the presented algorithm to solve the three examples of SFP is compared with that of the recent damped projection method [26]. The result shows that the presented algorithm is more efficient and more consistent than the damped algorithm. In the first three examples, we assign the parameters in both algorithms to be , , , and . Let be their stop criterion. All codes were written in Matlab R2011a and ran on laptop ASUS ZenbookUX31E with i7-2677M CPU.

Example 11. Let , , and The metric projections for and are Then, we can use both the presented algorithm and the damped projection algorithm to approach a point such that From Table 1, we observe that the presented algorithm is more efficient than the damped projection algorithm.


The damped projection method in [26] The presented method
CPU (sec.) CPU (sec.)

67.4971 157248 34.8173 91018
125.352 328067 35.0441 91018
411.5836 1052792 38.6464 91018

Example 12. Let all conditions be the same with those in Example 11 except to The result for solving Example 12 is shown in Table 2. We observe that the presented algorithm is still more efficient than the damped algorithm. From the columns for the runtime (CPU) and the approximate solution (), the result of the presented algorithm is consistent although it starts from different initial points.


The damped projection method in [26] The presented method
CPU (sec.) CPU (sec.)

31.3902 84818 35.5024 91018
142.6763 362480 37.0838 91018
448.3774 1042364 33.8532 91031

Example 13. In this example, we use in Example 11 but change its and . Let and . The metric projections for and are
The result is shown in Table 3. We also observe that the presented algorithm is more efficient and more consistent than the damped projection algorithm.


The damped projection method in [26] The presented method
CPU (sec.) CPU (sec.)

382.487 933580 100.475 247651
581.5485 1438799 101.4875 247960
>1000 100.0661 252832

The presented algorithm contains an arbitrary point and that is an advantage of the algorithm. Knowing any information about the solution of two-operator SCFP of interest, we can choose a better to enhance the performance of the presented algorithm. For instance, let which is different with related to the result in Table 3. From Table 4, we observe that the runtime of the presented algorithm is reduced by one-third.


The presented method
CPU (sec.)

64.0093 159081
66.2390 159477
66.0244 172465

Example 14. Minimizing a convex function is called a convex minimization problem. This example shows that the presented algorithm can be used to search the common optimal solutions of two convex minimization problems. Let and be two functions from to and define and . We know that both and are convex functions. Now, we would like to search a common minimal point of the two convex functions.
Let denote the partial derivative of function with respect to . Define two operators and from to by Since and are convex functions, and are maximal monotone operators and any one of their common zero points is the common minimal point of and . The resolvents of and are According to Corollary 9, our presented algorithm can be used to search a common zero point of and . Let , , , , and in the algorithm, and let be the stop criterion. We ran the algorithm and started from point . The algorithm stopped at point after iterations. We know that and . Finally, we use Figure 1 to show the behavior of sequence which converges to the common minimal point of and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Franćaise d'Informatique et de Recherche Opérationnelle, vol. 4, pp. 154–158, 1970. View at: Google Scholar | MathSciNet
  2. R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. O. A. Boikanyo and G. Moroşanu, “Inexact Halpern-type proximal point algorithm,” Journal of Global Optimization, vol. 51, no. 1, pp. 11–26, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  4. O. A. Boikanyo and G. Moroşanu, “Four parameter proximal point algorithms,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 2, pp. 544–555, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  5. O. A. Boikanyo and G. Moroşanu, “A proximal point algorithm converging strongly for general errors,” Optimization Letters, vol. 4, no. 4, pp. 635–641, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  6. S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,” Journal of Approximation Theory, vol. 106, no. 2, pp. 226–240, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. G. Marino and H. K. Xu, “Convergence of generalized proximal point algorithms,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 791–808, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  8. M. V. Solodov and B. F. Svaiter, “Forcing strong convergence of proximal point iterations in a Hilbert space,” Mathematical Programming, vol. 87, no. 1, pp. 189–202, 2000. View at: Google Scholar | MathSciNet
  9. S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 147, no. 1, pp. 27–41, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. F. Wang and H. Cui, “On the contraction-proximal point algorithms with multi-parameters,” Journal of Global Optimization, vol. 54, no. 3, pp. 485–491, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  11. H. K. Xu, “A regularization method for the proximal point algorithm,” Journal of Global Optimization, vol. 36, no. 1, pp. 115–125, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  12. H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  13. 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. View at: Publisher Site | Google Scholar | MathSciNet
  14. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  15. 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. View at: Publisher Site | Google Scholar | MathSciNet
  16. H. K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 17 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  17. Y. Y. Huang and C. C. Hong, “A unified iterative treatment for solutions of problems of split feasibility and equilibrium in Hilbert spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 613928, 13 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  18. Y. Y. Huang and C. C. Hong, “Approximating common fixed points of averaged self-mappings with applications to split feasibility problem and maximal monotone operators in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2013, article 190, 2013. View at: Publisher Site | Google Scholar
  19. E. Masad and S. Reich, “A note on the multiple-set split convex feasibility problem in Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 367–371, 2007. View at: Google Scholar | MathSciNet
  20. J. Quan, S. S. Chang, and X. Zhang, “Multiple-set split feasibility problems for κ-strictly pseudononspreading mapping in Hilbert spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 342545, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  21. F. Wang and H. Xu, “Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem,” Journal of Inequalities and Applications, Article ID 102085, 13 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  22. Y. Yao, J. Wu, and Y. C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 140679, 13 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  23. Y. Yao, Y. C. Liou, and N. Shahzad, “A strongly convergent method for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 125046, 15 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  24. Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587–600, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  25. A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems, vol. 26, no. 5, Article ID 055007, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  26. H. Cui, M. Su, and F. Wang, “Damped projection method for split common fixed point problems,” Journal of Inequalities and Applications, vol. 20113, article 123, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  27. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at: Publisher Site | MathSciNet
  28. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  29. R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  30. P. E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2014 Chung-Chien Hong and Young-Ye Huang. 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.


More related articles

673 Views | 514 Downloads | 1 Citation
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.