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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 414031, 5 pages
http://dx.doi.org/10.1155/2014/414031
Research Article

On an Iterative Method for Finding a Zero to the Sum of Two Maximal Monotone Operators

1School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang 453003, China
2Department of Mathematics, Luoyang Normal University, Luoyang 471022, China

Received 2 March 2014; Revised 18 August 2014; Accepted 19 August 2014; Published 21 August 2014

Academic Editor: Luigi Muglia

Copyright © 2014 Hongwei Jiao and Fenghui Wang. 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.

Linked References

  1. J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Mathematical Programming, vol. 55, no. 3, pp. 293–318, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. P. L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM Journal on Numerical Analysis, vol. 16, no. 6, pp. 964–979, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  3. T. Pennanen, “A splitting method for composite mappings,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 875–890, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. E. Spingarn, “Applications of the method of partial inverses to convex programming: decomposition,” Mathematical Programming, vol. 32, no. 2, pp. 199–223, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. P. Tseng, “Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,” Mathematical Programming, vol. 48, pp. 249–263, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. P. Tseng, “Applications of a splitting algorithm to decomposition in convex programming and variational inequalities,” SIAM Journal on Control and Optimization, vol. 29, no. 1, pp. 119–138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. D. W. Peaceman and J. Rachford, “The numerical solution of parabolic and elliptic differential equations,” Society for Industrial and Applied Mathematics, vol. 3, no. 1, pp. 28–41, 1955. View at Google Scholar · View at MathSciNet
  8. J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two or three space variables,” Transactions of the American Mathematical Society, vol. 82, pp. 421–439, 1956. View at Google Scholar
  9. G. B. Passty, “Ergodic convergence to a zero of the sum of monotone operators,” Journal of Mathematical Analysis and Applications, vol. 72, no. 2, pp. 383–390, 1979. View at Google Scholar · View at Scopus
  10. P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,” Optimization, vol. 53, no. 5-6, pp. 475–504, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. G. López, V. Martín-Márquez, F. Wang, and H.-K. Xu, “Forward-backward splitting methods for accretive operators in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 109236, 25 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Butnariu, Y. Censor, and S. Reich, Quasi-Fejérian Analysis of Some Optimization Algorithms, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Elsevier Science Publishers, Amsterdam, The Netherlands, 2001.
  14. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977. View at Publisher · View at Google Scholar · View at Scopus