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Abstract and Applied Analysis
Volume 2016, Article ID 2371857, 10 pages
http://dx.doi.org/10.1155/2016/2371857
Research Article

The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Private Bag Box 16, Palapye, Botswana

Received 8 September 2015; Accepted 8 December 2015

Academic Editor: Sergei V. Pereverzyev

Copyright © 2016 Oganeditse Aaron Boikanyo. 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 B. F. Svaiter, “A family of projective splitting methods for the sum of two maximal monotone operators,” Mathematical Programming B, vol. 111, no. 1-2, pp. 173–199, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J. Eckstein and B. F. Svaiter, “General projective splitting methods for sums of maximal monotone operators,” SIAM Journal on Control and Optimization, vol. 48, no. 2, pp. 787–811, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. 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
  4. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. G.-J. Tang and N.-J. Huang, “Strong convergence of a splitting proximal projection method for the sum of two maximal monotone operators,” Operations Research Letters, vol. 40, no. 5, pp. 332–336, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. 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
  7. G. B. Passty, “Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 72, no. 2, pp. 383–390, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. P. Tseng, “A modified forward-backward splitting method for maximal monotone mappings,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 431–446, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. B. Mercier, Inéquations Variationnnelles de la Mécanique, Publications Mathématiques d'Orsay 80.01, Université Paris-Sud, Orsay, France, 1980.
  10. D. Gabay, “Applications of the method of multipliers to variational inequalities,” in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, M. Fortin and R. Glowinski, Eds., pp. 229–331, North-Holland Publishing, Amsterdam, The Netherlands, 1983. View at Google Scholar
  11. G. H.-G. Chen, Forward-backward splitting techniques: theory and applications [Ph.D. thesis], Department of Applied Mathematics, University of Washington, Seattle, Wash, USA, 1994.
  12. B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Francaise d'Informatique et de Recherche Opérationnelle, vol. 3, pp. 154–158, 1970. View at Google Scholar · View at MathSciNet
  13. 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
  14. O. Güler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 403–419, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  16. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  17. N. Lehdili and A. Moudafi, “Combining the proximal algorithm and Tikhonov regularization,” Optimization, vol. 37, no. 3, pp. 239–252, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  18. 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 Publisher · View at Google Scholar
  19. W. Takahashi, “Approximating solutions of accretive operators by viscosity approximation methods in Banach spaces,” in Applied Functional Analysis, pp. 225–243, Yokohama Publishers, Yokohama, Japan, 2007. View at Google Scholar
  20. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Y. Yao and M. A. Noor, “On convergence criteria of generalized proximal point algorithms,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 46–55, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 6, no. 4, pp. 621–628, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. J. Eckstein and M. C. Ferris, “Operator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control,” INFORMS Journal on Computing, vol. 10, no. 2, pp. 218–235, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. F. Wang and H. Cui, “Convergence of the generalized contraction-proximal point algorithm in a Hilbert space,” Optimization, vol. 64, no. 4, pp. 709–715, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. C. Tian and F. Wang, “The contraction-proximal point algorithm with square-summable errors,” Fixed Point Theory and Applications, vol. 2013, article 93, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  28. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  29. G. Moroşanu, Nonlinear Evolution Equations and Applications, vol. 26 of Mathematics and Its Applications, Reidel, Dordrecht, The Netherlands, 1988. View at 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 · View at Google Scholar · View at MathSciNet · View at Scopus