Skip to main content
SpringerLink
Log in
Book cover

Applications of Nonlinear Analysis pp 175–191Cite as

General Inertial Mann Algorithms and Their Convergence Analysis for Nonexpansive Mappings

  • Chapter
  • First Online:

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 134))

Abstract

In this article, we introduce general inertial Mann algorithms for finding fixed points of nonexpansive mappings in Hilbert spaces, which includes some other algorithms as special cases. We reanalyze the accelerated Mann algorithm, which actually is an inertial type Mann algorithm. We investigate the convergence of the general inertial Mann algorithm, based on which, the strict convergence condition on the accelerated Mann algorithm is eliminated. Also, we apply the general inertial Mann algorithm to show the existence of solutions of the minimization problems by proposing a general inertial type gradient-projection algorithm. Finally, we give preliminary experiments to illustrate the advantage of the accelerated Mann algorithm.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14, 773–782 (2004)

    Article  MathSciNet  Google Scholar 

  2. F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)

    Google Scholar 

  3. H.H. Bauschke, J.M. Borwein, On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  Google Scholar 

  4. H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, Berlin, 2011)

    Google Scholar 

  5. A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  Google Scholar 

  6. R.I. Bot, E.R. Csetnek, A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36, 951–963 (2015)

    Article  MathSciNet  Google Scholar 

  7. R.I. Bot, E.R. Csetnek, C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)

    Article  MathSciNet  Google Scholar 

  8. A. Chambolle, C. Dossal, On the convergence of the iterates of the “fast iterative shrinkage/thresholding algorithm”. J. Optim. Theory Appl. 166, 968–982 (2015)

    Article  MathSciNet  Google Scholar 

  9. P. Chen, J. Huang, X. Zhang, A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Prob. 29, 025011 (33 pp.) (2013)

    Article  MathSciNet  Google Scholar 

  10. P. Chen, J. Huang, X. Zhang, A primal-dual fixed point algorithm for minimization of the sum of three convex separable functions. Fixed Point Theory Appl. 2016, 54 (2016)

    Google Scholar 

  11. Q.L. Dong, Y.Y. Lu, A new hybrid algorithm for a nonexpansive mapping. Fixed Point Theory Appl. 2015, 37 (2015)

    Google Scholar 

  12. Q.L. Dong, H.Y. Yuan, Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping. Fixed Point Theory Appl. 2015, 125 (2015)

    Google Scholar 

  13. Q.L. Dong, S. He, Y.J. Cho, A new hybrid algorithm and its numerical realization for two nonexpansive mappings. Fixed Point Theory Appl. 2015, 150 (2015)

    Google Scholar 

  14. Q.L. Dong, Y.Y. Lu, J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65(12), 2217–2226 (2016)

    Article  MathSciNet  Google Scholar 

  15. B. Halpern, Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  Google Scholar 

  16. S. He, C. Yang, Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory Appl. 2014, 56 (2014)

    Google Scholar 

  17. H. Iiduka, Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation. SIAM J. Optim. 22, 862–878 (2012)

    Article  MathSciNet  Google Scholar 

  18. H. Iiduka, Fixed point optimization algorithms for distributed optimization in networked systems. SIAM J. Optim. 23, 1–26 (2013)

    Article  MathSciNet  Google Scholar 

  19. S. Ishikawa, Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)

    Article  MathSciNet  Google Scholar 

  20. M.A. Krasnoselskii, Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123–127 (1955)

    Google Scholar 

  21. D.A. Lorenz, T. Pock, An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)

    Article  MathSciNet  Google Scholar 

  22. W.R. Mann, Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  Google Scholar 

  23. P.E. Mainge, Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)

    Article  MathSciNet  Google Scholar 

  24. P.E. Mainge, Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure. Comput. Math. Appl. 72, 720–728 (2016)

    Article  MathSciNet  Google Scholar 

  25. P.E. Mainge, M.L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)

    Article  MathSciNet  Google Scholar 

  26. Y. Malitski, Projected reflected gradient method for variational inequalities. SIAM J. Optim. 25, 502–520 (2015)

    Article  MathSciNet  Google Scholar 

  27. C.A. Micchelli, L. Shen, Y. Xu, Proximity algorithms for image models: denoising. Inverse Prob. 27, 45009–45038 (2011)

    Article  MathSciNet  Google Scholar 

  28. A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method formonotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)

    Article  MathSciNet  Google Scholar 

  29. J. Nocedal, S.J. Wright, Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, Berlin, 2006)

    Google Scholar 

  30. P.M. Pardalos, T.M. Rassias, Contributions in Mathematics and Engineering; In Honor of Constantin Caratheodory (Springer, Berlin, 2016)

    Google Scholar 

  31. P.M. Pardalos, P.G. Georgiev, H.M. Srivastava, Nonlinear Analysis, Stability, Approximation, and Inequalities; In Honor of Themistocles M. Rassias on the Occasion of his 60th Birthday (Springer, Berlin, 2012)

    Google Scholar 

  32. E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pures et Appl. 6, 145–210 (1890)

    Google Scholar 

  33. B.T. Polyak, Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Math. Phys. 4, 1–17 (1964)

    Article  Google Scholar 

  34. B.T. Polyak, Introduction to Optimization (Optimization Software Inc., Publications Division, New York, 1987)

    Google Scholar 

  35. T.M. Rassias, L. Toth, Topics in Mathematical Analysis and Applications (Springer, Berlin, 2014)

    Book  Google Scholar 

  36. K. Sakurai, H. Liduka, Acceleration of the Halpern algorithm to search for a fixed point of a nonexpansive mapping. Fixed Point Theory Appl. 2014, 202 (2014)

    Article  MathSciNet  Google Scholar 

  37. H.K. Xu, Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)

    Article  MathSciNet  Google Scholar 

  38. C. Yang, S. He, General alternative regularization methods for nonexpansive mappings in Hilbert spaces. Fixed Point Theroy Appl. 2014, 203 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

The first author would like to thank Professor Chunlin Wu for the helpful discussion and valuable suggestions. We wish to express our thanks to Professor Mihai Turinici for reading the paper and providing very helpful comments.

Author information

Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, China

    Qiao-Li Dong

  2. Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, South Korea

    Yeol Je Cho

  3. Center for General Education, China Medical University, Taichung, Taiwan

    Yeol Je Cho

  4. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Authors
  1. Qiao-Li Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yeol Je Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Themistocles M. Rassias
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yeol Je Cho .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dong, QL., Cho, Y.J., Rassias, T.M. (2018). General Inertial Mann Algorithms and Their Convergence Analysis for Nonexpansive Mappings. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation