Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Modified forward–backward splitting method for variational inclusions

  • Research Paper
  • Published:
4OR Aims and scope Submit manuscript

Abstract

In this paper we propose an explicit algorithm for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the other is monotone and Lipschitz continuous. The algorithm uses the variable stepsizes which are updated over each iteration by some cheap comptutations. These stepsizes are found without the prior knowledge of the Lipschitz constant of operator as well as without using lineseach procedure. The algorithm thus can be implemented easily. The convergence and the convergence rate of the algorithm are established under mild conditions. Several preliminary numerical results are provided to demonstrate the theoretical results and also to compare the new algorithm with some existing ones.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Anh PK, Vinh NT, Dung VT (2018) A new self-adaptive CQ algorithm with an application to the LASSO problem. J Fixed Point Theory Appl 142:1–19

    Google Scholar 

  • Attouch H, Peypouquet J, Redont P (2018) Backward–forward algorithms for structured monotone inclusions in Hilbert spaces. J Math Anal Appl 457:1095–1117

    Article  Google Scholar 

  • Brézis H, Chapitre II (1973) Operateurs maximaux monotones. North-Holland Math Stud 5:19–51

    Article  Google Scholar 

  • Bruck R (1977) On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J Math Anal Appl 61:159–164

    Article  Google Scholar 

  • Censor Y, Elfving T (1994) A multi-projection algorithm using Bregman projections in a product space. Numer Algorithms 8:221–239

    Article  Google Scholar 

  • Chen GH-G, Rockafellar T (1997) Convergence rates in forward–backward splitting. SIAM J Optim 7:421–444

    Article  Google Scholar 

  • Chen S, Donoho DL, Saunders M (1998) Atomic decomposition by basis pursuit. SIAM J Sci Comput. 20:33–61

    Article  Google Scholar 

  • Combettes PL, Wajs V (2005) Signal recovery by proximal forward–backward splitting. SIAM Multisc Model Simul 4:1168–1200

    Article  Google Scholar 

  • Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57:1413–1457

    Article  Google Scholar 

  • Davis D, Yin WT (2017) A three-operator splitting scheme and its optimization applications. Set-Valued Var Anal 25:829–858

    Article  Google Scholar 

  • Dong YD, Fischer A (2010) A family of operator splitting methods revisited. Nonlinear Anal 72:4307–4315

    Article  Google Scholar 

  • Duchi J, Singer Y (2009) Efficient online and batch learning using forward–backward splitting. J Mach Learn Res 10:2899–2934

    Google Scholar 

  • Facchinei F, Pang JS (2002) Finite-dimensional variational inequalities and complementarity problems. Springer, Berlin

    Google Scholar 

  • Gibali A, Hieu DV (2019) A new inertial double-projection method for solving variational inequalities. J Fixed Point Theory Appl. https://doi.org/10.1007/s11784-019-0726-7

    Article  Google Scholar 

  • Gibali A, Thong DV (2018) Tseng type methods for solving inclusion problems and its applications. Calcolo. https://doi.org/10.1007/s10092-018-0292-1

    Article  Google Scholar 

  • Goldstein AA (1964) Convex programming in Hilbert spaces. Bull Am Math Soc 70:709–710

    Article  Google Scholar 

  • Hieu DV, Quy PK (2019) An inertial modified algorithm for solving variational inequalities. RAIRO Oper Res. https://doi.org/10.1051/ro/2018115

    Article  Google Scholar 

  • Hieu DV, Anh PK, Muu LD (2017) Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl 66:75–96

    Article  Google Scholar 

  • Hieu DV, Anh PK, Muu LD (2019a) Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl 73:913–932

    Article  Google Scholar 

  • Hieu DV, Cho YJ, Xiao Y-B (2019b) Golden ratio algorithms with new stepsize rules for variational inequalities. Math Methods Appl Sci. https://doi.org/10.1002/mma.5703

    Article  Google Scholar 

  • Hieu DV, Cho Xiao YB, Kumam P (2019c) Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces. Optimization. https://doi.org/10.1080/02331934.2019.1683554

    Article  Google Scholar 

  • Hieu DV, Vy LV, Quy PK (2019) Three-operator splitting algorithm for a class of variational inclusion problems. Bull Iran Math Soc. https://doi.org/10.1007/s41980-019-00312-5

    Article  Google Scholar 

  • Hieu DV, Strodiot JJ, Muu LD (2020a) An explicit extragradient algorithm for solving variational inequalities. J Optim Theory Appl. https://doi.org/10.1007/s10957-020-01661-6

    Article  Google Scholar 

  • Hieu DV, Strodiot JJ, Muu LD (2020b) Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. J Comput Appl Math. https://doi.org/10.1016/j.cam.2020.112844

    Article  Google Scholar 

  • Huang YY, Dong YD (2014) New properties of forward–backward splitting and a practical proximal descent algorithm. Appl Math Comput 237:60–68

    Google Scholar 

  • Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, New York

    Google Scholar 

  • Lions PL, Mercier B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal 16:964–979

    Article  Google Scholar 

  • Malitsky Y, Tam MK (2018) A Forward-Backward splitting method for monotone inclusions without cocoercivity. arXiv:1808.04162

  • Monteiro RDC, Svaiter BF (2011) Complexity of variants of Tseng’s modified F–B splitting and Korpelevich’s methods for hemivariational inequalities with applications to saddle-point and convex optimization problems. SIAM J Optim 21:1688–1720

    Article  Google Scholar 

  • Passty GB (1979) Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J Math Anal Appl 72:383–390

    Article  Google Scholar 

  • Raguet H, Fadili J, Peyré G (2013) A generalized forward–backward splitting. SIAM J Imaging Sci 6:1199–1226

    Article  Google Scholar 

  • Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14:877–898

    Article  Google Scholar 

  • Ryu EK, Boyd S (2016) A primer on monotone operator methods. Appl Comput Math 15:3–43

    Google Scholar 

  • Takahashi W (2000) Nonlinear functional analysis-fixed point theory and its applications. Yokohama Publishers, Yokohama

    Google Scholar 

  • Tibshirami R (1996) Regression shrinkage and selection via lasso. J R Stat Soc Ser B 58:267–288

    Google Scholar 

  • Tseng P (2000) A modified forward–backward splitting method for maximal monotone mappings. SIAM J Control Optim 38:431–446

    Article  Google Scholar 

  • Zong C, Tang Y, Cho YJ (2018) Convergence analysis of an inexact three-operator splitting algorithm. Symmetry. https://doi.org/10.3390/sym10110563

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first-named and last-named authors are supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project: 101.01-2020.06.

Author information

Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Dang Van Hieu

  2. Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Pham Ky Anh

  3. TIMAS, Thang Long University, Hanoi, Vietnam

    Le Dung Muu

Authors
  1. Dang Van Hieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Ky Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Le Dung Muu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Van Hieu.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Hieu, D., Anh, P.K. & Muu, L.D. Modified forward–backward splitting method for variational inclusions. 4OR-Q J Oper Res 19, 127–151 (2021). https://doi.org/10.1007/s10288-020-00440-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10288-020-00440-3

Keywords

Mathematics Subject Classification

Search

Navigation