Skip to main content
SpringerLink
Log in

Inertial projection methods for finding a minimum-norm solution of pseudomonotone variational inequality and fixed-point problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this work, we introduce two new iterative methods for finding the common element of the set of fixed points of a demicontractive mapping and the set of solutions of a pseudomonotone variational inequality problem in real Hilbert spaces. It is shown that the proposed algorithms converge strongly under mild conditions. The advantage of the proposed algorithms is that it does not require prior knowledge of the Lipschitz-type constants of the variational inequality mapping and only requires computing one projection onto a feasible set per iteration as well as without the sequentially weakly continuity of the variational inequality mapping. The novelty of the proposed algorithm may improve the efficiency of algorithms. Our results improve and extend some known results existing in the literature.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Bot RI, Csetnek ER, Vuong PT (2020) The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces. Eur J Oper Res 287:49–60

    MathSciNet  MATH  Google Scholar 

  • Ceng LC, Hadjisavvas N, Wong NC (2010a) Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Glob Optim 46:635–646

    MathSciNet  MATH  Google Scholar 

  • Ceng LC, Teboulle M, Yao JC (2010b) Weak convergence of an iterative method for pseudomonotone variational inequalities and fixedpoint problems. J Optim Theory Appl 146:19–31

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S (2011) The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl 148:318–335

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S (2011) Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw 26:827–845

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Algor 59:301–323

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S (2012) Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61:1119–1132

    MathSciNet  MATH  Google Scholar 

  • Chidume CE, Maruster S (2010) Iterative methods for the computation of fixed points of demicontractive mappings. J Comput Appl Math 234:861–882

    MathSciNet  MATH  Google Scholar 

  • Cottle RW, Yao JC (1992) Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl 75:281–295

    MathSciNet  MATH  Google Scholar 

  • Denisov SV, Semenov VV, Chabak LM (2015) Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal 51:757–765

    MathSciNet  MATH  Google Scholar 

  • Dong Q, Jiang D, Gibali A (2018) A modified subgradient extragradient method for solving the variational inequality problem. Numer Algor 79:927–940

    MathSciNet  MATH  Google Scholar 

  • Goebel K, Reich S (1984) Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York

    MATH  Google Scholar 

  • Hicks TL, Kubicek JD (1997) On the Mann iteration process in a Hilbert space. J Math Anal Appl 59:498–504

    MathSciNet  MATH  Google Scholar 

  • Iiduka H (2015) Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping. Math Prog Ser A 149:131–165

    MathSciNet  MATH  Google Scholar 

  • Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12:747–756

    MathSciNet  MATH  Google Scholar 

  • Kraikaew R, Saejung S (2014) Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl 163:399–412

    MathSciNet  MATH  Google Scholar 

  • Li M, Yao Y (2010) Strong convergence of an iterative algorithm for \(\lambda \)-strictly pseudo-contractive mappings in Hilbert spaces. An St Univ Ovidius Constanta 18:219–228

    MathSciNet  MATH  Google Scholar 

  • Linh HM, Reich S, Thong DV et al (2022) Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems. Numer Algor 89:1695–1721

    MathSciNet  MATH  Google Scholar 

  • Maingé PE (2008) A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim 47:1499–1515

    MathSciNet  MATH  Google Scholar 

  • Malitsky YV (2015) Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim 25:502–520

    MathSciNet  MATH  Google Scholar 

  • Malitsky YV, Semenov VV (2015) A hybrid method without extrapolation step for solving variational inequality problems. J Glob Optim 61:193–202

    MathSciNet  MATH  Google Scholar 

  • Mann WR (1953) Mean value methods in iteration. Proc Amer Math Soc 4:506–510

    MathSciNet  MATH  Google Scholar 

  • Mongkolkeha C, Cho YJ, Kumam P (2013) Convergence theorems for \(k\)-dimeicontactive mappings in Hilbert spaces. Math Inequal Appl 16:1065–1082

    MathSciNet  MATH  Google Scholar 

  • Nadezhkina N, Takahashi W (2006) Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 128:191–201

    MathSciNet  MATH  Google Scholar 

  • Nadezhkina N, Takahashi W (2006) Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J Optim 16:1230–1241

    MathSciNet  MATH  Google Scholar 

  • Reich S (1979) Constructive techniques for accretive and monotone operators. Applied nonlinear analysis. Academic Press, New York, pp 335–345

    Google Scholar 

  • Reich S, Thong DV, Cholamjiak P et al (2021) Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer Algor 88:813–835

    MathSciNet  MATH  Google Scholar 

  • Saejung S, Yotkaew P (2012) Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal 75:742–750

    MathSciNet  MATH  Google Scholar 

  • Shehu Y, Cholamjiak P (2016) Another look at the split common fixed point problem for demicontractive operators. Rev R Acad Cienc Exactas Fís Nat Ser A Math RACSAM 110:201–218

    MathSciNet  MATH  Google Scholar 

  • Shehu Y, Iyiola OS (2018) Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer Algor 79:529–553

    MathSciNet  MATH  Google Scholar 

  • Shehu Y, Dong QL, Jiang D (2019) Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68:385–409

    MathSciNet  MATH  Google Scholar 

  • Solodov MV, Svaiter BF (1999) A new projection method for variational inequality problems. SIAM J Control Optim 37:765–776

    Article  MathSciNet  Google Scholar 

  • Takahashi W, Toyoda M (2003) Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 118:417–428

    MathSciNet  MATH  Google Scholar 

  • Thong DV (2017) Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J Fixed Point Theory Appl 19:1481–1499

    MathSciNet  MATH  Google Scholar 

  • Thong DV, Hieu DV (2018) Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67:83–102

    MathSciNet  MATH  Google Scholar 

  • Vuong PT (2018) On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theory Appl 176:399–409

    MathSciNet  MATH  Google Scholar 

  • Vuong PT, Shehu Y (2019) Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer Algor 81:269–291

    MathSciNet  MATH  Google Scholar 

  • Yamada I, Ogura N (2004) Hybrid steepest descent method for variational inequality operators over the problem certain fixed point set of quasi-nonexpansive mappings. Numer Funct Anal Optim 25:619–655

    MathSciNet  MATH  Google Scholar 

  • Yang J, Liu H (2019) Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algor 80:741–752

    MathSciNet  MATH  Google Scholar 

  • Yao Y, Postolache M (2012) Iterative methods for pseudomonotone variational inequalities and fixed point problems. J Optim Theory Appl 155:273–287

    MathSciNet  MATH  Google Scholar 

  • Yao Y, Marino G, Muglia L (2014) A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63:559–569

    MathSciNet  MATH  Google Scholar 

  • Zhao X, Yao Y (2020) Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems. Optimization 69:1987–2002

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two referees for their valuable comments and suggestions which helped us very much in improving and presenting the original version of this paper.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong, Vietnam

    Duong Viet Thong

  2. Department of Mathematics, University of Science, Vietnam National University, 34 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Vu Tien Dung

  3. Faculty of Mathematical Economics, National Economics University, Hanoi, Vietnam

    Luong Van Long

Authors
  1. Duong Viet Thong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vu Tien Dung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luong Van Long
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duong Viet Thong.

Additional information

Communicated by Andreas Fischer.

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

Thong, D.V., Dung, V.T. & Long, L.V. Inertial projection methods for finding a minimum-norm solution of pseudomonotone variational inequality and fixed-point problems. Comp. Appl. Math. 41, 254 (2022). https://doi.org/10.1007/s40314-022-01958-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01958-4

Keywords

Mathematics Subject Classification

Search

Navigation