Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

A MODIFIED KRASNOSELSKII-TYPE SUBGRADIENT EXTRAGRADIENT ALGORITHM WITH INERTIAL EFFECTS FOR SOLVING VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEM

  • Araya Kheawborisut (Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang) ;
  • Wongvisarut Khuangsatung (Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi)
  • Received : 2023.05.15
  • Accepted : 2024.02.12
  • Published : 2024.06.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we propose a new inertial subgradient extragradient algorithm with a new linesearch technique that combines the inertial subgradient extragradient algorithm and the KrasnoselskiiMann algorithm. Under some suitable conditions, we prove a weak convergence theorem of the proposed algorithm for finding a common element of the common solution set of a finitely many variational inequality problem and the fixed point set of a nonexpansive mapping in real Hilbert spaces. Moreover, using our main result, we derive some others involving systems of variational inequalities. Finally, we give some numerical examples to support our main result.

Keywords

Acknowledgement

The second author would like to thank Rajamangala University of Technology Thanyaburi (RMUTT) under The Science, Research and Innovation Promotion Funding (TSRI) (Contract No. FRB660012/0168 and under project number FRB66E0635) for financial support.

References

  1. J. A. Abuchu, G. C. Ugunnadi and O. K. Narain, inertial proximal and contraction methods for solving monotone variational inclusion and fixed point problems, Nonlinear Funct. Anal. Appl., 28(1) (2023), 175-203.
  2. T.O. Alakoya, L.O. Jolaoso and O.T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70(3) (2021), 545-574. https://doi.org/10.1080/02331934.2020.1723586
  3. A. Auslender, M. Teboulle and S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Opti. Appl., 12 (1999), 31-40. https://doi.org/10.1023/A:1008607511915
  4. H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Math. Stud., (Amsterdam: North-Holand), 5 (1973), 759-775.
  5. L.C. Ceng and M. Shang, Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings, Optimization, 70(4) (2021), 715-740. https://doi.org/10.1080/02331934.2019.1647203
  6. L.C. Ceng and Q. Yuan, Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems, J. Ineq. Appl., 274 (2019).
  7. Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Opti. Theory Appl. 148 (2011), 318-335. https://doi.org/10.1007/s10957-010-9757-3
  8. W. Cholamjiak, P. Cholamjiak, and S. Suantai, An inertial forwardbackward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl, 20(42) (2018).
  9. A. Dixit, D.R. Sahu, P. Gautam and T. Som, Convergence analysis of two-step inertial Douglas-Rachford algorithm and application, J. Appl. Math. Comput,, 68 (2022), 953-977. https://doi.org/10.1007/s12190-021-01554-5
  10. Y. Dong, New inertial factors of the Krasnoselski-Mann iteration, Set-Valued Var. Anal, 29 (2021), 145-161. https://doi.org/10.1007/s11228-020-00541-5
  11. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
  12. D.V. Hieu, P.K. Anh and L.D. Muu, Strong convergence of subgradient extragradient method with regularization for solving variational inequalities, Opti. Eng., (2020), https://doi.org/10.1007/s11081-020-09540-9.
  13. O.S. Iyiola and Y. Shehu, New Convergence Results for Inertial KrasnoselskiiMann Iterations in Hilbert Spaces with Applications, Results Math. 76(75) (2021).
  14. A. Kangtunyakarn, A new iterative scheme for fixed point problems of infinite family of ki-pseudo contractive mappings, equilibrium problem, variational inequality problems, J. Glob. Optim., 56 (2013), 1543-1562. https://doi.org/10.1007/s10898-012-9925-0
  15. A. Kangtunyakarn, An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and the set of solutions for a modified system of variational inequalities, Fixed Point Theory Appl., 143 (2013).
  16. C. Kanzow and Y. Shehu, Generalized Krasnoselskii-Mann-type iterations for nonexpansive mappings in Hilbert spaces, Comput. Opti. Appl., 67 (2017), 595-620. https://doi.org/10.1007/s10589-017-9902-0
  17. A. Kheawborisut and A. Kangtunyakarn, Modified subgradient extragradient method for system of variational inclusion problem and finite family of variational inequalities problem in real Hilbert space, J Ineq. Appl., 53 (2021).
  18. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, SIAM, Soc. Indust. Appl. Math.Philadelphia, 2000.
  19. P. Kitisak, W. Cholamjiak, D. Yambangwai and R. Jaidee, A modified parallel hybrid subgradient extragradient method for finding common solutions of variational inequality problems, Thai J. Math., 18(1),(2020), 261-274 .
  20. G.M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody, 12 (1976), 747-756.
  21. R. Lotfikar, G. Zamani Eskandani and J. K. Kim, The subgradient extragradient method for solving monotone bilevel equilibrium problems using Bregman distance, Nonlinear Funct. Anal. Appl., 28(2) (2023), 337-363.
  22. S. Migrski, Ch. Fang and Sh. Zeng, A new modified subgradient extragradient method for solving variational inequalities,, Applicable Analysis, 100(1) (2021), 135-144. https://doi.org/10.1080/00036811.2019.1594202
  23. A.E. Ofem, A.A. Mebawondu, C. Agbonkhese, G.C. Ugwunnadi and O.K. Narain, Alternated inertial relaxed Tseng method for solving fixed point and quasi-monotone variational inequality problems, Nonlinear Funct. Anal. Appl., 29(1) (2024), 13-164
  24. K. Saechou and A. Kangtunyakarn, The subgradient extragradient method for approximation of fixed-point problem and modification of equilibrium problem, Optimization, 72(2) (2023), 283-410. https://doi.org/10.1080/02331934.2021.1970751
  25. Y. Shehu, Convergence Rate Analysis of Inertial KrasnoselskiiMann Type Iteration with Applications, Nume. Funct. Anal. Opti., 39(10) (2018), 1077-1091. https://doi.org/10.1080/01630563.2018.1477799
  26. Y. Shehu, O.S. Iyiola and S. Reich, A modified inertial subgradient extragradient method for solving variational inequalities, Opti. Eng., 23 (2022), 421-449. https://doi.org/10.1007/s11081-020-09593-w
  27. Y. Shehu, O.S. Iyiola, D.V. Thong and N.T.C. Van, An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems, Math. Meth. Oper. Res., 93 (2021), 213-242. https://doi.org/10.1007/s00186-020-00730-w
  28. G. Stampacchia, Formes bilineaires coercitives surles ensembles convexes, C. R. Acad. Sci, 258 (1964), 4413-4416.
  29. B. Tan, P. Sunthrayuth, P. Cholamjiak and Y. Je Cho, Modified inertial extragradient methods for finding minimum-norm solution of the variational inequality problem with applications to optimal control problem, Int. J. Comput. Math., 100(3) (2023), 525-545. https://doi.org/10.1080/00207160.2022.2137672
  30. D.V. Thong, P. Cholamjiak, M.T. Rassias, Y.J. Cho, Strong convergence of inertial subgradient extragradient algorithm for solving pseudomonotone equilibrium problems, Opti. Lett., 16 (2022), 545-573. https://doi.org/10.1007/s11590-021-01734-z
  31. D.V. Thong and D.V. Hieu, Inertial subgradient extragradient algorithms with linesearch process for solving variational inequality problems and fixed point problems, Numer Algor., 80 (2019), 1283-1307. https://doi.org/10.1007/s11075-018-0527-x
  32. D.V. Thong, D.V. Hieu and T.M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Opti. Lett., 14 (2020), 115-144 . https://doi.org/10.1007/s11590-019-01511-z
  33. M. Tian and M. Tong, Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems, J. Ineq. Appl., 19 (2019).
  34. N. Wairojjana and N. Pakkaranang, Halpern Tseng's Extragradient Methods for Solving Variational Inequalities Involving Semistrictly Quasimonotone Operator, Nonlinear Funct. Anal. Appl., 27(1) (2022), 121-140. https://doi.org/10.22436/jmcs.027.01.04
  35. J. Yang, Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities, Applicable Anal., 100(5) (2021), 1067-1078. https://doi.org/10.1080/00036811.2019.1634257
  36. J. Yang, H. Liu and G. Li, Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities, Numer Algor., 84 (2020), 389-405. https://doi.org/10.1007/s11075-019-00759-x
  37. Y. Yonghong and Postolache, Mihai and Yao, Jen-Chih, Strong convergence of an extragradient algorithm for variational inequality and fixed point problems, UPB Sci. Bull., Ser. A, 82(1) (2020), 3-12.
  38. Y.C. Zhang, K. Guo, and T. Wang, Generalized KrasnoselskiiMann-Type Iteration for Nonexpansive Mappings in Banach Spaces, J. Oper. Res. Soc. China, 9 (2021), 195-206. https://doi.org/10.1007/s40305-018-0235-1
  39. X. Zhao and Y. Yao, Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems, Optimization, 69(9) (2020), 1987-2002. https://doi.org/10.1080/02331934.2019.1711087