A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES*

  • Duong Viet Thong ,
  • Vu Tien Dung
Expand
  • 1. Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam;
    2. Department of Mathematics, University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
Duong Viet Thong, E-mail: duzngvt@gmail.com

Received date: 2021-07-06

  Revised date: 2021-11-03

  Online published: 2023-03-01

Abstract

In this paper, we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. For solving this problem, we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method. Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in [0; 1]. The purpose of this work is to continue working in this direction, we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be $1$. Under suitable mild conditions, we establish the weak convergence of the proposed algorithm. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions. Finally, some numerical illustrations are given to confirm the theoretical analysis.

Cite this article

Duong Viet Thong , Vu Tien Dung . A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES*[J]. Acta mathematica scientia, Series B, 2023 , 43(1) : 184 -204 . DOI: 10.1007/s10473-023-0112-9

References

[1] Alvarez F, Attouch H.An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal, 2001, 9: 3-11
[2] Anh P K, Thong D V, Vinh N T.Improved inertial extragradient methods for solving pseudo-monotone variational inequalities. Optimization, 2020, 71(3): 505-528
[3] Antipin A S.On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat Metody, 1976, 12: 1164-1173
[4] Avriel M, Diewert W E, Schaible S, Zang I. Generalized Concavity. Society for Industrial and Applied Mathematics, 2010
[5] Baiocchi C, Capelo A.Variational and Quasivariational Inequalities, Applications to Free Boundary Prob- lems. New York: Wiley, 1984
[6] Cai X, Gu G, He B.On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput Optim Appl, 2014, 57: 339-363
[7] Cai G, Shehu Y, Iyiola O S.Inertial Tseng’s extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. J Ind Manag Optim, 2022, 18(4): 2873-2902
[8] Cegielski A.Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, Vol 2057. Berlin: Springer, 2012
[9] Censor Y, Gibali A, Reich S.The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl, 2011, 148: 318-335
[10] Censor Y, Gibali A, Reich S.Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw, 2011, 26: 827-845
[11] Censor Y, Gibali A, Reich S.Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization, 2011, 61: 1119-1132
[12] Cottle R W, Yao J C.Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl, 1992, 75: 281-295
[13] Denisov S V, Semenov V V, Chabak L M.Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal, 2015, 51: 757-765
[14] Dong L Q, Cho J Y, Zhong L L, Rassias M Th.Inertial projection and contraction algorithms for variational inequalities. J Glob Optim, 2018, 70: 687-704
[15] Dong Q L, Gibali A, Jiang D.A modified subgradient extragradient method for solving the variational inequality problem. Numer Algorithms, 2018, 79: 927-940
[16] Gibali A, Reich S, Zalas R.Iterative methods for solving variational inequalities in Euclidean space. J Fixed Point Theory Appl, 2015, 17: 775-811
[17] Gibali A, Reich S, Zalas R.Outer approximation methods for solving variational inequalities in Hilbert space. Optimization, 2017, 66: 417-437
[18] Gibali A, Iyiola O S, Akinyemi L, Shehu Y.Projected-reflected subgradient extragradient method and its real-world applications. Symmetry, 2021, 13(3): 489
[19] Facchinei F, Pang J S.Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols I and II. New York: Springer, 2003
[20] Fichera G.Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincei, VIII Ser, Rend Cl Sci Fis Mat Nat, 1963, 34: 138-142
[21] Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry,Nonexpansive Mappings. New York: Marcel Dekker, 1984
[22] Hadjisavvas N, Komlosi S, Schaible S, eds. Handbook on Generalized Convexity and Generalized Mono- tonicity, Nonconvex optimization and its Applications, Vol 76. New York: Springer, 2005
[23] He B S.A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim, 1997, 35: 69-76
[24] Karamardian S, Schaible S.Seven kinds of monotone maps. J Optim Theory Appl, 1990, 66: 37-46
[25] Kim D S, Vuong P T, Khanh P D.Qualitative properties of strongly pseudomonotone variational inequal- ities. Optim Lett, 2016, 10: 1669-1679
[26] Kinderlehrer D, Stampacchia G.An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980
[27] Konnov I V.Combined Relaxation Methods for Variational Inequalities. Berlin: Springer-Verlag, 2001
[28] Korpelevich G M.The extragradient method for finding saddle points and other problems. Ekonomika i Mat Metody, 1976, 12: 747-756
[29] Liu H, Yang J.Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput Optim Appl, 2020, 77(2): 491-508
[30] Shehu Y, Iyiola O S.Projection methods with alternating inertial steps for variational inequalities: weak and linear convergence. Appl Numer Math, 2020, 157: 315-337
[31] Shehu Y, Iyiola O S, Reich S.A modified inertial subgradient extragradient method for solving variational inequalities. Optim Eng, 2021, 23(1): 421-449
[32] Shehu Y, Iyiola O S, Yao J C.New projection methods with inertial steps for variational inequalities. Optimization, 2021, DOI:10.1080/02331934.2021.1964079
[33] Opial Z.Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc, 1967, 73: 591-597
[34] Ortega J M, Rheinboldt W C.Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press, 1970
[35] Reich S, Thong D V, Cholamjiak P, et al.Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer Algor, 2021, 88(2): 813-835
[36] Shehu Y, Liu L, Mu X, Dong Q L.Analysis of versions of relaxed inertial projection and contraction method. Applied Numerical Mathematics, 2021, 165: 1-21
[37] Stampacchia G.Formes bilineaires coercitives sur les ensembles convexes. C R Acad Sci, 1964, 258: 4413-4416
[38] Sun D F.A class of iterative methods for solving nonlinear projection equations. J Optim Theory Appl, 1996, 91: 123-140
[39] Thong D V, Hieu D V.Modified subgradient extragradient method for variational inequality problems. Numer Algorithms, 2018, 79: 597-610
[40] Thong D V, Vuong P T.Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities. Optimization, 2019, 68: 2207-2226
[41] Thong D V, Vuong P T.Improved subgradient extragradient methods for solving pseudomonotone varia- tional inequalities in Hilbert spaces. Applied Numerical Mathematics, 2021, 163: 221-238
[42] Vuong P T.On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theory Appl, 2018, 176: 399-409
[43] Vuong P T, Shehu Y.Convergence of an extragradient-type method for variational inequality with appli- cations to optimal control problems. Numer Algorithms, 2019, 81: 269-291
Options
Outlines

/