STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES

doi:10.1007/s10473-022-0224-7

Abstract

Abstract: In this work, we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space, and present a projection-type approximation method for solving this problem. Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping. A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions. Finally, we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.

Key words: Inertial method, Tseng's extragradient, viscosity method, variational inequality problem, pseudomonotone mapping, strong convergence

CLC Number:

65Y05

Nguyen Xuan LINH, Duong Viet THONG, Prasit CHOLAMJIAK, Pham Anh TUAN, Luong Van LONG. STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES[J].Acta mathematica scientia,Series B, 2022, 42(2): 795-812.

Trendmd

References

[1] Bauschke H H, Combettes P L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. New York:Springer, 2011
[2] Cai G, Dong Q L, Peng Y. Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators. J Optim Theory Appl, 2021, 188:447-472
[3] Cottle R W, Yao J C. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl, 1992, 75:281-295
[4] 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
[5] 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
[6] Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optimization, 2012, 61:1119-1132
[7] Censor Y, Gibali A, Reich S. Algorithms for the split variational inequality problem. Numer Algorithms, 2012, 59:301-323
[8] Ceng L C, Hadjisavvas N, Wong N C. Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Glob Optim, 2010, 46:635-646
[9] Ceng L C, Teboulle M, Yao J C. Weak convergence of an iterative method for pseudo-monotone variational inequalities and fxed-point problems. J Optim Theory Appl, 2010, 146:19-31
[10] Cegielski A. Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics. Vol 2057. Berlin:Springer, 2012
[11] 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
[12] 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
[13] Fichera G. Problemi elastostatici con vincoli unilaterali:il problema di Signorini con ambigue condizioni al contorno. Atti Accad Naz Lincei, Mem, Cl Sci Fis Mat Nat, Sez I, VIII Ser, 1964, 7:91-140
[14] Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York:Marcel Dekker, 1984
[15] Harker P T, Pang J S. A damped-Newton method for the linear complementarity problem. Lect Appl Math, 1990, 26:265-284
[16] He S N, Tian H L. Selective projection methods for solving a class of variational inequalities. Numer Algorithms, 2019, 80:617-634
[17] Hieu D V, Cholamjiak P. Modified extragradient method with Bregman distance for variational inequalities. Appl Anal, 2020. DOI:10.1080/00036811.2020.1757078
[18] Khanh P D. A modified extragradient method for infinite-dimensional variational inequalities. Acta Math Vietnam, 2016, 41:251-263
[19] Kraikaew R, Saejung S. Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl, 2014, 163:399-412
[20] Korpelevich G M. The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody, 1976, 12:747-756
[21] Maingé P E. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim, 2008, 47:1499-1515
[22] Maingé P E, Gobinddass M L. Convergence of one-step projected gradient methods for variational inequalities. J Optim Theory Appl, 2016, 171:146-168
[23] Malitsky Y V, Semenov V V. A hybrid method without extrapolation step for solving variational inequality problems. J Glob Optim, 2015, 61:193-202
[24] Malitsky Y V. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim, 2015, 25:502-520
[25] Moudafi A. Viscosity approximation methods for fixed points problems. J Math Anal Appl, 2000, 241:46-55
[26] Liu H, Yang J. Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput Optim Appl, 2020, 77:491-508
[27] Nadezhkina N, Takahashi W. Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J Optim, 2006, 16:1230-1241
[28] Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl, 2006, 128:191-201
[29] Shehu Y, Cholamjiak P. Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo, 2019, 56. https://doi.org/10.1007/s10092-018-0300-5
[30] Shehu Y, Dong Q L, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization, 2019, 68:385-409
[31] Solodov M V, Svaiter B F. A new projection method for variational inequality problems. SIAM J Control Optim, 1999, 37:765-776
[32] Thong D V, Vuong P T. Modified Tseng's extragradient methods for solving pseudo-monotone variational inequalities. Optimziation, 2019, 68:2207-2226
[33] Thong D V, Hieu D V. Weak and strong convergence theorems for variational inequality problems. Numer Algorithms, 2018, 78:1045-1060
[34] Thong D V, Hieu D V. Modified Tseng's extragradient algorithms for variational inequality problems. J Fixed Point Theory Appl, 2018, 20:152. https://doi.org/10.1007/s11784-018-0634-2
[35] Thong D V, Vinh N T, Cho Y J. A strong convergence theorem for Tseng's extragradient method for solving variational inequality problems. Optim Lett, 2020, 14:1157-1175
[36] Thong D V, Shehu Y, Iyiola O S. Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings. Numer Algorithms, 2020, 84:795-823
[37] Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim, 2000, 38:431-446
[38] 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
[39] Vuong P T, Shehu Y. Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer Algorithms, 2019, 81:269-291
[40] Wang F H, Xu H K. Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng's extragradient method. Taiwanese J Math, 2012, 16:1125-1136
[41] Xiao Y B, Huang N J, Cho Y J. A class of generalized evolution variational inequalities in Banach space. Appl Math Lett, 2012, 25:914-920
[42] Yang J, Liu H. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algorithms, 2019, 80:741-752
[43] Yao Y, Postolache M. Iterative methods for pseudomonotone variational inequalities and fixed point problems. J Optim Theory Appl, 2012, 155:273-287