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Modified Tseng’s extragradient algorithms for variational inequality problems. (English) Zbl 06969145

Summary: In this work, our interest is in investigating the monotone variational inequality problems in the framework of real Hilbert spaces. For solving this problem, we introduce two modified Tseng’s extragradient methods using the inertial technique. The weak convergence theorems are established under the standard assumptions imposed on cost operators. Finally, numerical results are reported to illustrate the behavior of the new algorithms and also to compare with others.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65Y05 Parallel numerical computation
47H05 Monotone operators and generalizations
47J25 Iterative procedures involving nonlinear operators
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[1] Alber, Ya; Iusem, AN, Extension of subgradient techniques for nonsmooth optimization in Banach spaces, Set Valued Anal., 9, 315-335, (2001) · Zbl 1049.90123 · doi:10.1023/A:1012665832688
[2] Alvarez, F., Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14, 773-782, (2004) · Zbl 1079.90096 · doi:10.1137/S1052623403427859
[3] Alvarez, F.; Attouch, H., An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set Valued Anal., 9, 3-11, (2001) · Zbl 0991.65056 · doi:10.1023/A:1011253113155
[4] Attouch, H.; Goudon, X.; Redont, P., The heavy ball with friction. I. The continuous dynamical system, Commun. Contemp. Math., 2, 1-34, (2000) · Zbl 0983.37016
[5] Attouch, H.; Czarnecki, MO, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Differ. Equ., 179, 278-310, (2002) · Zbl 1007.34049 · doi:10.1006/jdeq.2001.4034
[6] Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) · Zbl 1218.47001 · doi:10.1007/978-1-4419-9467-7
[7] Bot, RI; Csetnek, ER; Laszlo, SC, An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions, EURO J. Comput. Optim., 4, 3-25, (2016) · Zbl 1338.90311 · doi:10.1007/s13675-015-0045-8
[8] Bot, RI; Csetnek, ER, An inertial Tsengs type proximal algorithm for nonsmooth and nonconvex optimization problems, J. Optim. Theory Appl., 171, 600-616, (2015) · Zbl 1349.90688 · doi:10.1007/s10957-015-0730-z
[9] Bot, RI; Csetnek, ER, An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algorithms, 71, 519-540, (2016) · Zbl 1338.47076 · doi:10.1007/s11075-015-0007-5
[10] Bot, RI; Csetnek, ER; Hendrich, C., Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256, 472-487, (2015) · Zbl 1338.65145
[11] Bot, RI; Csetnek, ER, An inertial alternating direction method of multipliers, Minimax Theory Appl., 1, 29-49, (2016) · Zbl 1337.90082
[12] Bot, RI; Csetnek, ER, A hybrid proximal-extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36, 951-963, (2015) · Zbl 06514880 · doi:10.1080/01630563.2015.1042113
[13] Censor, Y.; Gibali, A.; Reich, S., Algorithms for the split variational inequality problem, Numer. Algorithms, 59, 301-323, (2012) · Zbl 1239.65041 · doi:10.1007/s11075-011-9490-5
[14] Censor, Y.; Gibali, A.; S, Reich, The subgradient extragradientmethod for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148, 318-335, (2011) · Zbl 1229.58018 · doi:10.1007/s10957-010-9757-3
[15] Censor, Y.; Gibali, A.; Reich, S., Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26, 827-845, (2011) · Zbl 1232.58008 · doi:10.1080/10556788.2010.551536
[16] Censor, Y.; Gibali, A.; Reich, S., Extensions of Korpelevichs extragradient method for the variational inequality problem in Euclidean space, Optimization, 61, 1119-1132, (2012) · Zbl 1260.65056 · doi:10.1080/02331934.2010.539689
[17] Ceng, LC; Hadjisavvas, N.; Wong, NC, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46, 635-646, (2010) · Zbl 1198.47081 · doi:10.1007/s10898-009-9454-7
[18] Chen, C.; Ma, S.; Yang, J., A general inertial proximal point algorithm for mixed variational inequality problem, SIAM J. Optim., 25, 2120-2142, (2015) · Zbl 1327.65106 · doi:10.1137/140980910
[19] Dong, LQ; Cho, JY; Zhong, LL; Rassias, MTh, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70, 687-704, (2017) · Zbl 1390.90568 · doi:10.1007/s10898-017-0506-0
[20] Gibali, A., A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces, J. Nonlinear Anal. Optim., 6, 41-51, (2015) · Zbl 1413.65252
[21] Gibali, A.; Reich, S.; Zalas, R., Iterative methods for solving variational inequalities in Euclidean space, J. Fixed Point Theory Appl., 17, 775-811, (2015) · Zbl 1332.47044 · doi:10.1007/s11784-015-0256-x
[22] Gibali, A.; Reich, S.; Zalas, R., Outer approximation methods for solving variational inequalities in Hilbert space, Optimization, 66, 417-437, (2017) · Zbl 1367.58006 · doi:10.1080/02331934.2016.1271800
[23] Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001
[24] Hieu, DV; Anh, PK; Muu, LD, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 66, 75-96, (2017) · Zbl 1368.65103 · doi:10.1007/s10589-016-9857-6
[25] Hieu, DV; Muu, LD; Anh, PK, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, 73, 197-217, (2016) · Zbl 1367.65089 · doi:10.1007/s11075-015-0092-5
[26] Hieu, DV, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21, 478-501, (2016) · doi:10.3846/13926292.2016.1183527
[27] Hieu, DV, Halpern subgradient extragradient method extended to equilibrium problems, RACSAM, 111, 823-840, (2017) · Zbl 1378.65136 · doi:10.1007/s13398-016-0328-9
[28] Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malays. Math. Sci. Soc. (2017). https://doi.org/10.1007/s40840-017-0474-z
[29] Kanzow, C.; Shehu, Y., Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaces, J. Fixed Point Theory Appl., 20, 24, (2018) · Zbl 1491.47065 · doi:10.1007/s11784-018-0531-8
[30] Kraikaew, R.; Saejung, S., Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163, 399-412, (2014) · Zbl 1305.49012 · doi:10.1007/s10957-013-0494-2
[31] Korpelevich, GM, The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody., 12, 747-756, (1976) · Zbl 0342.90044
[32] Maingé, PE, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47, 1499-1515, (2008) · Zbl 1178.90273 · doi:10.1137/060675319
[33] Maingé, PE; Gobinddass, ML, Convergence of one step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171, 146-168, (2016) · Zbl 06661393 · doi:10.1007/s10957-016-0972-4
[34] Maingé, PE, Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure, Comput. Math. Appl., 3, 720-728, (2016) · Zbl 1359.65102 · doi:10.1016/j.camwa.2016.05.028
[35] Maingé, PE, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set Valued Anal., 15, 67-79, (2007) · Zbl 1129.47054 · doi:10.1007/s11228-006-0027-3
[36] Maingé, PE, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl., 34, 876-887, (2008) · Zbl 1146.47042 · doi:10.1016/j.jmaa.2008.03.028
[37] Malitsky, YV; Semenov, VV, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61, 193-202, (2015) · Zbl 1366.47018 · doi:10.1007/s10898-014-0150-x
[38] Malitsky, YV, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25, 502-520, (2015) · Zbl 1314.47099 · doi:10.1137/14097238X
[39] Moudafi, A.; Elisabeth, E., An approximate inertial proximal method using enlargement of a maximal monotone operator, Int. J. Pure Appl. Math., 5, 283-299, (2003) · Zbl 1069.90077
[40] Moudafi, A.; Oliny, M., Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155, 447-454, (2003) · Zbl 1027.65077 · doi:10.1016/S0377-0427(02)00906-8
[41] Nadezhkina, N.; Takahashi, W., Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 16, 1230-1241, (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[42] Pang, JS; Gabriel, SA, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Program., 60, 295-337, (1993) · Zbl 0808.90123 · doi:10.1007/BF01580617
[43] Polyak, BT, Some methods of speeding up the convergence of iterarive methods, Zh. Vychisl. Mat. Mat. Fiz., 4, 1-17, (1964)
[44] Shehu, Y.; Iyiola, OS, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19, 2483-2510, (2017) · Zbl 1493.47100 · doi:10.1007/s11784-017-0435-z
[45] Solodov, MV; Svaiter, BF, A new projection method for variational inequality problems, SIAM J. Control Optim., 37, 765-776, (1993) · Zbl 0959.49007 · doi:10.1137/S0363012997317475
[46] Takahashi, W.: Nonlinear Functional Analysis-Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000) · Zbl 0997.47002
[47] Thong, DV, Viscosity approximation methods for solving fixed point problems and split common fixed point problems, J. Fixed Point Theory Appl., 19, 1481-1499, (2017) · Zbl 1453.47021 · doi:10.1007/s11784-016-0323-y
[48] Thong, DV; Hieu, DV, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78, 1045-1060, (2017) · Zbl 1398.65376 · doi:10.1007/s11075-017-0412-z
[49] Thong, Duong Viet; Van Hieu, Dang, Modified subgradient extragradient method for variational inequality problems, Numerical Algorithms, 79, 597-610, (2017) · Zbl 06945630 · doi:10.1007/s11075-017-0452-4
[50] Thong, DV; Hieu, DV, An inertial method for solving split common fixed point problems, J. Fixed Point Theory Appl., 19, 3029-3051, (2017) · Zbl 1482.65084 · doi:10.1007/s11784-017-0464-7
[51] Thong, DV; Hieu, DV, Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems, Optimization, 67, 83-102, (2018) · Zbl 1398.90184 · doi:10.1080/02331934.2017.1377199
[52] Thong, DV; Hieu, DV, New extragradient methods for solving variational inequality problems and fixed point problems, J. Fixed Point Theory Appl., 20, 129, (2018) · doi:10.1007/s11784-018-0610-x
[53] Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms. (2018) https://doi.org/10.1007/s11075-018-0527-x · Zbl 1398.90184
[54] Tseng, P., A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38, 431-446, (2000) · Zbl 0997.90062 · doi:10.1137/S0363012998338806
[55] Wang, FH; Xu, HK, Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng’s extragradient method, Taiwan. J. Math., 16, 1125-1136, (2012) · Zbl 1515.47107 · doi:10.11650/twjm/1500406682
[56] Xiu, NH; Zhang, JZ, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math., 152, 559-587, (2003) · Zbl 1018.65083 · doi:10.1016/S0377-0427(02)00730-6
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