×

A modified subgradient extragradient method for solving the variational inequality problem. (English) Zbl 06967417

Summary: The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. (J. Optim. Theory Appl. 148, 318–335, 2011), replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.

MSC:

47H05 Monotone operators and generalizations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anh, PN; An, LTH, The subgradient extragradient method extended to equilibrium problems, Optimization, 64, 225-248, (2015) · Zbl 1317.65149 · doi:10.1080/02331934.2012.745528
[2] Antipin, AS, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekon. Mat. Metody, 12, 1164-1173, (1976)
[3] Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984) · Zbl 0551.49007
[4] 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., 57, 339-363, (2014) · Zbl 1304.90203 · doi:10.1007/s10589-013-9599-7
[5] Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011) · Zbl 1218.47001 · doi:10.1007/978-1-4419-9467-7
[6] Censor, Y.; Gibali, A.; Reich, S., The subgradient extragradient method 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
[7] Censor, Y.; Gibali, A.; Reich, S., Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Method Soft., 6, 827-845, (2011) · Zbl 1232.58008 · doi:10.1080/10556788.2010.551536
[8] Censor, Y.; Gibali, A.; Reich, S., Extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space, Optimization, 61, 1119-1132, (2012) · Zbl 1260.65056 · doi:10.1080/02331934.2010.539689
[9] Dang, VH, New subgradient extragradient methods for common solutions to equilibrium problems, Comput. Optim. Appl., 67, 1-24, (2017) · Zbl 1401.90142 · doi:10.1007/s10589-016-9886-1
[10] Dang, VH, Halpern subgradient extragradient method extended to equilibrium problems, Racsam Rev. R. Acad. A., 111, 1-18, (2016)
[11] Denisov, SV; Semenov, VV; Chabak, LM, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51, 757-765, (2015) · Zbl 1331.49010 · doi:10.1007/s10559-015-9768-z
[12] Dong, QL; Lu, YY; Yang, J., The extragradient algorithm with inertial effects for solving the variational inequality, Optimization, 65, 2217-2226, (2016) · Zbl 1358.90139 · doi:10.1080/02331934.2016.1239266
[13] Dong, Q. L.; Cho, Y. J.; Zhong, L. L.; Rassias, Th. M., Inertial projection and contraction algorithms for variational inequalities, Journal of Global Optimization, 70, 687-704, (2017) · Zbl 1390.90568 · doi:10.1007/s10898-017-0506-0
[14] Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York and Basel (1984) · Zbl 0537.46001
[15] Goldstein, AA, Convex programming in Hilbert space, Bull. Am. Math. Soc., 70, 709-710, (1964) · Zbl 0142.17101 · doi:10.1090/S0002-9904-1964-11178-2
[16] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003) · Zbl 1062.90002
[17] Fang, C.; Chen, S., A subgradient extragradient algorithm for solving multi-valued variational inequality, Appl. Math. Comput., 229, 123-130, (2014) · Zbl 1364.65134
[18] Harker, P.T., Pang, J.-S.: A damped-Newton method for the linear complementarity problem. In: Allgower, G., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math, vol. 26, pp 265-284. AMS, Providence (1990)
[19] He, S.; Wu, T., A modified subgradient extragradient method for solving monotone variational inequalities, J. Inequal. Appl., 2017, 89, (2017) · Zbl 1382.47022 · doi:10.1186/s13660-017-1366-3
[20] 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
[21] Van Hieu, Dang; Thong, Duong Viet, New extragradient-like algorithms for strongly pseudomonotone variational inequalities, Journal of Global Optimization, 70, 385-399, (2017) · Zbl 1384.65041 · doi:10.1007/s10898-017-0564-3
[22] He, BS, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim., 35, 69-76, (1997) · Zbl 0865.90119 · doi:10.1007/BF02683320
[23] He, S.; Yang, C.; Duan, P., Realization of the hybrid method for Mann iterations, Appl. Math. Comput., 217, 4239-4247, (2010) · Zbl 1207.65065
[24] Khobotov, EN, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comput. Math. Math. Phys., 27, 120-127, (1987) · Zbl 0665.90078 · doi:10.1016/0041-5553(87)90058-9
[25] Korpelevich, GM, The extragradient method for finding saddle points and other problems, Ekon. Mate. Metody, 12, 747-756, (1976) · Zbl 0342.90044
[26] Kraikaew, R.; Saejung, S., Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optimiz. Theory App., 163, 399-412, (2014) · Zbl 1305.49012 · doi:10.1007/s10957-013-0494-2
[27] Levitin, ES; Polyak, BT, Constrained minimization problems, USSR Comput. Math. Math. Phys., 6, 1-50, (1966) · doi:10.1016/0041-5553(66)90114-5
[28] Malitsky, YV, Projected reflected gradient method for variational inequalities, SIAM J. Optim., 25, 502-520, (2015) · Zbl 1314.47099 · doi:10.1137/14097238X
[29] 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
[30] Malitsky, YV; Semenov, VV, An extragradient algorithm for monotone variational inequalities, Cybernet. Syst. Anal., 50, 271-277, (2014) · Zbl 1311.49024 · doi:10.1007/s10559-014-9614-8
[31] Noor, MA, Some developments in general variational inequalities, Appl. Math. Comput., 152, 199-277, (2004) · Zbl 1134.49304
[32] Popov, LD, A modification of the Arrow-Hurwicz method for searching for saddle points, Mat. Zametki, 28, 777-784, (1980) · Zbl 0456.90068
[33] Rockafellar, RT, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898, (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[34] Solodov, MV; Svaiter, BF, A new projection method for variational inequality problems, SIAM J. Control Optim., 37, 765-776, (1999) · Zbl 0959.49007 · doi:10.1137/S0363012997317475
[35] Sun, DF, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91, 123-140, (1996) · Zbl 0871.90091 · doi:10.1007/BF02192286
[36] 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
[37] Vinh, NT; Hoai, PT, Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems, Math. Method Appl. Sci., 39, 3808-3823, (2016) · Zbl 1351.90164 · doi:10.1002/mma.3826
[38] Yang, Q., On variable-step relaxed projection algorithm for variational inequalities, J. Math. Anal. Appl., 302, 166-179, (2005) · Zbl 1056.49018 · doi:10.1016/j.jmaa.2004.07.048
[39] Yao, Y.; Marino, G.; Muglia, L., A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality, Optimization, 63, 559-569, (2014) · Zbl 1524.47105 · doi:10.1080/02331934.2012.674947
[40] Zhou, H.; Zhou, Y.; Feng, G., Iterative methods for solving a class of monotone variational inequality problems with applications, J. Inequal. Appl., 2015, 68, (2015) · Zbl 1314.47116 · doi:10.1186/s13660-015-0590-y
[41] Zeidler, E.: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York (1985) · Zbl 0583.47051 · doi:10.1007/978-1-4612-5020-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.