A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities. (English) Zbl 1451.49014
Summary: In this paper, we introduce a new algorithm by incorporating inertial terms in a subgradient extragradient algorithm for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The strong convergence of the algorithm is obtained under mild assumptions. We also provide some numerical examples to illustrate that the acceleration of our algorithm is effective.
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
Keywords:
gradient projection; maximal monotone operator; inertial extrapolation; variational inequalitySoftware:
iPiascoReferences:
| [1] | Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications (1980), New York (NY): Academic Press, New York (NY) · Zbl 0457.35001 |
| [2] | Konnov, IV., Equilibrium models and variational inequalities (2007), Amsterdam: Elsevier, Amsterdam · Zbl 1140.91056 |
| [3] | Rockafellar, RT; Wets, RJB., Variational analysis (2009), New York: Springer Science & Business Media, New York |
| [4] | Cho, SY., Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J Appl Anal Comput, 8, 19-31 (2018) · Zbl 07303041 |
| [5] | Cho, SY; Kang, SM., Approximation of common solutions of variational inequalities via strict pseudocontractions, Acta Math Sci, 32, 1607-1618 (2012) · Zbl 1271.47055 · doi:10.1016/S0252-9602(12)60127-1 |
| [6] | Ceng, LC; Yao, JC., A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal, 72, 1922-1937 (2010) · Zbl 1179.49003 · doi:10.1016/j.na.2009.09.033 |
| [7] | Zhao, X.; Ng, KF; Li, C., Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems, Appl Math Optim, 78, 613-641 (2018) · Zbl 1492.47089 · doi:10.1007/s00245-017-9417-1 |
| [8] | Ansari, QH; Babu, F.; Yao, JC., Regularization of proximal point algorithms in Hadamard manifolds, J Fixed Point Theory Appl, 21, 25 (2019) · Zbl 1412.49037 · doi:10.1007/s11784-019-0658-2 |
| [9] | Takahashi, W.; Wen, CF; Yao, JC., The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Fixed Point Theory, 19, 407-419 (2018) · Zbl 1462.47043 · doi:10.24193/fpt-ro.2018.1.32 |
| [10] | Cho, SY; Li, W.; Kang, SM., Convergence analysis of an iterative algorithm for monotone operators, J Inequal Appl, 2013 (2013) · Zbl 1364.47017 |
| [11] | Ceng, LC; Petrusel, A.; Yao, JC, Systems of variational inequalities with hierarchical variational equality constraints for Lipschitzian pseudocontractions, Fixed Point Theory, 20, 113-133 (2019) · Zbl 1430.49004 · doi:10.24193/fpt-ro.2019.1.07 |
| [12] | Chang, SS; Wen, CF; Yao, JC., Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces, Optimization, 67, 1183-1196 (2018) · Zbl 1402.90119 · doi:10.1080/02331934.2018.1470176 |
| [13] | Korpelevich, GM., The extragradient method for finding saddle points and other problem, Ekon Mat Metody, 12, 747-756 (1976) · Zbl 0342.90044 |
| [14] | 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 |
| [15] | Hieu, DV., New extragradient method for a class of equilibrium problems in Hilbert spaces, Appl Anal, 97, 811-824 (2018) · Zbl 1391.90587 · doi:10.1080/00036811.2017.1292350 |
| [16] | Thong, DV; Hieu, DV., Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J Comput Appl Math, 34, 80-98 (2018) · Zbl 1524.65240 · doi:10.1016/j.cam.2018.03.019 |
| [17] | 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 |
| [18] | 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 |
| [19] | 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 |
| [20] | Polyak, BT., Some methods of speeding up the convergence of iterative methods, Zh Vychisl Mat Mat Fiz, 4, 1-17 (1964) · Zbl 0147.35301 |
| [21] | 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 |
| [22] | Bot, RI; Csetnek, ER., An inertial alternating direction method of multipliers, Minimax Theory Appl, 1, 29-49 (2016) · Zbl 1337.90082 |
| [23] | Attouch, H.; Peypouquet, J.; Redont, P., A dynamical approach to an inertial forward-backward algorithm for convex minimization, SIAM J Optim, 24, 232-256 (2014) · Zbl 1295.90044 · doi:10.1137/130910294 |
| [24] | Ochs, P.; Brox, T.; Pock, T., iPiasco: Inertial proximal algorithm for strongly convex optimization, J Math Imaging Vision, 53, 171-181 (2015) · Zbl 1327.90219 · doi:10.1007/s10851-015-0565-0 |
| [25] | Dong, QL; Lu, YY; Yang, JF., 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 |
| [26] | Malitsky, YV., Projected reflected gradient methods for monotone variational inequalities, SIAM J Optim, 25, 502-520 (2015) · Zbl 1314.47099 · doi:10.1137/14097238X |
| [27] | Goebel, K.; Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings (1984), New York (NY): Marcel Dekker, New York (NY) · Zbl 0537.46001 |
| [28] | Ofoedu, EU., Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in a real Banach space, J Math Anal Appl, 321, 722-728 (2006) · Zbl 1109.47061 · doi:10.1016/j.jmaa.2005.08.076 |
| [29] | 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 |
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