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Original Articles

Modified accelerated algorithms for solving variational inequalities

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Pages 2233-2258 | Received 26 Apr 2019, Accepted 23 Oct 2019, Published online: 12 Nov 2019

References

  • F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal. 9 (2001), pp. 3–11. doi: 10.1023/A:1011253113155
  • H. Attouch, X. Goudon, and P. Redont, The heavy ball with friction. I. The continuous dynamical system, Commun. Contemp. Math. 2 (2000), pp. 1–34. doi: 10.1142/S0219199700000025
  • H. Attouch and M.O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Differ. Equat. 179 (2002), pp. 278–310. doi: 10.1006/jdeq.2001.4034
  • H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
  • R.I. Bot, E.R. Csetnek, and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput. 256 (2015), pp. 472–487.
  • Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2011), pp. 318–335. doi: 10.1007/s10957-010-9757-3
  • Y. Censor, A. Gibali, and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Meth. Softw. 26 (2011), pp. 827–845. doi: 10.1080/10556788.2010.551536
  • Y. Censor, A. Gibali, and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization 61 (2012), pp. 1119–1132. doi: 10.1080/02331934.2010.539689
  • Q.L. Dong, Y.J. Cho, L.L. Zhong, and Th.M Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim. 70 (2018), pp. 687–704. doi: 10.1007/s10898-017-0506-0
  • Q.L. Dong, H.B. Yuan, Y.J. Cho, and Th.M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett. 12 (2018), pp. 87–102. doi: 10.1007/s11590-016-1102-9
  • Q.L. Dong, Y.J. Cho, and Th.M. Rassias, The projection and contraction methods for finding common solutions to variational inequality problems, Optim. Lett. 12 (2017), pp. 1871–1896. doi:10.1007/s11590-017-1210-1.
  • F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin, 2003.
  • A. Gibali, A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces, J. Nonlinear Anal. Optim. 6 (2015), pp. 41–51.
  • K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
  • P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), pp. 271–310. doi: 10.1007/BF02392210
  • D.V. Hieu, P.K. Anh, and L.D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl. 66 (2017), pp. 75–96. doi: 10.1007/s10589-016-9857-6
  • D.V. Hieu, P.K. Anh, and L.D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl. 73 (2019), pp. 913–932. doi: 10.1007/s10589-019-00093-x
  • D.V. Hieu, Y.J. Cho, and Y.B. Xiao, Modified extragradient algorithms for solving equilibrium problems, Optimization 67 (2018), pp. 2003–2029. doi: 10.1080/02331934.2018.1505886
  • D.V. Hieu and D.V. Thong, New extragradient-like algorithms for strongly pseudomonotone variational inequalities, J. Glob. Optim. 70 (2018), pp. 385–399. doi: 10.1007/s10898-017-0564-3
  • D.V. Hieu, Y.J. Cho, and Y.B. Xiao, Golden ratio algorithms with new stepsize rules for variational inequalities, Math. Meth. Appl. Sci. (2019). doi:10.1002/mma.5703.
  • D.V. Hieu and P.K. Quy, An inertial modified algorithm for solving variational inequalities, RAIRO Oper. Res. (2019). doi:10.1051/ro/2018115.
  • C. Kanzow, Some equation-based methods for the nonlinear complemantarity problem, Optim. Meth. Software 3 (1994), pp. 327–340. doi: 10.1080/10556789408805573
  • D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
  • I.V. Konnov, Equilibrium Models and Variational Inequalities, Elsevier, Amsterdam, 2007.
  • G.M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomikai Mat. Metody 12 (1976), pp. 747–756.
  • R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 163 (2014), pp. 399–412. doi: 10.1007/s10957-013-0494-2
  • Y.V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim. 25 (2015), pp. 502–520. doi: 10.1137/14097238X
  • P.E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47 (2008), pp. 1499–1515. doi: 10.1137/060675319
  • P.E. Maingé, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set. Valued Anal. 15 (2007), pp. 67–79. doi: 10.1007/s11228-006-0027-3
  • G.J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J. 29 (1962), pp. 341–346. doi: 10.1215/S0012-7094-62-02933-2
  • L.D. Popov, A modification of the Arrow-Hurwicz method for searching for saddle points, Mat. Zametki 28 (1980), pp. 777–784.
  • M. Sofonea, Y.B. Xiao, and M. Couderc, Optimization problems for elastic contact models with unilateral constraints, Z. Angew. Math. Phys. 70 (2019), p. 1. doi:10.1007/s00033-018-1046-2.
  • M. Sofonea, Y.B. Xiao, and M. Couderc, Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, Nonlinear Analysis: Real World Appl. 50 (2019), p. 86–103. doi: 10.1016/j.nonrwa.2019.04.005
  • M.V. Solodov and B.F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim. 37 (1999), pp. 765–776. doi: 10.1137/S0363012997317475
  • D. Sun, A projection and contraction method for the nonlinear complementarity problems and its extensions, Math. Numer. Sinica 16 (1994), pp. 183–194.
  • D.V. Thong and D.V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algor. 78 (2017), pp. 1045–1060. doi:10.1007/s11075-017-0412-z.
  • D.V. Thong, N.T. Vinh, and D.V. Hieu, Accelerated hybrid and shrinking projection methods for variational inequality problems, Optimization 68 (2019), pp. 981–998. doi: 10.1080/02331934.2019.1566825
  • D.V. Thong and D.V. Hieu, Modied Tsengs extragradient algorithms for variational inequality problems, J. Fixed Point Theory Appl. 20 (2018), p. 152. doi: 10.1007/s11784-018-0634-2
  • D.V. Thong and D.V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer. Algor. 80 (2019), pp. 1283–1307. doi: 10.1007/s11075-018-0527-x
  • Y.M. Wang, Y.B. Xiao, X. Wang, and Y.J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl. 9 (2016), pp. 1178–1192. doi: 10.22436/jnsa.009.03.44
  • Y.B. Xiao, N.J. Huang, and Y.J. Cho, A class of generalized evolution variational inequalities in Banach space, Appl. Math. Lett. 25 (2012), pp. 914–920. doi: 10.1016/j.aml.2011.10.035
  • Y.B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl. 475 (2019), pp. 364–384. doi:10.1016/j.jmaa.2019.02.046.
  • H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), pp. 109–113. doi: 10.1017/S0004972700020116

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