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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 73, 2024 - Issue 7
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Research Article

An inertial projection and contraction algorithm for pseudomonotone variational inequalities without Lipschitz continuity

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Pages 2033-2051 | Received 27 Jun 2022, Accepted 28 Feb 2023, Published online: 15 Mar 2023

References

  • Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York (NY): Academic Press; 1980.
  • Dafermos S. Traffic equilibrium and variational inequalities. Transp Sci. 1980;14(1):42–54.
  • Harker PT, Pang JS. Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math Program. 1990;48:161–220.
  • Bertsekas DP, Tsitsiklis JN. Parallel and distributed computation. numerical methods. Englewood Cliffs (NJ): Prentice-Hall; 1989.
  • Goldstein AA. Convex programming in Hilbert space. Bull Amer Math Soc. 1964;70:709–710.
  • Levitin ES, Polyak BT. Constrained minimization problems. USSR Comput Math Math Phys. 1966;6:1–50.
  • Korpelevich GM. The extragradient method for finding saddle points and other problems. Matecon. 1976;17(10):747–756.
  • Khobotov EN. Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput Math Math Phys. 1987;27:120–127.
  • Xiu NH, Zhang JZ. Some recent advances in projection-type methods for variational inequalities. J Comput Appl Math. 2003;152:559–585.
  • Solodov MV, Svaiter BF. A new projection method for variational inequality problems. SIAM J Control Optim. 1999;37(3):765–776.
  • He YR. A new double projection algorithm for variational inequalities. J Comput Appl Math. 2006;185(1):166–173.
  • Han DR, Lo HK. Two new self-adaptive projection methods for variational inequality problems. Comp Math Appl. 2002;43:1529–1537.
  • Solodov MV, Tseng P. Modified projection-type methods for monotone variational inequalities. SIAM J Control Optim. 1996;34(5):1814–1830.
  • He B. A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim. 1997;35:69–76.
  • Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38(2):431–446.
  • Ye ML. A half space projection method for solving generalized nash equilibrium problems. Optim. 2017;66(7):1119–1134.
  • Ye ML. An improved projection method for solving generalized variational inequality problems. Optim. 2018;67(9):1523–1533.
  • Konnov IV. On the rate of convergence of combined relaxation methods. Izv Vyssh Uchebn Zaved Mat. 1993;37(12):89–92.
  • Konnov IV. A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998;80:239–252.
  • Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer; 2001.
  • Konnov IV. Combined relaxation methods for generalized monotone variational inequalities, in: generalized convexity and related topics. Lect Notes Econ Math Syst. 2006;583:3–31.
  • Noor M. Some recent advances in variational inequalities, part I: basic concepts. New Zealand J Math. 1997;26:53–80.
  • Wang YJ, Xiu NH, Wang CY. Unified framework of extragradient-type methods for pseudomonotone variational inequalities. J Optim Theory Appl. 2001;111(3):641–656.
  • Facchinei F, Pang JS. Finite-Dimensional variational inequalities and complementarity problems. New York: Springer-Verlag; 2003.
  • Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optim. 2012;61:1119–1132.
  • Malitsky YV, Semenov VV. An extragradient algorithm for monotone variational inequalities. Cybern Syst Anal. 2014;50(2):271–277.
  • He BS, Liao LZ. Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl. 2002;112:111–128.
  • Polyak BT. Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math. 1964;4:1–17.
  • Lan G, Lu Z, Monteiro RDC. Primal-dual first-order methods with O(1/ϵ) iteration-complexity for cone programming. Math Program. 2011;126:1–29.
  • Nesterov Y. A method for solving the convex programming problem with convergence rate O(1/k2). Dokl Akad Nauk SSSR. 1983;269:543–547.
  • Nesterov Y. Gradient methods for minimizing composite objective function. Math Program. 2013;140:125–161.
  • Tseng P. Approximation accuracy, gradient methods and error bound for structured convex optimization. Math Program. 2010;125:263–295.
  • Ghadimi S, Lan G. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Math Program. 2016;156:59–99.
  • Pock T, Sabach S. Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. SIAM J Imaging Sci. 2016;9:1756–1787.
  • Wen B, Chen X, Pong TK. Aproximal difference-of-convex algorithm with extrapolation. Comput Optim Appl. 2018;69:297–324.
  • Ochs P, Chen Y, Brox T, et al. iPiano: inertial proximal algorithm for non-convex optimization. SIAM J Imaging Sci. 2014;7:1388–1419.
  • Dong QL, Lu YY, Yang J. The extragradient algorithm with inertial effects for solving the variational inequality. Optim. 2016;65:2217–2226.
  • Thong DV, Li XH, Dong QL, et al. An inertial popov's method for solving pseudomonotone variational inequalities. Optim Lett. 2020. DOI:10.1007/s11590-020-01599-8
  • Cao Y, Guo K. On the convergence of inertial two-subgradient extragradient method for variational inequality problems. Optim. 2020;69:1237–1253.
  • Fan JJ, Qin XL. Weak and strong convergence of inertial tseng's extragradient algorithms for solving variational inequality problems. Optim. 2020. DOI:10.1080/02331934.2020.1789129
  • Dong LQ, Cho YJ, Zhong LL, et al. Inertial projection and contraction algorithms for variational inequalities. J Glob Optim. 2018;70:687–704.
  • Karamardian S. Complementarity problems over cones with monotone and pseudomonotone maps. J Optim Theory Appl. 1976;18:445–454.
  • Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in hilbert spaces. 2nd ed. New York: Springer International Publishing AG; 2017.
  • Zarantonello EH. Projections on convex sets in Hilbert space and spectral theory, contributions to nonlinear functional analysis. New York (NY): Academic Press; 1971.
  • Gafni EM, Bertsekas DP. Two-metric projection problems and descent methods for asymmetric variational inequality problems. Math Program. 1984;53:99–110.
  • Ye ML, Pong TK. A subgradient-based approach for finding the maximum feasible subsystem with respect to a set. SIAM J Optim. 2020;30(2):1274–1299.
  • Sun D. A projection and contraction method for the nonlinear complementarity problem and its extensions. Math Numer Sin. 1994;16:183–200.
  • Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25(1):502–520.
  • Van Hieu D, Anh PK, Muu LD. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl. 2017;66(1):75–96.

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