Weak convergence of inertial proximal algorithms with self adaptive stepsize for solving multivalued variational inequalities

References

• Gibali A, Thong DV, Vinh NT. Three new iterative methods for solving inclusion problems and related problems. Comput. Appl. Math. 2020;39:187.
   Web of Science ®Google Scholar
• He X, Huang N, Li X. Modified projection methods for solving multi-valued variational inequality without monotonicity. Netw Spat Econ. 2022;22:361–377. doi:10.1007/s11067-019-09485-2 .
   Google Scholar
• Thong DV, Vinh NT, Cho YJ. New strong convergence theorem of the inertial projection and contraction method for variational inequality problems. Numer Algorithms. 2020;84:285–305.
   Web of Science ®Google Scholar
• Yang J, Liu H. Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algorithms. 2019;80:741–752.
   Web of Science ®Google Scholar
• Browder FE. Multivalued monotone nonlinear mappings and duality mappings in Banach spaces. Trans Am Math Soc. 1965;18:338–351.
   Google Scholar
• Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer-Verlag; 2000.
   Google Scholar
• Goebel K, Kirk WA. Topics in metric fixed point theory. Cambridge: Cambridge University Press; 1990.
   Google Scholar
• Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementary problems. NewYork: Springer-Verlag; 2003.
   Google Scholar
• Huang NJ. A new method for a class of nonlinear set-valued variational inequalities. Z Angew Math Mech. 1998;78:427–430.
   Web of Science ®Google Scholar
• Mashreghi J, Nasri M. Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory. Nonlinear Anal. 2010;72:2086–2099.
   Web of Science ®Google Scholar
• Nagurney A. Network economics: a variational inequality approach. Berlin: Springer; 1993.
   Google Scholar
• Oggioni G, Smeers Y, Allevi E, et al. A generalized Nash equilibrium model of market coupling in the European power system. Netw Spat Econ. 2012;12:503–560.
   Web of Science ®Google Scholar
• Polyak BT. Some methods of speeding up the convergence of iterates methods. USSR Comput Math Phys. 1964;4(5):1–17.
   Google Scholar
• Raciti F, Falsaperla P. Improved noniterative algorithm for the calculation of the equilibrium in the traffic network problem. J Optim Theory Appl. 2007;133:401–411.
   Web of Science ®Google Scholar
• Attouch H, Cabot A. Convergence of a relaxed inertial proximal algorithm for maximally monotone operators. Math Program. 2020;184:243–287.
   Web of Science ®Google Scholar
• Attouch H, Cabot A. Convergence of a relaxed inertial forward–backward algorithm for structured monotone inclusions. Appl Math Optim. 2019;80:547–598.
   Web of Science ®Google Scholar
• Dong QL, Cho YJ, Zhong LL, et al. Inertial projection and contraction algorithms for variational inequalites. J Glob Optim. 2018;70:687–704.
   Web of Science ®Google Scholar
• Dong QL, Lu YY, Yang J, et al. Approximately solving multivalued variational inequalities by using a projection and contraction algorithm. Numer Algorithms. 2017;76:799–812.
   Web of Science ®Google Scholar
• Dong QL, Jiang D, Cholamjiak P, et al. A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions. J Fixed Point Theory Appl. 2017;19(4):3097–3118.
   Web of Science ®Google Scholar
• Hieu D., Duong HN, Thai BH. Convergence of relaxed inertial methods for equilibrium problems. J Appl Numer Optim. 2021;3:215–229.
   Google Scholar
• Tian M, Jiang BN. Inertial hybrid algorithm for variational inequality problems in Hilbert spaces. J Inequal Appl. 2020;2020:12.
   Web of Science ®Google Scholar
• Anh PN, Kim JK. An interior proximal cutting hyperplane method for multivalued variational inequalities. J Nonlinear Convex Anal. 2010;11:491–502.
   Web of Science ®Google Scholar
• Anh PN, Kuno T. A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities. In: H.G. Bock et al. editors. Modeling, simulation and optimization of complex processes. Heidelberg: Springer-Verlag Berlin; 2012. p. 1–11. doi:10.1007/978-3-642-25707-0-1.
   Google Scholar
• Anh PN, Muu LD. Contraction mapping fixed point algorithms for multivalued mixed variational inequalities on network. In: Dempe S, and Vyacheslav K, editors. Optimization with multivalued mappings. Springer; 2006 (Printed in the United States of America).
   Google Scholar
• Shehu Y, Ezeora JN. Weak and linear convergence of a generalized proximal point algorithm with alternating inertial steps for a monotone inclusion problem. J Nonlinear Var Anal. 2021;5:881–892.
   Web of Science ®Google Scholar
• Fang CJ, Chen SL. A subgradient extragradient algorithm for solving multivalued variational inequality. Appl Math Comput. 2014;229:123–130.
   Web of Science ®Google Scholar
• Anh PN, Kim JK, Binh DT, et al. A proximal point-type algorithm for solving nonLipschitzian multivalued variational inequalities. Vietnam J Math. 2010;38:413–423.
   Google Scholar
• Xia FQ, Huang NJ. A projection proximal point algorithm for solving generalized variational inequalities. J Optim Theory Appl. 2011;150:98–117.
   Web of Science ®Google Scholar
• Anh PN, Le Thi HA. Outer-interior proximal projection methods for multivalued variational inequalities. Acta Math Vietnam. 2017;42:61–79.
   Web of Science ®Google Scholar
• Anh PN, Le Thi HA. Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces. Math Methods Appl Sci. 2020;43(6):3260–3279.
   Web of Science ®Google Scholar
• Anh PN, Muu LD, Strodiot JJ. Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math Vietnam. 2009;34:67–79.
   Google Scholar
• Anh PN, Thang TV, Thach HTC. Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces. Numer Algorithms. 2021;87:335–363.
   Web of Science ®Google Scholar
• Anh PN, Thang TV, Thach HTC. A subgradient proximal method for solving a class of monotone multivalued variational inequality problems. Numer Algorithms. 2022;89:409–430. doi:10.1007/s11075-021-01119-4.
   Web of Science ®Google Scholar
• Bao TQ, Khanh PQ. A projection-type algorithm for pseudomonotone nonLipschitzian multivalued variational inequalities. In: Generalized convexity, generalized monotonicity and applications. New York: Spring-Verlag; 2005. p. 113–129.
   Google Scholar
• Zhang X, Zhao W, Zhang M. A new algorithm for solving multivalued variational inequality problems. J Appl Math Comp. 2020;62:685–699.
   Google Scholar
• Cho YJ, Argyros IK, Petrot N. Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput Math Appl. 2010;60:2292–2301.
   Web of Science ®Google Scholar
• Khan AA, Migorski S, Sama M. Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data. Optim. 2019;68:1897–1931.
   Web of Science ®Google Scholar
• Ogbuisi FU, Iyiola OS, Ngnotchouye JMT, et al. On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems. J Nonlinear Funct Anal. 2021;4:2021.
   Google Scholar
• Yao JC. Multivalued variational inequalities with K-pseudomonotone operators. J Optim Theory Appl. 1994;83:391–403.
   Web of Science ®Google Scholar
• Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker; 1984.
   Google Scholar
• Alvarez F. Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces. SIAM J Optim. 2004;14:773–782.
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. New York: Springer; 2011.
   Google Scholar
• Anh PN, Muu LD, Nguyen NV, et al. Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J Optim Theory Appl. 2005;124:285–306.
   Web of Science ®Google Scholar
• Tseng P. A modified forward–backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38:431–446.
   Web of Science ®Google Scholar
• Bnouhachem A. A self-adaptive method for solving general mixed variational inequalities. J Math Anal Appl. 2005;309:136–150.
   Web of Science ®Google Scholar
• Shehu Y, Iyiola OS, Li XH, et al. Convergence analysis of projection method for variational inequalities. Comput Appl Math. 2019;38:161.
   Google Scholar