A projection and contraction method with adaptive step sizes for solving bilevel pseudo-monotone variational inequality problems

References

• Anh PK, Thong DV, Vinh NT. Improved inertial extragradient methods for solving pseudo-monotone variational inequalities. Optimization. 2020. doi:https://doi.org/10.1080/02331934.2020.1808644.
   Web of Science ®Google Scholar
• Dong QL, Cho YJ, Rassias TM. The projection and contraction methods for finding common solutions to variational inequality problems. Optim Lett. 2018;12:1871–1896.
   Web of Science ®Google Scholar
• Dong QL, Cho YJ, Zhong LL, et al. Inertial projection and contraction algorithms for variational inequalities. J Global Optim. 2018;70:687–704.
   Web of Science ®Google Scholar
• Khanh PQ, Thong DV, Vinh NT. Versions of the subgradient extragradient method for pseudomonotone variational inequalities. Acta Appl Math. 2020;170:319–345. doi:https://doi.org/10.1007/s10440-020-00335-9.
   Web of Science ®Google Scholar
• Reich S, Thong DV, Dong Q, et al. New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings. Numer Algorithms. 2020. doi:https://doi.org/10.1007/s11075-020-00977-8.
   Web of Science ®Google Scholar
• He BS, Liao LZ. Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl. 2002;112:111–128.
   Web of Science ®Google Scholar
• Korpelevich GM. The extragradient method for finding saddle points and other problems. Ekonom Mate Metod. 1976;12:747–756.
   Google Scholar
• Antipin AS. On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonom I Mate Metod. 1976;12:1164–1173.
   Google Scholar
• Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optimization. 2011;61:1119–1132.
   Web of Science ®Google Scholar
• Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw. 2011;26:827–845.
   Web of Science ®Google Scholar
• Censor Y, Gibali A, Reich S. The subgradient extragradientmethod for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335.
   PubMed Web of Science ®Google Scholar
• Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. New York: Springer; 2003. (Springer Series in Operations Research, Vols. I and II).
   Google Scholar
• Kraikaew R, Saejung S. Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl. 2014;163:399–412.
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Weak and strong convergence theorems for variational inequality problems. Numer Algorithms. 2017;78:1045–1060.
   Web of Science ®Google Scholar
• Thong DV, Li X, Dong Q, et al. An inertial Popov's method for solving pseudomonotone variational inequalities. Optim Lett. 2020. doi:https://doi.org/10.1007/s11590-020-01599-8.
   PubMed Web of Science ®Google Scholar
• Thong DV, Shehu Y, Iyiola OS, et al. New hybrid projection methods for variational inequalities involving pseudomonotone mappings. Optim Eng. 2020. doi:https://doi.org/10.1007/s11081-020-09518-7.
   Web of Science ®Google Scholar
• Thong DV, Shehu Y, Iyiola OS. A new iterative method for solving pseudomonotone variational inequalities with non-Lipschitz operators. Comp Appl Math. 2020;39:108. doi:https://doi.org/10.1007/s40314-020-1136-6.
   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
• Vuong PT. On the weak convergence of the extragradient method for solving pseudomonotone variational inequalities. J Optim Theory Appl. 2018;176:399–409.
   PubMed 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
• He BS. A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim. 1997;35:69–76.
   Web of Science ®Google Scholar
• Sun DF. A class of iterative methods for solving nonlinear projection equations. J Optim Theory Appl. 1996;91:123–140.
   Web of Science ®Google Scholar
• 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. 2014;57:339–363.
   Web of Science ®Google Scholar
• Luo ZQ, Pang JS, Ralph D. Mathematical programs with equilibrium constraints. Cambridge: Cambridge University Press; 1996.
   Google Scholar
• Solodov MV. An explicit descent method for bilevel convex optimization. J Convex Anal. 2007;14:227–237.
   Web of Science ®Google Scholar
• Daniele P, Giannessi F, Maugeri A. Equilibrium problems and variational models. Dordrecht: Kluwer Academic; 2003.
   Google Scholar
• Giannessi F, Maugeri A, Pardalos PM. Equilibrium problems: nonsmooth optimization and variational inequality models. Dordrecht: Kluwer Academic; 2004.
   Google Scholar
• Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer; 2001.
   Google Scholar
• Yao Y, Marino G, Muglia L. A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality. Optimization. 2014;63:559–569.
   Web of Science ®Google Scholar
• Zegeye H, Shahzad N, Yao Y. Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization. 2015;64:453–471.
   Web of Science ®Google Scholar
• Trujillo CR, Zlobec S. Bilevel convex programming models. Optimization. 2009;58:1009–1028.
   Web of Science ®Google Scholar
• Glackin J, Ecker JG, Kupferschmid M. Solving bilevel linear programs using multiple objective linear programming. J Optim Theory Appl. 2009;140:197–212.
   Web of Science ®Google Scholar
• Dempe S. Foundations of bilevel programming. Dordrecht: Kluwer Academic Publishers; 2002.
   Google Scholar
• Dempe S, Kalashnikov V, Prez-Valdés GA, et al. Bilevel programming problems: theory algorithms and applications to energy networks. Berlin Heidelberg: Springer-Verlag; 2015.
   Google Scholar
• Dinh BV, Muu LD. Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities. Acta Math Vietnam. 2013;38:529–540.
   Google Scholar
• Dinh BV, Hung PG, Muu LD. Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems. Numer Funct Anal Optim. 2014;35:539–563.
   Web of Science ®Google Scholar
• Dinh BV, Muu LD. A projection algorithm for solving pseudomonotone equilibrium problems and its application to a class of bilevel equilibria. Optimization. 2015;64:559–575.
   Web of Science ®Google Scholar
• Kalashnikov VV, Klashnikova NI. Solving two-level variational inequality. J Glob Optim. 1996;8:289–294.
   Web of Science ®Google Scholar
• Moudafi A. Proximal methods for a class of bilevel monotone equilibrium problems. J Glob Optim. 2010;47:287–292.
   Web of Science ®Google Scholar
• Thong DV, Triet NA, Li X et al.. Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer Algorithms. 2020;83:1123–1143.
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. A strong convergence of modified subgradient extragradient method for solving bilevel pseudomonotone variational inequality problems. Optimization. 2020;69:1313–1334.
   Web of Science ®Google Scholar
• Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker; 1984.
   Google Scholar
• Cegielski A. Iterative methods for fixed point problems in Hilbert spaces. Berlin: Springer; 2012. (Lecture Notes in Mathematics; 2057).
   Google Scholar
• Muu LD, Quy NV. On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J Math. 2015;43:229–238.
   Web of Science ®Google Scholar
• Cottle RW, Yao JC. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl. 1992;75:281–295.
   Web of Science ®Google Scholar
• Yamada I. The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S, editors. Inherently parallel algorithm for feasibility and optimization. Amsterdam: Elsevier; 2001. p. 473–504.
   Google Scholar
• Saejung S, Yotkaew P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 2012;75:742–750.
   Web of Science ®Google Scholar
• Denisov SV, Semenov VV, Chabak LM. Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal. 2015;51:757–765.
   Web of Science ®Google Scholar
• Anh PN, Kim JK, Muu LD. An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J Glob Optim. 2012;52:627–639.
   Web of Science ®Google Scholar
• Gibali A, Shehu Y. An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces. Optimization. 2019;68:13–32.
   Web of Science ®Google Scholar
• Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25:502–520.
   Web of Science ®Google Scholar
• Shehu Y, Dong QL, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization. 2019;68:385–409.
   Web of Science ®Google Scholar
• Harker PT, Pang JS. A damped-Newton method for the linear complementarity problem. Lectures Appl Math. 1990;26:265–284.
   Google Scholar
• Solodov MV, Svaiter BF. A new projection method for variational inequality problems. SIAM J Control Optim. 1999;37:765–776.
   Web of Science ®Google Scholar
• Hu X, Wang J. Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network. IEEE Trans Neural Netw. 2006;17:1487–1499.
   PubMed Web of Science ®Google Scholar
• Bot RI, Csetnek ER, Vuong PT. The forward–backward–forward method from discrete and continuous perspective for pseudo-monotone variational inequalities in Hilbert spaces. Eur J Oper Res. 2020;287:49–60.
   Web of Science ®Google Scholar