A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

References

• Dempe S. Foundations of Bilevel programming. Dordrecht: Kluwer Academic Publishers; 2002.
   Google Scholar
• Luo JQ, Pang JS, Ralph D. Mathematical programs with equilibrium constraints. Cambridge, MA: Cambridge University Press; 1996.
   Google Scholar
• Liduka H, Yamada I. A subgradient algorithm for the equilibrium problem over the fixed point set and its application. Optimization. 2008;1–11.
   Google Scholar
• Vuong PT, Strodiot J-J, Nguyen VH. On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization. 2015: 64:429–431.
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PH. Convex analysis and monotone operator in Hilbert spaces. New York, NY: Springer; 2010.
   Google Scholar
• Bigi G, Castellani M, Pappalardo M, et al. Existence and solution methods for equilibria. European J. Oper. Res. 2013;227:1–11.
   Web of Science ®Google Scholar
• Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52:301–316.
   Web of Science ®Google Scholar
• Konnov IV. Regularization methods for nonmonotone equilibrium problems. J. Nonlinear Convex Anal. 2009;10:93–101.
   Web of Science ®Google Scholar
• Muu LD, Quoc TD. Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash--Cournot equilibrium model. J. Optim. Theory Appl. 2009;142:185–204.
   Web of Science ®Google Scholar
• Quoc TD, Anh PN, Muu LD. Dual extragradient lgorithms extended to equilibrium problems. J. Glob. Optim. 2012;52:139–159.
   Web of Science ®Google Scholar
• Santos P, Scheimberg S. An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 2011;30:91–107.
   Web of Science ®Google Scholar
• Bello Cruz JY, Millan RD. A direct splitting method for nonsmooth variational inequalities. J. Optim. Theory Appl. 2014;161:729–737.
   Web of Science ®Google Scholar
• Eckstein J, Svaiter AF. General projective splitting methods for sums of maximal monotone operators. SIAM Control Optim. 2009;48:787–811.
   Web of Science ®Google Scholar
• Moudafi A. On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 2009;359:508–513.
   Web of Science ®Google Scholar
• Tseng P. A modified forward--backward splitting method for maximal monotone mappings. SIAM Control Optim. 2000;38:431–446.
   Web of Science ®Google Scholar
• Zhang H, Cheng L. Projective splitting methods for sums of maximal monotone operators with applications. J. Math. Anal. Appl. 2013;406:321–334.
   Web of Science ®Google Scholar
• Buong N, Duong LT. An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2011;151:513–524.
   Web of Science ®Google Scholar
• Buong N, Anh NTQ. An implicit iterative method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011. doi:10.1155/2011/276859.
   Google Scholar
• Zhang C, Yang C. A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings. Fixed Point Theory Appl. 2014;2014:60.
   Google Scholar
• Xu HK. Iterative algorithms for nonlinear operators. J. London Math. Soc. 2002;66:240–256.
   Web of Science ®Google Scholar
• Deutsch F, Yamada I. Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal, Optim. 1998;19:33–56.
   Web of Science ®Google Scholar
• Anh PN, Muu LD, Nguyen VH, et al. Using the Banach contraction principle to implement the proximal point method fo multivalued monotone variational inequalities. J. Optim. Theory Appl. 2005;124:285–306.
   Web of Science ®Google Scholar
• Zhu D, Marcotte P. Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Optim. 1996;6:714–726.
   Web of Science ®Google Scholar
• Hung PG, Muu LD. The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. TMA. 2011;74:6121–6129.
   Web of Science ®Google Scholar
• Combettes PL, Hirstoaga SA. Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005;6:117–136.
   Web of Science ®Google Scholar
• Rockafellar RT. Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976;5:877–898.
   Web of Science ®Google Scholar
• Smith JL. Monotone dynamical systems. Mathematical survey and monographs. Providence, RI: American Mathematical Society. 1995.
   Google Scholar