Iterative regularization methods for solving equilibrium problems

References

• Y.I. Alber and I. Ryazantseva, Nonlinear Ill-posed Problems of Monotone Type, Springer, Dordrecht, 2006.
   Google Scholar
• P.N. Anh, J.K. Kim, and L.D. Muu, An extragradient algorithm for solving bilevel pseudomonotone variational inequalities, J. Glob. Optim. 52 (2012), pp. 627–639.
   Web of Science ®Google Scholar
• A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers Group, Dordrecht, 1994.
   Google Scholar
• H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
   Google Scholar
• M. Bianchi and S. Schaible, Generalized monotone bifuntions and equilibrium problems, J. Optim. Theory Appl. 90 (1996), pp. 31–43.
   Web of Science ®Google Scholar
• G. Bigi, M. Castellani, M. Pappalardo, and M. Passacantando, Nonlinear Programming Techniques for Equilibria, Springer, Switzerland, 2019.
   Google Scholar
• E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), pp. 123–145.
   Google Scholar
• R.W. Cottle and J.C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl. 75 (1992), pp. 281–295.
   Web of Science ®Google Scholar
• S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, New York, 2002.
   Google Scholar
• S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization 52 (2003), pp. 333–359.
   Web of Science ®Google Scholar
• S. Dempe, V. Kalashnikov, G.A. Pérez-Valdés, and N. Kalashnykova, Bilevel Programming Problems, Springer, Berlin, 2015.
   Google Scholar
• B.V. Dinh, P.G. Hung, and L.D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim. 35 (2014), pp. 539–563.
   Web of Science ®Google Scholar
• F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin, 2002.
   Google Scholar
• K. Fan, A minimax inequality and applications, in Inequality III, O. Shisha ed. Academic Press, New York, 1972, pp. 103–113.
   Google Scholar
• S.D. Flam and A.S. Antipin, Equilibrium programming and proximal-like algorithms, Math. Program.78 (1997), pp. 29–41.
   Web of Science ®Google Scholar
• K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
   Google Scholar
• D.V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal. 21 (2016), pp. 478–501.
   Web of Science ®Google Scholar
• D.V. Hieu, Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems, Math. Model. Anal. 24 (2019), pp. 1–19.
   Web of Science ®Google Scholar
• D.V. Hieu, An explicit parallel algorithms for variational inequalities, Bull. Malays. Math. Sci. Soc.42 (2019), pp. 201–221.
   Web of Science ®Google Scholar
• D.V. Hieu and A. Moudafi, Regularization projection method for solving bilevel variational inequality problem, Optim. Lett. 15 (2021), pp. 205 –229. DOI:https://doi.org/10.1007/s11590-020-01580-5.
   Google Scholar
• D.V. Hieu and J.J. Strodiot, Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl. 20(131) (2018), pp. 1–32.
   Google Scholar
• D.V. Hieu, P.K. Anh, and L.D. Muu, Strong convergence of subgradient extragradient method with regularization f or solving variational inequalities, Optim. Eng. (2020). DOI:https://doi.org/10.1007/s11081-020-09540-9.
   Google Scholar
• D.V. Hieu, Y.J. Cho, and Y.B. Xiao, Modified extragradient algorithms for solving equilibrium problems, Optimization 67 (2018), pp. 2003–2029.
   Web of Science ®Google Scholar
• D.V. Hieu, Y.J. Cho, and Y.B. Xiao, Golden ratio algorithms with new stepsize rules for variational inequalities, Math. Methods Appl. Sci. 42 (2019), pp. 6067–6082. DOI:https://doi.org/10.1002/mma.5703.
   Google Scholar
• D.V. Hieu, P.K. Quy, and L.V. Vy, Explicit iterative algorithms for solving equilibrium problems, Calcolo 56 (2019). 11 pp.
   Web of Science ®Google Scholar
• D.V. Hieu, L.D. Muu, H.N. Duong, and B.H. Thai, Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces, Math. Methods Appl. Sci. 43 (2020), pp. 9745–9765. DOI:https://doi.org/10.1002/mma.6647.
   Google Scholar
• D.V. Hieu, L.D. Muu, P.K. Quy, and L.V. Vy, Explicit extragradient-like method with regularization for variational inequalities, Res. Math. 74 (2019), pp. 137.
   Web of Science ®Google Scholar
• D.V. Hieu, J.J. Strodiot, and L.D. Muu, Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems, J. Comput. Appl. Math. 376 (2020), pp. 112844. DOI:https://doi.org/10.1016/j.cam.2020.112844.
   Web of Science ®Google Scholar
• D.V. Hieu, Y.J. Cho, Y-B. Xiao, and P. Kumam, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math. (2020). DOI: https://doi.org/10.1007/s10013-020-00447-7.
   Web of Science ®Google Scholar
• P.G. Hung and L.D. Muu, The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions, Nonlinear Anal. 74 (2011), pp. 6121–6129.
   Web of Science ®Google Scholar
• P.G. Hung and L.D. Muu, On inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems, Vietnam J. Math. 40 (2012), pp. 255–274.
   Google Scholar
• E.V. Khoroshilova, Extragradient-type method for optimal control problem with linear constraints and convex objective function, Optim. Lett. 7 (2013), pp. 1193–1214.
   Web of Science ®Google Scholar
• I.V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001.
   Google Scholar
• I.V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems, J. Optim. Theory Appl. 119 (2003), pp. 317–333.
   Web of Science ®Google Scholar
• I.V. Konnov, Equilibrium Models and Variational Inequalities, Elsevier, Amsterdam, 2007.
   Google Scholar
• I.V. Konnov, Regularization methods for nonmonotone equilibrium problems, J. Nonlinear Convex Anal. 10 (2009), pp. 93–101.
   Web of Science ®Google Scholar
• P.E. Maingé and A. Moudafi, Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, J. Nonlinear Convex Anal. 9 (2008), pp. 283–294.
   Web of Science ®Google Scholar
• G. Mastroeni, On auxiliary principle for equilibrium problems, Publ. Dipart. Math. Univer. Pisa 3 (2000), pp. 1244–1258.
   Google Scholar
• A. Moudafi, Proximal point algorithm extended to equilibrium problem, J. Nat. Geom. 15 (1999), pp. 91–100.
   Google Scholar
• L.D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. 18 (1992), pp. 1159–1166.
   Web of Science ®Google Scholar
• T.T.V. Nguyen, J.J. Strodiot, and V.H. Nguyen, Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, J. Optim. Theory Appl. 160 (2014), pp. 809–831.
   Web of Science ®Google Scholar
• A. Pietrus, T. Scarinci, and V. Veliov, High order discrete approximations to Mayer's problems for linear systems, SIAM J. Control Optim. 56 (2016), pp. 102–119.
   Web of Science ®Google Scholar
• T.D. Quoc, L.D. Muu, and V.H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008), pp. 749–776.
   Web of Science ®Google Scholar
• P. Santos and S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Comput. Appl. Math. 30 (2011), pp. 91–107.
   Web of Science ®Google Scholar
• M. Seydenschwanz, Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions, Comput. Optim. Appl. 629 (2015), pp. 731–760.
   Google Scholar
• Q. Shu, M. Sofonea, and Y.B. Xiao, Tykhonov well-posedness of split problems. J. Inequal. Appl. 153 (2020). DOI:https://doi.org/10.1186/s13660-020-02421-w.
   Google Scholar
• M. Sofonea and Y.B. Xiao, Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electron. J. Differ. Equ. 2019(64) (2019), pp. 1–19.
   Google Scholar
• M. Sofonea, Y.B. Xiao, and S.D. Zeng, Generalized penalty method for history-dependent variational-hemivariational inequalities, submitted.
   Google Scholar
• J.J. Strodiot, T.T.V. Nguyen, and V.H. Nguyen, A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems, J. Glob. Optim. 56 (2013), pp. 373–397.
   Web of Science ®Google Scholar
• J.J. Strodiot, P.T. Vuong, and T.T.V. Nguyen, A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces, J. Glob. Optim. 64 (2016), pp. 159–178.
   Web of Science ®Google Scholar
• P.T. Vuong, J.J. Strodiot, and V.H. Nguyen, On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space, Optimization 64 (2015), pp. 429–451.
   Web of Science ®Google Scholar
• H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), pp. 109–113.
   Web of Science ®Google Scholar
• I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. I. Yamada: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithms for Feasibility and Optimization and their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., Elsevier, Amsterdam, 2001, pp. 473–504.
   Google Scholar