Regularized penalty method for non-stationary set valued equilibrium problems in Banach spaces
In this research works, we consider the general regularized penalty method for non-stationary set valued equilibrium problem in a Banach space. We define weak coercivity conditions and show that the weak and strong convergence problems of the regularized penalty method.
A. S. Antipin and F. P. Vasilev, A stabilization method for equilibrium programming problems with an approximately given set, Comput. Math. Math. Phys. 39 (1999), 1707–1714.
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Ap- plications to Free Boundary Problems, Wiley, New York, 1984.
A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Ill-Posed Problems, Nauka, Moscow, 1989, [in Russian].
M. Bianchi and R. Pini, Coercivity conditions for equilibrium problems, J. Optim. Theory Appl. 124 (2000), 79–92.
E. Blum and W. Oettli, From optimization and variational inequalities to equi- librium problems, Math. Stud. 63 (1994), 123–145.
Ky. Fan, A minimax inequality and applications, In: O. Shisha, (ed.) Inequalities III, pp. 103–113. Academic Press, New York, 1972.
J. Gwinner, On the regularization of monotone variational inequalities, Oper. Res. Verfahren 28 (1978), 374–386.
J. Gwinner, On the penalty method for constrained variational inequalities, In: J.-B. Hiriart-Urruty, W. Oettli, J. Stoer, (eds.) Optimization: Theory and Al- gorithms, pp. 197-211. Marcel Dekker, New York, 1981.
L. D. Muu and W. Oettli, A Lagrangian penalty function method for monotone variational inequalities, Numer. Funct. Anal. Optim. 10 (1989), 1003–1017.
I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001.
I. V. Konnov, Combined relaxation methods for generalized monotone variational inequalities, In: I. V. Konnov, D. T. Luc, A. M. Rubinov, (eds.) Generalized Convexity and Related Topics, pp. 3-31, Springer, Berlin, 2007.
I. V. Konnov, Regularization method for nonmonotone equilibrium problems, J. Nonlinear Convex Anal. 10 (2009), 93–101.
I. V. Konnov, On penalty methods for non monotone equilibrium problems, J. Glob. Optim. 59 (2014), 131–138.
I. V. Konnov, Regularized penalty method for general equilibrium problems in Banach spaces, J. Optim. Theory Appl. 164 (2015), 500–513.
I. V. Konnov and D. A. Dyabilkin, Nonmonotone equilibrium problems: coercivity conditions and weak regularization, J. Glob. Optim. 49 (2011), 575–587.
I. V. Konnov and Z. Liu, Vector equilibrium problems on unbounded sets, Lobachevskii J. Math. 31 (2010), 232–238.
B. S. Lee, M. F. Khan and Salahuddin, Generalized nonlinear quasi-variational inclusions in Banach spaces, Comput. Maths. Appl. 56 (5) (2008), 1414–1422.
B. S. Lee and Salahuddin, A general system of regularized nonconvex variational inequalities, Appl. Comput. Math. 3 (4) (2014).
J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–517.
H. Nikaido and K. Isoda, Note on noncooperative convex games, Pac. J. Math. 5 (1955), 807–815.
B. T. Polyak, Introduction to Optimization, Nauka, Moscow (1983) (Engl. transl. in Optimization Software, New York, 1987.
V. V. Podinovskii and V. D. Nogin, Pareto-Optimal Solutions of Multiple Objective Problems, Nauka, Moscow, 1982.
Salahuddin, Regularization techniques for inverse variational inequalities involving relaxed cocoercive mapping in Hilbert spaces, Nonlinear Anal. Forum 19 (2014), 65–76.
Salahuddin, Convergence analysis for hierarchical optimizations, Nonlinear Anal. Forum 20 (2) (2015), 229–239.
Salahuddin, Regularized equilibrium problems in Banach spaces, Korean J. Math. 24 (1) (2016), 51–53.
Salahuddin, On penalty method for non-stationary general set valued equilibrium problems, Commun. Appl. Nonlinear Anal. 23 (4) (2016), 82–92.
Salahuddin, Regularized equilibrium problems on Hadamard manifolds, Nonlinear Anal. Forum, 21 (2)(2016), 91–101.
Salahuddin and R. U. Verma, System of nonlinear generalized regularized nonconvex variational inequalities in Banach spaces, Adv. Nonlinear Var. Inequal. 19 (2) (2016), 27–40.
A. H. Siddiqi, M. K. Ahmad and Salahuddin, Existence results for generalized nonlinear variational inclusions, Appl. Maths. Letts. 18 (8) (2005), 859–864.
F. P. Vasilev, Methods for Solving Extremal Problems, Nauka, Moscow, 1981, [in Russian].
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