Penalty Method for Non-Stationary General Variational Like Inequalities
DOI:
https://doi.org/10.5890/JAND.2021.09.002Abstract
We suggest a general variational like inequalities in an infinite dimensional space, where only approximation sequences are known instead of an exact value of the cost mapping and feasible set, and to apply a sequence of inexact solutions of auxiliary problems involving general penalty functions. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under certain coercivity type assumptions.References
[1] Baiocchi, C. and Capelo, A. (1984), Variational and quasivariational inequalities: Applications to free boundary problems, Wiley, New York.
[2] Konnov, I.V. and Salahuddin (2017), Two-level iterative method for non-stationary mixed variational inequalities, Russian Math., 61(10), 44-53.
[3] Lee, B.S. and Salahuddin (2003), Generalized nonlinear mixed $eta$-type variational inequality problem, Nonlinear Analy. Forum., 8(1), 11-21.
[4] Ahmad, R. and Salahuddin (2003), On Generalized Multivalued Nonlinear Mixed Variational-Like Inclusions, Soochow J. Mathematics, 29(3), 235-247.
[5] Khan, M.F. and Salahuddin (2006), Generalized multivalued quasi variational-Like inclusions, Adv. Nonlinear Var. Inequal., 9 (1), 37-50.
[6] Konnov, I.V. (2015), An inexact penalty method for non stationary generalized variational inequalities, Set-Valued Var. Anal., 23, 239-248.
[7] Konnov, I.V. (2007), Equilibrium models and variational inequalitities, Elsevier Amsterdam.
[8] Alart,P. and Lemaire, B. (1991), Penalization in non-classical convex programming via variational convergence, Math. Program., 51, 307-331.
[9] Vasilev, F.P. (1981), Methods for solving external problems, Nauka, Moscow.
[10] Antipin, A.S. and Vasilev, F.P. (1999), A stabilization method for equilibrium programming problems with an approximately given set, Comput. Math. Math. Phys., 39, 1707-1714.
[11] Courant, R. (1943), Variational methods for the solution of problems in equilibrum and vibrations, Bul. of the Amer. Mathem. Soc., 49, 1-23.
[12] Konnov, I.V. (2013), Application of penalty methods to non-stationary variational inequalities, Nonl. Anal.: Theory, Methods and Appl., 92, 177-182.
[13] Salmon, G., Nguyen, V.H., and Strodiot, J.J. (2000), Coupling the auxiliary problem principle and epiconvergence theory for solving general variational inequalities, J. Optim. Theory Appl., 104, 629-657.
[14] Parida, J., Sahoo, M., and Kumar, A. (1989), A variational like inequality problem, Bull. Austral. Math. Soc. 39, 225-231.
[15] Siddiqi, A.H., Khan, M.F., and Salahuddin (1998), On vector variational like inequalities, Far East J. Math. Sci., Special volume, part III, 319-329.
[16] Kinderleher D. and Stampacchia, G. (1980), An introduction to variational inequalities and their applications, Academic Press, New York.
[17] Attouch, H. (1984), Variational convergence for functions and operators, Pitman Advanced Publishing Program, Boston.
[18] Ky Fan (1972), A minimax inequality and applications, In: Shisha, O. (ed) Inequalities III, pp. 103-113, Academic Press, New York.
[19] Aubin, J.P. (1998), Optima and equilibria, Springer-Verlag, berlin.
[20] Kneser, H. (1952), Sur le theoreme foudamental dela theorie des jeus, Compt. Rend. LAcad. Sci. Pans., 234, 2418--2420.
[21] Konnov, I.V. and Dyabilkin, D.A. (2011), Non-monotone equilibrium problems, coercivity conditions and weak regularization, J. Global Optim., 49, 575-587.
[22] Gwinner, J. (1981), On the penalty method for constrained variational inequalities: In Hiriart Urruty, J.-B., Oettli, W., Store, J. (eds) Optimization: Theory and Applications 197-211, Marcel Dekker, New York.
[23] Muu, L.D. and Oettle, W. (1989), A Lagrangian penalty function methods for monotone variational inequalitilities, Nummer. Funct. Anal. Optim., 10, 1003-1017.
[24] Konnov, I.V. (2003), Application of the proximal point method to non monotone equilibrium problems, J. Optim. Theo. Appl., 119, 319-333.
[25] Salahuddin (2016), On penalty method for non-stationary general set valued equilibrium problems, Commun. Appl. Nonlinear Anal., 23(4), 82-92.
[26] Panagiotopoulos, P.D. (1985), Inequality problems in mechanics and their applications, BirKhauser, Boston.
[27] Yu, C., Toe, K.L., Zhang, L., and Bai, Y. (2010), A new exact penalty function method for continuous inequality constrained optimization problems, J. Indust. Manag. Optim., 6, 895-910.
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