Abstract
We consider a set-valued (generalized) variational inequality problem in a finite-dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We suggest 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 conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer-Verlag, Berlin (2001)
Konnov, I.V.: Generalized monotone equilibrium problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Applications, vol. 76, pp 559–618. Springer, New York (2005)
Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)
Alart, P., Lemaire, B.: Penalization in non-classical convex programming via variational convergence. Math. Program 51, 307–331 (1991)
Antipin, A.S., Vasil’ev, F.P.: A stabilization method for equilibrium programming problems with an approximately given set. Comput. Math. Math. Phys. 39, 1707–1714 (1999)
Salmon, G., Nguyen, V.H., Strodiot, J.J.: Coupling the auxiliary problem principle and epiconvergence theory for solving general variational inequalities. J. Optim. Theory Appl. 104, 629–657 (2000)
Kaplan, A., Tichatschke, R.: A general view on proximal point methods for variational inequalities in Hilbert spaces. J. Nonl. Conv. Anal. 2, 305–332 (2001)
Konnov, I.V.: Application of penalty methods to non-stationary variational inequalities. Nonl. Anal. Theory Methods Appl. 92, 177–182 (2013)
Konnov, I.V., Dyabilkin, D.A.: Nonmonotone equilibrium problems: Coercivity conditions and weak regularization. J. Global Optim. 49, 575–587 (2011)
Attouch, H.: Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston (1984)
Fan Ky.: A minimax inequality and applications. In: Shisha, O (ed.) Inequalities III, pp 103–113. Academic Press, New York (1972)
Aubin, J.-P.: Optima and Equilibria. Springer-Verlag, Berlin (1998)
Kneser, H.: Sur le théorème fondamental de la théorie des jeux. Compt. Rend. L’Acad. Sci. Paris 234, 2418–2420 (1952)
Gwinner, J.: On the penalty method for constrained variational inequalities. In: Hiriart-Urruty, J.-B., Oettli, W., Stoer, J (eds.) Optimization: Theory and Algorithms, pp 197–211. Marcel Dekker, New York (1981)
Muu, L.D., Oettli, W.: A Lagrangian penalty function method for monotone variational inequalities. Num. Funct. Anal. Optim. 10, 1003–1017 (1989)
Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Their Applications. Birkhauser, Boston (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Konnov, I.V. An Inexact Penalty Method for Non Stationary Generalized Variational Inequalities. Set-Valued Var. Anal 23, 239–248 (2015). https://doi.org/10.1007/s11228-014-0293-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-014-0293-4
Keywords
- Variational inequality
- Non-stationarity
- Non-monotone mappings
- Set-valued mappings
- Approximate solutions
- Penalty method
- Coercivity conditions