Abstract
In this paper, we study a class of general monotone equilibrium problems in a real Hilbert space which involves a monotone differentiable bifunction. For such a bifunction, a skew-symmetric type property with respect to the partial gradients is established. We suggest to solve this class of equilibrium problems with the modified combined relaxation method involving an auxiliary procedure. We prove the existence and uniqueness of the solution to the auxiliary variational inequality in the auxiliary procedure. Further, we prove also the weak convergence of the modified combined relaxation method by virtue of the monotonicity and the skew-symmetric type property.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
1. Baiocchi, C., and Capelo, A., Variational and Quasivariational Inequalites: Applications to Free Boundary Problems, John Wiley and Sons, New York, NY, 1984.
2. Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994.
3. Hadjisavvas, N., and Schaible, S., Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems, Generalized Convexity, Generalized Monotonicity, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 257–275, 1998.
4. Demyanov, V. F., and Pevnyi, A. B., Numerical Methods for Finding Saddle Points, USSR Computational Mathematics and Mathematical Physics, Vol. 12, pp. 11–52, 1972.
5. Golshtein, E. G., and Tretyakov, N. V., Augmented Lagrange Functions, Nauka, Moscow, 1989 (in Russian).
6. Polyak, B. T., Introduction to Optimization, Optimization Software, New York, NY, 1987.
7. Konnov, I. V., Combined Subgradient Methods for Finding Saddle Points, Russian Mathematics, Vol. 36, pp. 30–33, 1992.
8. Konnov, I. V., A Two-Level Subgradient Method for Finding Equilibrium Points and Solving Related Problems, USSR Computational Mathematics and Mathematical Physics, Vol. 33, pp. 453–459, 1993.
9. Konnov, I. V., Combined Relaxation Methods for Finding Equilibrium Points and Solving Related Problems, Russian Mathematics, Vol. 37, pp. 44–51, 1993.
10. Konnov, I. V., Combined Relaxation Method for Monotone Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 111, pp. 327–340, 2001.
11. Zoutendijk, G., Methods of Feasible Directions, Elsevier, Amsterdam, Netherlands, 1960.
12. More, J., and Toraldo, G., Algorithms for Bound-Constrained Quadratic Programming Problems, Numerische Mahematik, Vol. 55, pp. 377–400, 1989.
13. Madsen, K., Nielsen, H. B., and Pinar, M. C., Bound-Constrained Quadratic Programming via Piecewise-Quadratic Functions, Mathematical Programming, Vol. 85, pp. 135–156, 1999.
14. Madsen, K., Nielsen, H. B., and Pinar, M. C., A Finite Continuation Algorithm for Bound-Constrained Quadratic Programming, SIAM Journal on Optimization, Vol. 9, pp. 62–83, 1999.
15. Konnov, I. V., Methods for Solving Finite-Dimensional Variational Inequalities, DAS, Kazan, Russia, 1998 (in Russian).
16. Fan, K., A Generalization of Tychonoff's Fixed-Point Problem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.
17. Tan, K. K., and Xu, H. K., Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process, Journal of Mathematical Analysis and Applications, Vol. 178, pp. 301–308, 1993.
Author information
Authors and Affiliations
Additional information
Communicated by F. Giannessi
His research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in Shanghai.
His research was partially supported by a grant from the National Science Council of Taiwan.
Rights and permissions
About this article
Cite this article
Zeng, L.C., Yao, J.C. Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces. J Optim Theory Appl 131, 469–483 (2006). https://doi.org/10.1007/s10957-006-9162-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-006-9162-0