A combined method of regularization and descent along a primal gap function for solving nonsmooth monotone equilibrium problems

Authors

Keywords:

nonsmooth monotone equilibrium problem, primal gap function, descent method, inexact line search, uniformly convex function

Abstract

A combined method of regularization and descent along a primal gap (merit) function for solving nonsmooth monotone equilibrium problems is proposed. The same auxiliary uniformly convex function is used for the construction of regularized problems and gap functions. The descent method along a gap function with an inexact line search is applied to solve the regularized problems.

Author Biography

O.V. Pinyagina

References

  1. Patriksson M. Nonlinear programming and variational inequality problems: a unified approach. Dordrecht: Kluwer Academic Publ., 1999.
  2. Konnov I.V., Pinyagina O.V. D-gap functions for a class of equilibrium problems in Banach spaces // Computational Methods in Applied Mathematics. 2003. 3. 274-286.
  3. Тихонов А.Н., Арсенин В.Я. Методы решения некорректных задач. М.: Наука, 1979.
  4. Browder F.E. Existence and approximation of solutions of nonlinear variational inequalities // Proc. Nat. Acad. Sci. USA. 1966. 56. 1080-1086.
  5. Konnov I.V., Kum S. Descent methods for mixed variational inequalities in a Hilbert space // Nonlinear Analysis: Theory, Methods and Applications. 2001. 47. 561-572.
  6. Konnov I.V., Kum S., Lee G.M. On convergence of descent methods for variational inequalities in a Hilbert space // Math. Meth. Oper. Res. 2002. 55. 371-382.
  7. Konnov I.V., Pinyagina O.V. D-gap functions and descent methods for a class of monotone equilibrium problems // Lobachevskii J. of Mathematics. 2003. 13. 57-65.
  8. Kaplan A., Tichatschke R. Auxiliary problem principle and the approximation of variational inequalities with non-symmetric multi-valued operators // CMS Conf. Proc. 2000. 27. 185-209.
  9. Pinyagina O.V., Ali M.S. S. Descent method for monotone mixed variational inequalities // Calcolo. 2008. 45. 1-15.
  10. Коннов И.В., Пинягина О.В. Метод решения монотонных смешанных вариационных неравенств // Уч. зап. Казанск. ун-та. 2011. 153, кн. 1. 221-230.
  11. Байокки К., Капело А. Вариационные и квазивариационные неравенства. Приложения к задачам со свободной границей. М.: Наука, 1988.
  12. Blum E., Oettli W. From optimization and variational inequalities to equilibrium problems // The Mathem. Student. 1994. 63. 123-145.
  13. Konnov I.V. Combined relaxation methods for variational inequalities. Berlin: Springer, 2001.
  14. Васильев Ф.П. Численные методы решения экстремальных задач. М.: Наука, 1988.
  15. Chadli O., Konnov I.V., Yao J.C. Descent method for equilibrium problems in a Banach space // Comp. Mathem. Appl. 2004. 48. 609-616.
  16. Бакушинский А.Б., Гончарский А.В. Итерационные методы решения некорректных задач. М.: Наука, 1989.
  17. Konnov I.V. Iterative solution methods for mixed equilibrium problems and variational inequalities with non-smooth functions // Game Theory: Strategies, Equilibria, and Theorems. Ed. by I.N. Haugen and A.S. Nilsen. Hauppauge: NOVA, 2008. Chapter 4. 117-160.
  18. Демьянов В.Ф., Рубинов А.И. Основы негладкого анализа и квазидифференциальное исчисление. М.: Наука, 1990.
  19. Коннов И.В. Метод спуска с неточным линейным поиском для смешанных вариационных неравенств // Известия вузов. Математика. 2009. № 8. 37-44.

Published

11-05-2012

How to Cite

Пинягина О. A Combined Method of Regularization and Descent Along a Primal Gap Function for Solving Nonsmooth Monotone Equilibrium Problems // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2012. 13. 316-323

Issue

Section

Section 1. Numerical methods and applications