Skip to main content
SpringerLink
Log in
Book cover

Recent Advances in Optimization pp 87–102Cite as

Application of the Proximal Point Method to a System of Extended Primal-Dual Equilibrium Problems

  • Conference paper

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 563))

Summary

We consider a general system of equilibrium type problems which can be viewed as an extension of Lagrangean primal-dual equilibrium problems. We propose to solve the system by an inexact proximal point method, which converges to a solution under monotonicity assumptions. In order to make the method implementable, we suggest to make use of a dual descent algorithm and utilize gap functions for ensuring satisfactory accuracy of certain auxiliary problems. Some examples of applications are also given.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. John Wiley and Sons, New York, 1984.

    Google Scholar 

  2. S.C. Billups and M.C. Ferris. QPCOMP: A quadratic programming based solver for mixed complementarity problems. Math. Programming, 76: 533–562, 1997.

    Article  MathSciNet  Google Scholar 

  3. E. Blum and W. Oettli. From optimization and variational inequalities to equilibrium problems. The Mathematics Student, 63: 127–149, 1994.

    MathSciNet  Google Scholar 

  4. N. El Farouq. Pseudomonotone variational inequalities: Convergence of proximal methods. J. Optim. Theory Appl., 109:311–326, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Gabay. Application of the method of multipliers to variational inequalities. In: M. Fortin and R. Glowinski (eds), Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam, 299–331, 1983.

    Google Scholar 

  6. F. Giannessi (ed). Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.

    Google Scholar 

  7. P.T. Harker and J.S. Pang. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Programming, 48:161–220, 1990.

    Article  MathSciNet  Google Scholar 

  8. I.V. Konnov. A combined method for variational inequalities with monotone operators. Comp. Math. and Math. Physics, 39:1051–1056, 1999.

    MATH  MathSciNet  Google Scholar 

  9. I.V. Konnov. Approximate methods for primal-dual mixed variational inequalities. Russian Math. (Iz. VUZ), 44:55–66, n.12, 2000.

    MATH  MathSciNet  Google Scholar 

  10. I.V. Konnov. Combined Relaxation Methods for Variational Inequalities. Springer-Verlag, Berlin, 2001.

    Google Scholar 

  11. I.V. Konnov. Combined relaxation method for monotone equilibrium problems. J. Optim. Theory Appi., 111:327–340, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. I.V. Konnov. Dual approach to one class of mixed variational inequalities. Comp. Math. and Math. Physics, 42:1276–1288, 2002.

    MATH  MathSciNet  Google Scholar 

  13. I.V. Konnov. The splitting method with linear searches for primal-dual variational inequalities. Comp. Math. and Math. Physics, 43:494–507, 2003.

    MATH  MathSciNet  Google Scholar 

  14. I.V. Konnov. Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl., 119: 317–333, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  15. H.W. Kuhn. On a theorem of Wald. In: H.W. Kuhn and A.W. Tucker(eds), Linear Inequalities and Related Topics. Annals of Mathem. Studies 38, Princeton University Press, Princeton, 265–273, 1956.

    Google Scholar 

  16. M.A. Noor. General algorithm for variational inequalities. Math. Japonica, 38:47–53, 1993.

    MATH  MathSciNet  Google Scholar 

  17. M. Patriksson. Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer Academic Publishers, Dordrecht, 1999.

    Google Scholar 

  18. R.T. Rockafellar and R.J.B. Wets. Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time. SIAM J. Control Optim., 28:810–822, 1990.

    Article  MathSciNet  Google Scholar 

  19. M. Sibony. Méthodes itératives pour les equations et inequations aux dérivées partielles non lin’eares de type monotone. Calcolo, 7:65–183, 1970.

    MATH  MathSciNet  Google Scholar 

  20. P. Tseng. Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim., 29:119–138, 1991.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Kazan University, Kazan, Russia

    Igor V. Konnov

Authors
  1. Igor V. Konnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Avignon, 33, rue Louis Pasteur, 84000, Avignon, France

    Alberto Seeger

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Konnov, I.V. (2006). Application of the Proximal Point Method to a System of Extended Primal-Dual Equilibrium Problems. In: Seeger, A. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28258-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation