Skip to main content
SpringerLink
Log in

Geometrically convergent projection method in matrix games

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We propose a method which evaluates the solution of a matrix game. We reduce the problem of the search for the solution to a convex feasibility problem for which we present a method of projection onto an acute cone. The algorithm converges geometrically. At each iteration, we apply a combinatorial algorithm in order to evaluate the projection onto the standard simplex.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Karlin, S.,Mathematical Methods and Theory in Games, Programming, and Economics, Vol. 1, Addison-Wesley Publishing Company, Reading, Massachusetts, 1962.

    Google Scholar 

  2. Owen, G.,Game Theory, W. B. Saunders Company, Philadelphia, Pennsylvania, 1968.

    Google Scholar 

  3. Szép, J., andForgö, F.,Einführung in die Spieltheorie, Akademiai Kiado, Budapest, Hungary, 1983.

    Google Scholar 

  4. Robinson, J.,An Iterative Method of Solving a Game, Annals of Mathematics, Vol. 54, pp. 296–301, 1951.

    Google Scholar 

  5. Danskin, J. M.,Fictitious Play for Continuous Games, Naval Research Logistic Quarterly, Vol. 1, pp. 313–320, 1954.

    Google Scholar 

  6. Belenky, A. S., andVolkonsky, V. A., Editors,Iterative Processes in the Theory of Games and Programming, Izdatelstvo Nauka, Moscow, Russia, 1974 (in Russian).

    Google Scholar 

  7. Cegielski, A.,Toepliz-Type Theorems for Double Sequences and Their Applications to Some Iterative Process in Zero-Sum Continuous Games, Optimization, Vol. 21, pp. 433–443, 1990.

    Google Scholar 

  8. Cegielski, A.,A Subgradient Projection Method in Matrix Games, Discussiones Mathematicae, Vol. 11, pp. 143–149, 1991.

    Google Scholar 

  9. Giorgobiani, G. D.,An Iterative Algorithm for the Solution of a Rectangular Game, Akademija Nauk Gruzinskoj SSR, Institut Vychislitel'noj Matematiki, Trudy, Vol. 24, pp. 31–34, 1984.

    Google Scholar 

  10. Cegielski, A.,An Ellipsoid Projection Method in Matrix Games, Optimization, Vol. 23, pp. 117–123, 1991.

    Google Scholar 

  11. Shor, N. Z.,Minimization Methods for Nondifferentiable Functions, Springer Verlag, Berlin, Germany, 1985.

    Google Scholar 

  12. Polyak, B. T.,Minimization of Nonsmooth Functionals, USSR Computational Mathematics and Mathematical Physics, Vol. 9, pp. 14–39, 1969.

    Google Scholar 

  13. Cegielski, A.,The Polyak Subgradient Projection Method in Matrix Games, Discussiones Mathematicae, Vol. 13, pp. 155–165, 1993.

    Google Scholar 

  14. Goffin, J. L.,The Relaxation Method for Solving Systems of Linear Inequalities, Mathematics of Operations Research, Vol. 5, pp. 388–414, 1980.

    Google Scholar 

  15. Cegielski, A.,Relaxation Methods in Convex Optimization Problems, Monograph No. 67, Higher College of Engineering, Zielona Góra, Poland, 1993 (in Polish).

    Google Scholar 

  16. Cegielski, A.,Projection onto an Acute Cone and Convex Feasibility Problems, Lecture Notes in Control and Information Sciences, Edited by J. Henry and J. P. Yvon, Vol. 197, pp. 187–194, 1994.

  17. Agmon, S.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 382–392, 1954.

    Google Scholar 

  18. Motzkin, T., andSchoenberg, I. J.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 393–404, 1954.

    Google Scholar 

  19. Oettli, W.,An Iterative Method Having Linear Rate of Convergence for Solving a Pair of Dual Linear Programs, Mathematical Programming, Vol. 3, pp. 302–311, 1972.

    Google Scholar 

  20. Flåm, S. D., andZowe, J.,Relaxed Outer Projections, Weighted Averages, and Convex Feasibility, BIT, Vol. 30, pp. 289–300, 1990.

    Google Scholar 

  21. Yang, K., andMurty, K. G.,New Iterative Method for Linear Inequalities, Journal of Optimization Theory and Applications, Vol. 72, pp. 163–185, 1992.

    Google Scholar 

  22. Schott, D.,A General Iterative Scheme with Applications to Convex Optimization and Related Fields, Optimization, Vol. 22, pp. 885–902, 1991.

    Google Scholar 

  23. Hoffman, A. J.,On Approximate Solutions of Systems of Linear Inequalities, Journal of Research of the National Bureau of Standards, Vol. 49, pp. 263–265, 1952.

    Google Scholar 

  24. Li, W.,A J. Hoffman's Theorem and Metric Projection in Polyhedral Spaces, Journal of Approximation Theory, Vol. 75, pp. 107–111, 1993.

    Google Scholar 

  25. Michelot, C., A Finite Algorithm for Finding the Projection of a Point onto the Canonical Simplex of ℝn, Journal of Optimization Theory and Applications, Vol. 50, pp. 195–200, 1986.

    Google Scholar 

  26. Cegielski, A.,An Algorithm of the Projection onto a Standard Simplex, Technical Report, Higher College of Engineering, Zielona Góra, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Higher College of Engineering, Zielona Góra, Poland

    A. Cegielski (Assistant Professor)

Authors
  1. A. Cegielski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by F. Zirilli

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cegielski, A. Geometrically convergent projection method in matrix games. J Optim Theory Appl 85, 249–264 (1995). https://doi.org/10.1007/BF02192226

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02192226

Key Words

Search

Navigation