Geometrically convergent projection method in matrix games

  • A. Cegielski
Contributed Papers
  • 69 Downloads

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.

Key Words

Projection methods matrix games optimal solutions geometric convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Karlin, S.,Mathematical Methods and Theory in Games, Programming, and Economics, Vol. 1, Addison-Wesley Publishing Company, Reading, Massachusetts, 1962.Google Scholar
  2. 2.
    Owen, G.,Game Theory, W. B. Saunders Company, Philadelphia, Pennsylvania, 1968.Google Scholar
  3. 3.
    Szép, J., andForgö, F.,Einführung in die Spieltheorie, Akademiai Kiado, Budapest, Hungary, 1983.Google Scholar
  4. 4.
    Robinson, J.,An Iterative Method of Solving a Game, Annals of Mathematics, Vol. 54, pp. 296–301, 1951.Google Scholar
  5. 5.
    Danskin, J. M.,Fictitious Play for Continuous Games, Naval Research Logistic Quarterly, Vol. 1, pp. 313–320, 1954.Google Scholar
  6. 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. 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. 8.
    Cegielski, A.,A Subgradient Projection Method in Matrix Games, Discussiones Mathematicae, Vol. 11, pp. 143–149, 1991.Google Scholar
  9. 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. 10.
    Cegielski, A.,An Ellipsoid Projection Method in Matrix Games, Optimization, Vol. 23, pp. 117–123, 1991.Google Scholar
  11. 11.
    Shor, N. Z.,Minimization Methods for Nondifferentiable Functions, Springer Verlag, Berlin, Germany, 1985.Google Scholar
  12. 12.
    Polyak, B. T.,Minimization of Nonsmooth Functionals, USSR Computational Mathematics and Mathematical Physics, Vol. 9, pp. 14–39, 1969.Google Scholar
  13. 13.
    Cegielski, A.,The Polyak Subgradient Projection Method in Matrix Games, Discussiones Mathematicae, Vol. 13, pp. 155–165, 1993.Google Scholar
  14. 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. 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. 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.Google Scholar
  17. 17.
    Agmon, S.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 382–392, 1954.Google Scholar
  18. 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. 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. 20.
    Flåm, S. D., andZowe, J.,Relaxed Outer Projections, Weighted Averages, and Convex Feasibility, BIT, Vol. 30, pp. 289–300, 1990.Google Scholar
  21. 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. 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. 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. 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. 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. 26.
    Cegielski, A.,An Algorithm of the Projection onto a Standard Simplex, Technical Report, Higher College of Engineering, Zielona Góra, 1991.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • A. Cegielski
    • 1
  1. 1.Institute of MathematicsHigher College of EngineeringPoland

Personalised recommendations