Advertisement

Max Plus Algebra, Optimization and Game Theory

  • Dipti Dubey
  • S. K. NeogyEmail author
  • Sagnik Sinha
Conference paper
  • 40 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 302)

Abstract

Max-plus algebra has been applied to several fields like matrix algebra, cryptography, transportation, manufacturing, information technology and study of discrete event systems like subway traffic networks, parallel processing systems, telecommunication networks for many years. In this paper, we discuss various optimization problem using methods based on max-plus algebra, which has maximization and addition as its basic arithmetic operations. We present some sub-classes of mathematical optimization problems like linear programming, convex quadratic programming problem, fractional programming problem, bimatrix game problem and some classes of stochastic game problem in max algebraic framework and discuss various connections between max-plus algebra and optimization.

Keywords

Max-plus algebra Linear programming Convex quadratic programming problem Fractional programming problem Bimatrix game problem 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their constructive suggestions which considerably improve the overall presentation of the paper. The authors would like to thank Professor R. B. Bapat, Indian Statistical Institute, Delhi Centre for his valuable comments and suggestions. The first author wants to thank the Science and Engineering Research Board, Department of Science & Technology, Government of India, for financial support for this research.

References

  1. 1.
    Akian, M., Bapat, R.B., Gaubert, S.: Max-plus algebra. Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds) Handbook of linear algebra, discrete mathematics and its applications, Vol. 39, 2nd edn, Chapman and Hall (2014)Google Scholar
  2. 2.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and linearity. Wiley, New York (1992)Google Scholar
  3. 3.
    Bapat, R.B., Stanford, D., van den Driessche, P.: The eigenproblem in max algebra, DMS-631-IR, University of Victoria, British Columbia (1993)Google Scholar
  4. 4.
    Bapat, R.B.: Max algebra and graph theory. In: Krishnamurthy, Ravichandran, N. (eds) Proceedings of the Advance Workshop and Tutorial in Operations Research, Allied Publisher Pvt Ltd. (2012)Google Scholar
  5. 5.
    Bapat, R.B.: Pattern properties and spectral inequalities in max algebra. SIAM J. Matrix Anal. Appl. 16, 964–976 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bouquard, J.-L., Lent, C., Billaut, J.-C.: Application of an optimization problem in max-plus algebra to scheduling problems. Discrete Appl. Math. 154, 2064–2079 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burkard, R.E., Butkovic, P.: Max algebra and the linear assignment problem. Math. Program. Ser. B 98, 415–429 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Butkovic, P.: Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. 367, 313–335 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Butkovic, P.: Max-linear systems: theory and algorithms. Springer, Berlin (2010)CrossRefGoogle Scholar
  10. 10.
    Cuninghame-Green, R.A.: Minimax algebra, lecture notes in economics and mathematical systems (1979)CrossRefGoogle Scholar
  11. 11.
    Cuninghame-Green, R.A.: Minimax algebra. Lecture notes in economics and mathematical systems, pp. 166. Springer, Berlin, New Yotk (1979)CrossRefGoogle Scholar
  12. 12.
    De Schutter, B., De Moor, B.: The extended linear complementarity problem. Math. Program. 71(3), 289–325 (1995)Google Scholar
  13. 13.
    Filar, J.A.: Orderfield property for stochastic games when the player who controls transitions changes from state to state. JOTA 34, 503–515 (1981)CrossRefGoogle Scholar
  14. 14.
    Filar, J.A., Vrieze, O.J.: Competitive markov decision processes. Springer, New York (1997)zbMATHGoogle Scholar
  15. 15.
    Fink, A.M.: Equilibrium in a stochastic \(n\)-person game. J. Sci. Hiroshima Univ. Ser. A 28, 89–93 (1964)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gillette, D.: Stochastic game with zero step probabilities. In: Tucker, A.W., Dresher, M., Wolfe, P. (eds) Theory of games. Princeton University Press, Princeton, New Jersey (1957)Google Scholar
  17. 17.
    Heidergott, B., Jan Olsder, G., van der Woude, J.: Max plus at work modeling and analysis of synchronized systems: a course on max-plus algebra and its applications, Princeton University Press (2006)Google Scholar
  18. 18.
    Mohan, S.R., Raghavan, T.E.S.: An algorithm for discounted switching control games. OR Spektrum 9, 41–45 (1987)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mohan, S.R., Neogy, S.K., Parthasarathy, T.: Pivoting algorithms for some classes of stochastic games: a survey. Int. Game Theory Rev. 3, 253–281 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Raghavan, T.E.S., Filar, J.A.: Algorithms for stochastic games, a survey. Zietch. Oper. Res. 35, 437–472 (1991)Google Scholar
  21. 21.
    Shapley, L.S.: Stochastic games. Proc. Natl. Acad. Sci. 39, 1095–1100 (1953)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sobel, M.J.: Noncooperative stochastic games. Ann. Math. Stat. 42, 1930–1935 (1971)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vrieze, O.J., Tijs, S.H., Raghavan, T.E.S., Filar, J.A.: A finite algorithm for the switching controller stochastic game. OR Spektrum 5, 15–24 (1983)CrossRefGoogle Scholar
  24. 24.
    Zimmermann, U.: Linear and combinatorial optimization in ordered algebraic structures. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Jadavpur UniversityKolkataIndia

Personalised recommendations