Max Plus Algebra, Optimization and Game Theory
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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.
KeywordsMax-plus algebra Linear programming Convex quadratic programming problem Fractional programming problem Bimatrix game problem
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.
- 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.Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and linearity. Wiley, New York (1992)Google Scholar
- 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.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
- 12.De Schutter, B., De Moor, B.: The extended linear complementarity problem. Math. Program. 71(3), 289–325 (1995)Google Scholar
- 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.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
- 20.Raghavan, T.E.S., Filar, J.A.: Algorithms for stochastic games, a survey. Zietch. Oper. Res. 35, 437–472 (1991)Google Scholar