In this paper, we propose a new formulation of Nash games for solving a general multi-objectives optimization problems. The objective of this approach is to split the optimization variables, allowing us to determine numerically the strategies between two players. The first player minimizes his function cost using the variables of the first table P and the second player, using the second table Q. The original contribution of this work concerns the construction of the two tables of allocations that lead to a Nash equilibrium on the Pareto front. The second proposition of this paper is to find a Nash Equilibrium solution, which coincides with the Kalai--Smorodinsky solution. Two algorithms that calculate P, Q and their associated Nash equilibrium, by using some extension of the normal boundary intersection approach, are tried out successfully. Then, we propose a search engine to look for similar images of a given image based on multiple image representations using Color, Texture and Shape Features.
- Ehrgott M. Multicriteria optimization. Springer Verlag (2005).
- Steuer R. E. Multiple criteria optimization: Theory, computation, and application. John Wiley and Sons (1986).
- Meskine D., Moussaid N., Berhich S. Blind image deblurring by game theory. NISS19: Proceedings of the 2nd International Conference on Networking, Information Systems & Security. Article No.: 31 (2019).
- Miettinen K. Nonlinear multiobjective optimization. Vol. 12. Springer Science & Business Media (2012).
- Pareto V. The New Theories of Economics. Journal of Political Economy. 5 (4), 485–502 (1897).
- Aubin J. P. Mathematical methods of game and economic theory. North-Holland Publishing Co. Amsterdam, New York (1979).
- Périaux J. Genetic algorithms and evolution strategy in engineering and computer science: recent advances and industrial applications. John Wiley and Son Ltd (1998).
- Ramos R. G., Périaux J. Nash equilibria for the multiobjective control of linear partial differential equations. Journal of Optimization Theory and Applications. 112 (3), 457–498 (2002).
- Désidéri J.-A., Duvigneau R., Abou El Majd B., Tang Z. Algorithms for efficient shape optimization in aerodynamics and coupled disciplines. 42nd AAAF Congress on Applied Aerodynamics. Sophia-Antipolis, France (2007).
- Abou El Majd B., Desideri J.-A., Habbal A. Aerodynamic and structural optimization of a business-jet wingshape by a Nash game and an adapted split of variables. Mécanique & Industries. 11 (3–4), 209–214 (2010).
- Abou El Majd B., Ouchetto O., Désidéri J.-A., Habbal A. Hessian transfer for multilevel and adaptive shape optimization. International Journal for Simulation and Multidisciplinary Design Optimization. 8, Article Number: A9 (2017).
- Habbal A., Petersson J., Thellner M. Multidisciplinary topology optimization solved as a Nash game. Int. J. Numer. Meth. Engng. 61, 949–963 (2004).
- Aboulaich R., Habbal A., Moussaid N. Split of an optimization variable in game theory. Math. Model. Nat. Phenom. 5 (7), 122–127 (2010).
- Smorodinsky M., Kalai E. Other Solution to Nash's Bargaining Problem. Econometrica. 43 (3), 513–518 (1975).
- Das I., Dennis J. E. Normal Boundary Intersection, A New methode for Generating the Pareto Surface in Nonlinear Multicreteria Optimization problems. SIAM Journal on Optimization. 8 (3), 631–657 (1998).
- Aboulaich R., Habbal A., Moussaid N. Optimisation multicrit\`ere : une approche par partage des variables. ARIMA. 13, 77–89 (2010).
- Aboulaich R., Ellaia R., Elmoumen S., Habbal A., Moussaid N. The Mean-CVaR Model for Portfolio Optimization Using a Multi-Objective Approach and the Kalai-Smorodinsky Solution. MATEC Web of Conferences. 105, Article Number: 00010 (2017).
- Bencharef O., Jarmouni B., Moussaid N., Souissi A. Image retrieval using global descriptors and multiple clustering in Nash game. Annals of the University of Craiova, Mathematics and Computer Science Series. 42 (1), 202–210 (2015).