Solutions of Nonlinear Multilevel Programming Based on Particle Swarm Optimization
ZHANG GuoFu, JIANG JianGuo, QI MeiBin, SU ZhaoPin
School of Computer and Information, Hefei University of Technology, Hefei 230009 Engineering Research Center of Safety Critical Industrial Measurement and Control Technology, Ministry of Education, Hefei University of Technology, Hefei 230009
Abstract:An optimization algorithm for nonlinear multilevel programming problems is presented based on the analysis of the standard particle swarm optimization. The search for StackelbergNash equilibrium of nonlinear multilevel programming problems is implemented. The dynamic region is used to search the whole solution space, therefore, the algorithm has good performance to achieve the global convergence. An adaptive disturbance factor is adopted to make swarms jump out of local optimums, and a constrained fitness value is added to ensure the feasibility of the solutions. The effectiveness of the algorithm has been proved by experiments.
[1] Walker R C. Introduction to Mathematical Programming. Upper Saddle River, USA: Prentice Hall, 1998 [2] Liu Baoding. Uncertain Programming. New York, USA: Wiley, 1999 [3] BenAyed O, Blair C E. Computational Difficulties of Bilevel Linear Programming. Operations Research, 1990, 38(3): 556560 [4] Savard G, Gauvin J. The Steepest Descent Direction for Nonlinear Bilevel Programming Problem. Operations Research Letters, 1994, 15(5): 265272 [5] Jan R H, Chern M S. Nonlinear Integer Bilevel Programming. European Journal of Operational Research, 1994, 72(3): 574587 [6] Liu Baoding. StackelbergNash Equilibrium for Multilevel Programming with Multiple Followers Using Genetic Algorithms. Computers & Mathematics with Applications, 1998, 36(7): 7989 [7] Shih H S, Wen U P, Lee E S, et al. A Neural Network Approach to Multiobjective and Multilevel Programming Problems. Computers and Mathematics with Applications, 2004, 48(1/2): 95108 [8] Kennedy J, Eberhart R C. Particle Swarm Optimization // Proc of the IEEE International Conference on Neural Networks. Piscataway, USA, 1995: 19421948 [9] Eberhart R C, Kennedy J. A New Optimizer Using Particle Swarm Theory // Proc of the 6th International Symposium on Micro Machine and Human Science. Nagoya, Japan, 1995: 3943 [10] Vlachogiannis J G, Lee K Y. A Comparative Study on Particle Swarm Optimization for Optimal SteadyState Performance of Power Systems. IEEE Trans on Power Systems,2006, 21(4): 17181728 [11] Bayraktar Z, Werner P L, Werner D H. The Design of Miniature ThreeElement Stochastic YagiUda Arrays Using Particle Swarm Optimization. IEEE Antennas and Wireless Propagation Letters, 2006, 5(1): 2226 [12] Laskari E C, Parsopoulos K E, Vrahatis M N. Particle Swarm Optimization for Integer Programming // Proc of the IEEE Congress on Evolutionary Computation. Honolulu, USA, 2002, Ⅱ: 15821587 [13] Shi Y, Eberhart R C. Parameter Selection in Particle Swarm Optimization // Proc of the 7th International Conference on Evolutionary Programming. Washington, USA, 1998: 591600 [14] Clerc M, Kennedy J. The Particle Swarm-Explosion, Stability, and Convergence in a Multidimensional Complex Space. IEEE Trans on Evolutionary Computation, 2002, 6(1): 5873 [15] Cui Zhihua, Zeng Jianchao, Cai Xingjuan. A Guaranteed Convergence Dynamic Double Particle Swarm Optimizer // Proc of the 5th World Congress on Intelligent Control and Automation. Hangzhou, China, 2004, Ⅲ: 21842188 [16] Solis F J, Wets J B. Minimization by Random Search Techniques. Mathematics of Operations Research, 1981, 6(1): 1930