We propose an inter-disciplinary approach to particle swarm optimization (PSO) by establishing a molecular dynamics (MD) formulation of the algorithm, leading to a physical theory for the swarm environment. The physical theory provides new insights on the operational mechanism of the PSO method. In particular, a thermodynamic analysis, which is based on the MD formulation, is introduced to provide deeper understanding of the convergence behavior of the basic classical PSO algorithm. The thermodynamic theory is used to propose a new acceleration technique for the PSO. This technique is applied to the problem of synthesis of linear array antennas and very good improvement in the convergence performance is observed. A macroscopic study of the PSO is conducted by formulating a diffusion model for the swarm environment. The Einstein's diffusion equation is solved for the corresponding probability density function (pdf) of the particles trajectory. The diffusion model for the classical PSO is used, in conjunction with Schr¨odinger's equation for the quantum PSO, to propose a generalized version of the PSO algorithm based on the theory of Markov chains. This unifies the two versions of the PSO, classical and quantum, by eliminating the velocity and introducing position-only update equations based on the probability law of the method.
1. Levin, F. S., An Introduction to Quantum Theory, Cambridge University Press, 2002.
2. Sijher, T. S. and A. A. Kishk, "Antenna modeling by infinitesimal dipoles using genetic algorithms," Progress In Electromagnetics Research, Vol. 52, 225-254, 2005. doi:10.2528/PIER04081801
3. Mahanti, G. K., A. Chakrabarty, and S. Das, "Phase-only and amplitude-phase only synthesis of dual-beam pattern linear antenna arrays using floating-point genetic algorithms," Progress In Electromagnetics Research, Vol. 68, 247-259, 2007.
4. Meng, Z., "Autonomous genetic algorithm for functional optimization," Progress In Electromagnetics Research, Vol. 72, 253-268, 2007. doi:10.2528/PIER07031506
5. Riabi, M. L., R. Thabet, and M. Belmeguenai, "Rigorous design and efficient optimizattion of quarter-wave transformers in metallic circular waveguides using the mode-matching method and the genetic algorithm," Progress In Electromagnetics Research, Vol. 68, 15-33, 2007.
6. Kennedy, J. and R. C. Eberhart, Particle swarm optimization, Proc. IEEE, 1995.
7. Kenedy, J. and R. C. Eberhart, Swarm Intelligence, Morgan Kaufmann Publishers, 2001.
8. Clerc, M. and J. Kennedy, "The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space," IEEE Trans. Evolutionary Computation, Vol. 6, No. 1, 58-73, 2002. doi:10.1109/4235.985692
9. Kadirkamanathan, V., K. Selvarajah, and P. J. Fleming, "Stability analysis of the particle swarm optimizer," IEEE Trans. Evolutionary Computation, Vol. 10, No. 3, 245-255, 2006. doi:10.1109/TEVC.2005.857077
10. Ciuprina, G., D. Ioan, and I. Munteanu, "Use of intelligentparticle swarm optimization in electromagnetics," IEEE Trans. Magn., Vol. 38, No. 2, 1037-1040, 2002. doi:10.1109/20.996266
11. Robinson, J. and Yahya Rahmat-Samii, "Particle swarm optimization in electromagnetics," IEEE Trans. Antennas Progat., Vol. 52, No. 2, 397-407, 2004. doi:10.1109/TAP.2004.823969
12. Boeringer, D. and D.Werner, "Particle swarm optimization versus genetic algorithms for phased array synthesis," IEEE Trans. Antennas Progat., Vol. 52, No. 3, 771-779, 2004. doi:10.1109/TAP.2004.825102
13. Sun, J., B. Feng, and W. B. Xu, Particle swarm optimization with particles having quantum behavior, Proc. Cong. Evolutionary Computation, Vol. 1, No. 6, 325-331, 2004.
14. Mikki, S. M. and A. A. Kishk, Investigation of the quantum particle swarm optimization technique for electromagnetic applications, IEEE Antennas and Propagation Society International Symposium, Vol. 2A, 3-8, 2005.
15. Mikki, S. M. and A. A. Kishk, "Quantum particle swarm optimization for electromagnetics," IEEE Trans. Antennas Progat., Vol. 54, No. 10, 2764-2775, 2006. doi:10.1109/TAP.2006.882165
16. Haile, J. M., Molecular Dynamics Simulation, Wiley, New York, 1992.
17. Schommers, W., "Structures and dynamics of surfaces I," Topics in Current Physics, Vol. 41, 1986.
18. Rieth, M., Nano-Engineering in Science and Technology, World Scientific, 2003.
19. Kreyszig, E., Advanced Engineering Mathematics, 8th edition, Wiley, 1999.
20. Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd edition, McGraw-Hill, 1991.
21. Penrose, R., The Emperor's New Mind, 1989., 1989.
22. Gossick, B. R., Hamilton's Principle and Physical Systems, Academic Press, 1967.
23. Iwasaki, N. and K. Yasuda, "Adaptive particle swarm optimization using velocity feedback," International Journal ofInnovative Computing, Vol. 1, No. 3, 369-380, 2005.
24. Bhattacharyya, A. K., Phased Array Antennas, Wiley- Interscience, 2006.
25. Einestien, A., "On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat," Ann. Phys. (Leipzig), Vol. 17, 549-560, 1905.
26. Kennedy, J., Probability and dynamics in the particle swarm, Proc. Cong. Evolutionary Computation, Vol. 1, 19-23, 2004.
27. Nelsson, E., "Derivation of Schrödinger equation from Newtonian mechanics," Phys. Rev., Vol. 150, 1079-1085, 1966. doi:10.1103/PhysRev.150.1079
28. Smolin, L., Could quantum mechanics be approximation to another theory?, http://arxiv.org/abs/quant-ph/0609109, 2006.
29. Smolin, L., Life of the Cosmos, Oxford University Press, 1999.