volume | PIER Journals

Abstract

Linear antenna array design is one of the most important electromagnetic optimization problems of current interest. In contrast to a plethora of recently published articles that formulate the design as the optimization of a single cost function formed by combining distinct and often conflicting design-objectives into a weighted sum, in this work, we take a Multi-objective Optimization (MO) approach to solve the same problem. We consider two design objectives: the minimum average Side Lobe Level (SLL) and null control in specific directions that are to be minimized simultaneously in order to achieve the optimal spacing between the array elements. Our design method employs a recently developed and very competitive multi-objective evolutionary algorithm called MOEA/D-DE that uses a decomposition approach for converting the problem of approximation of the Pareto Fronts (PF) into a number of single objective optimization problems. This algorithm employs Differential Evolution (DE), one of the most powerful real parameter optimizers in current use, as the search method. As will be evident from the shape of the approximated PFs obtained with MOEA/D-DE, the two design-objectives are in conflict and usually, performance cannot be improved significantly for one without deteriorating the other. Unlike the single-objective approaches, the MO approach provides greater flexibility in the design by yielding a set of equivalent final solutions from which the user can choose one that attains a suitable trade-off margin as per requirements. We illustrate that the best compromise solution attained by MOEA/D-DE can comfortably outperform state-of-the-art single-objective algorithms like Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Tabu Search Algorithm (TSA), and Memetic Algorithm (MA). In addition, we compared the results obtained by MOEA/D-DE with those obtained by one of the most widely used MO algorithm called NSGA-2 and another generic multi-objective DE variant that uses non-dominated sorting, on the basis of the Rindicator, hypervolume indicator, and quality of the best trade-off solutions obtained.

1. Godara, L. C., "Handbook of Antennas in Wireless Communications," CRC, Boca Raton, FL, 2002. Google Scholar

2. Bucci, O. M., D. D'Elia, G. Mazzarella, and G. Panatiello, "Antenna pattern synthesis: A new general approach," Proc. IEEE, Vol. 82, 358-371, 1994.
doi:10.1109/5.272140 Google Scholar

3. Rahmat-Samii, Y. and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms, Wiley, New York, 1999.

4. Lebret, H. and S. Boyd, "Antenna array pattern synthesis via convex optimization," IEEE Transactions on Signal Processing, Vol. 45, No. 3, 526-532, March 1997.
doi:10.1109/78.558465 Google Scholar

5. Khodier, M. M. and C. G. Christodoulou, "Linear array geometry synthesis with minimum side lobe level and null control using particle swarm optimization ," IEEE Transactions on Antennas and Propagation, Vol. 153, No. 8, 2674-2679, August 2005.
doi:10.1109/TAP.2005.851762 Google Scholar

6. Holland, J. H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Harbor, 1975.

7. Back, T., D. Fogel, and Z. Michalewicz, Handbook of Evolutionary Computation, Oxford Univ. Press, 1997.

8. Eiben, A. E. and J. E. Smith, Introduction to Evolutionary Computing, Springer, 2003.

9. Kirkpatrik, S., C. Gelatt, and M. Vecchi, "Optimization by simulated annealing," Science, Vol. 220, 671-680, 1983.
doi:10.1126/science.220.4598.671 Google Scholar

10. Glover, F. and M. Laguna, Tabu Search, Kluwer, Norwell, MA, 1997.

11. Taguchi, G., S. Chowdhury, and Y. Wu, Taguchi's Quality Engineering Handbook, Wiley, New York, 2005.

12. Ong, Y.-S. and A. J. Keane, "Meta-lamarckian learning in memetic algorithms," IEEE Transactions on Evolutionary Computation, Vol. 8, No. 2, 99-110, 2004.
doi:10.1109/TEVC.2003.819944 Google Scholar

13. Kennedy, J. and R. Eberhart, "Particle swarm optimization," Proc. IEEE Int. Conf. Neural Networks, 1942-1948, 1995.

14. Kennedy, J., R. C. Eberhart, and Y. Shi, "Swarm Intelligence," Morgan Kaufmann, San Francisco, CA, 2001. Google Scholar

15. Udina, A., N. M. Martin, and L. C. Jain, "Linear antenna array optimization by genetic means," Third International Conference on Knowledge-based Intelligent Information Engineering Systems Adelaide , Australia, September 1999.

16. Cengiz, Y. and H. Tokat, "Linear antenna array design with use of genetic, memetic and tabu search optimization algorithms," Progress In Electromagnetics Research C, Vol. 1, 63-72, 2008.
doi:10.2528/PIERC08010205 Google Scholar

17. Weng, W.-C., F. Yang, and A. Z. Elsherbeni, "Linear antenna array synthesis using taguchi's method: A novel optimization technique in electromagnetics ," IEEE Transactions on Antennas and Propagation, Vol. 55, No. 3, 723-730, March 2007.
doi:10.1109/TAP.2007.891548 Google Scholar

18. Ares-Pena, F. J., A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, "Genetic algorithms in the design and optimization of antenna array patterns," IEEE Transactions on Antennas and Propagation, Vol. 47, 506-510, March 1999.
doi:10.1109/8.768786 Google Scholar

19. Tian, Y. B. and J. Qian, "Improve the performance of a linear array by changing the spaces among array elements in terms of genetic algorithm ," IEEE Transactions on Antennas and Propagation, Vol. 53, 2226-2230, July 2005. Google Scholar

20. Storn, R. and K. V. Price, "Differential evolution --- A simple and efficient adaptive scheme for global optimization over continuous spaces,", Technical Report TR-95-012, ICSI, http://http.icsi.berkeley.edu/»storn/litera.html, 1995. Google Scholar

21. Storn, R. and K. Price, "Differential evolution --- A simple and efficient heuristic for global optimization over continuous spaces," Journal of Global Optimization, Vol. 11, No. 4, 341-359, 1997.
doi:10.1023/A:1008202821328 Google Scholar

22. Price, K., R. Storn, and J. Lampinen, Differential Evolution --- A Practical Approach to Global Optimization, Springer, Berlin, 2005.

23. Li, H. and Q. Zhang, "Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II," IEEE Trans. on Evolutionary Computation, Vol. 12, No. 2, 284-302, 2009.
doi:10.1109/TEVC.2008.925798 Google Scholar

24. Zhang, Q., W. Liu, and H. Li, "The performance of a New MOEA/D on CEC09 MOP test instances," Proceedings of the Eleventh Conference on Congress on Evolutionary Computation, 203-208, Trondheim, Norway, May 18-21, 2009.

25. Zhang, Q., A. Zhou, S. Z. Zhao, P. N. Suganthan, W. Liu, and S. Tiwari, Multiobjective optimization test instances for the CEC 2009 special session and competition, Technical Report CES-887, University of Essex and Nanyang Technological University, 2008.

26. Abido, M. A., "A novel multiobjective evolutionary algorithm for environmental/economic power dispatch," Electric Power Systems Research, Vol. 65, 71-81, Elsevier 2003. Google Scholar

27. Deb, K., A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 182-197, 2002.
doi:10.1109/4235.996017 Google Scholar

28. Panduroa, M. A., D. H. Covarrubiasa, C. A. Brizuelaa, and F. R. Maranteb, "A multi-objective approach in the linear antenna array design ," Int. J. Electron. Commun., (AEU), Vol. 59, 205-212, 2005. Google Scholar

29. Reklaitis, G. V., A. Ravindran, and K. M. Ragsdell, "Engineering Optimization: Methods and Applications," John Wiley, New Jersey, 2006. Google Scholar

30. Abraham, A., L. C. Jain, and R. Goldberg, "Evolutionary Multiobjective Optimization: Theoretical Advances and Applications," Springer Verlag, London, 2005. Google Scholar

31. Carlos, A. C. C. and G. B. Lamont, "Applications of Multi-Objective Evolutionary Algorithms," World Scientific, 2004. Google Scholar

32. Nedjah, N. and L. D. M. Mourelle, Real-world Multi-objective System Engineering, Illustrated Edition, Nova Science Publishers, Deccermber 2005.

33. Deb, K., "Multi-objective Optimization Using Evolutionary Algorithms," John Wiley & Sons, 2001. Google Scholar

34. Xue, F., A. C. Sanderson, and R. J. Graves, "Pareto-based multi-objective differential evolution," Proceedings of the 2003 Congress on Evolutionary Computation (CEC'2003), Vol. 2, 862-869, IEEE Press, Canberra, Australia, 2003. Google Scholar

35. Robic, T. and B. Filipic, "DEMO: Differential evolution for multiobjective optimization," Evolutionary Multi-Criterion Optimization, Third International Conference, EMO 2005, C. A. Coello Coello, A. H. Aguirre, and E. Zitzler, Vol. 3410, 520-533, Springer, Guanajuato, Mexico, 2005. Google Scholar

36. Iorio, A. W. and X. Li, "Solving rotated multi-objective optimization problems using differential evolution," AI 2004: Advances in Artificial Intelligence, Proceedings, Vol. 3339, 861-872, Springer-Verlag, 2004. Google Scholar

37. Huang, V. L., A, K. Qin, P. N. Suganthan, and M. F. Tasgetiren, "Multi-objective optimization based on self-adaptive differential evolution algorithm ," Congress on Evolutionary Computation (CEC'2007), 3601-3608, Singapore, September 2007. Google Scholar

38. Zamuda, A., J. Brest, B. Boskovic, and V. Zumer, "Differential evolution for multiobjective optimization with self adaptation," Congress on Evolutionary Computation (CEC'2007), 3617-3624, Singapore, September 2007. Google Scholar

39. Zhang, Q. and H. Li, "MOEA/D: A multi-objective evolutionary algorithm based on decomposition," IEEE Trans. on Evolutionary Computation, Vol. 11, No. 6, 712-731, 2007.
doi:10.1109/TEVC.2007.892759 Google Scholar

40. Miettinen, K., Nonlinear Multiobjective Optimization, Kuluwer Academic Publishers, 1999.

41. Powell, D. and M. M. Skolnick, "Using genetic algorithms in engineering design optimization with non-linear constraints," Proceedings of the Fifth International Conference on Genetic Algorithms, San Mateo, CA, USA, 1993. Google Scholar

42. Deb, K., "An efficient constraint handling method for genetic algorithms," Computer Methods in Applied Mechanics and Engineering, Vol. 186, 311-338, 2000.
doi:10.1016/S0045-7825(99)00389-8 Google Scholar

43. Van Veen, B. D. and K. M. Buckley, "Beamforming: A versatile approach to spatial filtering," IEEE Acoust. Speech Signal Process. Mag., Vol. 5, 4-24, 1988. Google Scholar

44. Knowles, J., L. Thiele, and E. Zitzler, "A tutorial on the performance assessment of stochastic multiobjective optimizers," Computer Engineering and Networks Laboratory, ETH Zurich, Switzerland, Feb. 2006. Google Scholar

45. Dhillon, J. S., S. C. Parti, and D. P. Kothari, "Stochastic economic emission load dispatch," Electric Power Syst. Res., 179-186, 1993.
doi:10.1016/0378-7796(93)90011-3 Google Scholar

46. Tapia, C. G. and B. A. Murtagh, "Interactive fuzzy programming with preference criteria in multiobjective decision making," Comput. Oper. Res., Vol. 18, 307-316, 1991.
doi:10.1016/0305-0548(91)90032-M Google Scholar

Dr. Siddharth Pal

Dr. Boyang Qu

Dr. Swagatam Das

Dr. P. N. Suganthan