An Adaptive Differential Evolution Algorithm to Solve Constrained Optimization Problems in Engineering Design

ABSTRACT

Differential evolution (DE) algorithm has been shown to be a simple and efficient evolutionary algorithm for global optimization over continuous spaces, and has been widely used in both benchmark test functions and real-world applications. This paper introduces a novel mutation operator, without using the scaling factor F, a conventional control parameter, and this mutation can generate multiple trial vectors by incorporating different weighted values at each generation, which can make the best of the selected multiple parents to improve the probability of generating a better offspring. In addition, in order to enhance the capacity of adaptation, a new and adaptive control parameter, i.e. the crossover rate CR, is presented and when one variable is beyond its boundary, a repair rule is also applied in this paper. The proposed algorithm ADE is validated on several constrained engineering design optimization problems reported in the specialized literature. Compared with respect to algorithms representative of the state-of-the-art in the area, the experimental results show that ADE can obtain good solutions on a test set of constrained optimization problems in engineering design.

Differential evolution (DE) algorithm has been shown to be a simple and efficient evolutionary algorithm for global optimization over continuous spaces, and has been widely used in both benchmark test functions and real-world applications. This paper introduces a novel mutation operator, without using the scaling factor F, a conventional control parameter, and this mutation can generate multiple trial vectors by incorporating different weighted values at each generation, which can make the best of the selected multiple parents to improve the probability of generating a better offspring. In addition, in order to enhance the capacity of adaptation, a new and adaptive control parameter, i.e. the crossover rate CR, is presented and when one variable is beyond its boundary, a repair rule is also applied in this paper. The proposed algorithm ADE is validated on several constrained engineering design optimization problems reported in the specialized literature. Compared with respect to algorithms representative of the state-of-the-art in the area, the experimental results show that ADE can obtain good solutions on a test set of constrained optimization problems in engineering design.

KEYWORDS

Differential Evolution, Constrained Optimization, Engineering Design, Evolutionary Algorithm, Constraint Handling

Differential Evolution, Constrained Optimization, Engineering Design, Evolutionary Algorithm, Constraint Handling

Cite this paper

nullY. AO and H. CHI, "An Adaptive Differential Evolution Algorithm to Solve Constrained Optimization Problems in Engineering Design,"*Engineering*, Vol. 2 No. 1, 2010, pp. 65-77. doi: 10.4236/eng.2010.21009.

nullY. AO and H. CHI, "An Adaptive Differential Evolution Algorithm to Solve Constrained Optimization Problems in Engineering Design,"

References

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[19] Y. Wang and Z. X. Cai, “A hybrid multi-swarm particle swarm optimization to solve constrained optimization problems,” Frontiers of Computer Science in China, Vol. 3, No. 1, pp. 38–52, 2009.

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[23] K. Hans Raj, R. S. Sharma, G. S. Mishra, A. Dua, and C. Patvardhan, “An evolutionary computational technique for constrained optimisation in engineering design,” Journal of the Institution of Engineers India Part Me Mechanical Engineering Division, Vol. 86, pp. 121–128, 2005.

[24] T. Ray and P. Saini, “Engineering design optimization using a swarm with intelligent information sharing among individuals,” Engineering Optimization, Vol. 33, No. 33, pp. 735–748, 2001.

[25] A. R. Hedar and M. Fukushima, “Derivative-free filter simulated annealing method for constrained continuous global optimization,” Journal of Global Optimization, Vol. 35, No. 4, pp. 521–549, 2006.

[26] C. A. Coello, “Self-adaptive penalties for GA- based optimization,” Proceedings of the Congress on Evolutionary Computation 1999 (CEC'99), Vol. 1, pp. 573–580, 1999.

[27] E. Mezura-Montes, C. A. Coello, and J. V. Reyes, “Increasing successful offspring and diversity in differential evolution for engineering design,” Proceedings of the Seventh International Conference on Adaptive Computing in Design and Manufacture (ACDM 2006), pp. 131– 139, 2006.

[28] A. E. Mu?oz Zavala, A. Hernández Aguirre, E. R. Villa Diharce, and S. Botello Rionda, “Constrained optimization with an improved particle swarm optimization algorithm,” International Journal of Intelligent Computing and Cybernetics, Vol. 1, No. 3, pp. 425–453, 2008.

[29] X. H. Hu, R. C. Eberhart, and Y. H. Shi, “Engineering optimization with particle swarm,” Proceedings of the 2003 IEEE Swarm Intelligence Symposium, pp. 53–57, 2003.

[30] A. Homaifar, S. H. Y. Lai, and X. Qi, “Constrained optimization via genetic algorithms,” Simulation, Vol. 62, No. 4, pp. 242–254, 1994.

[1] K. Deb, “An efficient constraint handling method for genetic algorithms,” Computer Methods in Applied Mechanics and Engineering, Vol. 186, No. 2, pp. 311–338, 2000.

[2] E. Mezura-Montes and A. G. Palomeque-Qrtiz, “Parameter control in differential evolution for constrained optimization,” 2009 IEEE Congress on Evolutionary Computation (CEC'2009), pp. 1375–1382, 2009.

[3] E. Mezura-Montes, C. A. Coello Coello, J. Velázquez- Reyes, and L. Mu?oz-Dávila, “Multiple trial vectors in differential evolution for engineering design,” Engineering Optimization, Vol. 39, No. 5, pp. 567–589, 2007.

[4] C. A. Coello Coello, “Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: A survey of the state of the art,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 11-12, pp. 1245–1287, 2002.

[5] R. Landa-Becerra and C. A. Coello Coello, “Cultured differential evolution for constrained optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 33–36, pp. 4303–4322, 2006.

[6] T. Ray and K. M. Liew, “Society and civilization: An optimization algorithm based on the simulation of social behavior,” IEEE Transactions on Evolutionary Computation, Vol. 7, No. 4, pp. 386–396, 2003.

[7] S. He, E. Prempain, and Q. H. Wu, “An improved particle swarm optimizer for mechanical design optimization problems,” Engineering Optimization, Vol. 36, No. 5, pp. 585–605, 2004.

[8] M. Zhang, W. J. Luo, and X. F. Wang, “Differential evolution with dynamic stochastic selection for constrained optimization,” Information Sciences, Vol. 178, pp. 3043–3074, 2008.

[9] S. Akhtar, K. Tai, and T. Ray, “A socio-behavioural simulation model for engineering design optimization,” Engineering Optimization, Vol. 34, No. 4, pp. 341–354, 2002.

[10] Q. He and L. Wang, “An effective co-evolutionary particle swarm optimization for constrained engineering design problems,” Engineering Applications of Artificial Intelligence, Vol. 20, No. 1, pp. 89–99, 2007.

[11] J. H. Wang and Z. Y. Yin, “A ranking selection-based particle swarm optimizer for engineering design optimization problems,” Structural and Multidisciplinary Optimization, Vol. 37, No. 2, pp. 131–147, 2008.

[12] R. Storn and K. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, Vol. 11, pp. 341–359, 1997.

[13] K. Price, R. Storn, and J. Lampinen, “Differential evolution: A practical approach to global optimization,” Berlin: Springer-Verlag, 2005.

[14] Z. Y. Yang, K. Tang, and X. Yao, “Self-adaptive differential evolution with neighborhood search,” 2008 Congress on Evolutionary Computation (CEC'2008), pp. 1110–1116, 2008.

[15] H. A. Abbass, R. Sarker, and C. Newton, “PDE: A Pareto-frontier differential evolution approach for multiobjective optimization problems,” Proceedings of IEEE Congress on Evolutionary Computation, Vol. 2, pp. 971– 978, 2001.

[16] Y. W. Leung and Y. P. Wang, “An orthogonal genetic algorithm with quantization for global numerical optimization,” IEEE Transactions on Evolutionary Computation, Vol. 5, No. 1, pp. 40–53, 2001.

[17] S. Tsutsui, M. Yamamure, and T. Higuchi, “Multi-parent recombination with simplex crossover in real coded genetic algorithms,” Proceedings of the Genetic and Evolutionary Computation Conference, pp. 657–664, 1999.

[18] J. Brest, V. Zumer, and M. S. Maucec, “Self-adaptive differential evolution algorithm in constrained real-parameter optimization,” 2006 IEEE Congress on Evolutionary Computation (CEC'2006), pp. 919–926, 2006.

[19] Y. Wang and Z. X. Cai, “A hybrid multi-swarm particle swarm optimization to solve constrained optimization problems,” Frontiers of Computer Science in China, Vol. 3, No. 1, pp. 38–52, 2009.

[20] L. C. Cagnina, S. C. Esquivel, and C. A. Coello Coello, “Solving engineering optimization problems with the simple constrained particle swarm optimizer,” Informatica, Vol. 32, pp. 319–326, 2008.

[21] C. A. Coello Coello, “Use of a self-adaptive penalty approach for engineering optimization problems,” Computers in Industry, Vol. 41, No. 2, pp. 113–127, 2000.

[22] T. P. Runarsson and X. Yao, “Stochastic ranking for constrained evolutionary optimization,” IEEE Transactions on Evolutionary Computation, Vol. 4, No. 3, pp. 284–294, 2000.

[23] K. Hans Raj, R. S. Sharma, G. S. Mishra, A. Dua, and C. Patvardhan, “An evolutionary computational technique for constrained optimisation in engineering design,” Journal of the Institution of Engineers India Part Me Mechanical Engineering Division, Vol. 86, pp. 121–128, 2005.

[24] T. Ray and P. Saini, “Engineering design optimization using a swarm with intelligent information sharing among individuals,” Engineering Optimization, Vol. 33, No. 33, pp. 735–748, 2001.

[25] A. R. Hedar and M. Fukushima, “Derivative-free filter simulated annealing method for constrained continuous global optimization,” Journal of Global Optimization, Vol. 35, No. 4, pp. 521–549, 2006.

[26] C. A. Coello, “Self-adaptive penalties for GA- based optimization,” Proceedings of the Congress on Evolutionary Computation 1999 (CEC'99), Vol. 1, pp. 573–580, 1999.

[27] E. Mezura-Montes, C. A. Coello, and J. V. Reyes, “Increasing successful offspring and diversity in differential evolution for engineering design,” Proceedings of the Seventh International Conference on Adaptive Computing in Design and Manufacture (ACDM 2006), pp. 131– 139, 2006.

[28] A. E. Mu?oz Zavala, A. Hernández Aguirre, E. R. Villa Diharce, and S. Botello Rionda, “Constrained optimization with an improved particle swarm optimization algorithm,” International Journal of Intelligent Computing and Cybernetics, Vol. 1, No. 3, pp. 425–453, 2008.

[29] X. H. Hu, R. C. Eberhart, and Y. H. Shi, “Engineering optimization with particle swarm,” Proceedings of the 2003 IEEE Swarm Intelligence Symposium, pp. 53–57, 2003.

[30] A. Homaifar, S. H. Y. Lai, and X. Qi, “Constrained optimization via genetic algorithms,” Simulation, Vol. 62, No. 4, pp. 242–254, 1994.