Abstract
While research on constrained optimization using evolutionary algorithms has been actively pursued, it has had to face the problem that the ability to solve multi-modal problems is insufficient, that the ability to solve problems with equality constraints is inadequate, and that the stability and efficiency of searches is low. We have proposed the εDE, defined by applying the ε constrained method to differential evolution (DE). It is shown that the εDE is a fast and stable algorithm that is robust to multi-modal problems and it can solve problems with many equality constraints by introducing a gradient-based mutation which finds a feasible point using the gradient of constraints. In this chapter, an improved εDE is proposed, in which faster reduction of the relaxation of equality constraints in the ε constrained method and higher gradient-based mutation rate are adopted in order to solve problems with many equality constraints and to find feasible solutions faster and very stably. Also, cutting off and reflecting back solutions outside of search space are adopted to improve the efficiency in finding optimal solutions. The improved εDE realizes stable and efficient searches, and can solve difficult constrained optimization problems with equality constraints. The advantage of the improved εDE is shown by applying it to twenty four constrained problems of various types.
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References
Camponogara, E., Talukdar, S.N.: A genetic algorithm for constrained and multiobjective optimization. In: Alander, J.T. (ed.) 3rd Nordic Workshop on Genetic Algorithms and Their Applications (3NWGA), August 1997, pp. 49–62. University of Vaasa, Finland (1997)
Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Heidelberg (2008)
Chootinan, P., Chen, A.: Constraint handling in genetic algorithms using gradient-based repair method. Computers & Operations Research 33(8), 2263–2281 (2006)
Coath, G., Halgamuge, S.K.: A comparison of constraint-handling methods for the application of particle swarm optimization to constrained nonlinear optimization problems. In: Proc. of IEEE Congress on Evolutionary Computation, Canberra, Australia, pp. 2419–2425 (2003)
Coello, C.A.C.: 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 191(11-12), 1245–1287 (2002)
Deb, K.: An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering 186(2/4), 311–338 (2000)
Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Springer, Heidelberg (2003)
Hu, X., Eberhart, R.C.: Solving constrained nonlinear optimization problems with particle swarm optimization. In: Proc. of the Sixth World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida (2002)
Kukkonen, S., Lampinen, J.: Constrained real-parameter optimization with generalized differential evolution. In: Proceedings of the 2006 IEEE Congress on Evolutionary Computation, July 16-2, 2006, pp. 207–214. IEEE Press, Vancouver (2006)
Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello, C.A.C., Deb, K.: Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization (2006), http://www.ntu.edu.sg/home/EPNSugan/cec2006/technical_report.pdf
Michalewicz, Z.: A survey of constraint handling techniques in evolutionary computation methods. In: Proceedings of the 4th Annual Conference on Evolutionary Programming, pp. 135–155. The MIT Press, Cambridge (1995)
Michalewicz, Z., Schoenauer, M.: Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation 4(1), 1–32 (1996)
Parsopoulos, K.E., Vrahatis, M.N.: Particle swarm optimization method for constrained optimization problems. In: Sincak, P., Vascak, J., et al. (eds.) Intelligent Technologies — Theory and Application: New Trends in Intelligent Technologies. Frontiers in Artificial Intelligence and Applications, vol. 76, pp. 214–220. IOS Press, Amsterdam (2002)
Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: A Practical Approach to Global Optimization. Springer, Heidelberg (2005)
Ray, T., Liew, K.M., Saini, P.: An intelligent information sharing strategy within a swarm for unconstrained and constrained optimization problems. Soft Computing – A Fusion of Foundations, Methodologies and Applications 6(1), 38–44 (2002)
Runarsson, T.P., Yao, X.: Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation 4(3), 284–294 (2000)
Sakai, S., Takahama, T.: Constrained optimization by the ε constrained differential evolution. In: Numerical Optimization methods, theory and applications, RIMS Kokyuroku, February 2008, vol. 1584, pp. 90–101 (2008) (in Japanese)
Campbell, S.L., Meyer, C.J.: Generalized Inverses of Linear Transformations. Dover Publications (1979)
Storn, R., Price, K.: Minimizing the real functions of the ICEC 1996 contest by differential evolution. In: Proc. of the International Conference on Evolutionary Computation, pp. 842–844 (1996)
Storn, R., Price, K.: Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11, 341–359 (1997)
Surry, P.D., Radcliffe, N.J.: The COMOGA method: Constrained optimisation by multiobjective genetic algorithms. Control and Cybernetics 26(3), 391–412 (1997)
Takahama, T., Sakai, S.: Tuning fuzzy control rules by α constrained method which solves constrained nonlinear optimization problems. The Transactions of the Institute of Electronics, Information and Communication Engineers J82-A(5), 658–668 (1999) (in Japanese)
Takahama, T., Sakai, S.: Learning fuzzy control rules by α-constrained simplex method. The Transactions of the Institute of Electronics, Information and Communication Engineers J83-D-I(7), 770–779 (2000) (in Japanese)
Takahama, T., Sakai, S.: Tuning fuzzy control rules by the α constrained method which solves constrained nonlinear optimization problems. Electronics and Communications in Japan, Part3: Fundamental Electronic Science 83(9), 1–12 (2000)
Takahama, T., Sakai, S.: Constrained optimization by α constrained genetic algorithm (αGA). The Transactions of the Institute of Electronics, Information and Communication Engineers J86-D-I(4), 198–207 (2003) (in Japanese)
Takahama, T., Sakai, S.: Learning fuzzy control rules by α-constrained simplex method. Systems and Computers in Japan 34(6), 80–90 (2003)
Takahama, T., Sakai, S.: Constrained optimization by α constrained genetic algorithm (αGA). Systems and Computers in Japan 35(5), 11–22 (2004)
Takahama, T., Sakai, S.: Constrained optimization by combining the α constrained method with particle swarm optimization. In: Proc. of Joint 2nd International Conference on Soft Computing and Intelligent Systems and 5th International Symposium on Advanced Intelligent Systems (2004), http://www.chi.its.hiroshima-cu.ac.jp/~takahama/eng/papers/aPSO0409c.pdf
Takahama, T., Sakai, S.: Constrained optimization by applying the α constrained method to the nonlinear simplex method with mutations. IEEE Transactions on Evolutionary Computation 9(5), 437–451 (2005)
Takahama, T., Sakai, S.: Constrained optimization by the α constrained particle swarm optimizer. Journal of Advanced Computational Intelligence and Intelligent Informatics 9(3), 282–289 (2005)
Takahama, T., Sakai, S.: Constrained optimization by ε constrained particle swarm optimizer with ε-level control. In: Proc. of the 4th IEEE International Workshop on Soft Computing as Transdisciplinary Science and Technology (WSTST 2005), May 2005, pp. 1019–1029 (2005)
Takahama, T., Sakai, S.: Constrained optimization by the ε constrained differential evolution with gradient-based mutation and feasible elites. In: Proc. of the 2006 IEEE Congress on Evolutionary Computation, July 2006, pp. 308–315 (2006)
Takahama, T., Sakai, S.: Constrained optimization by the ε constrained genetic algorithm. IPSJ Journal 47(6), 1861–1871 (2006) (in Japanese)
Takahama, T., Sakai, S.: Solving constrained optimization problems by the ε constrained particle swarm optimizer with adaptive velocity limit control. In: Proc. of the 2nd IEEE International Conference on Cybernetics & Intelligent Systems, June 2006, pp. 683–689 (2006)
Takahama, T., Sakai, S., Iwane, N.: Constrained optimization by the e constrained hybrid algorithm of particle swarm optimization and genetic algorithm. In: Zhang, S., Jarvis, R. (eds.) AI 2005. LNCS, vol. 3809, pp. 389–400. Springer, Heidelberg (2005)
Takahama, T., Sakai, S., Iwane, N.: Solving nonlinear constrained optimization problems by the ε constrained differential evolution. In: Proc. of the 2006 IEEE Conference on Systems, Man, and Cybernetics, October 2006, pp. 2322–2327 (2006)
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Takahama, T., Sakai, S. (2009). Solving Difficult Constrained Optimization Problems by the ε Constrained Differential Evolution with Gradient-Based Mutation. In: Mezura-Montes, E. (eds) Constraint-Handling in Evolutionary Optimization. Studies in Computational Intelligence, vol 198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00619-7_3
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DOI: https://doi.org/10.1007/978-3-642-00619-7_3
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