# A Penalty Function Based Differential Evolution Algorithm for Constrained Optimization

## Abstract

*Differential evolution (DE) and its various dialects are basically designed for solving unconstrained optimization problems and have been widely used .Adaptive differential evolution with optional external archive (JADE)is one of the efficient and updated versions of DE. This paper enhances the capability of JADE to solve constrained optimization problems (COPs). The enhancement is based on introducing a static penalty function in the selection scheme of JADE to handle constraints. The performance of the modified algorithm, abbreviated as CJADE-S is tested on a well-known test suit of COPs, CEC2006. The experimental results show the better performance of CJADE-S on most of the test problems of CEC2006. *

## References

Goldberg, D.E “Genetic Algorithms in Search, Optimization, and Machine Learning”, Addison-Wesley, Reading, Massachusetts, 1998.

D.V. Arnold, “Noisy Optimization With Evolution Strategies”, Norwell, MA. Kluwer. 2002.

C. A. C. Coello, D. A. Van Veldhuizen, and G. B. Lamont, “Evolutionary Algorithms for Solving Multi-Objective Problems”, Norwell, MA:Kluwer, 2002.

A. Homaifar, S.H.Y. Lai and X. Qi, “Constrained optimization via genetic algorithms”, Simulation, vol. 62, pp. 242-254, 1994.

Z. Michalewicz, K. Deb, M. Schmidt, and T. Stidsen, “Test-case generator for constrained parameter optimization techniques”, IEEE Trans. Evol. Comput., vol. 4, no. 3, pp. 197-215. 2000.

Z. Michalewicz and M. Schoenauer, “Evolutionary algorithm for con-strained parameter optimization problems”, Evol. Comput., vol. 4, no.1, pp. 1-32, 1996.

T. Beack (Ed.), “Proceedings of the Seventh International Conference on Genetic Algorithms”, Morgan Kaufmann, San Mateo, CA, 1997.

D.B. Fogel, “Evolutionary Computation. Toward a New Philosophy of Machine Intelligence”, The Institute of Electrical and Electronic Engineers, New York, 1995.

M. Gen, R. Cheng, “Genetic Algorithms Engineering Design”, Wiley, New York, 1997.

D.E. Goldberg, “Genetic Algorithms in Search, Optimization and Machine Learning”, Addison-Wesley, Reading, MA, 1989.

Z. Michalewicz, “Genetic Algorithms + Data Structures Evolution Programs”, 2nd Ed. Springer, Berlin, 1992.

M. Mitchell, “An Introduction to Genetic Algorithms”, MIT Press, Cambridge, MA, 1996.

I. Parmee (Ed.), “The Integration of Evolutionary and Adaptive Computing Technologies with Product/System Design and Realisation”, Springer, Plymouth, UK, 1998.

V.W. Porto, N. Saravanan, D. Waagen, A.E. Eiben (Eds.), “Evolutionary Programming” VII”, Proceedings of the Seventh Annual Conference on Evolutionary Programming, Lecture Notes in Computer Science,1447, Springer, San Diego, CA, 1998.

R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces”, International Computer Science Institute, Berkeley, Tech. Rep. TR-95-012, 1995.

R. Storn and K. V. Price, “Differential evolution-A simple and efficient heuristic for global optimization over continuous Spaces”, J. Global optim., vol. 11, pp. 341-359, 1997.

J. Zhang and A. C.Sanderson, “JADE: Adaptive Differential Evolution with Optional External Archive”, IEEE Transactions on Evolutionary Computation, vol 13, no 5, pp. 945-958, 2009 .

R. Gamperle, S. D. Muller, and P. Koumoutsakos, “A Parameter study for differential evolution, ”Proceedings of the International Conference on Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation (WSEAS '02), Interlaken, Switzerland, pp. 11–15, February 2002.

E. Mezura-Montes, J. Velzquez-Reyes, and C. A. Coello Coello, “A Comparative study of Differential Evolution variants for Global Optimization”, Proceedings of the 8th Annual Conference on Genetic EvolutionaryComputation. C Seattle, Washington., USA, pp. 485-492, 2006..

K. V. Price, R. M. Storn and J. A. Lampinen, “Differential Evolution: A Practical Approach to Global Optimization”. 1st Ed. New York: Springer-Verlag, 2005

H. A. Abbass, “The self-adaptive pareto differential evolution algorithm”, Proc. IEEE Congr. Evol. Comput., 1. Honolulu, HI, pp. 831-836, 2002

Z. Yang, K. Tang, and X. Yao, “Self-adaptive Differential Evolution with Neighbourhood Search”, Proc. of the 2008 IEEE Congress on Evolutionary Computing , Hong Kong, China, pp. 1110-1116, 2008.

Er. Anuj Kumar Parashar, Dr. BDK Patro, Dr. C Patvardhan, “Constraint-Handling techniques for optimization using Differential Evolution”, International Journal of Scientific & Engineering Research, vol. 4, no. 7, ISSN 2229-5518, 2013.

Ozgu Yeniay, “Penalty Function Methods for con-strained optimization with genetic algorithms”, Mathematical and Computational Applications, vol. 10, no. 1, pp. 45-56, 2005.

R. Courant, “Variational methods for the solution of problems of equilibrium and vibrations”, Bull. Am. Math. Soc., vol. 49, pp. 1-23, 1943.

C.W. Carroll, “The created response surface technique for optimizing nonlinear restrained systems”, Operations Research, vol. 9, pp. 169-184,1961.

A.V. Fiacco, G.P. McCormick, “Extensions of SUMT for nonlinear programming: Equality constraints and extrapolation”, Manage. Sci., vol. 12, no. 11, pp. 816-828, 1968.

J. J. Liang, T. P. Runarssaon and P.N. Suganthan, Problems definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization”, J. Appl. Mechanics, Technical Report, vol. 41, no. 8, 2006.

E. M. Montes and C. A. C. Coello, “A Simple Multimembered Evolution Strategy to Solve Constrained Optimization Problems”, IEEE Trans. Evol. Comput, vol. 9, no. 1, pp. 1-15, 2005.

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## How to Cite

*The Nucleus*, vol. 53, no. 2, pp. 155–161, Nov. 2017.