American Institute of Mathematical Sciences

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October  2014, 10(4): 1279-1296. doi: 10.3934/jimo.2014.10.1279

A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization

Qiang Long 1, and  Changzhi Wu 2,

1. 

School of Science, Information, Technology and Engineering, University of Ballarat, Mt Helen, 3350, Victoria
2. 

School of Built Environment, Curtin University, Perth 4845, WA, Australia

Received  August 2012 Revised  October 2013 Published  February 2014

A new global optimization method combining genetic algorithm and Hooke-Jeeves method to solve a class of constrained optimization problems is studied in this paper. We first introduce the quadratic penalty function method and the exact penalty function method to transform the original constrained optimization problem with general equality and inequality constraints into a sequence of optimization problems only with box constraints. Then, the combination of genetic algorithm and Hooke-Jeeves method is applied to solve the transformed optimization problems. Since Hooke-Jeeves method is good at local search, our proposed method dramatically improves the accuracy and convergence rate of genetic algorithm. In view of the derivative-free of Hooke-Jeeves method, our method only requires information of objective function value which not only can overcome the computational difficulties caused by the ill-condition of the square penalty function, but also can handle the non-differentiability by the exact penalty function. Some well-known test problems are investigated. The numerical results show that our proposed method is efficient and robust.
Keywords: Hooke-Jeeves method, Hybrid method, penalty function method, constrained global optimization., genetic algorithm.
Mathematics Subject Classification: Primary: 90c46, 90c90; Secondary: 90c2.
Citation: Qiang Long, Changzhi Wu. A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1279-1296. doi: 10.3934/jimo.2014.10.1279
References:
[1]

A. M. Bagirov and A. N. Ganjehlou, A quasisecant method for minimizing nonsmooth functions, Optimization Methods & Software, 25 (2010), 3-18. doi: 10.1080/10556780903151565.  Google Scholar

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithm (Third Edition), Wiley Online Library, 2006. doi: 10.1002/0471787779.  Google Scholar

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Z. Cai and Y. Wang, A multiobjective optimization-based evolutionary algorithm for constrained optimization, Evolutionary Computation, IEEE Transactions on, 10 (2006), 658-675. doi: 10.1109/TEVC.2006.872344.  Google Scholar

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C. Carroll and A. Fiacco, The created response surface technique for optimizing nonlinear restrained systems, Operations Research, 9 (1961), 169-185. doi: 10.1287/opre.9.2.169.  Google Scholar

[7]

R. Chelouah and P. Siarry, A hybrid method combining continuous tabu search and nelder-mead simplex algorithms for the global optimization of multiminima functions, European Journal of Operational Research, 161 (2005), 636-654. doi: 10.1016/j.ejor.2003.08.053.  Google Scholar

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Z. Chen, S. Qiu and Y. Jiao, A penalty-free method for equality constrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 391-409. doi: 10.3934/jimo.2013.9.391.  Google Scholar

[9]

F. E. Curtis and M. L. Overton, A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization, SIAM Journal on Optimization, 22 (2012), 474-500. doi: 10.1137/090780201.  Google Scholar

[10]

N. Durand and J. Alliot, A combined nelder-mead simplex and genetic algorithm, in Proceedings of the Genetic and Evolutionary Computation Conference GECCO, 99 (1999), 1-7. Google Scholar

[11]

R. Fletcher, An ideal penalty function for constrained optimization, IMA Journal of Applied Mathematics, 15 (1975), 319-342. doi: 10.1093/imamat/15.3.319.  Google Scholar

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H. Greenberg, The generalized penalty-function/surrogate model, Operations Research, 21 (1973), 162-178. doi: 10.1287/opre.21.1.162.  Google Scholar

[14]

A. Hedar, Global optimization methods and codes,, , ().   Google Scholar

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A. Hedar and M. Fukushima, Hybrid simulated annealing and direct search method for nonlinear global optimization, Department of Applied Mathematics & Physics Kyoto University, 17 (2002), 891-912. doi: 10.1080/1055678021000030084.  Google Scholar

[16]

A. Hedar and M. Fukushima, Derivative-free filter simulated annealing method for constrained continuous global optimization, Journal of Global Optimization, 35 (2006), 521-549. doi: 10.1007/s10898-005-3693-z.  Google Scholar

[17]

E. Karas, A. Ribeiro, C. Sagastizábal and M. Solodov, A bundle-filter method for nonsmooth convex constrained optimization, Mathematical Programming, 116 (2009), 297-320. doi: 10.1007/s10107-007-0123-7.  Google Scholar

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Y. Liu, An exterior point linear programming method based on inclusive nornal cone, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.  Google Scholar

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D. Luenberger, Introduction to linear, and nonlinear programming., ().   Google Scholar

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R. Mallipeddi and P. N. Suganthan, Ensemble of constraint handling techniques, Evolutionary Computation, IEEE Transactions on, 14 (2010), 561-579. doi: 10.1109/TEVC.2009.2033582.  Google Scholar

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E. Mezura-Montes and C. C. Coello, A simple multimembered evolution strategy to solve constrained optimization problems, Evolutionary Computation, IEEE Transactions on, 9 (2005), 1-17. doi: 10.1109/TEVC.2004.836819.  Google Scholar

[25]

W. Nakamura, Y. Narushima and H. Yabe, Nonlinear conjugrte gradient methods with sufficient descent properties for uniconstrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 595-619. doi: 10.3934/jimo.2013.9.595.  Google Scholar

[26]

W. Pierskalla, Mathematical programming with increasing constraint functions,, Management Science, 15 (): 416.   Google Scholar

[27]

T. P. Runarsson and X. Yao, Stochastic ranking for constrained evolutionary optimization, Evolutionary Computation, IEEE Transactions on, 4 (2000), 284-294. doi: 10.1109/4235.873238.  Google Scholar

[28]

J. Vasconcelos, J. Ramirez, R. Takahashi and R. Saldanha, Improvements in genetic algorithms, Magnetics, IEEE Transactions on, 37 (2001), 3414-3417. doi: 10.1109/20.952626.  Google Scholar

[29]

Y. Wang, Z. Cai, Y. Zhou and Z. Fan, Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique, Structural and Multidisciplinary Optimization, 37 (2009), 395-413. doi: 10.1007/s00158-008-0238-3.  Google Scholar

[30]

C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895. doi: 10.3934/jimo.2010.6.895.  Google Scholar

[31]

Q. H. Zhiqiang Meng and C. Dang, A penalty function algorithm with objective parameters for nonlinear mathematical programming, Journal of Industrial and Management Optimization, 5 (2009), 585-601. doi: 10.3934/jimo.2009.5.585.  Google Scholar

show all references

References:
[1]

A. M. Bagirov and A. N. Ganjehlou, A quasisecant method for minimizing nonsmooth functions, Optimization Methods & Software, 25 (2010), 3-18. doi: 10.1080/10556780903151565.  Google Scholar

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithm (Third Edition), Wiley Online Library, 2006. doi: 10.1002/0471787779.  Google Scholar

[3]

S. Ben Hamida and M. Schoenauer, Aschea: New results using adaptive segregational constraint handling, in Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on, 1, IEEE, (2002), 884-889. Google Scholar

[4]

Z. Cai and Y. Wang, A multiobjective optimization-based evolutionary algorithm for constrained optimization, Evolutionary Computation, IEEE Transactions on, 10 (2006), 658-675. doi: 10.1109/TEVC.2006.872344.  Google Scholar

[5]

G. Camp, Inequality-constrained stationary-value problems, Journal of the Operations Research Society of America, 3 (1955), 548-550. doi: 10.1287/opre.3.4.548a.  Google Scholar

[6]

C. Carroll and A. Fiacco, The created response surface technique for optimizing nonlinear restrained systems, Operations Research, 9 (1961), 169-185. doi: 10.1287/opre.9.2.169.  Google Scholar

[7]

R. Chelouah and P. Siarry, A hybrid method combining continuous tabu search and nelder-mead simplex algorithms for the global optimization of multiminima functions, European Journal of Operational Research, 161 (2005), 636-654. doi: 10.1016/j.ejor.2003.08.053.  Google Scholar

[8]

Z. Chen, S. Qiu and Y. Jiao, A penalty-free method for equality constrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 391-409. doi: 10.3934/jimo.2013.9.391.  Google Scholar

[9]

F. E. Curtis and M. L. Overton, A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization, SIAM Journal on Optimization, 22 (2012), 474-500. doi: 10.1137/090780201.  Google Scholar

[10]

N. Durand and J. Alliot, A combined nelder-mead simplex and genetic algorithm, in Proceedings of the Genetic and Evolutionary Computation Conference GECCO, 99 (1999), 1-7. Google Scholar

[11]

R. Fletcher, An ideal penalty function for constrained optimization, IMA Journal of Applied Mathematics, 15 (1975), 319-342. doi: 10.1093/imamat/15.3.319.  Google Scholar

[12]

D. E. Goldberg, Genetic algorithms in search,, optimization, ().   Google Scholar

[13]

H. Greenberg, The generalized penalty-function/surrogate model, Operations Research, 21 (1973), 162-178. doi: 10.1287/opre.21.1.162.  Google Scholar

[14]

A. Hedar, Global optimization methods and codes,, , ().   Google Scholar

[15]

A. Hedar and M. Fukushima, Hybrid simulated annealing and direct search method for nonlinear global optimization, Department of Applied Mathematics & Physics Kyoto University, 17 (2002), 891-912. doi: 10.1080/1055678021000030084.  Google Scholar

[16]

A. Hedar and M. Fukushima, Derivative-free filter simulated annealing method for constrained continuous global optimization, Journal of Global Optimization, 35 (2006), 521-549. doi: 10.1007/s10898-005-3693-z.  Google Scholar

[17]

E. Karas, A. Ribeiro, C. Sagastizábal and M. Solodov, A bundle-filter method for nonsmooth convex constrained optimization, Mathematical Programming, 116 (2009), 297-320. doi: 10.1007/s10107-007-0123-7.  Google Scholar

[18]

N. Karmitsa, A. Bagirov and M. Mäkelä, Comparing different nonsmooth minimization methods and software, Optimization Methods and Software, 27 (2012), 131-153. doi: 10.1080/10556788.2010.526116.  Google Scholar

[19]

S. Koziel and Z. Michalewicz, Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization, Evolutionary computation, 7 (1999), 19-44. doi: 10.1162/evco.1999.7.1.19.  Google Scholar

[20]

O. Kramer, A review of constraint-handling techniques for evolution strategies,, Applied Computational Intelligence and Soft Computing, 2010 ().  doi: 10.1155/2010/185063.  Google Scholar

[21]

Y. Liu, An exterior point linear programming method based on inclusive nornal cone, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.  Google Scholar

[22]

D. Luenberger, Introduction to linear, and nonlinear programming., ().   Google Scholar

[23]

R. Mallipeddi and P. N. Suganthan, Ensemble of constraint handling techniques, Evolutionary Computation, IEEE Transactions on, 14 (2010), 561-579. doi: 10.1109/TEVC.2009.2033582.  Google Scholar

[24]

E. Mezura-Montes and C. C. Coello, A simple multimembered evolution strategy to solve constrained optimization problems, Evolutionary Computation, IEEE Transactions on, 9 (2005), 1-17. doi: 10.1109/TEVC.2004.836819.  Google Scholar

[25]

W. Nakamura, Y. Narushima and H. Yabe, Nonlinear conjugrte gradient methods with sufficient descent properties for uniconstrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 595-619. doi: 10.3934/jimo.2013.9.595.  Google Scholar

[26]

W. Pierskalla, Mathematical programming with increasing constraint functions,, Management Science, 15 (): 416.   Google Scholar

[27]

T. P. Runarsson and X. Yao, Stochastic ranking for constrained evolutionary optimization, Evolutionary Computation, IEEE Transactions on, 4 (2000), 284-294. doi: 10.1109/4235.873238.  Google Scholar

[28]

J. Vasconcelos, J. Ramirez, R. Takahashi and R. Saldanha, Improvements in genetic algorithms, Magnetics, IEEE Transactions on, 37 (2001), 3414-3417. doi: 10.1109/20.952626.  Google Scholar

[29]

Y. Wang, Z. Cai, Y. Zhou and Z. Fan, Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique, Structural and Multidisciplinary Optimization, 37 (2009), 395-413. doi: 10.1007/s00158-008-0238-3.  Google Scholar

[30]

C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895. doi: 10.3934/jimo.2010.6.895.  Google Scholar

[31]

Q. H. Zhiqiang Meng and C. Dang, A penalty function algorithm with objective parameters for nonlinear mathematical programming, Journal of Industrial and Management Optimization, 5 (2009), 585-601. doi: 10.3934/jimo.2009.5.585.  Google Scholar

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