Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems

Dianzi Liu

doi:doi:10.11648/j.acm.20160503.13

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Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems

Dianzi Liu

Published in Applied and Computational Mathematics (Volume 5, Issue 3)

Received: 14 June 2016 Published: 15 June 2016

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Abstract

Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.

Published in	Applied and Computational Mathematics (Volume 5, Issue 3)
DOI	10.11648/j.acm.20160503.13
Page(s)	107-113
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Integer-Continuous Optimization, Multipoint Approximation Method, Metamodel, Direct Search, Hybrid Algorithm

References

[1]	R. T. Haftka, J. A Nachlas., L. A. Watson, T. Rizzo, R. Desai, “Two point constraint approximations in structural optimization,” Comput. Methods Appl. Mech. Engrg., vol. 60, pp. 289-301, 1987.
[2]	V. V. Toropov, “Simulation approach to structural optimization,” Struct. Multidisc. Optim., vol. 1, pp. 37-46, 1989.
[3]	G. M. Fadel, M. F. Riley, J. M. Barthelemy, “Two point exponential approximation method for structural optimization,” Struct. Multidisc. Optim., vol. 2, pp. 117-124, 1990.
[4]	L. P. Wang, R. V. Grandhi, “Improved two-point function approximations for design optimization,” AIAA J., vol. 33, pp. 1720-1727, 1995.
[5]	F. van Keulen, V. V. Toropov, “New development in structural optimization using adaptive mesh refinement and multipoint approximations,” Eng. Opt., vol. 29, pp. 217-234, 1997.
[6]	A. Polynkin, V. V. Toropov, “Mid-range metamodel assembly building based on linear regression for large scale optimization problems,” Struct. Multidisc. Optim., vol. 45, pp. 515-527, 2012.
[7]	F. A. C. Viana, R. T. Haftka, “Using multiple surrogates for metamodeling,” Proceedings of 7th ASMO-UK/ISSMO International Conference on Engineering Design Optimization, pp. 132-137, Bath, UK, July 7-8, 2008.
[8]	E. Acar, M. Rais-Rohani, “Ensemble of metamodels with optimized weight factors,” Struct. Multidisc. Optim., vol. 37, pp. 279-294, 2009.
[9]	V. O. Balabanov, G. Venter, “Response surface optimization with discrete variables,” Proc. 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, AIAA 2004–1872, April 19-22, 2004.
[10]	D. Liu, V. V. Toropov, “Multipoint approximation method for design optimization with discrete variables,” 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization (MAO) Conference, Indianapolis, IN, 17–19 September, 2012.
[11]	D. Liu, V. V. Toropov, “Implementation of discrete capability into the enhanced multipoint approximation method for solving mixed integer-continuous optimization problems,” International Journal for Computational Methods in Engineering Science & Mechanics, 17. pp. 22-35, 2016
[12]	T. G. Kolda, R. M. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev., vol. 45, pp. 385-482, 2003.
[13]	R. T. Haftka, Z. Gürdal, Elements of Structural Optimization, 3rd ed., Kluwer Academic Publishers, 1992.
[14]	G. H. Cheng, A. Younis, K. H. Hajikolaei, G. G. Wang, “Trust region based mode pursuing sampling method for global optimization of high dimensional design problems,” Journal of Mechanical Design 137 (2): 021407, 2015.
[15]	Y. Yuan, “Recent advances in trust region algorithms,” Mathematical Programming, 151 (1), pp 249-281, 2015.
[16]	A. Kamandi, K. Amini, M. Ahookhosh, “An improved adaptive trust-region algorithm,” Optimization Letters, pp 1-15, 2016, DOI: 10.1007/s11590-016-1018-4
[17]	G. E. P. Box, N. R. Draper, “Empirical model-building and response surfaces,” New York: John Wiley and Sons, 1987.
[18]	Y. S. Ong, P. Nair, A. Keane, “Evolutionary optimization of computationally expensive problems via surrogate modeling,” AIAA Journal, 41 (4), pp. 687-696, 2003
[19]	X. D. T. Garcia, F. Neri, G. L. Cascella, N. Salvatore, “A surrogate assisted hooke-jeeves algorithm to optimize the control system of a pmsm drive,” 2006 IEEE International Symposium on Industrial Electronics, pp 347 - 352 DOI: 10.1109/ISIE.2006.295618, 2006
[20]	HyperStudy V 12.0, Altair Engineering Inc, USA, 2012.
[21]	P. Hajela, J. Yoo, “Constraint handling in genetic search using expression strategies,” AIAA J., vol. 134, pp. 2414-2420, 1996.
[22]	S. Y. Mahfouz, “Design Optimization of Structural Steelwork,” PhD thesis, University of Bradford, Bradford, UK, 1999.

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Dianzi Liu. (2016). Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Applied and Computational Mathematics, 5(3), 107-113. https://doi.org/10.11648/j.acm.20160503.13

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Dianzi Liu. Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Appl. Comput. Math. 2016, 5(3), 107-113. doi: 10.11648/j.acm.20160503.13

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Dianzi Liu. Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Appl Comput Math. 2016;5(3):107-113. doi: 10.11648/j.acm.20160503.13

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@article{10.11648/j.acm.20160503.13,
  author = {Dianzi Liu},
  title = {Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {3},
  pages = {107-113},
  doi = {10.11648/j.acm.20160503.13},
  url = {https://doi.org/10.11648/j.acm.20160503.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.13},
  abstract = {Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.},
 year = {2016}
}

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TY  - JOUR
T1  - Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems
AU  - Dianzi Liu
Y1  - 2016/06/15
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160503.13
DO  - 10.11648/j.acm.20160503.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 113
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160503.13
AB  - Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.
VL  - 5
IS  - 3
ER  -

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Author Information

Dianzi Liu

School of Mathematics, Faculty of Science, University of East Anglia, Norwich, UK

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APA Style

Dianzi Liu. (2016). Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Applied and Computational Mathematics, 5(3), 107-113. https://doi.org/10.11648/j.acm.20160503.13

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ACS Style

Dianzi Liu. Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Appl. Comput. Math. 2016, 5(3), 107-113. doi: 10.11648/j.acm.20160503.13

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AMA Style

Dianzi Liu. Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems. Appl Comput Math. 2016;5(3):107-113. doi: 10.11648/j.acm.20160503.13

Copy | Download

@article{10.11648/j.acm.20160503.13,
  author = {Dianzi Liu},
  title = {Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {3},
  pages = {107-113},
  doi = {10.11648/j.acm.20160503.13},
  url = {https://doi.org/10.11648/j.acm.20160503.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.13},
  abstract = {Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems
AU  - Dianzi Liu
Y1  - 2016/06/15
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160503.13
DO  - 10.11648/j.acm.20160503.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 107
EP  - 113
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160503.13
AB  - Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.
VL  - 5
IS  - 3
ER  -

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