In this study, a hybrid approach combining trust region (TR) algorithm and particle swarm optimization (PSO) is proposed to solve multi-objective optimization problems (MOOPs). The proposed approach integrates the merits of both TR and PSO. Firstly, the MOOP converting by weighted method to a single objective optimization problem (SOOP) and some of the points in the search space are generated. Secondly, TR algorithm is applied to solve the SOOP to obtain a point on the Pareto frontier. Finally, all the points that have been obtained by TR are used as particles position for PSO; where homogeneous PSO is applied to get all nondominated solutions on the Pareto frontier. In addition, to restrict velocity of the particles and control it, a dynamic constriction factor is presented. Various kinds of multiobjective (MO) benchmark problems have been reported to show the importance of hybrid algorithm in generating Pareto optimal set. The results have demonstrated the superiority of the proposed algorithm to solve MOOPs.
| Published in | American Journal of Applied Mathematics (Volume 3, Issue 3) |
| DOI | 10.11648/j.ajam.20150303.11 |
| Page(s) | 81-89 |
| 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), 2015. Published by Science Publishing Group |
Multi-Objective Optimization, Trust Region algorithm, Particle Swarm Optimization, Pareto Optimal Set, Weighted Method
| [1] | Y. Ou, “A hybrid trust region algorithm for unconstrained optimization,” Applied Numerical Mathem., vol. 61, pp. 900–909, 2011. |
| [2] | M. Ahookhosh, K. Amini and M.R. Peyghami, “A nonmonotone trust-region line search method for large-scale unconstrained optimization.” Applied Mathem. Modell., vol. 36, pp. 478–487, 2012. |
| [3] | M. Ahookhosh, and K. Amini, “A Nonmonotone trust region method with adaptive radius for unconstrained optimization problems,” Comput Mathem. Applications, vol. 60, pp. 411–422, 2010. |
| [4] | J. Zhang, K. Zhang and S. Qu, “A nonmonotone adaptive trust region method for unconstrained optimization based on conic model,” Applied Mathem. Computation, vol. 217, pp. 4265–4273, 2010. |
| [5] | B. El-Sobky, “A multiplier active-set trust-region algorithm for solving constrained optimization problem,” Applied Mathem. and Computation, vol. 219, pp. 928–946, 2012. |
| [6] | S. Kim, and J. Ryu, “A trust-region algorithm for bi-objective stochastic optimization,” Procedia Comput. Sci., vol. 4, pp. 1422–1430, 2011. |
| [7] | A.A. El-Sawy, Z.M. Hendawy, M.A. El-Shorbagy, “Trust-Region Algorithm based local search for Multi-objective Optimization”, IEEE 1st International Conference on Innovative Engineering Systems (IEEE-RAS ICIES2012), Alexandria, Egypt, December, 7-9, 2012. |
| [8] | J. Kennedy, R.C. Eberhart and Y. Shi, “Swarm Intelligence,” Morgan Kaufmann, 2001. |
| [9] | K.E. Parsopoulos, and M.N. Vrahatis, “Particle swarm optimization method in multiobjective problems,” Proceedings of the ACM 2002 Symposium on Applied Computing, pp. 603–607, 2002. |
| [10] | A.A. Mousa, M.A. El-Shorbagy and W.F. Abd-El-Wahed, “Local search based hybrid particle swarm optimization algorithm for multiobjective optimization,” Swarm Evolutionary Computation, vol. 3, pp. 1–14, 2012. |
| [11] | M.R. Sierra, and C.C. Coello, “Multi-objective particle swarm optimizers: a survey of the state-of-the-art,” Int. J. Computational Intell. Res., vol. 2, pp. 287–308, 2006. |
| [12] | L. Tang, “A Hybrid Multiobjective Evolutionary Algorithm for Multiobjective Optimization Problems,” IEEE Transactions on Evolutionary Computation, vol. 17, pp. 20–45, 2013. |
| [13] | C.A. Coello, D.A.V. Veldhuizen, G.B. Lamount, “Evolutionary Algorithms for Solving Multi-Objective Problems,” Kluwer Academic Publishers, 2001. |
| [14] | T. Friedrich, T. Kroeger, and F. Neumann, “Weighted preferences in evolutionary multi-objective optimization,” Int. J. Mach. Learn. & Cyber, vol. 4, pp. 139–148, 2013. |
| [15] | J. Dennis, M. El-Alem and K. Williamson, “A trust-region approach to nonlinear systems of equalities and inequalities,” SIAM J. Optimization, vol. 9, pp. 291–315, 1999. |
| [16] | W.F. Abd-El-Wahed, A.A. Mousa and M.A. El-Shorbagy, “Integrating particle swarm optimization with genetic algorithms for solving nonlinear optimization problems,” J. Computational Applied Mathem., vol. 235, pp. 1446–1453, 2011. |
| [17] | K. Deb, “Multi-objective Using Evolutionary Algorithms,” 1 st ed., John Wiley & Sons, LTD, New York, 2001. |
APA Style
M. A. El-Shorbagy. (2015). Weighted Method Based Trust Region-Particle Swarm Optimization for Multi-Objective Optimization. American Journal of Applied Mathematics, 3(3), 81-89. https://doi.org/10.11648/j.ajam.20150303.11
ACS Style
M. A. El-Shorbagy. Weighted Method Based Trust Region-Particle Swarm Optimization for Multi-Objective Optimization. Am. J. Appl. Math. 2015, 3(3), 81-89. doi: 10.11648/j.ajam.20150303.11
AMA Style
M. A. El-Shorbagy. Weighted Method Based Trust Region-Particle Swarm Optimization for Multi-Objective Optimization. Am J Appl Math. 2015;3(3):81-89. doi: 10.11648/j.ajam.20150303.11
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Download@article{10.11648/j.ajam.20150303.11,
author = {M. A. El-Shorbagy},
title = {Weighted Method Based Trust Region-Particle Swarm Optimization for Multi-Objective Optimization},
journal = {American Journal of Applied Mathematics},
volume = {3},
number = {3},
pages = {81-89},
doi = {10.11648/j.ajam.20150303.11},
url = {https://doi.org/10.11648/j.ajam.20150303.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150303.11},
abstract = {In this study, a hybrid approach combining trust region (TR) algorithm and particle swarm optimization (PSO) is proposed to solve multi-objective optimization problems (MOOPs). The proposed approach integrates the merits of both TR and PSO. Firstly, the MOOP converting by weighted method to a single objective optimization problem (SOOP) and some of the points in the search space are generated. Secondly, TR algorithm is applied to solve the SOOP to obtain a point on the Pareto frontier. Finally, all the points that have been obtained by TR are used as particles position for PSO; where homogeneous PSO is applied to get all nondominated solutions on the Pareto frontier. In addition, to restrict velocity of the particles and control it, a dynamic constriction factor is presented. Various kinds of multiobjective (MO) benchmark problems have been reported to show the importance of hybrid algorithm in generating Pareto optimal set. The results have demonstrated the superiority of the proposed algorithm to solve MOOPs.},
year = {2015}
}
TY - JOUR T1 - Weighted Method Based Trust Region-Particle Swarm Optimization for Multi-Objective Optimization AU - M. A. El-Shorbagy Y1 - 2015/04/14 PY - 2015 N1 - https://doi.org/10.11648/j.ajam.20150303.11 DO - 10.11648/j.ajam.20150303.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 81 EP - 89 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20150303.11 AB - In this study, a hybrid approach combining trust region (TR) algorithm and particle swarm optimization (PSO) is proposed to solve multi-objective optimization problems (MOOPs). The proposed approach integrates the merits of both TR and PSO. Firstly, the MOOP converting by weighted method to a single objective optimization problem (SOOP) and some of the points in the search space are generated. Secondly, TR algorithm is applied to solve the SOOP to obtain a point on the Pareto frontier. Finally, all the points that have been obtained by TR are used as particles position for PSO; where homogeneous PSO is applied to get all nondominated solutions on the Pareto frontier. In addition, to restrict velocity of the particles and control it, a dynamic constriction factor is presented. Various kinds of multiobjective (MO) benchmark problems have been reported to show the importance of hybrid algorithm in generating Pareto optimal set. The results have demonstrated the superiority of the proposed algorithm to solve MOOPs. VL - 3 IS - 3 ER -
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