Journal Home > Volume 3 , Issue 1
Complex System Modeling and Simulation 2023, 3(1): 59-70 https://doi.org/10.23919/CSMS.2022.0024
Open Access | Issue | Published: 09 March 2023

A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives

Show Author's Information Fangqing Gu1( ),  Haosen Liu1,  Hailin Liu1
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China
Keywords:
evolutionary algorithm, decomposition, many-objective optimization, objective conflict, set covering model
Cite this article:
Gu F, Liu H, Liu H. A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives. Complex System Modeling and Simulation, 2023, 3(1): 59-70. https://doi.org/10.23919/CSMS.2022.0024
Download citation
EndNote(RIS)
BibTeX

632

Views

36

Downloads

Citations

3

Crossref

N/A

WoS

2

Scopus

N/A

CSCD

Abstract Full text About this article

Evolutionary algorithm is an effective strategy for solving many-objective optimization problems. At present, most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other. In some cases, however, the objectives are not always in conflict. It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance. The classical evolutionary many-objective algorithms may not be able to effectively solve such problems. Accordingly, we propose an objective set decomposition strategy based on the partial set covering model. It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible. An optimization subproblem is defined on each objective subset. A coevolutionary algorithm is presented to optimize all subproblems simultaneously, in which a nondominance ranking is presented to interact information among these sub-populations. The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems. Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.


menu
Abstract
Full text
Outline
About this article

A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives

Show Author's information Fangqing Gu1( ), Haosen Liu1, Hailin Liu1
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China

Abstract

Evolutionary algorithm is an effective strategy for solving many-objective optimization problems. At present, most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other. In some cases, however, the objectives are not always in conflict. It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance. The classical evolutionary many-objective algorithms may not be able to effectively solve such problems. Accordingly, we propose an objective set decomposition strategy based on the partial set covering model. It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible. An optimization subproblem is defined on each objective subset. A coevolutionary algorithm is presented to optimize all subproblems simultaneously, in which a nondominance ranking is presented to interact information among these sub-populations. The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems. Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.

Keywords: evolutionary algorithm, decomposition, many-objective optimization, objective conflict, set covering model

References(39)

[1]

X. Zhang, Y. Tian, R. Cheng, and Y. Jin, An efficient approach to nondominated sorting for evolutionary multiobjective optimization, IEEE Transactions on Evolutionary Computation, vol. 19, no. 2, pp. 201–213, 2015.
DOI Google Scholar
[2]

W. Fang, L. Zhang, S. Yang, J. Sun, and X. Wu, A multiobjective evolutionary algorithm based on coordinate transformation, IEEE Transactions on Cybernetics, vol. 49, no. 7, pp. 2732–2743, 2019.
DOI Google Scholar
[3]

H. -L. Liu, F. Gu, Y. -M. Cheung, S. Xie, and J. Zhang, On solving WCDMA network planning using iterative power control scheme and evolutionary multiobjective algorithm, IEEE Computational Intelligence Magazine, vol. 9, no. 1, pp. 44–52, 2014.
DOI Google Scholar
[4]

B. Li, J. Li, K. Tang, and X. Yao, Many-objective evolutionary algorithms: A survey, ACM Computing Surveys, vol. 48, no. 1, pp. 1–35, 2015.
DOI Google Scholar
[5]

H. Bai, T. Fan, Y. Niu, and Z. Cui, Multi-UAV cooperative trajectory planning based on many-objective evolutionary algorithm, Complex System Modeling and Simulation, vol. 2, no. 2, pp. 130–141, 2022.
DOI Google Scholar
[6]

L. Pan, L. Li, C. He, and K. C. Tan, A subregion division-based evolutionary algorithm with effective mating selection for many-objective optimization, IEEE Transactions on Cybernetics, vol. 50, no. 8, pp. 3477–3490, 2020.
DOI Google Scholar
[7]

K. Qiao, J. Liang, B. Qu, K. Yu, C. Yue, and H. Song, Differential evolution with level-based learning mechanism, Complex System Modeling and Simulation, vol. 2, no. 1, pp. 35–58, 2022.
DOI Google Scholar
[8]

K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182–197, 2002.
DOI Google Scholar
[9]
E. Zitzler, M. Laumanns, and L. Thiele, SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization, in Proc. Evolutionary Methods for Design Optimization and Control with Applications to Industrial Problems, Athens, Greece, 2001, pp. 95–100.
[10]

L. B. Said, S. Bechikh, and K. Ghedira, The r-dominance: A new dominance relation for interactive evolutionary multicriteria decision making, IEEE Transactions on Evolutionary Computation, vol. 14, no. 5, pp. 801–818, 2010.
DOI Google Scholar
[11]

S. Yang, M. Li, X. Liu, and J. Zheng, A grid-based evolutionary algorithm for many-objective optimization, IEEE Transactions on Evolutionary Computation, vol. 17, no. 5, pp. 721–736, 2013.
DOI Google Scholar
[12]

Q. Zhang and H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, vol. 11, no. 6, pp. 712–731, 2007.
DOI Google Scholar
[13]

H. Liu, F. Gu, and Q. Zhang, Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems, IEEE Transactions on Evolutionary Computation, vol. 18, no. 3, pp. 450–455, 2014.
DOI Google Scholar
[14]

L. Chen, H. -L. Liu, K. C. Tan, Y. -M. Cheung, and Y. Wang, Evolutionary many-objective algorithm using decomposition-based dominance relationship, IEEE Transactions on Cybernetics, vol. 49, no. 12, pp. 4129–4139, 2019.
DOI Google Scholar
[15]

J. Bader and E. Zitzler, Hype: An algorithm for fast hypervolume-based many-objective optimization, Evolutionary Computation, vol. 19, no. 1, pp. 45–76, 2011.
DOI Google Scholar
[16]

J. Yuan, H. -L. Liu, F. Gu, Q. Zhang, and Z. He, Investigating the properties of indicators and an evolutionary many-objective algorithm using promising regions, IEEE Transactions on Evolutionary Computation, vol. 25, no. 1, pp. 75–86, 2021.
DOI Google Scholar
[17]

Y. Sun, G. G. Yen, and Z. Yi, IGD indicator-based evolutionary algorithm for many-objective optimization problems, IEEE Transactions on Evolutionary Computation, vol. 23, no. 2, pp. 173–187, 2019.
DOI Google Scholar
[18]

J. G. Falcón-Cardona and C. A. Coello, Indicator-based multi-objective evolutionary algorithms: A comprehensive survey, ACM Computing Surveys, vol. 53, no. 2, pp. 1–35, 2021.
DOI Google Scholar
[19]
A. R. R. Freitas, P. J. Fleming, and F. G. Guimarães, A non-parametric harmony-based objective reduction method for many-objective optimization, in Proc. 2013 IEEE International Conference on Systems, Man, and Cybernetics, Manchester, UK, 2013, pp. 651–656.
DOI
[20]

H. K. Singh, A. Isaacs, and T. Ray, A Pareto corner search evolutionary algorithm and dimensionality reduction in many-objective optimization problems, IEEE Transactions on Evolutionary Computation, vol. 15, no. 4, pp. 539–556, 2011.
DOI Google Scholar
[21]

Y. Li, H. L. Liu, and E. D. Goodman, Hyperplane-approximation-based method for many-objective optimization problems with redundant objectives, Evolutionary Computation, vol. 27, no. 2, pp. 313–344, 2019.
DOI Google Scholar
[22]

D. Brockhoff and E. Zitzler, Objective reduction in evolutionary multiobjective optimization: Theory and applications, Evolutionary Computation, vol. 17, no. 2, pp. 135–166, 2009.
DOI Google Scholar
[23]

D. K. Saxena, J. A. Duro, A. Tiwari, K. Deb, and Q. Zhang, Objective reduction in many-objective optimization: Linear and nonlinear algorithms, IEEE Transactions on Evolutionary Computation, vol. 17, no. 1, pp. 77–99, 2013.
DOI Google Scholar
[24]
Y. -M. Cheung and F. Gu, Online objective reduction for many-objective optimization problems, in Proc. 2014 IEEE Congr. Evol. Comput., Beijing, China, 2014, pp. 1165–1171.
DOI
[25]

Y. -M. Cheung, F. Gu, and H. -L. Liu, Objective extraction for many-objective optimization problems: Algorithm and test problems, IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 755–772, 2016.
DOI Google Scholar
[26]

A. Zhou, Y. Wang, and J. Zhang, Objective extraction via fuzzy clustering in evolutionary many-objective optimization, Information Sciences, vol. 509, pp. 343–355, 2020.
DOI Google Scholar
[27]

X. Guo, Y. Wang, and X. Wang, An objective reduction algorithm using representative Pareto solution search for many-objective optimization problems, Soft Computing, vol. 20, no. 12, pp. 4881–4895, 2016.
DOI Google Scholar
[28]
R. C. Purshouse and P. Fleming, An adaptive divide-and-conquer methodology for evolutionary multi-criterion optimization, in Proc. the 2nd International Conference Evolutionary Multi-Criterion Optimization, Faro, Portugal, 2003, pp. 133–147.
DOI
[29]

N. Luo, X. Li, and Q. Lin, Objective reduction for many-objective optimization problems using objective subspace extraction, Soft Computing, vol. 22, no. 4, pp. 1159–1173, 2018.
DOI Google Scholar
[30]
H. Aguirre and K. Tanaka, Space partitioning with adaptive ϵ-ranking and substitute distance assignments: A comparative study on many-objective mnk-landscapes, in Proc. the 11th Annual Conference on Genetic and Evolutionary Computation, Montreal, Canada, 2009, pp. 547–554.
DOI
[31]

A. L. Jaimes, C. A. Coello, H. Aguirre, and K. Tanaka, Objective space partitioning using conflict information for solving many-objective problems, Information Sciences, vol. 268, pp. 305–327, 2014.
DOI Google Scholar
[32]

K. Deb and H. Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints, IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577–601, 2014.
DOI Google Scholar
[33]

R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, A reference vector guided evolutionary algorithm for many-objective optimization, IEEE Transactions on Evolutionary Computation, vol. 20, no. 5, pp. 773–791, 2016.
DOI Google Scholar
[34]

E. Zitzler, K. Deb, and L. Thiele, Comparison of multiobjective evolutionary algorithms: Empirical results, Evolutionary Computation, vol. 8, no. 2, pp. 173–195, 2000.
DOI Google Scholar
[35]

P. Bosman and D. Thierens, The balance between proximity and diversity in multiobjective evolutionary algorithms, IEEE Transactions on Evolutionary Computation, vol. 7, no. 2, pp. 174–188, 2003.
DOI Google Scholar
[36]

E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transactions on Evolutionary Computation, vol. 3, no. 4, pp. 257–271, 1999.
DOI Google Scholar
[37]

I. Das and J. Dennis, Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems, SIAM Journal on Optimization, vol. 8, no. 3, pp. 631–657, 1998.
DOI Google Scholar
[38]

K. Deb and R. B. Agrawal, Simulated binary crossover for continuous search space, Complex Systems, vol. 9, no. 3, pp. 115–148, 1995.
Google Scholar
[39]

K. Deb and M. Goyal, A combined genetic adaptive search (GeneAS) for engineering design, Computer Science and Informatics, vol. 26, no. 4, pp. 30–45, 1996.
Google Scholar
Publication history
Copyright
Acknowledgements
Rights and permissions

Publication history

Received: 08 September 2022
Revised: 04 October 2022
Accepted: 06 November 2022
Published: 09 March 2023
Issue date: March 2023

Copyright

© The author(s) 2023

Acknowledgements

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (No. 62172110), the Natural Science Foundation of Guangdong Province (Nos. 2021A1515011839 and 2022A1515010130), and the Programme of Science and Technology of Guangdong Province (No. 2021A0505110004).

Rights and permissions

The articles published in this open access journal are distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).

Return