Advertisement

Soft Computing

, Volume 22, Issue 12, pp 3997–4012 | Cite as

Adjust weight vectors in MOEA/D for bi-objective optimization problems with discontinuous Pareto fronts

  • Chunjiang Zhang
  • Kay Chen Tan
  • Loo Hay Lee
  • Liang Gao
Methodologies and Application

Abstract

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) is a recently proposed algorithm which is a research focus in the field of multi-objective evolutionary optimization. It decomposes a multi-objective problem into subproblems by mathematic programming methods and applies evolutionary algorithms to optimize the subproblems simultaneously. MOEA/D is good at finding Pareto solutions which are evenly distributed. However, it can be improved for problems with discontinuous Pareto fronts (PF). Many solutions will assemble in breakpoints in this situation. A method for adjusting weight vectors for bi-objective optimization problems with discontinuous PF is proposed. Firstly, this method detects the weight vectors which need to be adjusted using a property of MOEA/D. Secondly, the reserved vectors are divided into several subsets. Thirdly, after calculating the ideal number of vectors in each subset, vectors are adjusted evenly. Lastly, the corresponding solutions are updated by a linear interpolation. Numerical experiment shows the proposed method obtains good diversity and convergence on approached PF.

Keywords

MOEA/D Adjust weight vector Multi-objective evolutionary algorithm (MOEA) Discontinuous Pareto fronts 

Notes

Acknowledgements

This research work is supported by the National Key Technology Support Program under Grant No. 2015BAF01B04, and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51421062 and 61232008, and China Scholarship Council (CSC).

Compliance with ethical standards

Conflict of interest

All author declares that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76CrossRefGoogle Scholar
  2. Carvalho RD, Saldanha RR, Gomes B, Lisboa AC, Martins A (2012) A multi-objective evolutionary algorithm based on decomposition for optimal design of Yagi–Uda antennas. IEEE Trans Magn 48(2):803–806CrossRefGoogle Scholar
  3. Cheng R, Jin Y, Olhofer M, Sendhoff B (2016) A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 20(5):773–791CrossRefGoogle Scholar
  4. Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657MathSciNetCrossRefMATHGoogle Scholar
  5. Deb K, Jain H (2014) An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans Evol Comput 18(4):577–601CrossRefGoogle Scholar
  6. Deb K, Mohan M, Mishra S (2005) Evaluating the \(\varepsilon \)-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evol Comput 13(4):501–525CrossRefGoogle Scholar
  7. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  8. Deb K, Pratap A, Meyarivan T (2001) Constrained test problems for multi-objective evolutionary optimization. Paper presented at the evolutionary multi-criterion optimizationGoogle Scholar
  9. Dipama J, Teyssedou A, Aubé F, Lizon-A-Lugrin L (2010) A grid based multi-objective evolutionary algorithm for the optimization of power plants. Appl Therm Eng 30(8):807–816CrossRefGoogle Scholar
  10. García S, Molina D, Lozano M, Herrera F (2009) A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J Heuristics 15(6):617–644CrossRefMATHGoogle Scholar
  11. Gu F-Q, Liu H-L (2010) A novel weight design in multi-objective evolutionary algorithm. Paper presented at the 2010 international conference on computational intelligence and security (CIS)Google Scholar
  12. Hillermeier C (2001) Nonlinear multiobjective optimization: a generalized homotopy approach, vol 135. Springer, BerlinCrossRefMATHGoogle Scholar
  13. Horn J, Nafpliotis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. Paper presented at proceedings of the first IEEE conference on evolutionary computation, 1994. IEEE world congress on computational intelligence. Orlando, FLGoogle Scholar
  14. Jiang S, Cai Z, Zhang J, Ong Y-S (2011) Multiobjective optimization by decomposition with Pareto-adaptive weight vectors. Paper presented at 2011 seventh international conference on natural computation (ICNC)Google Scholar
  15. Konstantinidis A, Yang K (2011) Multi-objective energy-efficient dense deployment in Wireless Sensor Networks using a hybrid problem-specific MOEA/D. Appl Soft Comput 11(6):4117–4134. doi: 10.1016/j.asoc.2011.02.031 CrossRefGoogle Scholar
  16. Kukkonen S, Deb K (2006) A fast and effective method for pruning of non-dominated solutions in many-objective problems Parallel Problem Solving from Nature-PPSN IX. Springer, Berlin, pp 553–562Google Scholar
  17. Li H, Landa-Silva D (2011) An adaptive evolutionary multi-objective approach based on simulated annealing. Evol Comput 19(4):561–595CrossRefGoogle Scholar
  18. Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302CrossRefGoogle Scholar
  19. Liu Y, Gong D, Sun X, Zhang Y (2017) Many-objective evolutionary optimization based on reference points. Appl Soft Comput 50:344–355CrossRefGoogle Scholar
  20. Ma X, Liu F, Qi Y, Li L, Jiao L, Deng X et al. (2015) MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem. Soft Comput. doi: 10.1007/s00500-015-1789-z
  21. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscip Optim 25(2):86–98MathSciNetCrossRefMATHGoogle Scholar
  22. Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, BostonMATHGoogle Scholar
  23. Phan DH, Suzuki J (2013) R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization. Paper presented at 2013 IEEE congress on evolutionary computation (CEC)Google Scholar
  24. Qi Y, Ma X, Liu F, Jiao L, Sun J, Wu J (2014) MOEA/D with adaptive weight adjustment. Evol Comput 22(2):231–264. doi: 10.1162/EVCO_a_00109 CrossRefGoogle Scholar
  25. Schaffer JD (1985) Some experiments in machine learning using vector evaluated genetic algorithms. Vanderbilt University, NashvilleGoogle Scholar
  26. Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248CrossRefGoogle Scholar
  27. Takahama T, Sakai S (2006) Constrained optimization by the \(\varepsilon \) constrained differential evolution with gradient-based mutation and feasible elites. Paper presented at the 2006 IEEE congress on evolutionary computation (CEC), Vancouver, BCGoogle Scholar
  28. Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 17(5):721–736CrossRefGoogle Scholar
  29. Zhang CJ, Lin Q, Gao L (2015) A novel adaptive \(\varepsilon \)-constrained method for constrained problem. Paper presented at the proceedings of the 18th Asia Pacific symposium on intelligent and evolutionary systems, vol 1. SingaporeGoogle Scholar
  30. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731CrossRefGoogle Scholar
  31. Zhang Q, Liu W, Li H (2009) The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances. Paper presented at the IEEE congress on evolutionary computation, TrondheimGoogle Scholar
  32. Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2008) Multiobjective optimization test instances for the CEC 2009 special session and competition. technical report. University of Essex, Colchester, UK and Nanyang technological University, Singapore Google Scholar
  33. Zhang Y, Yang R, Zuo J, Jing X (2015) Enhancing MOEA/D with uniform population initialization, weight vector design and adjustment using uniform design. J Syst Eng Electron 26(5):1010–1022CrossRefGoogle Scholar
  34. Zhu Y, Wang J, Qu B (2014) Multi-objective economic emission dispatch considering wind power using evolutionary algorithm based on decomposition. Int J Electr Power Energy Syst 63:434–445. doi: 10.1016/j.ijepes.2014.06.027 CrossRefGoogle Scholar
  35. Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. Paper presented at the parallel problem solving from nature-PPSN VIIIGoogle Scholar
  36. Zitzler E, Laumanns M, Thiele L (2001) SPEA2: Improving the strength Pareto evolutionary algorithm. Technical Report. Zürich, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH), Institut für Technische Informatik und Kommunikationsnetze (TIK)Google Scholar
  37. Zitzler E, Thiele L (1998a) An evolutionary algorithm for multiobjective optimization: The strength pareto approach. Zürich, Switzerland: Technical Report 43, Eidgenössische Technische Hochschule Zürich (ETH), Institut für Technische Informatik und Kommunikationsnetze (TIK)Google Scholar
  38. Zitzler E, Thiele L (1998b) Multiobjective optimization using evolutionary algorithms—a comparative case study. Paper presented at the international conference on parallel problem solving from natureGoogle Scholar
  39. Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Chunjiang Zhang
    • 1
    • 2
  • Kay Chen Tan
    • 3
  • Loo Hay Lee
    • 4
  • Liang Gao
    • 1
  1. 1.The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and EngineeringHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.School of Electrical and Electronic EngineeringNanyang Technological UniversitySingaporeSingapore
  3. 3.Department of Computer ScienceCity University of Hong KongKowloon TongHong Kong
  4. 4.Department of Industrial and Systems EngineeringNational University of SingaporeSingaporeSingapore

Personalised recommendations