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.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76
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):803–806
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):773–791
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):631–657
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):577–601
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):501–525
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197
Deb K, Pratap A, Meyarivan T (2001) Constrained test problems for multi-objective evolutionary optimization. Paper presented at the evolutionary multi-criterion optimization
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):807–816
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):617–644
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)
Hillermeier C (2001) Nonlinear multiobjective optimization: a generalized homotopy approach, vol 135. Springer, Berlin
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, FL
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)
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
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 553–562
Li H, Landa-Silva D (2011) An adaptive evolutionary multi-objective approach based on simulated annealing. Evol Comput 19(4):561–595
Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302
Liu Y, Gong D, Sun X, Zhang Y (2017) Many-objective evolutionary optimization based on reference points. Appl Soft Comput 50:344–355
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
Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscip Optim 25(2):86–98
Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, Boston
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)
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
Schaffer JD (1985) Some experiments in machine learning using vector evaluated genetic algorithms. Vanderbilt University, Nashville
Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248
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, BC
Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 17(5):721–736
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. Singapore
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731
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, Trondheim
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
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):1010–1022
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
Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. Paper presented at the parallel problem solving from nature-PPSN VIII
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)
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)
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 nature
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):117–132
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).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
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.
Additional information
Communicated by V. Loia.
Rights and permissions
About this article
Cite this article
Zhang, C., Tan, K.C., Lee, L.H. et al. Adjust weight vectors in MOEA/D for bi-objective optimization problems with discontinuous Pareto fronts. Soft Comput 22, 3997–4012 (2018). https://doi.org/10.1007/s00500-017-2609-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-017-2609-4