Abstract
In recent years, the LLE manifold learning algorithm is innovatively introduced to the multi-objective evolutionary algorithm to solve continuous multi-objective problems. It is according to the regularity that the Pareto set of a continuous multi-objective problem is a piece-wise continuous (m − 1)-dimensional manifold, and m is the objective number. The goal of the LLE manifold learning algorithm is to reduce the dimension of data. However, the existing LLE-based algorithm directly applies LLE manifold learning algorithm to the MOP modeling for individual reproduction. It has the following weaknesses: (1) the (d − 1)-dimensional manifold, which is constructed by refactoring coefficient, is not necessarily the manifold of the optimal solution space; (2) when resampling, the original information of samples is basically lost, especially in the case of d = 2, only keeping the scope of a linear interval. The distribution of the original samples information is completely lost and the cost of repeated calculation is higher; (3) the neighborhood relationship of resampling does not mean the neighborhood relationship of samples in PS space. So, we propose a new LLE modeling algorithm which is called O2O-LLE approach. The new O2O-LLE approach is inspired by LLE manifold learning idea and makes full use of the mapping function known in the MOP that the decision space is considered as the high-dimensional space and the object space is regarded as a low-dimensional space. Thus, the new modeling algorithm is no longer to build the overall low-dimensional space of the sample and then reflected back to the high-dimensional space, but it replaces directly constructing new samples in high-dimensional space. Thereby, the above four weaknesses are effectively avoided. Its steps are as follows: (1) mapping the samples from PS space to PF space; (2) searching K neighbors in PF space; (3) calculating LLE refactoring coefficient according to the K neighbors in PF space; (4) producing offspring sample according to the refactoring coefficient in PS space. Also, different from the early algorithm framework HMOEDA_LLE, the new algorithm framework O2O-LLE-RM does not include the genetic operation, so its efficiency is improved. To verify the performance of O2O-LLE-RM, several widely used test problems are employed to conduct the comparison experiments with three state-of-the-art multi-objective evolutionary algorithms: NSGA-II, RM-MEDA and Firefly. The simulated results show that the proposed algorithm has better optimization performance.
Similar content being viewed by others
References
Srinivas N, Deb K (1994) Multi-objective optimization using non-dominated sorting in genetic algorithms. Evol Comput 2:221–248
Murata T, Ishibuchi H (1995) MOGA: multi-objective genetic algorithms. In: Proceedings of the 2nd IEEE international conference on evolutionary computing. IEEE, vol 2, pp 289–294
Zitzler E, Thiele L (1999) Multi-objective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3:257–271
Deb K, Pratap A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. Evol Comput 6:182–197
Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength Pareto evolutionary algorithm. Technical report 103. Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zutich,Gloriastrasse 35, CH-8092 Zurich, Switzerland
Zhou A, Zhang Q, Jin Y, Tsang E, Okabe T (2005) A model-based evolutionary algorithm for bi-objective optimization. In: Congress on evolutionary computation. IEEE, pp 2568–2575
Zhou A, Jin Y, Zhang Q, Sendhoff B, Tsang E (2006) Combining model-based and genetics-based offspring generation for multi-objective optimization using a convergence criterion. In: Congress on evolutionary computation. IEEE, pp 3234–3241
Zhou A, Zhang Q, Jin Y, Sendhoff B, Tsang E (2006) Modelling the population distribution in multiobjective optimization by generative topographic mapping. In: Parallel problem solving from nature-PPSN IX. Springer, Berlin, pp 443–452
Zhang Q, Zhou A, Jin Y (2008) RM-MEDA: a regularity model-based multiobjective estimation of distribution algorithm. IEEE Trans Evol Comput 12:41–63
Zhou A, Zhang Q, Jin Y (2009) Approximating the set of Pareto-optimal solutions in both the decision and objective spaces by an estimation of distribution algorithm. IEEE Trans Evol Comput 13:1167–1189
Yuan Q, Dai G (2015) Research on the features of NSGA-II and RM-MEDA. J Comput Inf Syst 11:6735–6745
Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323–2326
Chen J, Liu Y (2011) Locally linear embedding: a survey. Artif Intell Rev 36:29–48
Chahooki MAZ, Charkari NM (2014) Shape classification by manifold learning in multiple observation spaces. Inf Sci 262:46–61
Chahooki MAZ, Charkari NM (2014) Unsupervised manifold learning based on multiple feature spaces. Mach Vis Appl 25:1053–1065
Hettiarachchi R, Peters JF (2015) Multi-manifold LLE learning in pattern recognition. Pattern Recognit 48:2947–2960
Zhang Y, Dai* G et al (2014) HMOEDA_LLE: a hybrid multi-objective estimation of distribution algorithm combining locally linear embedding. In: IEEE congress on evolutionary computation, 2014. CEC’14, IEEE, pp 707–714
Yang XS (2013) Multiobjective firefly algorithm for continuous optimization. Eng Comput 29:175–184
Ding Shifei, Fulin Wu, Nie Ru, Junzhao Yu, Huang Huajuan (2013) Twin support vector machines based on quantum particle swarm optimization. J Softw 8:1743–1750
Ding S, Yu J, Huang H, Zhao H (2013) Twin support vector machines based on particle swarm optimization. J Comput 8:2296–2303
Ding S, Zhang Y, Chen J, Jia W (2013) Research on using genetic algorithms to optimize Elman neural networks. Neural Comput Appl 23:293–297
Ding S, Zhang X, Yu J (2015) Twin support vector machines based on fruit fly optimization algorithm. Int J Mach Learn Cybern. doi:10.1007_s13042-015-0424-8
Yang X-S, Deb S (2013) Multiobjective cuckoo search for design optimization. Comput Oper Res 40:1616–1624
Fister I, Yang X-S, Brest J (2013) Modified firefly algorithm using quaternion representation. Expert Syst Appl 40:7220–7230
Yang X-S (2014) Swarm intelligence based algorithms: a critical analysis. Evol Intell 7:17–28
Gandomia AH, Yang X-S (2014) Chaotic bat algorithm. J Comput Sci 5:224–232
Cayton L (2005) Algorithms for manifold learning. University of California, San Diego. Tech. Rep. CS2008-0923
Seung HS, Lee DD (2000) The manifold ways of perception. Science 290:2268–2269
Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290:2319–2323
Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15:1373–1396
Donoho DL, Grimes C (2003) Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci 100:5591–5596
Zhang Z, Zha H (2005) Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J Sci Comput 26:313–338
Laumanns M, Ocenasek J (2002) Bayesian optimization algorithms for multi-objective optimization. Parallel problem solving from nature PPSN VII. Springer, Berlin, pp 298–307
Okabe T, Jin Y, Sendoff B et al (2004) Voronoi-based estimation of distribution algorithm for multi-objective optimization. In: Congress on evolutionary computation, 2004. CEC2004. IEEE, vol 2. pp 1594–1601
Bosman PAN, Thierens D (2005) The naive MIDEA: a baseline multi-objective EA. In: Evolutionary multi-criterion optimization: Third international conference, EMO 2005, Guanajuato, Mexico, March 9–11, 2005, Proceedings, vol 3410. Springer, p 428
Pelikan M, Sastry K, Goldberg DE (2005) Multiobjective hBOA, clustering, and scalability. In: Proceedings of the 2005 conference on genetic and evolutionary computation. ACM, pp 663–670
Dong W, Yao X (2008) Unified eigen analysis on multivariate Gaussian based estimation of distribution algorithms. Inf Sci 178:3000–3023
Zhang D, Gong X, Dai G (2011) Multi-objective evolutionary algorithm for principal curve model based on multifractal. J Comput Res Dev 48:1729–1739
Farhang-Mehr A, Azarm S (2002) Diversity assessment of Pareto optimal solution sets: an entropy approach. In: Proceedings of the 2002 congress on evolutionary computation, 2002. CEC’02. IEEE, vol. 1. pp 723–728
Gunawan S, Farhang-Mehr A, Azarm S (2003) Multi-level multi-objective genetic algorithm using entropy to preserve diversity. In: Fonseca CM, Fleming PJ, Zitzler E, Thiele L, Deb K (eds) Evolutionary multi-criterion optimization. Springer, Berlin, pp 148–161
Ocenasek J (2006) Entropy-based convergence measurement in discrete estimation of distribution algorithms. Towards a new evolutionary computation. Springer, Berlin, pp 39–50
Wang YN, Wu LH, Yuan XF (2010) Multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure. Soft Comput 14:193–209
Qin Y, Ji J, Liu C (2012) An entropy-based multiobjective evolutionary algorithm with an enhanced elite mechanism. Appl Comput Intell Soft Comput 2012:91–96
Wang LL, Chen Y (2012) Diversity based on entropy: a novel evaluation criterion in multi-objective optimization algorithm. Int J Intell Syst Appl (IJISA) 4:113
Li H, Zhang Q (2009) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13:284–302
Bosman T, Thierens D (2003) The balance between proximity and diversity in multi objective evolutionary algorithms. IEEE Trans Evol Comput 7:174–188
Sheskin DJ (2006) Handbook of parametric and nonparametric statistical procedures, 4th edn [s. l.]. Chapman & Hall/CRC, London
Zhao H, Li M, Weng X, Zhou H (2015) Performance evaluation for biology-inspired optimization algorithms based on nonparametric statistics. J Air Force Eng Univ (Nat Sci Edn) 1:89–94
Acknowledgments
This paper is supported by National Natural Science Foundation of China (Nos. 61472375, 61103144, 61379101).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yuan, Q., Dai, G. & Zhang, Y. A novel multi-objective evolutionary algorithm based on LLE manifold learning. Engineering with Computers 33, 293–305 (2017). https://doi.org/10.1007/s00366-016-0473-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-016-0473-y