IEEE/CAA Journal of Automatica Sinica
Citation:  Y. Tian, H. W. Chen, H. P. Ma, X. Y. Zhang, K. C. Tan, and Y. C. Jin, “Integrating conjugate gradients into evolutionary algorithms for largescale continuous multiobjective optimization,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 10, pp. 1801–1817, Oct. 2022. doi: 10.1109/JAS.2022.105875 
[1] 
L. M. Antonio and C. A. C. Coello, “Use of cooperative coevolution for solving large scale multiobjective optimization problems,” in Proc. IEEE Congr. Evolutionary Computation, Cancun, Mexico, 2013, pp. 2758–2765.

[2] 
C. He, R. Cheng, C. J. Zhang, Y. Tian, Q. Chen, and X. Yao, “Evolutionary largescale multiobjective optimization for ratio error estimation of voltage transformers,” IEEE Trans. Evol. Comput., vol. 24, no. 5, pp. 868–881, Oct. 2020. doi: 10.1109/TEVC.2020.2967501

[3] 
G. L. Li and D. Z. Cao, “A multiobjective particle swarm algorithm for the optimization of IMRT inverse planning,” in Proc. 3rd Int. Conf. Biomedical Engineering and Informatics, Yantai, China, 2010, pp. 1327–1330.

[4] 
J. Dias, H. Rocha, T. Ventura, B. Ferreira, and M. Do Carmo Lopes, “Automated fluence map optimization based on fuzzy inference systems,” Med. Phys., vol. 43, no. 3, pp. 1083–1095, Mar. 2016. doi: 10.1118/1.4941007

[5] 
R. Cheng, Y. C. Jin, M. Olhofer, and B. Sendhoff, “Test problems for largescale multiobjective and manyobjective optimization,” IEEE Trans. Cybern., vol. 47, no. 12, pp. 4108–4121, Dec. 2017. doi: 10.1109/TCYB.2016.2600577

[6] 
Y. Tian, L. C. Si, X. Y. Zhang, R. Cheng, C. He, K. C. Tan, and Y. C. Jin, “Evolutionary largescale multiobjective optimization: A survey,” ACM Comput. Surv., vol. 54, no. 8, pp. 174, Nov. 2022.

[7] 
H. Zille, H. Ishibuchi, S. Mostaghim, and Y. Nojima, “A framework for largescale multiobjective optimization based on problem transformation,” IEEE Trans. Evol. Comput., vol. 22, no. 2, pp. 260–275, Apr. 2018. doi: 10.1109/TEVC.2017.2704782

[8] 
Y. Tian, C. Lu, X. Y. Zhang, K. C. Tan, and Y. C. Jin, “Solving largescale multiobjective optimization problems with sparse optimal solutions via unsupervised neural networks,” IEEE Trans. Cybern., vol. 51, no. 6, pp. 3115–3128, Jun. 2021. doi: 10.1109/TCYB.2020.2979930

[9] 
Y. Tian, X. T. Zheng, X. Y. Zhang, and Y. C. Jin, “Efficient largescale multiobjective optimization based on a competitive swarm optimizer,” IEEE Trans. Cybern., vol. 50, no. 8, pp. 3696–3708, Aug. 2020. doi: 10.1109/TCYB.2019.2906383

[10] 
R. Cheng, Y. C. Jin, K. Narukawa, and B. Sendhoff, “A multiobjective evolutionary algorithm using Gaussian processbased inverse modeling,” IEEE Trans. Evol. Comput., vol. 19, no. 6, pp. 838–856, Dec. 2015. doi: 10.1109/TEVC.2015.2395073

[11] 
S. S. Petrova and A. D. Solovév, “The origin of the method of steepest descent,” Hist. Math., vol. 24, no. 4, pp. 361–375, Nov. 1997. doi: 10.1006/hmat.1996.2146

[12] 
D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in Proc. 3rd Int. Conf. Learning Representations, San Diego, USA, 2015.

[13] 
M. Avriel, Nonlinear Programming: Analysis and Methods, Dover Publications, New York, USA, 2003.

[14] 
D. F. Shanno, “Conditioning of quasiNewton methods for function minimization,” Math. Comp., vol. 24, no. 111, pp. 647–656, Jul. 1970. doi: 10.1090/S0025571819700274029X

[15] 
K. G. Murty, Linear Programming. John Wiley & Sons, UK, 1983.

[16] 
M. M. Li, “Generalized Lagrange multiplier method and KKT conditions with an application to distributed optimization,” IEEE Trans. Circuits Syst. II: Express Briefs, vol. 66, no. 2, pp. 252–256, Feb. 2019. doi: 10.1109/TCSII.2018.2842085

[17] 
D. W. Marquardt, “An algorithm for leastsquares estimation of nonlinear parameters,” J. Soc. Indust. Appl. Math., vol. 11, no. 2, pp. 431–441, Jun. 1963. doi: 10.1137/0111030

[18] 
P. T. Boggs and J. W. Tolle, “Sequential quadratic programming,” Acta Numer., vol. 4, pp. 1–51, Jan. 1995. doi: 10.1017/S0962492900002518

[19] 
B. O’Donoghue, E. Chu, N. Parikh, and S. Boyd, “Conic optimization via operator splitting and homogeneous selfdual embedding,” J. Optim. Theory Appl., vol. 169, no. 3, pp. 1042–1068, Feb. 2016. doi: 10.1007/s1095701608923

[20] 
S. S. Yang, Y. Tian, C. He, X. Y. Zhang, K. C. Tan, and Y. C. Jin, “A gradientguided evolutionary approach to training deep neural networks,” IEEE Trans. Neural Networks Learn. Syst., 2021,

[21] 
Q. F. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposition,” IEEE Trans. Evol. Comput., vol. 11, no. 6, pp. 712–731, Dec. 2007. doi: 10.1109/TEVC.2007.892759

[22] 
Y. D. Sergeyev, D. E. Kvasov, and M. S. Mukhametzhanov, “On the efficiency of natureinspired metaheuristics in expensive global optimization with limited budget,” Sci. Rep., vol. 8, no. 1, pp. 453, Jan. 2018.

[23] 
Y. C. Jin and B. Sendhoff, “Paretobased multiobjective machine learning: An overview and case studies,” IEEE Trans. Syst.,Man,Cybern.,Part C (Appl. Rev.)

[24] 
Y. Tian, X. Y. Zhang, C. Wang, and Y. C. Jin, “An evolutionary algorithm for largescale sparse multiobjective optimization problems,” IEEE Trans. Evol. Comput., vol. 24, no. 2, pp. 380–393, Apr. 2020. doi: 10.1109/TEVC.2019.2918140

[25] 
K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGAII,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, Apr. 2002. doi: 10.1109/4235.996017

[26] 
Y. Tian, R. Cheng, X. Y. Zhang, F. Cheng, and Y. C. Jin, “An indicatorbased multiobjective evolutionary algorithm with reference point adaptation for better versatility,” IEEE Trans. Evol. Comput., vol. 22, no. 4, pp. 609–622, Aug. 2018. doi: 10.1109/TEVC.2017.2749619

[27] 
M. H. Li and J. X. Wei, “A cooperative coevolutionary algorithm for largescale multiobjective optimization problems,” in Proc. Genetic and Evolutionary Computation Conf. Companion, Kyoto, Japan, 2018, pp. 1716–1721.

[28] 
X. Y. Zhang, Y. Tian, R. Cheng, and Y. C. Jin, “A decision variable clusteringbased evolutionary algorithm for largescale manyobjective optimization,” IEEE Trans. Evol. Comput., vol. 22, no. 1, pp. 97–112, Feb. 2018. doi: 10.1109/TEVC.2016.2600642

[29] 
C. He, L. H. Li, Y. Tian, X. Y. Zhang, R. Cheng, Y. C. Jin, and X. Yao, “Accelerating largescale multiobjective optimization via problem reformulation,” IEEE Trans. Evol. Comput., vol. 23, no. 6, pp. 949–961, Dec. 2019. doi: 10.1109/TEVC.2019.2896002

[30] 
Y. L. Feng, L. Feng, S. Kwong, and K. C. Tan, “A multivariation multifactorial evolutionary algorithm for largescale multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 26, no. 2, pp. 248–262, Apr. 2021.

[31] 
H. Qian and Y. Yu, “Solving highdimensional multiobjective optimization problems with low effective dimensions,” in Proc. 31st AAAI Conf. Artificial Intelligence, San Francisco, USA, 2017, pp. 875–881.

[32] 
Y. Tian, C. Lu, X. Y. Zhang, F. Cheng, and Y. C. Jin, “A pattern miningbased evolutionary algorithm for largescale sparse multiobjective optimization problems,” IEEE Trans. Cybern., vol. 52, no. 7, pp. 6784–6797, Jul. 2022. doi: 10.1109/TCYB.2020.3041325

[33] 
Y. Tian, Y. D. Feng, X. Y. Zhang, and C. Y. Sun, “A fast clustering based evolutionary algorithm for superlargescale sparse multiobjective optimization,” IEEE/CAA J. Autom. Sinica. 2021.

[34] 
Y. J. Zhang, Y. Tian, and X. Y. Zhang, “Improved SparseEA for sparse largescale multiobjective optimization problems,” Complex Intell. Syst. 2021,

[35] 
X. Y. Wang, K. Zhang, J. Wang, and Y. C. Jin, “An enhanced competitive swarm optimizer with strongly convex sparse operator for largescale multiobjective optimization,” IEEE Trans. Evol. Comput., 2021, DOI: 10.1109/TEVC.2021.3111209.

[36] 
H. K. Chen, R. Cheng, J. M. Wen, H. F. Li, and J. Weng, “Solving largescale manyobjective optimization problems by covariance matrix adaptation evolution strategy with scalable small subpopulations,” Inf. Sci., vol. 509, pp. 457–469, Jan. 2020. doi: 10.1016/j.ins.2018.10.007

[37] 
C. He, S. H. Huang, R. Cheng, K. C. Tan, and Y. C. Jin, “Evolutionary multiobjective optimization driven by generative adversarial networks (GANs),” IEEE Trans. Cybern., vol. 51, no. 6, pp. 3129–3142, Jun. 2021. doi: 10.1109/TCYB.2020.2985081

[38] 
X. J. Yao, Q. Zhao, D. W. Gong, and S. Zhu, “Solution of largescale manyobjective optimization problems based on dimension reduction and solving knowledge guided evolutionary algorithm,” IEEE Trans. Evol. Comput., 2021, DOI: 10.1109/TEVC.2021.3110780.

[39] 
W. J. Hong, K. Tang, A. M. Zhou, H. Ishibuchi, and X. Yao, “A scalable indicatorbased evolutionary algorithm for largescale multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 23, no. 3, pp. 525–537, Jun. 2019. doi: 10.1109/TEVC.2018.2881153

[40] 
Y. Nesterov, “A method for solving a convex programming problem with convergence rate o(1/k^{2}),” Sov. Math. Dokl., vol. 27, pp. 372–376, 1983.

[41] 
T. Tieleman and G. Hinton, “Lecture 6.5RMSProp: Divide the gradient by a running average of its recent magnitude,” COURSERA: Neural Networks Mach. Learn., vol. 4, no. 2, pp. 26–31, 2012.

[42] 
C. G. Broyden, “QuasiNewton methods and their application to function minimisation”, Mathematics of Computation, vol. 21, no. 99, pp. 368–381, Jul. 1967.

[43] 
M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand., vol. 49, no. 6, pp. 409–436, Dec. 1952. doi: 10.6028/jres.049.044

[44] 
R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J., vol. 7, no. 2, pp. 149–154, Jan. 1964. doi: 10.1093/comjnl/7.2.149

[45] 
R. T. Marler and J. S. Arora, “The weighted sum method for multiobjective optimization: New insights,” Struct. Multidiscip. Optim., vol. 41, no. 6, pp. 853–862, Jun. 2010. doi: 10.1007/s0015800904607

[46] 
A. P. Wierzbicki, “A mathematical basis for satisficing decision making,” Math. Modell., vol. 3, no. 5, pp. 391–405, 1982. doi: 10.1016/02700255(82)900380

[47] 
K. Miettinen, Nonlinear Multiobjective Optimization. New York, USA: Springer, 1999.

[48] 
J. Fliege and B. F. Svaiter, “Steepest descent methods for multicriteria optimization,” Math. Methods Oper. Res., vol. 51, no. 3, pp. 479–494, Aug. 2000. doi: 10.1007/s001860000043

[49] 
X. Liu and A. C. Reynolds, “A multiobjective steepest descent method with applications to optimal well control,” Comput. Geosci., vol. 20, no. 2, pp. 355–374, Mar. 2016. doi: 10.1007/s1059601695627

[50] 
J. Fliege, L. M. G. Drummond, and B. F. Svaiter, “Newton’s method for multiobjective optimization,” SIAM J. Optim., vol. 20, no. 2, pp. 602–626, Jul. 2009. doi: 10.1137/08071692X

[51] 
K. Izui, T. Yamada, S. Nishiwaki, and K. Tanaka, “Multiobjective optimization using an aggregative gradientbased method,” Struct. Multidisc. Optim., vol. 51, no. 1, pp. 173–182, Jan. 2015. doi: 10.1007/s0015801411258

[52] 
Y. Sato, K. Izui, T. Yamada, and S. Nishiwaki, “Gradientbased multiobjective optimization using a distance constraint technique and point replacement,” Eng. Optim., vol. 48, no. 7, pp. 1226–1250, Jul. 2016. doi: 10.1080/0305215X.2015.1111068

[53] 
M. M. Noel, “A new gradient based particle swarm optimization algorithm for accurate computation of global minimum,” Appl. Soft Comput., vol. 12, no. 1, pp. 353–359, Jan. 2012. doi: 10.1016/j.asoc.2011.08.037

[54] 
K. Harada, K. Ikeda, and S. Kobayashi, “Hybridization of genetic algorithm and local search in multiobjective function optimization: Recommendation of GA then LS,” in Proc. 8th Annu. Conf. Genetic and Evolutionary Computation, Seattle, USA, 2006, pp. 667–674.

[55] 
R. Santos, G. Borges, A. Santos, M. Silva, C. Sales, and J. C. W. A. Costa, “A semiautonomous particle swarm optimizer based on gradient information and diversity control for global optimization,” Appl. Soft Comput., vol. 69, pp. 330–343, Aug. 2018. doi: 10.1016/j.asoc.2018.04.027

[56] 
R. Salomon, “Evolutionary algorithms and gradient search: Similarities and differences,” IEEE Trans. Evol. Comput., vol. 2, no. 2, pp. 45–55, Jul. 1998. doi: 10.1109/4235.728207

[57] 
C. K. Goh, Y. S. Ong, K. C. Tan, and E. J. Teoh, “An investigation on evolutionary gradient search for multiobjective optimization,” in Proc. IEEE Congr. Evolutionary Computation, Hong Kong, China, 2008, 3741–3746.

[58] 
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. Wadsworth Publ. Co., Belmont, USA, 1989.

[59] 
R. Martí, P. M. Pardalos, and M. G. C. Resende, Handbook of Heuristics. Cham, Germany: Springer, 2018.

[60] 
Y. Tian, X. S. Xiang, X. Y. Zhang, R. Cheng, and Y. C. Jin, “Sampling reference points on the Pareto fronts of benchmark multiobjective optimization problems,” in Proc. IEEE Congr. Evolutionary Computation, Rio de Janeiro, Brazil, 2018, 1–6.

[61] 
G. Taguchi, S. Chowdhury, and S. Taguchi, Robust Engineering. New York: McGrawHill, 2000.

[62] 
K. Deb and R. B. Agrawal, “Simulated binary crossover for continuous search space,” Complex Syst., vol. 9, no. 2, pp. 115–148, 1995.

[63] 
K. Deb and M. Goyal, “A combined genetic adaptive search (GeneAS) for engineering design,” Comput. Sci. Inf., vol. 26, no. 4, pp. 30–45, 1996.

[64] 
R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proc. 6th Int. Symp. Micro Machine and Human Science, Nagoya, Japan, 1995, pp. 39–43.

[65] 
E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective evolutionary algorithms: Empirical results,” Evol. Comput., vol. 8, no. 2, pp. 173–195, Jun. 2000. doi: 10.1162/106365600568202

[66] 
K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable test problems for evolutionary multiobjective optimization,” in Evolutionary Multiobjective Optimization, A. Abraham, L. Jain, and R. Goldberg, Eds. London, UK: Springer, 2005, pp. 105–145.

[67] 
C. A. C. Coello and N. C. Cortes, “Solving multiobjective optimization problems using an artificial immune system,” Genet. Program. Evolv. Mach., vol. 6, no. 2, pp. 163–190, Jun. 2005. doi: 10.1007/s107100056164x

[68] 
K. Shang, H. Ishibuchi, L. J. He, and L. M. Pang, “A survey on the hypervolume indicator in evolutionary multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 25, no. 1, pp. 1–20, Feb. 2021. doi: 10.1109/TEVC.2020.3013290

[69] 
Y. Tian, R. Cheng, X. Y. Zhang, and Y. C. Jin, “PlatEMO: A MATLAB platform for evolutionary multiobjective optimization [educational forum],” IEEE Comput. Intell. Mag., vol. 12, no. 4, pp. 73–87, Nov. 2017. doi: 10.1109/MCI.2017.2742868

[70] 
Y. Tian, T. Zhang, J. H. Xiao, X. Y. Zhang, and Y. C. Jin, “A coevolutionary framework for constrained multiobjective optimization problems,” IEEE Trans. Evol. Comput., vol. 25, no. 1, pp. 102–116, Feb. 2021. doi: 10.1109/TEVC.2020.3004012

[71] 
F. Cheng, Q. Q. Zhang, Y. Tian, and X. Y. Zhang, “Maximizing receiver operating characteristics convex hull via dynamic reference pointbased multiobjective evolutionary algorithm,” Appl. Soft Comput., vol. 86, p. 105896, Jan. 2020.

All_Data20220315.rar 