Abstract To improve the diversity maintenance ability of evolutionary multi-objective optimization algorithms and obtain a set of better distributed non-dominated solutions, a co-evolutionary multi-objective optimization algorithm with polymorphous populations is proposed. Firstly, a co-evolutionary frame of polymorphous populations is designed. Next, by introducing the minimum vectorial angle which is capable of measuring the similarity between different Pareto-ranked solutions, a selection strategy for suboptimum non-dominated solutions is proposed to enhance the diversity of populations. Finally, a population removal strategy based on an ordered link-list is put forward. Thus, the uniformity and the spread of the solutions are improved. Compared with some typical algorithms, the proposed algorithm has good convergence and remains a better diversity and uniformity.
[1] Jiao L C, Gong M G, Wang S, et al. Advances in Natural Computation, Machine Learning and Image Understanding. Xi'an, China: Xidian University Press, 2008 (in Chinese) (焦李成,公茂果,王 爽,等.自然计算、机器学习与图像理解前沿.西安:西安电子科技大学出版社, 2008) [2] Gong M G, Jiao L C, Yang D D, et al. Research on Evolutionary Multi-objective Optimization Algorithms. Journal of Software, 2009, 20(2): 271-289 (in Chinese) (公茂果,焦李成,杨咚咚,等.进化多目标优化算法研究.软件学报, 2009, 20(2): 271-289) [3] Schaffer J D. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms // Proc of the 1st International Conference on Genetic Algorithms. Pittsburgh, USA, 1985: 93-100 [4] Fonseca C M, Fleming P J. Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization // Proc of the 5th International Conference on Genetic Algorithm. San Francisco, USA: Morgan Kaufmann, 1993: 416-423 [5] Srinivas N, Deb K. Multiobjective Optimization Using Nondomina-ted Sorting in Genetic Algorithms. Evolutionary Computation, 1994, 2(3): 221-248 [6] Horn J, Nafpliotis N, Goldberg D E. A Niched Pareto Genetic Algorithm for Multiobjective Optimization // Proc of the 1st IEEE Conference on Evolutionary Computation. Orlando, USA, 1994, I: 82-87 [7] Deb K, Pratap A, Agarwal S, et al. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Trans on Evolutionary Computation, 2002, 6(2): 182-197 [8] Nebro A J, Durillo J J, Garcia-Nieto J, et al. SMPSO: A New POS-Based Metaheuristic for Multi-objective Optimization // Proc of the IEEE Symposium on Computational Intelligence in Multi-criteria Decision-Making. Nashville, USA, 2009: 66-73 [9] Gong M G, Jiao L C, Du H F, et al. Multiobjective Immune Algorithm with Nondominated Neighbor-Based Selection. Evolutionary Computation, 2008, 16(2): 225-255 [10] Potter M A, de Jong K A. A Cooperative Coevolutionary Approach to Function Optimization // Proc of the 3rd Conference on Parallel Problem Solving from Nature. London, UK: Springer, 1994: 249-257 [11] Tan K C, Yang Y J, Goh C K. A Distributed Cooperative Coevolutionary Algorithm for Multiobjective Optimization. IEEE Trans on Evolutionary Computation, 2006, 10(5): 527-549 [12] Liu Y, Yao X, Zhao Q F, et al. Scaling up Fast Evolutionary Programming with Cooperative Coevolution // Proc of the Congress on Evolutionary Computation. Seoul, Republic of Korea, 2001,Ⅱ: 1101-1108 [13] van den Bergh F, Engelbrecht A P. A Cooperative Approach to Particle Swarm Optimization. IEEE Trans on Evolutionary Computation, 2004, 8(3): 225-239 [14] Li X D, Yao X. Cooperatively Coevolving Particle Swarms for Large Scale Optimization. IEEE Trans on Evolutionary Computation, 2012, 16(2): 210-224 [15] Zhang Q F, Li H. MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition. IEEE Trans on Evolutionary Computation, 2007, 11(6): 712-731 [16] Zhang Q F, Suganthan P N. Final Report on CEC'09 MOEA Competition // Proc of the Congress on Evolutionary Computation. Trondheim, Norway, 2009: 1-11 [17] Jiao L C, Wang H D, Shang R H, et al. A Co-evolutionary Multi-objective Optimization Algorithm Based on Direction Vectors. Information Sciences, 2013, 228: 90-112 [18] Coello Coello C A, Lamont G B, Van Veldhuizen D A . Evolutionary Algorithms for Solving Multi-objective Problems. New York, USA: Springer, 2007 [19] Liu R C, Jiao L C, Ma Y J. A Differential Multi-objective Optimization Algorithm. Pattern Recognition and Artificial Intelligence, 2011, 24(6): 748-755 (in Chinese) (刘若辰,焦李成,马亚娟.一种差分多目标优化算法.模式识别与人工智能, 2011, 24(6): 748-755) [20] Zhou Z H, Wu J X, Tang W. Ensembling Neural Networks: Many Could Be Better Than All. Artificial Intelligence, 2002, 137(1/2): 239-263 [21] Shang R H, Jiao L C, Liu F, et al. A Novel Immune Clonal Algorithm for MO Problems. IEEE Trans on Evolutionary Computation, 2012, 16(1): 35-50 [22] Tang L X, Wang X P. A Hybrid Multiobjective Evolutionary Algorithm for Multiobjective Optimization Problems. IEEE Trans on Evolutionary Computation, 2013, 17(1): 20-45 [23] Deb K. Multi-objective Optimization Using Evolutionary Algorithms. New York, USA: John Wiley & Sons, 2001 [24] Cheng S, Shi Y H, Qin Q D. On the Performance Metrics of Multiobjective Optimization // Proc of the 3rd International Conference on Swarm Intelligence. Shenzhen, China, 2012: 504-512 [25] Mersmann O, Trautmann H, Naujoks B, et al. On the Distribution of EMOA Hypervolumes // Blum C, Battiti R, eds. Learning and Intelligent Optimization. Berlin, Germany: Springer, 2010: 333-337 [26] Zitzler E, Thiele L. Multiobjective Optimization Using Evolutionary Algorithms-A Comparative Case Study // Eiben A E, Bck T, Schoenauer M, et al., eds. Parallel Problem Solving from Nature-PPSN V. Berlin, Germany: Springer, 1998: 292-301 [27] McGill R, Tukey J W, Larsen W A. Variations of Box Plots. The American Statistician, 1978, 32(1): 12-16
2014, Vol. 27 
