Estimation of distribution algorithms are a class of evolutionary optimization algorithms based on probability distribution model. In this article, a Pareto-based multi-objective estimation of distribution algorithm with multivariate T-copulas is proposed. The algorithm employs Pareto-based approach and multivariate T-copulas to construct probability distribution model. To estimate joint distribution of the selected solutions, the correlation matrix of T-copula is firstly estimated by estimating Kendall’s tau and using the relationship of Kendall’s tau and correlation matrix. After the correlation matrix is estimated, the degree of freedom of T-copula is estimated by using the maximum likelihood method. Afterwards, the Monte Carte simulation is used to generate new individuals. An archive with maximum capacity is used to maintain the non-dominated solutions. The Pareto optimal solutions are selected from the archive on the basis of the diversity of the solutions, and the crowding-distance measure is used for the diversity measurement. The archive gets updated with the inclusion of the non-dominated solutions from the combined population and current archive, and the archive which exceeds the maximum capacity is cut using the diversity consideration. The proposed algorithm is applied to some well-known benchmark. The relative experimental results show that the algorithm has better performance and is effective.
Cite this paper
Y. Gao, L. Peng, F. Li, M. Liu and X. Hu, "Estimation of Distribution Algorithm with Multivariate T-Copulas for Multi-Objective Optimization," Intelligent Control and Automation, Vol. 4 No. 1, 2013, pp. 63-69. doi: 10.4236/ica.2013.41009.
 C. A. Coello, “A Comprehensive Survey of Evolutionary-Based Multi-objective Optimization Techniques,” International Journal of Knowledge and Information Systems, Vol. 1, No. 3, 1999, pp. 269-308.
 E. Zitzler, M. Laumanns and L. Thiele, “SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multi-Objective Optimization,” Proceedings of the EUROGEN Conference, Lake Como, 2001, pp. 95-100.
 K. Deb, S. Agrawal, A. Pratap and T. Meyarivan, “A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 2002, pp. 182-197. doi:10.1109/4235.996017
 H. Eskandari, C. D. Geiger and G. B. Lamont, “FastPGA: A Dynamic Population Sizing Approach for Solving Expensive Multi-Objective Optimization Problems,” In: Evolutionary Multi-objective Optimization Conference EMO, Springer-Verlag, Berlin, Heidelberg, 2007, pp. 141-155.
 P. Larranaga and J. A. Lozano, “Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation,” Kluwer Academic Publishers, Dordrecht, 2002.
 S. Shakya, “DEUM: A Framework for an Estimation of Distribution Algorithm Based on Markov Random Fields,” Ph.D. thesis, The Robert Gordon University, Aberdeen, 2006.
 S. Shakya and J. McCall, “Optimisation by Estimation of Distribution with DEUM Framework Based on Markov Random Fields,” International Journal of Automation and Computing, Vol. 4, No. 3, 2007, pp. 262-272.
 R. B. Nelsen, “An Introduction to Copula,” Springer-Verlag, New York, 1998.
 U. Cherubini, E. Luciano and W. Vecchiato, “Copula Methods in Finance,” John Wiley & Sons Ltd., Chichester, 2004.
 Y. Gao, “Multivariate Estimation of Distribution Algorithm with Laplace Transform Archimedean Copula,” IEEE International Conference on Information Engineering and Computer Science, Vol. 1, 2009, pp. 273-277.
 Y. Gao, X. Hu, H. L. Liu, F. F. Li and L. X. Peng, “Opposition-based Learning Estimation of Distribution Algorithm with Gaussian Copulas and Its Application to Placement of RFID Readers,” International Conference on Artificial Intelligence and Computational Intelligence, Taiyuan, 24-25 September 2011, pp. 219-227.
 E. Zitzler, K. Deb and L. Thiele, “Comparison of MultiObjective Evolutionary Algorithms: Empirical Results,” Evolutionary Computation, Vol. 8, No. 2, 2000, pp. 173195. doi:10.1162/106365600568202