Abstract To get better solution of the differential evolution (DE) algorithm, the mutation strategy of DE is proposed and divided into two parts to reflect the changes of the target population trends and their random variation. Fractal mutation factor differential evolution (FMDE) algorithm is put forward and it consists of an additional mutation factor simulated by a different Hurst index fractal Brownian motion. FMDE is tested on 25 benchmark functions presented at 2005 IEEE congress on evolutionary computation. The optimization results of at least 10 benchmark functions are better than the results obtained by other differential evolution algorithms, and the rest of the test results are approximate. Experimental results show that FMDE significantly improves the accuracy and adaptability of the optimization.
[1] Price K, Storn R, Lampinen J A. Differential Evolution: A Practical Approach to Global Optimization. New York, USA: Springer, 2005 [2] Storn R, Price K. Differential Evolution-A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization, 1997, 11(4): 341-359 [3] Ye H T, Luo F, Xu Y G. Differential Evolution for Solving Multi-objective Optimization Problems: A Survey of the State-of-the-Art. Control Theory & Applications, 2013, 30(7): 922-928 [4] Eiben A E, Hinterding R, Michalewicz Z. Parameter Control in Evo-lutionary Algorithms. IEEE Trans on Evolutionary Computation, 1999, 3(2): 124-141 [5] Ronkkonen J, Kukkonen S, Price K V. Real-Parameter Optimization with Differential Evolution // Proc of the IEEE Congress on Evo-lutionary Computation. Edinburgh, UK, 2005, I: 506-513 [6] Qin A K, Huang V L, Suganthan P N. Differential Evolution Algorithm with Strategy Adaptation for Global Numerical Optimization. IEEE Trans on Evolutionary Computation, 2009, 13(2): 398-417 [7] Rahnamayan S, Tizhoosh H R, Salama M M A. Opposition-Based Differential Evolution. IEEE Trans on Evolutionary Computation, 2008, 12(1): 64-79 [8] Das S, Abraham A, Chakraborty U K, et al. Differential Evolution Using a Neighborhood-Based Mutation Operator. IEEE Trans on Evo-lutionary Computation, 2009, 13(3): 526-553 [9] Fan H Y, Lampinen J. A Trigonometric Mutation Operation to Di-fferential Evolution. Journal of Global Optimization, 2003, 27(1): 105-129 [10] Liu G, Li Y X. Novel Oppositional Differential Evolution Algorithm Based on Theory of Molecular Motion. Journal of Chinese Computer Systems, 2012, 33(1): 115-120 (in Chinese) (刘 罡,李元香.分子动理论的新型反向差分演化算法.小型微型计算机系统, 2012, 33(1): 115-120) [11] Feoktistov V, Janaqi S. Generalization of the Strategies in Differential Evolution [EB/OL].[2014-01-02]. http:// ieeexplore.ieee.org/stamp/stamp.jsp？tp=&arnumber=1303160 &tag=1 [12] Mezura-Montes E, Velázquez-Reyes J, Coello Coello C A. A Comparative Study of Differential Evolution Variants for Global Optimization // Proc of the 8th Annual Conference on Genetic and Evolutionary Computation. Seattle, USA, 2006: 485-492 [13] Zhang J Q, Sanderson A C. JADE: Adaptive Differential Evolution with Optional External Archive. IEEE Trans on Evolutionary Computation, 2009, 13(5): 945-958 [14] Wang Y, Cai Z X, Zhang Q F. Differential Evolution with Composite Trial Vector Generation Strategies and Control Parameters. IEEE Trans on Evolutionary Computation, 2011, 15(1): 55-66 [15] Xiang W L, Ma S F, An M Q. Adaptive Multiple Strategy Diffe-rential Evolution Algorithm with Guiding Scheme of Pbest. Pattern Recognition and Artificial Intelligence, 2013, 26(8): 711-721 (in Chinese) (向万里,马寿峰,安美清.具有Pbest引导机制的适应性多策略差分进化算法.模式识别与人工智能, 2013, 26(8): 711-721) [16] Mandelbrot B B. The Fractal Geometry of Nature. New York, USA: W H Freeman, 1982 [17] Florindo J B, Bruno O M. Closed Contour Fractal Dimension Estimation by the Fourier Transform. Chaos, Solitons and Fractals, 2011, 44(10): 851-861 [18] Shaghaghian M R. Fractal Dimension: An Index to Quantify Parameters in Genetic Algorithms[EB/OL]. [2014-01-01]. http://waset.org/publications/6312/fractal-dimension-an-index-to-quantify-parameters-in-genetic-algorithms [19] Hu Y Z, Oksendal B. Fractional White Noise Calculus and Applications to Finance. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2003, 6(1): 1-32 [20] Abry P, Sellan F. The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F Sellan and Y Meyer: Remarks and Fast Implementation. Applied and Computational Harmonic Analysis, 1996, 3(4): 377-383 [21] Suganthan P N, Hansen N, Liang J J, et al. Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization. Technical Report, KanGAL #2005005. Singapore, Singapore: Nanyang Technological University, 2005 [22] Siegel S. Nonparametric Statistics for the Behavioral Sciences. New York, USA: McGraw-Hill Humanities, 1956
2015, Vol. 28 
