非线性函数的混沌优化方法比较研究

Comparative study on chaos optimization algorithm for nonlinear function

已有的混沌优化方法几乎都是利用Logistic映射作为混沌序列发生器,而Logistic映射产生的混沌序列的概率密度函数服从两头多、中间少的切比雪夫型分布,不利于搜索的效率和能力。为此,首先根据Logistic映射混沌轨道点密度函数的特点,建立改进的混沌-BFGS混合优化算法。之后,考虑到Kent映射混沌轨道点密度为均匀分布,建立了基于Kent映射的混沌-BFGS混合优化算法。然后对五种混合优化方法——不加改进的和改进的基于Logistic映射的混沌-BFGS法,基于Kent映射的混沌-BFGS法,MonteCarlo试验-BFGS法,网格-BFGS法进行了研究,分别对3个低维和2个高维非线性复杂测试函数进行优化计算,对它们的全局优化计算效率和寻优能力做了比较,并探讨了混合优化方法全局优化性能差异的原因。结果表明,混沌优化方法是与MonteCarlo方法类似的一种随机性试验优化方法。而且,这类优化方法的计算性能至少与以下因素有关:混沌/随机序列的统计性质,优化问题全局最优点位置。

The chaos optimization algorithms in the published papers are almost based on Logistic map. The probability density function of chaotic sequence for Logistic map is Chebyshev-type function, which may affect the global searching capacity and computational efficiency of chaos optimization algorithm severely. Considering the statistical property of chaotic sequence of Logistic map, the improved hybrid chaos-BFGS optimization algorithm is presented by eliminating the bad design points during chaos searching. Since the probability density function of chaotic sequence for Kent map is the uniform function on interval (0,1), the hybrid chaos optimization-BFGS algorithm is established on basis of Kent map. Five nonlinear functions (three low-dimensional and two high-dimensional functions) are employed to test the performance and efficiency of five hybrid optimization algorithms, which are improved Logistic map based chaos-BFGS algorithm, Kent map based chaos-BFGS algorithm, unimproved Logistic map based chaos-BFGS algorithm, Monte Carlo-BFGS algorithm, and mesh-BFGS algorithm. The global optimization performance of these algorithms is compared, and the performance discrepancy of optimization algorithms is discussed. It is concluded that the chaos optimization algorithm is one kind of stochastic method similar to Monte Carlo method, and the computational performance of hybrid optimization algorithms is affected by the statistical property of chaotic/stochastic sequences generated from optimization algorithms, and the position of global optimum of nonlinear functions.