差分演化的收敛性分析与算法改进

为了分析差分演化(differential evolution，简称 DE) 的收敛性并改善其算法性能，首先将差分算子 (differential operator，简称 DO)定义为解空间到解空间的笛卡尔积的一种随机映射，利用随机泛函理论中的随机压缩映射原理证明了 DE 的渐近收敛性；然后，在“拟物拟人算法”的启发下，通过对 DE 各进化模式的共性特征与性能差异的分析，提出了一种具有多进化模式协作的差分演化算法 (differential evolution with multi-strategy cooperating evolution，简称 MEDE)，分析了它所具有的隐含特性，并在多模式差分算子(multi-strategy differential operator，简称 MDO)定义的基础上证明了它的渐进收敛性。对 5 个经典测试函数的仿真计算结果表明，与原始的 DE,DEfirDE 和 DEfirSPX 等算法相比，MEDE 算法在求解质量、适应性和鲁棒性方面均具有较明显的优势，非常适于求解复杂高维函数的数值最优化问题。

To analyze the convergence of differential evolution (DE) and enhance its capability and stability, this paper first defines a differential operator (DO) as a random mapping from the solution space to the Cartesian product of solution space, and proves the asymptotic convergence of DE based on the random contraction mapping theorem in random functional analysis theory. Then, inspired by "quasi-physical personification algorithm", this paper proposes an improved differential evolution with multi-strategy cooperating evolution (MEDE) is addressed based on the fact that each evolution strategy of DE has common peculiarity but different characteristics. Its asymptotic convergence is given with the definition of multi-strategy differential operator (MDO), and the connotative peculiarity of MEDE is analyzed. Compared with the original DE, DEfirDE and DEfirSPX, the simulation results on 5 classical benchmark functions show that MEDE has obvious advantages in the convergence rate, solution-quality and adaptability. It is suitable for solving complex high-dimension numeral optimization problems.