Differential Evolution Algorithm for Multimodal Optimization Based on Abstract Convex Underestimation

Abstract

针对多模态优化问题,提出了基于广义凸下界估计模型的改进差分进化算法.首先,基于模型变换方法将原优化问题转变为单位单纯形约束条件下的严格递增射线凸优化问题;其次,基于广义凸理论,利用差分进化算法中更新个体的适应度知识,建立原优化问题广义凸下界估计模型,设计实现了基于N-叉树的估计模型快速计算方法;进而,综合考虑原问题目标值与其估计值之间的差异,提出一种基于有偏采样的小生境指标,并设计区域进化树更新策略来保证算法的局部搜索能力.数值实验结果表明,提出的算法能够有效地发现并维持一定数量的满意解模态,动态地实现全局模态搜索到模态内局部增强的自适应平滑过渡.对于给出的测试问题,能够发现所有的全局最优解以及一些较好的局部极值解.;In this paper, a modified differential evolution algorithm, which is based on abstract convex lower approximation, is proposed for multimodal optimization. First, the original bound constrained optimization problem is converted to an increasing convex along rays (ICAR) function over a unit simplex by using the projection transformation method. Second, based on abstract convex theory, the study builds a lower approximation to original optimization problem by using a finite subset of biased sampling points comes from the population replacement scheme in the basic DE algorithm. Some properties of underestimation model are analyzed theoretically, and an N-ary tree data structure have also been designed and implemented to solve them. Furthermore, considering the difference between the original and its underestimated function values, the paper proposes a niche identify indicator based on biased DE sampling procedure, and also design a regional phylogenetic tree replacement strategy to enhance the exploitation capacity in niche. Experimental results confirm that the proposed algorithm can distinguish between the different attraction basins, and safeguard the consequently discovered solutions effectively. For the given benchmark problems, the proposed algorithm can find all the global optimal solutions and some good local minimum solutions.