摘要：

求解非线性方程组的难点是在一次运行中获取问题的多个根，常规求解方法难以同时满足解的精度和解的数量要求。提出一种基于知识学习的目标空间头脑风暴优化（LBSOOS）算法，通过将非线性方程组问题建模为多模态优化问题进行求解，在求解过程融合算法的求解特性和待求解问题的领域知识，采用求解问题学习和求解算法学习两种学习方式解决求解精度和解集合多样性的冲突。从算法层面改进算子的学习方式，将随机解的扰动算子替换为最差解的解间学习，提高算法的整体寻优能力。通过对多模态问题进行分析，在算法中增加额外的档案集，保证输出解集合的多样性。将LBSOOS算法与5种群体智能优化算法在7个非线性方程组问题上进行性能测试，实验结果表明，LBSOOS算法在保证求解精度的条件下，在绝大多数测试问题上的求解多样性优于BSO、BSOOS、PIO等对比算法。

关键词: 群体智能, 头脑风暴优化算法, 探索与利用, 非线性方程组, 多模态优化

Abstract:

The difficulty of solving a Nonlinear Equation System(NES) involves the multiple roots obtained in one search round. Traditional methods can not simultaneously meet the requirements of accuracy and finding all available solutions. An algorithm of Learning-based Brain Storm Optimization in Objective Space (LBSOOS) is proposed. The NES problem is modeled as a multimodal optimization problem, whereby algorithm-solving characteristics are fused with the domain knowledge of the problem to be solved. Two learning methods, problem-solving and algorithmic learning, are used to solve the conflict between solution accuracy and solution set diversity. To improve the learning method of the operator, the disruption operator of the random solution is replaced by the inter-solution learning of the worst solution. This modification enhances the optimization ability of the algorithm. Through an analysis of multimodal problems, an archive is added to ensure the diversity of output solutions. LBSOOS and five swarm intelligence optimization algorithms were tested on seven nonlinear equation systems. The experimental results demonstrate that in ensuring a certain solution accuracy, the LBSOOS algorithm displays better solution diversity than Brain Strom Optimization(BSO), BSO in Objective Space(BSOOS), Pigeon-Inspired Optimization(PIO), and other comparative algorithms for the vast majority of testing problems.

Key words: swarm intelligence, Brain Storm Optimization(BSO) algorithm, exploration and exploitation, Nonlinear Equation System(NES), multimodal optimization