一种基于指数模型的非线性方程组问题求解算法

A Nonlinear Equations Problem Solving Algorithm Based on Exponential Model

摘要: 针对使用经典优化方法较难以进行求解的多元非线性方程组问题，首先我们使用群智能优化算法来对其进行求解，其次，在进行问题转化的过程中，提出了一种全新的目标函数构造模型——指数模型来实现对问题的转换；经过验证，相较于传统模型，指数模型有效的增强了算法在求解单目标优化问题上的全局搜索能力；此外，我们采用了带有档案集的差分进化算法，以解决在求解非线性方程组问题的过程中根的留存问题，同时采用随机指数的方式以增强算法的多样性，在所选的六个非线性方程组测试集上，指数模型在相关指标上对比原模型均表现出优势或绝对优势，其中在四个非线性方程组上表现为绝对优势，实验结果表明，所提出的模型能有效搜索到非线性方程组系统的多个根，并与传统的转换模型相比，所提出的指数模型在找根率(root ratio)和成功率(success rate)上更具优越性。

Abstract: For the problem of multivariate nonlinear equations, which is difficult to be solved by using classical optimization methods, firstly, we use swarm intelligent optimization algorithm to solve it. Secondly, in the process of problem transformation, a new objective function construction-exponential model is proposed to realize the transformation of the problem. It has been verified that compared with the traditional model, the exponential model effectively enhances the global search ability of the algorithm in solving the single objective optimization problem. In addition, we adopted a differential evolution algorithm with file set to solve the problem of root retention in the process of solving nonlinear equations, and adopted a random exponential method to enhance the diversity of algorithms. On the six selected test sets of nonlinear equations, the exponential model showed advantages or absolute advantages compared with the original model in relevant indicators. The experimental results show that the proposed model can effectively search multiple roots of the nonlinear equations system, and compared with the traditional transformation model, the proposed exponential model has more advantages in terms of root ratio and success rate.