Research On Conjugate Gradient Algorithm For Nonlinear Equations And Smooth Model

With the development of science and technology and the need for massive information analysis and extraction in Big Data,more and more practical problems are transformed into large-scale mathematical problems.Optimization problems,an important research field of mathematics,provide the cost-effective solution for social,economic and other fields,thus it is equipped to save natural resources and costs.Among them,large-scale nonlinear equations have attracted the attention of scholars due to their wide application in climate prediction,engineering design,physics,and finance.Algorithms such as conjugate gradient,Newton series and trust region are effective tools for solving nonlinear equations and smoothing problems.Based on the existingresearch results,a modified conjugate gradient method and nonlinear conjugate gradient method are proposed to solve the large-scale nonlinear equations model and smoothing problems respectively.The important mathematical properties of the algorithm are introduced and proved.For large-scale nonlinear equations,a modified three-term conjugate gradient algorithm is proposed based on the classical LS formula,three-term gradient and projection technique.Among them,the algorithm does not ne-ed other conditions to automatically have sufficient descent and trust region properties,and has global convergence under general assumptions.The numerical experiments show that the algorithm has higher efficiency than similar optimization algorithms.Aiming at large-scale smoothing problem,a nonlinear conjugate gradient method is proposed based on the characteristics of steepest descent method,which combines the popular line search technology and new search direction.The algorithm not only has sufficient descent and trust region characters automatically,but also has global convergence when certain conditions are satisfied.Then it has larger descent amount in each iteration step and has a higher convergence rate.To some extent,the new algorithm not only enriches the knowledge of optimization theories and the numeral results pro-ve the new algorithm is more competitive than similar algorithms.