Some Three-term Nonlinear Conjugate Gradient Algorithms Involving Special Parameters With Their Applications

In this paper,we mainly study the three-term nonlinear conjugate gradient methods with special parameters and their generalized form,and apply them to solve large-scale nonlinear monotone equations,noise removal and compressed sensing.In the first chapter,we introduce the academic backgroundand achievement of the issues to be studied,and some related knowledge.In the second chapter,we propose a three-term derivative-free projection algorithm with spectral quotient parameter.Firstly,we propose a three-term conjugate gradient method involving spectral quotient.This method satisfies the Dai-Liao conjugacy condition,quasi-Newton secant equation,and sufficient descent condition,and these properties are independent of any line search.Secondly,by combining the projection technique proposed by Solodov and Svaiter,we obtain a derivative-free three-term projection algorithm involving spectral quotient.Under suitable assumptions,the global convergence and R-linear convergence rate of this algorithm are proved.Finally,the algorithm is applied to solve large-scale nonlinear monotone equations and achieve the better experimental results.In the third chapter,we propose a three-term conjugate gradient method involving an adaptive parameter and prove that this method is sufficient descent under any line search.The adaptive parameter is obtained by minimizing the maximum eigenvalue and an upper bound of the condition number of relative matrix.Under the Wolfe line search,the global convergence of this method is proved.160 standard test functions are tested to show the effectiveness of our algorithm.Finally,we apply this method to solve a noise removal model and satisfactory results are obtained.In the fourth chapter,we construct two three-term derivative-free projection algorithms with a single adaptive parameter.Firstly,we construct a three-term conjugate gradient method with an adaptive parameter,the adaptive parameter is obtained by minimizing the distance between the relative matrix and BFGS iteration matrix.Secondly,by projection technique,the two three-term derivative-free projection methods are proposed.Under suitable assumptions,the global convergence and R-linear convergence rate of this algorithms are proved.Finally,the two algorithms are also applied to solve large-scale nonlinear monotone equations with convex constraints and compressed sensing and better results are obtained.In the fifth chapter,we propose a three-term derivative-free projection method with double parameters.Firstly,we construct a three-term conjugate gradient method with double parameters.This search direction is descent and satisfies the dynamic modified adaptive conjugacy condition.The two parameters are obtained by minimizing the distance between the symmetric correlation matrix and the Perry matrix.Secondly,by the projection technique,a three-term projection algorithm is proposed.Under proper assumptions,the global convergence and R-linear convergence rate of this algorithm are proved.Finally,this algorithms are also applied to solve large-scale nonlinear monotone equations with convex constraints and compressed sensing and obtain better results.