Research On Spectral Conjugate Gradient Methods For Unconstrained Optimization And Conjugate Gradient Projection Methods For Constrained Equations

The Conjugate gradient method and the spectral conjugate gradient method derived from it are very important numerical iterative methods for solving unconstrained optimization and nonlinear equations with convex constraints.Due to these two methods have many advantages such as simple iteration,superior numerical performance and low storage requirements in algorithms,they are especially suitable for solving large-scale problems.In this dissertation,the spectral conjugate gradient method for unconstrained optimization and the conjugate gradient projection method for constrained nonlinear equations are studied respectively.In the first part of this dissertation researched the spectral conjugate gradient method for unconstrained optimization.Firstly,a modified PRP type conjugate parameter is proposed and a spectral parameter is designed according to the sufficient descent condition,then,two new spectral conjugate gradient methods are established based on the strong Wolfe line search and the Armijo line search.Sufficient descent condition of the new algorithms does not depend on any line search.Under the usual assumptions,using the strong Wolfe line search or the Armijo line search to generate the step-length,the global convergence of the presented method are proved.By testing 100 problems,numerical experiments and their corresponding performance profiles for the proposed methods and other four comparisons are reported,which show that the proposed methods is effective.In the second part of the dissertation researched the conjugate gradient projection method for nonlinear equations with convex constraints.By modifying the search direction,constructing a suitable adaptive line search strategy to generate the step-length,and combining with the hyperplane projection technology,proposed a conjugate gradient projection algorithm for solving nonlinear equations with convex constraints.The descent property and adaptability of the algorithm are analyzed.Then,the global convergence of the proposed algorithm is analyzed under mild conditions,Finally,the numerical results show that the proposed algorithm is robust and effective.