Conjugate Gradient Method With Complexity Guarantee For Solving Non-Convex Optimization Problems And Its Applications

Conjugate gradient method(CG)is a class of the effective method for solving the large scale unconstrained optimization problems due to its simplicity of the iterations and lower storage requirements.Current researches show that it has wide applications in image restoration,sparse signal recovery and portfolio problem.In this dissertation,we study the improved PRP type CG method for solving the large scale unconstrained optimization problems with its appliations.The research contents are as follows:Firstly,we propose an improved PRP type CG method.The search directions generated by it satisfy the sufficient descent property independent of any line search.Under the Wolfe line search and Armijo line search,the global convergence of this algorithm is proved,respectively.In addition,the complexity of the propose algorithm with Armijo line search is investigated without any restart condition.This complexity bound is the same as the gradient descent method.Numerical results illustrate that the propose method is very effective compared with some classical algorithms in both large scale unconstrained optimization problems and image restoration.Secondly,based on the inertial extrapolation technique,an accelerated derivative free projection method for solving the monotone nonlinear equations with convex constraints is presented.This method exploits the information of the previously two iterations to obtain a new iteration.Its global convergence is proved.Numerical results illustrate that it is efficient when solving the large scale monotone nonlinear equations with convex constraints and sparse signal recovery problems.