Three-term Conjugate Gradient Method Based On Least Square

As an important part of operational research and cybernetics,optimization has been widely used in practical applications,such as economic management,engineering design,optimal control,oil exploration,etc..Unconstrained optimization plays a basic and important role in the field of optimization.These methods that we usually used to solve unconstrained optimization problems include steepest descent method,Newton method,quasi-Newton method,conjugate gradient method and trust region method,etc..Due to the low storage,the conjugate gradient method is one of the most effective methods to solve large-scale unconstrained optimization problems.In the study of conjugate gradient methods,the three-term conjugate gradient method is one of the hotspots.In this thesis,we study the three conjugate gradient method for solving unconstrained optimization problem and nonlinear equations.For solving large-scale unconstrained optimization problems,we proposed a three-term conjugate gradient method based on the least squares technique.In this thesis,we first study some three-term conjugate gradient methods which are effective to solve unconstrained optimization problems,then approximate them by the least squares to propose a new formula of three-term conjugate gradient method.The algorithm has the following advantages:(1)Without considering the line search,the algorithm has the descent property,that is to say,the descent property does not depend on the choice of line search technique;(2)Under certain conditions,the algorithm has global convergence;(3)the numerical experiment is used to illustrate the algorithm for large-scale unconstrained optimization.The numerical results show that the algorithm is efficient.For the problems of nonlinear equations,we propose an improved Polak-Ribiere-Polyak(PRP)conjugate gradient algorithm.In our thesis we prove the global convergence of the algorithm.Since the proposed algorithm has the advantages of low storage,it can be used to solve the large-scale nonlinear equations.The numerical results show that the algorithm still has well numerical results even the dimension of the objective function is high.