Gradient Method Of Theoretical And Numerical Behavior Of Several Classes Of Conjugated Spectrum

Although there has existed a great contribution in the research of spectral conjugate gradient methods up to now, it is significant to further investigate how to choose a suitable spectral parameter and a conjugate parameter such that the obtained method has the advantages owned by the spectral method and the conjugate gradient method. On the other hand, a suitable line search technique is important for the numerical performance of the developed algorithm and the establishment of global convergence. For these issues, in this dissertation, we intend to conduct an extensive and deep research on some types of spectral conjugate gradient methods from the viewpoints of theory and numerical performance. The main contributions are as follows:Since it is possible that the obtained step size by the standard Armijo line search is coarse, we first employ approximate conditions of Wolfe line search to improve the Armijo line search rule. On the basis of this new line search, a new spectral PRP conjugate gradient algorithm is developed for solving unconstrained optimization problems, where the search direction is sufficiently descent. Numerical experiments show that the developed algorithm is promising.Then, a new nonmonotone spectral conjugate gradient method is proposed. By introducing a hybrid coefficient, the conjugacy parameter is determined based on the combination of PRP and HS methods. A spectral parameter is appropriately chosen such that each search direction is a sufficiently descent direction independent of the employed line search techniques. On the other hand, the nonomonotone line search technique proposed by Zhang and Hager is modified, and under more mild assumptions, the global convergence of the developed algorithm is proved. Numerical experiments are employed to demonstrate the efficiency of the algorithm.Thirdly, a new modified spectral FR conjugate gradient method is presented for solving unconstrained optimization problems. The direction generated by the method produces sufficiently descent search direction. By introducing a control parameter to regulate the spectral coefficient and conjugate coefficient, the new method is assured to have the superiority of spectral method and conjugate gradient method. Global convergence results is established for the conjugate gradient methods with a Wolfe type line search. The numerical results show that the new method is efficient for general unconstrained optimization problems.Finally, based on a series of methods with sufficiently descent direction, a generic iterative formula of spectral conjugate gradient method is summarized. To make sure that the search direction is sufficiently descent direction, the value range of spectral and conjugate parameters is presented. With any line search conditions based on standard Armijo line search, the global convergence theory is established for this generic iterative formula.