The Research Of The Spectral Conjugate Gradient Method And Three-Term Conjugate Gradient Method For Unconstrained Optimization

The theory and algorithm of unconstrained optimization is the basis of the research on optimization problems, and the important branch of the opti-mization fields. Conjugate gradient method is a class of the effective method and also a hot method in the research on unconstrained optimization. This thesis discusses spectral conjugate gradient method and three-term conjugate gradient method for unconstrained optimization respectively, and proposes two kinds of spectral conjugate gradient methods and a three-term conjugate gradient method.Firstly, combined the idea of spectral conjugate gradient method with an effective selecting technology for conjugate parameters, a new approach is presented in this thesis, which can generate a descendant direction inde-pendent of the line search. Under the strong Wolfe line search, the global convergence of the proposed method is proved from the uniform convexity of objective function. The conjugate parameter and the spectral parameter are modified to weaken the conditions of convergence. It will generate a suf-ficiently descendant direction, which also irrelevant to the line search, and be convergent to the general functions. The selections of the relevant parame-ters are discussed in the numerical experiments. The numerical results and performance profiles show the effective of the methods.Secondly, a new three-term conjugate gradient method is proposed in the thesis, and the generated direction is sufficiently descendant. Under the standard Wolfe line search, the proposed method is always globally conver-gent. Additionally, to improve the efficiency of the algorithm, the thesis ap-plies the accelerated strategy [Appl. Math. Comput.,213 (2009)] to the algorithm, and then get an accelerate method. The numerical experiments show that the proposed methods are effective.