Two Modified Gradient Methods Based Quasi-Newton Update Technique

It is well-known that the conjugate gradient method and the quasi-Newton method are two kinds of very important and effective gradient methods for solving unconstrained optimization problems. The advantage of the conjugate gradient method is of less storage space and simple calculation, and hence it is suitable for solving middle-large-scale optimization problems, while the advantage of the quasi-Newton method is of rapid convergence, and after improvement it can effectively solve middle-scale optimization problems. In recent decades, there has been growing interest in the development of these two kinds of methods. Therefore, this thesis is concerned with the modification to the above two kinds of gradient methods based on the quasi-Newton principle and present a modified HS conjugate gradient method and a mixed spectrum scale BFGS method. In the first chapter, we briefly introduce some basic knowledge, the structure of descent algorithm and the previous works on the conjugate gradient method and the quasi-Newton method.In chapter2, based on the MBFGS update technique, we firstly modify the calculation formula of the search direction in the HS conjugate gradient method. The advantage of this modification is that it can ensure the generated search direction to be descent and can employ weak line search to calculate step length. Under mild conditions, we prove the global convergence of the proposed modified HS conjugate gradient method with the Armijo line search. Finally, the numerical results show that the proposed method is effective.In chapter3, based on the existing MBFGS method, CBFGS method and spectral-scaling strategy, we present a mixed spectrum scale BFGS method. Its advantage is that it can effectively reduce the condition number to prevent the quasi-Newton matrices from ill-condition and quasi-Newton matrix is always updated effectively. Under mild conditions, we prove the global convergence of the presented mixed spectrum scale BFGS method when it uses the Armijo line search or the Wolf-Powell line search. At last, we compare its performance with the MBFGS method, the CBFGS method and the classical BFGS method. The numerical results show that the proposed method is practical effective.