Some Researches On Newton-type Algorithm For Solving Unconstrained Optimization Problems

Newton methods are the oldest and most efficient ones for solving unconstrained optimization problems by now. In this paper, we propose a new modified damped Newton algorithm for solving the objective function minimization problems. The algorithm is shown to converge globally. Quasi-Newton methods are also the most efficient ones for solving unconstrained optimization problems. Traditional DFP algorithm needs strong conditions to keep the approximate inverse matrix Hk of the Hessian matrix positive definite. We propose a new DFP algorithm. It helps Hk to maintain positive definite by a very good qualitative result. And it proves that the method with accurate line search converges globally. Quasi-Newton equations play a key role in Quasi-Newton methods for optimization problems. The original Quasi-Newton equations only employ the gradient of the objective function, but ignore the value information of the available objective function. In order to use more value information, many people study the Quasi-Newton correction formula and have made good progress in the field. In this paper, we propose a new BFGS algorithm for the no convex unconstrained optimization problems. And it prove that the method with Goldstein type line search converges globally.