Solving Unconstrained Optimization Problems, A New Quasi-newton Method

Quasi-Newton Methods are ones of the most effective methods for solving nonlinear unconstrained optimization problems, and many methods for solving minimization problems are variants of Newton method. Here, We make great efforts on researching the Quasi-Newton Methods for unconstrained optimization problems, the first half of this article gives a variety of commonly used methods for solving unconstrained optimization problems and the background of the Quasi-Newton Methods, the latter part of this paper gives a new quasi-Newton equation, and then a series of new algorithms corresponding to the new quasi-Newton equation are proposed.First, in the way of Zhang’s new quasi-Newton equation which was born in Tensor methods, We get our new quasi-Newton equation, It contains most of the nature of the usual quasi-Newton equation, Secondly based on the new quasi-Newton equation, We gives a series of new algorithms, the algorithm contains more forms of rank one and rank two formulas, At the same time, it’s rank two formula contains the rank two formula which is derived by the original quasi-Newton equation and Zhang’s new quasi-Newton equation with widely application. Again, two simple forms of Hessen matrix are given in this paper and the corresponding BFGS-TYPE algorithms are proved to be globally and super-linearly convergent. Finally, extensive numerical experiments verified our theory analysis, and showing that the algorithms have the better practicability.