A New Class Of Quasi-newton Algorithm And Its Convergence

Quasi-Newton methods are the most efficient ones for unconstrained optimization. Quasi-Newton equations play a key role in Quasi-Newton methods for optimization problems. The original Quasi-Newton equation employs only the gradient of the objective function, but ignores the available function value information. In this paper, a class of modified Quasi-Newton equations are derived, which apply both the gradient and function value. This class of modified Quasi-Newton equations include new Quasi-Newton equation proposed by Chengxian Xu, Quasi-Newton equation proposed by Yunhai Xiao and the original Quasi-Newton equation. Moreover, the Broyden Quasi-Newton methods based on the class of modified Quasi-Newton equations are proposed. Further observations are completed, such as quadratic termination, heredity of positive-definite updates, global convergence, and local superlinear convergence. In this paper, the global convergence analysis of restricted Broyden methods based on the class of modified Quasi-Newton equations is given. At last, this paper completes the local superlinear convergence of BFGS, DFP, PSB methods based on the class of quasi-Newton equations. The analysis of convergence extends the existed convergence.