An Improved Algorithms Of BFGS And Its Convergence Analysis

The select of parameter in quasi-Newton algorithms, same type of quasi-Newtonalgorithms would lead to huge difference in the performance of relative algorithm bychoosing different parameter, e.g., the literature [1-6]. While same quasi-Newtonalgorithms, by using different step size searching way, would directly affect theeffectiveness and valid of the algorithm, especially, by using inaccurate step sizesearching way, according to different conditions, there is huge difference for theeffectiveness of the algorithm.Firstly, a new revise formula of BFGS based on Wei’s newly quasi-Newtonequation is proposed, and a new BFGS algorithm by virtue of the Wolfe step sizeprinciple is established. Then, the global convergence of this algorithm is obtainedwhen the function in the considered problem is convex. Superlinear convergence is alsoproved under certain circumstance.Secondly, a new inexact line search rule to obtain steps inspired by the literature [2,34] is proposed, and the BFGS algorithm is analyzed under certain assumptions to getthe global convergence, superlinear global convergence and quadratic convergence ofthis algorithm.