基于修正割线方程的自适应BB法

An Adaptive Barzilai-Borwein Method Based on Modified Secant Equations

基于一类带单参数γ的修正割线方程,给出了带参数γ的修正BB(Barzilai-Borwein)步长α_(k)(γ),并在某种意义下获得了γ的一个最优取值8/3.进而,依据当前和上一次迭代点连线段上目标函数的凸性,对步长α_(k)(γ)进行修正,并结合Zhang-Hager非单调线搜索技术,给出了求解无约束优化问题的一类自适应修正BB算法-AMBB算法.在适当的假设下,AMBB算法具有全局收敛性,且当目标函数为强凸函数时,AMBB算法具有线性收敛率.数值试验表明,给出的对应于参数γ取值8/3的AMBB算法是十分有效的.

In this paper,we propose a class of modified BB(Barzilai-Borwein)stepsize α_(k)(γ) based on a class of modified secant equations with a parameter and obtain an optimal choice 8/3 for γ in a certain sense.Furthermore,the stepsizes α_(k)(γ) are modified according to the convexity of objective function f on the line segment connecting two successive iterative points.By embedding Zhang-Hager nonmonotonic line search technique in our methods,we propose a class of adaptive modified Barzilai-Borwein(AMBB)algorithms for unconstrained optimization problems.Under suitable assumptions,AMBB algorithms possess global convergence for general nonlinear nonconvex functions and possess linear convergence rate for strongly convex functions.Numerical experiments show that when parameter γ is chosen as 8/3,the corresponding AMBB algorithm is quite effective.