修正FR共轭梯度法的全局收敛性

Global convergence of a modified FR conjugate gradient method

众所周知,由求解无约束优化问题的共轭梯度法所产生的方向往往不是下降方向.本文对FR共轭梯度法做适当的修正,使得修正后的算法能始终保持充分下降性,这一性质与算法所采用的线性搜索无关.同时,在精确线性搜索条件下,该算法就是标准的FR共轭梯度法.在适当条件下证明了该算法在强Wolfe线性搜索下具有全局收敛性.数值结果展现了算法的可行性.

It is well-known that the direction generated by conjugate gradient method may not be a decent direction of the objective function.In this paper,a little modification is made to the FR conjugate gradient method such that the direction generated by the modified method is always descent direction for the objective function.This property is independent of the line search used.Moreover,if exact line search is used,the method reduces to the ordinary FR method.Under appropriate conditions,the modified FR method with strong Wolfe line search is globally convergent.At the end of this paper,some numerical experiment are presented to show the efficiency of the proposed method.