修正的Dai-Liao三项共轭梯度方法

Modified Dai-Liao Three Term Conjugate Gradient Method

共轭梯度方法是求解大规模无约束优化问题最有效的方法之一.近年来提出的共轭梯度法具有良好的收敛性和数值结果.然而,这些方法并不总是产生下降方向.Dai-Liao方法虽然不会产生上升的搜索方向,但是也不满足充分下降条件.大量学者对Dai-Liao方法的参数进行研究使得Dai-Liao方法获得更好的收敛性和计算结果.因此,我们基于下降的三项共轭梯度方法和通过对Dai-Liao方法的迭代矩阵进行特征值分析,得到参数的一种自适应选择,并且利用HZ和DK方法的截断思想,提出了两种新的修正的Dai-Liao方法,使得修正的Dai-Liao方法满足充分下降条件和对一致凸函数全局收敛.在计算上优于HZ+、DK+方法和一种修正的Dai-Liao三项共轭梯度方法,有效可行.

Conjugate gradient method is one of the most effective methods to solve large-scale unconstrained optimization problems. In recent years, the conjugate gradient methods proposed have good convergence and numerical results. However, these methods do not always generate a descent direction. Dai-Liao method does not produce an ascending search direction, hut it does not satisfy the sufficient descent condition. A large number of scholars have studied the parameter of Dai-Liao method,which enables Dai-Liao method to obtain better convergence and computation result. Therefore, based on the descent three term conjugate gradient method and by means of the iterative matrix eigenvalue analysis of Dai-Liao method, an adaptive choice of the parameters is obtained. Two new modified Dai-Liao methods are proposed by using the truncation ideas of HZ and DK methods, which satisfy the sufficient descent condition and global convergence of the unifonnly convex functions. It is more effective and feasible than HZ +, DK + method and a modified Dai-Liao three term conjugate gradient method.