摘要
利用非单调性，邓乃扬等[1]提出了一类具有强收敛性质的非单调信赖域型算法为了保证算法的收敛性，他们假定以下两个条件成立：（1）信赖域半径｛△k｝有上界；（2）对所有k，有，其中sk＝x（k＋1）－xk，gk为f（t）在xk处的梯度，随后，柯小伍，韩继业[2]从另一角度也提出了一类非单调信赖域型算法．尽管他们未利用条件（1），但仍假定条件（2）成立．在本文中，我们提出了一类新的非单调信赖域算法，在没有假定条件（1）和（2）成立的情况下，我们仍得到了算法的全局收敛性及其超线性收敛性．本文也可看作[3]中NMTR2的一个更合理的变形，二者的主要区别是接受试探点和放大信赖域的策略不同另外，我们在没有假定二次模型的海色阵一致有界的条件下，证明了算法的全局收敛性，从而扩大了信赖域型方法的应用范围．
By exploring nonmonotonicity, Deng, Xiao and Zhou[1] proposed one nonmonotonic trust-region algorithm for unconstrained minimization with strong global convergence properties. TO ensure the algorithm's convergence, they assume that the following two conditions hold: (1) the sequence of trust-region radii △k is bounded from above; (2) for all k, where sk=xk＋1- xk, gk is the gradient of f(x) at xk, c > 0. Later, Ke and Han[2] also proposed a nonmonotonic trust-region algorithm. Although they do not make use of the assupmtion (1), the assumption (2) is still necessary in their algorithm. As an improvement of the above two algorithms, a new family of nonmonotonic trust-region algorithms is presented in this paper, we demonstrate that the new algorithm is globally convergent and the rate of convergence is superlinear without making use of the assumption (1) and (2). Our algorithm can also be considered as a variant of [3], hilt without requiring the uniform boundedness of the Hessian A.latricies of the quadtatic models, we show that our algorithm is globally convergent. This enlarges the applications of nonmonotomic trust region methods.
出处
《应用数学学报》
CSCD
北大核心
1999年第3期457-465,共9页
Acta Mathematicae Applicatae Sinica
基金
国家自然科学基金
北京市自然科学基金
关键词
无约束最优化
信赖域算法
非单调算法
收敛性
Unconstrained minimization, trust-region methods, nonmonotonicity, global convergence
