解一类Hessian矩阵亏秩的修正BFGS算法及其局部Q—超线性收敛性  

Solving a Type of Modifed BFGS Algorithm with any Rank Defects and the Local Q-superlinear Convergence Properties

作　　者：葛仁东[1] 赵岩[1] 刘建国[2] 刘胜蓝[1] 

机构地区：[1]大连民族学院理学院,大连116605 [2]大连理工大学管理学院,大连116024

出　　处：《运筹学学报》2007年第3期51-64,共14页Operations Research Transactions

基　　金：国家自然科学基金项目(10001007).

摘　　要：本文对凸函数在极值点的Hessian矩阵是秩亏一的情况下,给出了一类求解无约束优化问题的修正BFGS算法.算法的思想是对凸函数加上一个修正项,得到一个等价的模型,然后简化此模型得到一个修正的BFGS算法.文中证明了该算法是一个具有超线性收敛的算法,并且把修正的BFGS算法同Tensor方法进行了数值比较,证明了该算法对求解秩亏一的无约束优化问题更有效.An modified BFGS algorithm to solve the unconstrained optimization, whose Hessian matrix of the minimum point of the convex function is rank one defect, is presented in this paper. The idea of the algorithm is to give a modified part of the convex function to obtain an equivalent model, then simply the model to obtain the modified BFGS algorithm. The superlinear convergence property of the algorithm is proved in this paper, and compared with the Tensor algorithm. It is proved that this method is more efficient for solving unconstrained optimization whose object function is of rank one defect.

关 键 词：运筹学 凸函数 无约束优化 BFGS算法 局部收敛性 Tensor方法 

分 类 号：O175.25[理学—数学]