Global Convergence of a Class of Collinear Scaling Algorithms with Inexact Line Searches on Conves Functions

摘要

Ariyawansa[2] has presented a class of collinear scaling algorithms for unconstrained minimization. A certain family of algorthms contained in this class may be considered as an extension of quasi-Newton methods with the Broyden family [11] of approximants of the objective function Hessian. Byrd, Nocedal and Yuan[7] have shown that all members except the DFP [11]method of the Broyden convex family of quasi-Newton methods with Armijo[1] and Goldstein[12] line search termination criteria are globally and q-superlinearly convergent on uniformly convex functions. Extensionof this result to the above class of collinear scaling algorithms of Ariyawansa[2] has been impossible because line search termination criteria for collinearscaling algorithms werenot known until recently. Ariyawansa [4] has recently proposed such line search temination criteria. In this paper, we prove an analogue of the resuit of Byrd, Nocedal and Yuan[7]for the family of collinear scaling algorithms of Ariyawansa [2] with the line search termination criteria of Ariyawansa[4].