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

Abstract

Ariyawansa [2] has presented a class of collinear scaling algorithms for unconstrained minimization. A certain family of algorithms 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. Extension of this result to the above class of collinear scaling algorithms of Ariyawansa [2] has been impossible because line search termination criteria for collinear scaling algorithms were not known until recently. Ariyawansa [4] has recently proposed such line search termination criteria. In this paper, we prove an analogue of the result 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].