Local andQ-superlinear convergence of a class of collinear scaling algorithms that extends quasi-newton methods with broyden's bounded-⊘ class of updates† ‡

Abstract

Ariyawansa [3] has presented a class of coilinear scaling algorithms for unconstrained minimization. A certain family of algorithms contained in this class may be considered as an extension of the family of quasi-Newton methods with the Broyden family of approximants of the objective function Hessian. Let ⊘ k be the parameter that specifies the Broyden family member used at the (k+1)th iterate, .In [3], the members in this family of collinear scaling algorithms corresponding to ⊘ k := 0 for all k(which extends the DFP method) and ⊘ k :=1 for all k(which extends the BFGS method) were shown to be locally and q-superlinearly convergent. In this paper, we extend that result to other members in the above family of coilinear scaling algorithms in the following sense:all the members in the family of algorithms with {⊘ k } chosen so that for some and all kare locally and q-superlinearly convergent