Local and Q-Superlinear Convergence of a Class of Collinear Scaling Algorithms That Extends Quasi-Newton Methods with Broyden's Bounded-/phi/ Class of Updates

摘要

A derivation of collinear scaling algorithms for unconstrained minimization was presented. The local conic approximants to the objective function underlying these algorithms are forced to interpolate the value and gradient of the objective function at the two most recent iterates. The class of algorithms derived therein has a free parameter sequence /l brace/b/sub k//r brace/ and for a fixed choice of /l brace/b/sub k//r brace/ it contains collinear scaling algorithms that may be treated as extensions of quasi-Newton methods with Broyden family of updates. In this paper, under standard assumptions, it is shown that if b/sub k/ is set equal to the gradient of the objective function for all k and if /l brace/chemically bond1 /minus/ theta/sub k/chemically bond/r brace/ (where theta/sub k/ is the parameter in the Broyden family) is uniformly bounded, then these collinear scaling algorithms related to the Broyden family are locally and q-superlinearly convergent. 13 refs. (ERA citation 13:050299)