Analogues of Dixon’s and Powell’s Theorems for Unconstrained Minimization with Inexact Line Searches

Abstract

Dixon’s theorem (Math. Programming, 2 (1972), PP. 383–387) states that all variable metric methods in the Broyden class develop identical iterates when line searches are exact. Powell’s theorem (Rep. TP 495, AERE, Harwell, England, 1972) is a variant on this, which states that under similar conditions, the Hessian approximation developed by a BFGS update at any step is independent of the updates used at earlier steps. By modifying the way in which search directions are defined, we show how to remove the restrictive assumption on line searches in these two theorems. We show also that the BFGS algorithm, modified in this way, is equivalent to the three-term-recurrence (TTR) method on quadratic functions. Algorithmic implications are discussed and the results of some numerical experimentation are reported.