A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization

The BFGS update formula is shown to have an important property that is independent of the algorithmic context of the update, and that is relevant to both constrained and unconstrained optimization. The BFGS method for unconstrained optimization, using a variety of line searches, including backtracking, is shown to be globally and superlinearly convergent on uniformly convex problems. The analysis is particularly simple due to the use of some new tools introduced in this paper.