On the Local Super-Linear Convergence of a Matrix Secant Implementation of the Variable Metric Proximal Point Algorithm for Monotone Operators

Abstract

Interest in the variable metric proximal point algorithm (VMPPA) is fueled by the desire to accelerate the local convergence of the proximal point algorithm without requiring the divergence of the proximation parameters. In this paper, the local convergence theory for matrix secant versions of the VMPPA is applied to a known globally convergent version of the algorithm. It is shown under appropriate hypotheses that the resulting algorithms are locally super-linearly convergent when executed with the BFGS and the Broyden matrix secant updates. This result unifies previous work on the global and local convergence theory for this class of algorithms. It is the first result applicable to general monotone operators showing that a globally convergent VMPPA with bounded proximation parameters can be accelerated using matrix secant techniques. This result clears the way for the direct application of these methods to constrained and non-finite-valued convex programming. Numerical experiments are included illustrating the potential gains of the method and issues for further study.