A variable metric proximal point algorithm for monotone operators

The proximal point algorithm (PPA) is a method for solving inclusions of the form 0 is an element of 2 T(z), where T is a monotone operator on a Hilbert space. The algorithm is one of the most powerful and versatile solution techniques for solving variational inequalities, convex programs, and convex-concave mini-max problems. It possesses a robust convergence theory for very general problem classes and is the basis for a wide variety of decomposition methods called splitting methods. Yet the classical PPA typically exhibits slow convergence in many applications. For this reason, acceleration methods for the PPA algorithm are of great practical importance. In this paper we propose a variable metric implementation of the proximal point algorithm. In essence, the method is a Newton-like scheme applied to the Moreau-Yosida resolvent of the operator T. In this article, we establish the global and linear convergence of the proposed method. In addition, we characterize the superlinear convergence of the method. In a companion work, we establish the superlinear convergence of the method when implemented with Broyden updating (the nonsymmetric case) and BFGS updating (the symmetric case).