On the optimal linear convergence factor of the relaxed proximal point algorithm for monotone inclusion problems

TL;DR

This paper derives tight linear convergence bounds for the relaxed proximal point algorithm across all relaxation factors in (0,2), clarifying the optimal convergence behavior and improving understanding of its asymptotic performance.

Contribution

It establishes the first comprehensive set of optimal linear convergence bounds for the relaxed PPA for all relaxation factors in (0,2), extending previous results.

Findings

Convergence bounds are tight and optimal for all relaxation factors in (0,2).

The results clarify the asymptotic behavior of the relaxed PPA.

Previous bounds were suboptimal outside the interval [1,2).

Abstract

Finding a zero of a maximal monotone operator is fundamental in convex optimization and monotone operator theory, and \emph{proximal point algorithm} (PPA) is a primary method for solving this problem. PPA converges not only globally under fairly mild conditions but also asymptotically at a fast linear rate provided that the underlying inverse operator is Lipschitz continuous at the origin. These nice convergence properties are preserved by a relaxed variant of PPA. Recently, a linear convergence bound was established in [M. Tao, and X. M. Yuan, J. Sci. Comput., 74 (2018), pp. 826-850] for the relaxed PPA, and it was shown that the bound is optimal when the relaxation factor $\gamma$ lies in $[1,2)$. However, for other choices of $\gamma$, the bound obtained by Tao and Yuan is suboptimal. In this paper, we establish tight linear convergence bounds for any choice of $\gamma\in(0,2)$ and make the whole picture about optimal linear convergence bounds clear. These results sharpen our understandings to the asymptotic behavior of the relaxed PPA.