A unified and tight linear convergence analysis of the relaxed proximal point algorithm

Abstract

<p style='text-indent:20px;'>Finding a zero of a maximal monotone operator is fundamental in convex optimization and monotone operator theory, and <i>proximal point algorithm</i> (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 tight when the relaxation factor <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> lies in <inline-formula><tex-math id="M2">\begin{document}$ [1,2) $\end{document}</tex-math></inline-formula>. However, for other choices of <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, the bound obtained by Tao and Yuan is suboptimal. In this paper, we establish tight linear convergence bounds for any choice of <inline-formula><tex-math id="M4">\begin{document}$ \gamma\in(0,2) $\end{document}</tex-math></inline-formula> using a unified and much simplified analysis. These results sharpen our understandings to the asymptotic behavior of the relaxed PPA and make the whole picture for <inline-formula><tex-math id="M5">\begin{document}$ \gamma\in(0,2) $\end{document}</tex-math></inline-formula> clear.</p>