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

The first author was supported by NSFC grant 11671195. The second author was supported by NSFC grants 11922111 and 12126337

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  • Finding a zero of a maximal monotone operator is fundamental in convex optimization and monotone operator theory, and 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 tight 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) $ 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 $ \gamma\in(0,2) $ clear.

    Mathematics Subject Classification: Primary: 65K10, 9C025.


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  • Figure 1.  Illustration of the bounds. Left: results in [27]. Right: results of this work

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