2019/05/11 by Guoyong Gu, Junfeng Yang, Gu, Guoyong +1
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1905.04537
9 pages and 1 figure
arxiv created 2019/05/11 · arxiv updated 2019/05/14
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 optimal when the relaxation factor γ lies in [1,2). However, for other choices of γ, the bound obtained by Tao and Yuan is suboptimal. In this paper, we establish tight linear convergence bounds for any choice of γ∈(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.