A Double Extrapolation Primal-Dual Algorithm for Saddle Point Problems

The first-order primal-dual algorithms have received much considerable attention in the literature due to their quite promising performance in solving large-scale image processing models. In this paper, we consider a general saddle point problem and propose a double extrapolation primal-dual algorithm, which employs the efficient extrapolation strategy for both primal and dual variables. It is remarkable that the proposed algorithm enjoys a unified framework including several existing efficient solvers as special cases. Another exciting property is that, under quite flexible requirements on the involved extrapolation parameters, our algorithm is globally convergent to a saddle point of the problem under consideration. Moreover, the worst case O(1/t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}(1/t)$$\end{document} convergence rate in both ergodic and nonergodic senses, and the linear convergence rate can be established for more general cases, where t counts the iteration. Some computational results on solving image deblurring, image inpainting and the nearest correlation matrix problems further show that the proposed algorithm is efficient, and performs better than some existing first-order solvers in terms of taking less iterations and computing time in some cases.