A primal–dual prediction–correction algorithm for saddle point optimization

In this paper, we introduce a new primal–dual prediction–correction algorithm for solving a saddle point optimization problem, which serves as a bridge between the algorithms proposed in Cai et al. (J Glob Optim 57:1419–1428, 2013) and He and Yuan (SIAM J Imaging Sci 5:119–149, 2012). An interesting byproduct of the proposed method is that we obtain an easily implementable projection-based primal–dual algorithm, when the primal and dual variables belong to simple convex sets. Moreover, we establish 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 result in an ergodic sense, where t represents the number of iterations.