Two New Customized Proximal Point Algorithms Without Relaxation for Linearly Constrained Convex Optimization

In this paper, we propose a new customized proximal point algorithm for linearly constrained convex optimization problem, and further extend the proposed method to separable convex optimization problem. Unlike the existing customized proximal point algorithms, the proposed algorithms do not involve relaxation step, but still ensure the convergence. We obtain the particular iteration schemes and the unified variational inequality perspective. The global convergence and O(1/k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(1/k)$$\end{document}-convergence rate of the proposed methods are investigated under some mild assumptions. Numerical experiments show that compared to some state-of-the-art methods, the proposed methods are effective.