A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization

The convergence of the alternating direction method of multipliers (ADMMs) algorithm to convex/nonconvex combinational optimization has been well established in the literature. Due to the extensive applications of a weakly convex function in signal processing and machine learning, in this paper, we investigate the convergence of an ADMM algorithm to the strongly and weakly convex combinational optimization (SWCCO) problem. Specifically, we firstly show the convergence of the iterative sequences of the SWCCO-ADMM under a mild regularity condition; then we establish the 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}$o(1/k)$\end{document} sublinear convergence rate of the SWCCO-ADMM algorithm using the same conditions and the linear convergence rate by imposing the gradient Lipschitz continuity condition on the objective function. The techniques used for the convergence analysis in this paper are fundamental, and we accomplish the global convergence without using the Kurdyka–Łojasiewicz (KL) inequality, which is common but complex in the proof of nonconvex ADMM.