An adaptive primal-dual framework for nonsmooth convex minimization

We propose a new self-adaptive and double-loop smoothing algorithm to solve composite, nonsmooth, and constrained convex optimization problems. Our algorithm is based on Nesterov’s smoothing technique via general Bregman distance functions. It self-adaptively selects the number of iterations in the inner loop to achieve a desired complexity bound without requiring to set the accuracy a priori as in variants of augmented Lagrangian methods (ALM). We prove O1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( \frac{1}{k}\right) $$\end{document}-convergence rate on the last iterate of the outer sequence for both unconstrained and constrained settings in contrast to ergodic rates which are common in ALM as well as alternating direction method-of-multipliers literature. Compared to existing inexact ALM or quadratic penalty methods, our analysis does not rely on the worst-case bounds of the subproblem solved by the inner loop. Therefore, our algorithm can be viewed as a restarting technique applied to the ASGARD method in Tran-Dinh et al. (SIAM J Optim 28(1):96–134, 2018) but with rigorous theoretical guarantees or as an inexact ALM with explicit inner loop termination rules and adaptive parameters. Our algorithm only requires to initialize the parameters once, and automatically updates them during the iteration process without tuning. We illustrate the superiority of our methods via several examples as compared to the state-of-the-art.