#### Non-Ergodic Alternating Proximal Augmented Lagrangian Algorithms with Optimal Rates

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

#### Authors

*Quoc Tran Dinh*

#### Abstract

We develop two new non-ergodic alternating proximal augmented Lagrangian algorithms (NEAPAL) to solve a class of nonsmooth constrained convex optimization problems. Our approach relies on a novel combination of the augmented Lagrangian framework, alternating/linearization scheme, Nesterov's acceleration techniques, and adaptive strategy for parameters. Our algorithms have several new features compared to existing methods. Firstly, they have a Nesterov's acceleration step on the primal variables compared to the dual one in several methods in the literature.
Secondly, they achieve non-ergodic optimal convergence rates under standard assumptions, i.e. an $\mathcal{O}\left(\frac{1}{k}\right)$ rate without any smoothness or strong convexity-type assumption, or an $\mathcal{O}\left(\frac{1}{k^2}\right)$ rate under only semi-strong convexity, where $k$ is the iteration counter.
Thirdly, they preserve or have better per-iteration complexity compared to existing algorithms. Fourthly, they can be implemented in a parallel fashion.
Finally, all the parameters are adaptively updated without heuristic tuning.
We verify our algorithms on different numerical examples and compare them with some state-of-the-art methods.