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 rate without any smoothness or strong convexity-type assumption, or an 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.