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.

Neural Information Processing Systems http://nips.cc/

Summary: The paper proposed 4 new variants of Augmented Lagrangian methods, which called NAPALA (non-ergodic alternating proximal augmented Lagrangian algorithms) to solve non-smooth constrained convex optimization problems under different problem structure assumptions. The first algorithm only requires f and g to be convex but neither necessarily strongly convex nor smooth. Especially, its parallel variant (8) has more advantages. The second algorithm requires either f or g to be strongly convex. Its variant (12) also allows to have parallel steps.

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.

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.

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.