Adaptive inexact fast augmented Lagrangian methods for constrained convex optimization

TL;DR

This paper develops and analyzes inexact fast augmented Lagrangian methods with adaptive smoothing for constrained convex optimization, achieving improved computational complexity bounds for obtaining epsilon-optimal solutions.

Contribution

It introduces adaptive inexact augmented Lagrangian algorithms combining smoothing and inexact oracle frameworks, with proven complexity bounds.

Findings

Adaptive method achieves $ ilde{O}(1/\epsilon)$ complexity.

Basic method has $\mathcal{O}(1/\epsilon^{5/4})$ complexity.

Algorithms rely on projections onto primal set for convergence.

Abstract

In this paper we analyze several inexact fast augmented Lagrangian methods for solving linearly constrained convex optimization problems. Mainly, our methods rely on the combination of excessive-gap-like smoothing technique developed in [15] and the newly introduced inexact oracle framework from [4]. We analyze several algorithmic instances with constant and adaptive smoothing parameters and derive total computational complexity results in terms of projections onto a simple primal set. For the basic inexact fast augmented Lagrangian algorithm we obtain the overall computational complexity of order $\mathcal{O}\left(\frac{1}{\epsilon^{5/4}}\right)$, while for the adaptive variant we get $\mathcal{O}\left(\frac{1}{\epsilon}\right)$, projections onto a primal set in order to obtain an $\epsilon-$optimal solution for our original problem.