Adaptive inexact fast augmented Lagrangian methods for constrained convex optimization

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.