Application of a smoothing technique to decomposition in convex optimization

TL;DR

This paper introduces the proximal center algorithm, a new decomposition method for convex optimization that uses Nesterov's smoothing technique to improve efficiency and preserve problem separability.

Contribution

It develops a novel smoothing-based decomposition method that enhances convergence bounds for convex optimization problems with separable structure.

Findings

Improves iteration bounds by an order of magnitude over classical dual gradient schemes.

Uses Nesterov's smoothing to maintain separability in the dual function.

Provides a more stable and efficient approach for convex optimization decomposition.

Abstract

Dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure. Although the Augmented Lagrangian is computationally more stable than the ordinary Lagrangian, the prox-term destroys the separability of the given problem. In this paper we use another approach to obtain a smooth Lagrangian, based on a smoothing technique developed by Nesterov, which preserves separability of the problem. With this approach we derive a new decomposition method, called proximal center algorithm, which from the viewpoint of efficiency estimates improves the bounds on the number of iterations of the classical dual gradient scheme by an order of magnitude. This can be achieved with the new decomposition algorithm since the resulting dual function has good smoothness properties and since we make use of the particular structure of the given problem.