Combining Lagrangian Decomposition and Excessive Gap Smoothing Technique for Solving Large-Scale Separable Convex Optimization Problems

TL;DR

This paper introduces a novel algorithm that combines Lagrangian decomposition, excessive gap, and smoothing techniques to efficiently solve large-scale separable convex optimization problems, with proven convergence and robustness.

Contribution

It presents a new algorithm that dynamically updates smoothness parameters, improving robustness and convergence for large-scale convex problems, and couples it with a dual scheme for enhanced performance.

Findings

Algorithm achieves $O(1/k)$ convergence rate.

Numerical examples confirm theoretical robustness.

Coupled dual scheme enhances solution quality.

Abstract

A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main advantage of this algorithm is that it dynamically updates the smoothness parameters which leads to numerically robust performance. The convergence of the algorithm is proved under weak conditions imposed on the original problem. The rate of convergence is $O(\frac{1}{k})$, where $k$ is the iteration counter. In the second part of the paper, the algorithm is coupled with a dual scheme to construct a switching variant of the dual decomposition. We discuss implementation issues and make a theoretical comparison. Numerical examples confirm the theoretical results.