On the convergence properties of a majorized ADMM for linearly constrained convex optimization problems with coupled objective functions

TL;DR

This paper analyzes the convergence of a majorized ADMM for linearly constrained convex problems with coupled objectives, showing convergence under mild conditions and providing iteration complexity results.

Contribution

It establishes convergence properties for a majorized ADMM in the presence of coupled objective functions, extending the applicability of ADMM methods.

Findings

Convergence of 2-block ADMM with large step length for coupled objectives.

Use of generalized Mean-Value Theorem to control cross terms.

Iteration complexity results for the proposed algorithm.

Abstract

In this paper, we establish the convergence properties for a majorized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization problems whose objectives contain coupled functions. Our convergence analysis relies on the generalized Mean-Value Theorem which plays an important role to properly control the cross terms due to the presence of coupled objective functions. Our results in particular show that directly applying 2-block ADMM with a large step length to the linearly constrained convex optimization problem with a quadratically coupled objective function is convergent under mild conditions. We also provide several iteration complexity results for the algorithm.