Iteration complexity analysis of a partial LQP-based alternating direction method of multipliers

TL;DR

This paper introduces a novel partial LQP-based ADMM algorithm for convex optimization with multi-block variables, proving its global convergence and sublinear rate, and extending it to nonsmooth problems.

Contribution

The paper develops a new ADMM variant with LQP regularization, providing convergence analysis and extending it to nonsmooth convex optimization.

Findings

Proves global convergence of ADMM-LQP.

Establishes an $O(1/T)$ convergence rate.

Extends the method to nonsmooth problems with similar guarantees.

Abstract

In this paper, we consider a prototypical convex optimization problem with multi-block variables and separable structures. By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice with relatively larger stepsizes than the classical region $(0,\frac{1+\sqrt{5}}{2})$. Using a prediction-correction approach to analyze properties of the iterates generated by ADMM-LQP, we establish its global convergence and sublinear convergence rate of $O(1/T)$ in the new ergodic and nonergodic senses, where $T$ denotes the iteration index. We also extend the algorithm to a nonsmooth composite convex optimization and establish {similar convergence results} as our ADMM-LQP.