Convergence Analysis and Design of Multi-block ADMM via Switched Control Theory

TL;DR

This paper introduces a switched control framework for multi-block ADMM, providing convergence conditions, sequence selection, and parameter design using switched Lyapunov functions and semidefinite programming.

Contribution

It develops a novel switched control approach to analyze and design multi-block ADMM with arbitrary sequences and parameters, ensuring convergence and stability.

Findings

Convergence conditions for any block sequence are established.

A method to find convergent block sequences is proposed.

Parameter controllers are designed to guarantee linear convergence.

Abstract

We consider three challenges in multi-block Alternating Direction Method of Multipliers (ADMM): building convergence conditions for ADMM with any block (variable) sequence, finding available block sequences to be fit for ADMM, and designing useful parameter controllers for ADMM with unfixed parameters. To address these challenges, we develop a switched control framework for studying multi-block ADMM. First, since ADMM recursively and alternately updates the block-variables, it is converted into a discrete-time switched dynamical system. Second, we study exponential stability and stabilizability of the switched system for linear convergence analysis and design of ADMM by employing switched Lyapunov functions. Moreover, linear matrix inequalities conditions are proposed to ensure convergence of ADMM under arbitrary sequence, to find convergent sequences, and to design the fixed parameters. These conditions are checked and solved by employing semidefinite programming. Numerical experiments further verify the effectiveness of our proposed theories.