# Convergence and Applications of ADMM on the Multi-convex Problems

## TL;DR

This paper introduces a generic ADMM framework for multi-convex problems, providing convergence guarantees and demonstrating effectiveness through real-world experiments.

## Contribution

It proposes a new ADMM framework with convergence guarantees for multi-convex problems, extending its applicability and providing theoretical and empirical validation.

## Key findings

- Proven convergence to a Nash point with sublinear rate
- Effective on ten real-world datasets
- Demonstrates scalability and robustness

## Abstract

In recent years, although the Alternating Direction Method of Multipliers (ADMM) has been empirically applied widely to many multi-convex applications, delivering an impressive performance in areas such as nonnegative matrix factorization and sparse dictionary learning, there remains a dearth of generic work on proposed ADMM with a convergence guarantee under mild conditions. In this paper, we propose a generic ADMM framework with multiple coupled variables in both objective and constraints. Convergence to a Nash point is proven with a sublinear convergence rate $o(1/k)$. Two important applications are discussed as special cases under our proposed ADMM framework. Extensive experiments on ten real-world datasets demonstrate the proposed framework's effectiveness, scalability, and convergence properties. We have released our code at \url{https://github.com/xianggebenben/miADMM}.