  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}.
