  This paper introduces a novel approach to solving multi-block nonconvex
composite optimization problems through a proximal linearized Alternating
Direction Method of Multipliers (ADMM). This method incorporates an Increasing
Penalization and Decreasing Smoothing (IPDS) strategy. Distinguishing itself
from existing ADMM-style algorithms, our approach (denoted IPDS-ADMM) imposes a
less stringent condition, specifically requiring continuity in just one block
of the objective function. IPDS-ADMM requires that the penalty increases and
the smoothing parameter decreases, both at a controlled pace. When the
associated linear operator is bijective, IPDS-ADMM uses an over-relaxation
stepsize for faster convergence; however, when the linear operator is
surjective, IPDS-ADMM uses an under-relaxation stepsize for global convergence.
We devise a novel potential function to facilitate our convergence analysis and
prove an oracle complexity $\mathcal{O}(\epsilon^{-3})$ to achieve an
$\epsilon$-approximate critical point. To the best of our knowledge, this is
the first complexity result for using ADMM to solve this class of nonsmooth
nonconvex problems. Finally, some experiments on the sparse PCA problem are
conducted to demonstrate the effectiveness of our approach.
