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In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, ϕ (x₀, , xₚ, y), subject to coupled linear equality constraints. Our ADMM updates each of the primal variables x₀, , xₚ, y, followed by updating the dual variable. We separate the variable y from xᵢ's as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, q quasi-norm, Schatten-q quasi-norm (0
Wang et al. (Wed,) studied this question.