A proximal alternating direction method of multipliers with a proximal-perturbed Lagrangian function for nonconvex and nonsmooth structured optimization

Building on Yin et al. (J Glob Optim 89:899-926, 2024), we continue to focus on solving a nonconvex and nonsmooth structured optimization problem with linear and closed convex set constraints, where its objective function is the sum of a convex (possibly nonsmooth) function and a smooth (possibly nonconvex) function. Based on the traditional augmented Lagrangian construction, we introduce a proximal-perturbed Lagrangian function and propose a proximal alternating direction method of multipliers that leverages this new Lagrangian-based formulation. We establish that the iterative subsequence obtained by the proposed method converges to a stationary point under standard assumptions.