Perturbed Proximal Gradient ADMM for Nonconvex Composite Optimization

This paper proposes a Perturbed Proximal Gradient ADMM (PPG-ADMM) framework for general nonconvex composite optimization problems. The core of PPG-ADMM is a perturbation mechanism that enhances convergence stability and, more importantly, relaxes two restrictive assumptions common in existing ADMM-based methods: smoothness of the final objective block and range inclusion conditions on the constraint matrices. Convergence is established through a technically constructed Lyapunov function, which guarantees sufficient descent and has a well-defined lower bound. With appropriate parameters, PPG-ADMM converges to an$\epsilon$-approximate stationary point with a sublinear convergence rate of$\mathcal {O}(1/\sqrt{K})$. Furthermore, with a suitably decaying perturbation parameter, the algorithm can attain an$\epsilon$-stationary point, ensuring stronger optimality guarantees. The framework is extended to fully decentralized settings, and its effectiveness is validated on a distributed nonconvex composite optimization task.