Two Types of Two-Step Inertial ADMM Algorithms and Their Applications in Multi-Block Optimization Problems

In this paper, we propose two types of algorithms, named the “multi-block relaxed-dual two-step inertial proximal ADMM” and the “multi-block two-step inertial ADMM with separable updates”, which combine ADMM with a two-step inertial force in different frameworks to solve multi-block minimization problems composed of smooth and nonsmooth functions (possibly nonconvex), respectively. Sufficient conditions are presented to ensure the algorithm iterates remain bounded, and the global convergence of the algorithms is further verified by leveraging the Kurdyka–Łojasiewicz inequality. Additionally, to demonstrate the applicability of our proposed model and algorithms, we apply them to practical problems in the domains of compressed sensing, Lasso regression, and image processing. Experiments show that both types of algorithms exhibit superior performance with faster convergence than other related ADMM-based algorithms.