Sequential inertial linear ADMM algorithm for nonconvex and nonsmooth multiblock problems with nonseparable structure

The alternating direction method of multipliers (ADMM) has been widely used to solve linear constrained problems in signal processing, matrix decomposition, machine learning, and many other fields. This paper introduces two linearized ADMM algorithms, namely sequential partial linear inertial ADMM (SPLI-ADMM) and sequential complete linear inertial ADMM (SCLI-ADMM), which integrate linearized ADMM approach with inertial technique in the full nonconvex framework with nonseparable structure. Iterative schemes are formulated using either partial or full linearization while also incorporating the sequential gradient of the composite term in each subproblem’s update. This adaptation ensures that each iteration utilizes the latest information to improve the efficiency of the algorithms. Under some mild conditions, we prove that the sequences generated by two proposed algorithms converge to the critical points of the problem with the help of KŁ property. Finally, some numerical results are reported to show the effectiveness of the proposed algorithms.