Alternating direction method of multipliers with Bregman proximal gradient for non-convex and non-smooth optimization beyond Lipschitz

In this paper, we propose an algorithmic framework for solving a class of non-convex, non-smooth, multiblock composite optimization problems with linear constraints, dubbed the Alternating Direction Method of Multipliers based on Bregman Proximal Gradients (BPADMM). The key feature of this framework is the introduction of a kernel distance function during the iteration process, which frees it from the usual and restrictive assumption of global Lipschitz gradient continuity. Additionally, by employing the Minimization–Majorization principle and adopting a holistic block update strategy, we have designed a more efficient ADMM algorithm, thereby expanding its practical application scope. Under standard assumptions, we have proven the global convergence of the algorithm and demonstrated its effectiveness through application to the non-negative matrix factorization problem.