A generalized inertial proximal alternating linearized minimization method for nonconvex nonsmooth problems

We propose a general inertial version of the proximal alternating linearized minimization (PALM) (denoted by NiPALM) for a class of nonconvex and nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. NiPALM is general in the sense that it contains the popular PALM, the inertial PALM (iPALM) and Gauss-Seidel type inertial PALM (GiPALM) as special cases. Under mild assumptions, namely, the underlying functions satisfy the Kurdyka-Lojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by NiPALM globally converges to a critical point. We also apply NiPALM to nonnegative matrix factorization, sparse principal component analysis, and weighted low-rank matrix restoration problems. Comparing results with those by PALM, iPALM, and GiPALM demonstrate the robustness and effectiveness of the proposed algorithm.(c) 2023 Published by Elsevier B.V. on behalf of IMACS.