%0 Journal Article
%@ 0025-5610
%A Bolte, Jérôme
%A Sabach, Shoham
%A Teboulle, Marc
%D 2014
%F publications:16696
%I Springer
%J Mathematical Programming
%P 459-494
%R 10.1007/s10107-013-0701-9
%T Proximal alternating linearized method for nonconvex and nonsmooth problems
%U https://publications.ut-capitole.fr/id/eprint/16696/
%V 146
%X We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward–backward algorithms with semi-algebraic problem’s data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.