Proximal alternating linearized minimization for nonconvex and nonsmooth problems

We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful KurdykaLojasiewicz 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 semialgebraic 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. Jérôme Bolte: This research benefited from the support of the FMJH Program Gaspard Monge in optimization and operation research (and from the support to this program from EDF) and it was co-funded by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-ITN”, grant agreement number 264735-SADCO. Shoham Sabach: Supported by a Tel Aviv University postdoctoral fellowship. Marc Teboulle: Partially supported by the Israel Science Foundation, ISF Grant 998-12. J. Bolte TSE (GREMAQ, Université Toulouse I), Manufacture des Tabacs 21 allée de Brienne, 31015 Toulouse, France E-mail: [email protected] S. Sabach School of Mathematical Sciences, Tel-Aviv University Ramat-Aviv 69978, Israel E-mail: [email protected] M. Teboulle School of Mathematical Sciences, Tel-Aviv University Ramat-Aviv 69978, Israel E-mail: [email protected] 2 Jérôme Bolte et al.