Proximal alternating linearized method for nonconvex and nonsmooth problems

Résumé

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.

Référence

Jérôme Bolte, Shoham Sabach et Marc Teboulle, « Proximal alternating linearized method for nonconvex and nonsmooth problems », Mathematical Programming, vol. 146, août 2014, p. 459–494.