Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Jérôme Bolte, Shoham Sabach, Marc Teboulle

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)459-494
Number of pages36
JournalMathematical Programming
Volume146
Issue number1-2
DOIs
StatePublished - Aug 2014

Keywords

  • Alternating minimization
  • Block coordinate descent
  • Gauss-Seidel method
  • Kurdyka-Łojasiewicz property
  • Nonconvex-nonsmooth minimization
  • Proximal forward-backward
  • Sparse nonnegative matrix factorization

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