Mathematical Programming

, Volume 146, Issue 1–2, pp 459–494 | Cite as

Proximal alternating linearized minimization for nonconvex and nonsmooth problems

  • Jérôme Bolte
  • Shoham Sabach
  • Marc TeboulleEmail author
Full Length Paper Series A


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.


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

Mathematics Subject Classification (2000)

90C26 90C30 49M37 65K10 47J25 49M27 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.TSE (GREMAQ, Université Toulouse I), Manufacture des TabacsToulouseFrance
  2. 2.School of Mathematical SciencesTel-Aviv UniversityTel AvivIsrael

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