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Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Jerome Bolte 1 Shoham Sabach 2 Marc Teboulle 2
1 C&O - Equipe combinatoire et optimisation
IMJ-PRG - Institut de Mathématiques de Jussieu - Paris Rive Gauche
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.
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Submitted on : Monday, December 9, 2013 - 5:28:55 PM
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Jerome Bolte, Shoham Sabach, Marc Teboulle. Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Mathematical Programming, Springer Verlag, 2014, 146 (1-2), pp.459-494. ⟨10.1007/s10107-013-0701-9⟩. ⟨hal-00916090⟩



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