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 Kurdyka–Łojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded se...

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Bibliographic Details
Published inMathematical programming Vol. 146; no. 1-2; pp. 459 - 494
Main Authors Bolte, Jérôme, Sabach, Shoham, Teboulle, Marc
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014
Springer Nature B.V
Springer Verlag
Subjects
Algorithms
Analysis
Byproducts
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Construction
Data smoothing
Full Length Paper
Functions (mathematics)
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Minimization
Numerical Analysis
Optimization
Optimization and Control
Palm
Studies
Theoretical
Gauss-Seidel method
Kurdyka–Łojasiewicz property
Block coordinate descent
Alternating minimization
90C26
Nonconvex-nonsmooth minimization
Proximal forward-backward
90C30
65K10
49M27
49M37
47J25
Sparse nonnegative matrix factorization
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Summary: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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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-013-0701-9