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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Published in	Mathematical programming Vol. 146; no. 1-2; pp. 459 - 494
Main Authors	Bolte, Jérôme, Sabach, Shoham, Teboulle, Marc
Format	Journal Article
Language	English
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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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.
AbstractList	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. 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-ojasiewicz 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.[PUBLICATION 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-Aaojasiewicz 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.
Author	Bolte, Jérôme Teboulle, Marc Sabach, Shoham
Author_xml	– sequence: 1 givenname: Jérôme surname: Bolte fullname: Bolte, Jérôme organization: TSE (GREMAQ, Université Toulouse I), Manufacture des Tabacs – sequence: 2 givenname: Shoham surname: Sabach fullname: Sabach, Shoham organization: School of Mathematical Sciences, Tel-Aviv University – sequence: 3 givenname: Marc surname: Teboulle fullname: Teboulle, Marc email: teboulle@post.tau.ac.il organization: School of Mathematical Sciences, Tel-Aviv University
BackLink	https://inria.hal.science/hal-00916090$$DView record in HAL
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Cites_doi	10.1007/s10107-007-0133-5 10.1137/050644641 10.1007/3-540-31247-1 10.1016/S0167-6377(99)00074-7 10.1007/978-1-4419-9467-7 10.1007/BF00940050 10.1162/neco.2007.19.10.2756 10.5802/aif.1638 10.1016/j.csda.2006.11.006 10.1137/080716542 10.1016/S0022-0000(73)80033-9 10.1287/moor.1100.0449 10.1007/s10107-011-0484-9 10.1007/BF01584660 10.1137/110839072 10.1137/060670080 10.1007/BF01397082 10.1090/S0002-9947-09-05048-X 10.1002/9780470747278 10.1038/44565 10.1109/ICIP.2010.5652173 10.1007/978-3-642-45780-7_3
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Keywords	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
Language	English
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References	Bolte, Daniilidis, Ley, Mazet (CR15) 2010; 362 Auslender (CR4) 1971; 18 Palomar, Eldar (CR29) 2010 Zangwill (CR34) 1969 Sra, Nowozin, Wright (CR32) 2011 Attouch, Bolte, Svaiter (CR3) 2013; 137 CR14 Ortega, Rheinboldt (CR28) 1970 Kurdyka (CR22) 1998; 48 CR10 Grippo, Sciandrone (CR19) 2000; 26 Tseng (CR33) 2001; 109 Attouch, Bolte (CR1) 2009; 116 Attouch, Bolte, Redont, Soubeyran (CR2) 2010; 35 Bolte, Daniilidis, Lewis (CR16) 2006; 17 Auslender (CR5) 1976 Beck, Teboulle (CR9) 2009; 2 Hoyer (CR21) 2004; 5 CR7 Rockafellar, Wets (CR31) 1998 Bolte, Daniilidis, Lewis, Shiota (CR17) 2007; 18 Lin (CR24) 2007; 19 Powell (CR30) 1973; 4 Bauschke, Combettes (CR8) 2011 CR25 Luss, Teboulle (CR26) 2013; 55 Mordukhovich (CR27) 2006 Heiler, Schnorr (CR20) 2006; 7 Blum, Floyd, Pratt, Rivest, Tarjan (CR13) 1973; 7 Berry, Browne, Langville, Pauca, Plemmons (CR11) 2007; 52 Cichocki, Zdunek, Phan, Amari (CR18) 2009 Bertsekas, Tsitsiklis (CR12) 1989 Auslender (CR6) 1992; 73 Lee, Seung (CR23) 1999; 401 JM Ortega (701_CR28) 1970 RT Rockafellar (701_CR31) 1998 CJ Lin (701_CR24) 2007; 19 HH Bauschke (701_CR8) 2011 WI Zangwill (701_CR34) 1969 H Attouch (701_CR3) 2013; 137 MJD Powell (701_CR30) 1973; 4 701_CR25 H Attouch (701_CR1) 2009; 116 H Attouch (701_CR2) 2010; 35 701_CR7 A Beck (701_CR9) 2009; 2 M Heiler (701_CR20) 2006; 7 J Bolte (701_CR17) 2007; 18 M Berry (701_CR11) 2007; 52 A Auslender (701_CR5) 1976 M Blum (701_CR13) 1973; 7 B Mordukhovich (701_CR27) 2006 DD Lee (701_CR23) 1999; 401 (701_CR29) 2010 K Kurdyka (701_CR22) 1998; 48 A Cichocki (701_CR18) 2009 J Bolte (701_CR16) 2006; 17 DP Bertsekas (701_CR12) 1989 R Luss (701_CR26) 2013; 55 J Bolte (701_CR15) 2010; 362 (701_CR32) 2011 701_CR33 A Auslender (701_CR6) 1992; 73 701_CR10 701_CR14 A Auslender (701_CR4) 1971; 18 PO Hoyer (701_CR21) 2004; 5 L Grippo (701_CR19) 2000; 26
References_xml	– volume: 116 start-page: 5 year: 2009 end-page: 16 ident: CR1 article-title: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features publication-title: Math. Program. doi: 10.1007/s10107-007-0133-5 – volume: 17 start-page: 1205 year: 2006 end-page: 1223 ident: CR16 article-title: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems publication-title: SIAM J. Optim. doi: 10.1137/050644641 – year: 2006 ident: CR27 publication-title: Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren der Mathematischen Wissenschaften, vol. 330 doi: 10.1007/3-540-31247-1 – volume: 109 start-page: 475 year: 2001 end-page: 494 ident: CR33 article-title: Convergence of a block coordinate descent method for nondifferentiable minimization publication-title: J. Optim. Theory Appl. – volume: 26 start-page: 127 year: 2000 end-page: 136 ident: CR19 article-title: On the convergence of the block nonlinear Gauss-Seidel method under convex constraints publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(99)00074-7 – year: 1989 ident: CR12 publication-title: Parallel and Distributed Computation: Numerical Methods – year: 1998 ident: CR31 publication-title: Variational Analysis Grundlehren der Mathematischen Wissenschaften, vol. 317 – year: 1969 ident: CR34 publication-title: Nonlinear Programming: A Unified Approach – ident: CR14 – year: 2011 ident: CR8 publication-title: Convex Analysis and Monotone Operator Theory in Hilbert Spaces doi: 10.1007/978-1-4419-9467-7 – ident: CR10 – volume: 7 start-page: 1385 year: 2006 end-page: 1407 ident: CR20 article-title: Learning sparse representations by non-negative matrix factorization and sequential cone programming publication-title: J. Mach. Learn. Res – volume: 73 start-page: 427 year: 1992 end-page: 449 ident: CR6 article-title: Asymptotic properties of the Fenchel dual functional and applications to decomposition problems publication-title: J. Optim. Theory Appl. doi: 10.1007/BF00940050 – volume: 19 start-page: 2756 year: 2007 end-page: 2779 ident: CR24 article-title: Projected gradient methods for nonnegative matrix factorization publication-title: Neural Comput. doi: 10.1162/neco.2007.19.10.2756 – year: 2010 ident: CR29 publication-title: Convex Optimization in Signal Processing and Communications – volume: 48 start-page: 769 year: 1998 end-page: 783 ident: CR22 article-title: On gradients of functions definable in o-minimal structures publication-title: Annales de l’institut Fourier doi: 10.5802/aif.1638 – volume: 52 start-page: 155 year: 2007 end-page: 173 ident: CR11 article-title: Algorithms and applications for approximation nonnegative matrix factorization publication-title: Comput. Stat. Data Anal. doi: 10.1016/j.csda.2006.11.006 – year: 1970 ident: CR28 publication-title: Iterative Solution of Nonlinear Equations in Several Variables – volume: 2 start-page: 183 year: 2009 end-page: 202 ident: CR9 article-title: A fast iterative shrinkage-thresholding algorithm for linear inverse problems publication-title: SIAM J. Imaging Sci doi: 10.1137/080716542 – ident: CR25 – year: 2011 ident: CR32 publication-title: Optimization for Machine Learning – volume: 7 start-page: 448 year: 1973 end-page: 461 ident: CR13 article-title: Time bounds for selection publication-title: J. Comput. Syst. Sci. doi: 10.1016/S0022-0000(73)80033-9 – year: 1976 ident: CR5 publication-title: Optimisation—Méthodes numériques – volume: 35 start-page: 438 year: 2010 end-page: 457 ident: CR2 article-title: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka–Łojasiewicz inequality publication-title: Math. Oper. Res. doi: 10.1287/moor.1100.0449 – volume: 137 start-page: 91 year: 2013 end-page: 129 ident: CR3 article-title: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods publication-title: Math. Program. Ser. A doi: 10.1007/s10107-011-0484-9 – volume: 4 start-page: 193 year: 1973 end-page: 201 ident: CR30 article-title: On search directions for minimization algorithms publication-title: Math. Program. doi: 10.1007/BF01584660 – volume: 55 start-page: 65 year: 2013 end-page: 98 ident: CR26 article-title: Conditional gradient algorithms for rank-one matrix approximations with a sparsity constraint publication-title: SIAM Rev. doi: 10.1137/110839072 – volume: 18 start-page: 556 year: 2007 end-page: 572 ident: CR17 article-title: Clarke subgradients of stratifiable functions publication-title: SIAM J. Optim. doi: 10.1137/060670080 – volume: 18 start-page: 213 year: 1971 end-page: 223 ident: CR4 article-title: Méthodes numériques pour la décomposition et la minimisation de fonctions non différentiables publication-title: Numerische Mathematik doi: 10.1007/BF01397082 – ident: CR7 – volume: 362 start-page: 3319 year: 2010 end-page: 3363 ident: CR15 article-title: Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-09-05048-X – volume: 5 start-page: 1457 year: 2004 end-page: 1469 ident: CR21 article-title: Non-negative matrix factorization with sparseness constraints publication-title: J. Mach. Learn. Res. – year: 2009 ident: CR18 publication-title: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation doi: 10.1002/9780470747278 – volume: 401 start-page: 788 year: 1999 end-page: 791 ident: CR23 article-title: Learning the part of objects from nonnegative matrix factorization publication-title: Nature doi: 10.1038/44565 – volume: 48 start-page: 769 year: 1998 ident: 701_CR22 publication-title: Annales de l’institut Fourier doi: 10.5802/aif.1638 – ident: 701_CR33 – ident: 701_CR10 – volume: 7 start-page: 1385 year: 2006 ident: 701_CR20 publication-title: J. Mach. Learn. Res – volume-title: Optimisation—Méthodes numériques year: 1976 ident: 701_CR5 – volume: 7 start-page: 448 year: 1973 ident: 701_CR13 publication-title: J. Comput. Syst. Sci. doi: 10.1016/S0022-0000(73)80033-9 – volume: 362 start-page: 3319 year: 2010 ident: 701_CR15 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-09-05048-X – volume: 401 start-page: 788 year: 1999 ident: 701_CR23 publication-title: Nature doi: 10.1038/44565 – volume-title: Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren der Mathematischen Wissenschaften, vol. 330 year: 2006 ident: 701_CR27 doi: 10.1007/3-540-31247-1 – volume-title: Parallel and Distributed Computation: Numerical Methods year: 1989 ident: 701_CR12 – volume: 5 start-page: 1457 year: 2004 ident: 701_CR21 publication-title: J. Mach. Learn. Res. – ident: 701_CR25 – volume: 52 start-page: 155 year: 2007 ident: 701_CR11 publication-title: Comput. Stat. Data Anal. doi: 10.1016/j.csda.2006.11.006 – volume: 18 start-page: 556 year: 2007 ident: 701_CR17 publication-title: SIAM J. Optim. doi: 10.1137/060670080 – volume: 19 start-page: 2756 year: 2007 ident: 701_CR24 publication-title: Neural Comput. doi: 10.1162/neco.2007.19.10.2756 – volume: 55 start-page: 65 year: 2013 ident: 701_CR26 publication-title: SIAM Rev. doi: 10.1137/110839072 – volume: 137 start-page: 91 year: 2013 ident: 701_CR3 publication-title: Math. Program. Ser. A doi: 10.1007/s10107-011-0484-9 – volume-title: Optimization for Machine Learning year: 2011 ident: 701_CR32 – volume: 26 start-page: 127 year: 2000 ident: 701_CR19 publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(99)00074-7 – volume-title: Nonlinear Programming: A Unified Approach year: 1969 ident: 701_CR34 – volume: 4 start-page: 193 year: 1973 ident: 701_CR30 publication-title: Math. Program. doi: 10.1007/BF01584660 – volume: 17 start-page: 1205 year: 2006 ident: 701_CR16 publication-title: SIAM J. Optim. doi: 10.1137/050644641 – volume: 18 start-page: 213 year: 1971 ident: 701_CR4 publication-title: Numerische Mathematik doi: 10.1007/BF01397082 – ident: 701_CR14 doi: 10.1109/ICIP.2010.5652173 – volume-title: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation year: 2009 ident: 701_CR18 doi: 10.1002/9780470747278 – volume: 2 start-page: 183 year: 2009 ident: 701_CR9 publication-title: SIAM J. Imaging Sci doi: 10.1137/080716542 – volume-title: Variational Analysis Grundlehren der Mathematischen Wissenschaften, vol. 317 year: 1998 ident: 701_CR31 – volume-title: Convex Optimization in Signal Processing and Communications year: 2010 ident: 701_CR29 – ident: 701_CR7 doi: 10.1007/978-3-642-45780-7_3 – volume-title: Convex Analysis and Monotone Operator Theory in Hilbert Spaces year: 2011 ident: 701_CR8 doi: 10.1007/978-1-4419-9467-7 – volume: 116 start-page: 5 year: 2009 ident: 701_CR1 publication-title: Math. Program. doi: 10.1007/s10107-007-0133-5 – volume-title: Iterative Solution of Nonlinear Equations in Several Variables year: 1970 ident: 701_CR28 – volume: 35 start-page: 438 year: 2010 ident: 701_CR2 publication-title: Math. Oper. Res. doi: 10.1287/moor.1100.0449 – volume: 73 start-page: 427 year: 1992 ident: 701_CR6 publication-title: J. Optim. Theory Appl. doi: 10.1007/BF00940050
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Snippet	We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems....
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SubjectTerms	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
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Title	Proximal alternating linearized minimization for nonconvex and nonsmooth problems
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