Proximal Newton-Type Methods for Minimizing Composite Functions

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type meth...

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Bibliographic Details
Published inSIAM journal on optimization Vol. 24; no. 3; pp. 1420 - 1443
Main Authors Lee, Jason D., Sun, Yuekai, Saunders, Michael A.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects
Approximation
Bioinformatics
Composite functions
Convergence
Convex analysis
Handles
Learning
Mathematical analysis
Mathematical models
Methods
Optimization
Signal processing
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Summary:We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when search directions are computed inexactly. Many popular methods tailored to problems arising in bioinformatics, signal processing, and statistical learning are special cases of proximal Newton-type methods, and our analysis yields new convergence results for some of these methods. [PUBLICATION ABSTRACT]
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ISSN:1052-6234
1095-7189
DOI:10.1137/130921428