Inexact proximal memoryless quasi-Newton methods based on the Broyden family for minimizing composite functions

This study considers a proximal Newton-type method to solve the minimization of a composite function that is the sum of a smooth nonconvex function and a nonsmooth convex function. In general, the method uses the Hessian matrix of the smooth portion of the objective function or its approximation. Th...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 79; no. 1; pp. 127 - 154
Main Authors: Nakayama, Shummin, Narushima, Yasushi, Yabe, Hiroshi
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2021
Springer Nature B.V
Subjects:
Approximation
Composite functions
Convergence
Convex and Discrete Geometry
Hessian matrices
Management Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Quasi Newton methods
Statistics
Memoryless quasi-Newton method
Nonsmooth optimization
Inexact proximal method
Proximal Newton-type method
Global and local convergence properties
Broyden family
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:This study considers a proximal Newton-type method to solve the minimization of a composite function that is the sum of a smooth nonconvex function and a nonsmooth convex function. In general, the method uses the Hessian matrix of the smooth portion of the objective function or its approximation. The uniformly positive definiteness of the matrix plays an important role in establishing the global convergence of the method. In this study, an inexact proximal memoryless quasi-Newton method is proposed based on the memoryless Broyden family with the modified spectral scaling secant condition. The proposed method inexactly solves the subproblems to calculate scaled proximal mappings. The approximation matrix is shown to retain the uniformly positive definiteness and the search direction is a descent direction. Using these properties, the proposed method is shown to have global convergence for nonconvex objective functions. Furthermore, the R -linear convergence for strongly convex objective functions is proved. Finally, some numerical results are provided.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-021-00264-9