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

Shummin Nakayama (), Yasushi Narushima () and Hiroshi Yabe ()
Additional contact information
Shummin Nakayama: Chuo University
Yasushi Narushima: Keio University
Hiroshi Yabe: Tokyo University of Science

Computational Optimization and Applications, 2021, vol. 79, issue 1, No 5, 127-154

Abstract: Abstract 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.

Keywords: Nonsmooth optimization; Proximal Newton-type method; Inexact proximal method; Memoryless quasi-Newton method; Broyden family; Global and local convergence properties (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10589-021-00264-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:79:y:2021:i:1:d:10.1007_s10589-021-00264-9

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-021-00264-9

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().