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

Shummin Nakayama, Yasushi Narushima, Hiroshi Yabe

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)127-154
Number of pages28
JournalComputational Optimization and Applications
Issue number1
Publication statusPublished - 2021 May


  • Broyden family
  • Global and local convergence properties
  • Inexact proximal method
  • Memoryless quasi-Newton method
  • Nonsmooth optimization
  • Proximal Newton-type method

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


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