Regularized methods via cubic model subspace minimization for nonconvex optimization

Stefania Bellavia (), Davide Palitta (), Margherita Porcelli () and Valeria Simoncini ()
Additional contact information
Stefania Bellavia: Università degli Studi di Firenze
Davide Palitta: Alma Mater Studiorum - Università di Bologna
Margherita Porcelli: Università degli Studi di Firenze
Valeria Simoncini: Alma Mater Studiorum - Università di Bologna

Computational Optimization and Applications, 2025, vol. 90, issue 3, No 7, 837 pages

Abstract: Abstract Adaptive cubic regularization methods for solving nonconvex problems need the efficient computation of the trial step, involving the minimization of a cubic model. We propose a new approach in which this model is minimized in a low dimensional subspace that, in contrast to classic approaches, is reused for a number of iterations. Whenever the trial step produced by the low-dimensional minimization process is unsatisfactory, we employ a regularized Newton step whose regularization parameter is a by-product of the model minimization over the low-dimensional subspace. We show that the worst-case complexity of classic cubic regularized methods is preserved, despite the possible regularized Newton steps. We focus on the large class of problems for which (sparse) direct linear system solvers are available and provide several experimental results showing the very large gains of our new approach when compared to standard implementations of adaptive cubic regularization methods based on direct linear solvers. Our first choice as projection space for the low-dimensional model minimization is the polynomial Krylov subspace; nonetheless, we also explore the use of rational Krylov subspaces in case where the polynomial ones lead to less competitive numerical results.

Keywords: Adaptive regularization methods; Nonconvex optimization; Secant methods; Krylov subspaces; Worst-case iteration complexity; 49M37; 65K05; 68W40 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-025-00655-2 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:90:y:2025:i:3:d:10.1007_s10589-025-00655-2

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

DOI: 10.1007/s10589-025-00655-2

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 ().