Implementing and modifying Broyden class updates for large scale optimization

Martin Buhmann () and Dirk Siegel
Additional contact information
Martin Buhmann: Justus-Liebig University, Mathematics Department
Dirk Siegel: University of Cambridge, Pembroke College

Computational Optimization and Applications, 2021, vol. 78, issue 1, No 6, 203 pages

Abstract: Abstract We consider Broyden class updates for large scale optimization problems in n dimensions, restricting attention to the case when the initial second derivative approximation is the identity matrix. Under this assumption we present an implementation of the Broyden class based on a coordinate transformation on each iteration. It requires only $$2nk + O(k^{2}) + O(n)$$ 2 n k + O ( k 2 ) + O ( n ) multiplications on the kth iteration and stores $$nK+ O(K^2) + O(n)$$ n K + O ( K 2 ) + O ( n ) numbers, where K is the total number of iterations. We investigate a modification of this algorithm by a scaling approach and show a substantial improvement in performance over the BFGS method. We also study several adaptations of the new implementation to the limited memory situation, presenting algorithms that work with a fixed amount of storage independent of the number of iterations. We show that one such algorithm retains the property of quadratic termination. The practical performance of the new methods is compared with the performance of Nocedal’s (Math Comput 35:773--782, 1980) method, which is considered the benchmark in limited memory algorithms. The tests show that the new algorithms can be significantly more efficient than Nocedal’s method. Finally, we show how a scaling technique can significantly improve both Nocedal’s method and the new generalized conjugate gradient algorithm.

Keywords: Nonlinear optimization; Broyden class; BFGS method; Norcedal’s method; Conjugate gradient method (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-020-00239-2 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:78:y:2021:i:1:d:10.1007_s10589-020-00239-2

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

DOI: 10.1007/s10589-020-00239-2

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 ().