Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

John Birge, L. Qi and Z. Wei
Additional contact information
L. Qi: University of New South Wales
Z. Wei: University of New South Wales

Journal of Optimization Theory and Applications, 1998, vol. 97, issue 2, No 6, 357-383

Abstract: Abstract In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function $$f:\Re^{\mathfrak{n}} \to \Re \cup {\infty}$$ . Instead of the original objective function f, we employ a convex approximation f k + 1 at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate $$f\left( {x_k } \right) - \min _{x \in \Re ^n } f\left( x \right) = O\left( {1/\left( {\Sigma _{j = 0}^{k - 1} \sqrt {\lambda _j } } \right)^2 } \right)$$ even it the iteration points are calculated approximately, where $${\lambda_k}_{k = 0}^\infty$$ are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.

Keywords: Nonsmooth convex optimization; proximal point method; bundle algorithm; stochastic programming (search for similar items in EconPapers)
Date: 1998
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

Downloads: (external link)
http://link.springer.com/10.1023/A:1022630801549 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:joptap:v:97:y:1998:i:2:d:10.1023_a:1022630801549

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1023/A:1022630801549

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

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