Convergence analysis of some methods for minimizing a nonsmooth convex function

J. R. Birge, Liqun Qi, Z. Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

17 Citations (Scopus)


In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function f: R-fraktur signn→R-fraktur sign ∪ {∞}. Instead of the original objective function f, we employ a convex approximation fk+1at 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(Xk)-minx∈R-fraktur signnf(x) = O(1/(∑k+1j=0√λj)2)even if the iteration points are calculated approximately, where {λk}∞k=0 are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.
Original languageEnglish
Pages (from-to)357-383
Number of pages27
JournalJournal of Optimization Theory and Applications
Issue number2
Publication statusPublished - 1 Jan 1998
Externally publishedYes


  • Bundle algorithm
  • Nonsmooth convex optimization
  • Proximal point method
  • Stochastic programming

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics


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