A general approach to convergence properties of some methods for nonsmooth convex optimization

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

Research output: Journal article publicationJournal articleAcademic researchpeer-review

15 Citations (Scopus)


Based on the notion of the ε-subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {∥Xk∥} and {f(xk)} when {xk} is unbounded and {f(xk)} is bounded for the nonsmooth minimization methods (i), (ii), and (iii).
Original languageEnglish
Pages (from-to)141-158
Number of pages18
JournalApplied Mathematics and Optimization
Issue number2
Publication statusPublished - 1 Jan 1998
Externally publishedYes


  • Convergence rate
  • Global convergence
  • Nonsmooth convex minimization

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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