Convergence Analysis of Douglas-Rachford Splitting Method for “Strongly + Weakly” Convex Programming

Ke Guo, Deren Han, Xiaoming Yuan

Research output: Contribution to journalJournal articlepeer-review

30 Citations (Scopus)
92 Downloads (Pure)


We consider the convergence of the Douglas-Rachford splitting method (DRSM) for minimizing the sum of a strongly convex function and a weakly convex function; this setting has various applications, especially in some sparsity-driven scenarios with the purpose of avoiding biased estimates which usually occur when convex penalties are used. Though the convergence of the DRSM has been well studied for the case where both functions are convex, its results for some nonconvexfunction- involved cases, including the "strongly + weakly" convex case, are still in their infancy. In this paper, we prove the convergence of the DRSM for the "strongly + weakly" convex setting under relatively mild assumptions compared with some existing work in the literature. Moreover, we establish the rate of asymptotic regularity and the local linear convergence rate in the asymptotical sense under some regularity conditions.

Original languageEnglish
Pages (from-to)1549-1577
Number of pages29
JournalSIAM Journal on Numerical Analysis
Issue number4
Publication statusPublished - 6 Jul 2017

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Convergence
  • Convergence rate
  • Douglas-Rachford splitting method
  • Fejér monotone
  • Rate of asymptotic regularity
  • Weakly convex penalty


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