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On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

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Abstract

We present two modified versions of the primal-dual splitting algorithm relying on forward–backward splitting proposed in V\(\tilde{\mathrm{u}}\) (Adv Comput Math 38(3):667–681, 2013) for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of \(\mathcal{{O}}(\frac{1}{n})\) and \(\mathcal{{O}}(\omega ^n)\), for \(\omega \in (0,1)\), respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and pattern recognition in cluster analysis.

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Acknowledgments

The authors are grateful to anonymous reviewers for remarks and suggestions which improved the quality of the paper.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 , Vienna, Austria

    Radu Ioan Boţ & Ernö Robert Csetnek

  2. Department of Mathematics, Chemnitz University of Technology, 09107 , Chemnitz, Germany

    André Heinrich & Christopher Hendrich

Authors
  1. Radu Ioan Boţ
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  2. Ernö Robert Csetnek
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  3. André Heinrich
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  4. Christopher Hendrich
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Correspondence to Radu Ioan Boţ.

Additional information

Radu Ioan Boţ: Research partially supported by DFG (German Research Foundation), project BO 2516/4-1.

Ernö Robert Csetnek: Research supported by DFG (German Research Foundation), project BO 2516/4-1.

André Heinrich: Research supported by the European Union, the European Social Fund (ESF) and prudsys AG in Chemnitz.

Christopher Hendrich: Research supported by a Graduate Fellowship of the Free State Saxony, Germany.

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Boţ, R.I., Csetnek, E.R., Heinrich, A. et al. On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems. Math. Program. 150, 251–279 (2015). https://doi.org/10.1007/s10107-014-0766-0

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  • DOI: https://doi.org/10.1007/s10107-014-0766-0

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