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A Tseng’s Type Penalty Scheme for Solving Inclusion Problems Involving Linearly Composed and Parallel-Sum Type Monotone Operators

Abstract

In this paper we consider the inclusion problem involving a maximally monotone operator, a monotone and Lipschitz continuous operator, linear compositions of parallel-sum type monotone operators as well as the normal cone to the set of zeros of another monotone and Lipschitz continuous operator. We propose a forward–backward–forward type algorithm for solving it that assumes an individual evaluation of each operator. Weak ergodic convergence of the sequence of iterates generated by the algorithmic scheme is guaranteed under a condition formulated in terms of the Fitzpatrick function associated to one of the monotone and Lipschitz continuous operators. We also discuss how the proposed penalty scheme can be applied to convex minimization problems and present some numerical experiments in TV-based image inpainting.

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Acknowledgements

The authors are thankful to the anonymous reviewers for their recommendations which improved the quality of the paper.

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Correspondence to Radu Ioan Boţ.

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Dedicated to Professor Boris Mordukhovich on the occasion of his 65th birthday.

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

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Boţ, R.I., Csetnek, E.R. A Tseng’s Type Penalty Scheme for Solving Inclusion Problems Involving Linearly Composed and Parallel-Sum Type Monotone Operators. Vietnam J. Math. 42, 451–465 (2014). https://doi.org/10.1007/s10013-013-0050-2

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  • DOI: https://doi.org/10.1007/s10013-013-0050-2

Keywords

  • Maximally monotone operator
  • Fitzpatrick function
  • Resolvent
  • Lipschitz continuous operator
  • Parallel-sum
  • Forward–backward–forward algorithm
  • Subdifferential
  • Fenchel conjugate
  • Infimal-convolution
  • Convex minimization problem

Mathematics Subject Classification (2010)

  • 47H05
  • 65K05
  • 90C25