Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space

Abstract

By using our own approach, we study the strong convergence of an inexact proximal point algorithm with possible unbounded errors for a maximal monotone operator in a Banach space. We give a necessary and sufficient condition for the zero set of the operator to be nonempty and show that, in this case, this iterative sequence converges strongly to a zero of the operator. We present also some applications of our results to equilibrium problems and optimization. Our proximal point algorithm contains, as a special case, the one considered in Hilbert space by Djafari Rouhani and Moradi in (J Optim Theory Appl 172:222–235, 2017) and solves the open problem of extending it to a Banach space, which was stated in that paper and in Djafari Rouhani and Moradi in (J Optim Theory Appl 181:864–882, 2019) . Since the nonexpansiveness of the resolvent operator, which holds in Hilbert space, is not valid anymore in Banach space, our results require new methods of proofs, and significantly improve upon the previous results, both in theory and in applications.

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Acknowledgements

The authors are grateful to the Editor and the referees for their constructive comments leading to the improvement of the paper. This work is done while the second author was visiting the University of Texas at El Paso. The second author would like to thank Professor Djafari Rouhani and the Department of Mathematical Sciences for their kind hospitality at the University of Texas at El Paso during his visit.

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Correspondence to Behzad Djafari Rouhani.

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Communicated by Akhtar A. Khan.

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Djafari Rouhani, B., Mohebbi, V. Strong Convergence of an Inexact Proximal Point Algorithm in a Banach Space. J Optim Theory Appl 186, 134–147 (2020). https://doi.org/10.1007/s10957-020-01695-w

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Keywords

  • Equilibrium problems
  • Maximal monotone operator
  • Proximal point method
  • Resolvent
  • Strong convergence

Mathematics Subject Classification

  • 47H10
  • 47H05
  • 47J25