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Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space

Abstract

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided.

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Acknowledgments

All the authors are grateful to an anonymous referee for several helpful comments and useful suggestions

Funding

The first author was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.

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Correspondence to Simeon Reich or Duong Viet Thong.

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Cite this article

Reich, S., Thong, D.V., Cholamjiak, P. et al. Inertial projection-type methods for solving pseudomonotone variational inequality problems in Hilbert space. Numer Algor 88, 813–835 (2021). https://doi.org/10.1007/s11075-020-01058-6

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Keywords

  • Inertial method
  • Non-Lipschitz continuity
  • Pseudomonotone mapping
  • Tseng’s extragradient method
  • Variational inequality
  • Weak convergence

Mathematics subject classification (2010)

  • 47H09
  • 47H10
  • 47J20
  • 47J25
  • 65Y05
  • 65K15
  • 68W10