New extragradient-like algorithms for strongly pseudomonotone variational inequalities

Abstract

The paper considers two extragradient-like algorithms for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily than the regularized method. The construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of the cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with those of several previously known algorithms.

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Acknowledgements

The authors would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.315. The second author was partially funded by NAFOSTED under Grant No. 101.02-2017.15 and by Vietnam Institute for Advanced Study in Mathematics (VIASM).

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Correspondence to Dang Van Hieu.

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Van Hieu, D., Thong, D.V. New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J Glob Optim 70, 385–399 (2018). https://doi.org/10.1007/s10898-017-0564-3

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Keywords

  • Variational inequality problem
  • Monotone operator
  • Pseudomonotone operator
  • Strongly monotone operator
  • Strongly pseudomonotone operator
  • Extragradient method
  • Subgradient extragradient method
  • Projection method

Mathematics Subject Classification

  • 65Y05
  • 65K15
  • 68W10
  • 47H05
  • 47H10