Numerical Algorithms

, Volume 81, Issue 3, pp 983–1001 | Cite as

Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators

  • Pham Ky AnhEmail author
  • Nguyen The Vinh
Original Paper


In this paper, we introduce a self-adaptive inertial gradient projection algorithm for solving monotone or strongly pseudomonotone variational inequalities in real Hilbert spaces. The algorithm is designed such that the stepsizes are dynamically chosen and its convergence is guaranteed without the Lipschitz continuity and the paramonotonicity of the underlying operator. We will show that the proposed algorithm yields strong convergence without being combined with the hybrid/viscosity or linesearch methods. Our results improve and develop previously discussed gradient projection-type algorithms by Khanh and Vuong (J. Global Optim. 58, 341–350 2014).


Variational inequality Monotone operator Gradient projection algorithm Extragradient algorithm Subgradient extragradient algorithm Projected reflected gradient method Inertial-type algorithm 

Mathematics Subject Classification (2010)

47J20 90C25 


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The authors would like to thanks the editor and the referee for valuable remarks and helpful suggestions which improved the quality of the paper.

Funding information

The second named author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.08. His research is also partially supported by the Vietnam Institute for Advanced Study in Mathematics and by UTC under Grant T2019-CB-014.


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Authors and Affiliations

  1. 1.College of ScienceVietnam National UniversityThanh XuanVietnam
  2. 2.Department of MathematicsUniversity of Transport and CommunicationsHanoiVietnam

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