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Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces

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Abstract

In this paper, we introduce and study some low computational cost numerical methods for finding a solution of a variational inequality problem over the solution set of an equilibrium problem in a real Hilbert space. The strong convergence of the iterative sequences generated by the proposed algorithms is obtained by combining viscosity-type approximations with projected subgradient techniques. First a general scheme is proposed, and afterwards two practical realizations of it are studied depending on the characteristics of the feasible set. When this set is described by convex inequalities, the projections onto the feasible set are replaced by projections onto half-spaces with the consequence that most iterates are outside the feasible domain. On the other hand, when the projections onto the feasible set can be easily computed, the method generates feasible points and can be considered as a generalization of Maingé’s method to equilibrium problem constraints. In both cases, the strong convergence of the sequences generated by the proposed algorithms is proven.

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Authors and Affiliations

  1. Institute for Computational Science and Technology (ICST), Ho Chi Minh, Vietnam

    Phan Tu Vuong, Jean Jacques Strodiot & Van Hien Nguyen

  2. University of Namur, Namur, Belgium

    Jean Jacques Strodiot & Van Hien Nguyen

Authors
  1. Phan Tu Vuong
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  2. Jean Jacques Strodiot
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  3. Van Hien Nguyen
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Correspondence to Jean Jacques Strodiot.

Additional information

This research is funded by the Department of Science and Technology at Ho Chi Minh City. Computing resources and support provided by the Institute for Computational Science and Technology at Ho Chi Minh City is gratefully acknowledged.

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Vuong, P.T., Strodiot, J.J. & Nguyen, V.H. Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces. J Glob Optim 59, 173–190 (2014). https://doi.org/10.1007/s10898-013-0084-8

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  • DOI: https://doi.org/10.1007/s10898-013-0084-8

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