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

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  • Phan Vuong
  • Jean Strodiot
  • Van Nguyen

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. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Phan Vuong & Jean Strodiot & Van Nguyen, 2014. "Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces," Journal of Global Optimization, Springer, vol. 59(1), pages 173-190, May.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:1:p:173-190
    DOI: 10.1007/s10898-013-0084-8
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    4. H. Iiduka, 2009. "Hybrid Conjugate Gradient Method for a Convex Optimization Problem over the Fixed-Point Set of a Nonexpansive Mapping," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 463-475, March.
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    Cited by:

    1. Jean Strodiot & Phan Vuong & Thi Nguyen, 2016. "A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 64(1), pages 159-178, January.
    2. Le Quang Thuy & Trinh Ngoc Hai, 2017. "A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 411-431, November.
    3. Pham Ky Anh & Trinh Ngoc Hai, 2019. "Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications," Journal of Global Optimization, Springer, vol. 73(3), pages 637-657, March.

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