TY - JOUR
T1 - Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces
AU - Vuong, Phan Tu
AU - Strodiot, Jean-Jacques
AU - Nguyen, Van Hien
PY - 2013/6/19
Y1 - 2013/6/19
N2 - 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.
AB - 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.
KW - Equilibrium problem
KW - Projected subgradient method
KW - Strong convergence
KW - Variational inequality
KW - Viscosity approximation
UR - http://www.scopus.com/inward/record.url?scp=84878943525&partnerID=8YFLogxK
U2 - 10.1007/s10898-013-0084-8
DO - 10.1007/s10898-013-0084-8
M3 - Article
SN - 0925-5001
SP - 1
EP - 18
JO - Journal of Global Optimization
JF - Journal of Global Optimization
ER -