An extragradient algorithm for solving bilevel pseudomonotone variational inequalities

Abstract

We present an extragradient-type algorithm for solving bilevel pseudomonone variational inequalities. The proposed algorithm uses simple projection sequences. Under mild conditions, the convergence of the iteration sequences generated by the algorithm is obtained.

Composite relaxed extragradient method for triple hierarchical variational inequalities with constraints of systems of variational inequalities

“…Under some suitable conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results announced by some others, e.g., Iiduka [14], Zeng et al [33], Anh et al [1], and Yao et al [27].…”

“…Note that Anh et al [1] studied the above BVIP with H = R n . BVIP includes the classes of mathematical programs with equilibrium constraints ( [18]), bilevel minimization problems ( [23]), variational inequalities ( [3,31,32]) and complementarity problems as special cases.…”

“…In 2012, Anh et al [1] introduced the following extragradient iterative algorithm for solving the above bilevel variational inequality.…”

Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

“…The EP includes, as special cases, many mathematical models such as: variational inequality problems, fixed point problems, optimization problems, Nash equilirium problems, complementarity problems, etc., see [8,21] and the references therein. In recent years, many algorithms have been proposed for solving EPs [1,2,3,4,5,6,13,16,17,18,20,25,28]. In the case, the bifunction f is monotone, solution approximations of EPs are based on a regularization equilibrium problem, i.e., at the step n, known x n , the next approximation x n+1 is the solution of the following problem: (2) Find x ∈ C such that: f (x, y) + 1 r n y − x, x − x n ≥ 0, ∀y ∈ C, where r n is a suitable parameter.…”

A hybrid extragradient method for bilevel pseudomonotone variational inequalities with multiple solutions

“…In particular, whenever H = R n , the BVIP was recently studied by Anh, Kim and Muu [1]. Bilevel variational inequalities are special classes of quasivariational inequalities (see [2,3,14,31]) and of equilibrium with equilibrium constraints considered in [19,24].…”