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

Abstract

In this paper, we introduce and analyze a hybrid extragradient algorithm for solving bilevel pseudomonotone variational inequalities with multiple solutions in a real Hilbert space. The proposed algorithm is based on Korpelevich's extragradient method, Mann's iteration method, hybrid steepest-descent method, and viscosity approximation method (including Halpern's iteration method). Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is derived.

Inexact simultaneous projection method for solving bilevel equilibrium problems

“…By choosing f (x, y) := F (x), y − x , g j (x, y) = G(x), y − x and S i = 0 for all i ∈ I, j ∈ J, (x, y) ∈ C × C, we can easily see that (1.3) is equivalent to (1.2). The simplest approach for solving this problem is projection method in which only two projections on the feasible set C is performed per each iteration such as the projection method of Xu et al in [32] for neural network models, the extragradient methods in [12,13,29,1,7,21].…”