Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems

Abstract

The purpose of this paper is to introduce and study the bi-level split fixed point problems in the setting of infinite-dimensional Hilbert spaces. For solving this kind problems, some new simultaneous iterative algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in the paper to study bi-level split equilibrium problem, bi-level split optimization problems and the bi-level split variational inequality problems. The results presented in the paper are new which also extend and improve many recent results. c 2016 All rights reserved.Keywords: Bi-level split fixed point problem, bi-level split equilibrium problem, bi-level split optimization problem, bi-level split variational inequality problem, split feasibility problem. 2010 MSC: 47J25, 47H09, 65K10.

Proximal Gradient Method for Solving Bilevel Optimization Problems

“…Since Γ n − Γ n+1 → 0 (Γ n−1 − Γ n → 0) and using (C5), (C6), and Remark 1 (noting α n → 0, θ n x n − x n−1 → 0, {x n } is bounded and lim inf n→∞ ρ n (4 − ρ n ) > 0); we have, from (24) and 25,…”

“…The problem is also called the leader's and follower's problem where the problem (5) is called the leader's problem and (6) is called the follower's problem, meaning, the first player (which is called the leader) makes his selection first and communicates it to the second player (the so-called follower). There are many studies for several type bilevel problems, see, for example, [15,[17][18][19][20][21][22][23][24]. The bilevel optimization problem is a bilevel problem when the hierarchical structure involves the optimization problem.…”