Inertial Method for Bilevel Variational Inequality Problems with Fixed Point and Minimizer Point Constraints

Abstract

In this paper, we introduce an iterative scheme with inertial effect using Mann iterative scheme and gradient-projection for solving the bilevel variational inequality problem over the intersection of the set of common fixed points of a finite number of nonexpansive mappings and the set of solution points of the constrained optimization problem. Under some mild conditions we obtain strong convergence of the proposed algorithm. Two examples of the proposed bilevel variational inequality problem are also shown through numerical results.

Proximal Gradient Method for Solving Bilevel Optimization Problems

“…The result (i) follows from (14) and (15), and, in view of (C2)-(C6), the result (ii) follows from (14) and (15).…”

“…A bilevel problem is a two-level hierarchical problem such that the solution of the lower level problem determines the feasible space of the upper level problem. In general, Yimer et al [15] presented a bilevel problem as an archetypal model given by findx ∈ S ⊂ X that solves problem P1 installed in a space X,…”

“…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.…”

An inertial parallel CQ subgradient extragradient method for variational inequalities application to signal-image recovery

“…However, the convergence of this method requires slightly strong assumptions that operators are strongly monotone or inverse strongly monotone. Many algorithms have been proposed and studied for solving VIP(1) of these algorithms involve projection methods [5,6,10,11,39,40,43,46,47,51]. The VIP(1) serves as a powerful mathematical tool and generalizes many mathematical methods, in the sense that, it includes many special problems [29] such as convex feasibility problems, linear programming problem, minimizer problem, saddle -point problems, Hierarchical variational inequality problems.…”