Extragradient subgradient methods for solving bilevel equilibrium problems

Abstract

In this paper, we propose two algorithms for finding the solution of a bilevel equilibrium problem in a real Hilbert space. Under some sufficient assumptions on the bifunctions involving pseudomonotone and Lipschitz-type conditions, we obtain the strong convergence of the iterative sequence generated by the first algorithm. Furthermore, the strong convergence of the sequence generated by the second algorithm is obtained without a Lipschitz-type condition.

A self-adaptive inertial subgradient extragradient algorithm for solving bilevel equilibrium problems

“…It is clear that f satisfies condition (A1) -(A4); see, e.g. [59]. We choose the following parameters and compare the performance of our Algorithm 3.2 with Algorithms 1.2 and 1.3 respectively.…”

“…Recently, Yujing et al [59] introduced an extragradient method for solving the BEP when f is strongly monotone and g satisfies pseudomonotone and Lipschitz-like condition as follows:…”

“…Moreover, the stepsize λ n of Algorithm 1.2 depends on the prior estimate of the Lipschitz-like constants L 1 and L 2 which are very difficult to determine. In order to improve the efficiency of Algorithm 1.2, the authors in [59] also introduced the following extragradient method with line search:…”

Proximal Gradient Method for Solving Bilevel Optimization Problems

“…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.…”

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

“…Usually, (1) is called the upper level problem and (2) is called the lower level problem. Many real life problems can be modeled as a bilevel problem and some studies have been performed towards solving different kinds of bilevel problems using approximation theory-see, for example, for bilevel optimization problem [1][2][3], for bilevel variational inequality problem [4][5][6][7][8][9], for bilevel equilibrium problems [10][11][12], and [13,14] for its practical applications. In [14], application of bilevel problem (bilevel optimization problem) in transportation (network design, optimal pricing), economics (Stackelberg games, principal-agent problem, taxation, policy decisions), management (network facility location, coordination of multi-divisional firms), engineering (optimal design, optimal chemical equilibria), etc.…”