Bilevel Equilibrium Problems Research Articles

Forward–backward–forward dynamics for bilevel equilibrium problem

Forward–backward–forward dynamics for bilevel equilibrium problem

Inertial subgradient projection techniques for solving a class of bilevel equilibrium problems

ABSTRACT In this paper, we introduce a new algorithm for solving bilevel equilibrium problems with nonexpansive and pseudocontractive mappings in a real Hilbert space. The proposed algorithm is based on subgradient method with Armijo-type linesearch process, subgradient method, and inertial projection techniques. By choosing the suitable regular parameters, we prove strong convergence of the algorithm under classical conditions of the cost bifunctions.

Convergence of inertial prox-penalization and inertial forward–backward algorithms for solving monotone bilevel equilibrium problems

The main focus of this paper is on bilevel optimization on Hilbert spaces involving two monotone equilibrium bifunctions. We present a new achievement consisting on the introduction of inertial methods for solving these types of problems. Indeed, two several inertial type methods are suggested: a proximal algorithm and a forward–backward one. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the both iterative methods are established. Two particular cases illustrating the proposed methods are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under a saddle point constraint. Furthermore, numerical examples are given to demonstrate the implementability of our algorithms. The algorithms and their convergence results improve and develop previous results in the field.

Inexact simultaneous projection method for solving bilevel equilibrium problems

Inexact simultaneous projection method for solving bilevel equilibrium problems

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

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

One-Step iterative method for bilevel equilibrium problem in Hilbert space

One-Step iterative method for bilevel equilibrium problem in Hilbert space

Two New Weak Convergence Algorithms for Solving Bilevel Pseudomonotone Equilibrium Problem in Hilbert Space

In this paper, we introduce two new subgradient extragradient algorithms to find the solution of a bilevel equilibrium problem in which the pseudomonotone and Lipschitz‐type continuous bifunctions are involved in a real Hilbert space. The first method needs the prior knowledge of the Lipschitz constants of the bifunctions while the second method uses a self‐adaptive process to deal with the unknown knowledge of the Lipschitz constant of the bifunctions. The weak convergence of the proposed algorithms is proved under some simple conditions on the input parameters. Our algorithms are very different from the existing related results in the literature. Finally, some numerical experiments are presented to illustrate the performance of the proposed algorithms and to compare them with other related methods.

A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

On iterative methods for bilevel equilibrium problems

We use the notion of Halpern-type sequence recently introduced by the present authors to conclude two strong convergence theorems for solving the bilevel equilibrium problems proposed by Yuying et al. and some authors. Our result excludes some assumptions as were the cases in their results.

Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

New Explicit Extragradient Methods for Solving a Class of Bilevel Equilibrium Problems

The paper proposes a new explicit extragradient algorithm for solving an equilibrium problem over the solution set of another generalized equilibrium problem which usually is called bilevel equilibrium problems in a real Hilbert space. The algorithm combines extragradient methods with auxiliary problem techniques. Theorem of the strong convergence of the algorithm is established under standard assumptions imposed on the cost bifunctions. Finally, several of fundamental experiments are provided to illustrate the numerical behavior of the algorithm and also to compare with others.

Levitin–Polyak well-posedness of generalized bilevel equilibrium problem with perturbations

In this paper, a new class of problem known as the generalized bilevel mixed equilibrium problem with perturbations (GBMEPP) has been introduced on a real Banach space. Based on the diameter and the measure of non-compactness of the approximate solution set of the above stated problem, criteria and characterizations have been derived for Levitin–Polyak (LP) well-posedness. A sufficient condition for the LP well-posedness has also been derived assuming the boundedness of the approximate solution set.

An inertial extragradient method for solving bilevel equilibrium problems

In this paper, we propose an algorithm with two inertial term extrapolation steps for solving bilevel equilibrium problem in a real Hilbert space. The inertial term extrapolation step is introduced to speed up the rate of convergence of the iteration process. 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 proposed algorithm. A numerical experiment is performed to illustrate the numerical behavior of the algorithm and also comparison with some other related algorithms in the literature.

Subgradient projection methods extended to monotone bilevel equilibrium problems in Hilbert spaces

In this paper, by basing on the inexact subgradient and projection methods presented by Santos et al. (Comput. Appl. Math. 30: 91–107, 2011), we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms.

Bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces

In this paper, we introduce a new class of bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces and establish some conditions for the existence of solutions to these problems using the Kakutani-Fan-Glicksberg fixed-point theorem. Then, we establish generic stability of set-valued mappings and we show the set of essential points of a map is a dense residual subset of a (Hausdorff) metric space of set-valued maps for bilevel equilibrium problems with lower and upper bounds. The results presented in the paper are new and extend the main results given by some authors in the literature.

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

ABSTRACTIn this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.

Stability of approximating solutions to parametric bilevel vector equilibrium problems and applications

In this paper, we establish sufficient conditions for the approximate solution mappings of parametric bilevel equilibrium problems with stability properties such as upper semicontinuity, lower semicontinuity, Hausdorff lower semicontinuity, continuity and Hausdorff continuity. Moreover, we also apply these results to parametric traffic network problems with equilibrium constraints. Many examples are provided to ensure the essentialness of the assumptions.

A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications

In this paper, we propose a new algorithm for solving a bilevel equilibrium problem in a real Hilbert space. In contrast to most other projection-type algorithms, which require to solve subproblems at each iteration, the subgradient method proposed in this paper requires only to calculate, at each iteration, two subgradients of convex functions and one projection onto a convex set. Hence, our algorithm has a low computational cost. We prove a strong convergence theorem for the proposed algorithm and apply it for solving the equilibrium problem over the fixed point set of a nonexpansive mapping. Some numerical experiments and comparisons are given to illustrate our results. Also, an application to Nash–Cournot equilibrium models of a semioligopolistic market is presented.

Market Equilibrium in Active Distribution System With $\mu $ VPPs: A Coevolutionary Approach

To further liberalize the retail electricity market, this paper establishes a novel active distribution system market (ADSM) to encourage the new entries of micro virtual power plants ( $\mu $ VPPs) as prosumers. $\mu $ VPPs compete with traditional retailers by submitting price–quantity bids/offers for energy and reserve resources. The joint operation of energy and reserve market is modeled as a bilevel equilibrium problem with equilibrium constraints (EPEC) with an upper-level objective that maximizes all $\mu $ VPPs’ profit and a lower level objective that maximizes social welfare of the market clearing process. A coevolutionary approach is successfully employed to determine the pure strategy Nash equilibrium of the EPEC model. The case studies demonstrate the effectiveness of the coevolutionary method and show that $\mu $ VPPs’ bidding/offering strategies depend significantly on the penetration level of distributed energy resources and renewable energy sources and they can be considerably influenced by rivals’ strategies. This paper then compares $\mu $ VPPs’ performances under different market structures and addresses the advantages of the proposed ADSM in terms of higher asset utilization rate, higher economic profit, and more secured return on investment.

A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.