Equilibrium Problem Research Articles (Page 1)

In this paper, we study the split mixed equilibrium problem which includes the equilibrium problem, split convex minimization problem and split equilibrium problem, to mention a few. In addition, we propose a Halpern iterative method for solving split mixed equilibrium problem with multiple output sets and fixed point of a finite family of multi-valued strictly pseudo-contractive mappings in the framework of real Hilbert spaces. We prove a strong convergence theorem without imposing any compactness condition. Lastly, we present some consequences and give applications of our main result to split mixed variational inequality and split convex minimization problems. The result discussed in this article extends and complements many related results in literature.

In this paper, a forward–backward–forward method is employed for a dynamical system to solve the bilevel equilibrium problems over a real Hilbert space, and the convergence of the dynamic system's trajectory is established. Furthermore, quantitative stability analysis has been extensively studied for perturbed bilevel equilibrium problems by considering different metrics. As applications, stability analysis is derived for convex minimization problems and for optimal control problems where a variational inequality gives the state-control relation.

Inertial subgradient splitting projection methods for solving equilibrium problems and applications

Abstract This paper focuses on the study of an economic equilibrium problem for an electricity market model in a multistage-stochastic framework, where,stage by stage, the uncertainty evolves with continuity. We analyze the point of view of a finite number of power companies in a sequence of competitive markets.Each of them produces electricity, both with conventional and renewable-based plants, participates in the trade in the spot markets that open after the uncertainty is revealed, and signs bilateral and forward contracts. Moreover, we capture the risk attitude of each power company by considering a suitable coherent risk measure in the problem’s formulation. In order to prove the existence of at least one equilibrium solution, we introduce a suitable quasi-variational inequality formulation. In this light, we also investigate suitable regularity properties of the involved superdifferential operator in the presence of certain parameter perturbations in Banach spaces.

A new double relaxed inertial viscosity-type algorithm for split equilibrium problems and its application to detecting osteoporosis health problems

This paper considers a linear system of partial differential equations (PDEs) to describe the stress-strain state of a two-phase body under static load, such as water-saturated soil. It investigates the basic properties of a new general differential operator Lame. The equations differ from the classical Lame equations by including first derivatives, which account for the influence of pore water on soil mineral particles. The properties of the generalized Lamé operator are investigated for the application of variational methods to solve the problem. It also describes alternative of the Betti and Clapeyron formulas using strain energy results. The calculus of variations of the Galerkin method is used to solve the minimum functional problem. Properties of bilinear forms are established and a theorem on the existence and uniqueness of the solution of the two-phase equilibrium problem is proved. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The adequacy of the mathematical model of a water-saturated foundation for a natural experiment is shown. The use of Korn's inequality implies limitations on elastic properties (homogeneity, anisotropy) and the geometry of the region (requiring regularity and smooth boundaries). The study illustrates that the methodology of mechanics of a deformable solid can be adapted with appropriate modifications to a two-phase body in a stabilized state. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The finite element method is tested on the Flamand problem. The adequacy of the mathematical model of a water-saturated foundation for a full-scale experiment is shown. The problem of the action of distributed load on a water-saturated heterogeneous foundation was solved using the finite element method and the results were compared with experimental data. The effect of mesh partitioning on the accuracy of the numerical solution is also studied in the finite element method. The maximum discrepancy was no more than 26%.

Equilibrium problems with trifunctions and applications to hemivariational inequalities

The paper addresses the solution existence of equilibrium problem for a layered composite containing cracks and asymptotic behavior of solutions. Boundary conditions considered at the crack faces have an inequality type and describe a mutual nonpenetration and adhesion. At the external boundary of the composite, we impose the Neumann condition, which implies a non-coercivity of the boundary value problem. A solution existence of the problem is proved, and conditions imposed on the external forces guaranteeing the solution existence are found. An asymptotic analysis with respect to the parameter characterizing the elasticity tensor is provided, and limit models are analysed. In particular, the limit models describe the composite with a volume rigid inclusion and with a cavity.

Quasi-equilibrium problems generalize equilibrium problems, as introduced in the framework of Blum and Oettli. This study focuses on quasi-equilibrium problems in which the constraint map is not a self-map. We present various reformulations that establish existence results for projected solutions in Banach spaces. Furthermore, we apply these findings to quasi-variational inequalities and quasi-optimization problems.

A usual problem in chemical engineering and fuel processing is to achieve the highest possible efficiency concerning the target products. In this paper, we consider the inverse problem of chemical equilibrium and propose mathematical methods to obtain conditions under which the equilibrium state of the reacting system achieves the required characteristics. For the case of maximising the aim component yield, a new two-step algorithm is developed based on the inverse problem solution. The methods are tested using the methane reforming example.

Proximal methods for equilibrium problems over the set of fixed points of enriched nonexpansive mappings

ABSTRACTIn this article, we proposed enhanced extragradient methods for solving equilibrium problems in Hilbert spaces. The constraint set is defined as the solution set for a fixed‐point problem with demicontractive mappings. The proposed methodologies center on a subgradient extragradient approach that combines an inertial technique, a self‐adaptive procedure, and a viscosity approximation scheme. These algorithms use variable step sizes that are dynamically updated in accordance with previous iteration outcomes. A key advantage of the proposed methods is their ability to function eliminating the need for prior knowledge of Lipschitz constants or line search techniques. Instead, the step sizes are determined through simple calculations at each iteration. The convergence analysis is established under relaxed assumptions. Furthermore, we present a numerical study that compares the effectiveness of the proposed techniques to existing approaches.

Solving equilibrium and fixed-point problems in Hilbert spaces: A class of strongly convergent Mann-type dual-inertial subgradient extragradient methods

An inertial iterative method for generalized mixed equilibrium problem and fixed point problem of nonexpansive semigroups

Sequential splitting algorithms with Bregman distance for solving equilibrium problems

In this paper, we study the generalized mixed equilibrium problem and the fixed point problem. We propose an inertial iterative method for approximating the common solution of a generalized mixed equilibrium problem of a monotone mapping and a fixed point problem for a Bregman strongly nonexpansive mapping in the framework of real reflexive Banach spaces. Under certain mild conditions, we obtain a strong convergence result of the proposed method. Finally, we present numerical examples to illustrate the applicability of our method.

An Armijo−Type viscosity algorithm for continuous equilibrium problems without monotonicity on Hadamard manifolds

Abstract We consider numerical aspects of finding classical J. Nash’s equilibrium in concave n-persons game, nonlinear equilibrium (NE), as an alternative to primal and dual linear programming (LP) problems, and recently introduced nonlinear production-consumption equilibrium (NPCE). The problems are particular cases of a general nonlinear equilibrium problem, which is equivalent to a variational inequality (VI). The corresponding VIs have simple feasible sets, that the projection on them is a low cost operation. Therefore, we apply two projection methods for finding the equilibrium: pseudo-gradient projection (PGP) and extra pseudo-gradient (EPG). We present and analyze results obtained on random generated sets of these three classes of problems. The obtained results show expected advantages of the EPG over PGP. What is most important: the number of iterations requited by EPG method to find an approximation for the equilibrium with a given accuracy grows linearly with the number of products in case of NE and NPCE, or with the number of active strategies in case of J. Nash’s equilibrium. The number of operations, or solution time grows as a cube of the corresponding parameters. These results corroborate the complexity bounds established in [18–20] under reasonable assumptions on the input data.

ABSTRACTIn this article, we propose a new iterative scheme to solve equilibrium problems in the framework of Banach spaces. This study presents an enhanced convergence framework through the development of a new algorithm that integrates a novel control sequence with an improved iterative scheme. We prove a theorem that converges strongly. Finally, we present a few optimal comparisons of numerical examples in real line and that support our claim of the superior convergence behavior of the proposed algorithm over the existing ones in the literature.

Optimizing the regional business environment plays a crucial role in improving the market supply structure, enhancing market dynamism, and boosting consumer welfare. Investigating how the government can effectively improve the business environment and promote consumer welfare through scientific and strategic investment allocation is a topic that warrants comprehensive and in-depth research. This paper proposes a bi-level programming model based on consumer welfare, with the upper-level model focusing on optimizing the government’s investment allocation strategy to maximize consumer welfare, and the lower-level model addressing the spatial price equilibrium problem after improving the business environment. The experimental results confirm the effectiveness and practicality of the proposed algorithm. The findings reveal that the bi-level programming model, integrating simulated annealing and projection algorithms, provides support for governments in accurately determining investment allocation strategies, enabling the simultaneous maximization of consumer welfare and optimization of the business environment. Additionally, increased government investment significantly improves both the business environment and consumer welfare, while appropriately managing the intensity of investment further enhances consumer welfare. This study offers valuable theoretical insights and practical guidance for governments to refine investment decisions, foster business environment development, and improve societal well-being.