Equilibrium Problem Research Articles

Approximate Benson properly efficient solutions for set-valued equilibrium problems

Approximate Benson properly efficient solutions for set-valued equilibrium problems

Modified inertial subgradient extragradient with auxiliary parameters and parallel viscosity algorithm for minimization problem induced by bounded linear operator over common solution of fixed points of nonexpansive mappings and pseudomonotone equilibrium problems

Modified inertial subgradient extragradient with auxiliary parameters and parallel viscosity algorithm for minimization problem induced by bounded linear operator over common solution of fixed points of nonexpansive mappings and pseudomonotone equilibrium problems

Vector Equilibrium Problems—A Unified Approach and Applications

We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some “interiority” properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given.

An approach of equilibrium problems based on selection theorems

In the study of the existence of solutions to equilibrium problems, the intersection theorems are useful tools of investigation. In this paper, the proofs combine selection results with intersection theorems. An important feature of our results is that they are obtained without the standard equilibrium condition.

Tseng-type splitting projection algorithms for equilibrium problems in Hilbert spaces

In this work, based on the subgradient-type projection method considered by Santos and Scheimberg (Optimization, 2017), we propose some Tseng-type splitting algorithms for solving equilibrium problems whose bifunction is decomposed into the sum of two bifunctions in Hilbert spaces. We establish the weak and strong convergence of the proposed algorithm under standard assumptions. Numerical results in Nash–Cournot oligopolistic electricity market equilibrium model and comparisons with other related methods demonstrate the effectiveness of the proposed algorithm.

Jump–diffusion productivity models in equilibrium problems with heterogeneous agents

In this paper we adopt a rational expectations framework to formulate general equilibrium models with heterogeneous agents. The productivity dynamics are characterized by a jump–diffusion model, thus allowing to account for sudden and impactful events. The modelling approach utilizes Hamilton–Jacobi–Bellman (HJB) formulations to represent the endogenous decision-making of firms to remain or exit the industry. When firms decide to exit, they are instantaneously replaced by new entrants. This dynamic leads to the development of a probability density function for firms, which satisfies a Kolmogorov–Fokker–Planck (KFP) equation with a source term. Both HJB and KFP formulations involve partial-integro differential operators due to the presence of jumps. Equilibrium models are completed with the household problem and feasibility conditions. Since (semi-)analytical solutions are not available, a numerical methodology is considered. This approach involves a Crank–Nicolson scheme for the time discretization, an augmented Lagrangian active set method and a finite difference discretization for the HJB formulation, and an appropriate finite difference method for the KFP problem. Moreover, Adams–Bashforth schemes are employed to handle integral terms explicitly. For the global equilibrium problem, we introduce a Steffensen algorithm. Numerical examples are provided to showcase the performance of our proposed numerical methodologies and to illustrate the expected behaviour of computed economic variables.

Non-additive network pricing with non-cooperative mobility service providers

This study addresses a mobility network pricing problem in a competitive environment. We consider a multimodal transportation network where the links are operated by multiple profit-maximizing, mobility service providers (MSPs). We take the perspective of a network regulator that aims to increase ridership in a target mobility network by providing non-additive, path-based subsidies to travelers. We model paths’ attractiveness using generalized cost functions that combine path travel time and path cost, and we use linear elastic travel demand functions to capture the proportion of demand served by a path. MSPs are non-cooperative and adjust link fares according to the subsidy policy implemented by the regulator. The goal of the network regulator is to solve a budget-constrained mobility network pricing problem under MSP competition. This game-theoretical framework is modeled as a single-leader multi-follower game (SLMFG) wherein the leader player represents the network regulator and multiple follower players represent the MSPs. We conduct a theoretical analysis of this SLMFG by identifying necessary and sufficient conditions for the existence of solutions to the parameterized generalized Nash equilibrium problem (GNEP) that is played amongst MSPs. We show that this GNEP is jointly convex and we use this property to develop an exact numerical approach to solve the SLMFG based on customized branch-and-bound algorithms. Numerical results reveal the impact of MSP competition in this mobility network pricing problem and shed novel insights into the design of optimal path-based subsidy policies.

A note on the Finite intersection property in “Existence of Nash equilibria for generalized multiobjective games through the vector extension of Weierstrass and Berge maximum theorems”

This note qualifies some results in Cotrina and Flores-Bazán (2024) on the ‘Finite intersection property’. We remind the essential role of this property for proving our results. In this note, we recall the statement of this property. We then specify the restrictive conditions that should be provided. An example proves the interest of this note revising some points in Cotrina and Flores-Bazán. Furthermore, it is showed that a class of generalized Nash equilibrium problems can be viewed as a particular case of a vector optimization problem, whose vector-valued function always possesses the Finite intersection property on its natural domain. Thus, the existence of a generalized Nash equilibrium will be a consequence of our Berge-type theorem and the Kakutani fixed point theorem, being it more general than those existing in recent literature, as shown by our examples.

Electricity market equilibria analysis on the value of demand-side flexibility portfolios’ mix and the strategic demand aggregators’ market power

Electricity market equilibrium analysis is becoming increasingly important within the today’s liberalized electricity market and regulatory context for both market participants and policy makers. The former ones want to make informed business decisions to optimize their market position, while the latter need to promote efficient equilibria that satisfy both supply and demand-side requirements and optimize social welfare. In this paper, we focus on modeling the strategic demand aggregators’ (DA) behavior via an Equilibrium Problem with Equilibrium Constraints (EPEC) formulation. We model typical demand flexibility portfolios that comprise of several types of distributed flexibility assets and renewable energy resources. We then quantify the value of demand flexibility portfolios’ mix and respective value of market power that a strategic DA may exercise with respect to each DA’s cost decrease and social welfare as a function of time for a day-ahead market use case. Simulation results show that: (i) there is an optimal equilibrium point where overall DAs’ costs decrease, while social welfare’s deteriorates negligibly, (ii) there may be a trade-off between the rate of a certain DA’s cost decrease and the rate of market power (or else % share of total demand flexibility) that this DA possesses, so the strategic DA should take this into account in order to find its optimal CAPEX/OPEX balance for its portfolio, (iii) competition among strategic DAs at the demand-side can mitigate grid congestion problems and serve as a counter-balance to the strategic behavior of the supply-side market participants.

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.

Approximate weak optimality conditions in multiobjective generalized Nash equilibrium problems

A multiobjective generalized Nash equilibrium problem (MGNEP) is a Nash equilibrium problem with constraints that include multiobjective games. We focus in this paper on examining the approximate Karush–Kuhn–Tucker (KKT) conditions for MGNEPs and their impact on the global convergence of algorithms. To begin, we define standard approximate weak KKT (standard-AWKKT) conditions for MGNEPs. We demonstrate that every weak efficient solution meets the standard-AWKKT condition. As these optimality conditions are strong, we propose a new set of approximate weak KKT conditions, specifically tailored for MGNEPs, and show their convergence towards weak KKT points provided a cone-continuity property is satisfied. It is important to note that while these newly introduced approximate weak KKT conditions do not serve as optimality conditions for general MGNEPs, we illustrate that they hold true for the special case of a weak variational equilibrium point in a jointly convex MGNEP. Finally, we propose an enhanced Lagrangian-type algorithm for estimating an approximate weak KKT of an MGNEP and demonstrate its global convergence.

Multiscale modelling of composite laminates with voids through the direct FE2 method

A concurrent multiscale method, namely the direct FE2, is developed to analyse the mechanical response of fibre-reinforced composites with several void fractions in a nested manner, one at the macro and another one at the microscale. The FE2 is developed through the finite element (FE) mesh at the macro scale, wherein there is a representative volume element (RVE) at every Gauss point where the strains and stresses are evaluated. The RVEs, located at macroscopic FE mesh, with linear boundary conditions applied to the boundary nodes, are linked to macro-scale nodes via Multi-Point Constraints (MPCs). This procedure enables solving a single equilibrium problem, with boundary conditions and loadings at the macro scale and provides data for all defined RVEs. The methodology is applied to a composite laminate under transverse tension. Initially, the method is evaluated to compare the results from the FE2 with those from analyses where the entire domain is discretised (conventional FE full-scale modelling). The stresses obtained from the FE2 method match traditional FE analyses but utilise a significantly lower number of finite elements, resulting in substantial computational cost savings. It follows a parametric study of void distribution effects within the RVE. Afterwards, the stress distribution within the RVE is assessed and then failure of the composites is predicted at the macro scale through the stresses and strains concurrently calculated at the micro-scale.

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection technique for determining an approximate common solution to three significant problems. The first is the system of generalized mixed equilibrium problems with relaxed monotone mappings, the second is the problem of fixed points of a countable family of equally continuous and asymptotically demicontractive mappings in the intermediate sense, and the third is of determining a point in a null space of a countable family of inverse strongly monotone mappings in Hilbert space. Based on these problems, we formulate a theorem and establish its strong convergence to their common solution. Additionally, we studied the applications of our algorithm to variational inequality problems and convex optimization problems. Finally, we numerically demonstrate the efficiency and robustness of our scheme. Several results available in the literature can be obtained as special cases of our result.

Strict Vector Equilibrium Problems of Multi-Product Supply–Demand Networks with Capacity Constraints and Uncertain Demands

This paper considers a multi-product, multi-criteria supply–demand network equilibrium model with capacity constraints and uncertain demands. Strict network equilibrium principles are proposed both in the case of a single criterion and multi-criteria, respectively. Based on a single criterion, it proves that strict network equilibrium flows are equivalent to vector variational inequalities, and the existence of strict network equilibrium flows is derived by virtue of the Fan–Browder fixed point theorem. Based on multi-criteria, the scalarization of strict network equilibrium flows is given by using Gerstewitz’s function without any convexity assumptions. Meanwhile, the necessary and sufficient conditions of strict network equilibrium flows are derived in terms of vector variational inequalities. Finally, an example is given to illustrate the application of the derived theoretical results.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty

Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty

Hybrid Inertial Iterative Method for Fixed point, Variational Inequality and Generalized Mixed Equilibrium Problems in Banach Space

In this paper, we introduced a hybrid inertial iterative method which converges strongly to a common element of solution of generalized mixed equilibrium, variational inequality and fixed point problems in a two uniformly smooth and uniformly convex Banach space. Our hybrid inertial iterative method, techniques of proof and corollaries improves, extends and generalizes many results in the literature.

Finite termination of the optimal solution sequence in parametric optimization

This paper investigates the finite termination of the optimal solution sequence in parametric optimization as a subproblem of the equilibrium problems. By introducing an augmented mapping on the solution set of the equilibrium problems, we establish the concept of augmented weak sharpness for this set relative to the sequence of optimal solutions in parametric optimization. Augmented weak sharpness is an extension of the concepts of weak sharpness and strongly non-degeneracy. Under the condition that the solution set of the equilibrium problems is augmented weak sharpness, sufficient conditions for the finite termination of the sequence of optimal solutions in parametric optimization are provided. For some special cases of the equilibrium problems (Mathematical Programming (MP), Variational Inequality Problems (VIP), etc.), these results extend the corresponding results under weak sharpness or strongly non-degeneracy conditions in the existing literature. Additionally, they also provide weaker sufficient conditions for the finite termination of many optimization algorithms.

Distributed Nash Equilibrium Seeking and Disturbance Rejection for High-Order Integrators Over Jointly Strongly Connected Switching Networks.

In this article, we study the problem of Nash equilibrium (NE) seeking of N -player games with high-order integrator agents over jointly strongly connected networks. The same problem was studied recently over static, undirected, and connected networks. Since a jointly strongly connected network can be directed and disconnected at every time instant, our result strictly includes the existing results as special cases. Moreover, in addition to some analytic conditions on the payoff functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. By adopting a different approach, our result does not rely on the satisfaction of this inequality. Furthermore, our result can also be extended to the case where the additive disturbances are imposed on the input of each agent. Our design is illustrated by two numerical examples.

Communication‐resilient and convergence‐fast peer‐to‐peer energy trading scheme in a fully decentralized framework

AbstractThe wide deployment of distributed energy resources, combined with a more proactive demand‐side management, is boosting the emergence of the peer‐to‐peer market. In the present study, an innovative peer‐to‐peer energy trading model is introduced, enabling a group of price‐setting prosumers to engage in direct negotiations via a straightforward best‐response approach. A Nash equilibrium problem (NEP) is initially formulated and a sufficient condition for the unique solution of the NEP is derived. Afterwards, an asynchronous and convergence‐fast solving method is employed to determine the trading quantity and price. The efficiency and resilience of the presented method are demonstrated through a comprehensive case study.