Equilibrium Problem Research Articles

Weak set-equilibrium problems: existence and stability conditions

We formulate four types of set-equilibrium problems based on some well-known set-order relations. Then, we study the existence of solutions for these problems by imposing standard generalized convexity and Lipschitz conditions. A comparison with some widely studied vector equilibrium problems is performed. Finally, we are interested in an issue concerning the stability of solutions of some related approximate set-optimization problems.

A Novel Method for Solving Nonmonotone Equilibrium Problems

A Novel Method for Solving Nonmonotone Equilibrium Problems

Forward–backward–forward dynamics for bilevel equilibrium problem

Forward–backward–forward dynamics for bilevel equilibrium problem

Strong Convergence of an Iterative Method for Solving Generalized Mixed Equilibrium Problems and Split Feasibility Problems

In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.

Bidding Strategy for the Alliance of Prosumer Aggregators in the Distribution Market

Photovoltaic energy storage system (PV-ESS) prosumer aggregators are characterized by a large number but small scale in the distribution system and are not competitive enough to participate in market transactions. For this reason, a prosumer aggregator alliance is proposed to participate in the distribution market bidding strategy. Firstly, based on the framework for prosumer aggregator alliances participating in distribution market trading, a bilevel bidding model is constructed. The upper level represents the optimal decision-making model for the prosumer aggregators, while the lower level constitutes the distribution market-clearing model. Secondly, the additional benefits obtained by the alliance are distributed more fairly using the improved Shapley value based on the PV self-consumption rate. Given the problem that the traditional diagonalization algorithm (DA) has an excessive number of iterations when solving the game equilibrium problem of multiple subjects, the DA is improved by optimizing the initial value of the inputs. Finally, case studies are conducted based on the improved IEEE-33 bus distribution system to validate the feasibility and economic viability of the proposed strategy. The case study results show that forming cooperative alliances to participate in market bidding can significantly increase overall profits. The improved DA reduces the number of bids and computation time by 75% and 80%, respectively. Additionally, the improved Shapley value facilitates compensation for some of the aggregators.

Agricultural product harvest equilibrium with transportation bottleneck and random disasters

The decentralized planting practices of farmers often result in non-cooperative harvesting and transportation of agricultural products. The high concentration of harvesting demand can cause transportation and processing bottlenecks, resulting in queuing congestion in planting, which in turn affects farmers’ decisions on harvest timing. Thus, the fully competed decision-making process leads to a game equilibrium. To solve this harvest equilibrium problem, a nonlinear equation model with transportation bottleneck capacity constraints is developed. Assumptions to be made are that the value of the pre-harvest product is a non-negative continuously differentiable strictly concave function about the harvest time, and that post-harvest losses are a linear increasing function of queuing time. Subsequently, the model is further extended to incorporate the effects of random natural disasters. Additionally, this paper derives optimal harvest schedules for comparison. A numerical example is used to analyze how the related parameters and the probability of disasters influence farmers’ harvest decisions and the unit benefits of agricultural produce. The analysis results show that a larger total volume of agricultural products prompts earlier harvesting and reduces unit benefits in both equilibrium and optimal harvesting scenarios. Expanding bottleneck capacity allows farmers to harvest closer to the optimal time and increases unit benefits, although the marginal increase diminishes at the margin as bottleneck capacity increases. Increasing the queuing penalty within a range leads to an earlier harvest and reduces the unit benefits in equilibrium and optimality until a critical value is reached, beyond which it has no effect. A higher probability of natural disasters will incentivize farmers to harvest earlier, which in turn will reduce unit benefits.

New opportunities in produced water management: A market-based approach to produced water trading

Produced water (PW) is a byproduct of oil and gas (O&G) production. Obtained alongside the more valuable energy products, PW is usually characterized by high levels of salinity and often contains many contaminants (chemicals, soluble and insoluble oil, organics, etc.) making it unsuitable for release without substantial treatment. Couple this with the fact that PW is typically obtained at multiple times the rate of oil or gas, and the added transport, treatment, and disposal costs become a serious challenge for operators. These realities have led to ad-hoc practices including cooperation between industry competitors to recycle, share, or otherwise mitigate PW costs. The National Energy Technology Laboratory (NETL) in partnership with the Ground Water Protection Council (GWPC) is pursuing novel technology solutions to address PW issues that complement or improve ad-hoc practices adopted by operators. In this paper, we observe that well-established market management practices used in electrical power generation have natural analogues in the PW supply chain. These parallels open up a new line of research where we view PW management as a market equilibrium problem, and explore solutions that foster active and data-based collaboration among operators through market structures similar to power markets, with the ultimate objective of improving PW management costs and recycling rates. We make a case for our observations, present a PW market clearing optimization model that shows how such a market system could operate in the O&G space, and provide an illustrative case study for demonstration.

Multi-scale ceramic TiC solves the strength-plasticity equilibrium problem of high entropy alloy

High entropy alloys (HEAs) with single FCC structure exhibits unique structure and properties. However, lack of strength hinds the engineering application ability seriously. Therefore, it is urgent to propose effective methods and theories to enhance the strength while maintaining favorable plasticity. Introducing reinforced phases is one crucial research approach for improving the mechanical properties of HEAs. However, the resulting reduction in ductility has been neglected. In this work, we propose a novel multi-scale TiC coupling reinforced alloy which preserves the high plasticity while increasing the strength. The results demonstrate that TiC (μm) hinders the dislocations movement and improves strength through the second phase strengthening mechanism. Apart from acting as barriers to dislocation motion, the TiC (nm) also provides more dislocation sources in the distortion area at the junction with the matrix, which increasing the number of movable dislocations and enhancing the plastic strain capacity. Compared with the Al0.4 alloy, the tensile yield strength of the Al0.4-TiC (μm + nm) alloy is increased by 145 %, the ultimate tensile strength is up to 574 MPa, while maintaining a high plastic strain by 30.1 %. The addition of multi-scale ceramic phase TiC provides a novel approach to obtain high strength and high plasticity HEAs.

A nonlinear split problem regarding variational inequalities and equilibrium problems

Split problems under linear transformations have been studied extensively in the literature. In this paper, we investigate a split problem under a nonlinear transformation, which consists of two variational inequalities and an equilibrium problem. We propose an iterative sequence for computing a solution to this split problem. Strong convergence theorem is achieved under additional standard conditions.

Jacobian-free Newton-Krylov method for multilevel nonlocal thermal equilibrium radiative transfer problems

Context. The calculation of the emerging radiation from a model atmosphere requires knowledge of the emissivity and absorption coefficients, which are proportional to the atomic level population densities of the levels involved in each transition. Due to the intricate interdependence of the radiation field and the physical state of the atoms, iterative methods are required in order to calculate the atomic level population densities. A variety of different methods have been proposed to solve this problem, which is known as the nonlocal thermodynamical equilibrium (NLTE) problem. Aims. Our goal is to develop an efficient and rapidly converging method to solve the NLTE problem under the assumption of statistical equilibrium. In particular, we explore whether the Jacobian-Free Newton-Krylov (JFNK) method can be used. This method does not require an explicit construction of the Jacobian matrix because it estimates the new correction with the Krylov-subspace method. Methods. We implemented an NLTE radiative transfer code with overlapping bound-bound and bound-free transitions. This solved the statistical equilibrium equations using a JFNK method, assuming a depth-stratified plane-parallel atmosphere. As a reference, we also implemented the Rybicki & Hummer (1992) method based on linearization and operator splitting. Results. Our tests with the Fontenla, Avrett and Loeser C model atmosphere (FAL-C) and two different six-level Ca II and H I atoms show that the JFNK method can converge faster than our reference case by up to a factor 2. This number is evaluated in terms of the total number of evaluations of the formal solution of the radiative transfer equation for all frequencies and directions. This method can also reach a lower residual error compared to the reference case. Conclusions. The JFNK method we developed poses a new alternative to solving the NLTE problem. Because it is not based on operator splitting with a local approximate operator, it can improve the convergence of the NLTE problem in highly scattering cases. One major advantage of this method is that it is expected to allow for a direct implementation of more complex problems, such as overlapping transitions from different active atoms, charge conservation, or a more efficient treatment of partial redistribution, without having to explicitly linearize the equations.

Analysis of Students' Difficulties in Solving Physics Problems Using Polya's Theory

The ability of students in solving physics problems used Polya's theory is still at a low level, in general students have difficulty in solving physics problems at the stages of planning and implementing problem solving. This study aims to determine the level of difficulty of students in solving physics problems used Polya's theory and find out what are the difficulties of students in solving physics problems used Polya's theory. The research approach used is mixed method, with the explanatory sequential design. Data collection techniques used tests and interviews. The results of data analysis show that the very high level of difficulty of students in solving equilibrium problems of rigid bodies is at the stage of seeing / checking back, but the difficulties experienced by students at this stage are very dependent on the other three stages of Polya, namely understanding the problem, planning problem solving and implementing problem solving. Students' difficulties in solving physics problems used Polya's theory, among others, students had difficulty in translating or changing the problem in physics symbols, students had difficulty in making plans because they did not know the formula that should be used, students had difficulty in completing mathematical operations and students did not re-examine the solution steps and answers that had been obtained so that some answers were still not correct.

Multilevel interpolations for the generalized Nikishin system on a tree graph

We study a multilevel interpolation problem for a system of Markov functions defined by a tree graph. The normality of all indices has been proven. Asymptotic properties are studied in terms of the vector equilibrium problem for logarithmic potential.

Multiscale contact homogenisation: A novel perspective through the method of multiscale virtual power

The interaction between deformable bodies and rigid foundations undergoing finite strains is explored in this work with the Method of Multiscale Virtual Power, unlocking novel insights into contact homogenisation theories. The focus lies in establishing the foundational kinematical links across scales and achieving seamless homogenisation of the traction vector through rigorous duality arguments. Two distinct families of contact homogenisation models are formulated: surface–surface and surface–volume models. These families emerge from the virtual power balance established between the target macroscopic surface and the corresponding microscopic contact surface or volume. They not only represent fundamental kinematical entities at each scale but also enable the replacement of the heterogeneous contact traction distribution with a smooth, homogenised version. Microscale equilibrium problems and homogenisation relations, along with the relevant macro- and microscale quantities, are derived by means of straightforward variational arguments. Furthermore, potential finite size effects are addressed in the homogenised response, enhancing the reliability and applicability of the strategy across diverse scenarios. Promising submodels have been identified within both families, broadening the understanding of multiscale contact phenomena and including existing models as particular cases. The proposed continuum-based variational families of models naturally lead to generic finite element-based frameworks for contact homogenisation of heterogeneous solids, as described in a forthcoming paper.

Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints via Generalized Subdifferentials

Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints via Generalized Subdifferentials

On α-Fractional Bregman Divergence to study α-Fractional Minty’s Lemma

In this paper fractional variational inequality problems (FVIP) and dual fractional variational inequality problems (DFVIP), Fractional minimization problems are defined with the help of fractional Caputo differential operator and Reimann-Liouville differential operator. The equivalence theorem of Caputo FVIP and fractional minimization problem is established. The equivalence between FVIP and DFVIP is studied with the help of fractional Bregman divergence to establish the fractional Minty’s lemma. Introduction: The theory of variational inequality problems is one of the best tool to solve the equilibrium and non-equilibrium problems arises in the branches of Applied Mathematics, Applied Physics, Engineering, Medical, Financial and many more. On the other hand, fractional calculus manages to study various types of dynamic problems arises in real and complex analysis It is certain for the development of fractional calculus because the fractional derivative has global correlation, which can reflect the historical process of the systematic function.With regard to the definition of fractional derivatives, many mathematicians studied fractional derivatives from different aspects and advanced different definitions for them, for example Riemann-Liouville (RL) derivative, Caputo derivative and other derivatives. Conclusions: The concept of -fractionally differential inequality problem is defined and studied its existence theorem with the help of -fractionally convex function. The concept of -fractionally Bregman divergence is developed and using it the existence of -fractionally minimization problem and -fractionally differential inequality problem are studied. Equivalence theorem in between -fractionally differential inequality problem and -fractionally dual differential inequality problem is studied to establish the Minty’s lemma in fractional calculus.

Necessary and sufficient optimality conditions for nonsmooth nonconvex generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP), which allows the strategies and the objective function of each player to depend on the decision variables of all other players. In this paper, we consider a GNEP where the objective and constraint functions are not necessarily differentiable or convex. By proposing an alternative approach based on the equivalence with a single constrained optimization problem, we then present new sharp KKT-type necessary optimality conditions for a solution of this problem under a weak constraint qualification. We also offer an example to show that these conditions are more efficient for such problems than the previous ones. Further under weak convexity assumptions, we prove that these necessary conditions are also sufficient for optimality. Finally, we extend the definitions of some constraint qualifications from classical optimization to GNEP and study their interrelations. Throughout the paper, several examples are provided to illustrate the results.

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

AbstractThis article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

On a Differential Generalized Nash Equilibrium Problem with Mean Field Interaction

On a Differential Generalized Nash Equilibrium Problem with Mean Field Interaction

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces