Theory Of Variational Inequalities Research Articles

Mixed finite element method for multi-layer elastic contact systems

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is provided. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter H is verified.

A Ramsey-Type Equilibrium Model with Spatially Dispersed Agents

AbstractWe present a spatial and time-continuous Ramsey-type equilibrium model for households and firms that interact on a spatial domain to model labor mobility in the presence of commuting costs. After discretization in space and time, we obtain a mixed complementarity problem that represents the spatial equilibrium model. We prove existence of equilibria using the theory of finite-dimensional variational inequalities and derive a tailored diagonalization method to solve the resulting large-scale instances. Finally, we present a case study that highlights the influence of commuting costs and show that the model allows to analyze transitory effects of industrial agglomeration that emerge and vanish over time as in the real economy.

Responding to consumer environmental awareness: A variational inequality approach to green manufacturing optimization

In response to increasing consumer environmental awareness, our study introduces a sustainable supply chain optimization model that stimulates competition among multiple green manufacturers within an uncertain market context. Each manufacturer crafts a selection of products showcasing various degrees of greenness, purposefully designed to meet the expanding demand of environmentally conscious consumers. This strategic model emphasizes profit maximization for green manufacturers, necessitating the definition of ideal greenness levels and production quantities for an array of product types. By employing variational inequality theory, we transmute the preliminary optimization model into an interconnected variational inequality problem. We propose an improved Euler algorithm, a proficient tool for attaining equilibrium solutions. Empirical scenarios inspired by the electric vehicle industry lend credibility to our model, indicating that the surge in consumer environmental awareness compels manufacturers to diversify their offerings and accentuate their products’ greenness levels. Intriguingly, the research suggests that escalating competition can incentivize manufacturers to elevate their products’ greenness levels, strategically countering potential profit erosion. In addition, the study reveals that intensifying consumers’ green utility sensitivity doesn’t invariably augment demand for manufacturers, even with significant improvements in product greenness. This paradox highlights the necessity for manufacturers to strategically adjust their production approaches in light of varying consumer preferences. Moreover, when the level of consumer environmental awareness achieves a certain level, the fierce competition might not always be harmful to some manufacturers and the industry if they can adopt effective strategies to improve green utility sensitivity.

Pivoting and pandemics: A game-theoretic framework for agile personal protective equipment supply chains

Supply chain models frequently tackle manufacturing issues but must also account for the distinctive nature of the disease. Conversely, most epidemiological models solely concentrate on the disease’s spread but must address logistical challenges. The medical supply chain encounters numerous problems during a pandemic, requiring adaptation through pivoting strategies. For instance, when the COVID-19 outbreak began, several nations prohibited the export of medical supplies, including personal protective equipment (PPE). Consequently, in times of crisis, many countries adopt a localization strategy that encourages domestic companies to adapt their operations and produce medical items. Nevertheless, an interconnected system is essential to align suppliers with the actual demand for medical supplies. This study focuses on the design of a game model for the supply chain that considers manufacturers’ equilibrium behaviors in response to the real demand for medical items. We propose a game model that incorporates both the medical supply chain and the unique characteristics of pandemics. Various decisions are taken into account, such as production volume, actual demand for medical products, price, distribution of medical supplies, and investment costs in manufacturing technologies. To determine the Nash Equilibrium solutions for the proposed game model, the Variational Inequality (VI) theory is implemented.

A Game-Theoretic Approach for Electric Vehicle Aggregators Participating in Phase Balancing Considering Network Topology

With the increasing popularity of electric vehicles (EVs), uncontrolled charging of single-phase plug-in EVs will result in severe three-phase unbalance problem in distribution networks. To solve the problem, a phase balancing (PB) scheme performed by EV aggregators (EVAs) based on the linear multiphase power flow model is proposed in this paper. In the scheme, the PB compensations are provided to encourage EVAs to help the distribution system operator reduce the phase unbalance of the entire grid. Considering all EVAs are selfish and rational, the PB scheme is formulated as a game problem, in which each EVA competes with other EVAs to determine its charging strategy to minimize the charging cost minus the PB compensation. The consideration of coupling voltage constraints leads this game model to a challenging generalized Nash equilibrium problem (GNEP). By employing the theory of variational inequality, the properties of solutions for the GNEP are investigated. Furthermore, a two-level distributed algorithm is designed to find the unique variational solution, a fair and stable solution. Finally, comprehensive case studies are conducted on the IEEE-13 and IEEE-123 test systems to corroborate that the proposed PB game model can effectively mitigate voltage unbalance and cut EV users’ costs.

Research on the Network Equilibrium of Fresh Agricultural Product Supply Chain with Third Party Logistics Participation in Decision Making

Introducing Third-party logistics into the supply chain of fresh agricultural products can reduce the logistics cost of the supply chain and improve the freshness and quality of fresh agricultural products. Taking the fresh agricultural product supply chain network, which includes three decision-making entities: fresh agricultural product suppliers, 3PL service providers, and retailers, as the research object, a fresh agricultural product supply chain network equilibrium model is constructed using variational inequality theory. The impact of 3PL service providers' logistics service level on their equilibrium transaction volume and overall profit is studied. The research results show that when Third-party logistics is involved in the supply chain of fresh agricultural products, considering the logistics service level of 3PL service providers, with the improvement of service level, the overall profit of the supply chain of fresh agricultural products increases. It is a beneficial attempt to realize the profit of the supply chain of fresh agricultural products to improve the freshness and quality of fresh agricultural products through Third-party logistics; When the logistics service level rises to a certain level, and the service level continues to improve, the overall profit growth of the supply chain decreases. It is necessary to improve the impact of logistics service level on the freshness and quality of fresh agricultural products through other forms.

Equilibrium decisions of electricity and ancillary services for energy storage in a renewable power system

Mitigating the power supply fluctuations and maintaining profitability is essential for the operation of the renewable power system (RPS). This study examines, from a supply chain perspective, how the decisions of generators with energy storage technologies (ESTs) in the electricity market (EM) and ancillary services market (ASM) will affect the volatility and profitability of the RPS. The supply chain network equilibrium approach is adopted to investigate the optimal transaction prices and volumes among various decision-makers in the network with the objectives of maximizing profits. The equilibrium conditions are presented with the variational inequality theory and a modified projection algorithm. A case study based on a real situation is carried out and the results show that: (1) When ESTs are used in the EM only, it works well on smoothing the fluctuations of electricity output and price. (2) ESTs, when used in the ASM, can significantly improve the profitability of both the generator and the RPS. (3) When ESTs are used in both the EM and ASM, ancillary services account for approximately 84 % of storage capacity. In addition, ESTs can also promote the consumption of renewable power. This study provides valuable insights for informed decisions to enhance the economic and stable operation of the RPS.

Multicommodity international agricultural trade network equilibrium: Competition for limited production and transportation capacity under disaster scenarios with implications for food security

The number of people affected by disasters, including man-made ones, is on the rise globally, with rising food insecurity being one of the most critical impacts. Disasters, both sudden-onset and slow-onset ones, can cause disruptions to the production and transportation of agricultural commodities. Having the tools that can quantitatively assess the changes in agricultural commodity shipment volumes and their prices under disruptions caused by disaster scenarios is of major importance. In this paper, we utilize the theory of variational inequalities as the methodology to construct a multicommodity international agricultural trade network equilibrium model, which contains novel features of capacities on the production and transportation of multiple agricultural commodities to capture competition. The model includes exchange rates and accounts for multiple routes and possibly distinct transportation modes and combinations. Theoretical results are given and an algorithm is proposed. A series of numerical examples, both illustrative and algorithmically solved ones, inspired by Russia’s war on Ukraine, highlight the effects of reduced production and transportation capacities on food security in the Middle Eastern and North African (MENA) countries of Lebanon and Egypt. We also include sensitivity analysis results for exchange rates. The solutions reveal insights into the importance of the production and transportation capacities regarding food security, along with having multiple transportation routes that are cost-efficient as well as the importance of the magnitude of exchange rates.

Polynomial inclusions: Definitions, applications, and open problems

Predictive modeling in physical science and engineering is mostly based on solving certain partial differential equations where the complexity of solutions is dictated by the geometry of the domain. Motivated by the broad applications of explicit solutions for spherical and ellipsoidal domains, in particular, the Eshelby’s solution in elasticity, we propose a generalization of ellipsoidal shapes called polynomial inclusions. A polynomial inclusion (or p-inclusion for brevity) of degree k is defined as a smooth, connected and bounded body whose Newtonian potential is a polynomial of degree k inside the body. From this viewpoint, ellipsoids are identified as the only p-inclusions of degree two; many fundamental problems in various physical settings admit simple closed-form solutions for general p-inclusions as for ellipsoids. Therefore, we anticipate that p-inclusions will be useful for applications including predictive materials models, optimal designs, and inverse problems. However, the existence of p-inclusions beyond degree two is not obvious, not to mention their explicit algebraic parameterizations.In this work, we explore alternative definitions and properties of p-inclusions in the context of potential theory. Based on the theory of variational inequalities, we show that p-inclusions do exist for certain polynomials, though a complete characterization remains open. We reformulate the determination of surfaces of p-inclusions as nonlocal geometric flows which are convenient for numerical simulations and studying geometric properties of p-inclusions. In two dimensions, by the method of conformal mapping we find an explicit algebraic parameterization of p-inclusions. We also propose a few open problems whose solution will deepen our understanding of relations between domain geometry, Newtonian potentials, and solutions to general partial differential equations. We conclude by presenting examples of applications of p-inclusions in the context of Eshelby inclusion problems and magnet designs.

A Spatial Equilibrium Problem for the European Pulp and Paper Industry Under the Emission Trading System

AbstractWe propose a spatial equilibrium problem for a pulp and paper supply chain network under the Emission Trading System (ETS) and the circular economy principles. The developed closed-loop supply chain model, based on the variational inequality theory, accounts for the forest deployment and the production of paper using both virgin pulp and recyclable paper waste in an area that is subject to the ETS. In addition, it considers the exchanges of virgin pulp, paper, recyclable and not-recyclable paper waste between this regulated area and geographical zones where no environmental policies apply. The aim of this paper is to investigate the ETS impacts on the pulp and paper industries located in an environmentally regulated area in terms of operating costs, production and trading volumes of the aforementioned commodities, and emissions generated. The case study focuses on the European pulp and paper companies, which are among the energy-intensive industries deemed to be exposed to the carbon leakage risk as a consequence of the ETS application. Sensitivity analyses on the carbon price value and on the trade flexibility are conducted. Our results show that the ETS implies additional costs for the European pulp and paper companies and induces them to resort more on imports of virgin pulp and paper from not-regulated zones.

The Impact of R&D Subsidy and IPP on Global Supply Chain Networks System—A Technology Spillover Perspective

In the current globalized business environment, multinational competition has become the norm for companies. This paper considers technology spillovers among manufacturers and develops a global supply chain network equilibrium model. Firstly, the optimal decision-making behaviors of manufacturers, retailers, and demand markets are characterized separately. Secondly, based on the variational inequality theory, the optimal decision-making behaviors of global supply chain members are transformed. Finally, the model is solved and analyzed using the Euler algorithm. The primary objective is to explore the impact of research and development (R&D) subsidies and intellectual property protection (IPP) strategies on manufacturers’ research and development technological levels. Furthermore, the study delves into their effects on the production and transactions of the global supply chain network and social welfare. The following conclusions are drawn: (1) Technology spillovers have a positive effect on the technological level achieved by manufacturers through research and development investment and social welfare. However, intense technological competition may harm manufacturers’ profits. (2) Under the symmetric subsidy policy, higher subsidies may lead to a decrease in social welfare. (3) Under symmetric intellectual property protection policies, increasing the intensity of intellectual property protection benefits manufacturers but is detrimental to retailers and social welfare. However, under an asymmetric intellectual property protection strategy, implementing high-intensity intellectual property protection by high-technology countries is advantageous for retailers and social welfare. This conclusion has contributed to the technical research and development and production operation decision making of global supply chain members, as well as government policy formulation, and has also provided a new perspective for theoretical research in the field of global supply networks.

Multi-period network equilibrium in power system with energy storage on generation side

Due to the intermittency and unpredictability of wind and photovoltaic power, a power system with high penetration of renewable sources is always imbalanced. Energy storage on generation side can enhance the quality and reliability of such power systems. To study the impact of energy storage on power system networks, this study proposes a framework that regards the renewable energy power system with storage as a multi-period power supply chain network (PSCN). The equilibrium conditions are presented using the variational inequality theory and solved with a modified projection algorithm, followed by several examples based on real-world data to illustrate these findings. It is shown that energy storage enhances the reliability of power systems and increases the penetration of renewable power. Notably, the utilization efficiency of energy storage is lower than 50%, which is limited by the single utilization of energy storage facilities.

Analysis and Numerical Approach of a Coupled Thermo-Electro-Mechanical System for Nonlinear Hencky-Type Materials with Nonlocal Coulomb’s Friction

The effects of an included temperature field in the contact process between a piezoelectric body and a rigid foundation with thermal and electrical conductivity are discussed. The constitutive relation of the material is assumed to be thermo-electro-elastic and involves the nonlinear elastic constitutive Hencky’s law. The frictional contact is modeled with Signorini’s conditions, the regularized Coulomb law, and the regularized electrical and thermal conductivity conditions. The resulting thermo-electromechanical model includes the temperature field as an additional state variable to take into account thermal effects alongside with those of the piezoelectric. The existence of the unique weak solution to the problem is established by using Schauder’s fixed point theorem combined with arguments from the theory of variational inequalities involving nonlinear strongly monotone Lipschitz continuous operators. A successive iteration technique to solve the problem numerically is proposed, and its convergence is established. A variant of the Augmented Lagrangian, the so-called Alternating Direction Method of multipliers (ADMM), is used to decompose the original problem into two sub-problems, solve them sequentially and update the dual variables at each iteration. The influence of the thermal boundary condition on the behavior of contact forces and electrical potential is shown through graphical illustrations.

Equilibrium pricing mechanism of low-carbon supply chain considering carbon cap-and-trade policy

In the low-carbon supply chain (LSC) network, because of the different carbon emission reduction investment costs and product exhibition costs, members of the same level will show certain competitive behavior. This paper aims to realize the low-carbon development of the supply chain network based on the carbon quota trading system and the risk aversion characteristics of retailers and uses variational inequality theory to construct the network equilibrium model under the constraint of inventory capacity, and designs an improved projection contraction algorithm to solve the model. The results mean that when the retailer is risk averse, the improvement of the risk aversion coefficient (RAC) reduces the manufacturer's carbon emission rate and profit and increases the product sales price and retailer's profit. When the retailer has an inventory capacity constraint, the enhancement of the inventory capacity constraint increases the profit of the manufacturer and the retailer but reduces the emission reduction rate of the manufacturer. If the retailer has risk aversion characteristics and inventory capacity constraints, the manufacturers' profits will increase as RAC increases, while retailers' profits will decrease as RAC increases. In addition, the carbon trading mechanism can promote enterprises to invest in carbon emission reduction and raise the low-carbon level of products.

Online Learning for Rate-Adaptive Task Offloading Under Latency Constraints in Serverless Edge Computing

We consider the interplay between latency constrained applications and function-level resource management in a serverless edge computing environment. We develop a game theoretic model of the interaction between rate adaptive applications and a load balancing operator under a function-oriented pay-as-you-go pricing model. We show that under perfect information, the strategic interaction between the applications can be formulated as a generalized Nash equilibrium problem, and use variational inequality theory to prove that the game admits an equilibrium. For the case of imperfect information, we propose an online learning algorithm for applications to maximize their utility through rate adaptation and resource reservation. We show that the proposed algorithm can converge to equilibria and achieves zero regret asymptotically, and our simulation results show that the algorithm achieves good system performance at equilibrium, ensures fast convergence, and enables applications to meet their latency constraints.

Aggregative Variational Inequalities

We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten–Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies.

Neuro-adaptive control for searching generalized Nash equilibrium of multi-agent games: A two-stage design approach

This paper investigates the generalized Nash equilibrium searching problem of multi-agent games over weight-unbalanced directed networks. In this problem, the feasible action set of each agent is not only constrained by private convex sets and shared coupling equality, but also constrained by its own uncertain dynamics. Moreover, the local cost function is related to both his own actions and others, and each agent is only allowed access to neighbor information. To address this problem, we propose a neuro-adaptive control strategy based on two-stage design. In the first stage of design process, an approximator with adaptive-gain is designed to generate a virtual trajectory that converges to a necessary Nash equilibrium, the compensator based on consensus-tracking is used to obtain non-neighbor information and several auxiliary state variables are used to exchange information with local neighborhoods on a digraphs to deal with equality constraints. In the second stage of the strategy, we further developed a neuro-adaptive tracking controller based on one-step design for tracking the virtual trajectory. In the controller design, the virtual control variables and the actual control laws are obtained in a collective way, without repeating the design process. We introduce a variable called the minimum learning parameter to reduce the number of online learning parameters of the neural network. By using tools from variational inequality theory and Lyapunov stability theory, we prove that the proposed control strategy can drive all agents reach Nash equilibrium. Finally, the effectiveness of control strategy is verified by simulation example.

A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

NUMERICAL SIMULATION OF ELASTIC-PLASTIC DEFORMATION OF TORSION SHAFTS OF VEHICLE SPRING SYSTEMS, TAKING INTO ACCOUNT CONTACT INTERACTION

The paper develops a model of stress-strain state of torsion shafts. A variable formulation of the problem on the basis of the theory of variational inequalities was made. Numerical modeling of the stress-head-deformed state was made with the involvement of the finite element method. In the model of the stress-head-deformed state of torsion shafts, its generalized parametric description is integrated. It contains both varied significant design and technological indicators, in particular, the structural design of the torsion head, the galley transition and the diameter of the stem torsion shaft stem, and the parameters of the technological process of double winding: twist angles and features of the shaft material. It is established that the stresses that occur in the torsion shaft are mainly concentrated in the contact interaction zone of the spline shaft head with the spline coupling. The parametric dependencies of the level of equivalent stresses by Mises on the level of double windings and on the ratio of the head diameter and the stem of the torsion shaft stem have been determined. Recommendations have been developed regarding the improvement of strength characteristics of torsion shafts. Contact pressure distribution on the slots of the torsion bar head is also determined. Its characteristic features depend on the type of boundary conditions. If bilateral interaction is enabled on the boundary, the contact pressure may become negative. Furthermore, the contact pressure gets a sharp spike. This non-physical feature disappears if unilateral contact conditions are imposed. Therefore, this version of the boundary conditions is more adequate. It was used to compute the stress-strain state of a torsion bar, taking into account the plastic deformation of the material and the contact interaction with conjugate bodes. Keywords: torsional shaft; stress-strain state; finite element method; theory of variational inequalities; contact interaction

NEW PHYSICAL FACTORS IN CONTACT INTERACTION OF ELASTIC BODIES OVER CLOSELY MATCHED SURFACES

The paper contains generalization and development of approaches, models and research methods for contact interaction of complex shaped bodies. In the case of contact along the surfaces of a close shape there are a number of problems. Firstly, analytical methods are not universally applied. Secondly, with a small gap between the surfaces of the contacting bodies the contribution of the deformation of the intermediate and surface layers to the balance of displacements becomes substantial. Thirdly, the application of traditional numerical methods also sharply increases the contribution of errors in contact surface shape approximation. All these factors bring on the need to develop new approaches, models and methods for the analysis of contact interaction of complex shaped bodies. For this purpose, the apparatus of the theory of variational inequalities is applied to deliver Kalker’s variational principle. In addition, a new principle for deriving multiple models of contact interaction of complex shaped bodies has been developed. It consists in gradually increasing the number of physical factors that are taken into account when building these models. This makes it possible to form a sequence of varied parametric models by taking into account new factors while naturally avoiding conflict situations. Furthermore it ensures inheritance in the sequence of models of different levels. The proposed improved approaches, models and methods are illustrated by the case study of the contact interaction of bodies with the intermediate layers located between them. Keywords: contact interaction, complex shaped body, finite element method, boundary element method, Kalker’s variational principle, variational inequalities