Theory Of Variational Inequalities Research Articles

A variational inequality theory for constrained problems in reflexive Banach spaces

‎‎Let $X$ be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space $X^*$‎, ‎and let $K$ be a nonempty‎, ‎closed‎, ‎and convex subset of $X$‎. ‎Let $T‎: ‎X\supseteq D(T)\to 2^{X^*}$ be maximal monotone‎, ‎let $S‎: ‎K\to 2^{X^*}$ be bounded and of type $(S_+)$‎, ‎and let $C‎: ‎X\supseteq D(C)\to X^*$ with $D(T)\cap D(\partial \phi)\cap K\subseteq D(C)$‎. ‎Let $\phi‎ : ‎X\to (-\infty‎, ‎\infty]$ be a proper‎, ‎convex‎, ‎and lower semicontinuous function‎. ‎New existence theorems are proved for solvability of variational inequality problems of the type $\rm{VIP}(T+S+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $C$ is compact and $\rm{VIP}(T+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $T$ is of compact resolvent and $C$ is bounded and continuous‎. ‎Various improvements and generalizations of the existing results for $T+S$ and $\phi$ are obtained‎. ‎The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems‎.

Dynamic Modeling of Sphere, Cylinder, Cone, and their Assembly

Dynamic simulation of revolved solids plays an important role in many fields. Aiming at the lacks of solutions in some key aspects, this study establishes governing equation of motion based on theory of variational inequality; designs a compatibility iteration algorithm for solving contact forces; deduces parametric equations of arbitrary cylinder and cone in three-dimensional space; provides corresponding analytical methods to identify contact points between bodies and to calculate volume integral over bodies; proposes a rotation matrix modification approach to conserve volume and shape of rigid body in the case of large rotation. The accuracy, availability, competence, robustness, and application prospects of the presented methodology are demonstrated by several interesting and challenging problems.

A Supply Chain-Logistics Super-Network Equilibrium Model for Urban Logistics Facility Network Optimization

The logistics facility decisions may be the most critical and most difficult of the decisions needed to realize an efficient supply chain since these decisions have significant effects on the logistics costs generated in the logistics network. We establish a logistics super network equilibrium integrating urban logistics facilities with members of traditional supply chain network, using the variational inequality theory. This model takes into account the behavior of logistics facilities and the transactions between retailers and logistics facilities are examined in this paper. Furthermore, we obtain the equilibrium condition of the system, and the economic explanation and algorithm are given. Finally, some verification examples are provided to verify the solution and decision-making application.

Multitiered Blood Supply Chain Network Competition: Linking Blood Service Organizations, Hospitals, and Payers

In this paper, we present a multi-tiered competitive supply chain network model for the blood banking industry, with a focus on the United States, that captures the economic interactions between three tiers of stakeholders; namely, the blood service organizations, the hospitals or medical centers, which transfuse blood to patients, and the payer groups that patients belong to. In addition, the supply chain framework for this life-saving product includes the competition among blood service organizations and their various supply chain activities. We model the behavior of each category of stakeholder and use the theory of variational inequalities to derive the equilibrium conditions for the entire supply chain. Illustrative examples are provided, along with qualitative properties, followed by an algorithm, accompanied by convergence results, that is used to solve simulated numerical examples. Results from these examples demonstrate that such a model can be effectively used to determine the prices and blood pathways from blood service organizations to hospitals to payers.

Supply chain network equilibrium with strategic supplier investment: A real options perspective

This article develops a network equilibrium model that studies competing supply chain firms who can strategically invest in new capacity at suppliers under cost and demand uncertainty. We model the decision-making process of each supply chain firm as a stochastic optimization problem. We formulate the equilibrium conditions of all supply chain firms using the variational inequality theory. Analytical results establish connections from the supplier investment decisions to real put and call options. Option pricing theory is applied to show that the value of supplier capacity investment will increase if the demand uncertainty or the cost uncertainty increases, and will decrease if the price sensitivity or the cost factor increases. Numerical experiments are conducted to confirm and extend the analytical results, and to investigate the decision-making and profits of different supply chain firms in more complex and realistic settings. Our results also suggest that manufacturers should strategically invest in a portfolio of complementary suppliers with different risk characteristics that serve different purposes.

3D DDA Based on Variational Inequality Theory and Its Solution Scheme

The advantage of 3D discontinuous deformation analysis (3D DDA) is the rigorous contact conditions on the interaction of 3D blocks. These conditions are enforced by the penalty function convention; however, inappropriate penalty parameters easily generate numerical instability. To avoid the introduction of the penalty parameters, the contact conditions in 3D DDA are described as variational inequalities in this study, and the extra-gradient method is employed to solve this new formulation of 3D DDA. The proposed computation scheme is more flexible and dispenses with large scale matrix inversion. Some practical examples originally designed by Shi are analysed, verifying the effectiveness and precision of the new scheme.

Closed‐loop supply chain network static equilibrium and dynamics under pollution permits system

A closed‐loop supply chain network including multi‐manufacturing/remanufacturing firms and multi‐demand markets is considered. We present equilibrium conditions for decision‐makers in the network and develop a static model with marketable pollution permits and noncompliant behavior, using the variational inequality theory. We provide qualitative analysis of the model. Then, using the projected dynamical system, we propose a dynamic model. The results show that the set of stationary points of dynamic model coincides with the set of solutions of static model. Finally, numerical examples are proposed. The equilibrium results including production output of new products, the forward and reverse product flows, the prices of product, the prices of license, the number of licenses, and the possible noncompliant overflows and underflows of the waste are obtained. We find that increase in the unit penalty cost contributes to decrease in waste emission from firms in the network, and when the unit penalty cost increase to a certain level, they no longer choose excess emission of waste. © 2018 American Institute of Chemical Engineers Environ Prog, 38:e13021, 2019

The mathematical foundations of dynamic user equilibrium

This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.

Supply chain network equilibrium with strategic financial hedging using futures

In this paper, we develop a network equilibrium model for supply chain networks with strategic financial hedging. We consider multiple competing firms that purchase multiple materials and parts to manufacture their products. The supply chain firms’ procurement activities are exposed to commodity price risk and exchange rate risk. The firms can use futures contracts to hedge the risks. Our research studies the equilibrium of the entire network where each firm optimizes its own operation and hedging decisions. We use variational inequality theory to formulate the equilibrium model, and provide qualitative properties. We provide analytical results for a special case with duopolistic competition, and use simulations to study an oligopolistic case. The analytical and simulation studies reveals interesting managerial insights.

Power Penalty Approach to American Options Pricing Under Regime Switching

This work aims at studying a power penalty approach to the coupled system of differential complementarity problems arising from the valuation of American options under regime switching. We introduce a power penalty method to approximate the differential complementarity problems, which results in a set of coupled nonlinear partial differential equations. By virtue of variational inequality theory, we establish the unique solvability of the system of differential complementarity problems. Moreover, the convergence property of this power penalty method in an appropriate infinite-dimensional space is explored, where an exponential convergence rate of the power penalty method is established and the monotonic convergence of the penalty method with respect to the penalty parameter is shown. Finally, some numerical experiments are presented to verify the convergence property of the power penalty method.

Power allocation for SWIPT in K-user interference channels using game theory

A simultaneous wireless information and power transfer system in interference channels of multi-users is considered. In this system, each transmitter sends one data stream to its targeted receiver, which causes interference to other receivers. Since all transmitter-receiver links want to maximize their own average transmission rate, a power allocation problem under the transmit power constraints and the energy-harvesting constraints is developed. To solve this problem, we propose a game theory framework. Then, we convert the game into a variational inequalities problem by establishing the connection between game theory and variational inequalities and solve the variational inequalities problem. Through theoretical analysis, the existence and uniqueness of Nash equilibrium are both guaranteed by the theory of variational inequalities. A distributed iterative alternating optimization water-filling algorithm is derived, which is proved to converge. Numerical results show that the proposed algorithm reaches fast convergence and achieves a higher sum rate than the unaided scheme.

Efficiency and Vulnerability Analysis for Congested Networks with Random Data

In this note, we combine two theories that have been proposed in the last decade: the theory of vulnerability and efficiency of a congested network, and the theory of stochastic variational inequalities. As a result, we propose a model that describes the performance and vulnerability of a congested network with random traffic demands and where the travel time can be affected by uncertainty. As an application, we investigate in detail the famous Braess’ network.

Resource Optimization in Heterogeneous Cloud Radio Access Networks

This letter tackles the problem of sum-rate maximization via joint user association and power allocation in heterogeneous cloud radio access networks, while guaranteeing the quality-of-service requirements of all users. A generalized Stackelberg game is applied to this problem with the coupled strategy set of power allocation, and a centralized-distributed method is designed to achieve the optimal solution. Specifically, the user association problem is efficiently solved in a centralized manner and a distributed power allocation algorithm is proposed by using variational inequality theory.

Homogenization of Cahn–Hilliard-type equations via evolutionary $$\varvec{\Gamma }$$-convergence

In these notes we discuss two approaches to evolutionary \(\Gamma \)-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ \(\Gamma \)-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals \(\Gamma \)-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

The equilibrium model of dual channel closed-loop supply chain network based on carbon trading and carbon tax

In this paper, on the basis that carbon trading and carbon tax are brought into the category of the carbon management, an equilibrium model of a dual channel closed-loop supply chain network is developed. Many decision-makers and their independent behaviours are fully processed based on the variational inequality theory and Lagrange duality theory. Based on the above analysis, a finite-dimensional variational inequality formulation is established. Besides, qualitative properties of the equilibrium model and a modified projection algorithm are given.

The equilibrium model of dual channel closed-loop supply chain network based on carbon trading and carbon tax

In this paper, on the basis that carbon trading and carbon tax are brought into the category of the carbon management, an equilibrium model of a dual channel closed-loop supply chain network is developed. Many decision-makers and their independent behaviours are fully processed based on the variational inequality theory and Lagrange duality theory. Based on the above analysis, a finite-dimensional variational inequality formulation is established. Besides, qualitative properties of the equilibrium model and a modified projection algorithm are given.

Decentralized Algorithm for Randomized Task Allocation in Fog Computing Systems

Fog computing is identified as a key enabler for using various emerging applications by battery powered and computationally constrained devices. In this paper, we consider devices that aim at improving their performance by choosing to offload their computational tasks to nearby devices or to an edge cloud. We develop a game theoretical model of the problem and use a variational inequality theory to compute an equilibrium task allocation in static mixed strategies. Based on the computed equilibrium strategy, we develop a decentralized algorithm for allocating the computational tasks among nearby devices and the edge cloud. We use the extensive simulations to provide insight into the performance of the proposed algorithm and compare its performance with the performance of a myopic best response algorithm that requires global knowledge of the system state. Despite the fact that the proposed algorithm relies on average system parameters only, our results show that it provides a good system performance close to that of the myopic best response algorithm.

Consumer learning of product quality with time delay: Insights from spatial price equilibrium models with differentiated products

In this paper, we present spatial price equilibrium network models, both static and adaptive, with differentiated products under perfect quality information for producers and consumers and under quality information asymmetry with consumer learning of product quality with a time delay. The adaptive model with information asymmetry is able to adapt to the uncertainty in consumer learning as well as in supply, demand, transportation cost, and product quality over time. In addition, we provide measures of consumer welfare under perfect quality information and under information asymmetry as well as the value of perfect quality information for consumers. The models are formulated and qualitatively analyzed using variational inequality theory. We establish theoretically and illustrate computationally that, under appropriate assumptions, the equilibrium solution, consisting of supply and demand markets prices, quality levels, and product flows, of the adaptive spatial price equilibrium model with information asymmetry converges to that of the corresponding static model with perfect quality information. The models are especially relevant to agricultural products where spatial price equilibrium models have found wide application. We also present several numerical examples with practical insights provided.

Evaluating the effects of environmental regulations on a closed-loop supply chain network: a variational inequality approach

Global climate change has encouraged international and regional adoption of pollution taxes and carbon emission reduction policies. Europe has taken the leadership in environmental regulations by introducing the European Union Emissions Trading System (EU-ETS) in 2005 and by promoting a set of policies destined to lower carbon emissions from energy, industrial, and transport sectors. These environmental policies have significantly affected the production choices of these European sectors. Considering this framework, the objective of this paper is to evaluate the effects of the application of environmental policies in a multitiered closed-loop supply chain network where raw material suppliers, manufacturers, consumers, and recovery centers operate. In particular, we assume that manufacturers are subject to the EU-ETS and a carbon tax is imposed on truck transport. In this way, the developed model captures carbon emission regulations, recycling, transportation and technological factors within a unified framework. In particular, it allows for evaluating the impacts of the considered environmental regulations on carbon emissions, product flows, and prices. The proposed model is optimized and solved by using the theory of variational inequalities. Our analysis shows that the combined application of the EU-ETS at the manufacturers’ tier and the carbon tax on truck transport implies additional costs for producers that reduce their good provisions. On the other side, this has a positive outcome for the environment since $$\hbox {CO}_2$$ emissions reduce. Moreover, an increase of the efficiency level of the recycling process increments the availability of reusable raw material in the reverse supply chain. Finally, the distance between a couple of CLSC tiers plays a very important role. The lower is the distance covered by vehicles, the higher is the production of goods and the lower is the amount of $$\hbox {CO}_2$$ emitted.

A Variational Inequality Theory with Applications to P $P$ -Laplacian Elliptic Inequalities

The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous $\psi_{p}$ -dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for $p$ -Laplacian elliptic inequalities are obtained under suitable conditions.