Variational Inequality Framework Research Articles

A Neural Network Based on a Nonsmooth Equation for a Box Constrained Variational Inequality Problem

The variational inequality framework holds significant prominence across various domains including economic finance, network transportation, and game theory. In addition, a novel approach utilizing a neural network model is introduced in the current work to address a box constrained variational inequality problem. Initially, the original problem is reformulated into a nonsmooth equation, following which the neural network model is meticulously devised to tackle this reformulated equation. This study comprehensively investigated inherent characteristics and properties of this neural network model. In addition, employing the Lyapunov function method, stability analysis of the neural network model proposed is rigorously demonstrated in the Lyapunov sense. Furthermore, the efficacy of the proposed technique is substantiated through numerical simulations, providing empirical support for its applicability and effectiveness.

Edge-based distributed primal–dual algorithms for seeking generalized Nash equilibria

This paper introduces two novel distributed algorithms aimed at finding the generalized Nash equilibria (GNE) in noncooperative games. To tackle this, we utilize the variational inequality framework, transforming the noncooperative game into the challenge of identifying the zeros of a sum of monotone operators. Within noncooperative games, the decisions made by individual players are interlinked via shared affine constraints. Our proposed approach involves two edge-based distributed algorithms, differentiated by their access to players’ actions, whether it is full-decision information or partial-decision information. We establish the parameter range and demonstrate the convergence of both algorithms, showcasing their efficacy under local constant step-sizes. Additionally, we validate the effectiveness and advantages of these algorithms through two numerical experiments.

Modeling of Covid‐19 trade measures on essential products: a multiproduct, multicountry spatial price equilibrium framework

In this paper, we develop a unified variational inequality framework in the context of spatial price network equilibrium problems that handles multiple products with multiple demand and supply markets in multiple countries as well as multiple transportation routes. The model incorporates a plethora of distinct trade measures, which is particularly important in the pandemic, as PPEs and other essential products are in high demand, but short in supply globally. In the model, product flows as well as prices at the supply markets and the demand markets in different countries are variables that allows us to seamlessly introduce various trade measures, including tariffs, quotas, as well as price floors and ceilings. Qualitative properties are analyzed. Numerical examples are provided to illustrate the impacts of the trade measures on equilibrium product path and link flows, and on prices, and demand and supply quantities. Given the relevance of the trade measures in the world today and discussions concerning the impacts, the framework constructed in this paper is especially timely.

Inertial generalized proximal Peaceman–Rachford splitting method for separable convex programming

The Peaceman–Rachford splitting method (PRSM) is a preferred method for solving the two-block separable convex minimization problems with linear constraints at present. In this paper, we propose an inertial generalized proximal PRSM (abbreviated as IGPRSM) to improve computing efficiency, which unify the ideas of inertial proximal point and linearization technique. Both subproblems are linearized by positive semi-definite proximal matrices, and we explain why the matrix cannot be indefinite. The global convergence and the worst-case asymptotic iteration complexity are derived theoretically via the variational inequality framework. Numerical experiments on LASSO, total variation (TV) based denoising models and image decomposition problems are presented to show the effectiveness of the introduced method even compared with the state-of-the-art methods.

Refugee migration networks and regulations: a multiclass, multipath variational inequality framework.

In this paper, we take up the timely topic of the modeling, analysis, and solution of refugee migration networks. We construct a general, multiclass, multipath model, determine the governing equilibrium conditions, and provide alternative variational inequality formulations in path flows and in link flows. We also demonstrate how governmental imposed regulations associated with refugees can be captured via constraints. We provide qualitative properties and then establish, via a supernetwork transformation, that the model(s) are isomorphic to traffic network equilibrium models with fixed demands. Illustrative examples are given, along with numerical examples, inspired by a refugee crisis from Mexico to the United States, which are solved using the Euler method embedded with exact equilibration. The work sets the foundation for the development of additional models and algorithms and also provides insights as to who wins and who loses under certain refugee regulations.

A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.

Tariffs and quotas in world trade: A unified variational inequality framework

In this paper, we develop a unified variational inequality framework in the context of spatial price network equilibrium problems that captures world trade policies in the form of tariff rate quotas, which are two-tiered tariffs, and that have been applied in practice to numerous commodities. The spatial price network equilibrium model allows for multiple supply markets and multiple demand markets in different countries as well as multiple transportation routes. The model is qualitatively analyzed and also related to models with trade policies such as ad valorem tariffs and strict quotas. A case study on the dairy industry focusing on the United States and France is presented to illustrate the modeling and computational framework with impacts of stricter quotas, higher over quota tariffs, as well as additional transportation routes assessed. The numerical examples reveal that although domestic producers can gain under tariff rate quotas, consumers may experience higher prices. Given the relevance of tariffs to global trade in the world today and discussions concerning the impacts, the framework constructed in this paper is especially timely.

Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity

In 1968, H. Brezis introduced a notion of operator pseudomonotonicity which provides a unified approach to monotone and nonmonotone variational inequalities. A closely related notion is that of Ky–Fan hemicontinuity, a continuity property which arises if the famous Ky–Fan minimax inequality is applied to the variational inequality framework. It is clear from the corresponding definitions that Ky–Fan hemicontinuity implies Brezis pseudomonotonicity, but quite surprisingly, a recent publication by Sadeqi and Paydar (J Optim Theory Appl 165(2):344–358, 2015) claims the equivalence of the two properties. The purpose of the present note is to show that this equivalence is false; this is achieved by providing a concrete example of a nonlinear operator which is Brezis pseudomonotone but not Ky–Fan hemicontinuous.

Hospital competition in prices and quality: A variational inequality framework

In this paper, we construct a game theory model to capture competition among hospitals for patients for their medical procedures. The utility functions of the hospitals contain a revenue component and a component due to altruism benefit. The hospitals compete in prices charged to paying patients as well as in the quality levels of their procedures. Both prices and quality levels are subject to lower and upper bounds. We state the governing Nash equilibrium conditions and provide the variational inequality formulation. We establish existence of an equilibrium price and quality pattern and also present a Lagrange analysis of the equilibrium solutions. An algorithm is proposed and then applied to numerical examples comprising a case study focusing on four major hospitals in Massachusetts.

Distributed Power Optimization for Security-Aware Multi-Channel Full-Duplex Communications: A Variational Inequality Framework

In this paper, we consider the physical layer security issue for the multi-channel full-duplex (FD) communications in the presence of eavesdroppers. There co-exist multiple FD pairs, where the two users in each pair perform bi-directional transmissions. The secure communication is then challenged by the users’ self-interference, external interference from other pairs, and threats from the eavesdroppers. We investigate the problem from a distributed perspective and formulate the problem as a non-cooperative game, where each user optimizes their power allocation over the channels to maximize their own secrecy rate. Confirming the existence of the Nash equilibrium, we introduce an equivalent variational inequality (VI) formulation to derive the sufficient condition for the equilibrium to be unique. We then develop the iterative security-aware water-filling (ISWF) algorithm that can be implemented at each individual user in a distributed manner and prove that the condition for the unique equilibrium also claims the convergence of ISWF algorithm. Furthermore, we extend our formulation to the heterogeneous cases that there co-exist FD and half-duplex users with different security requirements in the networks, and demonstrate that they can all be covered as special cases under our formulated VI framework. Finally, we present simulation results to validate our theoretical findings.

QoS‐driven spectrum sharing game in cognitive radio systems: a variational inequality approach

AbstractCognitive radio (CR) technology aspires to the efficient usage of the limited and underutilised radio spectrum resources. Constituting a flexible and intelligent network, the unlicensed secondary users (SUs) may coexist with the licensed primary users (PUs), under the prerequisite not to disturb their communication. A spectrum interweave CR system is considered, where SUs coexist with a single PU pair in a fading environment. The SUs are allowed to transmit their data ‘peacefully’, only with the absence of PU's transmission, by exploiting spectrum holes of the licensed band. A quality of service (QoS)‐driven spectrum sharing scheme is proposed. Specifically, the problem of licensed spectrum sharing among SUs is formulated as a novel non‐cooperative game with a view of maximising the effective capacity of each SU that has specific delay QoS requirements, subject to the coupled constraint of the available bandwidth. Because this game constitutes a generalised Nash equilibrium problem, we apply the variational inequality framework to solve it. The variational solution is proven to be unique, and it is obtained through a distributed iterative algorithm. The performance of the proposed scheme is evaluated through numerical simulations, and the results confirm the importance of incorporating the delay QoS factor in spectrum sharing studies. Copyright © 2014 John Wiley & Sons, Ltd.

Interior projection-like methods for monotone variational inequalities

We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.

Marketable Pollution Permits in Oligopolistic Markets with Transaction Costs

In this paper, we present a variational inequality framework for the modeling, qualitative analysis, and computation of equilibrium patterns in multiproduct, multipollutant oligopolistic markets with marketable pollution permits in the presence of transaction costs. The model deals explicitly with spatial differentiation and also guarantees that the imposed environmental quality standards are met through the initial allocation of licenses. An algorithm is proposed, with convergence results, to compute the profit-maximized quantities of the oligopolistic firms' products and the quantities of emissions, along with the equilibrium allocation of licenses and their prices. Numerical examples are included to illustrate this approach.

Variational inequalities for marketable pollution permits with technological investment opportunities: The case of oligopolistic markets

In this paper, we develop a variational inequality framework for the modeling, qualitative analysis, and computation of equilibrium patterns in multiproduct, multipollutant oligopolistic markets with marketable pollution permits along with opportunities for investment in production technology and/or emission-abatement technology. The model deals explicitly with spatial differentiation and also guarantees that the imposed environmental standards are met through the initial allocation of licenses. An algorithm is proposed, with convergence results, to compute the profitmaximized quantities of oligopolistic firms' products and the quantities of emissions, along with the equilibrium allocation of licenses and their prices, as well as the optimal investments in the technologies. Numerical examples are presented to illustrate the model and the algorithm.

A variational inequality approach for marketable pollution permits

In this paper, we present a variational inequality framework for the modeling, qualitative analysis, and computation of equilibria in markets in pollution permits. The models developed herein allow for different modes of competitive behavior, including oligopolistic behavior, while taking into account the cost of polluting plus the prices of the licenses to pollute at different receptor points. The models deal explicitly with spatial differentiation and also ensure that the imposed environmental quality standards are met through the initial allocation of licenses. An algorithm is proposed, along with convergence results, to compute the profit-maximized quantities of the firms' products and quantities of emissions, as well as the equilibrium allocation of licenses and their prices. Numerical examples are included to illustrate the approach. This is the first time that this methodology is utilized in environmental economics.

Variational inequality model for the simulation of gas transmission networks

The steady-state simulation of a gas transmission network involves the solution of a system of nonlinear algebraic equations subject to a set of inequality constraints. The purpose of this paper is to consider how the problem can be cast into a variational inequality framework using the dual extremum principles described by Noble and Sewell in 1972. The reliability of the model is tested on two networks supplied by the British Gas Corporation.