Variational Inequality Technique Research Articles

Comparing climate pledges and eco-taxation in a networked agricultural supply chain organisation

Abstract This paper examines the effectiveness of climate pledges and eco-taxation as strategies for mitigating climate change within a networked agricultural supply chain organisation. We utilise variational inequality techniques within a multicriteria decision-making framework and validate our theoretical findings through numerical simulations using a machine learning augmented algorithm. By employing this approach, we position the Agricultural Sector Roadmap, aimed at capping global warming at 1.5°C, within the wider agricultural sector’s climate action framework. Our results demonstrate that environmental taxation emerges as the most effective approach for addressing climate change. Eco-taxation leads to a 57.87 per cent reduction in global emissions, whereas climate pledges only account for a 20.59 per cent reduction at the same level of production. Furthermore, eco-taxation results in a 45.68 per cent greater reduction in emission intensity compared to climate pledges. In contrast to climate commitments, an eco-fiscal policy is capable of achieving the objectives established by the European Union.

Equilibrium characterization and shock propagation in conflict networks

We establish the general properties of equilibria (existence and uniqueness) in a very general model of interconnected multiplayer conflicts. In particular, under mild conditions on the cost function and the contest technology, we show that a pure-strategy Nash equilibrium always exists and the set of Nash equilibria is convex. Furthermore, under the strong monotonicity of the cost function, the equilibrium is unique, regardless of the conflict structure. To establish these properties of equilibria, we use variational inequality techniques to obtain equivalent equilibrium characterizations. After conducting comparative statics analysis with respect to the model primitives, we explore, in several simple examples of conflict structures, how shocks that originated locally propagate to the rest of the conflict network. Our general framework subsumes a number of models studied in the contest literature, as we are able to generalize many results obtained in these papers.

Distributed constrained aggregative games of uncertain Euler-Lagrange systems under unbalanced digraphs

In this paper, the constrained Nash equilibrium seeking problem of aggregative games is investigated for uncertain nonlinear Euler-Lagrange (EL) systems under unbalanced digraphs, where the cost function for each agent depends on its own decision variable and the aggregate of all other decisions. By embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the digraph Laplacian matrix, a dynamic adaptive average consensus protocol is employed to estimate the aggregate function in the unbalanced case. To solve the constrained Nash equilibrium seeking problem, an integrated distributed protocol based on output-constrained nonlinear control and projected dynamics is proposed for uncertain EL players to reach the Nash equilibrium. The convergence analysis is established by using variational inequality technique and Lyapunov stability analysis. Finally, a numerical example in electricity market is provided to validate the effectiveness of the proposed method.

Game-Based Multitype Task Offloading Among Mobile-Edge-Computing-Enabled Base Stations

The widely used Internet of Things (IoT) mobile devices (MDs) require fast processing capability to handle a large volume of computing tasks. Mobile-edge computing (MEC) can augment the capability of IoT MDs through offloading their computing tasks to the MEC-enabled base station (MEC-BS) that covers them. Most of the existing research works only focus on the computation offloading problems for a single MEC-BS. However, the load of a MEC-BS will rise as the increase of the scale of the offloaded tasks, especially during rush hours, and further it will result in deterioration of system performance. In this article, we propose a game-based multitype task offloading scheme among MEC-BSs. The tasks offloaded from IoT MDs can be further offloaded among MEC-BSs to alleviate high-load MEC-BSs. Aiming at balancing the computing delays of the tasks on each MEC-BS, a noncooperative game is formulated to model the computation offloading for the tasks with different types, indicated by computation amount, data size, and delay tolerance. The existence and convergence of the Nash equilibrium of the game are first proved using the variational inequality and regularization techniques. Then, we design a distributed iterative algorithm to efficiently solve the game problem. Simulation results show the fast convergence of our algorithm. The reduction of total computing delay optimized by our scheme can reach 45%–50% on average in multiple scenarios, and the superiority of our scheme is also demonstrated in comparisons with reference schemes.

Stability analysis of partial differential variational inequalities in Banach spaces

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

A Stackelberg-population competition model via variational inequalities and fixed points

In this paper, we introduce and study a new Stackelberg-population competition model which captures the desired features of both population games and Stackelberg competition model within the same framework. We obtain some characterization results for the Stackelberg-population equilibrium response set and the Stackelberg-population equilibrium leader set by using the variational inequality technique and Brouwer’s fixed point theorem. We also show an existence theorem of Nash equilibrium for Stackelbergpopulation competition model under some mild conditions. Finally, we give an example to illustrate our main results.

Distributed Nash Equilibrium Seeking for Aggregative Games via Derivative Feedback

In this paper, we investigate a continuous-time distributed Nash equilibrium seeking algorithm for a class of aggregative games, with application to the real-time pricing demand response. To seek the Nash equilibrium via local communication among neighbors, by combining projected gradient dynamics and consensus tracking dynamics, we propose a novel distributed algorithm for the players. We prove the convergence of the distributed algorithm via a constructed Lyapunov function and the variational inequality technique, and show an illustrative simulation related to the energy consumption control in smart grids.

Convergence analysis of positive-indefinite proximal ADMM with a Glowinski’s relaxation factor

In this paper, we propose a modified positive-indefinite proximal linearized ADMM (PIPL-ADMM) with a larger Glowinski’s relaxation factor for solving two-block linearly constrained separable convex programming by variational inequality technique. We investigate the internal relationships between the step size coefficient and the penalty coefficient to identify the convergence of PIPL-ADMM. The convergence of PIPL-ADMM and its convergence rate measured by the iteration complexity are established in the ergodic case. Numerical experiments are reported to illustrate the efficiency of the proposed methods.

Confidence Intervals and Regions for the Lasso by Using Stochastic Variational Inequality Techniques in Optimization

Summary Sparse regression techniques have been popular in recent years because of their ability in handling high dimensional data with built-in variable selection. The lasso is perhaps one of the most well-known examples. Despite intensive work in this direction, how to provide valid inference for sparse regularized methods remains a challenging statistical problem. We take a unique point of view of this problem and propose to make use of stochastic variational inequality techniques in optimization to derive confidence intervals and regions for the lasso. Some theoretical properties of the procedure are obtained. Both simulated and real data examples are used to demonstrate the performance of the method.

Variational inequality formulation of circular cone eigenvalue complementarity problems

In this paper, we study the circular cone eigenvalue complementarity problem (CCEiCP) by variational inequality technique, prove the existence of a solution to CCEiCP, and investigate different nonlinear programming formulations of the symmetric and asymmetric CCEiCP, respectively. We reduce CCEiCP to a variational inequality problem on a compact convex set, which guarantees that CCEiCP has at least one solution. Based on the variational inequality formulation of CCEiCP, the symmetric CCEiCP can be reformulated as a nonlinear programming problem NLP1, and solved by computing a stationary point of the Rayleigh quotient function on a compact set. We formulate the asymmetric CCEiCP as another nonlinear programming problem NLP2, and show that any global minimum of NLP2 with an objective function value equal to zero is a solution of the asymmetric CCEiCP. Moreover, a stationary point of NLP2 is a solution of the asymmetric CCEiCP, if and only if the Lagrange multipliers associated with the equalities in NLP2 are equal to zero. The different formulations of CCEiCP provide alternative approaches for solving CCEiCP, which will play an important role in designing efficient algorithms to solve CCEiCP.

Dynamic and disequilibrium analysis of interdependent infrastructure systems

There is a growing awareness in recent years that the interdependencies among the civil infrastructure systems have significant economic, security and engineering implications that may influence their resiliency, efficiency and effectiveness. To capture the various types of infrastructure interdependencies and incorporate them into decision-making processes in various application domains, Zhang and Peeta (2011) propose a generalized modeling framework that combines a multilayer infrastructure network (MIN) concept and a market-based economic approach using computable general equilibrium (CGE) theory and its spatial extension (SCGE) to formulate a static equilibrium infrastructure interdependencies problem. This paper extends the framework to address the dynamic and disequilibrium aspects of the infrastructure interdependencies problems. It briefly reviews the static model, and proposes an alternative formulation for it using the variational inequality (VI) technique. Based on this equivalent VI formulation, a within-period equilibrium-tending dynamic model is proposed to illustrate how these systems evolve towards an equilibrium state within a short duration after a perturbation. To address a longer time scale, a multi-period dynamic model is proposed. This model explicitly considers the evolution of infrastructure interdependencies over time and the temporal interactions among the various systems through dynamic parameters that link the different time periods. Using this model, numerical experiments are conducted for a special case with a single region to analyze the sensitivity of the model to the various parameters, and demonstrate the ability of the modeling framework to formulate and solve practical problems such as cascading failures, disaster recovery, and budget allocation in a dynamic setting.

Stock Loans in Incomplete Markets

A stock loan is a contract whereby a stockholder uses shares as collateral to borrow money from a bank or financial institution. In Xia and Zhou (2007, Stock loans, Mathematical Finance, 17(2), pp. 307–317), this contract is modelled as a perpetual American option with a time-varying strike and analysed in detail within a risk-neutral framework. In this paper, we extend the valuation of such loans to an incomplete market setting, which takes into account the natural trading restrictions faced by the client. When the maturity of the loan is infinite, we use a time-homogeneous utility maximization problem to obtain an exact formula for the value of the loan fee to be charged by the bank. For loans of finite maturity, we characterize the fee using variational inequality techniques. In both cases, we show analytically how the fee varies with the model parameters and illustrate the results numerically.

Variational Inequalities and Vector Optimization

As very powerful and important tools in the study of nonlinear sciences, variational inequalities and vector optimization have attracted so much attention. Over the last decades, variational inequality and vector optimization techniques have been applied extensively in such diverse fields as biology, chemistry, economics, engineering, game theory, management science, physics, and so on. The thorough study of both theory and methods about variational inequalities and vector optimization contained in the literature will help us to find new variational inequalities and vector optimization techniques for solving the practical problems. The aim of this special issue is to present new approaches and theories for variational inequalities and vector optimization problems arising in mathematics and applied sciences. This special issue includes 24 high-quality peer-reviewed papers that deal with different aspects of variational inequalities and vector optimization problems.These papers contain some new, novel, and innovative techniques and ideas. We hope that all the papers published in this special issue can motivate and foster further scientific works and development of the research in the area of the theory, algorithms, and applications of variational inequalities and vector optimization problems.

Real and Complex Monotone Communication Games

Noncooperative game-theoretic tools have been increasingly used to study many important resource allocation problems in communications, networking, smart grids, and portfolio optimization. In this paper, we consider a general class of convex Nash Equilibrium Problems (NEPs), where each player aims to solve an arbitrary smooth convex optimization problem. Differently from most of current works, we do not assume any specific structure for the players' problems, and we allow the optimization variables of the players to be matrices in the complex domain. Our main contribution is the design of a novel class of distributed (asynchronous) best-response- algorithms suitable for solving the proposed NEPs, even in the presence of multiple solutions. The new methods, whose convergence analysis is based on Variational Inequality (VI) techniques, can select, among all the equilibria of a game, those that optimize a given performance criterion, at the cost of limited signaling among the players. This is a major departure from existing best-response algorithms, whose convergence conditions imply the uniqueness of the NE. Some of our results hinge on the use of VI problems directly in the complex domain; the study of these new kind of VIs also represents a noteworthy innovative contribution. We then apply the developed methods to solve some new generalizations of SISO and MIMO games in cognitive radios and femtocell systems, showing a considerable performance improvement over classical pure noncooperative schemes.

GLOBAL EXPONENTIAL STABILITY OF REACTION-DIFFUSION TIME-VARYING DELAYED CELLULAR NEURAL NETWORKS WITH DIRICHLET BOUNDARY CONDITIONS

This paper is devoted to global exponential stability of reaction-diffusion time-varying delayed cellular neural networks with Dirichlet boundary conditions. Without assuming the monotonicity and differentiability of activation functions, nor symmetry of synaptic interconnection weights, the authors present some delay independent and easily verifiable sufficient conditions to ensure the global exponential stability of the equilibrium solution by using the method of variational parameter and inequality technique. These conditions obtained have important leading significance in the designs and applications of global exponential stability for reaction-diffusion neural circuit systems with delays. Lastly, one example is given to illustrate the theoretical analysis.

Economic equilibrium through variational inequalities

The purpose of this paper is to present an alternative proof of the existence of the Walrasian equilibrium for the Arrow–Debreu–McKenzie model by the variational inequality technique. Moreover, examples of the generalized Arrow–Debreu&#8

General iterative algorithms for solving mixed quasi-variational-like inclusions

In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857–869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan–KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233–247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.

Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions

In the present paper, we introduce the concept of η -relaxed strong convexity of a differentiable functional and extend Ding and Yao’s auxiliary variational inequality technique [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Computers and Mathematics with Applications, 49 (2005), 857–869] to develop iterative algorithms for finding the approximate solutions to the mixed quasi-variational-like inclusion problem (in short, MQVLIP) in a Banach space. On the one hand, we establish a result on the existence of a solution to the equilibrium problem by virtue of well-known Brouwer’s fixed-point theorem. Moreover, by using this result we derive the existence and uniqueness of a solution to the MQVLIP and the existence of the approximate solutions generated by the algorithm for the MQVLIP. On the other hand, we use the concepts of η -relaxed strong convexity of a differentiable functional and η -cocoercivity of a composite map to prove the strong convergence of the approximate solutions to the unique solution of the MQVLIP.

Analysis of exercise boundary of American interest rate option

By applying the variational inequality technique, we analyzed the behavior of the exercise boundary of the American-style interest rate option under the assumption that the interest rates obey a mean-reverting random walk as given by the Vasicek model. The monotonicity, boundedness and C ∞-smoothness of the exercise boundary are proved in this paper.

EXPONENTIAL STABILITY OF REACTION–DIFFUSION FUZZY RECURRENT NEURAL NETWORKS WITH TIME-VARYING DELAYS

Exponential stability of reaction–diffusion fuzzy recurrent neural networks (RDFRNNs) with time-varying delays are considered. By using the method of variational parameters, M-matrix properties and inequality technique, some delay-independent or delay-dependent sufficient conditions for guaranteeing the exponential stability of an equilibrium solution are obtained. One example is given to demonstrate the theoretical results.