Inverse Variational Inequality Research Articles

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Notes on random optimal control equilibrium problem via stochastic inverse variational inequalities

The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

<abstract><p>Compared to the standard variational inequalities, inverse variational inequalities are more suitable for pricing American options with indefinite payoff. This paper investigated the initial-boundary value problem of inverse variational inequalities constituted by a class of non-divergence type parabolic operators. We established the existence and Hölder continuity of weak solutions. Since the comparison principle in the case of standard variational inequalities is no longer applicable, we constructed an integral inequality using differential inequalities to determine the global upper bound of the solution. By combining it with the continuous method, we obtained the existence of weak solutions. Additionally, by employing truncation factors, we obtained the lower bound of weak solutions in the cylindrical subdomain, thereby obtaining the Hölder continuity.</p></abstract>

A Fixed-Time Noise-Tolerance Neurodynamic Approach for Inverse Variational Inequalities

A novel projection neurodynamic approach (PNA) possessed self-adaptive dynamic stepsize is proposed to deal with inverse variational inequality problems (IVIPs) in this paper. It is proved that the proposed PNA converges to the solution of the corresponding IVIPs within fixed-time under some mild conditions, and also proved that the convergence time upper bound of fixed-time neurodynamic approach does not depend on initial states. It is different from existing results with exponential convergence that the fixed-time range can be theoretically calculated. The proposed PNA has the advantage that they are all robust against bounded vanishing perturbations. Numerical simulation results expectably conform to the theoretical analysis of the proposed PNA.

Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods.

Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

A neural network for solving the generalized inverse mixed variational inequality problem in Hilbert Spaces

<abstract><p>In this paper, we study and analyze the generalized inverse mixed variational inequality. The existence and uniqueness of the solution of such problem are proposed. The neural network associated with the generalized inverse mixed variational inequality is presented, and moreover, the Wiener-Hopf equation which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality, is considered. The stability and existence of solution of such neural network are proved. Finally, we introduce some algorithms which are constructed by the concept of the neural network and display a numerical example for understanding our results.</p></abstract>

Inverse Tensor Variational Inequalities and Applications

Inverse Tensor Variational Inequalities and Applications

Stability for a Class of Differential Set-Valued Inverse Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained.

A random time-dependent noncooperative equilibrium problem

The paper deals with the random time-dependent oligopolistic market equilibrium problem. For such a problem the firms’ point of view has been analyzed in Barbagallo and Guarino Lo Bianco (Optim. Lett. 14: 2479–2493, 2020) while here the policymaker’s point of view is studied. The random dynamic optimal control equilibrium conditions are expressed by means of an inverse stochastic time-dependent variational inequality which is proved to be equivalent to a stochastic time-dependent variational inequality. Some existence and well-posedness results for optimal regulatory taxes are obtained. Moreover a numerical scheme to compute the solution to the stochastic time-dependent variational inequality is presented. Finally an example is discussed.

Global Exponential Stability of a Neural Network for Inverse Variational Inequalities

We investigate the convergence properties of a projected neural network for solving inverse variational inequalities. Under standard assumptions, we establish the exponential stability of the proposed neural network. A discrete version of the proposed neural network is considered, leading to a new projection method for solving inverse variational inequalities, for which we obtain the linear convergence. We illustrate the effectiveness of the proposed neural network and its explicit discretization by considering applications in the road pricing problem arising in transportation science. The results obtained in this paper provide a positive answer to a recent open question and improve several recent results in the literature.

Stability analysis of set-valued inverse variational inequalities in reflexive Banach spaces

Stability analysis of set-valued inverse variational inequalities in reflexive Banach spaces

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

A proximal neurodynamic model for solving inverse mixed variational inequalities

This paper proposes a proximal neurodynamic model (PNDM) for solving inverse mixed variational inequalities (IMVIs) based on the proximal operator. It is shown that the PNDM has a unique continuous solution under the condition of Lipschitz continuity (L-continuity). It is also shown that the equilibrium point of the proposed PNDM is asymptotically stable or exponentially stable under some mild conditions. Finally, three numerical examples are presented to illustrate effectiveness of the proposed PNDM.

A minimax approach for inverse variational inequalities

In this work, we characterize the existence of a solution for a certain variational inequality by means of a classical minimax theorem. In addition, we propose a numerical algorithm for the solution of an inverse problem associated with a variational inequality. To this end we state a collage-type result in this variational framework.

An inertial projection neural network for solving inverse variational inequalities

A novel inertial projection neural network (IPNN) is proposed for solving inverse variational inequalities (IVIs) in this paper. It is shown that the IPNN has a unique solution under the condition of Lipschitz continuity and that the solution trajectories of the IPNN converge to the equilibrium solution asymptotically if the corresponding operator is co-coercive. Finally, several examples are presented to illustrtae the effectiveness of the proposed IPNN.

Notes on a neural network approach to inverse variational inequalities

We consider a neural network approach to an inverse variational inequality which is assumed to have a non-empty set of solutions. In the case of gradient mappings, we prove that every trajectory of the network converges to the solution set in the case of convex potentials and if the solution set is singleton, the network is globally asymptotically stable at the equilibrium point. We also prove that if the network has a strongly convex potential, then the network is globally exponentially stable at the equilibrium point. Another purpose of this paper is to point out certain fatal mistakes in the paper by Zou et al. [A novel method to solve inverse variational inequality problems based on neural networks. Neurocomputing. 2016;173:1163–1168].

Solving policy design problems: Alternating direction method of multipliers-based methods for structured inverse variational inequalities

Inverse variational inequalities have broad applications in various disciplines, and some of them have very appealing structures. There are several algorithms (e.g., proximal point algorithms and projection-type algorithms) for solving the inverse variational inequalities in general settings, while few of them have fully exploited the special structures. In this paper, we consider a class of inverse variational inequalities that has a separable structure and linear constraints, which has its root in spatial economic equilibrium problems. To design an efficient algorithm, we develop an alternating direction method of multipliers (ADMM) based method by utilizing the separable structure. Under some mild assumptions, we prove its global convergence. We propose an improved variant that makes the subproblems much easier and derive the convergence result under the same conditions. Finally, we present the preliminary numerical results to show the capability and efficiency of the proposed methods.

Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces

In this paper, we employ the image space analysis to study constrained inverse variational inequalities by means of a nonlinear separation approach. We introduce such a nonlinear functional, which is based on the known Gerstewitz functional, and show its property as a weak separation function and a regular weak separation function under different parameter sets. In contrast to known results, we do not assume any topology on the considered spaces. Then, an alternative theorem is established, which leads directly to a sufficient and necessary optimality condition of the constrained inverse variational inequality. Finally, a gap function and an error bound are obtained for the constrained inverse variational inequality.