Parametric Variational Inequality Research Articles

New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities

The aim of this work is to study the stability for a class of fuzzy fractional differential variational inequalities modeled by variational inequalities coupled with fuzzy fractional differential inclusions. By introducing the parametric problem with the help of two parameters in the involved mapping and constraint set, we first establish an existence result for the parametric fuzzy fractional differential variational inequality (PFFDVI), and we also establish the compactness and continuous dependence on the initial value for the PFFDVI. Furthermore, when the parameters are perturbed, we obtain the closedness of solution mapping for the PFFDVI. Then we further investigate the upper semicontinuity of solution mapping for the PFFDVI under some mild conditions. Finally, in order to illustrate the obtained results, we give a numerical example at the end of this paper.

Continuous Selections of Solutions to Parametric Variational Inequalities

Continuous Selections of Solutions to Parametric Variational Inequalities

Lipschitz Continuity Results for a Class of Parametric Variational Inequalities and Applications to Network Games

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks.

Decision-Focused Learning for Inverse Noncooperative Games: Generalization Bounds and Convergence Analysis

Finding the equilibrium strategy of agents is one of the central problems in game theory. Perhaps equally intriguing is the inverse of the above problem: from the available finite set of actions at equilibrium, how can we learn the utilities of players competing against each other and eventually use the learned models to predict their future actions? Instead of following an estimate-then-predict approach, this work proposes a decision-focused learning (DFL) method that directly learns the utility function to improve prediction accuracy. The game's equilibrium is represented as a layer and integrated into an end-to-end optimization framework. We discuss the statistical bounds of covering numbers for the set of solution functions corresponding to the solution of a generic parametric variational inequality. Also, we establish the generalization bound for the set of solution functions with respect to the smooth loss function with an improved rate. Moreover, we proposed an algorithm based on the iterative differentiation strategy to forward and backpropagate through the equilibrium layer. The convergence analysis of the proposed algorithm is established. Finally, We numerically validate the proposed framework in the utility learning problem among the agents whose utility functions are approximated by partially input convex neural networks (PICNN).

Coderivatives and Aubin properties of solution mappings for parametric vector variational inequality problems

Coderivatives and Aubin properties of solution mappings for parametric vector variational inequality problems

The Lipschitz Properties of the Value Function and the Solution Map to a Parametric Discrete Optimal Control Problem

Motivated by the works of Toan and Kien (J. Nonlinear Convex Anal.12:635–650, 2011) and Thuy and Toan (Acta Math. Vietnam.43:175–199, 2018) on the stability and the solution sensitivity in parametric discrete optimal control problems, this paper is devoted to the study of the Lipschitz property of the value function and the Aubin property of the solution map to a parametric dynamic programming problem with linear constraints. By establishing abstract results on the Lipschitz property of the value function to a parametric programming and the Aubin property of the solution map to a parametric variational inequality, we obtain the Lipschitz property of the value function and the Aubin property of the solution map to a parametric discrete optimal control problem.

A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem

In this paper, we present a numerical method to describe the solution set of a generalized Nash equilibrium problem (GNEP). Previous approaches show how to reformulate the GNEP as a family of parametric variational inequalities in the special case where the game has shared constraints. We extend this result to generalized Nash problems by means of an umbrella shared constraint approximation of the game. We show the validity of our approach on numerical examples from the literature, and we provide new results that pinpoint the handling of the algorithm’s parameters for its implementation. Last but not least, we extend, solve, and discuss an applied example of a generalized Nash equilibrium problem of environmental accords between countries.

Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach

We consider the sample average approximation method for a stochastic multiobjective programming problem constrained by parametric variational inequalities. The first order necessary conditions for the original problem and its sample average approximation (SAA) problem are established under constraint qualifications. By graphical convergence of set-valued mappings, the stationary points of the SAA problem converge to the stationary points of the true problem when the sample size tends to infinity. Under proper assumptions, the convergence of optimal solutions of SAA problems is also obtained. The numerical experiments on stochastic multiobjective optimization problems with variational inequalities are given to illustrate the efficiency of SAA estimators.

On a class of fuzzy parametric variational inequality controlled differential equation problems in finite dimension spaces

This work is motivated by the fact that very little is known about the fuzzy variational inequalities controlled differential equation problems in finite dimension real numeral spaces, which are studied more difficult than differential variational inequalities. It is interesting and challenging that how to solve the fuzzy variational inequalities in a fuzzy environment. The purpose of this paper is to introduce and study a class of new fuzzy parametric variational inequality controlled initial-value differential equation problems in finite dimensional Euclidean spaces. We establish existence of Caratheodory weak solutions for the fuzzy parametric variational inequality controlled initial-value differential equation problem under suitable conditions. Further, using method of centres with entropic regularization techniques and time-stepping methods, we emerge convergence analysis on iterative process for solving the initial-value differential fuzzy parametric inequalities. Finally, we give some open questions for our future research.

On describing the solution sets of generalized Nash games with shared constraints

In this paper we use parametric variational inequality problems for the purpose of describing entire solution sets of generalized Nash games with shared constraints. We prove two theoretical results and we introduce a computational method that practitioners can implement in applied problems modeled as generalized Nash games, under assumptions present in the current literature. Further, we give illustrative examples of how our computational technique is used to derive solution sets of known generalized Nash games previously not solved by existing techniques. We close with the presentation of an applied problem formulated as a generalized Nash game, namely a model of a joint implementation environmental accord between countries. We discuss the possible advantages of modeling it within a generalized Nash game framework.

The Mordukhovich Coderivative and the Local Metric Regularity of the Solution Map to a Parametric Discrete Optimal Control Problem

In this paper, we study the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric dynamic programming problem with linear constraints and convex cost functions. By establishing abstract results on the coderivative and the local metric regularity of the solution map to a parametric variational inequality, we obtain the Mordukhovich coderivative and the local metric regularity in Robinson’s sense of the solution map to a parametric discrete optimal control problem.

On the Stability and Solution Sensitivity of a Consumer Problem

Various stability properties and a result on solution sensitivity of a consumer problem are obtained in this paper. Focusing on some nice features of the budget map, we are able to establish the continuity and the locally Lipschitz continuity of the indirect utility function, as well as the Lipschitz–Holder continuity of the demand map under a minimal set of assumptions. The recent work of Penot (J Nonlinear Convex Anal 15:1071–1085, 2014) is our starting point, while an implicit function theorem of Borwein (J Optim Theory Appl 48:9–52, 1986) and a theorem of Yen (Appl Math Optim 31:245–255, 1995) on solution sensitivity of parametric variational inequalities are the main tools in our proofs.

Solution Stability of a Linearly Perturbed Constraint System and Applications

Linear complementarity problems and affine variational inequalities have been intensively investigated by different methods. Recently, some authors have shown that solution stability of these problems with respect to total perturbations can be effectively studied via a generalized linear constraint system. The present paper focuses on characterizing stability properties of the solution map of a linearly perturbed generalized linear constraint system. The obtained results lead to several stability conditions for parametric linear complementarity problems and affine variational inequalities in explicit forms.

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.

Existence and stability of solutions to inverse variational inequality problems

In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-Kuratowski-Mazurkiewicz (KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem. Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.

Stability of solution mappings for parametric bilevel vector equilibrium problems

In this paper, we first revisit the parametric bilevel vector equilibrium problems in Hausdorff topological vector spaces. Then we study the stability conditions such as (Hausdorff) upper semicontinuity, (Hausdorff) lower semicontinuity, outer-continuity and outer-openness of solutions for such problems. Many examples are provided to illustrate the essentialness of the imposed assumptions. For the applications, we obtain the stability results for the parametric vector variational inequality problems with equilibrium constraints and parametric vector optimization problems with equilibrium constraints.

Linear Openness of the Composition of Set-Valued Maps and Applications to Variational Systems

In this paper we study the linear openness of the composition of set-valued maps carried out thanks to applications of Nadler’s fixed point theorem and Lim’s lemma. As a byproduct, we obtain the Lipschitz property of the solution map of a generalized parametric equation and of parametric approximate variational inequalities, as well.

Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

A note on the optimality condition for a bilevel programming

The equality type Mordukhovich coderivative rule for a solution mapping to a second-order cone constrained parametric variational inequality is derived under the constraint nondegenerate condition, which improves the result published recently. The rule established is then applied to deriving a necessary and sufficient local optimality condition for a bilevel programming with a second-order cone constrained lower level problem.

Upper Semicontinuity of Solution Mappings to Parametric Generalized Vector Quasiequilibrium Problems

We establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems. As applications, we obtain the upper semicontinuity of solution mappings to several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem.