Abstract

In this paper, we propose a new class of variational inequality problems, say, uncertain variational inequality problems based on uncertainty theory in finite Euclidean spaces $R^{n}$ . It can be viewed as another extension of classical variational inequality problems besides stochastic variational inequality problems. Note that both stochastic variational inequality problems and uncertain variational inequality problems involve uncertainty in the real world, thus they have no conceptual solutions. Hence, in order to solve uncertain variational inequality problems, we introduce the expected value of uncertain variables (vector). Then we convert it into a classical deterministic variational inequality problem, which can be solved by many algorithms that are developed on the basis of gap functions. Thus the core of this paper is to discuss under what conditions we can convert the expected value model of uncertain variational inequality problems into deterministic variational inequality problems. Finally, as an application, we present an example in a noncooperation game from economics.

Highlights

  • The variational inequality problem (VIP for short) is an important discipline of mathematics

  • The problem to find a solution to variational inequality ( ) is called a variational inequality problem associated with the mapping F and the subset S, Chen and Zhu Journal of Inequalities and Applications (2015) 2015:231 which is denoted by VIP(F, S)

  • In Section, we review some definitions and lemmas which are useful in the sequel; in Section, we present a class of uncertain variational inequalities, discuss its expected value model and convert it into a class of classical deterministic variational inequalities in detail; as an application, we investigate in Section an example developed from economics; we conclude the paper with Section

Read more

Summary

Introduction

The variational inequality problem (VIP for short) is an important discipline of mathematics. Definition (Liu [ ]) An uncertain variable is a function ξ from an uncertainty space ( , L, M) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B. Definition (Liu [ ]) The uncertainty distribution of an uncertain variable ξ is defined by (x) = M{ξ ≤ x}

Results
Conclusion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call