An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Wensheng Jia,Xiaoling Qiu,Dingtao Peng

doi:10.3390/math8010045

Abstract

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

Highlights

In the last two decades, the vector equilibrium problem (VEP) has received much attention because it provides a unified framework for many important particular problems such as vector variational inequality, vector optimization, vector saddle points, multiobjective games, multiobjective transportation equilibrium problems, and so forth
We propose an approximate theorem (Theorem 1) for vector equilibrium problems under bounded rationality
That means the bounded rationality is an approximate way to full rationality

Summary

Introduction

In the last two decades, the vector equilibrium problem (VEP) has received much attention because it provides a unified framework for many important particular problems such as vector variational inequality, vector optimization, vector saddle points, multiobjective games, multiobjective transportation equilibrium problems, and so forth. The approximation theorem of VEP has hardly been seen from the perspective of algorithms. The approximation theorems are central to many computing problems and the related theory provides insight as well as a foundation for algorithms. The VEP model is generally solved by numerical methods (iterative procedures or heuristic algorithms), producing approximations to the exact solutions. In general, obtaining exact solutions to many practical problems is not possible. The exact optimal solution is hardly obtained in the specific calculation process. It is very important that the approximate solution approaches the exact optimal solution which implies that the full rationality can be approached by the bounded rationality.

Preliminaries

Approximation Theorem

Generic Convergence

Vector Variational Inequality Problems

Vector Optimization Problems

Conclusions