On vector matrix game and symmetric dual vector optimization problem

Jeong Min Hong,Moon Hee Kim,Gue Myung Lee

doi:10.1186/1687-1812-2012-233

Abstract

Abstract A vector matrix game with more than two skew symmetric matrices, which is an extension of the matrix game, is defined and the symmetric dual problem for a nonlinear vector optimization problem is considered. Using the Kakutani fixed point theorem, we prove an existence theorem for a vector matrix game. We establish equivalent relations between the symmetric dual problem and its related vector matrix game. Moreover, we give an example illustrating the equivalent relations.

Highlights

A matrix game is defined by B of a real m × n matrix together with the Cartesian product Sn × Sm of all n-dimensional probability vectors Sn and all m-dimensional probability vectors Sm; that is, Sn := {x = (x, . . . , xn)T ∈ Rn : xi
When B is an n × n skew symmetric matrix, x ∈ Sn is called a solution of the matrix game B if Bx [ ]
A vector matrix game with more than two skew symmetric matrices, which is an extension of the matrix game, is defined and a nonlinear vector optimization problem is considered

Summary

Introduction

Kim and Noh [ ] established equivalent relations between a certain matrix game and symmetric dual problems. A vector matrix game with more than two skew symmetric matrices, which is an extension of the matrix game, is defined and a nonlinear vector optimization problem is considered. We formulate a symmetric dual problem for the nonlinear vector optimization problem and establish equivalent relations between the symmetric dual problem and the corresponding vector matrix game.

Results

Conclusion