# On vector matrix game and symmetric dual vector optimization problem

## 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.

## 1 Introduction

A matrix game is defined by B of a real $m×n$ matrix together with the Cartesian product ${S}_{n}×{S}_{m}$ of all n-dimensional probability vectors ${S}_{n}$ and all m-dimensional probability vectors ${S}_{m}$; that is, ${S}_{n}:=\left\{x={\left({x}_{1},\dots ,{x}_{n}\right)}^{T}\in {\mathbb{R}}^{n}:{x}_{i}\geqq 0,{\sum }_{i=1}^{n}{x}_{i}=1\right\}$, where the symbol T denotes the transpose. A point $\left(\overline{x},\overline{y}\right)\in {S}_{n}×{S}_{n}$ is called an equilibrium point of a matrix game B if ${x}^{T}B\overline{y}\leqq {\overline{x}}^{T}B\overline{y}\leqq {\overline{x}}^{T}By$ for all $x,y\in {S}_{n}$ and $\overline{x}B\overline{y}=v$, where v is value of the game. If $n=m$ and B is skew symmetric, then we can check that $\left(\overline{x},\overline{y}\right)\in {S}_{n}×{S}_{n}$ is an equilibrium point of the game B if and only if $B\overline{x}\leqq 0$ and $B\overline{y}\leqq 0$. When B is an $n×n$ skew symmetric matrix, $\overline{x}\in {S}_{n}$ is called a solution of the matrix game B if $B\overline{x}\leqq 0$ [1].

Consider the linear programming problem (LP) and its dual (LD) as follows:

(LP)  Minimize ${c}^{T}x$  subject to $Ax\geqq b$, $x\geqq 0$,

(LD)  Maximize ${b}^{T}y$  subject to ${A}^{T}y\leqq c$, $y\geqq 0$,

where $c\in {\mathbb{R}}^{n}$, $x\in {\mathbb{R}}^{n}$, $b\in {\mathbb{R}}^{m}$, $y\in {\mathbb{R}}^{m}$, $A=\left[{a}_{ij}\right]$ is an $m×n$ real matrix.

Now consider the matrix game associated with the following $\left(n+m+1\right)×\left(n+m+1\right)$ skew symmetric matrix B:

$B=\left[\begin{array}{ccc}0& {A}^{T}& -c\\ -A& 0& b\\ {c}^{T}& -{b}^{T}& 0\end{array}\right].$

Dantzig [1] gave the complete equivalence between the linear programming duality and the matrix game B. Many authors [25] have extended the equivalence results of Dantzig [1] to several kinds of scalar optimization problems. Very recently, Hong and Kim [6] defined a vector matrix game and generalized the equivalence results of Dantzig [1] to a vector optimization problem by using the vector matrix game.

Recently, Kim and Noh [4] established equivalent relations between a certain matrix game and symmetric dual problems. Symmetric duality in nonlinear programming, in which the dual of the dual is the primal, was first introduced by Dorn [7]. Dantzig, Eisenberg and Cottle [8] formulated a pair of symmetric dual nonlinear problems and established duality results for convex and concave functions with non-negative orthant as the cone. Mond and Weir [9] presented two pairs of symmetric dual vector optimization problems and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions.

In this paper, 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. Moreover, we give a numerical example for showing such equivalent relations.

## 2 Vector matrix game and existence theorem

Throughout this paper, we will denote the relative interior of ${S}_{p}$ by ${\stackrel{o}{S}}_{p}$, and we will use the following conventions for vectors in the Euclidean space ${\mathbb{R}}^{n}$ for vectors $x:=\left({x}_{1},\dots ,{x}_{n}\right)$ and $y:=\left({y}_{1},\dots ,{y}_{n}\right)$:

Consider the nonlinear programming problem (VOP):

where $X=\left\{x\in {\mathbb{R}}^{n}:g\left(x\right)\geqq b,x\geqq 0\right\}$, $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{p}$, $g:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ are continuously differentiable. The gradient $\mathrm{\nabla }f\left(x\right)$ is an $n×p$ matrix, and $\mathrm{\nabla }g\left(x\right)$ is an $n×m$ matrix.

Definition 2.1 [10]

A point $\overline{x}\in X$ is said to be an efficient solution for (VOP) if there exists no other feasible point $x\in X$ such that $\left({f}_{1}\left(x\right),\dots ,{f}_{p}\left(x\right)\right)\le \left({f}_{1}\left(\overline{x}\right),\dots ,{f}_{p}\left(\overline{x}\right)\right)$.

Now, we define solutions for a vector matrix game as follows.

Definition 2.2 [6]

Let ${B}_{i}$, $i=1,\dots ,p$, be real $n×n$ skew-symmetric matrices. A point $\overline{x}\in {S}_{n}$ is said to be a vector solution of the vector matrix game ${B}_{i}$, $i=1,\dots ,p$ if $\left({\overline{x}}^{T}{B}_{1}x,\dots ,{\overline{x}}^{T}{B}_{p}x\right)\nleqq \left({\overline{x}}^{T}{B}_{1}\overline{x},\dots ,{\overline{x}}^{T}{B}_{p}\overline{x}\right)\nleqq \left({x}^{T}{B}_{1}\overline{x},\dots ,{x}^{T}{B}_{p}\overline{x}\right)$ for any $x\in {S}_{n}$.

We proved the characterization of a vector solution of the vector matrix game in [6].

Lemma 2.1 [6]

Let ${B}_{i}$, $i=1,\dots ,p$, be an $n×n$ skew symmetric matrix. Then $\overline{y}\in {S}_{n}$ is a vector solution of the vector matrix game ${B}_{i}$, $i=1,\dots ,p$, if and only if there exists $\xi \in {\stackrel{o}{S}}_{p}$ such that $\left({\sum }_{i=1}^{p}{\xi }_{i}{B}_{i}\right)\overline{y}\leqq 0$.

Remark 2.1 Let ${B}_{i}$, $i=1,\dots ,p$, be an $n×n$ skew symmetric matrix. From Lemma 2.1, we can obtain the following remark saying that the vector matrix game can be solved by fixed point problems; $\overline{y}\in {S}_{n}$ is a vector solution of the vector matrix game ${B}_{i}$, $i=1,\dots ,p$, if and only if there exists $\xi \in {\stackrel{o}{S}}_{p}$ such that $\overline{y}\in {F}_{\xi }\left(\overline{y}\right)$, where ${F}_{\xi }\left(x\right)=\left\{y\in {S}_{n}\mid y\in x-\left({\sum }_{i=1}^{p}{\xi }_{i}{B}_{i}\right)x-{\mathbb{R}}_{+}^{n}\right\}$.

Noticing Remark 2.1, we can obtain an existence theorem for the vector matrix game.

Theorem 2.1 Let ${B}_{i}$, $i=1,\dots ,p$, be an $n×n$ skew symmetric matrix. Then there exists a vector solution of the vector matrix game ${B}_{i}$, $i=1,\dots ,p$.

Proof Let $\xi \in {\stackrel{o}{S}}_{p}$. Define a multifunction ${F}_{\xi }:{S}_{n}\to {S}_{n}$ by, for any $x\in {S}_{n}$,

${F}_{\xi }\left(x\right)=\left\{y\in {S}_{n}|y\in x-\left(\sum _{i=1}^{p}{\xi }_{i}{B}_{i}\right)x-{\mathbb{R}}_{+}^{n}\right\}.$

Then the multifunction ${F}_{\xi }$ is closed and hence upper semi-continuous, and so it follows from the well-known Kakutani fixed point theorem [11] that the multifunction ${F}_{\xi }$ has a fixed point. So, by Remark 2.1, there exists a vector solution of the vector matrix game ${B}_{i}$, $i=1,\dots ,p$. □

## 3 Equivalence relations

Now, we consider the nonlinear symmetric programming problem (SP) together with its dual (SD) as follows:

where $f:=\left({f}_{1},\dots ,{f}_{p}\right):{\mathbb{R}}^{n}×{\mathbb{R}}^{m}\to {\mathbb{R}}^{p}$ are continuously differentiable.

Consider the vector matrix game defined by the following $\left(n+m+1\right)×\left(n+m+1\right)$ skew symmetric matrix ${B}_{i}\left(x,y\right)$, $i=1,\dots ,p$, related to (SP) and (SD):

${B}_{i}\left(x,y\right)=\left[\begin{array}{ccc}0& -x{\mathrm{\nabla }}_{y}{f}_{i}{\left(x,y\right)}^{T}& -{\mathrm{\nabla }}_{x}{f}_{i}\left(x,y\right)\\ {\mathrm{\nabla }}_{y}{f}_{i}\left(x,y\right){x}^{T}& 0& {\mathrm{\nabla }}_{y}{f}_{i}\left(x,y\right)\\ {\mathrm{\nabla }}_{x}{f}_{i}{\left(x,y\right)}^{T}& -{\mathrm{\nabla }}_{y}{f}_{i}{\left(x,y\right)}^{T}& 0\end{array}\right].$

Now, we give equivalent relations between (SD) and the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,\dots ,p$.

Theorem 3.1 Let $\left(\overline{x},\overline{y},\overline{\xi }\right)$ be feasible for (SP) and (SD), with ${\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)={\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)×\left(\overline{x},\overline{y}\right)=0$. Let ${z}^{\ast }=1/\left(1+{\sum }_{i}{\overline{x}}_{i}+{\sum }_{j}{\overline{y}}_{j}\right)$, ${x}^{\ast }={z}^{\ast }\overline{x}$ and ${y}^{\ast }={z}^{\ast }\overline{y}$. Then $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)$ is a vector solution of the vector matrix game ${B}_{i}\left(\overline{x},\overline{y}\right)$, $i=1,\dots ,p$.

Proof Let $\left(\overline{x},\overline{y},\overline{\xi }\right)$ be feasible for (SP) and (SD). Then the following holds:

(3.1)
(3.2)
(3.3)
(3.4)

Multiplying (3.3) by $\overline{x}\geqq 0$ gives $-\overline{x}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{y}=0$ and from (3.2),

$-\overline{x}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{y}-{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0.$
(3.5)

Multiplying (3.1) by ${\overline{x}}^{T}\overline{x}\geqq 0$, ${\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right){\overline{x}}^{T}\overline{x}\leqq 0$. It implies that since ${\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0$,

${\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right){\overline{x}}^{T}\overline{x}+{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0.$
(3.6)

From (3.3) we have

${\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{x}-{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{y}=0.$
(3.7)

But ${z}^{\ast }>0$ by (3.4), from (3.5), (3.6) and (3.7), we get

(3.8)
(3.9)
(3.10)

From (3.8), (3.9) and (3.10), we have the following inequality:

$\left(\sum _{i=1}^{p}{\overline{\xi }}_{i}{B}_{i}\left(\overline{x},\overline{y}\right)\right)\left(\begin{array}{c}{x}^{\ast }\\ {y}^{\ast }\\ {z}^{\ast }\end{array}\right)\leqq 0.$

By Lemma 2.1, $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)$ is a vector solution of the vector matrix game ${B}_{i}\left(\overline{x},\overline{y}\right)$, $i=1,\dots ,p$. □

Theorem 3.2 Let $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)$ with ${z}^{\ast }>0$ be a vector solution of the vector matrix game ${B}_{i}\left(\overline{x},\overline{y}\right)$, $i=1,\dots ,p$, where $\overline{x}={x}^{\ast }/{z}^{\ast }$ and $\overline{y}={y}^{\ast }/{z}^{\ast }$. Then there exists $\overline{\xi }\in {\stackrel{o}{S}}_{p}$ such that $\left(\overline{x},\overline{y},\overline{\xi }\right)$ is feasible for (SP) and (SD), and ${\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)={\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)=0$. Moreover, if ${f}_{i}\left(\cdot ,y\right)$, $i=1,\dots ,p$, are convex for fixed y and ${f}_{i}\left(x,\cdot \right)$, $i=1,\dots ,p$, are concave for fixed x, then $\left(\overline{x},\overline{y}\right)$ is efficient for (SP) with fixed $\overline{\xi }$ and $\left(\overline{x},\overline{y}\right)$ is efficient for (SD) with fixed $\overline{\xi }$.

Proof Let $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)$ with ${z}^{\ast }>0$ be a vector solution of the vector matrix game ${B}_{i}\left(\overline{x},\overline{y}\right)$, $i=1,\dots ,p$. Then by Lemma 2.1, there exists $\overline{\xi }\in {\stackrel{o}{S}}_{p}$ such that

$\left(\sum _{i=1}^{p}{\overline{\xi }}_{i}{B}_{i}\left(\overline{x},\overline{y}\right)\right)\left(\begin{array}{c}{x}^{\ast }\\ {y}^{\ast }\\ {z}^{\ast }\end{array}\right)\leqq 0.$

Thus, we get

(3.11)
(3.12)
(3.13)
(3.14)

Dividing (3.11), (3.12) and (3.13) by ${z}^{\ast }>0$, we have

(3.15)
(3.16)
(3.17)

From (3.14),

$\overline{x}\geqq 0,\phantom{\rule{2em}{0ex}}\overline{y}\geqq 0.$
(3.18)

By (3.16), ${\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\left({\overline{x}}^{T}\overline{x}+1\right)\leqq 0$. It implies that since ${\overline{x}}^{T}\overline{x}+1>0$,

$-{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\geqq 0.$
(3.19)

From (3.15), $-\overline{x}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{y}\leqq {\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)$. Using (3.18) and (3.19), we obtain $0\leqq -\overline{x}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right){\left(\overline{x},\overline{y}\right)}^{T}\overline{y}\leqq {\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)$. It implies that $-{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0$. From (3.17), ${\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq {\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)$. But since $\overline{x}\geqq 0$ and ${\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\geqq 0$, ${\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\geqq 0$ and since $\overline{y}\geqq 0$ and ${\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0$, ${\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0$. Then we have

$0\leqq {\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq {\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\leqq 0.$

Hence, ${\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)={\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)$. Thus, $\left(\overline{x},\overline{y},\overline{\xi }\right)$ is feasible for (SP) and (SD) with ${f}_{i}\left(\overline{x},\overline{y}\right)-{\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)={f}_{i}\left(\overline{x},\overline{y}\right)-{\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)$, $i=1,\dots ,p$. Since $\left(\overline{x},\overline{y},\overline{\xi }\right)$ is feasible for (SD), by weak duality in [9], $\left({f}_{1}\left(x,y\right)-{y}^{T}{\mathrm{\nabla }}_{y}\left({\xi }^{T}f\right)\left(x,y\right),\dots ,{f}_{p}\left(x,y\right)-{y}^{T}{\mathrm{\nabla }}_{y}\left({\xi }^{T}f\right)\left(x,y\right)\right)\nleqq \left({f}_{1}\left(\overline{x},\overline{y}\right)-{\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right),\dots ,{f}_{p}\left(\overline{x},\overline{y}\right)-{\overline{y}}^{T}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\right)$ and $\left({f}_{1}\left(\overline{x},\overline{y}\right)-{\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right),\dots ,{f}_{p}\left(\overline{x},\overline{y}\right)-{\overline{x}}^{T}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)\right)\nleqq \left({f}_{1}\left(u,v\right)-{u}^{T}{\mathrm{\nabla }}_{u}\left({\xi }^{T}f\right)\left(u,v\right),\dots ,{f}_{p}\left(u,v\right)-{u}^{T}{\mathrm{\nabla }}_{u}\left({\xi }^{T}f\right)\left(u,v\right)\right)$ for any feasible $\left(u,v,\xi \right)$ of (SP) and (SD). Therefore, $\left(\overline{x},\overline{y}\right)$ is efficient for (SP) with fixed $\overline{\xi }$ and $\left(\overline{x},\overline{y}\right)$ is efficient for (SD) with fixed $\overline{\xi }$. □

Now, we give an example illustrating Theorems 3.1 and 3.2.

Example 3.1 Let ${f}_{1}\left(x,y\right)={x}^{2}-{y}^{2}$ and ${f}_{2}\left(x,y\right)=y-x$. Consider the following vector optimization problem (SP) together with its dual (SD) as follows:

Now, we determine the set of all vector solutions of the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,2$. Let

${B}_{i}\left(x,y\right)=\left(\begin{array}{ccc}0& -x{\mathrm{\nabla }}_{y}{f}_{i}{\left(x,y\right)}^{T}& -{\mathrm{\nabla }}_{x}{f}_{i}\left(x,y\right)\\ -{\mathrm{\nabla }}_{y}{f}_{i}\left(x,y\right){x}^{T}& 0& {\mathrm{\nabla }}_{y}{f}_{i}\left(x,y\right)\\ {\mathrm{\nabla }}_{x}{f}_{i}{\left(x,y\right)}^{T}& -{\mathrm{\nabla }}_{y}{f}_{i}{\left(x,y\right)}^{T}& 0\end{array}\right).$

Then

${B}_{1}\left(x,y\right)=\left(\begin{array}{ccc}0& 2xy& -2x\\ -2xy& 0& -2y\\ 2x& 2y& 0\end{array}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{B}_{2}\left(x,y\right)=\left(\begin{array}{ccc}0& -x& 1\\ x& 0& 1\\ -1& -1& 0\end{array}\right).$

Let $\left(x,y\right)\in {\mathbb{R}}^{2}$ and $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)\in {S}_{3}$ be a vector solution of the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,2$, if and only if there exist ${\xi }_{1}>0$, ${\xi }_{2}>0$, ${\xi }_{1}+{\xi }_{2}=1$ such that

$\left({\xi }_{1}\left(\begin{array}{ccc}0& 2xy& -2x\\ -2xy& 0& -2y\\ 2x& 2y& 0\end{array}\right)+{\xi }_{2}\left(\begin{array}{ccc}0& -x& 1\\ x& 0& 1\\ -1& -1& 0\end{array}\right)\right)\left(\begin{array}{c}{x}^{\ast }\\ {y}^{\ast }\\ {z}^{\ast }\end{array}\right)\leqq \left(\begin{array}{c}0\\ 0\\ 0\end{array}\right).$

there exist ${\xi }_{1}>0$, ${\xi }_{2}>0$, ${\xi }_{1}+{\xi }_{2}=1$ such that

$\left(\begin{array}{c}x\left(2y{\xi }_{1}-{\xi }_{2}\right){y}^{\ast }-\left(2x{\xi }_{1}-{\xi }_{2}\right){z}^{\ast }\\ -x\left(2y{\xi }_{1}-{\xi }_{2}\right){x}^{\ast }-\left(2y{\xi }_{1}-{\xi }_{2}\right){z}^{\ast }\\ \left(2x{\xi }_{1}-{\xi }_{2}\right){x}^{\ast }+\left(2y{\xi }_{1}-{\xi }_{2}\right){y}^{\ast }\end{array}\right)\leqq \left(\begin{array}{c}0\\ 0\\ 0\end{array}\right).$

Thus, we determine the set of all the vector solutions of the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,2$.

1. (I)

the case that $x>0$:

2. (a)

$2x{\xi }_{1}-{\xi }_{2}>0$, $2y{\xi }_{1}-{\xi }_{2}>0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(0,0,1\right)$.

3. (b)

$2x{\xi }_{1}-{\xi }_{2}>0$, $2y{\xi }_{1}-{\xi }_{2}=0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(0,\alpha ,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}$.

4. (c)

$2x{\xi }_{1}-{\xi }_{2}>0$, $2y{\xi }_{1}-{\xi }_{2}<0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(0,1,0\right)$.

5. (d)

$2x{\xi }_{1}-{\xi }_{2}=0$, $2y{\xi }_{1}-{\xi }_{2}>0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\alpha ,0,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}$.

6. (e)

$2x{\xi }_{1}-{\xi }_{2}=0$, $2y{\xi }_{1}-{\xi }_{2}=0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left({x}_{1},{x}_{2},{x}_{3}\right)\mid {x}_{1}\geqq 0,{x}_{2}\geqq 0,{x}_{3}\geqq 0,{x}_{1}+{x}_{2}+{x}_{3}=1\right\}$.

7. (f)

$2x{\xi }_{1}-{\xi }_{2}=0$, $2y{\xi }_{1}-{\xi }_{2}<0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(0,1,0\right)$.

8. (g)

$2x{\xi }_{1}-{\xi }_{2}<0$, $2y{\xi }_{1}-{\xi }_{2}>0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(1,0,0\right)$.

9. (h)

$2x{\xi }_{1}-{\xi }_{2}<0$, $2y{\xi }_{1}-{\xi }_{2}=0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}$.

10. (i)

$2x{\xi }_{1}-{\xi }_{2}<0$, $2y{\xi }_{1}-{\xi }_{2}<0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(0,1,0\right)$.

11. (II)

the case that $x=0$:

12. (a)

$2y{\xi }_{1}-{\xi }_{2}>0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(1-\alpha ,\alpha ,0\right)\mid \alpha \leqq \frac{{\xi }_{2}}{2y{\xi }_{1}},y>0,{\xi }_{1}>0,{\xi }_{2}>0,{\xi }_{1}+{\xi }_{2}=1\right\}$.

13. (b)

$2y{\xi }_{1}-{\xi }_{2}=0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}$.

14. (c)

$2y{\xi }_{1}-{\xi }_{2}<0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}$.

15. (III)

the case that $x<0$:

16. (a)

$2y{\xi }_{1}-{\xi }_{2}>0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\frac{2y{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},-\frac{2x{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},-\frac{2xy{\xi }_{1}-x{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}}\right)$: $2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}>0,2y{\xi }_{1}-{\xi }_{2}>0,2x{\xi }_{1}-{\xi }_{2}<0,{\xi }_{1}>0,{\xi }_{2}>0,{\xi }_{1}+{\xi }_{2}=1\right\}$.

17. (b)

$2y{\xi }_{1}-{\xi }_{2}=0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right):\left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}$.

18. (c)

$2y{\xi }_{1}-{\xi }_{2}<0$: $\left({x}^{\ast },{y}^{\ast },{z}^{\ast }\right)=\left(1,0,0\right)$.

Let $\left(x,y\right)\in {\mathbb{R}}^{2}$ and ${S}_{\left(x,y\right)}$ be the set of vector solutions of the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,2$. From (I), (II) and (III),

$\begin{array}{rcl}\bigcup _{\left(x,y\right)\in {\mathbb{R}}^{2}}S\left(x,y\right)& =& \left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}\cup \left\{\left(0,\alpha ,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}\\ \cup \left\{\left(\alpha ,0,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}\\ \cup \left\{\left(\alpha ,\beta ,\gamma \right)\mid \alpha \geqq 0,\beta \geqq 0,\gamma \geqq 0,\alpha +\beta +\gamma =1\right\}\\ \cup \left\{\left(\frac{2y{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},-\frac{2x{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},\\ -\frac{2xy{\xi }_{1}-x{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}}\right)|x<0,2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}>0,\\ 2y{\xi }_{1}-{\xi }_{2}>0,\phantom{\rule{0.25em}{0ex}}2x{\xi }_{1}-{\xi }_{2}<0,{\xi }_{1}>0,{\xi }_{2}>0,{\xi }_{1}+{\xi }_{2}=1\right\}.\end{array}$

Let $\left(\overline{x},\overline{y},\overline{\xi }\right)$ be feasible for (SP) and (SD) with $\overline{y}{\mathrm{\nabla }}_{y}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)=\overline{x}{\mathrm{\nabla }}_{x}\left({\overline{\xi }}^{T}f\right)\left(\overline{x},\overline{y}\right)=0$. We can easily check that

Thus,

Therefore, Theorem 3.1 holds.

Let $\left(x,y\right)\in {\mathbb{R}}^{2}$ and ${S}_{\left(x,y\right)}$ be the set of vector solutions of the vector matrix game ${B}_{i}\left(x,y\right)$, $i=1,2$. Then

$\begin{array}{rcl}\bigcup _{\left(x,y\right)\in {\mathbb{R}}^{2}}{S}_{\left(x,y\right)}& =& \left\{\left(\alpha ,1-\alpha ,0\right)\mid 0\leqq \alpha \leqq 1\right\}\cup \left\{\left(0,\alpha ,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}\\ \cup \left\{\left(\alpha ,0,1-\alpha \right)\mid 0\leqq \alpha \leqq 1\right\}\\ \cup \left\{\left(\alpha ,\beta ,\gamma \right)\mid \alpha \geqq 0,\beta \geqq 0,\gamma \geqq 0,\alpha +\beta +\gamma =1\right\}\\ \cup \left\{\left(\frac{2y{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},-\frac{2x{\xi }_{1}-{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}},\\ -\frac{2xy{\xi }_{1}-x{\xi }_{2}}{2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}}\right)|x<0,2y{\xi }_{1}-2x{\xi }_{1}-2xy{\xi }_{1}+x{\xi }_{2}>0,\\ 2y{\xi }_{1}-{\xi }_{2}>0,2x{\xi }_{1}-{\xi }_{2}<0,{\xi }_{1}>0,{\xi }_{2}>0,{\xi }_{1}+{\xi }_{2}=1\right\}.\end{array}$

So,

Let F be the set of all feasible solutions of (SP) and let G be the set of all feasible solutions of (SD). Then we can check that $\left\{\left(\frac{{\xi }_{2}}{2{\xi }_{1}},\frac{{\xi }_{2}}{2{\xi }_{1}},{\xi }_{1},{\xi }_{2}\right)\mid {\xi }_{1}>0,{\xi }_{2}>0,{\xi }_{1}+{\xi }_{2}=1\right\}\subset F\cap G$ and $\left(\frac{{\xi }_{2}}{2{\xi }_{1}}\right){\mathrm{\nabla }}_{y}\left({\xi }^{T}f\right)\left(\frac{{\xi }_{2}}{2{\xi }_{1}},\frac{{\xi }_{2}}{2{\xi }_{1}}\right)=\left(\frac{{\xi }_{2}}{2{\xi }_{1}}\right){\mathrm{\nabla }}_{x}\left({\xi }^{T}f\right)\left(\frac{{\xi }_{2}}{2{\xi }_{1}},\frac{{\xi }_{2}}{2{\xi }_{1}}\right)=0$. Therefore, Theorem 3.2 holds.

## References

1. Dantzig GB: A proof of the equivalence of the programming problem and the game problem. Cowles Commission Monograph 13. In Activity Analysis of Production and Allocation. Edited by: Koopmans TC. Wiley, New York; 1951:330–335.

2. Chandra S, Craven BD, Mond B: Nonlinear programming duality and matrix game equivalence. J. Aust. Math. Soc. Ser. B, Appl. Math 1985, 26: 422–429. 10.1017/S033427000000463X

3. Chandra S, Mond B, Duraga Prasad MV: Continuous linear programs and continuous matrix game equivalence. In Recent Developments in Mathematical Programming. Edited by: Kumar S. Gordan and Breach Science Publishers, New York; 1991:397–406.

4. Kim DS, Noh K: Symmetric dual nonlinear programming and matrix game equivalence. J. Math. Anal. Appl. 2004, 298: 1–13. 10.1016/j.jmaa.2003.12.049

5. Preda V: On nonlinear programming and matrix game equivalence. J. Aust. Math. Soc. Ser. B, Appl. Math 1994, 35: 429–438. 10.1017/S0334270000009528

6. Hong JM, Kim MH: On vector optimization problem and vector matrix game equivalence. J. Nonlinear Convex Anal. 2011, 12(3):651–662.

7. Dorn WS: A symmetric dual theorem for quadratic programs. J. Oper. Res. Soc. Jpn. 1960, 2: 93–97.

8. Dantzig GB, Eisenberg E, Cottle RW: Symmetric dual nonlinear programs. Pac. J. Math. 1965, 15: 809–812. 10.2140/pjm.1965.15.809

9. Mond B, Weir T: Symmetric duality for nonlinear multiobjective programming. Q. J. Mech. Appl. Math. 1965, 23: 265–269.

10. Sawaragi Y, Nakayama H, Tanino T: Theory of Multiobjective Optimization. Academic Press, Orlando; 1985.

11. Aubin JP: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam; 1979.

## Acknowledgements

The authors would like to thank the referees for giving valuable comments for the revision of the paper.

## Author information

Authors

### Corresponding author

Correspondence to Gue Myung Lee.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors, together discussed and solved the problems in the manuscript. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Hong, J.M., Kim, M.H. & Lee, G.M. On vector matrix game and symmetric dual vector optimization problem. Fixed Point Theory Appl 2012, 233 (2012). https://doi.org/10.1186/1687-1812-2012-233

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2012-233

### Keywords

• Symmetric Matrix
• Linear Programming Problem
• Probability Vector
• Vector Solution
• Vector Optimization Problem