On vector matrix game and symmetric dual vector optimization problem
© Hong et al.; licensee Springer. 2012
Received: 30 June 2012
Accepted: 26 November 2012
Published: 28 December 2012
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.
A matrix game is defined by B of a real matrix together with the Cartesian product of all n-dimensional probability vectors and all m-dimensional probability vectors ; that is, , where the symbol T denotes the transpose. A point is called an equilibrium point of a matrix game B if for all and , where v is value of the game. If and B is skew symmetric, then we can check that is an equilibrium point of the game B if and only if and . When B is an skew symmetric matrix, is called a solution of the matrix game B if .
Consider the linear programming problem (LP) and its dual (LD) as follows:
(LP) Minimize subject to , ,
(LD) Maximize subject to , ,
where , , , , is an real matrix.
Dantzig  gave the complete equivalence between the linear programming duality and the matrix game B. Many authors [2–5] have extended the equivalence results of Dantzig  to several kinds of scalar optimization problems. Very recently, Hong and Kim  defined a vector matrix game and generalized the equivalence results of Dantzig  to a vector optimization problem by using the vector matrix game.
Recently, Kim and Noh  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 . Dantzig, Eisenberg and Cottle  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  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
where , , are continuously differentiable. The gradient is an matrix, and is an matrix.
Definition 2.1 
A point is said to be an efficient solution for (VOP) if there exists no other feasible point such that .
Now, we define solutions for a vector matrix game as follows.
Definition 2.2 
Let , , be real skew-symmetric matrices. A point is said to be a vector solution of the vector matrix game , if for any .
We proved the characterization of a vector solution of the vector matrix game in .
Lemma 2.1 
Let , , be an skew symmetric matrix. Then is a vector solution of the vector matrix game , , if and only if there exists such that .
Remark 2.1 Let , , be an 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; is a vector solution of the vector matrix game , , if and only if there exists such that , where .
Noticing Remark 2.1, we can obtain an existence theorem for the vector matrix game.
Theorem 2.1 Let , , be an skew symmetric matrix. Then there exists a vector solution of the vector matrix game , .
Then the multifunction is closed and hence upper semi-continuous, and so it follows from the well-known Kakutani fixed point theorem  that the multifunction has a fixed point. So, by Remark 2.1, there exists a vector solution of the vector matrix game , . □
3 Equivalence relations
where are continuously differentiable.
Now, we give equivalent relations between (SD) and the vector matrix game , .
Theorem 3.1 Let be feasible for (SP) and (SD), with . Let , and . Then is a vector solution of the vector matrix game , .
By Lemma 2.1, is a vector solution of the vector matrix game , . □
Theorem 3.2 Let with be a vector solution of the vector matrix game , , where and . Then there exists such that is feasible for (SP) and (SD), and . Moreover, if , , are convex for fixed y and , , are concave for fixed x, then is efficient for (SP) with fixed and is efficient for (SD) with fixed .
Hence, . Thus, is feasible for (SP) and (SD) with , . Since is feasible for (SD), by weak duality in , and for any feasible of (SP) and (SD). Therefore, is efficient for (SP) with fixed and is efficient for (SD) with fixed . □
Now, we give an example illustrating Theorems 3.1 and 3.2.
the case that :
, : .
, : .
, : .
, : .
, : .
, : .
, : .
, : .
, : .
the case that :
the case that :
: : .
Therefore, Theorem 3.1 holds.
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 and . Therefore, Theorem 3.2 holds.
The authors would like to thank the referees for giving valuable comments for the revision of the paper.
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