\documentstyle [twocolumn,12pt]{article} \oddsidemargin 1 in \evensidemargin .9 in \textwidth 5 in \newcommand{\co}{\mbox{${\mbox{co}}$}} \newcommand{\cl}{\mbox{${\mbox{cl}}$}} \newcommand{\Inf}{\mbox{${\mbox{Inf}\,}$}} \newcommand{\Sup}{\mbox{${\mbox{Sup}\,}$}} \newcommand{\Liminf}{\mbox{${\mbox{Liminf}\,}$}} \newcommand{\Limsup}{\mbox{${\mbox{Limsup}\,}$}} \newcommand{\op}{\mbox{${\mbox{Int}}$}} \newcommand{\R}{\mbox{${\mbox{I} \! \mbox{R}}$}} \newcommand{\real}{I\!\! R} \newcommand{\thm}{\newtheorem} \begin{document}
\title{On Vector Equilibrium Problems and Vector Optimization} \author{Jen-Chih Yao \\ \\ NSC 90-2115-M-110-017}
\date{90/ 08 / 01 -- 91/ 07 / 31} \maketitle
\noindent
{\bf 1. Abstract} In this three-year project, we shall focus our attention on vector equilibrium problems and vector optimization. In the first year of the project, we shall study generalized vector equilibrium problems and their existence. For the second year of the project, we shall consider vector optimization problems. In particular, we will investigate the relationship between vector equilibrium problems and vector optimization problems, and we also want to study the variational principles for vector equilibrium
problems. Finally, in the last year of the project, we shall consider the system of vector equilibrium problems and its applications.
problems, vector optimization, system of vector equilibrium problems. \vspace{10pt}
\noindent
{\bf 2. Introduction} \vspace{10pt}
Let $X$ and $Y$ be topological vector spaces and let $C$ be a pointed closed convex proper cone in $Y$ with nonempty interior. Then the cone $C$ induces a vector ordering in $Y$ setting for all $x, y \in Y$,
$$x \leq y \hspace{5pt} {\rm if \ and \ only \ if } \hspace{5pt} y-x \in C;$$ $$x \not\leq y \hspace{5pt} {\rm if \ and \ only \ if } \hspace{5pt} y-x \not\in C;$$
$$x < \hspace{5pt} {\rm if \ and \ only \ if } \hspace{5pt} y-x \in {\rm int}C;$$ $$x \not< \hspace{5pt} {\rm if \ and \ only \ if } \hspace{5pt} y-x \not\in {\rm int}C.$$
Ler $K$ be a nonempty convex set in $X$ and $f: K \times K \to Y$ be a bifunction satisfying $f(x, x)=0$ for all $x \in K$. The vector equilibrium problem (VEP) is to find
$x \in K$ such that
$$f(x, y) \not< 0, \hspace{5pt} {\rm for \ all}\ y \in K.$$
(VEP) is a natural extension of the scalar equilibrium problems and it has important applications in many problems like optimization, mathematical economics, variational inequality, complementarity problems, mathematical physics and so on.
It is well known that the (VEP) is closely related to the following problem: find $y \in K$ such that
$$f(x, y) \not> 0, \hspace{5pt} {\rm for\ all}\ x \in K.$$
The above problem can be termed as the dual vector equilibrium problem (DVEP). In the
third year of the project, we shall study the system of vecror equilibrium problems. In
fact, we shall consider the system of generalized vector equilibrium problems. We shall
establish some existence results for a solution of the system of the generalized vector
equilibrium problems. Some interesting applications are given.
\vspace{10pt} \noindent
{\bf 3. Results and Discussion} \vspace{10pt}
Let $I$ be an index set and for each $i \in I$, let $X_i$ be a Hausdorff topological vector space. Consider a family of nonempty convex subsets
$\{K_{i}\}_{i \in I}$ with $K_{i}$ in $X_{i}$. Let $K=\Pi_{i \in I}K_{i}$ and $X=\Pi_{i \in I}X_{i}$. For each $i \in I$, let $Y_{i}$ be a topological vector space and let $C_{i}:K \to \Pi(Y_{i})$ be a multivalued map such that for each $x \in K$, $C_{i}(x)$
is a proper, closed and convex cone with nonempty interior. For each $i \in I$, let $F_{i}:K \times K_{i} \to \Pi(Y_{i})$ be a multivalued bifunction. We consider the following
system of generalzied vector equilibrium problems (in short, SGVEP) which is to find $x \in K$ such that for each $i \in I$,
$$F_{i}(x, y_{i}) \not\subset -{\rm int}C_{i}(x), \vspace{15pt} {\rm for\ all}\ y_i \in K_i.$$
If the index set $I$ is a singl eton, then the (SGVEP) reduces to a generalized vector equilibrium problem studied in (Oettli and Schl \"{a}ger, [14]; Ansari and Yao, [4]; Ansari et al., [1]) which contains generalized implicit vector
variational inequality problem (Lee and Kum, [12]), ge neralized vector variational and variational-like inequality problems and vector equilibrium problems as special cases;
see for example (Lee et al., [11]; Bianchi et al., [7]; Oettli, [13]; Hadjisavvas and
Schaible, [9, 10]; Tan and Tinh, 1998; Giannessi, [8]) and references therein. We now
state the main results of this project without proof. \vspace{10pt}
{\bf Definition 1.} (Ansari and Yao, [4]) Let $W$ and $Z$ be topological vector spaces and $M$ a nonempty convex subset of $W$ and let $P:M \to \Pi(Z)$ be a multivalued map
such that for each $x \in M$, $P(x)$ is a closed convex cone with nonempty interior. A
multivalued bifunction $F: M \times M \to \Pi(Z) \cup \{\emptyset\}$ is called $P(x)$-quasiconvex-like if for all $x, y_{1}, y_{2} \in M$ and $t \in [0,1]$, we have
either
$$F(x, ty_{1}+(1-t)y_{2}) \subset F(x, y_{1})-P(x)$$ or
$$F(x, ty_{1}+(1-t)y_{2}) \subset F(x, y_{2})-P(x).$$
\noindent
{\bf Theorem 2.} For each $i \in I$, let $K_{i}$ be a nonempty convex subset of a Hausdorff topological vector space $X_{i}$ and let $Y_{i}$ be a topological vector space. For each $i \in I$, let $F_{i}:K \times K_{i} \to \Pi(Y_{i})$ be a multivalued bifunction. For each $i \in I$, assume that
\begin{enumerate}
\item[{\rm (i)}] $C_{i}:K \to \Pi(Y_{i})$ is a multivalued map such that for each $x \in K$, $C_{i}(x)$ is a proper, closed and convex cone with nonempty interior; \item[{\rm (ii)}] for all $x \in K$, $F(x, x_i) \not\subset -{\rm int}C_{i}(x)$, where $x_i$ is the $i$th component of $x$;
\item[{\rm (iii)}] for all $x \in K$, the set $\{y_i \in K_i: F_{i}(x, y_i) \subset -{\rm int}C_{i}(x)\}$ is convex;
\item[{\rm (iv)}] for all $y_i \in K_i$, the set $\{x \in K: F_{i}(x, y_i) \not\subset -{\rm int}C_{i}(x)\}$ is closed in $K$;
\item[{\rm (v)}] there exist a nonempty compact subset $N$ of $K$ and a nonempty compact convex subset $B_i$ of $K_i$ for each $i \in I$ such that for each $x \in K
\backslash N$ there exist $i \in I$ and $y_i \in B_i$ such that $F_{i}(x, y_i) \subset -{\rm int}C_{i}(x)$.
\end{enumerate}
Then the (SGVEP) has a solution. \vspace{10pt}
\noindent
{\bf Theorem 3.} For each $i \in I$, let $K_i, X_i, Y_i, C_i$ and $F_i$ be the same as in
Theorem 2. For each $i \in I$, assume that \begin{enumerate}
\item[{\rm (i)}] $Q_i:K \to \Pi(Y_i)=Y_i \backslash {-{\rm int}C_{i}(x)}$ for all $x \in K$
such that its graph is closed in $K \times Y_i$;
\item[{\rm (ii)}] for all $x \in K$, $F_{i}(x, x_i) \not\subset -{\rm int}C_{i}(x)$; \item[{\rm (iii)}] $F_i$ is $C_{i}(x)$-quasiconvex-like;
\item[{\rm (iv)}] for all $y_i \in K_i$, the multivalued map $x \mapsto F_{i}(x, y_i)$ is
upper semicontinuous on $K$;
\item[{\rm (v)}] there exist a nonempty compact subset $N$ of $K$ and a nonempty compact convex subset $B_i$ of $K_i$ for each $i \in I$ such that for each $x \in K
\backslash N$ there exist $i \in I$ and $y_i \in B_i$ such that $F_{i}(x, y_i) \subset -{\rm int}C_{i}(x)$.
\end{enumerate}
Then the (SGVEP) has a solution. \vspace{10pt}
\noindent
{\bf 4. Project Evaluation} \vspace{10pt}
In the third year of this project, we introduce the system of generalized vector equilibrium problems which includes as special cases the system of generalized implicit vector variational inequality problems, the system of generalized vector variational and variational-like inequality problems and the system of vector equilibrium problems. By using a maximal element theorem, we establish existence results for a solution of these systems. We also give applications to general Nash equilibrium problem for vector-valued functions. The results are interesting and make significant contribution to the literature. Part of the results of this project has been written as a full paper and has been published in Journal of Global Optimization,
Vol. 22, pp. 3-16, 2002. \vspace{10pt} \noindent {\bf References} \vspace{10pt} \noindent
1. Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems, Journal of Nonlinear Analysis, 47, 543-554, 2001.
\noindent
2. Q. H. Ansari, W. Otelli and D. Schl\"{a}ger, A generalization of vectorial equilibria, Mathematical Mathods of Operations Research, 46, 147-152, 1997. \noindent
3. Q. H. Ansari, S. Schaible and J. C. Yao, The system of vector equilibrium problems and its applications, Journal of Optimization Theory and applications, 107(3), 547-557, 2000.
\noindent
4. Q. H. Ansari and J. C. Yao, An existence result for the generalized vector equilibrium problem, Applied Mathematics Letters, 12, 53-56, 1999.
\noindent
5. Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to the system of variational inequalities, Bulliten of the Australian Mathematical Society, 54,
433-442, 1999. \noindent
6. Q. H. Ansari and J. C. Yao, On nondifferentiable and nonconvex vector
optimization problems, Journal of Optimization Theory and Applications, 106, 487 -500, 2000.
\noindent
7. M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, Journal of Optimization Theory and Applications, 92, 527-542, 1997.
\noindent
8. F. Giannessi, Vector Variational Inequalities and Vector Equilibria.
Mathematical Theories, Kluwer Academic Publications, Dordrecht -Boston-London, 2000. \noindent
9. N. Hadjisavvas \& S. Schaible, From scalar to vector equilibrium problems in the quasimonotone case, Journal of Optimization Theory and Applications,
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and M. Volle, editors, Generalized Convexity, Generalized Monotonicity: Recent Results,
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\noindent
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\noindent
13. W. Otelli, A remark on vector-valued equilibria and generalized monotonicity, Acta Mathematica Vietnamica, 22, 213-221, 1997.
\noindent
14. W. Otelli and D. Schl\"{a}ger, Existence of equilibria for monotone
multivalued mappings, Mathematical Methods of Operations Research, 48, 219-228, 1998.
\noindent
15. N. X. Tan and P. N. Tinh, On the existence of equilibrium points of vector functions, Numerical Functional Analysis and Optimization, 19, 141-156, 1998.
