\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 set-valued vector optimization problems} \author{Jen-Chih Yao \\ \\ NSC 90-2218-E-110-002} \date{90/ 08 / 01 -- 91/ 07 / 31}
\maketitle \noindent
{\bf 1. Abstract} This project will be devoted to the study of vector optimization problems for set-valued mappings. Vector optimization problems are closely related to multiple objective programming and have been extensively studied and considered in the literature for last decade. Recently, vector optimization problems for set-valued
mappings have been noticed and investigated. But it is known that set-valued mappings are more complicated than single-valued mappings and there
are various different definitions of continuity and differentiability for set-valued mappings. We intend in this project to investigate and to study the vector optimization problems for set-valued mappings more deeply and thoroughly. We plan to study and introduce possibly new
definition of differentiability of set-valued mappings. Also, we hope to
derive optimality conditions as well as necessary and sufficient conditions for solutions. These results will cover the classical optimality condition (e.g., KKT conditions) as special cases.
{\bf Key words :} Vector optimization, weakly efficient solution, efficient solution, set-valued mappings.
\vspace{10pt} \noindent
{\bf 2. Introduction} \vspace{10pt}
Let $X$ and $Y$ be two real normed spaces and $C$ be a nonempty closed convex cone in $Y$ which introduces a partial order in $Y$. Let $K$ be a nonempty subset of $X$ and let $F:X \to 2^Y$ be a set-valued map. The set
$$G(F)=\{(x,y) \in X \times Y: y \in F(x)\}$$
is called the grapg of the map $F$. Let $y \in K$ be given. The contingent cone to $K$ at $y$ [4], denotd by $C(K;y)$, is the set of all $x \in K$ for which there exist sequences $\{t_n\} \subset (0, \infty)$ and $\{x_n\}$ converges to $x$ such that $y+t_{n}x_{n} \in K$ for all $n$ and $t_{n}x_{n} \to 0$. Let $(u, v) \in G(F)$ be given. The contingent derivative $D_{C}F(u,v)$ of $F$ at $(u,v)$ [1] is the set-valued map from $X$ into $Y$ defined by $y \in D_{C}F(u,v)(x)$ if and only if $(x,y) \in C(G(F);(u,v))$. Also, the radial drivative $D_{R}F(u,v)$ of $F$ at $(u,v) \in G(F)$ is the set-valued map from $X$ into $Y$ defined by $y \in D_{R}F(u,v)(x)$ if and only if $(x,y) \in R(G(F);(u,v))$ where $R(K;w)$ is the radial cone to $K$ at $w$ [4]. Let $(x, y) \in G(F)$ be given. A single -valued map $D^{e}_{C}F(x,y)$ from $X$ into $Y$ is called contingent epiderivative of $F$ at
$(x,y)$ if
$${\rm epi}(D^{e}_{C}F(x,y))=C({\rm epi}(F);(x,y))$$
where ${\rm epi}(F)$ is the epigraph of $F$. Also, a single-valued map $D^{e}_{R}F(x,y)$
from $X$ into $Y$ is called radial epiderivative of $F$ at $(x,y)$ if $${\rm epi}(D^{e}_{R}F(x,y))=R({\rm epi}(F);(x,y)).$$ Finally,
from $X$ into $Y$ is called $S$-epiderivative of $F$ at $(x,y)$ if $${\rm epi}(D^{e}_{S}F(x,y))=S({\rm epi}(F);(x,y))$$
where $S({\rm epi}(F);(x,y))$ is the $S$-cone to ${\rm epi}(F)$ at $(x,y)$ [8].
\vspace{10pt} \noindent
{\bf Theorem 1.} let $F:X \to 2^{Y}$ be a set-valued map and let $y \in F(x)$ be given. If the radial epiderivative $D^{e}_{R}F(x,y)$ exists, then it is unique. \vspace{10pt}
\noindent
{\bf Theorem 2.} Let $C$ be a pointed cone, $K$ be a convex set and let $F$ be a $C$-convex map on $K$. If the radial epiderivative $D^{e}_{R}F(x,y)$ exists, then it is positive homogenuous and subadditive.
\vspace{10pt} \noindent
{\bf Theorem 3.} Let $Y$ be the set of real numbers and assume that there are functions $f$ and $g$ from $X$ into $Y$ with
$${\rm epi}(g) \subset R({\rm epi}(F);(x,y)) \subset {\rm epi}(f).$$ Then the radial epiderivative $D^{e}_{R}F(x,y)$ is given by
$D^{e}_{R}F(x,y)(u)={\rm inf}\{v \in R:(u,v) \in R({\rm epi}(F);(x,y))\}$ \noindent
for all $x \in X$. \vspace{10pt} \noindent
{\bf Theorem 4.} Let $F:X \to R$ be a single -valued function and $x \in X$ be given. If $D^{e}_{R}F(x,y)$ exists, then it is lower semicontinuous.
\vspace{10pt} \noindent
{\bf 3. Results and Discussion} \vspace{10pt}
Consider the set-valued vector optimization problem (in short, VOP) $${\rm min}_{x \in K}F(x),$$
where $K$ is a nonempty subset of $X$ and $F$ is a set-valued map from $K$ into $Y$. Let $C$ be a closed convex cone in $Y$ with nonempty interior. A pair $(x,y)$ with $x \in K$ and $y \in F(x)$ is called a weak minimizer of the (VOP) if $y$ is a weakly minimal element of the set $F(K)=\cup_{u \in K}F(u)$, i.e.,
$$(\{y\}-{\rm int}C) \cap F(K) = \emptyset.$$
A pair $(x,y)$ with $x \in K$ and $y \in F(x)$ is called a strong minimizer of the (VOP) if $y$ is a strongly minimal element of the set $F(K)$, i.e.,
$$F(K) \subset \{ y \}+C.$$ \noindent
{\bf Proposition 1.} Let $(x,y) \in G(F)$ for which $D^{e}_{R}F(x,y)$ exists. Then for all $u \in K$,
$$F(u)-\{y\} \subset \{D^{e}_{R}F(x,y)(u-x)\} +C.$$ \vspace{10pt}
\noindent
{\bf Theorem 5.} Let $C$ have nonempty interior and $(x,y) \in G(F)$. If $D^{e}_{R}F(x,y)$ exists and for all $u \in K$,
$$D^{e}_{R}F(x,y)(u-x) \not\in -{\rm int}C,$$ then $(x,y)$ is a weak minimizer of the (VOP). \vspace{10pt}
\noindent
{\bf Theorem 6.} Let $C$ have nonempty interior and $(x,y) \in G(F)$. If $(x,y)$ is a weak minimizer of the (VOP) and the $S$-epiderivative $D^{e}_{S}F(x,y)$ exists, then for all $u \in K$,
$$D^{e}_{S}F(x,y)(u-x) \not\in -{\rm int}C.$$
\noindent
{\bf 4. Project Evaluation}
In this project, we first introduce the concepts of the $S$-epiderivative and radial epiderivative. Then we establish some optimality conditions for a set-valued vector optimization problem. These results are interesting in the sense that we can check some sufficient conditions to see if a vector optimizaiton problem has a weak minimizer. We also get some necessary
conditions for a weak minimizer. Overall speaking, the results of this project are interesting and make significant contribution to the literature.
\vspace{10pt} \vspace{10pt} \noindent {\bf References} \vspace{10pt} \noindent
1. J. P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, in Mathematical Analysis and Applications, Part A, Edited by Nachbin, Academic Press, New York, 160-229, 1981.\\
\noindent
2. J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.\\
\noindent
3. J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkauser, 1990.\\ \noindent
4. G. Bouligand, Sur l'existence des demi-tangents a une courbe de Jordan, Fund. Math., 15, 215-218, 1930.\\
\noindent
5. H. W. Corley, Optimality conditions for maximizations of set-valued
functions, Journal of Optimization Theory and Applications, 58. 1-10, 1988.\\ \noindent
6. J. Jahn and R. Rauh, Contingent epiderivatives and set-valued
optimization, Mathematical Methods of Operations Research, 46, 193-211, 1997.\\ \noindent
7. D. T. Luc, Contingent derivative of set-valued maps and applications to vector optimization, Mathematical Programming, 50, 99-111, 1991.\\
\noindent
8. D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, Journal of Optimization Theory and Applications, 70, 385 -395, 1991.\\ \noindent
9. A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numerical Functional Analysis and Optimization, 19(1 $\&$ 2), 121-140, 1998.\\
