Set optimization problems with objective set-valued maps are considered, and some criteria of solutions are defined. Also, cone lower semicontinuities of set-valued maps are introduced, and existence theorems of solutions of such problems are es-tablished. Moreover, some duality results of these problems are investigated

Kuroiwa, Daishi

Kyoto University Research Information Repository

TitleLagrange Duality of Set-Valued Optimization with NaturalCriteriaAuthor(s)Kuroiwa, DaishiCitation数理解析研究所講究録 (1998), 1068: 164-170Issue Date1998-10URL http://hdl.handle.net/2433/62514RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityLagrange Duality of Set-ValuedOptimization with Natural Criteria*(Daishi Kuroiwa)Department of Mathematics and Computer ScienceInterdisciplinary Faculty of Science and Engineering, Shimane University1060 Nishikawatsu, Matsue, Shimane 690-8504, JAPANAbstractSet optimization problems with objective set-valued maps are considered, andsome criteria of solutions are defined. Also, cone lower semicontinuities of set-valuedmaps are introduced, and existence theorems of solutions of such problems are es-tablished. Moreover, some duality results of these problems are investigated.1 IntroductionWe observe a set-valued optimization problem $(\mathrm{S}\mathrm{P})$ as follows:$(\mathrm{S}\mathrm{P})$ Minimize $F(x)$subject to $x\in S$where $X$ is a set, $(\}’,$ $\leq)$ an ordered vector space, $F$ a map from $X$ to $2^{1}$ , and $S$ anonempty subset of $\mathrm{D}\mathrm{o}\mathrm{m}(F)=\{x\in X|F(x)\neq\emptyset\}$ . This type of set-valued optimizationproblem has been developed as a generalization of vector-valued optimization problems foraround twenty years. In many paper concerned with set-valued optimization( $\mathrm{f}\mathrm{o}\mathrm{r}$ example[2, 5, 4, 6, 7, 11] $)$ , we can see that a minimal solution $x_{0}\in S$ is defined such as:$F(x_{0})\cap{\rm Min}\cup F(_{X)}x\in S\neq\emptyset$and this problem are often called (vector optimization with set-valued maps.’ Howeverthe criterion of solutions is sometimes not suitable for set-valued $\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}_{\mathrm{I}}\mathrm{n}\mathrm{i}_{\mathrm{Z}}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ because itis only based on simple comparisons between vectors though our problem is set-valuedoptimization.*This research is $\mathrm{P}^{\mathrm{a}\mathrm{l}\mathrm{t}}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$ supported by Grant-in-Aid for Scientific Research from the AIinistry ofEducation, Science and Culture of.Japan, No. 097401461068 1998 164-170 164Our aim of this paper is to introduce some natural, suitable, and proper criteria based oncomparisons between values of the (set-valued) objective map for set-valued optimization,and investigate some properties concerned with the problem. In this paper, we call suchcriteria based on the idea above natural criteria for set-valued optimization, see [9].The organization of this paper is as follows: in Section 2, we formulate our set-valued op-timization problem and define two types of notions of solutions. In Section 3, we introduce(natural) lower semicontinuities for set-valued maps, characterize such continuities, andderive some existence theorems of solutions by using the lower semicontinuities. Finally,we show some duality theorems for our set-valued optimization in Section 4.2 Natural Criteria of Set-Valued OptimizationFirst, we redefine our set-valued minimization problem $(\mathrm{S}\mathrm{P})$ . Let $X$ be a topologicalspace, $(Y, \leq_{K})$ an ordered topological space with an ordering convex cone $K,$ $F$ a mapfrom $X$ to $2^{Y}$ , and $S\subset \mathrm{D}\mathrm{o}\mathrm{m}(F)(=\{x\in X|F(x)\neq\emptyset\})$ . Our set-valued minimizationproblem is the following:$(\mathrm{S}\mathrm{P})$ Minimize $F(x)$subject to $x\in S$ .To define notions of solutions for our problem, we introduce some relations betweentwo nonempty sets which like the order relation in topological vector spaces; though thenumber types of such relations is six, we treat two important relations of them, see [8].In this paper, we define$A\leq^{l}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A+K\supset B$ ,$A\leq^{u}B\Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}A\subset B-K$ ,for nonempty subsets $A,$ $B$ of $Y$ . In these cases, $A$ is said to be smaller than $B$ with$l$-inequality(resp. $u$-inequality) if $A\leq^{l}B$ (resp. $A\leq^{u}B$ ).In the above notations, $l$ means lower and $u$ means upper. Note that $A\leq^{l}B$ is equivalentto ${\rm Min} A=\mathrm{b}\mathrm{I}\mathrm{i}\mathrm{n}B$ and $A\leq^{u}B$ is equivalent to ${\rm Max} A={\rm Max} B$ .By using the set relations above, we introduce two types criteria of minimal solutionsin the following definition. In this paper, when we consider $l$-minimal solution, we assumethat $F$ is $l$-closed map, that is $F(x)$ is $l$-closed for each $x\in X$ for simple consideration. Alsowe assume similar assumption, $u$-closedness of $F$ , when we consider $u$-minimal solution.Definition 2.1 [9] An element $x_{0}\in S$ is said to be(i) $l$-minimal solution of $(\mathrm{S}\mathrm{P})$ iffor any $x\in S$ with $F(x)\leq^{l}F(X_{0}),$ $F(X_{0})\leq^{l}F(x)$ is satisfied;(ii) $u$-minimal solution of $(\mathrm{S}\mathrm{P})$ iffor any $x\in S$ with $F(x)\leq^{u}F(x_{0}),$ $F(x_{0})\leq^{u}F(x)$ is satisfied.1653Semicontinuities of Set-Values Maps and ExistenceTheoremsTo consider existence of solutions of $(\mathrm{S}\mathrm{P})$ for our solutions, some cone semicontinuitieswere introduced in $[5, 9]$ .Definition 3.1 [9] A set-valued map $F$ is said to be $l$-type lower semicontinuous on $S$ iffor any $l$-closed subset $A$ of $Y,$ $\mathcal{L}^{l}(A)=\{x\in S|F(x)\leq^{l}A\}$ is closed.Definition 3.2 [9] A set-valued map $F$ is said to be $l$-type quasi lower semicontinuousat $x_{0}\in S$ if for each net $\{x_{\lambda}\}$ converges to $x_{0}$ with $\{F(x_{\lambda})\}$ is $l$-decreasing, that is,$F(x_{\lambda}’)\leq^{l}F(x_{\lambda})$ for $\lambda<\lambda’,$ $F(X_{0})\leq^{l}$ Lim $\sup_{\lambda}(F(x\lambda)+K)$ is satisfied. A set-valuedmap $F$ is said to be $l$ -type quasi lower semicontinuous on $S$ if it is $l$ -type quasi lowersemicontinuous at each point of $S$ .Definition 3.3 [5] A set-valued map $F$ is said to be upper $IC$-semicontinuous at $x_{0}\in S$ iffor any open set $V$ with $V\leq^{l}F(X_{0})$ , there exists a neighborhood $U$ of $x_{0}$ such that $x\in U$implies $V\leq^{l}F(x)$ ; A set-valued map $F$ is said to be upper $IC$-semicontinuous on $S$ if it isupper $IC$-semicontinuous at each point of $S$ .Now we can see some characterization with respect to these lower semicontinuities.Proposition 3.1 [9] Let $F$ be a $l$-closed set-valued map. Then we have the following:(i) upper $K$-semicontinuity on $S$ implies $l$ -type lower semicontinuity on $S$ ;(ii) $l$ -type lower semicontinuity on $S$ implies $l$ -type quasi lower semicontinuity on $S$ .Also, if $X$ and $Y$ are finite dimensional and $F$ is locally bounded, then we have(iv) $l$ -type lower semicontinuity on $S$ implies upper $K$-semicontinuity on $S$ .Moreover, $Y$ is the real-field, and $F$ is a singleton map, then $l$ -type lower semicontinuityand upper $K$-semicontinuity are equivalent to to the ordinary lower semicontinuity of real-valued functions.Note that quasi lower semicontinuity is more weaker than another semicontinuities.Now, we investigate $u$-type lower semicontinuities of set-valued maps.Definition 3.4 [9] A set-valued map $F$ is said to be $u$-type lower semicontinuous on $S$ iffor any $u$-closed subset $A$ of}’, $L^{u}(A)=\{x\in S|F(x)\leq^{u}A\}$ is closed.Definition 3.5 [9] A set-valued map $F$ is said to be $u$-type quasi lower semicontinuousat $x_{0}\in S$ if for each net $\{x_{\lambda}\}$ converges to $x_{0}$ with $\{F(x_{\lambda})\}$ is $u$-decreasing, that is,$F(x_{\lambda}’)\leq^{u}F(x_{\lambda})$ for $\lambda<\lambda’,$ $F(X_{0})\leq^{u}$ Lim $\sup_{\lambda}(F(x\lambda)+IC)$ is satisfied. A set-valuedmap $F$ is said to be $u$-type quasi lower sernicontinuous on $S$ if it is $u$-type quasi lowersemicontinuous at each point of $S$ .166Definition 3.6 [5] A set-valued map $F$ is said to be lower $K$-semicontinuous at $x_{0}\in S$ iffor any open set $V$ with $V\cap F(x_{0})\neq\emptyset$ , there exists a neighborhood $U$ of $x_{0}$ such that $x\in U$implies $V\cap(F(x)-K)\neq\emptyset$ ; A set-valued map $F$ is said to be lower $I\zeta$-semicontinuous on$S$ if it is lower $K$-semicontinuous at each point of $S$ .Now we can see some characterization with respect to these lower semicontinuities.Proposition 3.2 [9] Let $F$ be a $u$-closed set-valued map. Then we have the following:(i) $u$-type lower semicontinuity on $S$ implies $u$-type quasi lower semicontinuity on $S$ .Also, if $X$ and}’ are finite dimensional and $F$ is locally bounded, then we have(iii) $u$-type lower semicontinuity on $S$ is equivalent to lower $K$-semicontinuity on $S$ .Moreover, $Y$ is the real-field, and $F$ is a singleton map, then $u$-type lower semicontinuityand lower $I\zeta$-semicontinuity are equivalent to to the ordinary lower semicontinuity of real-valued functions.Now, we consider existence theorems for $l$-type and $u$-type minimal solutions.Theorem 3.1 [9] Let $X$ be a topological space and}’ an ordered topological vector space.If $S$ is a nonempty compact subset of $X$ and $F$ : $Sarrow 2^{1’}$ is a $l$-type quasi lower semicon-tinuous and $l$-closed set-valued map, then there exists a $l$ -type minimal solution of $(\mathrm{S}\mathrm{P})$ .Theorem 3.2 [9] Let $X$ be a topological space and $Y$ an ordered topological vector space.If $S$ is a nonempty compact subset of $X$ and $F$ : $Sarrow 2^{1^{r}}$ is a $u$-type quasi lower semi-.continuous and $u$-closed set-valued map, then there exists a $u$-type minimal solution of$(\mathrm{S}\mathrm{P})$ .By using one of the above theorems, we can prove the following:Corollary 3.1 Let $X$ be a topological space, $S$ a nonempty compact subset of $X$ , and$f$ : $Sarrow 2^{1’}$ is a lower semicontinuous, then there exists an element $x_{0}\in S$ such that$f(x_{0})= \inf_{x\in S}f(x)$ .Let $\}’*\mathrm{b}\mathrm{e}$ the topological dual space of 1 $r,$ $\theta^{*}$ the null vector of $Y^{*}$ . and $I\mathrm{c}’+$ the positivepolar cone of $IC$ , that is, $K^{+}=\{y^{*}\in\}’*|\langle y^{*}, k\rangle\geq 0,$ $\forall k\in K\}$ .Theorem 3.3 [9] Let $(x_{\text{ }}.d)$ be a complete metric space, 1 $r$ an ordered locally convexspace with the cone $I\iota’$ . Also, $F$ be a map from $X$ to $2^{1^{r}}$ satisfying the following conditions:$\bullet$ there exists $y^{*}\in I\mathrm{i}^{\prime+}\backslash \{\theta^{*}\}$ such thatinf $\langle y^{*}, F(.\mathcal{B})\rangle$ is finite for each $x\in S$$F(x_{1})\leq^{l}F(x_{2}),$ $X1,$ $X_{2}\in S\Rightarrow \mathrm{i}\mathrm{I}\mathrm{l}\mathrm{f}\langle y^{*}.F(x_{2})\rangle$ –inf $\langle y^{*}, F(x_{1})\rangle\geq d(x_{2,1}x)$167$\bullet$ $F:Sarrow 2^{Y}$ is $l$ -type lower semicontinuous and l-closed.Then, there exists a $l$ -type minimal solution of $(\mathrm{S}\mathrm{P})$ .Theorem 3.4 [9] Let (X, $d$) be a complete metric space, $Y$ an ordered locally convexspace with the cone $K$ . Also, $F$ be a map from $X$ to $2^{Y}$ satisfying the following conditions:$\bullet$ there exists $y^{*}\in K^{+}\backslash \{\theta^{*}\}$ such thatinf $\langle y^{*}, F(X)\rangle$ is finite for each $x\in S$$F(x_{1})\leq^{u}F(x_{2})jx_{1},$ $X2 \in S\Rightarrow\sup\langle y^{*}, F(x_{2})\rangle-\sup\langle y^{*}, F(x_{1})\rangle\geq d(x_{21}, x)$$\bullet$$F:Sarrow 2^{1^{\gamma}}$ is $u$-type lower semicontinuous and u-closed.Then, there exists a $u$-type minimal solution of $(\mathrm{S}\mathrm{P})$ .Using one of the above theorems, we can show Phelps’ theorem, see [1]:Corollary 3.2 Let $(\mathrm{I}^{\nearrow}, ||\cdot||)$ be a Banach space, $D$ a closed nonempty subset of $Y$ , and $I\zeta$a convex cone of $Y$ . If there exist $y^{*}\in K^{+}$ and $\alpha>0$ such that(i) $\langle y^{*}, \cdot\rangle$ is bounded from below on $D$ and(ii) $I\zeta\subset\{y\in \mathrm{I}’|\langle y^{*}, y\rangle+\alpha||y||\leq 0\}$ .Then ${\rm Min} D=\mathrm{E}\mathrm{x}\mathrm{t}_{K}D\neq\emptyset$ .4 Duality of Set-Valued OptimizationIn this section, we consider a $l$-type set-valued minimization problem with an inequalityconstraint $(\mathrm{S}\mathrm{P})$ and its dual problem $(\mathrm{S}\mathrm{D})$ .$(\mathrm{S}\mathrm{P})$$l-\mathrm{I}_{\mathrm{V}}\mathrm{I}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$ $F(x)$subject to $G(x)\leq^{l}\theta$$(\mathrm{S}\mathrm{D})$$l-\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}_{\mathrm{I}}\mathrm{n}\mathrm{i}_{\mathrm{Z}\mathrm{e}}$ $\Phi(T)$subject to $T\in \mathcal{L}_{+}(1’, Z)$where, $X$ is a nonempty set, $(Y, \leq_{K}),$ $(Z, \leq_{L})$ ordered vector spaces with ordering cones$K,$ $L$ , respectively, $F$ : $Xarrow 2^{Z},$ $G$ : $Xarrow 2^{1^{r}},$ $\mathcal{L}(Y, Z)=$ { $T$ : $Yarrow Z|T$ is iinear},$\mathcal{L}_{+}(1^{r}, Z)=\{T\in \mathcal{L}(]^{\nearrow}, Z)|T(IC)\subset L\},$ $\mathrm{G}\mathrm{r}(G)=\{(x, y)\in X\cross Y|y\in G(x)\}$ and$\Phi$ : $\mathcal{L}(1’, Z)arrow 2^{Z}$ defined by $\Phi(T)=l-{\rm Min}\{F(X)+T(y)|(x, y)\in \mathrm{G}\mathrm{r}(G)\}$ .Definition 4.1 (Solutions) $x_{0}$ is said to be(i) an $l$-feasible solution of $(\mathrm{S}\mathrm{P})$ if $G(x)\leq^{l}\theta$ :168(ii) an $l$-solution of $(\mathrm{S}\mathrm{P})$ if $x_{0}$ is $l$-feasible and$x\in X,$ $G(x)\leq^{l}\theta,$ $F(x)\leq^{l}F(X\mathrm{o})$ implies $F(X_{0})\leq^{l}F(x)$ .$T_{0}$ is said to be(i) a feasible solution of $(\mathrm{S}\mathrm{D})$ if$T_{0}\in \mathcal{L}_{+}(Y, Z)$ and $\Phi(T)\neq\emptyset$ ;(ii) an $l$-solution of $(\mathrm{S}\mathrm{D})$ if $T_{0}$ is feasible and there exists $A_{0}\in\Phi(T_{0})$ such that$T_{1}\in \mathcal{L}_{+}(Y, Z),$ $A_{1}\in\Phi(T),$ $A_{0}\leq^{l}A_{1}$ implies $A_{1}\leq^{l}A_{0}$Proposition 4.1 (Weak Duality)Let $x_{0}$ be an $l$-feasible solution of $(\mathrm{S}\mathrm{P}),$ $T_{1}$ an $l$-feasible solution of $(\mathrm{S}\mathrm{D})$ , and $(x_{1}, y_{1})$ anelement of $\mathrm{G}\mathrm{r}(G)$ satisfying $F(x_{1})+T_{1}(y_{1})\in\Phi(T_{1})$ . Then,$F(X_{0})\leq^{l}F(x_{1})+T_{1}(y_{1})$ implies $F(x_{1})+T_{1}(y_{1})\leq^{l}F(X\mathrm{o})$Definition 4.2 (Lagrangian Function) For $x\in X,$ $y\in Y,$ $T\in \mathcal{L}(Y, Z)$ ,$L(x, y, T)=F(x)+T(y)$ .In usual, $y$ is an element of $G(x)$ ..Definition 4.3 (Saddle Point. )$(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$ is said to be an $l$-saddle point of $L$ if(i) $L(x,$ $y,$ $\tau_{0}\mathrm{I}\leq^{l}L(x_{0}, y_{0}, \tau_{0}),$ $(x, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x0, y_{0}, \tau_{0})\leq^{l}L(x,$ $y,$ $\tau_{0)}$(ii) $L(x_{0}, y_{0}, T_{0})\leq^{l}L(X0, y0, T),$ $T\in \mathcal{L}_{+}(Y, Z)\Rightarrow L(x_{0,y_{0},)}T\leq^{l}L(X_{0,y_{0},T_{0})}$Theorem 4.1 Assume that $I\zeta$ is closed, $L$ is solid, and $F$ satisfies the following boundedcondition: for each $x\in \mathrm{D}\mathrm{o}\mathrm{m}(F)$ there exists $y^{*}\in IC^{+}$ such that$\bullet$ $\langle y^{*}, y\rangle>0$ for each $y\in I\zeta\backslash \{\theta\}$ ;$\bullet\inf_{y\in F(x)}\langle y^{*}, y\rangle>-\infty$.If $(.c_{0,}.y_{0,o} T)\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$ is an $l$-saddle point of $L$ , then we have(i) $y0\leq\theta$ and $T_{0}(y_{0})=\theta$ ;(ii) $x_{0}$ is an $l$-solution of $(\mathrm{s}\mathrm{P})$ ;(iii) $T_{0}$ is an $l$-solution of $(\mathrm{S}\mathrm{D})$ .Theorem 4.2 $(x_{0}, y_{0}, T_{0})\in \mathrm{G}\mathrm{r}(G)\cross \mathcal{L}_{+}(Y, Z)$ is an $l$-saddle point of $L$ if and only if(i) $L(x, y, \tau_{0})\leq^{l}L(x_{0}, y_{0}, T_{0}),$ $(x, y)\in \mathrm{G}\mathrm{r}(G)\Rightarrow L(x_{0}, y0, \tau 0)\leq^{l}L(x,$ $y,$ $T_{0)}$(ii) $y_{0}\leq\theta$ and $T_{0}(y_{0})=\theta$ .169References[1] H. Attouch and H. Riahi, Stability Results for Ekeland’s $\epsilon$-Variational Principle and ConeExtremal Solutions, Math. Oper. Res. 18 (1993), 173-201.[2] J. M. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J.Control Optim. 15 (1977), 57-63.[3] H. W. Corley, Existence and Lagrangian Duality for Maximizations of Set-Valued Functions,J. Optim. Theo. Appl. 54 (1987), 489-501.[4] H. W. Corley, Optimality Conditions for Maximizations of Set-Valued Functions, J. Optim.Theo. Appl. 58 (1988), 1-10.[5] D. T. Luc, “Theory of Vector Optimization,” Lecture Notes in Economics and MathematicalSystems 319, Springer-Verlag, 1989.[6] H. Kawasaki, Conjugate Relations and Weak Subdifferentials of Relations, Math. Oper. Re8.6 (1981), 593-607.[7] H. Kawasaki, A Duality Theorem in Multiobjective Nonlinear Programming, Math. Oper.Res. 7 (1982), 95-110.[8] D. Kuroiwa, On Cone Convexity of Set-Valued Maps, Nonlinear Analysis, Theory, MethodsEY Applications, 30 (1997).[9] D. Kuroiwa, Natural Criteria of Set-Valued Optimization Journal of Mathematical Analysisand Applications, submitted, 1998.[10] Z. F. Li and G. Y. Chen, Lagrangian Multipliers, Saddle Points, and Duality in VectorOptimization of Set-Valued Maps, Journal of Mathematical Analysis and Applications, 215(1997), 297-316.[11] P. L. Yu, Cone Convexity, Cone Extreme Points, and Nondominated Solutions in DecisionProblems with Multiobjectives, J. Optim. Theo. Appl. 14 (1974), 319-377.170

Lagrange Duality of Set-Valued Optimization with Natural Criteria

Daishi Kuroiwa

CiteSeerX

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/62514/1/1068-17.pdf

Lagrange Duality of Set-Valued Optimization with Natural Criteria

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository

CiteSeerX