Kong et al. Journal of Inequalities and Applications (2017) 2017:47 DOI 10.1186/s13660-017-1319-x
R E S E A R C H Open Access Optimality for set-valued optimization in the sense of vector and set criteria Xiangyu Kong, GuoLin Yu* and Wei Liu
*Correspondence: [email protected] Abstract Institute of Applied Mathematics, Beifang University of Nationalities, The vector criterion and set criterion are two defining approaches of solutions for the Yinchuan, Ningxia 750021, P.R. China set-valued optimization problems. In this paper, the optimality conditions of both criteria of solutions are established for the set-valued optimization problems. By using Studniarski derivatives, the necessary and sufficient optimality conditions are derived in the sense of vector and set optimization. MSC: Primary 90C29; 90C46; secondary 26B25 Keywords: vector optimization; set optimization; optimality conditions; pseudoconvexity; Studniarski derivative
1 Introduction Let F be a set-valued mapping from a set S to a real normed linear space Y and let D ⊂ Y be a convex cone. The set-valued optimization problem is formalized as follows: ⎧ ⎨ minimize F(x) (P) ⎩ subject to x ∈ S.
It is generally known that there are two types of criteria of solutions for problem (P): the vector criteria and set criteria. The vector criterion is the most commonly used in prob- lem (P), which is looking for efficient elements of the set F(S)= x∈S F(x). The problem (P) with vector criterion is often defined as set-valued vector optimization. Over the past decades, the set-valued vector optimization theory and its applications have been inves- tigated extensively by many authors. Fruitful results are produced in this field, and we recommend the excellent books [–] and the references therein to the reader. Another criterion, the set criterion, also called set optimization, was proposed by Kuroiwa [, ] in for the first time. Set optimization consists of looking for image sets F(x¯), with x¯ ∈ S, which satisfy some set-relations between the rest of the image sets F(x)withx ∈ S.In[], p. , Jahn states that the set relation approach opens a new and exciting field of research. Although the set criterion seems to be more natural and inter- esting than the traditional one, the study of set optimization is very limited, such as papers on existence conditions, see [–]; on duality theory, see [–]; on optimality condi- tions, see [–]; on scalarization [–]; on well-posedness properties, see [, ]; on Ekeland variational principles, see [, ].
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 2 of 11
The investigation of optimality conditions, especially as regards the vector criterion, has received tremendous attention in the research of optimization problems and has been studied extensively. As everyone knows most of the problems under consideration are nonsmooth, which leads to introducing many kinds of generalized derivatives by the au- thors. A meaningful concept is the Studniarski derivative [], which has some properties similar to the contingent derivative [] and the Hadamard derivative []. Recently, much attention has been paid to optimality conditions and related topics for vector optimization by using Studniarski derivatives; see [, ]. Inspired by the above observations, in this paper, the optimality conditions are estab- lished for problem (P) in both the vector criterion and set criterion by using Studniarski derivatives.Therestofthepaperisorganizedasfollows:InSection, some well-known definitions and results used in sequel are recalled. In Section and Section ,theneces- sary and sufficient optimality conditions are given in the sense of the vector criterion and set criterion, respectively.
2Preliminaries Throughout the rest of paper, it is assumed that X and Y are two real normed linear spaces, and C ⊂ Y is a convex closed pointed cones with nonempty interior, i.e., int C = ∅.The partial order of Y is deduced by cone C.TheorderinggivenbyC on Y is denoted by ≤, which is defined as follows:
y ≤ y if y – y ∈ C.
Let S ⊂ X be a nonempty set and x¯ ∈ S. The contingent cone to S at x¯ is defined by (see []): + T(S, x¯)= x ∈ X : ∃(tn) → ,(xn) → x with x¯ + tnxn ∈ S for all n ∈ N .
For a set-valued mapping F : X → Y ,theset dom(F):= x ∈ X : F(x) = ∅ ,
is called the domain of F,theset graph(F):= (x, y) ∈ X × Y : y ∈ F(x)
is called the graph of the map F,andtheset epi(F):= (x, y) ∈ X × Y : y ∈ F(x)+C
is termed the epigraph of F.Let(x¯, y¯) ∈ graph(F), based upon the definition of contingent cone, we can derive T(epi(F), (x¯, y¯)), which is formulated by + T epi(F), (x¯, y¯) := (x, y) ∈ X × Y : ∃(tn) → ,(xn, yn) → (x, y) with y¯ + tnyn ∈ F(x¯ + tnxn) for all n ∈ N .(.) Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 3 of 11
Definition . (see []) Let F : X → Y be a set-valued map. F is called C-pseudoconvex at (x¯, y¯) ∈ graph(F)if epi(F)–(x¯, y¯) ⊂ T epi(F), (x¯, y¯) .
For a set-valued mapping F : X → Y ,theupperandlowerStudniarskiderivativesofthe map F is defined as follows:
Definition . (see []) Let (x¯, y¯) ∈ graph(F). (i) The upper Studniarski derivatives of F at (x¯, y¯) is defined by
F(x¯ + tx )–y¯ DF(x¯, y¯)(x):= lim sup . (t,x )→(+,x) t
(ii) The lower Studniarski derivative of F at (x¯, y¯) is defined by
F(x¯ + tx )–y¯ DF(x¯, y¯)(x):= lim inf . (t,x )→(+,x) t
Remark . Regarding the upper and lower Studniarski derivatives of the map F,one has: (a) It is easily to see from Definition . that + DF(x¯, y¯)(x)= y ∈ Y : ∃(tn) → , ∃(xn, yn) → (x, y) s.t. ∀n, y¯ + tnyn ∈ F(x¯ + tnxn) , + DF(x¯, y¯)(x)= y ∈ Y : ∀(tn) → , ∀(xn) → x, ∃(yn) → y s.t. ∀n, y¯ + tnyn ∈ F(x¯ + tnxn) .
(b) The upper Studniarski derivative is an exactly contingent derivative [] or an upper Hadamard derivative [], and the lower Studniarski derivative is the lower Hadamard derivative introduced by Penot [].
Lemma . is an useful property associated with C-pseudoconvexity of a set-valued mapping, which will be used in the next two sections.
Lemma . Let S ⊂ X be a convex set and F : S → Y be C-pseudoconvex at (x¯, y¯) ∈ graph(F). One has
F(x)–y¯ ⊂ DF(x¯, y¯)(x – x¯)+C, for all x ∈ S. (.)
Proof Since F at (x¯, y¯)isC-pseudoconvex, it yields (x – x¯, y – y¯) ∈ T epi(F), (x¯, y¯) , for all x ∈ S, y ∈ F(x).
+ Therefore, we see from (.) that there exist (tn) → ,(xn, yn) → (x – x¯, y – y¯)suchthat
(x¯ + tnxn, y¯ + tnyn) ∈ epi(F). Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 4 of 11
Thus
y¯ + tnyn ∈ F(x¯ + tnxn)+C.
Then one has
F(x¯ + tnxn)–y¯ yn ∈ + C. tn
By taking the limits when n → +∞,wederive
y – y¯ ∈ DF(x¯, y¯)(x – x¯)+C, for all x ∈ S, y ∈ F(x).
So, we get (.), as desired.
Y Example . Let X = Y = R, C = R+ and F : X → be defined by F(x):= y ∈ Y : y ≥ x , for all x ∈ X.
Taking (, ) = (x¯, y¯) ∈ graph(F), we can derive that
DF(; )(x)=DF(; )(x)=R+, for all x ∈ X.
On the other hand, it is obviously that F is C-pseudoconvex at (x¯, y¯) = (, ) and satisfies
F(x)–y¯ ⊂ DF(x¯, y¯)(x – x¯)+C, for all x ∈ X.
3 Optimality conditions with vector criterion Let S ⊂ X be a nonempty set, F : S → Y be a set-valued mapping and (x¯, y¯) ∈ graph(F). From now on, for convention we always assume that the upper and lower Studniarski derivatives of the map F at (x¯, y¯) exist and dom DF(x¯, y¯)=dom DF(x¯, y¯)=S. Consider the set-valued optimization problem (P) formulated in Section .Weshalles- tablish the optimality conditions for vector criteria of solutions for problem (P). In order to distinguish from the case of set criteria, we rewrite the problem (P) as follows: ⎧ ⎨ min≤ F(x) (VP) ⎩ s.t. x ∈ S ⊂ X.
Definition . The pair (x¯, y¯) ∈ graph(F) is said to be a weak minimizer of (VP), denoted by y¯ ∈ WMin[F(S), C], if F(S)–y¯ ∩ (– int C)=∅.
Theorems . and . are necessary optimality conditions for the weak minimizer of problem (VP). Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 5 of 11
Theorem . Let (x¯, y¯) ∈ graph(F) be a weak minimizer of (VP). Then DF(x¯, y¯)(x) ∩ (– int C)=∅, ∀x ∈ S.(.)
+ Proof Let y ∈ DF(x¯, y¯)(x)withx ∈ S. Then there exist (tn) → ,(xn) → x¯ and yn ∈ F(x¯ +
tnxn)suchthat
y – y¯ n → y.(.) tn
Because (x¯, y¯) is a weak minimizer of (VP), one has F(x¯ + tnxn)–y¯ ∩ (– int C)=∅. (.)
Now, let us show that y ∈/ – int C. In fact, assuming that y ∈ – int C,since–int C is open, we obtain from (.)
yn – y¯ ∈ – int C,forlargen.
Hence, it follows from yn ∈ F(x¯ + tnxn)that yn – y¯ ∈ F(x¯ + tnxn)–y¯ ∩ – int C,forlargen,
which contradicts (.).
Theorem . Let (x¯, y¯) ∈ graph(F). If (x¯, y¯) is a weak minimizer of (VP), then DF(x¯, y¯)(x) ∩ (– int C)=∅, for all x ∈ T(S, x¯). (.)
Proof According to Theorem ., one shows that for all x ∈ S DF(x¯, y¯)(x) ∩ (– int C)=∅.(.)
We proceed by contradiction. If the conclusion in Theorem . does not hold, then there exists xˆ ∈ T(S, x¯)suchthat
DF(x¯, y¯)(xˆ) ∩ (– int C) = ∅.(.)
+ Noticing that xˆ ∈ T(S, x¯), there exist (tn) → and (xn) → x such that x¯ + tnxn ∈ S for all n. In addition, we see from (.) that there exists yˆ ∈ DF(x¯, y¯)(xˆ) ∩ (– int C). Hence, with the
above (tn)and(xn)thereis(yn) → yˆ such that
y¯ + tnyn ∈ F(x¯ + tnxn).
+ So, we see that there exist (tn) → ,(xn) → x,and(yn) → yˆ such that
y¯ + tnyn ∈ F(x¯ + tnxn). Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 6 of 11
Thus, yˆ ∈ DF(x¯, y¯)(xˆ) ∩ (– int C),
which is a contradiction to (.).
Now, we present a sufficient optimality condition for problem (VP) under the assump- tion of pesudoconvexity of F.
Theorem . Let S be a convex set and F be C-pseudoconvex at (x¯, y¯) ∈ graph(F). If DF(x¯, y¯)(x – x¯) ∩ (– int C)=∅, ∀x ∈ S,(.)
then (x¯, y¯) is a weak minimizer of (VP).
Proof We proceed by contradiction. Suppose that (x¯, y¯) is not a weak minimizer of (VP), then there exists x ∈ S such that F(x)–y¯ ∩ (– int C) = ∅.(.)
Since F is C-pseudoconvex at (x¯, y¯), one shows from Lemma . that
F(x)–y¯ ⊂ DF(x¯, y¯)(x – x¯)+C.
Therefore, one shows from (.)that DF(x¯, y¯)(x – x¯)+D ∩ (– int C) = ∅,
and as a consequence DF(x¯, y¯)(x – x¯) ∩ (– int C) = ∅.
This is a contradiction to (.).
4 Optimality conditions with set criterion This section works with the optimality conditions for problem (P) in the sense of the set criterion. Firstly, let us recall the concepts of set relations.
Definition . (see []) Let A and B be two nonempty sets of Y .WewriteA ≤l B,iffor all b ∈ B there exists a ∈ A such that a ≤ b. Here, ‘≤l’ is called a lower relation.
Remark . The above defined lower relation is equivalent to B ⊂ A + C and it is the generalization of the order induced by a pointed convex cone C in Y , in the following sense: a ≤ b if b ∈ a + C.
Definition . (see []) Let A and B be two nonempty sets of Y .WewriteA Definition . (see []) Let A be a family of subsets of Y . () A ∈ A is named a lower minimal of A,ifforeachB ∈ A such that B ≤l A,itsatisfies A ≤l B; () A ∈ A is called a lower weak minimal of A,ifforeachB ∈ A such that B A ∼ B ⇔ A ≤l B and B ≤l A whose classes it is denoted by [A]l. Analogously the relation ‘ In the sense of a set criterion, the problem (P) can be rewritten in the next form: ⎧ ⎨ min≤l {F(x)} (SP) ⎩ s.t. x ∈ S ⊂ X. In this case, a lower weak minimizer of the problem (SP) is a pair (x¯, F(x¯)) such that x¯ ∈ S and F(x¯) is a lower weak minimal of the family of images of F, i.e. the family F = F(x):x ∈ S . We say that F(x¯) is a lower weak minimal of (SP) instead of F. ⊂ R R R → R Example . Let S =[,] , Y = and C = +.LetF :[,] be a set-valued map and defined by ⎧ ⎪{ ∈ R } ⎨⎪ (y, y) :(y –) +(y –) = , x =, F(x)=⎪[(x,–x + ), (, –x + )], < x <, ⎩⎪ [(, ), (, )], x =, where [y, y ]withy, y ∈ R denotes the line segment, i.e. [y, y ]={λy +(–λ)y :≤ λ ≤ }. Considering the following problem (SP): ⎧ ⎨ min≤l {F(x)} (SP) ⎩ s.t. x ∈ [, ]. By computing, we can derive that (, F()) is the lower minimizer of (SP) and (x, F(x)) with < x ≤ is the lower weak minimizer of (SP). Definition . (see []) Let F(x¯) be a lower weak minimal of (SP). F(x¯)isnamedstrict lower weak minimal of (SP), if there exists a neighborhood U(x¯)ofx¯ such that F(x) ≮lw F(x¯), for all x ∈ U(x¯) ∩ S.Thenx¯ is called a strict lower weak minimum of (SP). Lemma . (see []) Let x¯ ∈ S. Then F(x¯) is a lower weak minimal of (SP) if and only if for each x ∈ S one of the conditions below is fulfilled: Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 8 of 11 () F(x) ∈ [F(x¯)]lw. () There exists y¯ ∈ F(x¯) such that (F(x)–y¯) ∩ (– int C)=∅. Definition . (see []) (Domination property). It is called that a subset A ⊂ Y has the D-weak minimal property, if for all y ∈ A there exists a weakly minimal a of A such that a – y ∈ (– int C) ∪{}. Lemma . Let x¯, x ∈ Sandy¯ ∈ F(x¯). If F(x) ≮lw F(x¯), WMin[F(x¯), C]={y¯} and F(x¯) has the C-weak minimal property, one has F(x)–y¯ ∩ (– int C)=∅.(.) Proof Assuming that (.) does not hold, that is, F(x)–y¯ ∩ (– int C) = ∅. Then there exist y ∈ F(x)andd ∈ int C such that y – y¯ =–d.So,ityieldsy¯ = y + d and y¯ ∈ F(x)+int C. On the other hand, it follows from WMin[F(x¯), C]={y¯} and the C-weak minimal property of F(x¯)that,forally¯ ∈ F(x¯), there is d ∈ int C ∪{} such that y¯ = y¯ + d ∈ F(x)+int C. Therefore, we derive that F(x¯) ⊂ F(x)+int C, which means F(x) Theorem . Let x¯ be a strict lower weak minimum of (SP). If WMin[F(x¯), C]={y¯} and F(x¯) has the C-weak minimal property, then DF(x¯, y¯)(x) ∩ (– int C)=∅, for all x ∈ S.(.) Proof Because x¯ is a strict lower weak minimum of (SP), we see from Definition . that there exists a neighborhood U(x¯)ofx¯ such that F(x ) ≮lw F(x¯) for all x ∈ U(x¯)∩S.Letx ∈ S. + Assuming that y ∈ DF(x¯, y¯)(x) then there exist (tn) → ,(xn) → x¯ and yn ∈ F(x¯ +tnxn)such that y – y¯ n → y. tn lw Therefore for large enough n,wecangetx¯ + tnxn ∈ U(x¯) ∩ S which verifies F(x¯ + tnxn) ≮ F(x¯). Thus, it follows from Lemma . that F(x¯ + tnxn)–y¯ ∩ (– int C)=∅,forlargen.(.) Let us prove that y ∈/ – int D.Otherwise,onehas yn – y¯ ∈ – int C,forlargen. Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 9 of 11 Noticing that yn ∈ F(x¯ + tnxn), we derive yn – y¯ ∈ F(x¯ + tnxn)–y¯ ∩ (– int C), for large n. This is a contradiction to (.). So, we derive (.), as desired. Theorem . Let x¯ be a strict lower weak minimum of (SP). If WMin[F(x¯), C]={y¯} and F(x¯) has the C-weak minimal property then DF(x¯, y¯)(x) ∩ (– int C)=∅, for all x ∈ T(S, x¯). (.) Proof Based upon Theorem ., we see that equation (.) holds. The rest of the proof can be followed by similar arguments to that of Theorem .. Theorem . Let S be convex set and F be C-pseudoconvex at (x¯, y¯) ∈ graph(F). If for every x ∈ S we have DF(x¯, y¯)(x – x¯) ∩ (– int C)=∅, ∀x ∈ S, then x¯ is a lower weak minimum of (SP). Proof Suppose that x¯ is not a lower weak minimum of (SP), it follows from Lemma . that there exists x ∈ S such that for each y¯ ∈ F(x¯)thereisy ∈ F(x )with y – y¯ ∈ – int C.(.) Since F is C-pseudoconvex at (x¯, y¯), we derive from Lemma . that F x – y¯ ⊂ DF(x¯, y¯) x – x¯ + C. Therefore, we obtain y – y¯ ∈ DF(x¯, y¯) x – x¯ + C. (.) Combining (.)with(.), we get DF(x¯, y¯) x – x¯ + D ∩ (– int C) = ∅, furthermore, DF(x¯, y¯) x – x¯ ∩ (– int D) = ∅, which is a contradiction. Remark . By comparing the results derived in Section , it can be found that the op- timality for the set criterion and vector criterion, with the suitable conditions, possesses the same forms in terms of Studniarshi derivatives. Kong et al. Journal of Inequalities and Applications (2017) 2017:47 Page 10 of 11 5Conclusions We have studied the optimality conditions for both the vector criterion and set criterion for a set-valued optimization problem in this note. We have presented two sufficient opti- mality conditions and a necessary condition for a weak minimizer in terms of Studniarski derivatives. In the set optimization criterion, utilizing the known results, we have proved two necessary optimality conditions for a strict lower weak minimum and a lower weak minimum. A sufficient optimality condition has been proved under the assumption of pseudoconvexity. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements This research was supported by Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101. The authors are grateful to the anonymous referees who have contributed to improve the quality of the paper. Received: 13 October 2016 Accepted: 10 February 2017 References 1. Aubin, JP, Frankowsa, H: Set-Valued Analysis. Birkhauser, Boston (1990) 2. Luc, DT: Theory of Vector Optimization. Springer, Berlin (1989) 3. Jahn, J: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2003) 4. Chen, G, Huang, X, Yang, X: Vector Optimization-Set-Valued and Variational Analysis. Springer, Berlin (2005) 5. Kuroiwa, D: Some duality theorems of set-valued optimization with natural criteria. 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