Optimality for set-valued optimization in the sense of vector and set criteria

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1 Kong et al. Journal of Inequalities and Applications :47 DOI /s 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: guolin_yu@126.com Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan, Ningxia , P.R. China Abstract The vector criterion and set criterion are two defining approaches of solutions for the 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: P minimize Fx 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 problem P, which is looking for efficient elements of the set FS= x S Fx. 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 investigated extensively by many authors. Fruitful results are produced in this field, and we recommendthe excellentbooks [1 4] and the references therein to the reader. Another criterion, the set criterion, also called set optimization, was proposed by Kuroiwa [5, 6] in 1999 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 Fx withx S. In[3], p. 378, 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 interesting than the traditional one, the study of set optimization is very limited, such as papers on existence conditions, see [7 12]; on duality theory, see [13 16]; on optimality conditions, see [17 19]; on scalarization [20 23];on well-posedness properties,see [24, 25];on Ekeland variational principles, see [26, 27]. The Authors This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Kong et al. Journal of Inequalities and Applications :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 authors. A meaningful concept is the Studniarski derivative [28], which hassome properties similar to the contingent derivative [1] and the Hadamardderivative [29]. Recently, much attention has been paid to optimality conditions and related topics for vector optimization by using Studniarski derivatives; see [30, 31]. Inspired by the above observations, in this paper, the optimality conditions are established for problem P in both the vector criterion and set criterion by using Studniarski derivatives.therestofthepaperisorganizedasfollows:insection2, some well-known definitions and results used in sequel are recalled. In Section 3 and Section 4, thenecessary and sufficient optimality conditions are given in the sense of the vector criterion and set criterion, respectively. 2 Preliminaries 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 1 y 2 if y 2 y 1 C. Let S X be a nonempty set and x S. The contingent cone to S at x is defined by see [1]: TS, x= { x X : t n 0 +,x n x with x + t n x n S for all n N }. For a set-valued mapping F : X 2 Y,theset domf:= { x X : Fx }, is called the domain of F,theset graphf:= { x, y X Y : y Fx } is called the graph of the map F,andtheset epif:= { x, y X Y : y Fx+C } is termed the epigraph of F. Let x, ȳ graphf, based upon the definition of contingent cone, we can derive TepiF, x, ȳ, which is formulated by T epif, x, ȳ := { x, y X Y : t n 0 +,x n, y n x, y with ȳ + t n y n F x + t n x n for all n N }. 2.1

3 Kong et al. Journal of Inequalities and Applications :47 Page 3 of 11 Definition 2.1 see [1] Let F : X 2 Y be a set-valued map. F is called C-pseudoconvex at x, ȳ graphfif epif x, ȳ T epif, x, ȳ. For a set-valued mapping F : X 2 Y,theupperandlowerStudniarskiderivativesofthe map F is defined as follows: Definition 2.2 see [28] Let x, ȳ graphf. i The upper Studniarski derivatives of F at x, ȳ is defined by DF x, ȳx:= lim sup t,x 0 +,x F x + tx ȳ. t ii The lower Studniarski derivative of F at x, ȳ is defined by DF x, ȳx:= lim inf t,x 0 +,x F x + tx ȳ. t Remark 2.3 Regarding the upper and lower Studniarski derivatives of the map F, one has: a It is easily to see from Definition 2.2 that DF x, ȳx= { y Y : t n 0 +, x n, y n x, y s.t. n, ȳ + t n y n F x + t n x n }, DF x, ȳx= { y Y : t n 0 +, x n x, y n y s.t. n, ȳ + t n y n F x + t n x n }. b The upper Studniarski derivative is an exactly contingent derivative [1] or an upper Hadamard derivative [29], and the lower Studniarski derivative is the lower Hadamard derivative introduced by Penot [29]. Lemma 2.4 is an useful property associated with C-pseudoconvexity of a set-valued mapping, which will be used in the next two sections. Lemma 2.4 Let S X be a convex set and F : S 2 Y be C-pseudoconvex at x, ȳ graphf. One has Fx ȳ DF x, ȳx x+c, for all x S. 2.2 Proof Since F at x, ȳisc-pseudoconvex, it yields x x, y ȳ T epif, x, ȳ, for all x S, y Fx. Therefore, we see from 2.1 that there exist t n 0 +,x n, y n x x, y ȳsuchthat x + t n x n, ȳ + t n y n epif.

4 Kong et al. Journal of Inequalities and Applications :47 Page 4 of 11 Thus ȳ + t n y n F x + t n x n +C. Then one has y n F x + t nx n ȳ t n + C. By taking the limits when n +,wederive y ȳ DF x, ȳx x+c, for all x S, y Fx. So, we get 2.2, as desired. Example 2.5 Let X = Y = R, C = R + and F : X 2 Y be defined by Fx:= { y Y : y x 2}, for all x X. Taking 0, 0 = x, ȳ graphf, we can derive that DF0; 0x=DF0; 0x=R +, for all x X. On the other hand, it is obviously that F is C-pseudoconvex at x, ȳ = 0, 0 and satisfies Fx ȳ DF x, ȳx x+c, for all x X. 3 Optimality conditions with vector criterion Let S X be a nonempty set, F : S 2 Y be a set-valued mapping and x, ȳ graphf. From now on, for convention we always assume that the upper and lower Studniarski derivatives of the map F at x, ȳ exist and dom DF x, ȳ =dom DF x, ȳ =S. Consider the set-valued optimization problem P formulated in Section 1.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 Fx VP s.t. x S X. Definition 3.1 The pair x, ȳ graphf is said to be a weak minimizer of VP, denoted by ȳ WMin[FS, C], if FS ȳ int C=. Theorems 3.2 and 3.3 are necessary optimality conditions for the weak minimizer of problem VP.

5 Kong et al. Journal of Inequalities and Applications :47 Page 5 of 11 Theorem 3.2 Let x, ȳ graphf be a weak minimizer of VP. Then DF x, ȳx int C=, x S. 3.1 Proof Let y DF x, ȳx withx S. Then there exist t n 0 +,x n x and y n F x + t n x n suchthat y n ȳ t n y. 3.2 Because x, ȳ is a weak minimizer of VP, one has F x + tn x n ȳ int C=. 3.3 Now, let us show that y / int C. In fact, assuming that y int C, since int C is open, we obtain from 3.2 y n ȳ int C, forlargen. Hence, it follows from y n F x + t n x n that y n ȳ F x + t n x n ȳ int C, forlargen, which contradicts 3.3. Theorem 3.3 Let x, ȳ graphf. If x, ȳ is a weak minimizer of VP, then DF x, ȳx int C=, for all x TS, x. 3.4 Proof According to Theorem 3.2, one shows that for all x S DF x, ȳx int C=. 3.5 We proceed by contradiction. If the conclusion in Theorem 3.3 does not hold, then there exists ˆx TS, xsuchthat DF x, ȳˆx int C. 3.6 Noticing that ˆx TS, x, there exist t n 0 + and x n x such that x + t n x n S for all n. In addition, we see from 3.6 that there exists ŷ DF x, ȳˆx int C. Hence, with the above t n andx n thereisy n ŷ such that ȳ + t n y n F x + t n x n. So, we see that there exist t n 0 +,x n x,andy n ŷ such that ȳ + t n y n F x + t n x n.

6 Kong et al. Journal of Inequalities and Applications :47 Page 6 of 11 Thus, ŷ DF x, ȳˆx int C, which is a contradiction to 3.5. Now, we present a sufficient optimality condition for problem VP under the assumption of pesudoconvexity of F. Theorem 3.4 Let S be a convex set and F be C-pseudoconvex at x, ȳ graphf. If DF x, ȳx x int C=, x S, 3.7 then x, ȳ is a weak minimizer of VP. Proof We proceed by contradiction. Suppose that x, ȳ is not a weak minimizer of VP, then there exists x S such that Fx ȳ int C. 3.8 Since F isc-pseudoconvexat x, ȳ, oneshowsfromlemma2.4that Fx ȳ DF x, ȳx x+c. Therefore, one shows from 3.8that DF x, ȳx x+d int C, and as a consequence DF x, ȳx x int C. This is a contradiction to 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 4.1 see [5] 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 4.2 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 4.3 see [14] Let A and B be two nonempty sets of Y.WewriteA < lw B,if B A + int C. Here, < lw is named a lower weak relation.

7 Kong et al. Journal of Inequalities and Applications :47 Page 7 of 11 Definition 4.4 see [14] Let A be a family of subsets of Y. 1 A A is named a lower minimal of A,ifforeachB A such that B l A,itsatisfies A l B; 2 A A is called a lower weak minimal of A,ifforeachB A such that B < lw A,it satisfies A < lw B. The relation l determinesina the equivalence relation: A B A l B and B l A whose classes it is denoted by [A] l. Analogously the relation < lw determinesina an equivalence relation whose classes are written [A] lw. In the sense of a set criterion, the problem P can be rewritten in the next form: SP min l {Fx} 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 = { Fx:x S }. We say that F x is a lower weak minimal of SP instead of F. Example 4.5 Let S =[0,1] R, Y = R 2 and C = R 2 +.LetF 0 :[0,1] 2 R2 be a set-valued map and defined by {y 1, y 2 R 2 :y y =1}, x =0, F 0 x= [x, x + 1, 1, x + 1], 0 < x <1, [0, 1, 1, 0], x =1, where [y, y ]withy, y R 2 denotes the line segment, i.e. [y, y ]={λy +1 λy :0 λ 1}. Considering the following problem SP 0 : SP 0 min l {F 0 x} s.t. x [0, 1]. By computing, we can derive that 1, F1 is the lower minimizer of SP 0 and x, Fx with 0 < x 1 is the lower weak minimizer of SP 0. Definition 4.6 see [19] 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 of x such that Fx lw F x, for all x U x S.Then x is called a strict lower weak minimum of SP. Lemma 4.7 see [19] 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:

8 Kong et al. Journal of Inequalities and Applications :47 Page 8 of 11 1 Fx [F x] lw. 2 There exists ȳ F x such that Fx ȳ int C =. Definition 4.8 see [19] 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 {0}. Lemma 4.9 Let x, x Sandȳ F x. If Fx lw F x, WMin[F x, C]={ȳ} and F x has the C-weak minimal property, one has Fx ȳ int C=. 4.1 Proof Assuming that 4.1 doesnothold, thatis, Fx ȳ int C. Then there exist y Fx andd int C such that y ȳ = d. So,ityieldsȳ = y + d and ȳ Fx+int C. On the other hand, it follows from WMin[F x, C]={ȳ} and the C-weak minimal property of F xthat,forallȳ F x, there is d int C {0} such that ȳ = ȳ + d Fx+int C. Therefore, we derive that F x Fx+int C, which means Fx< lw F x. This contradicts the hypothesis. Theorem 4.10 Let x be a strict lower weak minimum of SP. If WMin[F x, C]={ȳ} and F x has the C-weak minimal property, then DF x, ȳx int C=, for all x S. 4.2 Proof Because x is a strict lower weak minimum of SP, we see from Definition 4.6 that there exists a neighborhood U xof x such that Fx lw F x for all x U x S.Letx S. Assuming that y DF x, ȳx then there exist t n 0 +,x n x and y n F x+t n x n such that y n ȳ t n y. Therefore for large enough n,wecanget x + t n x n U x S which verifies F x + t n x n lw F x. Thus, it follows from Lemma 4.9 that F x + tn x n ȳ int C=, forlargen. 4.3 Let us prove that y / int D.Otherwise,onehas y n ȳ int C, forlargen.

9 Kong et al. Journal of Inequalities and Applications :47 Page 9 of 11 Noticing that y n F x + t n x n, we derive y n ȳ F x + t n x n ȳ int C, for large n. This is a contradiction to 4.3. So, we derive 4.2, as desired. Theorem 4.11 Let x be a strict lower weak minimum of SP. If WMin[F x, C]={ȳ} and F x has the C-weak minimal property then DF x, ȳx int C=, for all x TS, x. 4.4 Proof Based upon Theorem 4.10, we see that equation 4.2 holds. The rest of the proof can be followed by similar arguments to that of Theorem 3.3. Theorem 4.12 Let S be convex set and F be C-pseudoconvex at x, ȳ graphf. If for every x S we have DF x, ȳx 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 4.7 that there exists x S such that for each ȳ F xthereisy Fx with y ȳ int C. 4.5 Since F isc-pseudoconvexat x, ȳ, wederivefromlemma2.4that F x ȳ DF x, ȳ x x + C. Therefore, we obtain y ȳ DF x, ȳ x x + C. 4.6 Combining 4.5with4.6, we get DF x, ȳ x x + D int C, furthermore, DF x, ȳ x x int D, which is a contradiction. Remark 4.13 By comparing the results derived in Section 3, it can be found that the optimality for the set criterion and vector criterion, with the suitable conditions, possesses the same forms in terms of Studniarshi derivatives.

10 Kong et al. Journal of Inequalities and Applications :47 Page 10 of 11 5 Conclusions 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 optimality 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 ; Natural Science Foundation of Ningxia under Grant No. NZ 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 Luc, DT: Theory of Vector Optimization. Springer, Berlin Jahn, J: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin Chen, G, Huang, X, Yang, X: Vector Optimization-Set-Valued and Variational Analysis. Springer, Berlin Kuroiwa, D: Some duality theorems of set-valued optimization with natural criteria. In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp World Scientific, River Edge Kuroiwa, D: On the set-valued optimization. Nonlinear Anal. 47, Kuroiwa, D: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 241, Kuroiwa, D: Existence of efficient points of set optimization with weighted criteria. J. Nonlinear Convex Anal. 41, Hernández, E, Rodríguez-Marín, L, Sama, M: On solutions of set-valued optimization problems. Comput. Math. Appl. 60, Hernández, E, Rodríguez-Marín, L: Existence theorems for set optimization problems. Nonlinear Anal. 67, Hernández, E, Rodríguez-Marín, L, Sama, M: Some equivalent problems in set optimization. Oper. Res. Lett. 37, Maeda, T: On optimization problems with set-valued objective maps: existence and optimality. J. Optim. Theory Appl. 153, Hernández, E, Rodríguez-Marín, L: Duality in set optimization with set-valued maps. Pac. J. Optim. 32, Hernández, E, Rodríguez-Marín, L: Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134, Hernández, E, Löhne, A, Rodríguez-Marín, L, Tammer, C: Lagrange duality, stability and subdifferentials in vector optimization. Optimization 72, Hamel, AH, Löhne, A: Lagrange duality in set optimization. J. Optim. Theory Appl. 161, Alonso, M, Rodríguez-Marín, L: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63, Alonso-Durán, M, Rodríguez-Marín, L: On approximate solutions in set-valued optimization problems. J. Comput. Appl. Math. 236, Alonso-Durán, M, Rodríguez-Marín, L: Optimality conditions for set-valued maps with set optimization. Nonlinear Anal. 70, Hernández, E, Rodríguez-Marín, L: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, Araya, Y: Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Anal. 75, Yao, Y, Chen, R, Xu, HK: Schemes for finding minimum-norm solutions of variational inequalities. Nonlinear Anal. 72, Gutiérrez, C, Jiménez, B, Miglierina, E, Molho, E: Scalarization in set optimization with solid and nonsolid ordering cones. J. Optim. Theory Appl. 61, Zhang, WY, Li, SJ, Teo, KL: Well-posedness for set-optimization problem. Nonlinear Anal. 71, Gutiérrez, C, Miglierina, E, Molho, D, Novo, V: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, Ha, TXD: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124,

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