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A new kind of inner superefficient points
Journal of Inequalities and Applications volume 2017, Article number: 177 (2017)
Abstract
In this paper, some properties of the interior of positive dual cones are discussed. With the help of dilating cones, a new notion of inner superefficient points for a set is introduced. Under the assumption of near conesubconvexlikeness, by applying the separation theorem for convex sets, the relationship between inner superefficient points and superefficient points is established. Compared to other approximate points in the literature, inner superefficient points in this paper are really ‘approximate’.
1 Introduction
The approximate efficient solution is an important notion of vector optimization theory. Many interesting results have been obtained in recent years. For example, Loridan [1, 2] introduced the concept of ϵsolutions in general vector optimization problems. Rong and Wu [3] considered conesubconvexlike vector optimization problems with setvalued maps in general spaces and derived scalarization results, ϵsaddle point theorems, and ϵduality assertions using ϵLagrangian multipliers. Qiu and Yang [4] studied the approximate solutions for vector optimization problem with setvalued functions, and derived the scalar characterization without imposing any convexity assumption on the objective functions. The authors of [5] introduced the notion of ϵstrictly efficient solution for vector optimization with setvalued maps. Under the assumption of the icconeconvexlikeness for setvalued maps, the scalarization theorem, ϵLagrangian multiplier theorem, ϵsaddle point theorems and ϵduality assertions were established for ϵstrictly efficient solution. The concept of nondominated solutions with variable domination structures or variable orderings was introduced by Yu [6]. This is a generalization of the nondominated solution concept with fixed domination structure in multicriteria decision making problems. Chen [7] introduced a nonlinear scalarization function for a variable domination structure,and by applying this nonlinear function characterized the weakly nondominated solution of multicriteria decision making problems.
On the other hand, proper efficiency is a natural concept in vector optimization. Borwein [8] introduced a new kind of proper efficiency,namely super efficiency. Super efficiency refines the notions of efficiency and other kinds of proper efficiency. Fu [9] and Zheng [10] gave two different generalizations of super efficiency in a locally convex space. Xu and Zhu [11] investigated the setvalued optimization problem with constraints in the sense of super efficiency in locally convex linear topological spaces. Hu and Ling [12] studied the connectedness of the cone superefficient point set in locally convex topological vector spaces. In [3, 4, 11–14], the nonemptiness of the interior of the order cone must be satisfied when proving optimization results.
However, in many cases, the ordering cone has an empty interior. For example, for each \(1< p<+\infty\), the normed space \(l_{p}\) partially ordered by the positive cone is an important space in applications; the positive cone has an empty interior. Another example is the case when C is the Cartesian product \(C'\times C''\) of a trivial cone \(C'=\{0\}\) and a cone \(C''\) having a nonempty interior. Thus, to study the vector optimization problem under the condition that ordering cone has an empty interior has become an important topic [15, 16].
Let Y be a locally convex topological vector space; let \(C\subset Y\) be a pointed closed convex cone and B be a base of C. Let \(A\subset Y\) be nonempty.
Rong and Wu [3] showed that
The authors of [5] demonstrated that
and the inverse inclusion does not hold in general.
Naturally, we come up with the problem how to introduce the notion of approximate point and under what condition the set of inner points equals the set of primal points. When order cone C has an empty interior, how to investigate superefficient points?
Motivated by Yu [6], Zheng [10] and Gong [15], we will introduce a new kind of superefficient point (see Definition 2.2), under certain conditions, we will obtain the relationship between inner superefficient point and superefficient point (see Theorem 2.3). In the literature [1–5], approximate points were defined by adding ϵ or −ϵ to a set A, in this paper, inner superefficient points will be introduced by a variable domination structure.
2 Notation and preliminaries
In the remainder of the paper, let Y be a normable locally convex topological vector space; \(C\subset Y\) be a pointed closed convex cone; \(Y^{\ast}\) be the dual space of Y. The positive dual cone \(C^{\ast}\) of C is denoted by
For a set \(B\subset C\), we write
The closure of set B is denoted by clB. A convex subset B of a cone C is a base of C if \(0{\notin }\operatorname{cl}B\) and \(C=\operatorname{cone}B\).
Let B be a base of C, then \(0\notin\operatorname{cl}B\). By the separation theorem, there exists \(\bar{f}\in Y^{\ast}\backslash\{0\}\) such that
Let
Define the neighborhood family of 0 in Y as follows:
All bounded subsets of Y are denoted by Ω, and for any \(M\in\Omega\), let
Let \(\tau^{\ast}\) denote the locally convex topology induced by the quasinorm family \(\{P_{M}:M\in\Omega\}\) on \(Y^{\ast}\), \(\operatorname{int} C^{\ast}\) denote the interior set of \(C^{\ast}\) in the sense of \(\tau^{\ast}\).
Theorem 2.1
Let B be a bounded base of C. Then
Proof
Since \(C\subset\operatorname{cl}\,\operatorname {cone}B\subset\operatorname{cl}\,\operatorname{cone}(B+U)\), for any \(U\in N(0)\). we get \((\operatorname{cl}\,\operatorname{cone}(B+U))^{\ast}\subset C^{\ast}\), so \(\operatorname{int}(\operatorname{cl}\,\operatorname {cone}(B+U))^{\ast}\subset\operatorname{int}C^{\ast}\). Thus
In what follows, we prove
Take \(f_{0}\in\operatorname{int}C^{\ast}\), then there exists \(\epsilon _{0}\in(0,1)\) such that
it follows from \(B\subset C\) that
which implies, together with (2.1), that
Take \(M_{0}\in\Omega\), let
then \(U_{0}\in N(0)\). Denote
Since B and \(U_{0}\) are bounded, we conclude that \(V^{\ast}\) is a neighborhood of 0 in \(Y^{\ast}\).
In the following, we need to prove that
that is,
For each \(t\in B+U_{0}\), there exist \(b\in B\), \(u\in U_{0}\) such that \(t=b+u\), then,
Combining (2.5), (2.6) and (2.7), we see that
For each \(s\in\operatorname{cl}\,\operatorname{cone}(B+U_{0})\), there exist \(\lambda_{n}\geq0\), \(t_{n}\in B+U_{0}\), such that \(s=\lim_{n\rightarrow\infty}\lambda_{n}t_{n}\), then
Thus,
Hence,
Therefore,
Consequently,
The proof is complete. □
Definition 2.1
[10]
Let \(A\subset Y\) be nonempty, \(\bar {y}\in A\) is said to be a superefficient point of A, written as \(\bar{y}\in \operatorname{SE}(A, C)\), if, for each neighborhood V of 0 in Y, there exists a neighborhood W of 0 such that
Suppose \(U\in N(0)\), let
The notion \(C_{U}(B)\) will be used throughout this paper.
In the following, with variable ordering, we introduce the concept of inner superefficient point.
Definition 2.2
Let \(A\subset Y\) be nonempty, \(\bar{y}\in A\) is said to be an inner superefficient point, if \(\bar{y}\in \operatorname{SE}(A, C_{U}(B))\), for some \(U\in N(0)\). That is, for each neighborhood V of 0 in Y, there exists a neighborhood W of 0 such that
Remark 2.1
It is clear that
In the following, we give the existence theorem of inner superefficient points.
Theorem 2.2
Let Y be a normable locally convex topological vector space, B be a bounded base of C, and \(A\subset Y\) be a weakly compact set. Then for any \(U\in N(0)\), \(\operatorname{SE}(A,C_{U}(B))\neq\emptyset\).
Proof
Since U is bounded and Y is normable, in the same way as the proof of [8], Theorem 1.1, we can show that
Hence
By (2.2) and (2.3), we conclude \(\operatorname{cl}(B+U)\) is a bounded base of \(C_{U}(B)\). From [10], Corollary 3.1, we deduce \(\operatorname{SE}(A,C_{U}(B))\neq\emptyset\). □
Convexity plays a key role in optimization theory. Yang et al. [17] introduced a new class of generalized convexity termed near Csubconvexlikeness.
Definition 2.3
[17]
Suppose \(A\subset Y\) is nonempty, A is said to be nearly Csubconvexlike if \(\operatorname{cl}\,\operatorname{cone}(A+C)\) is convex.
Remark 2.2
Sach [16] introduced another convexity called icconeconvexlikeness. The authors of [18] obtain the following results:

(1)
when the ordering cone has nonempty interior, icconeconvexness is equivalent to near conesubconvexlikeness;

(2)
when the ordering cone has empty interior, icconeconvexness implies near conesubconvexlikeness, a counter example is given to show that the converse implication is not true.
Thus, near Csubconvexlikeness is a very generalized convexity.
Theorem 2.3
Suppose that for each \(y\in Y\), \(A\{\bar{y}\}\) is nearly Csubconvexlike, and B is a bounded base of C. Then
Proof
Since \(\operatorname{SE}(A, C_{U}(B))\subset \operatorname{SE}(A, C)\), \(\forall U\in N(0)\),
In what follows, we prove
Take \(\bar{y}\in \operatorname{SE}(A,C)\).
1^{∘} We show there exists \(U\in N(0)\) such that
Indeed, since \(0\notin\operatorname{cl}B\), there exists \(V\in N(0)\) such that \(B\cap V=\emptyset\), hence,
As \(\bar{y}\in \operatorname{SE}(A,C)\), for the above V there is a neighborhood W of 0 such that \(\operatorname{cone}(A\{\bar{y}\})\cap(WC)\subset\frac{V}{2}\). Taking \(U=W\cap\frac{V}{2}\cap V_{B}\), we get
By \(B\subset C\), we conclude
and from (2.10),
Since \(U\subset\frac{V}{2}\), it follows from (2.9) that \(\frac{V}{2}\cap (UB)=\emptyset\). In view of (2.11),
2^{∘} In the following, we demonstrate
By contradiction, suppose that \(\operatorname{cone}(A+C\{\bar{y}\})\cap (UB)\neq\emptyset\). Then there exist \(\lambda_{1}\geq0\), \(a_{1}\in A\), \(c_{1}\in C\), \(u_{1}\in U\), \(b_{1}\in B\) such that
Since B is the base of C, there exist \(\lambda_{2}\geq0, b_{2}\in B\) such that \(c_{1}=\lambda_{2}b_{2}\), hence,
Consequently,
Since B is convex, the above equal elements belong to \(\operatorname {cone}(A\{\bar{y}\})\cap(UB)\), this is a contradiction.
Since U is open, it follows from (2.12) that
3^{∘} Since \(A\{\bar{y}\}\) is nearly Csubconvexlike, \(\operatorname{cl}\,\operatorname{cone}(A+C\{\bar{y}\})\) is convex. By the separation theorem of convex sets, there exists \(\hat{f}\in Y^{\ast}\backslash\{0\}\) such that
Since \(0\in\operatorname{cl}\,\operatorname{cone}(A+C\{\bar{y}\})\), thus,
Consequently,
Since U is absorbed, there exists \(u_{0}\in U\) such that \(\hat {f}(u_{0})>0\), thus,
4^{∘} In what follows, we show
Let
In the following, we show
In fact, for each \(f\in V_{0}^{\ast}\), \(c\in C\), there exist \(\lambda\geq 0\), \(b\in B\) such that \(c=\lambda b\), then
it follows from (2.15) and (2.16) that
Thus,
That is
It means that
5^{∘} From Theorem 2.1, there exists \(\hat{U}\in N(0)\) such that
Thus there exists \(\hat{\epsilon}\in(0,1)\) such that
For each \(b\in B\), \(u\in\hat{U}\in N(0)\), it follows from (2.1)(2.3) that
Thus
From (2.17), we get
In what follows, we prove
where \(C_{\hat{U}}(B)=\operatorname{cl}\,\operatorname{cone}(B+\hat {U})\).
6^{∘} Since Û is bounded and Y is normable, in the same way as in the proof of [8], Theorem 1.1, we can show that
Consequently,
Since B is bounded and \(\hat{U}\in N(0)\), we see that \(\operatorname {cl}(B+\hat{U})\) is bounded. From (2.18), we get
Thus,
Therefore, it follows from (2.19) that \(\operatorname{cl}(B+\hat{U})\) is a bounded base of \(C_{\hat{U}}(B)\). Let
7^{∘} In the following, we prove
By (2.14), since \(\operatorname{cl}\,\operatorname{cone}(A+C\{\bar{y}\} )\) is a cone and on which f̂ has lower bound, we have
Since \(0\in C\), we have \(A\{\bar{y}\}\in\operatorname{cl}\,\operatorname{cone}(A+C\{ \bar{y}\})\), \(\forall a\in A\), thus,
We get
On the other hand, from (2.20) and (2.21), we see that
Thus,
This together with (2.23) leads to
8^{∘} Since Ũ is open, we have \(\tilde {U}+\operatorname{cl}(B+\hat{U})\) is open, we get
Since \(\operatorname{cl}\,\operatorname{cone}(A\{\bar{y}\})\) is a cone, we get
By [10], Definition 2.1, we have
Since \(\operatorname{cl}(B+\hat{U})\) is bounded, from [10], Proposition 3.2, we have
From [10], Proposition 3.5, we have
The proof is complete. □
3 Conclusions
In this paper, some properties for the interior of positive dual cones were studied. Using the dilating cones, we introduced a new notion of inner superefficient points, which has a nice property (see Theorem 2.3): suppose for each \(y\in Y\), \(A\{y\}\) is nearly Csubconvexlike, and B is a bounded base of C, then \(\bigcup_{U\in N(0)}\operatorname{SE}(A,C_{U}(B))=\operatorname{SE}(A,C)\). Hence it is really ‘approximate’. When the interior of C is empty, however, \(\operatorname{int}C_{U}(B)\neq\emptyset\), in this case, we can obtain the properties of \(\operatorname{SE}(A,C)\) by investigating \(\operatorname{SE}(A,C_{U}(B))\). The research on the inner points of a set is very important in the study of multiobjective optimization. Hence, further research on the inner superefficient solutions of the setvalued optimization problem seems to be of interest and value.
References
Loridan, P: Necessary conditions for ϵoptimality. Math. Program. Stud. 19, 140152 (1982)
Loridan, P: ϵSolutions in vector minimization problems. J. Optim. Theory Appl. 43, 265276 (1984)
Rong, WD, Wu, YN: ϵWeak minimal solutions of vector optimization problems with setvalued maps. J. Optim. Theory Appl. 106, 569579 (2000)
Qiu, QS, Yang, XM: Some properties of approximate solutions for vector optimization problem with setvalued functions. J. Glob. Optim. 47, 112 (2010)
Li, TY, Xu, YH: ϵStrictly efficient solutions of vector optimization problems with setvalued maps. AsiaPac. J. Oper. Res. 24(6), 841854 (2007)
Yu, PL: MultipleCriteria Decision Making: Concepts, Techniques and Extensions. Plenum, New York (1985)
Chen, GY, Yang, XQ: Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112(1), 97110 (2002)
Borwein, JM, Zhuang, DM: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105122 (1993)
Fu, WT, Cheng, YH: On the super efficiency in locally convex spaces. Nonlinear Anal., Theory Methods Appl. 44, 821828 (2001)
Zheng, XY: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469486 (1997)
Xu, YH, Zhu, CX: On super efficiency in setvalued optimization in locally convex spaces. Bull. Aust. Math. Soc. 71, 183192 (2005)
Hu, YD, Ling, C: Connectedness of cone super efficient point sets in locally convex topological spaces. J. Optim. Theory Appl. 107, 433446 (2000)
Li, ZF: Benson proper efficiency in the vector optimization of setvalued maps. J. Optim. Theory Appl. 98, 623649 (1998)
Chen, GY, Rong, WD: Characterizations of the Benson proper efficiency for nonconvex optimization. J. Optim. Theory Appl. 98, 365384 (1998)
Gong, XH: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 1231 (1998)
Sach, PH: New generalized convexity notion for setvalued maps and application to vector optimization. J. Optim. Theory Appl. 125, 157179 (2005)
Yang, XM, Li, D, Wang, SY: Nearsubconvexlikeness in vector optimization with setvalued functions. J. Optim. Theory Appl. 110, 413427 (2001)
Xu, YH, Song, XS: Relationship between icconeconvexness and nearly conesubconvexlikeness. Appl. Math. Lett. 24, 16221624 (2011)
Acknowledgements
This research was supported by the National Natural Science Foundation of China Grant 11461044 and the Natural Science Foundation of Jiangxi Province (20151BAB201027).
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Yihong Xu (1969), Professor, Doctor, major field of interest is in the area of setvalued optimization.
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Xu, Y., Wang, L. & Shao, C. A new kind of inner superefficient points. J Inequal Appl 2017, 177 (2017). https://doi.org/10.1186/s1366001714526
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DOI: https://doi.org/10.1186/s1366001714526