A new kind of inner superefficient points
- Yihong Xu^{1}Email author,
- Lei Wang^{1} and
- Chunhui Shao^{1}
https://doi.org/10.1186/s13660-017-1452-6
© The Author(s) 2017
Received: 14 March 2017
Accepted: 12 July 2017
Published: 1 August 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 cone-subconvexlikeness, 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’.
Keywords
MSC
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 ϵ-solutions in general vector optimization problems. Rong and Wu [3] considered cone-subconvexlike vector optimization problems with set-valued maps in general spaces and derived scalarization results, ϵ-saddle point theorems, and ϵ-duality assertions using ϵ-Lagrangian multipliers. Qiu and Yang [4] studied the approximate solutions for vector optimization problem with set-valued functions, and derived the scalar characterization without imposing any convexity assumption on the objective functions. The authors of [5] introduced the notion of ϵ-strictly efficient solution for vector optimization with set-valued maps. Under the assumption of the ic-cone-convexlikeness for set-valued maps, the scalarization theorem, ϵ-Lagrangian multiplier theorem, ϵ-saddle point theorems and ϵ-duality assertions were established for ϵ-strictly 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 set-valued 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.
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
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 \(C=\operatorname{cone}B\).
Theorem 2.1
Proof
Definition 2.1
[10]
In the following, with variable ordering, we introduce the concept of inner superefficient point.
Definition 2.2
Remark 2.1
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
Convexity plays a key role in optimization theory. Yang et al. [17] introduced a new class of generalized convexity termed near C-subconvexlikeness.
Definition 2.3
[17]
Suppose \(A\subset Y\) is nonempty, A is said to be nearly C-subconvexlike if \(\operatorname{cl}\,\operatorname{cone}(A+C)\) is convex.
Remark 2.2
- (1)
when the ordering cone has nonempty interior, ic-cone-convexness is equivalent to near cone-subconvexlikeness;
- (2)
when the ordering cone has empty interior, ic-cone-convexness implies near cone-subconvexlikeness, a counter example is given to show that the converse implication is not true.
Theorem 2.3
Proof
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 C-subconvexlike, 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 set-valued optimization problem seems to be of interest and value.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China Grant 11461044 and the Natural Science Foundation of Jiangxi Province (20151BAB201027).
Open Access 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, provided 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.
Authors’ Affiliations
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