A new kind of inner superefficient points

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’.


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 [, ] introduced the concept of -solutions in general vector optimization problems. Rong and Wu [] considered cone-subconvexlike vector optimization problems with set-valued maps in general spaces and derived scalarization results, -saddle point theorems, andduality assertions using -Lagrangian multipliers. Qiu and Yang [] 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 [] introduced the notion of -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, -Lagrangian multiplier theorem, -saddle point theorems and -duality assertions were established for -strictly efficient solution. The concept of nondominated solutions with variable domination structures or variable orderings was introduced by Yu []. This is a generalization of the nondominated solution concept with fixed domination structure in multicriteria decision making problems. Chen [] 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. However, in many cases, the ordering cone has an empty interior. For example, for each  < p < +∞, 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 × C of a trivial cone C = {} 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 [, ].
Let Y be a locally convex topological vector space; let C ⊂ Y be a pointed closed convex cone and B be a base of C. Let A ⊂ Y be nonempty. Rong  , we will introduce a new kind of superefficient point (see Definition .), under certain conditions, we will obtain the relationship between inner superefficient point and superefficient point (see Theorem .). In the literature [-], approximate points were defined by adding or -to a set A, in this paper, inner superefficient points will be introduced by a variable domination structure.

Notation and preliminaries
In the remainder of the paper, let Y be a normable locally convex topological vector space; C ⊂ Y be a pointed closed convex cone; Y * be the dual space of Y . The positive dual cone C * of C is denoted by Let B be a base of C, then  / ∈ cl B. By the separation theorem, there existsf ∈ Y * \{} such that Define the neighborhood family of  in Y as follows: All bounded subsets of Y are denoted by , and for any M ∈ , let Let τ * denote the locally convex topology induced by the quasi-norm family {P M : M ∈ } on Y * , int C * denote the interior set of C * in the sense of τ * .

Theorem . Let B be a bounded base of C. Then
In what follows, we prove which implies, together with (.), that In the following, we need to prove that that is, Combining (.), (.) and (.), we see that For each s ∈ cl cone(B + U  ), there exist λ n ≥ , t n ∈ B + U  , such that s = lim n→∞ λ n t n , then Thus, Hence, Therefore, Consequently, The proof is complete.

Definition . ([]) Let
A ⊂ Y be nonempty,ȳ ∈ A is said to be a superefficient point of A, written asȳ ∈ SE(A, C), if, for each neighborhood V of  in Y , there exists a neighborhood W of  such that 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.

Remark . It is clear that
In the following, we give the existence theorem of inner superefficient points.

Theorem . Let Y be a normable locally convex topological vector space, B be a bounded base of C, and A ⊂ Y be a weakly compact set. Then for any U ∈ N(), SE(A, C U (B)) = ∅.
Proof Since U is bounded and Y is normable, in the same way as the proof of [], Theorem ., we can show that

Remark . Sach [] introduced another convexity called ic-cone-convexlikeness. The authors of [] obtain the following results:
() when the ordering cone has nonempty interior, ic-cone-convexness is equivalent to near cone-subconvexlikeness; () 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. Thus, near C-subconvexlikeness is a very generalized convexity.

Theorem . Suppose that for each y ∈ Y , A -{ȳ} is nearly C-subconvexlike, and B is a bounded base of C. Then
In what follows, we prove By B ⊂ C, we conclude and from (.), In view of (.),  • In the following, we demonstrate By contradiction, suppose that cone Consequently, Since B is convex, the above equal elements belong to cone(A -{ȳ}) ∩ (U -B), this is a contradiction.
Since U is open, it follows from (.) that By the separation theorem of convex sets, there existsf ∈ Y * \{} such that Consequently, Since U is absorbed, there exists u  ∈ U such thatf (u  ) > , thus, In the following, we shoŵ it follows from (.) and (.) that Thus,f It means that Thus there existsˆ ∈ (, ) such that From (.), we get In what follows, we provē We get On the other hand, from (.) and (.), we see that The proof is complete.

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 .): suppose for each y ∈ Y , A -{y} is nearly C-subconvexlike, and B is a bounded base of C, then U∈N() SE(A, C U (B)) = SE(A, C). Hence it is really 'approximate' . When the interior of C is empty, however, int C U (B) = ∅, in this case, we can obtain the properties of SE(A, C) by investigating 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.