Globally proper efficiency of set optimization problems based on the certainly set less order relation

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results.


Introduction
Vector optimization is an important topic in optimization theory and has been investigated by many scholars.The model of the vector optimization problem is formed as follows: where f : X → Y is a vector-valued map, and X and Y are a linear space and a locally convex space, respectively.Many scholars studied different kinds of solutions of (VOP) (see [1][2][3][4] and the references therein), for example, properly minimal solutions, weakly minimal solutions, or approximately minimal solutions, see [2].
With the development of set-valued analysis, several scholars have been paying attention to the following set-valued optimization problem: where F : X ⇒ Y is a set-valued map.When F(x) is a singleton set for any x ∈ X, (SVOP) reduces to (VOP).Among the introduced minimality notions of (SVOP) are Benson properly efficient solution [5], Henig properly efficient solution [6] and super efficient solution [7].Not only the topological properties of these solutions such as the connectedness of the solution set have been investigated [8], but also optimality conditions of (SVOP) in the sense of proper efficiency have been established.
In order to define the solution x 0 of (VOP), we only need to compare f (x 0 ) with every element in f (X), where x 0 ∈ X.However, when we define the solution x 0 of (SVOP), we need to compare y 0 ∈ F(x 0 ) with every element in F(X).This described the so-called vector approach to a set-valued optimization problem.Note that when defining the solutions of (VOP) and (SVOP), we usually make some comparisons between two vectors.Recently, several scholars have been using the ordered convex cone to make comparisons between two sets and formed the following set optimization problem (SOP): When we define the solution x 0 of (SOP), we need to compare F(x 0 ) with every element in {F(x)|x ∈ X}.This is the so-called set approach, see [9].The set approach to a set-valued optimization problem has gained tremendous interest in the last years.Very recently, Ansari et al. [10] have studied the convergence of the solution set of set optimization problems.Moreover, it is interesting to mention that there exist several publications that deal with a vectorization approach of (SOP), see, for example, [11][12][13].The extremal value functions used in vectorization have been intensely studied in Gerlach and Rocktäschel [14].
To generalize the cone convexity of set-valued maps, Kuroiwa et al. [15] introduced six kinds of set-relations based on cone ordering.Hernández and Rodríguez-Marín [16] introduced the efficient solution and the weakly efficient solution of (SOP) and investigated duality, Lagrangian multiplier rule and saddle points of (SOP).Dhingra and Lalitha [17] investigated the set optimization involving improvement sets [18].Very recently, the existence of solutions [19,20], scalarizations [21,22] and the connectedness [23,24] of the solutions set for (SOP) have been studied.
In the theory of vector optimization, we always consider a partial order relation to compare two vectors or sets.However, the set order relation induced by the improvement set in [17] is not a partial order relation.On the other hand, it is well-known that the set of (weakly) efficient solutions for (VOP) or (SVOP) is usually too big, which poses difficulties for the decision maker.To overcome this defect, the notion of properly efficient solution of (VOP) and (SVOP) has been introduced.Therefore, using the partial order relation to introduce the notion of proper efficiency of (SOP) is an interesting point that we are tackling in this work.To the best of our knowledge, there are no publications introducing the notions of properly efficient solutions of (SOP).The purpose of this paper is to use the certainly set less order relation [25,26] to introduce a globally proper efficient solution of (SOP).
This paper is organized as follows.In Section 2, we give some notations and preliminaries.In Section 3, we obtain a Lagrange multiplier rule of (SOP).In Section 4, we establish duality theorems of (SOP) and finally, in Section 5, we give saddle point theorems of (SOP).

Notations and preliminaries
Let Y and Z be two real locally convex spaces.Y * and Z * are dual spaces of Y and Z, respectively.The zero element of every space is denoted by 0. Write The interior and the closure of C are denoted by intC and clC, respectively.Unless otherwise specified, we suppose that C and D are two nontrivial point convex cones of Y and Z with intC = ∅ and intD = ∅, respectively.

Example 2.1:
To compare two elements of P(Y), we establish the following the set order relation defined between two nonempty sets.

Definition 2.3 ([25,26]): Let A, B ∈ P(Y).
We define the following set order relation: Remark 2.2: According to Proposition 6.2 in Ansari [25], the set order relation Let C be replaced by C + H\{0} in the set order relation ≤ c C , we can introduce a new set order relation ≤ c C+H\{0} as follows:

Remark 2.3:
The set order relation Thus, we have C+H\{0} is a partial order relation.
Next, based on the above set order relation, we define the globally proper minimal element of S ⊆ P(Y).Proof: The sufficiency is clear.Now, we prove the necessity.We suppose that, for any H ∈ H C , there exists B ∈ S such that

Lagrangian multiplier
In this section, we will establish a Lagrange multiplier rule of the set optimization problem.From now on, we suppose that X is a linear space and A is a nonempty subset of X.

Definition 3.1 ([5]):
The set-valued map F : Let F : X ⇒ Y and G : X ⇒ Z be two set-valued maps on A. Now, we consider the following set optimization problem: (i) (SOP) satisfies generalized Slater constraint qualification; (v) There does not exist x and x 0 is a c-globally minimal solution of the following unconstrained set optimization problem: We assert that Otherwise, there exists and (9) implies x ∈ .By (8), there exist y ∈ F(x) and h ∈ H\{0} such that Clearly, B − H\{0} ⊆ B. Therefore, we have It follows from ( 10) and ( 11) that (12) shows that y ∈ y 0 − H\{0}, ∀y 0 ∈ F(x 0 ), i.e.
The following example is used to illustrate Theorem 3.1.

Duality
We define the Lagrangian set-valued map L : The dual set-valued map G : (Z, Y) ⇒ Y is write as The dual problem associated to (SOP) is the following optimization problem: Proof: By (23), the conclusion holds when According to (h • G)(x 0 ) ⊆ −C and (24), we obtain Because (x , h) is a feasible pair of (DSOP), it follows from Definition 4.2 that According to (25), (26) and Definition 2.4, we have By (27), we have Since (h • G)(x 0 ) ⊆ −C, it follows from (28) that is a feasible pair of (DSOP) and then x 0 is a c-globally minimal solution of (SOP) and h 1 is a c-globally maximal solution of (DSOP).
Proof: Let x ∈ and F(x ) ≤ c C+H\{0} F(x 0 ).x 0 is a c-globally minimal solution of (SOP) when We assert that x 0 is a c-globally minimal solution of (SOP).
Hence, h 1 is a c-globally maximal solution of (DSOP).
then u 0 is a c-globally maximal solution of (DSOP).

Definition 3 . 2 (Lemma 3 . 1 :
[29]): Let B be a nonempty subset of Y. B is said to be C-bounded iff, for any neighborhood U of zero in Y, there exists t > 0 such that B ⊆ tU + C. Let B ⊆ Y be an C-bounded set and H ∈ H C .Then, b∈B (b − intH) = ∅.

Theorem 4 . 2 :
Let x 0 ∈ be a c-globally minimal solution of (SOP) and H ∈ H C .There exists u0 ∈ L + (Z, Y) such that (u 0 • G)(x) ⊆ C with x ∈ and x 0 is a solution of c-GMin{F(x) + (u 0 • G)(x)|x ∈ A}.If there exists x ∈ such that globally maximal element of S (denoted by A ∈ c-GMaxS) iff there exists H ∈ H C such that B ∈ S and A ≤ c C+H\{0} B imply B ≤ c C+H\{0} A. When S ⊆ P(Y) becomes S ∈ P(Y), Definition 2.4(i) reduces to Definition 2.2.The following proposition shows a relationship between a c-globally minimal element of S and the set order relation B cC+H\{0} A for all B ∈ S.
Proposition 2.1: Let S ⊆ P(Y) and A ∈ S. We have A ∈ c-GMinS iff there exists H ∈ H C such that B c C+H\{0} A, ∀B ∈ S.