Connectedness and Path Connectedness of Weak Efﬁcient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods

: The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of the paper is to deal with the connectedness and the path connectedness for the weak efﬁcient solution set to a vector optimization problem by using the nonlinear scalarization method. Firstly, the union relationship between the weak efﬁcient solution set to the vector optimization problem and the solution sets to a series of parametric scalar minimization problems, is established. Then, some properties of the solution sets of scalar minimization problems are investigated. Finally, by using the union relationship, the connectedness and the path connectedness for the weak efﬁcient solution set of the vector optimization problem are obtained.


Introduction
Whether the decision is made by a team or an individual, it usually involves several conflicting goals. Problems in the real world must be solved optimally according to criteria, which leads to the development of vector (multi-criteria) optimization problems. Vector optimization theory is widely used in many fields such as economic management, financial insurance, engineering design, transportation, environmental protection, decision-making science and so on. The properties of solution sets are a very important research direction in optimization theories and applications; a lot of research results have been obtained on this aspect. Among the properties of solution sets, the connectedness that can provide the possibility of continuously moving from one solution to any other solution is of considerable interest (see, for example, [1][2][3][4][5][6][7][8][9][10]).
It is well known that the scalarization method is one effective approach to deal with the connectedness of the solution sets to vector optimization problems, vector variational inequalities and vector equilibrium problems. Recently, by means of the linear scalarization method, the authors in [11][12][13][14][15] established the connectedness of the solution set to the class of vector optimization, weak vector variational inequalities and weak vector equilibrium problems. However, to the best of our knowledge, there are very few results on the path connectedness of the solution sets of vector optimization problems. Very recently, in terms of the linear scalarization method, Han and Huang [16] investigated the path connectedness of the weak efficient solution set for a generalized vector quasi-equilibrium problem. Xu and Zhang [17] established the path connectedness for the Definition 3. A set-valued mapping G [19]: Λ ⇒ Y is said to be lower semicontinuous (l.s.c, for short) at λ 0 ∈ Λ iff for any open set V with G(λ 0 ) ∩ V = ∅, there exists a neighbourhood U of λ 0 such that G(λ) ∩ V = ∅, for all λ ∈ U. We say that G is l.s.c on Λ, if it is l.s.c at each points of λ ∈ Λ. Definition 4. Let A be a convex subset of X and ϕ : A → Y [20].

Lemma 2.
Let A be a nonempty subset of a Hausdorff topological vector space X and T [25]: A ⇒ X be a KKM mapping with closed values. If there exists y 0 ∈ A such that T(y 0 ) is compact, then y∈A T(y) = ∅.
Lemma 3. Let A be a paracompact Hausdorff path connected space and let Y be a Banach space [2]. Assume that (i) F : A ⇒ Y is a lower semicontinuous set-valued mapping; (ii) For each x ∈ A, F(x) is nonempty, closed and convex.
Then, F(A) is a path connected set.

Scalarization for VOP
In this section, we consider the following scalar minimization problem with the parametric q, which induced by the nonlinear scalarization function η e (·, q).

Definition 8.
A pointx ∈ A is called a solution for P q , written asx ∈ S(q), iff Proof. For anyx ∈ q∈Y S(q), there exists q ∈ Y such thatx ∈ S(q). Then we have It follows from the strict monotonicity of the function η that which contradicts (1). Therefore,x ∈ WE( f , A). Next, we claim that With the help of Proposition 1 (i), we can obtain that In terms of Proposition 2, we have This implies This shows thatx ∈ S( f (x)) and so WE( f , A) ⊆ q∈Y S(q). Therefore, WE( f , A) = q∈Y S(q).

Remark 2.
Theorem 1 gives the union relationship between the weak efficient solution set of VOP and the solution sets of a series for scalar minimization problems (P q ) without any convexity assumptions on the objective function and the feasible set. Hence, the result improves the corresponding ones in [14][15][16].

Connectedness and Path Connectedness of VOP
In this section, we shall apply the union relationship established in Section 3 to study the connectedness and the path connectedness of WE( f , A) for VOP.

Lemma 4.
Suppose that A is a closed subset of X and f is continuous on A. Then, for any q ∈ Y, S(q) is a closed set.
Now, we need to prove that x 0 ∈ S(q). Indeed, it follows from the closedness of A that x 0 ∈ A. Then, by (9), the continuity of η e (·, q) and f , we have η e ( f (x 0 ), q) ≤ m. This implies that So, x 0 ∈ S(q) and for any q ∈ Y, S(q) is a closed set.

Lemma 5.
Suppose that A is a nonempty convex set of X and f : A → Y is a properly quasiconvex function. Then, for any q ∈ Y, S(q) is convex. Then It is clear that Since η e (·, q) is monotone increasing for each q ∈ Y, and by (11), we obtain That is, x(λ) ∈ S(q) for each λ ∈ [0, 1]. Hence, S(q) is convex.

Lemma 6.
Assume that (i) A is a compact and convex subset of X; (ii) f is continuous on A; (iii) f is properly quasiconvex on A.
Then, q∈Y S(q) is nonempty.
Clearly, x ∈ T(x) and so it is nonempty for each x ∈ A. Since the continuity of f , it is easy to get that T(x) ⊆ A is a closed set. Furthermore, by the compactness of A, we obtain T(x) is compact. We now claim that T : A ⇒ A is a KKM mapping. Indeed, if not, then there exists a finite subset {x 1 , ..., x m } ⊆ A and x 0 ∈ conv({x 1 , ..., x m }) such that This shows that Since Noting that f is properly quasiconvex on A, there exists i 0 ∈ {1, 2, ..., m} such that This contradicts with (17). Therefore, T is a KKM mapping and x∈A T(x) = ∅ by Lemma 2. Letx ∈ x∈A T(x), then for any It follows from the monotonicity of h e (·, q) that for any x ∈ A, η e ( f (x), q) ≤ η e ( f (x), q), that is,x ∈ S(q). By the arbitrariness of x ∈ x∈A T(x), we have x∈A T(x) ⊆ q∈Y S(q). Hence, q∈Y S(q) is nonempty.

Lemma 7. Assume that (i) A is a compact subset of X; (ii) f is continuous on A; (iii)
f is strictly proper quasiconvex on A.
Proof. Assume that there exists q 0 ∈ Y such that S(·) is not l.s.c on q 0 . Then there exist x 0 ∈ S(q 0 ), a neighborhood W 0 of 0 ∈ X and a sequence {q n } with q n → q 0 , such that There are two cases to be considered. Case 1. S(q 0 ) is singleton. Let x n ∈ S(q n ). We have Clearly, x n ∈ A. By the compactness of A, without loss of generality, we can assume that x n →x. Now, we claim thatx ∈ S(q 0 ). Indeed, if not, then there exists x 0 ∈ A such that Since η e (·, ·) and f respectively are continuous on Y × Y and A, it follows from (20) that there exists N ∈ N, such that This contradicts (19). Therefore,x ∈ S(q 0 ). As S(q 0 ) is singleton, it follows thatx = x 0 and so x n → x 0 .
Hence, x n ∈ (x 0 + W 0 ) ∩ S(q n ) for n large enough, which contradicts (18). Case 2. S(q 0 ) is not singleton. Without loss of generality, we assume that x 0 , x ∈ S(q 0 ) with x 0 = x , that is, and Since f is strictly proper quasiconvex on A, for any λ ∈ (0, 1), one has where x(λ) = λx + (1 − λ)x 0 . It is easy to see that there exists λ 0 ∈ (0, 1) such that It follows from the strict monotonicity of η e (·, q 0 ) and (24) that Combining with (22) and (23), we can obtain that It follows from (18) and (25) that In terms of the continuity of η e (·, ·), one has This is a contradiction with (27). Therefore, S(·) is l.s.c on Y.
Theorem 2. Assume the following conditions are satisfied: (i) A is a compact and convex subset of X; (ii) f is continuous on A; (iii) f is properly quasiconvex on A.
Then, WE( f , A) is a connected set.
Proof. It follows from Lemmas 5 and 6 that for any q ∈ Y, S(q) is a convex and nonempty set. Obviously, S(q) is connected. By Lemma 1 and Theorem 1, we can see that is a connected set.
Theorem 3. Suppose that the following conditions are satisfied: (i) A is a compact and convex set of X; (ii) f is continuous on A; (iii) f is strictly proper quasiconvex on A.
Then, WE( f , A) is a path connected set.
Proof. By means of Lemmas 4-6, we have that for any q ∈ Y, S(q) is a closed, convex and nonempty set. With the help of Theorem 1, we can see that Therefore, it follows from Lemmas 3 and 7 that WE( f , A) is path connected. [6] prove the connectedness of the set of approximate solutions by using the upper semicontinuity of the solution sets of the following scalar minimization problem

Remark 3. Qiu and Yang
where ξ e ( f (x), q) = inf{t ∈ R : f (x) ∈ te + q − C} is the Gerstewitz function defined in [18,26]. Theorem 2 in this paper is different from Theorem 5.1 in [6]. In fact, on the one hand, we derive the connectedness by using the nonlinear scalarization function η e that is different of the Gerstewitz function. On the other hand, Theorem 2 of this paper is obtained without the upper semicontinuity of the solution sets of the scalar minimization problem P q . Furthermore, we also establish the path connectedness of the weak efficient solution set in Theorem 3, which is not established in [6].
Now, we give the following example to illustrate that Theorem 2 and Theorem 3.
It is easy to see that the constraint set A = [−1, 1] is a compact and convex set. The function f is continuous on A. And for each x 1 , That is, all assumptions in Theorems 2 and 3 are satisfied. It follows from a direct computation that WE( f , A) = [−1, 0] is connected. By Proposition 2, we know that By a direct computation, we see that S(q) = [−1, 0] for any q ∈ Y. Hence, WE( f , A) = q∈Y S(q) is connected and it is also path connected.
Remark 4. Now we give an example in economics. This is a strategic game with vector payoffs described in [27]. The bicriteria strategic game is a tuple Ψ = N, (Ω i ) i∈N , (ψ i ) i∈N , where N is the set of players, Ω i is the strategy set for player i ∈ N, Ω is the Cartesian product ∏ i∈N Ω i of the strategy sets (Ω i ) i∈N , and ψ i : Ω → R 2 is the utility function for player i. By [27],x ∈ ∏ i∈N Ω i is called a weak Pareto efficient solution of the game if, for each i ∈ N, Here, the set WPB(x −i ) of weak Pareto best answer tox −i is the set of the weak efficient solution x i to the following bi-objective optimization problem (for short, BOOP) min Obviously,x ∈ Ω is a weak Pareto solution of the bicriteria strategic game Ψ if and only if for each i ∈ N, Now, we give the following example to illustrate the above economic problem.
2) satisfy the assumptions in Theorems 2 and 3. It is easy to see that the functions f i (x) (i = 1, 2) are linear functions. Therefore, f is continuous and strictly proper quasiconvex on Ω i (i = 1, 2).

Conclusions
In this paper, we firstly established the union relationship between the weak efficient solution set to VOP and the solution sets to a series of parametric scalar minimization problems P q . Then, we applied the union relationship to obtain the connectedness and the path connectedness of VOP under suitable assumptions. The method may be viewed as a refinement and improvement of the linear scalarization method used in [6,[14][15][16]. However, we find that the parametric set Y of q in the union relationship of Theorem 1 is too large. Moreover, by the nonlinear scalarization method, the union relationship can be established only for the weak efficient solution set. Therefore, alteration of the parametric set Y of q by other parametric sets and the study for the connectedness of efficient solution sets of vector optimization problems, is a good direction for us.
It would be also interesting to investigate the connectedness and the path connectedness of Robust efficient solution sets to vector optimization problems under uncertain data by means of the nonlinear scalarization method. Support vector machine (SVM) and extreme learning machine (ELM) have gained increasing interest from various research fields recently (see, for example, [28][29][30]). If we can combine the knowledge of SVM and ELM with VOP, it may constitute very valuable research in the future.

Conflicts of Interest:
The authors declare no conflict of interest.