Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

and Applied Analysis 3 Remark 4. The oriented distance function Δ is not applicable for studying Hölder continuity of solution mappings to parametric vector equilibrium problems (PVEP) and (PVAEP) in this paper, unlike the Gerstewitz function ξ done in the next section. This is because the properties like Proposition 2 (ii) and (v) with r ̸ = 0 are not satisfied by the function Δ, while they play important roles in our study. Let B(x0, δ) be the closed ball centered at x0 and radius δ > 0. It is said that g : X → Y is Lipschitz around x0 ∈ X iff there exist L > 0 and δ > 0 such that ‖g(x) − g(x)‖ ≤ L‖x − x‖, ∀x, x ∈ B(x0, δ), and g is locally Lipschitz on A ⊂ X if and only if g is Lipschitz around each x0 ∈ A. g is called (globally) Lipschitz onA if and only if ‖g(x)−g(x)‖ ≤ L‖x − x‖, ∀x, x ∈ A. Proposition 5 (see [15, 16]). ξq is globally Lipschitz on Y, and its Lipschitz constant is L := sup λ∈Kq ‖λ‖ ∈ [1/‖q‖, +∞[. In particular, under the scalar case of Y = R and K = R+, the Lipschitz constant of ξq is L = (1/q) (q > 0). Remark 6. Let q ∈ intK and a ∈ Y. The nonlinear scalarization function ξq,a : Y → R is defined as ξq,a(y) := min{t ∈ R | y ∈ a + tq − K}. It is easy to see that ξq,a is still globally Lipschitz on Y with Lipschitz constant L = sup λ∈Kq ‖λ‖, because ξq,a(y) = maxλ∈Kq⟨λ, y − a⟩. Assume that Y is a separated locally convex space, and K ⊂ Y is a proper, closed, and convex cone with intK ̸ = 0. Let q ∈ intK. Then it follows from the proof of [8, Theorem 3.1(ii)] (see inequality (6) therein) that ξq is Lipschitz on Y; namely, 󵄨󵄨󵄨󵄨ξq (y) − ξq (y 󸀠 ) 󵄨󵄨󵄨󵄨 ≤ pV (y − y 󸀠 ) , ∀y, y 󸀠 ∈ Y, (3) where pV : Y → R is the Minkowski functional associated with V and V ⊂ Y is a symmetric closed and convex neighborhood of 0Y such that q + V ⊂ K. WhenY is a linear normed space andV := τK,qB for some τK,q > 0 (B denotes the closed unit ball), because pV (x) := inf {α > 0 | x


Introduction
In the development of vector optimization, the theory and the methods of scalarization have always played important roles [1][2][3][4][5].The linear scalarization is historically the first method proposed and the most widely known and used.Besides this, the nonlinear scalarization is also fully developed.Several well-known nonlinear scalarization functions were introduced, such as the Hiriart-Urruty function [6] and the Gerstewitz (Tammer) function [7,8].Among them, the function   (see Definition 1) commonly known as the Gerstewitz function in vector optimization [7,9,10] is a powerful tool, which was introduced in [11] and has further been mentioned in [12,13].It has many good properties, such as continuity, sublinearity, convexity, monotonicity, and Lipschitz property.These properties have been fully exploited in the literature [5,[7][8][9][10][14][15][16][17] to deal with various problems with vector objectives, such as existence and continuity of solutions, optimality conditions, gap functions, duality, vector variational principles, well posedness, vector minimax inequalities, and vector network equilibrium problems.
However, as far as we know, the locally and globally Lipschitz properties of   have not been noticed until recently.
Tammer and Zȃlinescu [8] studied Lipschitz continuity properties of such kind of functions and gave some applications for deriving necessary optimality conditions for vector optimization problems.For other close works about this aspect, one can refer to Durea and Tammer [14] and Nam and Zȃlinescu [18].Chen and Li [15,16] deduced the globally Lipschitz property of   by the dual space approach and applied it to discussing Hölder continuity of solutions to parametric vector (quasi)equilibrium problems.Motivated by the work reported in [8,15,16], in this paper we further discuss the globally Lipschitz property of   in linear normed spaces via the primal space approach (see Proposition 7).The equivalence of both expressions for globally Lipschitz constants  and   obtained by primal and dual spaces approaches is established (see Proposition 8).The expressions for   and  are related to a minimization problem () and a maximization problem (), respectively, and hence, the equivalence that   =  means that the property of strong duality (i.e., inf() =   =  = sup()) holds between primal and dual problems.Furthermore, the above discussions are extended to general Gerstewitz function  −V , and exact characterizations to the globally Lipschitz property for  −V are discussed, which would further complete the theory of [8,18].In addition, when the ordering cone is 2 Abstract and Applied Analysis polyhedral, the expression for calculating Lipschitz constant is also given.
Scalarization approaches have been used as efficient methods to study semicontinuity and Hölder continuity of parametric VEPs.Among them, scalarizing approaches were applied by using linear functionals [27,35] or nonlinear scalarization functions [15,16,24].We notice that nonlinear scalarization methods by virtue of several nonlinear scalarization functions to deal with solution stability have received some attention; for example, Sach and Tuan [36,37] have used Gerstewitz-like scalarization functions to study both upper and lower semicontinuities of solution mappings for parametric VEPs.The globally Lipschitz property of the nonlinear scalarization function   seems to be good at dealing with stability and sensitivity analysis of VEPs [15][16][17].It is necessary to further exploit applications of the globally Lipschitz property of   together with other useful properties for studying Hölder continuity of parametric VEPs.Motivated by the work reported in [25,31,38], this paper also aims to give some applications of the properties of   to the Hölder continuity of solutions for parametric VEPs.To our aim, the nonlinear scalarization function   as a fundamental tool will play key roles such that, its globally Lipschitz property, monotonicity, and sublinearity will be fully exploited.The results obtained are new and generalizations of known ones [25,31] for the corresponding scalar cases, and our approach is totally based on the techniques of nonlinear scalarization.
The rest of the paper is organized as follows.In Section 2, we first summarize basic properties of the nonlinear scalarization function   , then discuss the globally Lipschitz property of   in linear normed spaces via the primal space approach, establish the equivalence that   = , and finally, extend the discussions to the general case  −V .In Section 3, as applications of the Lipschitz property of   , we study Hölder continuity of both single-valued and setvalued solution mappings to parametric VEPs based on the nonlinear scalarization approach.The last section gives some conclusions.

Lipschitz Properties of Nonlinear Scalarization Functions
In this section, we first recall the nonlinear scalarization function   in vector optimization.Its main properties, especially, the globally Lipschitz property, are summarized.
Remark that the form of (, )-max scalarizing function () := max ∈  ⟨, ⟩ is also widely developed and has many applications, such as stability analysis of vector equilibrium problems [24] and optimality conditions for vector optimization problems [39].
Remark 4. The oriented distance function Δ is not applicable for studying Hölder continuity of solution mappings to parametric vector equilibrium problems (PVEP) and (PVAEP) in this paper, unlike the Gerstewitz function  done in the next section.This is because the properties like Proposition 2 (ii) and (v) with  ̸ = 0 are not satisfied by the function Δ, while they play important roles in our study.
Let B( 0 , ) be the closed ball centered at  0 and radius  > 0. It is said that  :  →  is Lipschitz around  0 ∈  iff there exist  > 0 and  > 0 such that ‖() − (  )‖ ≤ ‖ −   ‖, ∀, where   :  → R is the Minkowski functional associated with  and  ⊂  is a symmetric closed and convex neighborhood of 0  such that  +  ⊂ .
Note that   = 1/ max , = inf{1/ > 0 |  + B ⊂ }.Tammer and Zȃlinescu [8] recently have studied the Lipschitz property of the Gerstewitz function under more general settings than ours, using the primal space approach, which is different from the dual space approach adopted by us [15,16].In this paper, we limit our discussions in linear normed spaces to get more exact characterizations and more clear geometrical interpretations.
Whether the two Lipschitz constants  =   hold, we show it as follows.
Without loss of generality, we may assume that ⟨ λ, ⟩ = 1; that is, λ ∈   .Hence, we can deduce that Thus, By the arbitrariness of  > 0, we obtain that   ≤ .
Based on the above analysis, the equivalence that  =   holds.
We have calculated that the Lipschitz constant  = sup ∈  ‖‖ = ‖(2, −1)‖ = √ 5. Now we calculate another Lipschitz constant   = 1/ max , .Notice that the distance from a point ( 0 ,  0 ) ∈ R 2 to the line  =  +  (resp.,  +  +  = 0) is It is easy to verify that Thus, we also get that the Lipschitz constant Thus, we have Because  ∈ int ,     < 0,  = 1, . . ., .Whence, we obtain that This completes the proof.As a direct application to the proof of Proposition 8, we give a note on Lipschitz properties of the directional minimal time function [18].
Given a vector V ∈ , V ̸ = 0  , and a nonempty closed set Ω ⊂  and Ω ̸ = , the directional minimal time function with direction V and target set Ω is defined by This class of functions is similar to the class of nonlinear scalarization functions that has been extensively used to study vector optimization problems (see [7,8]): Obviously,   is a special but popular case of  V , by taking Ω := − and V := − ∈ − int .
We recall a result of the globally Lipschitz property for  V [18, Proposition 4.1].
Proposition 15 (see [18]).Suppose that V ∈ int Ω ∞ .Then  V is globally Lipschitz on  with Lipschitz constant Based on a similar proof to that of  =   , letting  := Ω ∞ , which is a closed and convex cone, and  := V, we can get where Ω * ∞ is the dual cone of Ω ∞ .Thus, we have the following equivalent proposition.
Similarly, the conclusion also follows by applying Corollary 20 with  := Ω ∞ .

Applications to the Hölder Continuity
The globally Lipschitz property of the nonlinear scalarization function   seems to be good at dealing with stability and sensitivity analysis of vector optimization problems, such as [15][16][17].In this section, we will give some direct applications of this property to the Hölder continuity of solutions for parametric vector equilibrium problems.The proofs of the results obtained are applications of the corresponding ones in [25,31] for the scalar problems, by using the mentioned scalarization function   .The results are new and generalizations of known ones for the corresponding scalar cases.

A Single-Valued Case.
In this subsection, let (,   ) be a linear metric space, let  be a linear normed space, let Λ and Ω be nonempty subsets of metric spaces, and let  ⊂  be a pointed, closed, and convex cone with int  ̸ = 0. Let  : Λ   be a set-valued mapping with nonempty, closed, and convex values and let  :  ×  × Ω →  be a vector-valued mapping.

Definition 23 (classical notion). A set-valued mapping 𝐺 :
Ω   is said to be ℓ ⋅ -Hölder continuous at  0 , if and only if there is a neighborhood where ℓ ≥ 0 and  > 0, and B  denotes the unit ball of .
We say that  (or Next, we introduce the concept of strong -convexity for a vector-valued mapping, which extends [25, Definition 2.1] from real-valued to vector-valued case. Definition 24.Let (, ) be a linear metric space.A vectorvalued mapping  :  →  is said to be ℎ ⋅ -strongly convex with respect to  ∈ int  on , if and only if there exists  ∈ int  such that for all ,  ∈  and all  ∈]0, 1[, where ℎ,  > 0.
Note that  ∈ int  plays the role of the "modulus of strong -convexity" of the mapping .Clearly, as in the scalar case, strong -convexity implies (strict) -convexity.As shown in [25], the strong -convexity of  plays important roles.Lemma 25.If  :  →  is ℎ ⋅ -strongly -convex with respect to  ∈ int  on , then the real-valued function   →   (()) is ℎ ⋅ -strongly convex on .
Proof.For any ,  ∈  and all  ∈]0, 1[, we have where the first and the third inequalities follow from the monotonicity and sublinearity of   , respectively, and the second equality follows from Proposition 2(v).Whence, the composite function   ∘  is ℎ ⋅ -strongly convex on .
Now we state and prove the following results.
Thus, we could apply Theorem 3.1 of [25] by replacing  therein with   ∘ .Now we need to check all conditions of   ∘ .
Obviously, it is a special case of the model of finding  ∈ () such that Stability for parametric variational problems of the Minty type has not received much attention so far.Very recently, Lalitha and Bhatia [38] have studied upper and lower semicontinuity of the solutions as well as the approximate solutions to a parametric quasivariational inequality of the Minty type.Chen and Li [30] have established upper Hölder continuity of the solutions to Minty-type parametric vector quasiequilibrium problems.In this subsection, by using nonlinear scalarization technique, we will study a special Mintytype parametric vector approximate equilibrium problem (PVAEP).
In this subsection, for fixed  ∈ int , we assume that   (, , ) ̸ = 0 for small positive  and (, ) in a neighborhood of the considered point ( 0 ,  0 ).In general,   (, , ) may not be a singleton.
As pointed out in [15], we will show how to establish the Hölder continuity of set-valued solution mappings to parametric vector equilibrium problems without using any priori information of the solution sets, by employing nonlinear scalarization approach.In [17], we have given a positive answer to this subject.Now we give another answer herein.
Corollary 32.For problem (SEP), assume that () = , a nonempty bounded convex subset of  and   (, ) is nonempty for small  > 0 and  in a neighborhood of the considered point  0 .Assume further that (i) there is a neighborhood  of  0 such that, for each  ∈  and  ∈ , (, ⋅, ) is -convex on ; (ii) for ,  ∈ , (, , ⋅) is ℎ ⋅ -Hölder continuous on .

Conclusions
In this paper, motivated by the work of Tammer and Zȃlinescu [8], we deduce the globally Lipschitz property of   in linear normed spaces via the primal space approach.This is different from the dual space approach adopted by our previous works [15,16].The equivalence between them is established, and primal-dual interpretations of both expressions for globally Lipschitz constants are explained.Furthermore, exact characterizations to the globally Lipschitz property for general Gerstewitz function  −V are discussed.We mention that the expression for calculating Lipschitz constant is given when the ordering cone is polyhedral.As simple applications, Hölder continuity of solutions for parametric vector equilibrium problems is also showed.Besides this, the globally Lipschitz property of   and  −V seems to have many potential applications, for example, the stability and sensitivity analysis of vector equilibrium problems [15][16][17] and optimality conditions for vector optimization problems [8,14].For more applications, we will exploit in future research.
sup()) holds between the following primal problem () and dual problem () for fixed  ∈ int :Clearly, the Lipschitz constants   and  are deduced via two approaches: the primal space and the dual space approaches, respectively.Remark 14.Relative to , the expression for Lipschitz constant   is natural and exhibits a clear geometrical interpretation.In the setting of linear normed space , if  > 0 is the largest radius such that the closed ball B(, ) centered at given  ∈ int  with radius  lies in the ordering cone  of , that is, B(, ) ⊂ , then 1/ is the Lipschitz constant   .Clearly, the value of  coincides with the distance from  to the boundary of .Based on the geometrical interpretation of   , we know that the choice of  ∈ int , namely, the location of , will directly confirm the modulus of Lipschitz continuity of   .It is obvious that the Lipschitz constant   becomes larger whenever  is closer to the boundary of .