Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization

We show that under a separation property, a Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\mathcal {Q}}}}$$\end{document}-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.


Introduction
Proper efficient points were introduced to eliminate efficient points exhibiting some abnormal properties.They can be described in terms of separations between the ordering cone and the considered set.Such points have been the object of many investigations, see for example [1][2][3][4][5][6][7][8][9].In [5], the author presented the notion of Q-minimal point and showed that several types of proper efficient points can be reduced in a unified form as Q-minimal points.The following kinds of proper efficient points were studied in [5]: Henig global proper efficient points, Henig proper efficient points, super efficient points, Benson proper efficient points, Hartley proper efficient points, Hurwicz proper efficient points, and Borwein proper efficient points; the latter three types considered for the first time.Optimality conditions for proper efficient points were obtained, and scalarization for Q-minimal points was established.Since the scalar optimization theory is widely developed, scalarization turns out to be of great importance for the vector optimization theory [8,[10][11][12][13][14][15][16][17].In this work, we show that under a separation property called SSP, a Q-minimal point in a normed space is the minimum of a given sublinear function.This fact provides sufficient conditions for the proper efficient points analysed in [5] and also for tangentially Borwein proper efficient points which were not considered there.The sufficient condition becomes a characterization in case of Benson proper efficient points.We note that SSP is a variation of a separation property introduced in [18].On one hand, our results complement those obtained in [5] making use of a different scalar function and also establishing conditions for tangentially Borwein proper efficient points.In our results, for every type of proper efficient point, we apply the separation property to a fixed Q-dilation of the ordering cone instead of to a sequence of ε-conic neigbhourhoods of the ordering cone (as it is done in [18]).This fact leads us to optimal conditions for nine types of proper efficient points (instead of two types in [18]) deriving scalarization results in the setting of normed spaces under weaker assumptions than those in [18] and making use of the same scalar function.In addition, our characterization of Benson proper efficient points shed light to the last question stated in the conclusions of [19].
Recently, several authors have been interested in introducing and studying approximate proper efficiency notions.The common idea in the concepts of approximate efficiency is to consider a set that approximates the ordering cone, that does not contain the origin in order to impose the approximate efficiency (or non-domination) condition.In [20,21], the notions of approximate proper efficient points in the senses of Benson and Henig were introduced extending and improving the most important concepts of approximate proper efficiency given in the literature at the moment.In addition, the authors characterized such approximate efficient points through scalarization assuming generalized convexity conditions.In this manuscript, we adapt the approach followed to obtain optimal conditions for Q-minimal points to establish new characterizations of Benson and Henig approximate proper efficient points through scalarizations.Again, our results are based on SSP and we do not impose any kind of convexity assumption.So, our results complement those in [20,21].
The paper is organized as follows.We introduce preliminary terminology in Section 2. In Section 3, we introduce SSP and establish two separation theorems, Theorems 3.1 and 3.3.The first one provides an extension of [18,Theorem 4.3] to normed spaces in which some assumptions for the equivalence have been relaxed, and the latter, that is our main separation result, provides some optimal conditions for proper and approximate proper efficient points.Theorem 3.6 shows that under SSP, Q-minimal points can be obtained minimizing a sublinear function explicitly defined.Corollary 3.7 particularizes the former necessary condition for each type of proper efficient point.Corollary

Notation and previous definitions
Throughout the paper X will denote a normed space, • the norm on X, X * the dual space of X, • * the norm on X * , and 0 X the origin of X.By B X (resp.B • X ) we denote the closed (resp.open) unit ball of X and by S X we denote the unit sphere, i.e., B X := {x ∈ X : x ≤ 1}, B • X = {x ∈ X : x < 1}, and S X := {x ∈ X : x = 1}.Given a subset S ⊂ X, we denote by S (resp.bd(S), int(S), co(S), co(S)) the closure (resp.the boundary, the interior, the convex hull, the closure of the convex hull) of S. Besides, for every f ∈ X * , we will denote by sup S f (resp.inf S f ) the supremum (resp.infimum) of f on the set S. By R + (resp.R ++ ) we denote the set of non negative real numbers (resp.strictly positive real numbers).A subset C ⊂ X is said to be a cone if λx ∈ C for every λ ≥ 0 and x ∈ C. Let C ⊂ X be a cone: and C is said to be solid if int(C) = ∅.All cones in this manuscript are assumed to be non-trivial unless stated otherwise.From now on, A 0 denotes A ∪ {0 X } for every subset An open cone Q is said to be pointed (resp.convex) if Q 0 is a pointed (resp.convex) cone on X.Fixed a subset S ⊂ X, we define the cone generated by S as cone(S) := {λs : λ ≥ 0, s ∈ S} and cone(S) stands for the closure of cone(S).A non-empty convex subset B of a convex cone C is said to be a base for C if 0 X ∈ B and for every x ∈ C \ {0 X } there exist unique λ x > 0, b x ∈ B such that x = λ x b x .Given a cone C ⊂ X, its dual cone is defined by C * := {f ∈ X * : f (x) ≥ 0, ∀x ∈ C} and the set of strictly positive functionals by C # := {f ∈ X * : f (x) > 0, ∀x ∈ C, x = 0 X }.In general, int(C * ) ⊂ C # .It is known that a convex cone C ⊂ X has a base if and only if C # = ∅, and C # = ∅ implies that C is pointed.In particular, for every f ∈ C # , the set B := {x ∈ C : f (x) = 1} is a base for C. A convex cone C is said to have a bounded base if there exists a base B for C such that it is a bounded subset of X.It is known that C has a bounded base if and only if int(C * ) = ∅ if and only if 0 X is a denting point for C (see [? , Theorem 3.8.4]for the first equivalence and [22][23][24] for further information about dentability and optimization).
A convex cone C is said to have a (weak) compact base if there exists a base B of C which is a (weak) compact subset of X.A pointed cone admits a compact base if and only if it is locally compact if and only if the cone satisfies the strong property (π) if and only if there exists f ∈ C # such that C ∩ {f ≤ λ} is compact, ∀λ > 0. We refer the reader to [25, p. 338] for the first equivalence, to [26,Definition 2.1] for the definition of strong property (π) and to [27,Remark 2.1] for the last two equivalences.The following two sets are called augmented dual cones of a given cone C and they were introduced in [18], Let C ⊂ X be a pointed convex cone, then C provides a partial order on X, say ≤, in the following way, x ≤ y ⇔ y − x ∈ C. In this situation, we say that X is a partially ordered normed space and C is the ordering cone.Let X be a partially ordered normed space, C ⊂ X the ordering cone, and A ⊂ X a subset.We say that x 0 ∈ A is an efficient (or Pareto minimal) point of A, written x 0 ∈ Min(A, C), if (x 0 − C) ∩ A = {x 0 }.Next, we define some types of proper efficient points.Note that (i)-(iii) and (v)-(viii) are taken from [5,Definition 21.3] (see also [8,Definition 2.4.4]),(iv) is taken from [28] adapting from maximal to minimal proper efficient point, and (ix) is taken from [29].The latter was obtained adapting [30, Definition 2] and was called Borwein proper efficient point, but we have changed the name to distinguish (v) and (ix) below.Recall that fixed a subset A ⊂ X and a point x ∈ A, the contingent cone to A at x is defined by T (A, x) := {lim n λ n (x n − x) ∈ X : (λ n ) ⊂ R ++ , (x n ) ⊂ A, and lim n x n = x}, see [31] for details.
Definition 1 Let X be a partially ordered normed space, C the ordering cone, and A ⊂ X a subset.
x 0 ∈ Min(A, C) and there exists M > 0 such that if f ∈ C * and f (x−x 0 ) < 0 for some x ∈ A, then there exists g ∈ C * , g = 0, satisfying g f Proper efficient points were introduced for two main reasons: first, to eliminate certain anomalous minimal points; second, to establish some equivalent scalar problems whose solutions provide at least most of the minimal points.We have the following chain of inclusions (see [5,Proposition 21.4] and [8,Proposition 2.4 In addition, by [12,Theorem 3.44], if we have Furthermore, under extra assumptions on A or C, we can find more inclusions (again [5,8]).

Separation theorems and scalarization for Q-minimal points
We begin this section introducing a separation property of cones called strict separation property (SSP for short).Later, we establish two theorems which separate two cones in a normed space by a sublevel set of a sublinear function.The second separation theorem will be key to determining optimality conditions for the proper efficient points introduced in section 2.
Let us recall the following.Throughout the paper, we denote by A 0 the set A ∪ {0 X }, for any A ⊂ X.On the other hand, every cone C ⊂ X we consider is assumed non-trivial, i.e., {0 X } C X.For any cone C, we define the following convex sets C ∧ := co(C ∩ S X ) and C ∨ := co((bd(C) ∩ S X ) 0 ) to be used in the following separation property.
Definition 2 Let X be a normed space and C, K cones on X.We say that the pair of cones (C, K) has the strict separation property (SSP for short) if 0 X ∈ C∧ − K∨.
Remark 1 Given two cones C, K ⊂ X we have the following.
for every non-trivial cone K ⊂ X, it follows that (C, K) has SSP if and only if (−C, −K) has SSP if and only if (C, (X \ K) 0 ) has SSP.
In [18,Definition 4.1] the author introduces the separation property SP for closed cones under the condition 0 X ∈ C ∧ − K ∨ .It is clear that, for closed cones, SSP implies SP.The reverse is true in reflexive Banach spaces because on such spaces the closure of the difference of two bounded convex sets equals the difference of the closures of the sets (referring to the Minkowski difference).
The following result provides a version of [18,Theorem 4.3] in the setting of general normed spaces.It is worth pointing out that we do not need extra assumptions to obtain an equivalence (such as the cones to be convex and closed or the space to be of finite dimension).
Theorem 3.1 Let X be a normed space and C, K cones on X.The following assertions are equivalent.
Corollary 3.2 Let X be a normed space and C, K cones on X.Then (C, K) has SSP if and only if (co(C), K) has SSP.
Theorem 3.1 establishes that the results in [18] obtained for reflexive Banach spaces can be extended to general normed spaces under SSP.On the other hand, the following result is our main separation theorem and it will provide optimal conditions for the proper minimal points introduced in Section 2 and for the approximate proper efficient points in Section 4.
Corollary 3.4 Let X be a normed space and C, K cones on X.
Definition 3 Let X be a normed space, Q ⊂ X an open cone, and A ⊂ X a subset.We say that The following notion is directly related to Q-minimal points.
Definition 4 Let X be a partially ordered normed space, C ⊂ X the ordering cone, and Q ⊂ X an open cone.We say that The following result shows that the proper efficient points introduced in Definition 1 are Q-minimal points with Q being appropiately chosen cones.Assertions (i)-(viii) below are Assertions (iii)-(x) from [5, Theorem 21.7], respectively, under minor adaptations (see also [8,Theorem 2.4.11]).On the other hand, assertion (ix) below can be proved in a similar way as (ii) and (iii) in [5,Theorem 21.7], so we omit the proof.We need more terminology.Let X be a partially ordered normed space, C ⊂ X the ordering cone, and B a base of C. Let δ B := inf{ b : b ∈ B} > 0, for every 0 < η < δ B we define a convex, pointed and open cone V η (B) := cone(B + ηB • X ).On the other hand, given 0 < ε < 1 we define another open cone C(ε) := {x ∈ X : d(x, C) < εd(x, −C)}.Theorem 3.5 Let X be a partially ordered normed space, C the ordering cone, A ⊂ X a subset, and x 0 ∈ A. The following statements hold.
and B a bounded base of C. (ix) x 0 ∈ TBo(A, C) if and only if x 0 ∈ QMin(A) for Q = −X \ T (A + C, x 0 ).
In [5], the author provides some necessary and sufficient conditions for a Q-minimal point to be a solution for a scalar optimization problem.In the following result, we provide a necessary condition in terms of SSP for Qminimal points to be a solution of another scalar problem.
Theorem 3.6 Let X be a partially ordered normed space, C the ordering cone, Q ⊂ X an open cone, and x 0 ∈ A ⊂ X. Assume that x 0 ∈ QMin(A) and Proof It is not restrictive to assume that x 0 = 0 X (by translation we obtain the general case).Since (C, Q 0 ) has SSP, Theorem 3.3 applies and there exists (f, α) In the following result, we establish a particular version of the former one for each type of proper efficient point introduced in Definition 1.
Corollary 3.7 Let X be a partially ordered normed space, C the ordering cone, and x 0 ∈ A ⊂ X. Assume that either of the following statements holds.
is attained only at x 0 .
Proof We begin noting the inclusions C \ {0 X } ⊂ Q for (i)-(ix).The inclusion for the cases (i) and (iv)-(ix) is trivial.For case (ii), we apply the following chain of equivalences: co Now assume that either of (i)-(ix) holds.By Theorem 3.5, x 0 ∈ QMin(A, C) and by the former paragraph, C ∩ Q = ∅.Then Theorem 3.6 applies.
The following result characterizes Benson proper minimal points via (1).
Corollary 3.8 Let X be a partially ordered normed space, C the ordering cone, and Proof It is not restrictive to assume that x 0 = 0 X .⇒ As (−C, cone(A + C)) has SSP, then (C, −(X \cone(A+C)) 0 ) has SSP too.Therefore, Remark 1 yields that (−C, (X \ cone(A + C)) 0 ) has SSP as well.Now, apply Theorem 3.5 (iii) and Corollary 3.7 (iii).
[18, Theorem 5.8] also characterizes Benson proper efficient points via ( 1), but under more restrictive assumptions than Corollary 3.8 and, in addition, applying the separation property to a sequence of ε-conic neighbourhoods instead of to an only cone −cone(A − x 0 + C).
In the following, we study sufficient conditions to have GHe(A, C) = Be(A, C).Such equality will lead to a characterization for Henig global proper efficient points via (1).The set GHe(A, C) is contained in the set Be(A, C) whenever C is a closed, convex, and pointed cone (see [4]).The next result establishes the equality of such sets under some extra assumptions.Theorem 3.9 Let X be a partially ordered normed space, C the ordering cone, and A ⊂ X a subset such that A + C is convex.If C has a weakly compact base, then GHe(A, C) = Be(A, C).
Proof ⊂ is provided by [8, Proposition 2.4.6 (i)] for separated topological vector spaces and non closed cones.⊃ Fix an arbitrary x 0 ∈ Be(A, C).It is not restrictive to assume that x 0 = 0 X .Then 0 X ∈ Min(A, C) and 0 X ∈ Min(cone(A + C), C).Hence (−C) ∩ cone(A + C) = {0 X }.On the other hand, A + C is convex.Then cone(A + C) is convex, implying that cone(A+C) is weak closed.Now, [18, Theorem 5.2] applies and there exists a convex cone The following result is a direct consequence of Corollary 3.8 and Theorem 3.9.
Corollary 3.10 Let X be a partially ordered normed space, C the ordering cone, and x 0 ∈ A ⊂ X such that A + C is convex.Assume that C has a weakly compact base.If (−C, cone(A − x 0 + C)) has SSP, then x 0 ∈ GHe(A, C) if and only if there exists (f, α) ∈ C a# + such that min x∈A {f (x − x 0 ) + α x − x 0 } is attained only at x 0 .
Corollary 3.8 provides a sufficient condition to find elements in TBo(A, C).In the following, we will establish that it becomes a characterization if we assume the following geometric condition on x 0 ∈ A. It is said that a set A ⊂ X is starshaped at some Consequently, we obtain the following characterization for tangentially Borwein proper efficient points.
Corollary 3.11 Let X be a partially ordered normed space, C the ordering cone, and x 0 ∈ A ⊂ X. Assume that A + C is starshaped at x 0 and that C has a weakly compact base.
We finish this section with the following problem for future research.
Problem 3.12 Is it possible to characterize Q-minimal points via SSP assuming any extra conditions?
In Theorem 3.6 we provide necessary conditions for Q-minimal points.On the other hand, in Corollaries 3.8, 3.10, and 3.11, we establish characterizations of Benson, global Henig, and tangentially Borwein proper efficient points, respectively, answering the former problem for such particular kinds of Qminimal points.So, it is of interest to solve Problem 3.12 for any of the other types of Q-minimal points.

Scalarization for approximate proper efficient points
In this section, we obtain optimal conditions through scalarization for approximate proper efficient points in the senses of Benson and Henig.We obtain our results after extending the approach for Benson and Henig proper efficient points in the precedent section.
Let us introduce the terminology of approximate proper efficiency.From now on, the ordering cone C ⊂ X is assumed to be closed, convex, and pointed.The notions of approximate efficiency are defined replacing the ordering cone C by a non-empty set D that it.For a non-empty set D ⊂ X \{0 X }, we define the set D(ε) := εD, for ε > 0, and D(0) := cone(D) \ {0 X }.We also introduce the family of sets for every ε ≥ 0. The following notion was introduced by Gutierrez, Huerga, and Novo in [20] for locally convex spaces.
Let us recall that a function g : X → R is strongly monotonically increasing if for each x, y ∈ X, y − x ∈ C \ {0 X } ⇒ g(x) < g(y).It is clear that g (f,α) is strongly monotonically increasing for every (f, α) ∈ C a# + .Monotonicity will be used in the proof of the following result showing that the necessary condition (i) in Theorem 4.1 is also sufficient.
As a consequence of the former result and Theorem 4.1 (i) we obtain the following characterization for approximate proper efficiency in the sense of Benson.
Corollary 4.3 Let X be a partially ordered normed space, C the ordering cone, Unfortunately, the necessary condition (ii) in Theorem 4.1 is not sufficient, as the following example shows.
The preceding example leads us to the following natural question.We devote the rest of this section to study approximate Henig proper efficiency.An easy adaptation of the proof of Theorem 4.1 gives the following necessary conditions for approximate proper solution in the sense of Henig.
Theorem 4.5 Let X be a partially ordered normed space, C the ordering cone, A ⊂ X a subset, ε ≥ 0, and D ∈ H. Let x 0 ∈ He(A, C, D, ε) and the corresponding Since He(A, C, D, ε) ⊂ Be(A, C, D, ε), Theorem 4.1 also provides necessary conditions for Henig approximate proper solutions.Furthermore, when the former inclusion becomes a set equality, Corollary 4.3 provides a characterization Henig approximate proper solutions.This leads to the last results in the work.Before stating them, we introduce the notion of approximating family of cones.
Definition 7 Let X be a partially ordered normed space and C the ordering cone.
(i) Let F = {C n ⊂ X : n ∈ N} be a family of decreasing (with respect to the inclusion) solid, closed, and pointed convex cones.We say that F approximates C if C \ {0 X } ⊂ int(C n ) eventually (i.e., there exists n 0 ∈ N such that C \ {0 X } ⊂ int(C n ) for every n ≥ n 0 ) and C = ∩ n C n .(ii) Let F be an approximating family of cones for C. We say that F separates C from a closed cone K ⊂ X if C ∩ K = {0 X } ⇒ C n ∩ K = {0 X } eventually.
Given D ⊂ X \ {0 X }, ε > 0, and x ∈ X, we denote by S(D(ε), x) the set of all families of cones that approximate C and separate C from the cone −cone(A − x + D(ε)).

Conclusions
In this work, we provide optimal sufficient conditions with a sublinear function for Henig global proper efficient points, Henig proper efficient points, super efficient points, Benson proper efficient points, Hartley proper efficient points, Hurwicz proper efficient points, Borwein proper efficient points, and tangentially Borwein proper efficient points; in the case of Benson proper efficiency the optimal condition becomes a characterization.The approach is done in a unified way considering such proper efficient points as Q-minimal points.For every type of proper efficient point we apply a separation property to a fixed Q-dilation of the ordering cone.For future research we ask if it is possible to characterize Q-minimal points in general via SSP assuming any extra conditions.In the last part of the work, we adapt our arguments to obtain new characterizations of Benson and Henig approximate proper efficient points through scalarizations.We also provide necessary conditions for approximate Benson proper efficient points via ε-approximate solutions and we ask if it is possible to extend such a result to a characterization assuming any extra condition.Our results are established in the setting of normed spaces and they do not impose any kind of convexity and boundedness assumption.

Theorem 4 . 2
Let X be a partially ordered normed space, C the ordering cone, x 0 ∈ A ⊂ X, ε ≥ 0, and D ∈ H.If there exists

Problem 4 . 4
Is it possible to characterize approximate Benson proper efficient points via ε-approximate solutions assuming any extra conditions?

Corollary 4 . 7
Let X be a partially ordered normed space, C the ordering cone, x 0 ∈ A ⊂ X, ε ≥ 0, and D ∈ H. Assume that (−C, cone(A − x 0 + D(ε))) has SSP and at least one of the following assertions hold:(i) X has finite dimension.(ii) C has a weakly compact base and cone(A − x 0 + D(ε)) is weakly closed.Then x 0 ∈ He(A, C, D, ε) if and only if there exists (f, α) ∈ C a# + such that min x∈(A+D(ε))∪{x0} f (x − x 0 ) + α x − x 0 ,is attained only at x 0 .
3.8 characterizes Benson proper efficient points via scalarization, and Corollaries 3.10 and 3.11 characterize, respectively, Henig global and tangentially Borwein proper efficient points under some extra assumptions.In Section 4, we recall the notions of approximate efficiency in the sense of Benson and of Henig.Theorems 4.1 and 4.2 provide, respectively, necessary and sufficient conditions through scalarization for approximate Benson proper efficient points; in a similar way but under extra assumptions, we establish Corollary 4.6 that characterizes approximate Henig proper points.