Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological / algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper eﬃciency concept based on generalized dilating cones which are relatively solid (i.e


Introduction
It is known that in vector optimization as well as in image space analysis (see Jahn [25], Giannessi [28,29] and references therein) in infinite dimensional linear spaces difficulties may arise because of the possible non-solidness of ordering cones (for instance in the fields of optimal control, approximation theory, duality theory). Thus, it is of increasing interest to derive optimality conditions and duality results for such vector optimization problems using generalized interiority conditions (see, e.g., Adan and Novo [1][2][3], Bagdasar and Popovici [4], Bao and Mordukhovich [5], Borwein and Goebel [7], Borwein and Lewis [7], Grad and Pop [8], Khazayel et al. [9], Zȃlinescu [10,11], Cuong et al. [12]). Such conditions can be formulated using the well-established generalized interiority notions given by quasi-interior, quasirelative interior, algebraic interior (also known as core), relative algebraic interior (also known as intrinsic core, pseudo relative interior or intrinsic relative interior). Moreover, it is known that for defining Pareto-type solution concepts of vector optimization problems, generalized interiority notions are also useful.
In recent works related to vector optimization in real linear spaces (see, e.g., Adan and Novo [1][2][3], Bao and Mordukhovich [5], Khazayel et al. [9], Novo and Zȃlinescu [13], Popovici [14], and Zhou, Yang and Peng [15]), the intrinsic core notion is studied in more detail. Having two real linear spaces X and E, a vector-valued objective function f : X → E, a certain set of constraints Ω ⊆ X, a convex (ordering) cone K ⊆ E (with possibly empty algebraic interior), a vector optimization problem is defined by f (x) → min w.r.t. K x ∈ Ω.
For this problem, a useful solution concept is to say that a pointx ∈ Ω is optimal if where icor K denotes the intrinsic core of K. It is important to know that in finite dimensional real linear spaces the intrinsic core of any convex set (cone) is nonempty but core could be empty (for instance recession cones of polyhedral sets in R n are convex cones which are not necessarily solid). Replacing icor K in (1) by any other (generalized) interior of K one can define other solution concepts. By involving an appropriate set S ⊆ E \ {0} with icor K ⊆ S one can define a stronger solution concept by replacing (1) by Notice that (2) implies (1) (if K is not a linear subspace). This leads to other solutions concepts such as the well-known concepts of Pareto efficiency (i.e.,x satisfies (2) for S := K \(−K)) or proper Pareto efficiency (i.e.,x satisfies (2) with S := cor C for some generalized dilating cone C).
In order to derive theoretical (duality assertions) and computational (algorithms based on scalarization) results in vector optimization, one needs strict monotonicity properties concerning the scalarizing functional. This is the reason that the solution concepts of weak and proper efficiency (based on generalized interiors of the ordering cone) are of big importance. Using linear scalarization, it is known that the set of properly (respectively, weakly) efficient solutions can be completely characterized in the (generalized) convex case (e.g., if f [Ω] + K is convex) if the ordering cone is pointed (respectively, solid).
In particular, the notion of proper Pareto efficiency is very important, not only from a theoretical point of view, but also from a practical point of view. In the literature, several notions of proper efficiency have been proposed. The concept of proper efficiency dates back to the work by Kuhn and Tucker [16]. Geoffrion [17] proposed a very useful concept for multiobjective optimization problems (i.e., E := R m and K := R m + ) for which the solutions have a bounded trade-off (which decision makers could prefer in view of applications). Some well-known generalizations of the mentioned proper efficiency concepts are given by Borwein [18], Benson [19] and Henig [20]. These concepts and corresponding generalizations are discussed, among others, by Durea, Florea and Strugariu [21], Eichfelder and Kasimbeyli [22], Gutierrez et al. [23], Hernández, Jiménez and Novo [24], Jahn [25], Khan, Tammer and Zȃlinescu [26], and Luc [27].
To attack vector optimization problems, it is known that the Image Space Analysis (ISA) approach by Giannessi [28] (see also [29] for some perspectives on vector optimization via ISA) is of great importance. In the ISA approach, one is constructing a certain convex cone using the original ordering cone (defining the solution concept) as well as the cone which is used to describe the constraints. So, the idea arises to consider generalized interiority notions within ISA. We will not focus on this ISA approach but like to highlight that the framework of this paper is useful for this field. Notice, also for the case X := R n , E := R m and K := R m + , the cone in the ISA approach is not solid if beside inequality constraints also equality constraints appear in the problem.
The outline of the article is as follows. First, in Section 2 we recall important algebraic properties of convex sets and convex cones in linear spaces. In our main results, we will deal with relatively solid, convex cones, and for proving them, we will use separation techniques in linear spaces that are based on the intrinsic core notion (see [9] and Proposition 2.2).
In Section 3, we study vector optimization problems involving relatively solid, convex cones which are not necessarily pointed. We will concentrate in this section on the concept of Pareto efficiency as well as on the concept of weak Pareto efficiency (based on the intrinsic core notion). For the sets of solutions of the given vector optimization problem w.r.t. these concepts, we derive some useful properties and relationships.
Henig-type proper efficiency concepts based on certain families of generalized dilating cones play the main role in Section 4. Our proposed concept uses generalized dilating cones which are relatively solid and convex (see Definition 4.7). Since our concept is based on the intrinsic core notion, we are able to find (in the case that K is not solid) a better inner approximation of the set of Pareto efficient solutions in comparison to most of the known (Henig-type) proper efficiency concepts (see also Remark 4.11).
In Section 5, we present scalarization results for vector optimization problems. For the linear scalarization case, we are able to state representations for the sets of (weakly, properly) efficient solutions under certain generalized convexity assump-tions.
The article concludes in Section 6 with a brief summary and an outlook to future work.

Preliminaries in Preordered Linear Spaces
Throughout the paper, let E = {0} be a real linear space, and let E be its algebraic dual space, which is given by It is well-known that E can be endowed with the strongest locally convex topology τ c , that is generated by the family of all the semi-norms defined on E (see Khan,Tammer and Zȃlinescu [26,Sec. 6.3 ]). In the literature, the topology τ c is known as the convex core topology. According to [26,Prop. 6.3.1 ], the topological dual space of E, namely (E, τ c ) * , is exactly the algebraic dual space E . In recent works (see, e.g., Khazayel et al. [9], Novo and Zȃlinescu [13]), the convex core topology τ c is used to derive properties for algebraic interiority notions (such as core and intrinsic core).

Algebraic Interiority Notions
Let us define, for any two points x and x in E, the closed, the open, the half-open line segments by Consider any set Ω ⊆ E. The smallest affine (respectively, linear) subspace of E containing Ω is denoted by aff Ω (respectively, span Ω). Two special subsets of Ω will be of interest (c.f. Holmes [30, pp. 7-8]): • the algebraic interior (or the core) of Ω, which is given as • the relative algebraic interior (or the intrinsic core) of Ω, which is defined by Notice, for any nonempty (not necessarily convex) set Ω ⊆ E, we have and if icor Ω = ∅, cor Ω = ∅ ⇐⇒ aff Ω = E.
If Ω is convex, then icor Ω is convex as well. The algebraic closure of Ω is defined using all linearly accessible points of Ω (c.f. Holmes [30, p. 9]) as For any d ∈ E, the vector closure of Ω in the direction d is denoted by Notice that cl τc Ω, int τc Ω and rint τc Ω denotes the closure, the interior and the relative interior of Ω with respect to the convex core topology τ c , respectively.
In contrast, if Ω is a not relatively solid, convex set (hence E has infinite dimension), then it may happen that acl Ω = cl τc Ω (see Novo and Zȃlinescu [10, Ex. 1.1]). To prove our main scalarization results for vector optimization problems in Section 5, we will apply well-known separation results for convex sets in linear spaces. Proposition 2.1 Assume that Ω 1 , Ω 2 ⊆ E are nonempty, convex sets, and Ω 1 is solid. Then, the following assertions are equivalent: For two relatively solid, convex sets we have the following separation result, which is a consequence of the well-known support theorem by Holmes [30, p. 21] (see also Khazayel et al. [9,Cor. 2.24]).
Proposition 2.2 Assume that Ω 1 , Ω 2 ⊆ E are relatively solid, convex sets. Then, the following assertions are equivalent:

Convex Cones
In what follows, R + denotes the set of nonnegative real numbers, while P := R ++ denotes the set of positive real numbers. Recall that a cone K ⊆ E (i.e., 0 The set (K) is called the lineality space of K. Notice that (K) ⊆ K ⊆ aff K, and K is a linear subspace of E if and only if K = (K).
In this paper, we assume that Then, according to Khazayel et al. [9,Lem. 2.9], the following hold: Moreover, if K is relatively solid, then cl τc K is not a linear subspace, i.e., cl τc K = (cl τc K). The following convex cone is called the (algebraic) dual cone of K. It is well-known that acl K + = K + = (acl K) + = (cl τc K) + . Moreover, if K is relatively solid, then If K (respectively, K + ) is solid, then K + (respectively, K) is pointed. Define It is obvious that P · K # = K # = K # + K # = K + + K # (hence K # is convex). Furthermore, if K # = ∅, then K is pointed. In particular, the following set will be of special interest. Since K = (K) we have Notice that the τ c -closedness assumption concerning K in Lemma 2.3 can not be omitted, as the example by Khazayel et al. [9,Ex. 4.3] shows. Lemma 2.4 ([9, Lem. 2.9] ) Suppose that K satisfies (3). Then, Q := K \ (K) is a nonempty, convex set and the following properties hold: Having a cone K that satisfies (3), we are also interested in the analysis of the subsets Q 0 := (K \ (K)) ∪ {0} and P 0 : The next two lemmata, which are direct consequences of the results by Khazayel et al. [9], show that the sets Q 0 and P 0 are actually nontrivial, pointed, convex cones.
The following lemma will play a key role for deriving characterizations of solution sets of vector optimization problems under certain generalized convexity assumptions (see Section 5).

Pareto Efficiency and Weak Pareto Efficiency in Vector Optimization
Given two real linear spaces X and E, a nonempty feasible set Ω ⊆ X, and a vectorvalued objective function f : X → E, we consider the following vector optimization problem: where the image space E is preordered by a cone K such that (3) is fulfilled. It is well-known that K induces on E a preorder relation K defined, for any two points y, y ∈ E, by y K y :⇐⇒ y ∈ y − K.
For notational convenience, we consider the binary relations ≤ 0 K , ≤ K and < K that are defined, for any two points y, y ∈ E, by One type of solutions of the problem (P) can be defined according to the next definition (see, e.g., Bagdasar The set of all Pareto efficient solutions of (P) is denoted by The following representations of Eff(Ω | f, K) are well-known.
Lemma 3.2 Suppose that K satisfies (3). The following assertions hold: Remark 3.3 Some authors are also interested to compute solutions of the set Eff 0 (Ω | f, K) from Lemma 3.2 (2 • ) when K is a (not necessarily pointed) convex cone (see, e.g., Bao and Mordukhovich [5, p. 302 . Then, the following assertions hold: it is easy to check that ( A kind of weak solution concept for the vector optimization problem (P) will be given in the next definition where the intrinsic core of the convex cone K is used.
The set of all weakly Pareto efficient solutions of (P) is denoted by Remark 3.6 The weak solution concept considered in Definition 3.5, which is based on the intrinsic core notion, is also studied by Adán

It is obvious that Eff
Next, we present some localization results for the image points of weakly Pareto efficient solutions: Lemma 3.7 Suppose that K is relatively solid and satisfies (3). Then, the following assertions hold: Lemma 3.9 Suppose that K is relatively solid and satisfies (3).
Lemma 3.11 Assume that K 1 , K 2 ⊆ E are convex cones. Then, the following assertions hold: , and so, 3 • follows also easily by 1 • .

Henig-type Proper Efficiency in Vector Optimization
As usual for Henig-type proper efficiency concepts, (generalized) dilating cones for the cone K (which satisfies (3)) will play an important role in our work. More precisely, our considered proper efficiency concepts will mainly be based on two specific families of cones, namely C(K) and D(K), that we introduce in the next section.

Generalized Dilating Cones
Let us define two specific families of convex cones related to K, It is easy to check that D(K) ⊆ C(K). Moreover, K ⊆ acl(K \ (K)) ⊆ acl(icor C) = acl C for C ∈ C(K) as well as K ⊆ acl D for D ∈ D(K). In Khan, Tammer and Zȃlinescu [26,Def. 2.4.14.] (applied for (E, τ c )), the cones from the set D(K) are called "generalized dilating cones". We will also use this name for the cones of the family C(K).
From Khan, Tammer and Zȃlinescu [26,Lem. 2.4.15.], we derive the following result, which states some important relationships between the cone K and cones from the set D(K).
Lemma 4.1 Suppose that K satisfies (3). Then, the following assertions hold: The next lemma, which includes also an intrinsic counterpart to Lemma 4.1, states relationships between the cone K and cones from C(K) and D(K).
the inclusion "⊇" is clear. In order to show "⊆", take some x ∈ (acl Q)+icor C. There exist k ∈ acl Q and c ∈ icor C such that x = k+c. For the case k ∈ Q, the inclusion holds. Now, assume that k ∈ (acl Q) \ Q. Consequently, 3 • Using similar ideas as in the proof of [26,Lem. 2.4.15], one gets taking into account Lemma 4.1 (4 • ).
For any C ∈ C(K), we have K & ⊇ C + \ (C + ), as the proof of 1 • shows.
is clear. Take some C ∈ C(K). First, notice that C is a nontrivial, convex cone. Because cor K ⊆ Q ⊆ icor C, we get icor C = cor C = ∅. Thus, we conclude that C ∈ D(K).

Henig Proper Efficiency
In the following, we study a well-known Henig-type proper efficiency concept.

An Extension of Henig Proper Efficiency
In the following, we will propose an extension of the concept of proper efficiency in the sense of Henig [20]. To our knowledge, it is a extended approach to use the family C(K) of generalized dilating cones of K in order to define a new Henig-type proper efficiency concept. A point x ∈ Ω is said to be a Henig properly efficient solution if there is a convex cone C ⊆ E with K \ (K) ⊆ icor C and C = (C) (i.e., C ∈ C(K)) such that x ∈ Eff(Ω | f, C). The set of all Henig properly efficient solutions of (P) is denoted by PEff(Ω | f, K). where the authors assume that K is a nontrivial, pointed, convex cone. Hence, this family C ZYP (K) is always contained in the family C(K) (however notice that C ∈ C(K) may not be pointed). In Remark 4.14, we will take a closer look on the relationships between our concept from Definition 4.7 and the concept proposed by Zhou, Yang and Peng [15,Def. 4.2].
First properties for the set of Henig properly efficient solutions (in the sense of Definition 4.7) are studied in the following lemma.

Scalarization Results
Let two real linear spaces X and E, a nonempty feasible set Ω ⊆ X, and a vectorvalued objective function f : X → E be given. Beside the vector optimization problem (P) from Section 3, we consider the following scalar optimization problem where ϕ : E → R is a real-valued function. Clearly, to get useful relationships between the problems (P) and (P ϕ ) one needs to impose certain properties on ϕ. By solving the scalar problem (P ϕ ) (with a specific function ϕ) one can also get some knowledge about the original vector problem (P). Applying such a strategy is called scalarization method in the literature of vector optimization. Within such methods, the function ϕ is called scalarization function. For more details, we refer also the reader to the standard books of vector/set optimization Jahn [ In this paper, we like to analyse the relationships between solutions of (P ϕ ) and the (weakly, properly) efficient solutions of (P) for the case that ϕ satisfies certain monotonicity properties. For doing this, we recall monotonicity concepts for the function ϕ (c.f. Jahn [25,Def. 5
Lemma 5.2 Consider a real-valued function ϕ : E → R. Then, the following assertions hold: Proof: Assertions 1 • − 3 • are given in [9,Lem. 5.6], while assertion 4 • is a consequence of 1 • and 2 • (applied for C ∈ C(K) in the role of K) taking into account The proof of 5 • is similar to the proof of 4 • .

Remark 5.3
Notice that the convex cone K considered in Lemma 5.2 is neither assumed to be pointed nor solid, in contrast to the known results by Jahn [25,Lem. 5.14 and 5.24].
To derive representations for the sets WEff(Ω | f, K), PEff c (Ω | f, K) and PEff(Ω | f, K) using linear scalarization, we need some well-known generalized convexity concepts. The vector function f : X → E is called • K-convex on the convex set Ω ⊆ X if, for any x,x ∈ Ω and λ ∈ (0, 1), we have Lemma 5.6 Consider f : X → E and Ω ⊆ X. The following assertions hold: 2 • If f is K-convexlike on Ω, then f is (acl C)-convexlike on Ω for any C ∈ C(K).
3 • If f is K-convex on Ω, then f is (acl C)-convex on Ω for any C ∈ C(K).
Proof: 1 • This fact is well-known.
For the case that K is nontrivial and solid, the following result is well-known (see, e.g., Bot, Grad and Wanka [32,Cor. 2.4.26] and Jahn [25,Cor. 5.29]): Remark 5.8 Consider any nontrivial, convex cone K with int τ K = ∅ in a real linear topological space (E, τ ). Let E * be the topological dual space of E, and K * be the topological dual cone (w.r.t. τ ) of K. It is known that cor K = int τ K (see Holmes [30, p. 59]), and whenever f is K-convexlike on Ω we have Indeed, the inclusion "⊇" in (6) follows directly by Proposition 5.7 taking into account that K * ⊆ K + . The proof of the inclusion "⊆" in (6)  Next, we present a counterpart to Proposition 5.7 for the case that K is relatively solid but not necessarily solid. Theorem 5.9 Suppose that K is relatively solid and satisfies (3). In addition, assume that the function f is K-convexlike on Ω, and f [Ω] + K is relatively solid. Then, the following assertions hold: Proof: Notice that assertion 2 • is a direct consequence of Theorem 5.4 (1 • ) and assertion 1 • . Let us show assertion 1 • .
Now, it is easy to see that the sets f [Ω]+K and f (x)−K are nonempty, relatively solid and convex, taking into account icor(f [Ω] + K) = ∅ and icor(f (x) − K) = f (x) − icor K = ∅ (by Lemma 2.7). By the separation result stated in Proposition 2.2, there exist x ∈ E \ {0} and α ∈ R such that for all y ∈ icor(f [Ω] + K) and c ∈ icor K. It is easy to check that x (c) ≥ 0 for all c ∈ icor K (hence also x ∈ K + \ {0}). Indeed, on the contrary assume that x (c) < 0 for some c ∈ icor K. Notice that λc ∈ icor K for all λ > 0, and for λ → +∞, a contradiction. By (7), we also get for all k, c ∈ K and x ∈ Ω. Finally, letting k = 0 and c = 0 in (9), it follows x (f (x)) ≤ x (f (x)) for all x ∈ Ω, which actually means that x ∈ argmin x∈Ω (x • f )(x). The proof of 1 • is complete.
The conclusion in Theorem 5.9 (4 • ) does not imply the inclusion f [WEff(Ω | f, K)] + icor K ⊆ icor(f [Ω] + K) in general, as the following example shows: Example 5.13 Consider the linear space X := E := R 2 with the maximum norm.
Let the function f , the convex cone K and its dual cone K + be given as in Example 5.12. Define Ω as the closed unit ball (denoted by B ∞ ) in the normed space X.
is a solid (hence relatively solid), convex set. Obviously, we have Notice that for x 1 := (−1, −1), From this example, we can also deduce that the inclusion in Theorem 5.9 (1 • ) does not need to be an equality in general. Hence, the pointedness assumption concerning K + in Theorem 5.9 (2 • ) is a not superfluous condition.

Remark 5.18
The representation (12) given in Theorem 5.17 was established by • Luc [27,Th. 4.2.11] for the case that E is a reflexive real linear topological space, K is a convex cone, and f is a K-convex function; • Makarov and Rachkovski [34,Th. 3.2] for the case that E is a separated real linear topological space, K is a nontrivial, pointed, closed, convex cone with K # ∩ E * = ∅, and f is a K-convexlike function; • El Maghri and Laghdir [33,Th. 3.1] for the case that E is a separated real linear topological space, K is a nontrivial, pointed, convex cone, and f is a K-convex function.
Our proof of Theorem 5.17 shows that for the validity of (12) no pointedness assumption related to K is needed.
The next theorem studies properties of the set of classical Henig properly efficient solutions of (P) (in the sense of Definition 4.4) where the cone K is not necessarily assumed to be pointed.
Finally, we study properties of the set of Henig properly efficient solutions of (P) where the cone K is not necessarily assumed to be pointed.

Conclusions
In this paper, we studied vector optimization problems involving not necessarily pointed and not necessarily solid, convex cones in real linear spaces. Based on the "intrinsic core" interiority notion (a well-known generalized interiority notion), we defined our solution concepts (proper/weak efficiency) for such vector optimization problems.
One the one hand, we proposed a Henig-type proper efficiency solution concept based on generalized dilating convex cones which have nonempty intrinsic cores (but cores could be empty). Notice that any convex cone has a nonempty intrinsic core in finite dimension, however this property may fail in infinite dimension. We showed certain useful properties of the new solution concept, pointed out that the set of solutions w.r.t. this concept is always between the set of classical Henig properly efficient solutions and the set of Pareto efficient solutions, and showed an example where all these three sets do not coincide, i.e., PEff c (Ω | f, K) PEff(Ω | f, K) Eff(Ω | f, K).
On the other hand, using linear functionals from the dual cone of the ordering cone K, we were able to characterize the sets WEff(Ω | f, K), PEff c (Ω | f, K) and PEff(Ω | f, K) under the assumption f [Ω] + K is convex. The analysis of the rule A + icor K ⊆ icor(A + K) for A ⊆ f [Ω] (see also Lemma 2.7) was essential for deriving the representations of the sets WEff(Ω | f, K) and PEff(Ω | f, K).
In forthcoming works, we aim to extend the scalarization results derived in Section 5. One idea could be to involve further generalized convexity concepts, another idea could be to consider nonlinear scalarization techniques (for instance based on the so-called Gerstewitz function, see [26,Sec. 5.2]). In this context, Arrow-Barankin-Blackwell type theorems for our considered Henig-type proper efficiency concepts are of interest.
The investigation of the Image Space Analysis (ISA) approach (in the sense of Giannessi [28,29]) involving a relatively solid (but not necessarily solid), convex cone could be a further interesting task.