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Directional Pareto efficiency: concepts and optimality conditions

We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.


Introduction and notation
This paper has two main motivations. On one hand, we are aiming at continuing the effort made by several authors in the last decade to investigate directional phenomena in mathematical programming and, on the other hand, we show the power of several tools related to directional regularities that have been developed recently. For detailed accounts on these topics we refer the reader to the following works and references therein: [15], [16], [1], [10], [12].
In this work, inspired by some ideas coming in vector optimization problems from location theory where some directions are privileged with respect to the others, we present a notion of directional minimality for mappings and we illustrate by examples its relevance even for the case of real-valued functions. Then, we observe on the simplest case of real-valued functions of a real variable that the natural necessary optimality conditions are given by the Fermat Theorem at an endpoint of an interval. This gives us the impetus to consider far-reaching generalization of this case, namely, problems where the objective is a set-valued map and the constraint is given by The topological interior, topological closure, the convex hull and conic hull of A are denoted, respectively, by int A, cl A, conv A, cone A. The negative polar of A is A − := {x * ∈ X * | x * (a) ≤ 0, ∀a ∈ A} .

The concepts under study
Let K ⊂ Y be a proper (that is, K = {0}, K = Y ) convex cone (we do not suppose that K is pointed, in general). For such a cone, its positive dual cone is K + := {y * ∈ Y * | y * (y) ≥ 0, ∀y ∈ K} .
Take F : X ⇒ Y as a set-valued mapping, and let us consider the following geometrically constrained optimization problem with multifunctions: where A ⊂ X is a closed nonempty set.
Usually, the minimality is understood in the Pareto sense given by the next definition.
The vectorial notion described by (2.1) covers as well the situation where f is a function (in which case y = f (x) will not be mentioned) and the situation of classical local minima in scalar case (in which case we drop the label "Pareto"). If K is pointed (that is, K ∩ −K = {0}) then Let L ⊂ S X be a nonempty closed set. Then it is not difficult to see that cone L is closed as well. Indeed, let us consider a sequence (u n ) ⊂ cone L converging towards u ∈ X. We have to show that u ∈ cone L. The case u = 0 ∈ cone L is clear. Otherwise, there are some sequences (t n ) ⊂ (0, ∞) and (ℓ n ) ⊂ L such that u n = t n ℓ n for every n. If (t n ) → 0 (on a subsequence), the boundedness of (ℓ n ) leads to u = 0, a situation avoided at this stage. If (t n ) is unbounded, then again the relation t n ℓ n → u leads to a contradiction. So, on a subsequence, (t n ) → t > 0 which means, by the closedness of L, that ℓ n = t −1 n t n ℓ n → t −1 u ∈ L, therefore u ∈ cone L, as claimed.
The main purpose of this paper is to introduce and to study the following concept. If one compares this relation to (2.1), then one observes that this concept corresponds to the situation where the restriction has the special form (depending on the reference point) A ∩ (x + cone L) . Of course, when A = X in (2.2) then one says that (x, y) ∈ Gr F is a local directional Pareto minimum point for F with respect to L. Now, the concept of local directional Pareto maximum is obtained in an obvious way.
If int K = ∅, one defines as well the weak counterpart of the above notion.
In all these notions, if one takes U = X, then we get the corresponding global concepts.
Remark 2.5 If L 1 , L 2 ⊂ S X are nonempty closed subsets such that L 1 ⊂ L 2 , then a local directional Pareto minimum point for F with respect to L 2 is a local directional Pareto minimum point for F with respect to L 1 .
It is obvious that (2.1) implies (2.2), but the converse is not true. To justify the latter affirmation, let us consider the following simple scalar example (when the output space is R we always consider K := [0, ∞)).
Example 2.6 Let f : R → R be a strictly increasing function. Then every x ∈ R is local directional minimum for f with respect to L := {+1}, but it is not a local minimum for f. Moreover, the minimality concept introduced here covers some interesting situations described by the next examples.
Example 2.7 Let f : R 2 → R be given by f (x, y) = x 2 − y 2 . It is well known that (0, 0) is a critical saddle point, whence it is not a minimum point. However, it is a directional minimum point for f with respect to L = {−1, 1} × {0} since for every (x, y) ∈ (0, 0) + cone L = R × {0}, one has f (x, y) ≥ f (0, 0). Similarly, (0, 0) is a directional maximum point for f with respect to Example 2.8 Let f : R 2 → R be given by f (x, y) = x 2 − y 3 . Again, (0, 0) is a critical saddle point. It is now easy to see that it is, however, a directional minimum point for f with respect to L = {−1, 1} × {0} and to respect to L = {0} × {−1}.
The next example emphasizes that there are points which are not directional minima with respect to any nonempty closed set L ⊂ S X . This applies also for critical points of smooth functions. Example 2.9 Let f : R → R be given by Then x = 0 is not directional minimum for f neither for L := {−1}, nor for L := {+1}. In the same manner, f : R → R given by The next example underlines the idea that for every prescribed set of directions one can define functions that achieve directional minimum with respect to the given set.
Using these basic examples of scalar-valued functions, we are able to easily build examples for vector-valued maps. Here are two such examples.
Example 2.12 Let f : R → R 2 be given by f (x) = (2x, x) and K = cone conv {(1, 0) , (1, 1)} . It is easy to see that x := 0 is a directional minimum for f with respect to L := {+1} , but x is not a local Pareto minimum point for f.
The concepts introduced in this section are studied in the sequel from the point of view of optimality conditions.

Optimality conditions for directional minima
In order to start with the necessary optimality conditions for directional minima, let us to observe that the obvious prototype for such an investigation is the Fermat Theorem for derivable real-valued functions with one variable at interval endpoints: if f : [a, b] → R is a function for which a is local minimum point (that is, a directional minimum with respect to L := {+1}), and f is derivable at a, then f ′ (a) ≥ 0, and, similarly, if b is a minimum point for f (that is, a directional minimum with respect to L := {−1}), and f is derivable at b, then f ′ (b) ≤ 0.
We approach this issue from two points of view, namely, making use of tangent cones (which are objects of generalized differentiation on primal spaces) and of normal cones (constructions that are defined on dual spaces).

Optimality conditions using tangent cones
Let us consider now several concepts that will help us in studying optimality conditions for the directional minima. Definition 3.1 Let A ⊂ X be a nonempty set and L ⊂ S X be a nonempty closed set. Then the Bouligand tangent cone to A at x ∈ A with respect to L is the set −→ 0 such that for all n, x + t n u n ∈ A , where (u n ) cone L −→ u means (u n ) −→ u and (u n ) ⊂ cone L, and similarly for (t n ) Obviously, this is a adaptation of the concept of Bouligand tangent cone to A at x defined as −→ 0 such that for all n, x + t n u n ∈ A .
Some remarks are in order.
Remark 3.2 As the usual Bouligand tangent cone, the set T L B (A, x) is a closed cone: the proof of this assertion can be made directly as for the classical concept (see [3]) or by observing that In view of the fact that cone L is closed, one has that T L B (A, x) ⊂ cone L. Moreover, However, the inclusion above does not hold as equality, in general. To see this, consider the set A ⊂ X := R 2 as the plane domain bounded by the curve (the cardioid), which has the parametric representation Another useful and easy-to-see inclusion is −→ 0 such that for all n, y + t n v n ∈ F (x + t n v n ).

Clearly,
Gr D L,M B F (x, y) ⊂ cone L × cone M. Again, this is an adaptation of the well-known Bouligand derivative of F at (x, y), which is the set-valued map D B F (x, y) : X ⇒ Y defined by Other derivability objects in primal spaces that can be adapted in directional setting in a similar manner are the Ursescu (adjacent) tangent cone and the Ursescu (adjacent) derivative (see [5]), and the Dini lower derivative of F at (x, y), which is the multifunction D D F (x, y) from X into Y given, for every u ∈ X, by When F := f is a single-valued map, for simplicity, we write D L,M B f (x) for D L,M B f (x, y), and similarly for D D .
We present now the first result of this work.
Proposition 3.4 In the above notation, if int K = ∅ and (x, y) ∈ Gr F is a local weak directional Pareto minimum point for F on A with respect to L then Proof. We prove only the second part, since the first part, on one hand, is similar, and, on the other hand, it follows from the definitions and [5, But, for n large enough, x + t n u n is close enough to x and belongs as well to x + cone L. Then, for such n, taking into account the minimality of (x, y), one gets t n k n / ∈ − int K which contradicts the fact that k n → k ∈ − int K.
In [9], by means of a special type of minimal time function, several directional regularity properties for set-valued maps are introduced and studied. In order to further investigate the directional minima we need to briefly point out the main aspects concerning the minimal time function and some related directional metric regularity.
Consider ∅ = L ⊂ S X and ∅ = Ω ⊂ X. Then the function is called the directional minimal time function with respect to L. Remark that, if L = S X , then T L (·, Ω) = d(·, Ω). Moreover, we add the convention that What we need in the sequel is the following concept of directional calmness. One says that F is directionally calm at (x, y) with respect to L and M if there are α > 0 and some neighborhoods U of x and V of y such that for every x ∈ U, We use the convention sup x∈∅ T L (x, Ω) := 0 for every nonempty set Ω ⊂ X.
As usual (see [4, Section 3H]), for a calmness concept for F , it is natural to have a metric subregularity notion such that the former property for F −1 to be equivalent to the latter property for F. In our setting, this corresponding concept reads as follows: one says that F is directionally metric subregular at (x, y) with respect to L and M if there exist α > 0 and some neighborhoods U of x and V of y such that for every x ∈ U, The expected equivalence is described in the following result.
Proposition 3.5 The set-valued map F is directionally metric subregular at (x, y) with respect to L and M iff F −1 is directionally calm at (y, x) with respect to M and L.
Proof. Suppose first that F is directionally metric subregular at (x, y) with respect to L and M.
Then, by hypothesis, for all y ∈ V, whence the conclusion. For the converse, suppose that F −1 is directionally calm at (y, x) with respect to M and L.
Therefore, y ε − y ∈ cone M, x ∈ F −1 (y ε ) ∩ U and from the hypothesis, Passing to the limit as ε → 0 we get the conclusion. Now, we use the directional calmness for getting an evaluation of the directional Bouligand tangent cone to a value of a set-valued mapping in terms of the image of 0 through the directional Bouligand derivative of the same application.
Moreover, if F is directionally calm at (x, y) with respect to L and M, and cone M is convex, then the equality holds.
But, the assumed calmness of F and the fact that (t n u n ) ⊂ cone L, mean that, for a positive α and for all n large enough, Therefore, for every n (large enough) there are w n ∈ M and τ n ≥ 0 such that β n := y+t n v n +τ n w n ∈ F (x) and τ n < αt n u n + t 2 n . So, for every n, Taking into account the convexity of cone M, for every n, Consider now the situation when G : X ⇒ Z is a set-valued map, Q ⊂ Z is a closed convex and pointed cone and the set of restrictions for (P ) is A := {x ∈ X | 0 ∈ G(x) + Q}. This is a standard situation which encompasses the classical case where one has equalities and inequalities constraints. The following result holds.
Proof. We remark that A = E −1 G (0), whence, by Propositions 3.5 and 3.6, −→ 0 such that for all n, But, w n + q → w and for every n, w n + q ∈ cone N + Q ⊂ cone N, whence w ∈ D L,N B E G (x, 0)(u).
Proposition 3.8 Suppose that int K = ∅ and (x, y) ∈ Gr F is a local weak directional Pareto minimum point for F on A := E −1 G (0) with respect to a closed nonempty set L ⊂ S X . Consider z ∈ G(x) ∩ −Q and ∅ = N ⊂ S Z a closed set. Moreover, suppose that Q ∩ S Z ⊂ N, cone L and cone N are convex, and E G is directionally metric subregular at (x, 0) with respect to L and N. Then Proof. The result follows by using successively Propositions 3.4 and 3.7.
Let us to specialize, in two steps, the ideas above to the classical smooth case of optimization problems with single-valued maps. First, suppose that F := f and G := g are continuously Fréchet differentiable functions. Then taking a point x ∈ A = E −1 G (0) it is easy to see that for all u ∈ X, Then we get the following Fritz John and Karush-Kuhn-Tucker type result.
Theorem 3.9 Suppose that int K = ∅ and x ∈ A := E −1 g (0) is a local weak directional Pareto minimum point for f on A with respect to L. Moreover, suppose that cone L is convex, and E g is directionally metric subregular at (x, 0) with respect to L and S Z . Then, in either of the following conditions: (ii) Y and Z are finite dimensional spaces, there exist y * ∈ K + , z * ∈ Q + , (y * , z * ) = 0 such that for every u ∈ cone L, If, moreover, there exists u ∈ cone L such that ∇g(x)(u) ∈ int Q = ∅ or ∇g(x)(cone L) = Z then y * = 0.
Proof. According to Proposition 3.8 and the subsequent discussion, Notice that {(∇f (x)(u), ∇g(x)(u)) | u ∈ cone L} is a convex set and both (i) and (ii) ensure the possibility to apply a separation result for convex sets. Therefore, there exist y * ∈ Y * , z * ∈ Z * , (y * , z * ) = 0 such that for every u ∈ cone L, k ∈ int K, q ∈ Q, one has Standard arguments yield y * ∈ K + , z * ∈ Q + and for every u ∈ cone L.
If one supposes that y * = 0 then the relation above and the either of the final assumptions give z * = 0, which contradicts (y * , z * ) = 0.
A similar but different result could be done taking into account the special structure of this case, using directly Proposition 3.4, and some results one can find in literature concerning the calculus of Bouligand tangent cone to the counter image of a set through a differentiable mapping. Let us recall some facts from [8]. Let f : X → Y be a function and D ⊂ X be a nonempty closed set. One says that f is metrically subregular at ( In fact, the above notion coincides with that of calmness of the set-valued map y ⇒ f −1 (y) ∩ D at (f (x), x) (see, for instance, [4, Section 3H]). One of the main results in [8] reads as follows.
Furthermore, we consider the case where Y = R k (k ≥ 1), Z = R p (p ≥ 1), Q = R m + × {0} n with m + n = p, and f, g are Fréchet differentiable. This means that we are dealing with a vectorial optimization problem with finitely many inequalities and equalities constraints. Let us denote by µ i with i ∈ 1, m the first m coordinates functions of g and by ν j with j ∈ 1, n the next n coordinates functions of g.
For the next step of our approach, we use the Gerstewitz functional in the special case when the ordering cone has nonempty interior. The next result combines [ Theorem 3.12 Let K ⊂ Y be a closed convex cone with nonempty interior. Then for every e ∈ int K the functional s K,e : Y → R given by is convex continuous and for every λ ∈ R, {y ∈ Y | s K,e (y) < λ} = λe − int K, and {y ∈ Y | s K,e (y) = λ} = λe − bd K.
Since cone L is convex, this is a convex problem, whence, from [19, Theorem 2.9.6], there exist λ i ≥ 0 for i ∈ I(x) and τ j ∈ R for j ∈ 1, n such that where ι denotes the indicator function. Finally, using (3.6), and taking λ i := 0 for i ∈ 1, m\I(x), we get the existence of y * ∈ K + \{0} such that whence the conclusion.
Remark 3.14 Observe that in the simplest case of a derivable real-valued function f : R → R, if x is a directional minimum with respect to L := {+1} (without constraints) the above theorem reduces to −f (x) ∈ L − which is exactly f ′ (x) ≥ 0, as discussed before.
Our aim now is to derive sufficient conditions for a point x ∈ g −1 (−Q) to be a local weak directional Pareto minimum point. In order to formulate such conditions we use, besides the convexity notion for scalar functions, a generalized convexity concept. Namely, we use the following well-known concept: one says that F : X ⇒ Y is K−convex if for any λ ∈ (0, 1) , and any x, y ∈ X, one has λF (x) + (1 − λ)F (y) ⊂ F (λx + (1 − λ)y) + K.

Proposition 3.15
Suppose that X is a Banach space, int K = ∅, cone L is convex, f is K−convex, µ i , i ∈ 1, m, are convex and ν j , j ∈ 1, n, are affine. If there exist (λ, τ ) ∈ R m + ×R n and y * ∈ K + \{0} such that (3.7) and (3.8) hold, then x is a global weak directional Pareto minimum point for f on g −1 (−Q) with respect to L.
Proof. By relation (3.7), we immediately get that

Consider the convex optimization problem
We hence obtain, by virtue of [19,Theorem 2.9.1], that x is a global minimum point for the above problem. Note that, for all feasible points Using (3.8), it follows that, given any x ∈ (x + cone L) ∩ g −1 (−Q) , i.e., the conclusion.

Optimality conditions using normal cones
In order to tackle the question of optimality conditions for directional minima in terms of generalized differentiation objects in dual spaces, we recall some notions and results concerning Fréchet and limiting (Mordukhovich) generalized differentiation (see [14] for details). Consider S a nonempty subset of a Banach space X and x ∈ S. Then for every ε ≥ 0, the set of ε−normals to S at x is defined by where u S → x means that u → x and u ∈ S. The set N 0 (S, x) is denoted by N (S, x) and it is called the Fréchet normal cone to S at x.
Let x ∈ S. The Mordukhovich normal cone to S at x is given by Up to the end of this section, we consider that all the involved spaces are Asplund, unless otherwise stated. In this context, if S ⊂ X is closed around x, the formula for the Mordukhovich normal cone takes the following form: For the set-valued map F : X ⇒ Y, its Fréchet coderivative at (x, y) ∈ Gr F is the set-valued map D * F (x, y) : Y * ⇒ X * given by D * F (x, y)(y * ) = {x * ∈ X * | (x * , −y * ) ∈ N (Gr F, (x, y))}.
As usual, when F = f is a function, since y ∈ F (x) means y = f (x) , we write D * f (x) for D * f (x, y) , and similarly for D * .
Notice that for a convex set S ⊂ X one has that and this cone coincides with the negative polar of T B (S, x). If S ⊂ X is closed around x ∈ S, one says that S is sequentially normally compact (SNC, for short) at x if In the case where S = C is a closed convex cone, the (SNC) property at 0 is equivalent to In particular, if int C = ∅, then C is (SNC) at 0. Let f : X → R ∪ {+∞} be finite at x ∈ X and lower semicontinuous around x; the Fréchet subdifferential of f at x is defined by where epi f denotes the epigraph of f. The Mordukhovich subdifferential of f at x is given by It is well-known that if f is a convex function, then ∂f (x) and ∂f (x) coincide with the Fenchel subdifferential. However, in general, ∂f (x) ⊂ ∂f (x), and the following generalized Fermat rule holds: if x ∈ X with f (x) < +∞ is a local minimum point for f : X → R ∪ {+∞}, then 0 ∈ ∂f (x).
Consider now some subsets C 1 , ..., C k of X (k ∈ N\{0, 1}). Take x ∈ C 1 ∩ ... ∩ C k and suppose that all the sets C i , i ∈ 1, k are closed around x. One says that C 1 , ..., C k are allied at x if for every (x in ) The concept of alliedness was introduced by Penot and his coauthors in [17] and [13] in order to get a calculus rule for the Fréchet normal cone to the intersection of sets. More precisely, if the subsets C 1 , ..., C k are allied at x, then there exists r > 0 such that, for every ε > 0 and every In what follows we use the results concerning the theory of generalized differentiation built on these objects directly at the places we need them, without separate quotation.
We discuss next a concept of directional openness at the reference point of a certain multifunction. We recall that the classical concept of openness proven to be useful for the announced aim by means of the incompatibility between this property and the Pareto minimality (see, e.g., [7] for details).
In fact, the directional openness we consider here is related to several other notions introduced in [9], and to the concept of directional calmness already used in the previous subsection.
Consider a multifunction F : X ⇒ Y, a point (x, y) ∈ Gr F, and ∅ = L ⊂ S X , ∅ = M ⊂ S Y . One says F is directionally open at (x, y) with respect to L and M if for any ε > 0, there exists r > 0 such that When F is single-valued, for simplicity, we sometimes omit y in the definition above and we say that F is directionally open at x, instead of directionally open at (x, f (x)).

Proposition 3.16
If (x, y) ∈ Gr F is a local directional Pareto minimum point for F with respect to L, then for every C ⊂ S Y with C ∩ (K \ −K) = ∅, the set-valued map E F : X ⇒ Y, given by E F (x) := F (x) + K is not directionally open at (x, y) with respect to L and C. In particular, F is not directionally open at (x, y) with respect to L and C.
Proof. Suppose, by contradiction, that for ε > 0 involved in the definition of the minimality of (x, y), there exists r > 0 such that By subtracting y on both sides, according to the hypothesis, one has that Passing to the conic hull, this yields − cone C ∩ −K ⊂ K, which contradicts the fact that C ∩ (K \ −K) = ∅. So E F is not directionally open at (x, y) with respect to L and C. Since F (x) ⊂ E F (x) for any x, the same conclusion holds for F as well.
Before obtaining necessary optimality conditions, we remark that a converse of Proposition 3.16 can be done if one considers a (generalized) convex framework. Proof. Suppose, by contradiction, that (x, y) is not a local directional Pareto minimum point of F with respect to L. Then for every r > 0, there is y r ∈ F (B(x, r) ∩ [x + cone L]) ∩ [y − K] such that y r / ∈ y + K. Denote k := y − y r ∈ K \ −K and consider x r ∈ B(x, r) ∩ (x + cone L) such that y r ∈ F (x r ).
Moreover, since E F is not directionally open with respect to L and k at (x, y), it follows that there is r > 0 such that, for every ε > 0 small enough, there is y ε ∈ B(y, ε) ∩ y − cone k ⊂ [y, y r ] such that y ε / ∈ E F (B(x, r) ∩ [x + cone L]) (hence, in particular, y ε = y and y ε = y r ). Then, there is λ ∈ (0, 1) such that a contradiction. Now, we use Proposition 3.16 to get optimality conditions. Theorem 3.18 Suppose that X and Y are finite dimensional spaces, (x, y) ∈ Gr F is a local directional Pareto minimum point for F with respect to L, cone L is convex, u ∈ int K ∩ S Y , and the set-valued map E F : X ⇒ Y has closed graph and is Lipschitz-like around (x, y). Then there exist x * ∈ X * , y * ∈ K + with x * (ℓ) ≥ 0 for all ℓ ∈ L, y * (u) = 1 and Proof. According to Proposition 3.16, E F is not directionally open at (x, y) with respect to L and {u}. Therefore, this is not directionally open around (x, y) with respect to L and {u} (in the sense of [9,Definition 2.2]) and, therefore, the sufficient condition for directional openness from [9,Theorem 4.3] does not hold. This means that for all natural numbers n = 0, there exist x * n ∈ X * , y * n ∈ Y * , (x n , y n ) Gr F −→ (x, y) such that y * n (u) = 1, n −1 > −x * n (ℓ) for all ℓ ∈ L and x * n ∈D * E F (x n , y n )(y * n ). Now [7,Lemma 3.2] ensures that y * n ∈ K + for any n. This, together with the condition u ∈ int K imply, by using [11,Lemma 2.2.17], that the sequence (y * n ) is bounded. The assumed Lipschitz property of E F ensures, by means of [14,Theorem 1.43], that the sequence (x * n ) is bounded too. Therefore, we can suppose, without loss of generality, that both these sequences are convergent to some x * ∈ X * and y * ∈ K + , respectively. Passing to the limit in the relations satisfied by (x * n ) and (y * n ) we get, x * (ℓ) ≥ 0 for all ℓ ∈ L, y * (u) = 1 and x * ∈ D * E F (x, y)(y * ), that is the conclusion.

Remark 3.19
Observe that, in the case L = S X (that is Pareto minimality) the necessary optimality condition given by the previous result is the generalized Fermat rule (see [7,Theorem 3.11]): there exists y * ∈ K + \ {0} with 0 ∈ D * E F (x, y)(y * ).
We tackle now the case of constrained problems and we have the following result.
Theorem 3.20 Let A ⊂ X and L ⊂ S X be nonempty closed sets and F : X ⇒ Y be a set-valued map with (x, y) ∈ Gr F ∩ (A × Y ) such that Gr F is closed around (x, y) . Suppose that the following assertions hold: (i) F is Lipschitz-like around (x, y) ; (ii) K\−K = ∅ and K is (SNC) at 0; (iii) the sets A and x + cone L are allied at x.
If (x, y) is a local directional Pareto minimum point for F on A with respect to the set of directions L, then there exists y * ∈ K + \ {0} such that Proof. From the hypothesis, there exists a neighborhood U ∈ V (x) such that and there exists c ∈ Y such that c ∈ K\−K. Consider the following two sets: and and the conclusion follows.
Now, we make one step forward by considering the vectorial optimization problem min f (x) , x ∈ A, (3.14) where f : X → Y is a vector-valued function and A ⊂ X is a closed set. As before, the ordering cone on Y is K. Consider the following vectorial Lipschitz property for f : following [18], one says that f is K−Lipschitz around x ∈ X of rank ℓ f > 0 if there exist a neighborhood U of x and an element e ∈ K ∩ S Y such that for every We record the following result.
Theorem 3.22 Let x ∈ A be a local directional Pareto minimum for f on A with respect to L ⊂ S X . Suppose that: (i) f is K−Lipschitz around x of rank ℓ f and let e be the element in K ∩ S Y given by the Lipschitz property of f ; and this is the conclusion.

Pareto directional minima for sets
As made clear in Definition 2.3 and the subsequent comments, the notion of directional Pareto minimum is motivated by the case of (generalized) mappings. However, it is possible to define such a notion for sets as well. In order to point out this aspect of directional minimality, in this section we define some appropriate notions and we give, only briefly, some examples and optimality conditions for them. Consider, as above, a closed nonempty set L ⊂ S X and take now K as a proper closed convex cone in X. Definition 4.1 Let M ⊂ X be a nonempty set. One says that x ∈ M is a local directional Pareto minimum point for M with respect to L if (4.1) It is simple to see that relation (4.1) is equivalent to    −→ 0 such that for all n, x + t n u n ∈ M. Clearly, for n large enough, t n u n ∈ (M − x) ∩ cone L ∩ − int K, which contradicts the minimality assumption.
(ii) Suppose, again by way of contradiction, that there exists x ∈ M such that x − x ∈ cone L ∩ − int K. Consider (t n ) (0,∞) −→ 0. Then, for every n large enough, whence using the fact that cl(M + int K) = cl(M + K) (which, in turn, is easy to prove using the closedness and the convexity of K which ensures K = cl int K) one can write:  Proof. The proof is similar to that of Theorem 4.3 (ii).

Conclusions
The directional efficiencies introduced in this paper generalize in a meaningful way the classical situation of Pareto optimality and require non-trivial adaptations of the usual techniques of investigation used in the latter case. Besides the results of this paper, we think that our approach opens new possibilities to model directional situations, especially arising in vector optimization problems dealing with location issues. We consider that our concept here introduced is able to capture the situation where some directions are more important than the others (hence which can be dropped) in the possible models under consideration. Another possible continuation for theoretical investigation of directional efficiency is to devise an adapted (directional) normal limiting cone with respect to a set of directions and to use it in order to write down more specific optimality conditions for our concept. All these ideas will be topics for future research.