Characterizations of the Set Less Order Relation in Nonconvex Set Optimization

For nonconvex set optimization problems based on the set less order relation, this paper presents characterizations of optimal sets and gives necessary conditions for set inequalities and non-optimal sets using directional derivatives. For specific order cones, the directional derivatives of known functionals describing the negative cones are also given.

l-type less and u-type less order relations (see Kuroiwa [23] for first results). The so-called minmax less order relation (introduced in [19]) is only investigated for a special result.
The results of this paper consist of conditions under which the inequalities in R are well-defined, characterizations of nonoptimal sets, necessary optimality conditions for the afore-mentioned sup inf problems, necessary conditions for a set inequality and a necessary condition for the non-optimality of a set. The necessary conditions of the main results work with the directional derivative of a functional being used for the description of the negative order cone in the considered real linear space. For different real linear spaces, the directional derivative is calculated for various standard functionals of this type.
This paper is organized as follows: Background material and first results are summarized in Sect. 2. Investigations of non-optimal sets and optimality conditions in nonconvex set optimization are given in Sect. 3. The last section contains the main results like necessary conditions for set inequalities.

Basic Results
Throughout this paper we use the following standard assumption. Assumption 2.1 Let Y be a real linear space, let C ⊂ Y be a convex cone, and let a functional ψ : Y → R be given with ψ(y) ≤ 0 ⇐⇒ y ∈ −C. (1) Such a functional ψ (compare [7,Remark 3.3]), which characterizes the cone −C, is not uniquely defined. With ψ the functional αψ also has the required properties for every α > 0.
For convenience, known examples of these functionals are now recalled for different spaces. And the functional ψ : S n → R with x T M x for all M ∈ S n is associated to the copositive cone is associated to the natural ordering cone (d) Let (Y , · ) be a real normed space and let some continuous linear functional ∈ Y * be arbitrarily chosen. Then the functional ψ : Y → R with ψ(y) = (y) + y for all y ∈ Y is associated to the so-called Bishop-Phelps cone [11,15] for Bishop-Phelps cones).
There are various order relations, which can be used for the comparison of sets in the real linear space Y (e.g. compare [19]). In this paper we restrict ourselves to the well-known set less order relation introduced by Young [ Next we characterize the set inclusions in Definition 2.1 by certain inequalities.
Proof For arbitrary nonempty sets A, B ⊂ Y we have In analogy, one can prove the second part of the assertion.
The advantage of this simple result is that one can check the validity of the aforementioned set inclusions using appropriate optimization problems. Such a rewriting of these set inclusions as inequalities has been already given by Hernández and Rodríguez-Marín [13, Thm. 3.10,(iii)] in 2007 using an extension of the Tammer (formerly Gerstewitz) scalarization approach [8]. Later such a rewriting of set inclusions as inequalities was also given in [22,Thm. 3.3 and 3.8] for scalarizing functionals introduced by Tammer. Proposition 2.1 is parameter free and it subsumes these scalarization approaches.
Next, we investigate under which conditions the sup min problems in Proposition 2.1 are solvable. Proposition 2.2 Let Assumption 2.1 be satisfied and, in addition, let (Y , · ) be a real normed space. Let A ⊂ Y be a nonempty weakly compact set, let B ⊂ Y be a nonempty set, let ψ be bounded on A − B, and let ψ(· − b) be weakly semicontinuous for every b ∈ B. Then the problem sup b∈B min a∈A ψ(a − b) is solvable.
Proof First, we show the solvability of the problem min a∈A ψ(a − b) for an arbitrary b ∈ B. Since ψ(·−b) is weakly semicontinuous for every b ∈ B and the set A is weakly compact, there is at least one minimal solution a b ∈ A with ψ(a b −b) = min a∈A ψ(a− b) (e.g., compare [18,Thm. 2.3]). Now we consider the problem sup b∈B ψ(a b − b). Because the functional ψ is assumed to be bounded on the set A − B, it is evident that sup b∈B ψ(a b − b) < ∞, which has to be shown. The following proposition is a direct consequence of the Propositions 2.1 and 2.2.

Proposition 2.3
Let Assumption 2.1 be satisfied and, in addition, let (Y , · ) be a real normed space. Let A, B ⊂ Y be nonempty weakly compact sets, let ψ be bounded on A − B, and let the functionals ψ(a − ·) and ψ(· − b) be weakly lower semicontinuous for every a ∈ A and b ∈ B, respectively. Then This proposition gives a sufficient condition for the set less order relation, which may be helpful in practice (see also [22,Corollary 3.11] for specific scalarizing functionals).

Optimality
We now turn our attention to problems of set optimization. Based on the set less order relation in a real linear space Y , we consider a family F of nonempty subsets of Y and we investigate optimal sets of F. Definition 3.1 Let Assumption 2.1 be satisfied, and let F be a family of nonempty subsets of Y . A setĀ ∈ F is called an optimal set of F, iff Under Assumption 2.1 recall for any nonempty subset A of Y that the set denotes the set of all minimal elements of A, and the set With the following proposition we investigate the question: What does it mean, if for some set A ∈ F the two inequalities A sĀ andĀ s A given in Definition 3.1 hold?

Proposition 3.1 Let Assumption 2.1 be satisfied and, in addition, let the convex cone C be pointed. Let F be a family of nonempty subsets of Y , for which the set of minimal elements and the set of maximal elements are nonempty. For every A ∈ F let the set equalities
be satisfied. For someĀ ∈ F, we then havē A optimal ⇐⇒ for every A ∈ F with A sĀ : min A = minĀ and max A = maxĀ.
Proof For someĀ ∈ F and an arbitrary A ∈ F it holds by definition

By [17, Lemma 2.4] this is equivalent to the inclusions
By the equalities (2) this can be written as Illustration of the set equalities (2) The first equality implies minĀ ⊂ (min A) + C, which means Since the first equality in (3) also implies min A ⊂ (minĀ) + C, there areâ ∈ minĀ andĉ ∈ C with a =â +ĉ. Hence, we getā = a + c =â +ĉ + c. Because of a,â ∈ minĀ we obtainĉ Consequently, we haveā = a and thereby minĀ ⊂ min A. By renaming we also get min A ⊂ minĀ, an so we have min A = minĀ. The equality max A = maxĀ can be proven in analogy. The assertion then follows with the definition of optimality. Figure 1 illustrates that the set equalities (2) may also be satisfied for nonconvex sets. For Y := R 2 , C := R 2 + and A given in Fig. 1 But the set inequalities (2) do not hold in general; for instance, if we choose Y := R 2 , i.e. the first set inequality in (2) is not satisfied.

Remark 3.1 Under the assumptions of Proposition 3.1 we havē
For instance, if one works with a descent method for the calculation of an optimal set, one can use this result in order to decide whether a setĀ is not optimal. LetĀ ∈ F be given and let some set A ∈ F with A sĀ be calculated. If one can check that min A = minĀ or max A = maxĀ, one knows that the setĀ cannot be optimal.
The assumption in Proposition 3.1 that the set equalities (2) are fulfilled for all A ∈ F can be avoided, if one works with a more strict order relation introduced in [19,Def. 3.5]. (b) Let F be a family of subsets of Y , for which the set of minimal elements and the set of maximal elements are nonempty. A setĀ ∈ F is called a minmax optimal set of F, iff Proof In analogy to the proof of Proposition 3.1 we obtain for someĀ ∈ F and an arbitrary A ∈ F This leads to the assertion.
The proof of this proposition follows the lines in [17, Lemma 2.8,(a)]. (2) because (min A) + C = min(min A) + C and the other set equalities follow similarly.

Necessary Conditions for Set Inequalities
The set less order relation is defined by certain set inclusions; by Proposition 2.1 these set inclusions are characterized by appropriate inequalities. And now we investigate necessary conditions for these inequalities. Under Assumption 2.1 we consider two arbitrary nonempty subsets A, B ⊂ Y . For an arbitrarily chosenb ∈ B we use the abbreviation Ab denotes the set of all minimal solutions of the optimization problem min a∈A ψ(a −b).
For the first result we need a technical lemma.
But in (4) we replace the argument a −b by a −b + λh, i.e. we consider elements with respect to the direction of h, and we require a stronger inequality. Besides the functional ψ the special choice of the set A plays a central role for this condition. For simplicity we use the following notation for the next result. Under Assumption 2.1 we consider nonempty subsets A, B ∈ Y and define the functional ϕ : B → R with provided that the min term exists. Next, we investigate the question under which conditions the functional ϕ has a directional derivative.

In addition, suppose that for all b
Then it follows Proof First, we determine the directional derivative of ϕ atb in every direction b −b with arbitrary b ∈ B. With Lemma 4.1 and the assumptions of this theorem, we then get for all b ∈ B Sinceb ∈ B is a maximal solution of the problem max b∈B ϕ(b) = − min b∈B −ϕ(b) and ϕ has a directional derivative atb in every direction b −b with arbitrary b ∈ B, by [18, Thm. 3.8,(a)] we obtain which leads to the assertion.
Early investigations on directional derivatives of sup inf functions can be found in [4] (compare also [2]).
With this necessary condition for certain min max problems, we then obtain necessary conditions for set inequalities as well. ψ(a − b). LetÂb = A andBā = B. Let min b∈B ψ(· − b) be directionally differentiable atā in every direction a −ā with arbitrary a ∈ A, and let min a∈A ψ(a − ·) be directionally differentiable atb in every directionb − b with arbitrary b ∈ B. For an arbitrary a ∈ A and an arbitraryb ∈Bā suppose that

Corollary 4.1 Let Assumption 2.1 be satisfied, and let A, B ⊂ Y be arbitrarily chosen. Letā ∈ A be a solution of the optimization problem max a∈A min b∈B ψ(a − b) and letb ∈ B be a solution of the optimization problem max b∈B min a∈A
Moreover, for an arbitrary b ∈ B and an arbitraryâ ∈Âb suppose that

If the inequality A s B is satisfied, then we have
provided that the arising min terms exist.
Proof Let the set inequality A s B hold for arbitrary ∅ = A, B ⊂ Y . By Proposition 2.1 and the assumptions, we then have Hence, the inequalities (5) and (6) follow from Theorem 4.1.

Remark 4.1
The inequalities (5) and (6)  By Corollary 4.2 we now present a necessary condition for sets to be non-optimal. The assumptions of Corollaries 4.1 and 4.2 concerning the considered sets and the functional ψ are very strong. For the two sets it seems to be helpful, if they consist of finitely many elements. But for the functional ψ we need the directional derivative, which depends on the order cone C. In the following proposition the directional derivative of ψ is investigated for various special real linear spaces and order cones. x T Ax = maximal eigenvalue of A for all A ∈ S n ( · 2 denotes the Euclidean norm in R n ) has the directional derivative at an arbitraryĀ ∈ S n given by x T M x for all M ∈ S n ( · 2 denotes the Euclidean norm in R n ) has the directional derivative at an arbitraryĀ ∈ S n given by (e) Let a real normed space (Y , · ) and the Bishop-Phelps cone for an arbitrary continuous linear functional ∈ Y * be chosen. Then the functional ψ : Y → R with ψ(y) = (y) + y for all y ∈ Y has the directional derivative at an arbitraryȳ ∈ Y given by Proof (b) For arbitrarily chosenȳ, h ∈ C[0, 1] we obtain Next, we prove the converse inequality. For every λ > 0 there is some  1] h(t).
Then there is a sequence (λ k ) k∈N of positive real numbers converging to 0 with lim k→∞ t λ k =:t ∈ [0, 1], and it is evident thatt ∈ M(ȳ). Hence, we conclude The inequalities (7) and (9) lead to the assertion. (d) For arbitrarily chosenĀ, H ∈ S n we have For the proof of the converse inequality choose an arbitrary λ > 0. Then there exists some x λ ∈ R n + with x λ 2 = 1 so that For an arbitraryx ∈ X (Ā) we obtain max x∈R n x TĀ x + λ max x T H x for all λ > 0 we conclude for some α ≥ 0 x TĀ x ≤ λα for all λ > 0.
Hence, we have x TĀ x.
So, there is a sequence (λ k ) k∈N of positive real numbers converging to 0 with lim k→∞ x λ k =:x ∈ X (Ā). And we conclude The inequalities (10) and (11) imply the assertion. (e) Since the norm · is continuous and convex, its directional derivative is given by [18,Theorem 3.28]. Then for arbitrarily chosenȳ, h ∈ Y we obtain where the subdifferential ∂ ȳ is calculated in [18, Example 3.24,(b)].
The proof of part (b) follows the lines of the proof of the directional derivative of the maximum norm (compare [18,Exercise 3.2]). Investigations of the directional derivative of the maximal eigenvalue of a symmetric matrix have been already given in [25] (compare also [1,6,24,26,28]).  If the maximal eigenvalue ofĀ is a double eigenvalue, then the set X (Ā) is a circle in a two dimensional subspace of R n spanned by the associated eigenvectors. Letx 1 andx 2 be orthogonal normed eigenvectors associated to the maximal eigenvalue of A. Then this subspace is spanned byx 1 andx 2 , and we have X (Ā) = {αx 1 + βx 2 | α, β ∈ R and αx 1 + βx 2 2 = 1}.
In this special case we obtain the directional derivative This maximization problem with only two real variables is very simple to solve.

Conclusions
The investigation of set inequalities generally leads to nontrivial necessary or sufficient conditions. Starting from known sup inf problems characterizations are given for set inequalities and optimal and non-optimal sets. The decisive key for the main results is the use of directional derivatives. For various standard cones the directional derivative of the functionals describing the negative cone is recalled or calculated. It seems to be that the presented technique of proof can also be applied to additional functionals used in practice. A simple example shows the usefulness of this theory but it also demonstrates that the decision, whether a set inequality holds or not, is a difficult task.