Representation of Set Relations in a Real Linear Space

The following theorem shows a first connection between the upper set less relation and the nonlinear scalarizing functionalz^D,k, where the spaceY is not a priori equipped with a topology.

Theorem 2.5.3 ([45, Theorem 3.2]). Let D ⊂ Y be a convex cone, A, B ∈ P(Y) and k∈Y \ {0}. Then

A^u_D,rlsB =⇒ sup

a∈A

z^D,k(a)≤sup

b∈B

z^D,k(b).

Proof. Because D is a convex cone, it holds D+D ⊆ D ⊆ vcl_kD = {0}+ vcl_kD ⊆ [0,+∞)k+ vclkD. Due to Proposition 2.5.1 (c), we obtain theD-monotonicity property ofz^D,k. Now let A⊆B−D. Then for all a∈A, there existsb∈B such thata∈b−D.

This immediately yieldssup_a∈Az^D,k(a)≤sup_b∈Bz^D,k(b).

The converse implication in Theorem 2.5.3 is not generally fulfilled, even if the un-derlying sets are convex, see [66, Example 3.2]. However, we have the following result.

Theorem 2.5.4 ([45, Theorem 3.3]). Let D ⊂ Y. For two sets A, B ∈ P(Y) and k∈Y \ {0}, it holds

A^u_D,rlsB =⇒ sup

a∈A

b∈Binf z^D,k(a−b)≤0.

Assume on the other hand that there exists ak₀ ∈Y \ {0} such that infb∈Bz^D,k0(a−b) is attained for all a∈A,D is k₀-vectorially closed and [0,+∞)k₀+D⊆D. Then

sup

a∈A

b∈Binf z^D,k0(a−b)≤0 =⇒ A^u_D,rlsB .

Proof. LetA⊆B−D. This corresponds to

∀a∈A ∃ b∈B: a−b∈ −D⊆ −vcl_kD . Due to Proposition 2.5.1 (a), we obtain

∀ a∈A ∃b∈B : z^D,k(a−b)≤0

and this directly implies the assertion, i.e.,sup_a∈A infb∈Bz^D,k(a−b)≤0. Conversely, let sup_a∈Ainfb∈Bz^D,k0(a−b)≤0. This means that for all a∈A, we have infb∈Bz^D,k0(a− b)≤0. Because for all a∈A,infb∈Bz^D,k0(a−b) is attained, we obtain

∀a∈A ∃¯b∈B : z^D,k0(a−¯b) = inf

b∈Bz^D,k0(a−b)≤0.

This implies

∀a∈A ∃¯b∈B : a−¯b∈(−∞,0]k₀−vcl_k0D.

Therefore,A⊆B+ (−∞,0]k₀−vcl_k0D⊆B−D.

Remark 2.5.5 ([45, Remark 3.4]). (1) Note that for any A, B ∈ P(Y), the set relation A^u_D,rls B by Theorem 2.5.4 also implies sup_k∈Y\{0}sup_a∈Ainfb∈Bz^D,k(a−b)≤0.

(2) Let A, B ∈ P(Y) and D ⊂ Y. If there exists an element k₀ ∈ D\ {0} such that infb∈Bz^D,k0(a−b)is attained for alla∈A,Disk0-vectorially closed and[0,+∞)k₀+D⊆ D, then it follows from Theorem 2.5.4 that

A^u_D,rlsB ⇐⇒ sup

a∈A

b∈Binfz^D,k0(a−b)≤0

⇐⇒ sup

k∈Y\{0}

sup

a∈A

b∈Binfz^D,k(a−b)≤0.

The following example shows that the attainment property in Theorem 2.5.4 cannot be omitted.

Example 2.5.6 ([45, Example 3.5]). Let Y = R², A := {(0,0)^T}, B := {(y₁, y₂) ∈ R² |y1, y2∈(−1,0)}, D=R²₊ and k0 = (1,1)^T. We havevclk0D=Dand[0,+∞)k₀+ D ⊆ D. It holds that sup_a∈A infb∈Bz^D,k0(a−b) ≤ 0, however, A 6^u_D,rls B. This is because infb∈Bz^D,k0(a−b) is not attained for a= (0,0)^T.

In the second part of Theorem 2.5.4, we need the assumption that there exists a k₀ ∈Y\ {0}such thatinfb∈Bz^D,k0(a−b)is attained for alla∈A. As already mentioned in Remark 2.3.8, sufficient conditions for such an attainment property, i.e., assertions concerning the existence of solutions of the corresponding optimization problems (ex-tremal principles) are given in the literature. Since the functional z^D,k0 is studied here in the context of real linear spaces that are not endowed with a particular topology, we cannot rely on continuity assumptions. Therefore, we propose the following theorem without any attainment property.

Theorem 2.5.7 ([45, Theorem 3.6]). Let D ⊂Y, A, B ∈ P(Y) and k0 ∈Y \ {0} such that (−∞,0)k0−vcl_k0D⊆ −D andvcl−k0(B−D)⊆B−D. Then

sup

a∈A

b∈Binf z^D,k0(a−b)≤0 =⇒ A^u_D,rlsB .

Proof. Letsup_a∈A infb∈Bz^D,k0(a−b)≤0. This means that for alla∈A, the inequality infb∈Bz^D,k0(a−b)≤0holds true. Therefore, for all >0and for all a∈A there exists b∈B such thatz^D,k(a−b)< . By means of Proposition 2.5.1 (d), we obtain

∀ >0, ∀a∈A, ∃b∈B : z^D,k0(a−b−k₀)<0.

This implies by Proposition 2.5.1 (b) that

∀ >0, ∀a∈A, ∃b∈B : a−b−k₀ ∈(−∞,0)k₀−vcl_k0D.

This results in

∀ >0 : A⊆B+k₀+ (−∞,0)k₀−vcl_k0D⊆B+k₀−D⊆vcl−k0(B−D)⊆B−D.

We illustrate by the example below that the assumption vcl−k0(B−D)⊆B−Din Theorem 2.5.7 cannot be dropped.

Example 2.5.8([45, Example 3.7]). We return to Example 2.5.6. We havevcl_k0D=D, and becausek₀ ∈corD, (−∞,0)k₀−vcl_k0D⊆ −Dholds true. Moreover, the inequality sup_a∈A infb∈Bz^D,k0(a−b)≤0 is fulfilled. Because A 6^u_D,rls B, due to Theorem 2.5.7, vcl−k₀(B−D)⊆B−Dcannot be satisfied. This is immediate, as vcl−k₀(B−D) =−R²+. BecauseA^u_D,rls Bis equivalent to−B ^l_D,rls−A, we obtain the following corollaries from Theorems 2.5.3, 2.5.4 and 2.5.7.

Corollary 2.5.9 ([45, Corollary 3.9]). Let D ⊂Y be a convex cone, A, B ∈ P(Y) and k∈Y \ {0}. Then

A^l_D,rls B =⇒ inf

a∈Az^D,k(a)≤ inf

b∈Bz^D,k(b).

Corollary 2.5.10 ([45, Corollary 3.10]). Let D ⊂ Y. For two sets A, B ∈ P(Y) and k∈Y \ {0}, it holds

A^l_D,rls B =⇒ sup

b∈B

a∈Ainf z^D,k(a−b)≤0. Assume on the other hand that there exists a k0 ∈Y \ {0} such that

infa∈Az^D,k0(a−b)is attained for allb∈B,Disk₀-vectorially closed and[0,+∞)k₀+D⊆ D. Then

sup

b∈B

a∈Ainf z^D,k0(a−b)≤0 =⇒ A^l_D,rls B .

Corollary 2.5.11 ([45, Corollary 3.11]). Let D ⊂ Y, A, B ∈ P(Y) and k0 ∈ Y \ {0}

such that (−∞,0)k0−vcl_k0D⊆ −D andvcl−k0(−A−D)⊆ −A−D. Then sup

b∈B

a∈Ainf z^D,k0(a−b)≤0 =⇒ A^l_D,rls B . For the set less relation, we immediately obtain the following results.

Corollary 2.5.12 ([45, Corollary 3.13]). Let D⊂Y be a convex cone,A, B∈ P(Y)and k∈Y \ {0}. Then

A^s_D,rlsB =⇒ sup

a∈A

z^D,k(a)≤sup

b∈B

z^D,k(b) and inf

a∈Az^D,k(a)≤ inf

b∈Bz^D,k(b).

Corollary 2.5.13 ([45, Corollary 3.14]). Let D ⊂ Y. For two sets A, B ∈ P(Y) and k∈Y \ {0}, it holds

A^s_D,rlsB =⇒ sup

a∈A

b∈Binfz^D,k(a−b)≤0 and sup

b∈B

a∈Ainf z^D,k(a−b)≤0. Assume on the other hand that there exists ak0 ∈Y \ {0} such that infb∈Bz^D,k0(a−b) is attained for all a∈ A, and there exists k₁ ∈ Y \ {0} such that infa∈Az^D,k1(a−b) is attained for all b ∈B, D is both k₀- and k₁-vectorially closed, [0,+∞)k₀+D⊆D and [0,+∞)k₁+D⊆D. Then

sup

a∈A

b∈Binfz^D,k0(a−b)≤0 and sup

b∈B

a∈Ainf z^D,k1(a−b)≤0 =⇒ A^s_D,rlsB . Corollary 2.5.14 ([45, Corollary 3.15]). Let D⊂Y,A, B ∈ P(Y) andk₀, k₁ ∈Y \ {0}

such that(−∞,0)k₀−vcl_k0D⊆ −D,(−∞,0)k₁−vcl_k1D⊆ −D,vcl−k₀(B−D)⊆B−D andvcl−k1(−A−D)⊆ −A−D. Then

sup

a∈A

b∈Binfz^D,k0(a−b)≤0 and sup

b∈B

a∈Ainf z^D,k1(a−b)≤0 =⇒ A^s_D,rlsB . Now we intend to study set optimization problems in terms of (1.10) with Y being a real linear space. By the definition of minimal solutions of the problem (1.10) w.r.t.

the relation , we know that ifx∈S is a minimal solution of the problem (1.10) w.r.t.

and F(˜x)F(x) for some x˜∈S, thenx˜ is a minimal solution of the problem (1.10) w.r.t. as well. Therefore, let us denote

[F(x)]⁻¹ :={x∈S : F(x)F(x), F(x)F(x)}.

We now consider problem (1.10) with =^u_D,rls, where D ⊂ Y is a nonempty set (not necessarily convex). We define a function g^u :S×S→R∪ {±∞}by

g^u(x, x) := sup

y∈F(x)

y∈Finf(x)z^D,k(y−y).

Assumption 2.5.15. For D⊂Y, k∈Y \ {0}, and x∈S we assume that

1. D is k-vectorially closed, [0,+∞)k+D⊆D, and for allx ∈S\[F(x)]u

D,rls and y∈F(x), inf_y∈F(x)z^D,k(y−y) is attained; or

2. (−∞,0)k−vcl_kD⊆ −D andvcl−k(F(x)−D) =F(x)−D.

The following proposition will be useful in the theorem below.

Proposition 2.5.16. x∈S is a minimal solution of the problem (1.10) w.r.t. if and only if for any x∈S\[F(x)]⁻¹ , we have F(x)6F(x).

Proof. First note that x ∈ S \[F(x)]⁻¹ means that x ∈ S such that F(x) 6 F(x) or F(x)6F(x). Letx∈S be a minimal solution of the problem (1.10) w.r.t. . Then we have to consider two cases:

Case 1: For x∈S and F(x)6F(x), there is nothing left to show.

Case 2: Forx∈S and F(x) 6F(x), we obtain F(x)6F(x) due to x’s minimality, as desired.

Conversely, assume that for all x ∈S\[F(x)]⁻¹ , F(x)6F(x) holds true. Suppose, by contradiction, thatx is not a minimal solution of the problem (1.10) w.r.t. . This implies the existence of somex∈S with the properties F(x)F(x) and F(x)6F(x), in contradiction to the assumption.

We next present a sufficient and necessary condition for minimal solutions of the problem (1.10) w.r.t. the relation^u_D,rls.

Theorem 2.5.17 ([45, Theorem 4.3]). Let Assumption 2.5.15 be satisfied. Then x is a minimal solution of the problem (1.10) w.r.t. ^u_D,rls if and only if the following system (in the unknown x)

g^u(x, x)≤0, x∈S\[F(x)]⁻¹u D,rls, is impossible.

Proof. First note that, due to Proposition 2.5.16, x ∈ S is a minimal solution of the problem (1.10) w.r.t. ^u_D,rls if and only if for x∈S\[F(x)]⁻¹u

D,rls, we haveF(x)6^u_D,rls F(x). Furthermore, we have

g^u(x, x)≤0, x∈S\[F(x)]⁻¹u

D,rls is impossible

⇐⇒ @x∈S\[F(x)]^−1u

D,rls : sup

y∈F(x)

y∈Finf(x)z^D,k(y−y)≤0

⇐⇒ ∀x∈S\[F(x)]⁻¹u

D,rls : sup

y∈F(x)

y∈Finf(x)z^D,k(y−y)>0

⇐⇒ ∀x∈S\[F(x)]⁻¹u

D,rls : F(x)6^u_D,rls F(x).

Furthermore, let us consider problem (1.10) with =^l_D,rls. We define the function g^l :S×S→R∪ {±∞}by

g^l(x, x) := sup

y∈F(x)

inf

y∈F(x)z^D,k(y−y).

Assumption 2.5.18. For D⊂Y, k∈Y \ {0}, and x∈S we assume that 1. D is k-vectorially closed, [0,+∞)k+D⊆D, and for allx ∈S\[F(x)]⁻¹l

D,rls

and y∈F(x), inf_y∈F(x)z^D,k(y−y) is attained; or

2. (−∞,0)k−vcl_kD ⊆ −D and for all x ∈ S\[F(x)]⁻¹l D,rls

, vcl−k(−F(x)−D) =

−F(x)−D.

In the following, we present a sufficient and necessary condition for minimal solutions of the problem (1.10) w.r.t. ^l_D,rls.

Corollary 2.5.19 ([45, Corollary 4.5]). Let Assumption 2.5.18 be satisfied. Then x is a minimal solution of the problem (1.10) w.r.t. ^l_D,rls if and only if the following system (in the unknown x)

g^l(x, x)≤0, x∈S\[F(x)]⁻¹_l

D,rls

, is impossible.

Finally, we have the following result for minimal solutions of the problem (1.10) w.r.t.

^s_D,rls.

Corollary 2.5.20 ([45, Corollary 4.6]). Let Assumptions 2.5.15 and 2.5.18 be satisfied for the same k ∈ Y \ {0}. Then x is a minimal solution of the problem (1.10) w.r.t.

^s_D,rls if and only if the following system (in the unknownx):

g^u(x, x)≤0 andg^l(x, x)≤0, x∈S\

[F(x)]⁻¹u D,rls

∪[F(x)]⁻¹_l

D,rls

, is impossible.

Variable Domination Structures in Set Optimization

3.1 Introduction

It is well known that for certain applications, the solution concept given in Definition 1.2.11 together with one of the set relations from Chapter 2 is not sufficient. As one possible resolution, variable domination structures have been introduced when defining a solution concept. This chapter, which is based on [65], presents a concept for dealing with variable ordering structures in set optimization by equipping the upper set less relation ^u_D with a variable cone D. Note that in this thesis, the notions variable domination structure and variable ordering structure are used simultaneously.

Going back to Yu [112], variable domination structures generalize the concept of ordering structures in vector optimization and have since been intensely studied in the field of vector optimization. Motivated by applications in medical image registration [24, 25], variable domination structures in vector optimization gained recognition as they allow to introduce a specification of the decision-maker’s preferences into the model. Due to these important applications, variable domination structures have gained increasing interest, compare Durea, Strugariu, Tammer [20], or Eichfelder, Bao, Soleimani, Tammer [7] for an analysis of Ekeland’s variational principle with variable ordering structures.

Note that Chen et al. [15] consider a vector approach to set optimization with a variable ordering structure. In addition, Bouza and Tammer [14] have introduced a nonlinear scalarizing functional to characterize and compute minimal points of a set with respect to a variable domination structure.

Variable domination structures play a crucial role, for example, in medical image reg-istration, which has been used widely in medical treatment, for instance in radiotherapy (treatment verification, treatment planning, treatment guidance), orthopaedic surgery, and surgical microscope. The problem of image registration is finding a transformation matching two given sets of data (images). The similarity of the transformed data set to the target set can then be measured by several distance measures. As a multitude of measures exist that evaluate distinct characteristics, such as the sum of square

differ-63

ences, mutual information or cross-correlation, it is necessary to decide which distance measure to use. It is well known that different measures can lead to different best trans-formations. According to [25], some measures fail on special data sets, i.e. they lead to mathematically correct, but useless results. Thus it is important to combine several measures. Possible approaches are a weighted sum of different measures. But difficulties appear, such as badly scaled or nonconvex functions. Instead, Wacker [103] proposed to collect all available distance measures in a vector-valued function and minimizes this function. This leads to a vector-valued optimization problem. This connection with vari-able domination structures in vector programming has first been analyzed in Wacker [103]

and further developed by Eichfelder [24] (see also [25, Section 10.3]). Recently, variable domination structures have been introduced to set optimization problems in [64, 65, 70].

This is particularly useful if uncertainties appear in the objective function, i.e., in the function that comprises the distance measures, for example due to inaccuracies of the data, or movements of the patient during the procedure. Then it is possible to convert the uncertain vector optimization problem into a set optimization problem and compute, for example, robust solutions. This is one of the main motivations why recently it has also been of great interest to consider set-valued optimization problems equipped with a variable domination structure by following a set approach. Recently, Köbis [64, 65]

and Eichfelder and Pilecka [27] have introduced several set relations for the case that the order is given by a cone-valued map. A very general scalarization scheme for solving set optimization problems w.r.t. variable domination structures has been proposed in [70]

(see also [71]).

In this chapter, we modify the upper set less relation given by Kuroiwa [77, 76] (see Definition 2.2.1, where D is a convex cone) in order to compare sets. We equip the upper set less relation with a variable domination structure in Section 3.2, formulate an optimality concept in Section 3.3 and we discuss optimal elements of sections of feasible elements in Section 3.4. Furthermore, Section 3.5 is concerned with providing scalarization results. We conclude this chapter with an application to image registration in medical engineering. Note that some of the results presented in this chapter can be formulated for other set relations, too; compare [64] for an overview.

Im Dokument Approaches to set optimization without convexity assumptions (Seite 62-69)