approximation of set-valued functions (adaptation of classical approximation operators) || on...

October 8, 2014 10:15 9in x 6in Approximation of Set-Valued Functions:. . . b1776-ch03

Chapter 3

On Set-Valued Functions (SVFs)

The notion of set-valued functions, which is central to the book, is presentedin this chapter together with several examples. Two important notionsrelated to set-valued functions, selections and representations, are thendiscussed. In particular two types of representations of F : [a, b]→ K(Rn)are considered, one based on a parametrization of the sets F (x), x ∈ [a, b]and one on selections of F . A representation of F allows the introductionof various moduli of continuity and notions of smoothness of F .

In Chapters 4, 8 and 11 we base the adaptation of classical approxima-tion operators to set-valued functions on representations of these functions.

3.1. Definitions and Examples

A set-valued function (SVF, multifunction) F is a mapping with valueswhich are sets. In this book F : [a, b] → K(Rn). Recall that K(Rn), end-owed with the Hausdorff metric, is a complete metric space. We investigatethe approximation of SVFs using the notions of continuity, boundedvariation and Holder regularity as defined in Section 1.1.

For F : [a, b]→ K(Rn) we denote by coF the SVF coF : [a, b] →Co(Rn) such that coF (x) = co(F (x)), and call it the convex hull of F .We call the sets F (x), x ∈ [a, b] images of F . The graph of F is

Graph(F ) = {(x, y) : y ∈ F (x), x ∈ [a, b]} .

It is known that Graph(F ) is closed if F is a continuous multifunction withcompact images defined on [a, b].

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32 Approximation of Set-Valued Functions

A single-valued function f : [a, b]→ Rn satisfying f(x) ∈ F (x),

∀x ∈ [a, b] is called a selection of F .A special interesting class of SVFs are multifunctions with images in

Co(Rn). This class has applications in linear control theory and in convexoptimization, and has been studied thoroughly.

A simple example of a multifunction with convex images is a segmentalfunction, with each image a segment in R, namely F : [a, b]→ Co(R).

A more sophisticated example of a multifunction with convex imagesis generated by the solutions of the linear control system

x(t) = A(t)x + B(t)u(t), x(t0) = x0, t ≥ t0, (3.1)

with x(t) ∈ Rn, A(t),B(t) matrices of order n× n and n×m respectively,

and u(t) ∈ Rm a piecewise continuous control function, satisfying

∀t ≥ t0, u(t) ∈ U, with U ∈ Co(Rm). (3.2)

Denoting the set of all possible solutions x(·) of (3.1) subject to (3.2) by S,the reachable SVF is defined as

R(t) = {x(t) : x ∈ S}, t ≥ t0.

It is not difficult to see that R(t) is a multifunction with convex images. Inthe non-linear case

x(t) = f(t, x(t), u(t)), x(t0) = x0, x(t) ∈ Rn, (3.3)

with u(t) as above, the images of R(t) are not necessarily convex.Another example of SVFs with general images is provided by regarding

a 3D object M as a univariate SVF

F (x) = {(y, z) ∈ R2 : (x, y, z) ∈M}, x ∈ R,

namely F (x0) is the cross-section of M with the plane x = x0 (which canbe the empty set).

3.2. Representations of SVFs

In this section we introduce the notion of a representation of multifunctionswhich belong to a given family F of SVFs, mapping [a, b] into K(Rn).

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On Set-Valued Functions (SVFs) 33

A collection of single-valued functions mapping [a, b] into Rm, for

some m,

RΞ(F ) = {fξ(·) : ξ ∈ Ξ}, (3.4)

with Ξ an index set, is called a representation of F if the correspondencebetween F ∈ F and RΞ(F ) is a bijection. In such a case we denote

F ∼= RΞ(F ) .

Parametrization-based representations

Here we consider representations which are inherently related toparametrizations of the family of sets containing the images of F .

Given a collection of sets A ⊂ K(Rn) and a parametrization G witha bijection T : A → G, the G-based representation of a multifunctionF : [a, b]→ A is

RG(F ) = {gF (·)(ξ) : ξ ∈ D}, (3.5)

with D the domain of definition of the parametrizing functions in G.Note that in this case Ξ in (3.4) is D, and fξ(x) = gF (x)(ξ), x ∈ [a, b].A representation based on a canonical parametrization is called canonical.

As an example of a canonical representation we consider the caseA = Co(Rn) and G the set of all support functions of sets in A (seeRemark 2.2.2). In this case

gF (x)(ξ) = δ∗(F (x), ξ), ξ ∈ Sn−1. (3.6)

It follows from Property 4 of support functions, that a boundedmultifunction F with convex images satisfies the equality

ω[a,b](F, δ) = supξ∈Sn−1

ω[a,b](δ∗(F (·), ξ), δ), δ > 0, (3.7)

with the modulus of continuity of F in the Hausdorff metric (seeSection 1.1).

Motivated by (3.7), in Section 3.3 we define moduli of continuity forSVFs in terms of any given parametrization-based representation of F .

Selection-based representations

We call the representation (3.4) of F selection-based, if fξ for any ξ ∈ Ξis a selection of F .

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The representation by selections is called Castaing representation iffor any x ∈ [a, b], F (x) = cl{fξ(x) : ξ ∈ Ξ} with Ξ a countable set.An important special case is when

F (x) = {fξ(x) : ξ ∈ Ξ} for all x ∈ [a, b]. (3.8)

We call such a representation complete.As an example we consider the collection of sets A = K(Rn), and define

the selections of F : [a, b]→ A as

fξ(x) ∈ ΠF (x)(ξ), ξ ∈ Ξ, x ∈ [a, b], (3.9)

with Ξ = ∪z∈[a,b]F (z) ⊂ Rn.

To see that this is a complete representation, we note that for anyz ∈ [a, b], and for all ξ ∈ F (z), all the selections fξ(x) satisfy fξ(z) = ξ.In case A = Co(Rn), ΠF (x)(ξ) is a singleton, and there is equality in thedefinition of fξ(x) in (3.9).

In Chapter 8 and in Section 10.3 we investigate two complete represen-tations, and derive approximation results based on these representations.Examples of selection-based representations which are not complete arestudied in Sections 10.1 and 10.2.

A representation is both selection-based and parametrization-based,when gF (x) satisfies both gF (x)(ξ) ∈ F (x) for all ξ ∈ D and gF (x) ∈ G forall x ∈ [a, b].

For example, such a representation is obtained by generalized Steinerselections of SVFs with convex images. In this example, D is a set ofprobability measures on the unit ball in R

n. For instance, D may be theset SM of probability measures with C1 density, or, alternatively, the setAM of atomic probability measures concentrated in a single point of theunit sphere in R

n.To introduce a generalized Steiner selection, we first recall a definition

of the Steiner point of a set A ∈ Rn

St(A) =1

vol(B1)

∫

B1

y(l, A)dl,

where B1 is the unit ball in Rn, and y(l, A) is a point in the set

Y (l, A) = {a ∈ A : 〈a, l〉 = δ∗(A, l)}.

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For a measure ξ ∈ D, the corresponding generalized Steiner point of theset A ∈ Co(Rn) is

gA(ξ) =∫

B1

St(Y (l, A))ξ(dl).

Taking as D a countable dense subset of SM, we get a canonical Castaingrepresentation

F (x) = cl{gF (x)(ξ) : ξ ∈ D}.

On the other hand, taking D = co(AM), we get a complete canonicalrepresentation.

It is interesting to note that for SVFs with 1D convex sets as images,the representation based on generalized Steiner selections coincides withthe representation based on the canonical parametrization (2.13).

In the next section we define notions of regularity of SVFs in terms oftheir representations.

3.3. Regularity Based on Representations

Any representation RΞ(F ) induces notions of regularity and smoothness.We define a modulus of continuity of F based on (3.4) by

ωR[a,b]

(F, δ

)= sup

ξ∈Ξω[a,b](fξ, δ), (3.10)

and term F to be R-continuous whenever limδ→0 ωR[a,b](F, δ) = 0, which,

in view of (3.10), implies that the family {fξ : ξ ∈ Ξ} is equicontinuous.Similarly we define the k-th order (k ≥ 2) modulus of smoothness

of F by

ωRk,[a,b]

(F, δ

)= sup

ξ∈Ξωk,[a,b](fξ, δ). (3.11)

Furthermore, we define F to be R-smooth of order k or R-Ck if fξ ∈ Ck

for all ξ ∈ Ξ and if the functions{

dk

dxk fξ : ξ ∈ Ξ}

are equicontinuous. Note

that if F is R-Ck then ωRk,[a,b]

(F, δ

)= O(δk).

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In case of a representation based on a parametrization G, we changethe notation above replacing R by G. Thus by (3.5)

ωG[a,b]

(F, δ

)= sup

ξ∈Dω[a,b](gF (·)(ξ), δ), (3.12)

and we call ωG[a,b] induced modulus of continuity of F . In the same way

we define ωGk,[a,b] and call it induced modulus of smoothness of F .

A multifunction F is called G-Holder-ν if ωG[a,b](F, δ) ≤ Cδν , with

ν ∈ (0, 1]. Similarly, F is called G-smooth of order k or G-Ck if thefunctions in

RG,kF ={

dk

d(·)kfξ : ξ ∈ D

}

are equicontinuous, and it is called G-Ck+ν if the functions in RG,kF areHolder-ν with the same Holder constant independent of ξ.In particular, ωG

[a,b] in (3.12) can be rewritten as

ωG[a,b](F, δ) = sup{dG

(F (x1), F (x2)

): x1, x2 ∈ [a, b], |x1 − x2| ≤ δ}

with dG the induced metric (see Section 2.2.1).For the example of SVFs with convex images, and the parametrization

by support functions, the induced metric is the Hausdorff metric, and (3.10)becomes (3.7).

For the case of complete representations we show in the next lemmathat ωR

[a,b]

(F, δ

)bounds the modulus of continuity in the Hausdorff metric.

Lemma 3.3.1 Let {fξ : ξ ∈ Ξ} be a complete representation of a multi-function F : [a, b]→ K(Rn). Then for every δ > 0

ω[a,b](F, δ) ≤ ωR[a,b]

(F, δ

). (3.13)

Proof Let x1, x2 ∈ [a, b] and y ∈ F (x1) be such that

haus(F (x1), F (x2)) = dist(y, ΠF (x2)(y)).

Then by (3.8), there exists ξ∗ such that fξ∗(x1) = y. Hence

haus(F (x1), F (x2)) = dist(fξ∗

(x1), ΠF (x2)(fξ∗

(x1)))

≤ |fξ∗(x1)− fξ∗

(x2)|.

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Finally we get

ω[a,b](F, δ) = sup|x1−x2|≤δ

haus(F (x1), F (x2))

≤ sup|x1−x2|≤δ

supξ∈Ξ|fξ(x1)− fξ(x2)|

= supξ∈Ξ

ω[a,b](fξ, δ).

�

Remark 3.3.2 The result of this lemma holds also for Castaing represen-tations.

It is well known that continuity of a multifunction in the Hausdorffmetric does not necessarily imply the existence of continuous selections. Yet,in Chapters 8 and 10 we construct for CBV multifunctions representationsby special selections, with modulus of continuity bounded by Cω[a,b](vF , δ),with C a constant. This guarantees the continuity of the selections in ourspecial representations.

3.4. Bibliographical Notes

For general information on SVFs the reader can consult [8]. Set-valuedfunctions and their selections play a central role in the theory of differentialinclusions and their numerical solution (see e.g. [7, 29, 38, 70]), as wellas in variational and non-smooth analysis, optimization and moderncontrol theory [26, 59, 71, 72, 83]. An example showing that continuity ofa multifunction in the Hausdorff metric does not necessarily imply theexistence of continuous selections can be found e.g. in [7], Section 1.6.Representations of SVFs by selections, and in particular Castaing repre-sentations, are discussed in [21]. For SVFs with convex images, generalizedSteiner selections are introduced in [30] where a Castaing representationbased on them is constructed. This construction is developed further andappied for numerical set-valued integration in [14]. Regularity notions withrespect to the induced metric are also introduced there, such as Lipschitzcontinuity and bounded variation.

Various moduli of continuity of multifunctions with values in themetric space (Co(Rn), haus(·, ·)) are defined in [36, 37, 76], and moduli ofsmoothness of such SVFs are introduced in [34], based on the canonicalrepresentation by support functions.

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