Technical Note on Sub Differentials of Set-Valued Maps

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J O U R N A L O F OPTIMIZATION THEORY AN D APPLICATIONS: Vol. 100. No. 1, pp. 233-240, JANUARY 1999

TECHNICAL NOTE

On Subdifferentials of Set-Valued Maps

J . B A I E R1

A N D J . J A H N2

Communicated by F. Giannessi

Abstract. Using the concept of contingent epiderivative, we generalizethe notion of subdifferential to a cone-convex set-valued map. Properties

of the subdifferential are presented and an optimality condition is

discussed.

K ey Words. Convex analysis, set-valued analysis, Subdifferentials,

vector optimization.

1. Introduction

It is well known from convex analysis that a subgradient of a convexfunctional f: X->R [denned on a real normed space (X, || • | | x ) ] at some xe X

is a continuous linear functional l on X with

1Graduate Student, Angewandte Mathematik I I , Universitat Erlangen-Nurnberg, Erlangen,

Germany.2AssociateProfessor, Angewandte Mathematik I I , Universitat Erlangen-Nurnberg, Erlangen,

Germany.

2 33

0022-3239/99/0100-0233$16.00/0 C 1999 Plenum Publishing Corporation

The set of all subgradients is commonly called subdifferential. Using the

notion of directional derivative f ' ( x ) ( • ) , this subdifferential ca n also becharacterized by the inequality

e.g., se e Lemma 3.25 in Ref. 1 . This inequality is the key for a generalizationto cone-convex set-valued maps. Instead of the directional derivative, w euse the concept of contingent epiderivative presented in Ref. 2, and the



2 34 JOTA: VOL. 100, NO. 1, JANUARY 1999

inequality sign has to be understood as a partial ordering. This is the actual

approach of this short paper.Although there are many papers generalizing the concept of subdiffer-

ential to the vector-valued (single-valued) case, there are only a few papers

investigating subdifferentials of set-valued maps. For instance, Yang (Ref.

3) introduced a weak subgradient of a convex relation and Chen and Jahn

(R e f . 4) considered a weak subgradient for a general set-valued map. Here,

w e present an approach different from that in Ref. 4 .For the investigation of set-valued maps, we use the following standard

assumption:

(Al) Let (X , | |- Ix) and (Y , ||- ||y) be real normed spaces, le t (Y ,|| • || Y) be partially ordered by a convex cone C< Y, let S be a

nonempty subset of X, let F: S->2y

be a set-valued map, and let

xeS and yeF(x) be given elements.

It is well known that the convex cone C induces a partial ordering ^c (i.e.,

a reflexive and transitive binary relation compatible with addition and scalar

multiplication) in the space Y; fo r instance, se e Ref. 5.

First, we recall the concept of contingent epiderivative given in Ref. 2,

which is based on the definition of contingent derivative introduced by Aubin

(R e f . 6).

Definition 1.1. Let Assumption (A1) be satisfied. Then:

(a) The set

is called the epigraph of F.(b) A single-valued map DF(x, y): X-> Y whose epigraph equals the

contingent cone to the epigraph of F at (x, y), i.e.,

is called the contingent epiderivative of F at (x, y).

Recall that the contingent cone T(epi(.F), (x, y)) consists of all tangent

vectors

with



and A n > 0 , n e N. Properties of the contingent epiderivative can be f o u n d in

R e f . 2 .Since convexity plays an important role in the following investigations,

recall the definition of cone-convex maps.

Definition 1.2. Let Assumption (A1) be satisfied, and in addition let

S be a convex set. F: S->2Y

is called C-convex if, for all x1, x2eS and

Ae[0,1],

2. Concept of Subdifferential

In this section, we generalize the concept of subdifferential of a convex

functional to the case of a cone-convex set-valued map.For these investiga-

tions, we extend Assumption (A1)as follows:

(A2) Let the set S be convex, let the set-valued map F: S->2y

be C-

convex, and let the contingent epiderivative DF(x, y) of F at

(x, y) exist.

On the basis of the concept of contingent epiderivatives, we introduce

in Section 2 a subdifferential for cone-convex set-valued maps and discuss

the properties of this concept in Section 3. In Section 4, we present a gen-

eralization of a well-known optimality condition to a convex set-valuedoptimization problem. The results are based on the presentation in Ref.7.

Definition 2.1. Let Assumptions (A1)and (A2)be satisfied. Then:

(a) A linear map L: X-+ Y, with

is called a subgradient of F at (x,y); see Fig. 1.

(b) The set

of all subgradients L of F at (x, y) is called the subdifferential of

F at (x, y).

The inequality (2) is a natural extension of the inequality (1) to the set-

valued case. Here, the directional derivative is replaced by the contingent

J O T A : VOL. 100, N O. 1 , J A N U A R Y 1999 235



2 36 J O T A : VOL. 100, NO. 1, J A N U A R Y 1999

Fig. 1. Subgradients of Fat (x, y).

epiderivative and the usual < ordering is replaced by the partial ordering

<c induced by the convex cone C.

Obviously, the subdifferential is not defined, if the contingent epideriv-

ative does not exist. Conditions ensuring the existence of the contingent

epiderivative can be found in Theorem 1 in Ref. 2 and Theorem 3 in Ref.

4. Notice also that the assumption of cone-convexity of F is actually not

needed in Definition 2.1.

3. Properties of the Subdifferential

In this section, we present basic properties of subdifferentials known

from the convex single-valued case. First, we remark under whichassumptions the subdifferential isnonempty.

Theorem 3.1. Let Assumptions (A1) and (A2) be satisfied, and in

addition let S=X, let C be pointed [i.e., C n (-C) = 0Y], and let Fbe order

complete. Then, the subdifferential dF(x, y) isnonempty.

Proof. By Theorem 4 in Ref. 2 the contingent epiderivative DF(x, y)

is sublinear. Then, by the Hahn-Banach theorem generalized by Zowe (Ref.

8), there is a linear map L: X-> Y with

Hence, the subdifferential dF(x, y) is nonempty. Q

Next, we show the convexity of the subdifferential.



JOTA: VOL. 100, N O. 1 , J A N U A R Y 1999 237

Theorem 3.2. Let Assumptions (A1) and (A2) be satisfied. Then, the

subdifferential is convex.

Proof. For an empty subdifferential, the assertion is trivial. Take two

arbitrary subgradients L1 , L2edF(x, y) and an arbitrary Ae[0, 1].Then, we

obtain

Hence,

The next result shows that the subdifferential is closed under appropri-

ate assumptions.


addition let C be closed. If all subgradients are bounded, then the subdiffer-ential is closed in the linear space of all linear bounded maps.

Proof. Choose an arbitrary sequence (Ln )ne N of subgradients converg-

in g to some linear bounded map L. Next, fix an arbitrary xeX. Then, we

obtain

where ||| • ||| denotes the operator norm. Since

the inequality (3) implies

B y the definition of the subgradients Ln, w e have

or

and with (4) and the assumption that C is closed we conclude that

Hence, L is a subgradient, and therefore the subdifferential is closed. D



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Notice that, for X=Rn

and Y=Rm

, linear maps are bounded, and in

this special case the subdifferential is closed whenever C is closed.The following result presents a condition under which the subdifferential

is a singleton.


addition le t C be pointed. If the contingent epiderivative DF(x, y) of F at

(x , y) is linear, then dF(x, y) = {DF(x, y)}.

Proof. Since DF(x, y) is linear, DF(x, y) is a subgradient. Assume that

there is another subgradient L = D F ( x , y). Then, we obtain

or

This inequality implies, by addition of L(x) +DF(x, y)(x),

Since C is pointed, we get with (2)

a contradiction to our assumption. Hence, we conclude that

Finally, we discuss the relationship of the presented definition of sub-

differential to the standard definition used in convex analysis.

Theorem 3.5. Let Assumptions (A1) and (A2) be satisfied. Then, every

subgradient L of F at (x, y) fulfills the inequality

Proof. B y Lemma 3 in Ref. 2 , we obtain

and with Inequality (2) we conclude that



JOTA : VOL. 100, N O. 1 , J A N U A R Y 1999 239

4. Optimality Conditions

U nder A s s umption ( A 1) , we now investigate the set-valued optimiza-tion problem

In this context, minimization means that we determine strong minimizers.

Definition 4.1. Let Assum ption (A 1 ) be satisfied, and let the problem

(5 ) be given. Le t F(S) := U x e s F ( x ) d e n o t e the image set of F. The pair(x , y) is called a strong minimizer of the set-valued optimization problem(5), if y is a strongly minimal element of the set F(S), i.e.,

Fo r strong minimizers, an optimality condition based on the subdiffer-ential can be given. This result extends the well-known result of convexanalysis that a point is a minimal point of a convex functional if and onlyif the null functional is a subgradient (e.g., see Theorem 3.27 in R e f . 1 ).

Theorem 4.1. Le t A s sumptions (A 1) an d ( A 2 ) be satisfied. Then:

(a) If the null map is a subgradient of F at (x, y), then the pair (x, y)

is a strong minimizer of the set-valued optimization problem (5).(b) In addition, let S equal X and let C be closed. If the pair (x , y) is

a strong minimizer of the set-valued optimization problem (5),then the null map is a subgradient of F at (x, y).

Proof, (a) By Theorem 3.5, w e conclude that

or

i.e., (x, y) is a strong minimizer of the set-valued optimization problem (5).(b) B y Theorem 9 in R ef. 2, we obtain for the strong minimizer (x, y)

or

Hence, the null map is a subgradient of F at (x, y). D

Th e preceding theorem implies immediately the following corollary.



Corollary 4.1. Let Assumptions (A1) and (A2) be satisfied, and in

addition let S equal X and let C be closed. The pair (x, y) is a strongminimizer of the set-valued optimization problem (5) if and only if the null

map is a subgradient of F at (x, y).

5. Conclusions

This short paper shows that it makes sense to introduce subdifferentials

in a set-valued setting because standard results known from convex analysis

can be generalized easily. The concept of so-called weak subgradients of set-

valued maps (investigated in Ref. 4) seems to be suitable for the investigation

of weak minimizers of a set-valued optimization problem, whereas the con-

cept presented in this paper is appropriate for the characterization of strong

minimizers. Both approaches have advantages, but further investigations arenecessary.

References

1 . J A H N , J. , Introduction to the Theory of Nonlinear Optimization, Springer, Berlin,

Germany, 1996.

2. J A H N , J., and R A U H , R., Contingent Epiderivatives and Set-Valued Optimization,

Mathematical Methods of OperationsResearch, Vol. 46, pp. 193-211, 1997.

3. Y A N G , Q . X., A Hahn-Banach Theorem in Ordered Linear Spaces and Its Applica-

tions, Optimization, Vol. 25, pp. 1-9, 1992.

4. C H E N , G. Y., and J A H N , J., Optimality Conditions for Set-Valued Optimization

Problems, Mathematical Methods of Operations Research, Vol.48, 1998.

5. J A H N , J. , Mathematical Vector Optimization in Partially-Ordered Linear Spaces,

Peter Lang, Frankfurt, Germany, 1986.

6. A U B I N , J. P., Contingent Derivatives of Set-Valued Maps an d Existence of Solu-

tions to Nonlinear Inclusions and Differential Inclusions, Mathematical Analysis

and Applications, Part A, Edited by L. Nachbin, Academic Press, New York,

N ew York, pp. 160-229, 1981.

7. B A I E R , J. , Subdifferentiale fur mengenwertige Abbildungen, Diplom Thesis, Univer-

sity of Erlangen-Nurnberg, 1997.

8. Z O W E , J. , Konvexe Funktionen undkonvexe Dualitatstheorie in geordneten Vektor-

raumen, Habilitation Thesis, University of Wurzburg, 1976.

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Technical Note on Sub Differentials of Set-Valued Maps