A Note on the Star-Kernel for Quasidifferentiable Functions

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

A NOTE ON THE &KERNEL FOE

QUASiDIPFEaSN'MABL6

FTJNCTIONS

Z. Q. Xia

July, 1987 WP-87-66

Working Papers a r e interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only lim- ited review. Views o r opinions expressed herein do not neces- sarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

The continuity of t h e star-kernel of quasidifferentials of a quasidif- ferentiable function with a star-equivalent bounded quasidifferential sub- family are studied and p a r t s of results are r e p r e s e n t e d in this note. I t also has been pointed out t h a t t h e directional subderivative and superderivative of a function can b e expressed as support functions of its star-kernel if t h e star-kernel c a n be generated by a quasidifferential of t h e function.

A. Kunhanski Program Leader System and Decision Sciences Program.

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A NOTE ON THE @KERNEL FOR QUASIDIFFERENTLAB FUNCTIONS

2.

9.

X i a

In t h i s s h o r t n o t e t h e demonstrations of some propositions r e l a t e d t o t h e u p p e r semicontinuity of t h e S k e r n e l f o r quasidifferentiable functions a n d some examples concerning t h e S k e r n e l will be given, [I], [2] a n d [3].

Suppose t h a t f ( z ) is a quasidifferentiable function, defined on S

c

Rn where S i s a n open s e t , with a @-equivalent bounded quasidifferential subfamily. The notations we will use c a n be found o u t in [3]. Their definitions will not be r e p e a t e d h e r e .

L.EIUMA 1

(a) u E a,f(z) - H ( z , u )

#

$ .

(b) w E _ W ( u )

- v ( u

@ d ) Z < w , d > , V d € R n . (c) u E a , p ( z ) - u E ( Z , ~ ) .

f ' ( z ; d ) S max <w , d >

-

w , u )

=

6 ~

I a,p(u

c ~ * ) ( o ) ) ,

F o r t h e s a k e of convenience, assume t h a t &' i s a closed set.

LEMMA

2 If T ( Z , u C3 d ) is u p p e r semicontinuous in ( z , u ) E S X

a,

^f ( z ) f o r e a c h d €IRn, t h e n t h e mapping a,f(*) and

r(*

^{, *)} are closed, i.e., u E a,f(z) and w E

H(z

, u ) ( o r

4 2

(u C3 *)(O)) whenever zi ⁺ x , ui -, u , w i ^-, w , a n d ui E a & ( z i ) , w i E ~ Y ( Z ~ , u i ) , i -,

=.

PROOF Suppose zi -, z ,ui -, u , w i -, w and ui E

BJ

(zi ) , _{w i} E _W(zi , ui ), i -,

= .

According t o t h e L E M 1 (b), o n e h a s

f o r e a c h d E Rn

.

S i n c e

-

~ (, u C3 z d ) is u p p e r semicontinuous in ( z , u ) f o r e a c h d E R , o n e h a s

In o t h e r words, w E _W(Z , u )

.

I t follows from t h e L E M 1 (a) t h a t

The proof is completed. O

THEOREM 3. Suppose

DM

f ( z ) i s bounded uniformly in a neighborhood of z ,

N, ( a ) ,

where 6 is a positive number. If ~ (, u C3 z d ) is u p p e r semicontinuous in ( z , u ) E S x

a,

^f ( z ) for e a c h d E Rn , t h e n

a,f

(*) is u p p e r semicontinuous.

PROOF By contradiction, suppose

a d ( . )

i s not u p p e r semicontinuous. Then t h e r e e x i s t s a n o p e n set 0, a n E

>

0 , a n d t h e r e e x i s t s sequences fzi

1

a n d f u i

1

such t h a t

S i n c e

y ^DM

^{f ( z )} is bounded, t h e r e e x i s t s a s u b s e q u e n c e

tuik 1

c o n v e r g e n t

2 E B ( O , E )

t o

u .

The

u

belongs t o OC b e c a u s e of 0 being a n o p e n set. Obviously,

u

!Z

a,

^f

^{( z}

^). However, in t e r m s of t h e LEM. 2, we h a v e

s i n c e t h e mapping

a,

^{f ( * )} i s closed. This c o n t r a d i c t s t h e f a c t t h a t

u

!Z

a,

f ( z ) . T h e r e f o r e , a , f ( * ) is a n u p p e r semi-continuous mapping. The t h e o r e m is p r o v e d . 3 LEXMA 4. S u p p o s e

-

f

' ( z

; d ) i s lower semi-continuous in

z

^E S f o r e a c h d E Rn a n d t h e mapping

_W(z

,

u )

is u p p e r semicontinuous in

( z

,

u )

^E S X a , f ( z ) . Then t h e mapping a , f ( * ) i s closed.

PROOF S u p p o s e t h a t

S i n c e f o r a n y

u

^E a,f

( z )

a n d f o r e a c h d E Rn ,

' ( z )

max < w , d > ,

-

_w

€ & ( I , ^{U )}

o n e h a s t h a t f o r

fzC 1

a n d

{ui 1

t h e r e e x i s t s a s e q u e n c e lwC ] s u c h t h a t

f '(zi ,

d C ) S max

- <w

, d > ,

w E l ( 2 1 , u )

According t o t h e assumption of t h e u p p e r semi-continuity of t h e mapping _W(. , *) t h e r e e x i s t s a s u b s e q u e n c e ,

{wik

{, c o n v e r g e n t t o

w

s u c h t h a t

w € _ W ( Z ,

u ) .

On t h e o t h e r hand, we h a v e

f ' ( z , d ) S < w , d > ,

-

b e c a u s e of t h e lower semicontinuity of t h e function

-

f ' ( . , d ) . From t h i s , o n e h a s f ' ( z , d ) S max < w , d > .

-

_w € & ( I , ^{U )}

Hence,

u

E

a , f ( z ) .

from LEY. 1 (d). The lemma h a s been p r o v e d .

THEOREM

5.

S u p p o s e f ' ( z , d ) i s lower semi-continuous in

z

E S f o r e a c h d E R n , a n d

DM

f ( z ) i s b k n d e d uniformly in a neighbourhood of

z

,

N,

(6), a n d

&(z

,

u

⁾ i s u p p e r semicontinuous in

( z

,

u

) E S x a e f ( z ) . Then

day(.)

i s a Gpper

semicontinuous mapping.

THEOREM

6. If t h e r e e x i s t s a quasidifferential

La,

Y ( z ) ^#

5 , l ( z ) l

€ D M ,P(z) such t h a t

and

then

a@s(z) =-a, s<z>

⁺

zo s<z>

and

- -

a@s(z) ⁼ a, s ( z ) - a, s ( z ) ,

and

J ' ( z ; d )

=

max <u , d

>

, V d E R~

u @ @ f ( x ) and

PROOF. S i n c e V @ P ( z ) , z f ( z ) l € D M l ( z ) 6(*

1

B P ( z )

+ Z P ( ~ ) )

^{2 6 ( .}

1 2,

J'(z> +

So

^J'(z))

one h a s

~ ' ( 2 ; ^0)s 6 ( .

1 3,

f ( z ) + Z o f ' ( z ) )

-

In o t h e r words,

j"(z ; 0 )

=

6(.

12,

f ( z )

+ z,

f ( z ) ) .

-

Theref o r e ,

a, s(z) ⁺ z,s(z)

^c

a,s(z).

According t o t h e definition of

3 ,

[3], w e have _u,(z> c_a, l ( z ) +

z,

^{f ( z )}

S o t h e equality (3) holds. Similarly, (4) c a n b e p r o v e d in t h e same way. In t h i s case where t h e conditions (1) a n d (2) a r e s a t i s f i e d o n e h a s t h a t f o r a n y

u

^E

a , f ( z )

a n d f o r a n y d E Rn t h e r e l a t i o n s

q(u LL d )

=

-

_u E e a f ( z ) ^max

^{< u}

^,

^{d >}

and

-

q(u LLd)

=

max <u

.

^d

^>

2

' s @ f ( z )

are t r u e . Hence, t h e e q u a l i t i e s (5) a n d

( 6 )

are t r u e . The t h e o r e m h a s b e e n proved. I7

EXAMPLE?

L e t f E

c'w").

Then

a , f ( z ) = t v f ( z ) 1

,

P f ( z ) = l o ] .

EXAMPLE 8 Let f b e a convex function defined in Rn

.

S u p p o s e 0,

f ( z > = La, f ( z )

*

[Oil

= [ a m )

^,

i o j i .

D f ( z > = L a f ( z )

^#

Z f ( z ) l

,

w h e r e

a ^{f ( z )}

i s t h e s u b d i f f e r e n t i a l of f at z in t h e convex s e n s e . S i n c e

a f ( z ) - - a f ( z ) = _ a p ( z ) ,

o n e h a s

a f ( z ) + Z P ( Z ) ⁼ a f ( z )

+

( Z f ( z ) - - a m ) ) -

=@, 4 ( z ) + 5, f ( z ) )

⁺

( Z f ( z ) - a f ( z ) )

a n d

- - -

a m ) - a f ( z ) = (5,

^{P ( Z )}

- a ,

~ ( 2 ) )

+ ( Z J ( Z ) - 5 ~ ( ~ ) ) .

and

From Th. 6, o n e h a s

a , f ( z ) = a f ( z )

^*

P f ( z ) = to1

EXAMPLE

9. Let f b e a c o n c a v e function defined in Rn a n d Do

f ( z ) = [I01 a, f ( z ) l

=[to{

^D

- a ( - f > ( z > l

and

I ) f ( z )

=

L a f ( z ) ^# Z f ( z ) l . Then

a , f ( z )

=a, -

^{f ( z )}

=

- a ( - f ) ( z )

and

- -

* f ( z )

= a, ^{f ( ~ )} - a,

f ( z ) .

EXAMPLE

10. Let f 1 be a convex function and f 2 b e a concave function defined in Rn , and f

=

f 1

+

^{f 2 .} Then

0, f ( z )

= Lao

f ( z )

, 5 ,

f ( z ) l

=

18 f l ( z ) ⁸ -8 ( - f 2 ) ( z

)I

= D,

f l ( z ) ,

- a,

^{f 2}

( z ) I .

For any

La

f ( z ) ,

5

^{f ( z}

)I

^E DM f ( z ) one h a s from (6) and Ex.9 t h a t BP(z) + Z . f ( z )

=

C B f l ( z ) + Z f l ( z ) ) + caf,<z> + Z f 2 ( z ) )

=

a f l ( z ) + ( B f l ( z )

- -

O f ( z ) )

+ w 2 ( z ) - 2 f 2 ( z ) )

-

a ( - f 2 ) ( z )

=

( Z f l ( Z )

-

Z f l ( 2 ) ) + CBP2(z) - B f 2 ( z ) ) + ( a f l ( z )

-

a ( - f 2 ) ( z ) ) ,

where

D f l ( z )

=

w l ( z )

.

5 f l ( z ) l and

9

D f 2 ( z >

=

W 2 ( z ) a f 2 ( z ) I

.

I t follows from (8) t h a t

# ( z ) +

~ P ( z )

3 2 d l ( z ) + 5 d 2 ( z )

=

apl(z)

-

a ( - p 2 ) ( z ) . On t h e o t h e r hand, since

and

9 9

8 f 2 ( z )

-

5 f 2 ( z )

=

( ~ Q P ~ ( z )

-

812(z))

-

(&9'2(z) - 2 d 2 ( z ) ) , one h a s

-

a f ( z )

- Z J ( Z ) =

( Z f 1 ( z ) - 5 f l ( z ) ) + w 2 ( z ) - u 2 ( z ) ) + ( b d 2 ( z ) - 5 d 2 ( z ) ) .

Hence

- - -

a f ( z )

- a f ( z )

³

3 d 2 ( z ) -

a d 2 ( z ) ) . From ( 9 ) , ( 1 0 ) and Th. 6 w e now have

a d ( z ) = a f l ( z ) - a f 2 ( z )

= B d l ( z ) + Bd2(z)

and

-

a @ f ( z

⁾

= 5,,.f2(z) - adZ(z 1) .

EXAMPLE

11Let f 1 and f 2 b e convex, and f

=

f 1

-

^{f 2 .}

T h e n w e h a v e

~ J ( z )

= a f i ( z ) - a f 2 ( z )

and

P ' ( z ) = a f 2 ( 2 ) - a f 2 ( z ) .

ACKNOWLEDGWENT

The a u t h o r i s deeply indebted t o Mrs. Adolfine Egleston f o r h e r g r e a t help.

REFERENCES

[ I ] Demyanov, V.F. a n d Rubinov, A.M. (1980). On Q u a s i d i f l e r e n t i a b L e f i n c t i o n - aLs Soviet Math. Dokl. Vol. 2, N o . 1 , pp. 14-17.

[ Z ] Demyanov, V.F. and Rubinov, A.M. (1985/1986). Q u a s i d i f l e r e n t i a L C a l c u l u s IIASA, Laxenburg Austria/ Springer-Verlag.

[ 3 ] Xia,

Z.Q.

(1986). The @Kernel for a Q u a s i d i f l e r e n t i a b L e f i n c t i o n . R e p o r t , SDS, IIASA, Laxenburg, Austria.