Sensitivity Analysis in Multiobjective Optimization

NOT FOR QUOTATION

WITHOUT THE PERMISSION OF THE AUTHOR

SENSITIVITY ANALYSIS

IN

MULTIOBJECTIVE OF'TIMIZATION

T e t s u z o T a n i n o

February 1986 WP-86-5

W o r k i n g P a p e r s a r e interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems Analysis and h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o not n e c e s s a r i l y r e p r e s e n t t h o s e of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREFACE

Sensitivity analysis is both theoretically and practically useful in optimiza- tion. However, only a few r e s u l t s in t h i s direction have been obtained f o r multiob- jective optimization. In t h i s p a p e r , t h e &sue of sensitivity analysis in multiobjec- Live optimization i s d e a l t with. Given a family of parametrized multiobjective optim- ization problems, t h e p e r t u r b a t i o n map is defined as t h e set-valued map which as- sociates t o e a c h p a r a m e t e r value t h e set of minimal points of t h e p e r t u r b e d feasi- ble set with r e s p e c t t o a fixed o r d e r i n g convex cone. The behavior of t h e p e r t u r - bation map is analyzed quantitatively by using t h e concept of contingent derivative f o r set-valued maps. P a r t i c u l a r l y i t i s shown that. t h e contingent derivative of t h e p e r t u r b a t i o n map f o r multiobjective programming problems with parametrized ine- quality c o n s t r a i n t s is closely r e l a t e d t o t h e corresponding Lagrange multipliers.

Alexander B. Kurzhanski Chairman

System and Decision Sciences Program

SENSITIVITY

ANALYSIS

IN MULTIOBJECITVE OPTIMIZATION

T e t s u z o T a n i n o

1. INTRODUCTION

Stability and sensitivity analysis i s not only theoretically interesting but a l s o practically important in optimization theory. A number of useful r e s u l t s have been obtained in usual s c a l a r optimization (see, f o r example, Fiacco [3] and Rockafellar [4]). Here, by stability w e mean t h e quantitative analysis, t h a t is, t h e study of various continuity p r o p e r t i e s of t h e p e r t u r b a t i o n ( o r marginal) function ( o r map) of a family of parametrized optimization problems. On t h e o t h e r hand, by sensitivi- t y w e mean t h e quantitative analysis, t h a t is, t h e study of d e r i v a t i v e s of t h e p e r - turbation function.

F o r multiobjective optimization, t h e "optimal" value of a problem i s not unique and hence w e must consider not a function but a set-valued p e r t u r b a t i o n map. The a u t h o r and Sawaragi [7] investigated some sufficient conditions f o r t h e semicon- tinuity of t h e p e r t u r b a t i o n map. However, t h e i r r e s u l t s are qualitative and t h e r e - f o r e provide no quantitative information. In t h i s p a p e r , t h e behavior of t h e p e r - turbation map will b e studied quantitatively via t h e concept of contingent deriva- tive introduced by Aubin

[I].

Though s e v e r a l o t h e r concepts of derivatives of set-valued maps were proposed (see Aubin and Ekeland [2]. p. 493), t h e concept of t h e contingent derivative i s t h e most adequate f o r o u r purpose. Because i t depends on t h e point in t h e g r a p h of a set-valued map and when w e discuss t h e sen- sitivity of t h e p e r t u r b a t i o n map, w e fix some point in i t s g r a p h .

The contents of t h i s p a p e r are as follows. In Section 2, w e introduce t h e con- c e p t of t h e contingent derivative of set-valued maps along with some basic p r o p e r - t i e s which are n e c e s s a r y in t h e l a t e r sections. Section 3 is devoted t o t h e analysis of t h e contingent derivative of t h e p e r t u r b a t i o n map, which i s defined from a feasi- b l e s e t map by taking t h e set of minimal points with r e s p e c t t o a given closed con- vex cone. In Section 4, w e analyze t h e sensitivity in g e n e r a l multiobjective optimi- zation problems specified by feasible decision sets and objective functions which

depend on a p a r a m e t e r v e c t o r . In Section 5, we c o n c e n t r a t e on multiobjective pro- gramming problems in which only t h e right-hand side of inequality c o n s t r a i n t s i s p e r t u r b e d . I t is shown t h a t t h e sensitivity of t h e p e r t u r b a t i o n map is closely re- lated with t h e Lagrange multipliers of t h e nominal problem.

2. CONTINGENT DERIVATJYES OF SET-VALUED MAPS

In t h i s section w e introduce t h e concept of t h e contingent d e r i v a t i v e of set- valued maps. Throughout t h i s section V and Z a r e two Banach s p a c e s and F is a set-valued map f r o m V t o 2.

D e p i n i t i o n 2.2. (Aubin and Ekeland [2]). Let C b e a nonempty s u b s e t of a Banach s p a c e V and

f

^E V. The s e t T C ( f )

c

V defined by

is called t h e contingent cone t o C at 6 , where B i s t h e unit ball in V. In o t h e r words, u E TC(6) if and only if t h e r e e x i s t sequences f h k

I ck+

a n d f u k { C V such t h a t hk + O+, u k + u and

v^ +

h k u k E

c

for V k ,

0

where

R+

is t h e set of positive real numbers.

I t i s well known t h a t T C ( f ) i s a closed (but not always convex) cone.

The g r a p h of a set-valued map F from V t o Z is defined and denoted by

The contingent d e r i v a t i v e of F is defined by considering t h e contingent cone to g r a p h F .

D e p i n i t i o n 2.2. (Aubin and Ekeland [2]) Let

(c,;)

^{b e a point in} g r a p h F . W e denote by ~ ~ ( 6 , s ) t h e set-valued map f r o m V t o Z whose g r a p h i s t h e contingent cone ~ ~ ~ ~ t o t h e g r a p h ~ ~ ( f ,of gF a t ( 6 ) ,;), and c a l l i t t h e contingent d e r i v a t i v e of F a t ( 6 , ) . In o t h e r words, 2 E L F

(6,;)

( v ) if and only if

(U , z ) E TgTaphF(f,;).

D F ( 6 , Z ) is a positively homogeneous set-valued map with closed g r a p h . Due t o Definition 2.1, z E D F ( 6 , z ^ ) ( v ) if and only if t h e r e e x i s t sequences

ihkj c d + , l v k j c V a n d i z k j c Z such t h a t hk + 0 + , v k ^-t

v . z k + z

^and z^ + h k z k E F ( G + h k v k ) f o r V k

.

Now w e consider a nonempty pointed closed convex cone P in Z . This cone P

t

induces a p a r t i a l o r d e r on Z . W e use t h e following notations: F o r z ,z ' E P

Y

SPY'

iff y'-y E P (2.3) y s p y 1 iff y O - y E P \

toj .

^(2.4)

W e consider t h e set-valued map F

+

P from V t o Z defined by ( F + P ) ( v ) = F ( v ) + P f o r V v E V

.

The g r a p h of F

+

P is often called t h e P-epigraph of F (Sawaragi et al. [6], p. 23).

The following r e s u l t , which shows a relationship between t h e contingent d e r i v a t i v e s of F

+

P and F , is useful.

Proposition 2.1. Let

( f

);, belong t o graphF. Then

~ ( 6 , ; )

( v )

+

P C D ( F

+

P )

( f

, % ) ( v ) f o r V V E V

.

(2.5) (Proof). Let z

~ m ( f

,%) ( v ) and ci E P . Then t h e r e e x i s t sequences

t h k

1

^{C R+1 *} l v k j c v a n d i r k ] C Z s u c h t h a t h , + O + , v k + v , z k

+ z

and

% + h k z k E F ( ~ + h k v k ) f o r v k

.

~ e t

zk ⁼

^{z k}

⁺

ci f o r a l l k . Then

zk

^{+ z}

⁺

^{d and}

% + h k z k = f + h k z k + h k c i ~ ~ ( G + h k v k ) + ~ f o r V k ,

Hence

z +

ci E D ( F

+

P )

(6

,%) ( v ) and t h e proof is complete.

The c o n v e r s e inclusion relation of t h i s proposition D ( F

+

^{P )}

^(6,s)

( v ) C f f ( G , z ^ ) ( v )

+

P

d o e s not generally hold. (See Proposition 3.1 and Examples 3.3 and 3.4).

A c o n e P is s a i d t o b e p o i n t e d if P n ( - P )

=

I O j .

Since we d e a l with multiobjective optimization, we must i n t r o d u c e t h e c o n c e p t of minimal points with r e s p e c t t o t h e c o n e P .

Depinition 2.3. Given a s u b s e t S of 2 , a point z^ E S is s a i d t o b e a P-minimal point of S if t h e r e e x i s t s no z E S s u c h t h a t z S p 2 . W e d e n o t e t h e set of a l l P- minimal points of S by MinpS, i.e.

Mi* E

IP

E S

!

t h e r e e x i s t s no z E S s u c h t h a t z S

p 2

j (2.6)

The following t h e o r e m i s fundamental.

m e o r e m 2.1. Let

( f

^);, belong t o graphF a n d s u p p o s e t h a t Z i s f i n i t e dimen- sional. Then, f o r a n y v E V.

MinpD(F

+

P )

( f

);, ( v )

c

_DF

( f

);, ( v )

( P r o o f ) Let z E M i n p D ( F + P ) ( f , g ) ( v ) . S i n c e z E D ( F + P ) ( f , ; ) ( v ) t h e r e e x i s t s e q u e n c e s ihk j

c d,.

^{i v k j}

^c

^{V a n d l z k j C Z s u c h t h a t}

hk ⁺ O+, v k + V , z k ⁺ z a n d

z^

+ h k z k E F ( ~ + h k v k ) + P f o r v k

.

T h e r e a l s o e x i s t s a s e q u e n c e i d k j C P s u c h t h a t

2 +

h k z k - d k E F ( ~

+

h k v k ) f o r v k

.

d k

W e s h a l l p r o v e t h a t

-

^-, 0 . I f t h i s w e r e not t h e case, t h e n f o r some E

>

0, we c a n

h k

c h o o s e a s u b s e q u e n c e of t h e n a t u r a l numbers Ilk j satisfying

I

dLk

6 >

^{E f o r V k}

^.

hlk

-

Taking a n d renumbering t h i s s u b s e q u e n c e , w e may assume from t h e f i r s t t h a t

~ h k

I d k '

? & f o r a l l k . S e t

zk ^=-

d k E P . Then

Zk

^{S p d k a n d}

h k

- B

dkl

I z k

!

S i n c e

- =

^E f o r a l l k , w e may assume without l o s s of g e n e r a l i t y t h a t t h e se-

h k

q u e n c e

Zk

c o n v e r g e s t o some v e c t o r d E 2 . S i n c e P i s closed, d E P a n d

I d

1 =

E

>

^0. Thus, zk-

Zk

-, z

-

^d and hence z

-

^{ti E D ( F}

⁺

^{P ) (b ,z^) (v ). How-}

hk

e v e r , this c o n t r a d i c t s t h e assumption z E Minp D ( F

+

P ) ( 6 ) ,; ( v ), since

z

-

^{t i}

^s

z . T h e r e f o r e w e c a n conclude t h a t

-

tik ^{-, 0.} This implies t h a t

hk

and

T h e r e f o r e z E D F ( 6 , ; ) ( v ) and this completes t h e proof of t h e theorem.

The c o n v e r s e inclusion of this theorem is not valid generally. (See Example 3.2.)

3. CONTINGENT DERIVATIYE OF THE PERTURBATION

M A P

In this section we consider a family of parametrized multiobjective optimiza- tion problems. Let Y b e a set-valued map from U t o

RP,

where U is t h e Banach s p a c e of a p e r t u r b a t i o n p a r a m e t e r v e c t o r ,

RP

i s t h e objective s p a c e and Y i s con- s i d e r e d as t h e feasible set map. Let P b e a nonempty pointed closed convex cone in

RP.

In t h e optimization problem corresponding t o e a c h p a r a m e t e r value u , w e aim t o find t h e set of P-minimal points of t h e feasible set Y(u ). Hence w e define a n o t h e r set-valued map W from U t o

RP

by

W(u )

=

MinpY(u ) f o r Vu E U (3.1)

and call i t t h e p e r t u r b a t i o n map. The p u r p o s e of t h i s section i s t o investigate re- lationships between t h e contingent derivative of W and t h a t of Y. H e r e a f t e r in t h i s p a p e r , w e fix a nominal value of u as u^ and consider a point y^ E W(u^).

In view of Theorem 2.1, w e have t h e following relationship:

MnpD(W + P ) ( 6 , c ) ( u ) c ~ ~ ( 6 . c ) ( u ) f o r V u E U

.

^(3.2)

Depinition 3.1. We say t h a t Y i s P-minicomplete n e a r

6

^if

Y ( u )

c

W(u) + P f o r Vu E N

where N i s some neighborhood of G .

Since W(u )

c

Y(u ), t h e P-minicompleteness of Y n e a r G implies t h a t

W ( u ) . + P = Y ( u ) + P f o r Vu E N

.

(3.4) Hence, if Y i s P-minicomplete n e a r 6 , t h e n D(Y

+

P ) ( 6 , y ^ )

=

D(W

+

P ) ( f , y ^ ) f o r a l l y^ E W(G ). Thus w e o b t a i n t h e following t h e o r e m from (3.2j.

m e o r e m 3.1. If Y i s P-minicomplete n e a r 6 , t h e n

MinpD(Y + P )

(G,y^)

( u ) C D W ( ~ , < ) ( u ) f o r Vu E U

.

(3.5) The following example i l l u s t r a t e s t h a t t h e P-minicompleteness of Y is e s s e n t i a l f o r t h e above theorem.

E z a m p l e 3.1. (Y i s not P-minicomplete n e a r 6 ) . L e t

U =

R , p

=

1, P

= ^{R +}

a n d Y b e defined by

Then

f O j if u = O W(u)

= #

i f u $ 0 ^'

Let

6 =

0. Then

~ ( ~ + ~ ) ( - ; i , y ^ ) ( u ) = ~ ~ ( - ; i , < ) ( u ) = ! y I ~ > = I u I { f o r vu E R

On t h e o t h e r hand

1 0 1 i f u = O

~ ( - ; i . y ^ ) ( u ) =

#

i f U + O *

Hence

MinpD(Y

+

P ) ( 7 2 , ~ ) ( u )

k

D W ( C , ~ ^ ) ( u ) f o r u # O

.

The c o n v e r s e inclusion of t h e theorem does n o t generally hold as i s shown in t h e following example.

Ezample 3.2. Let U

=

R , p

=

² and Y b e defined by

Let P

=

R : , C

=

0 and

=

(0,O). Then W ( u )

=

Y(u) f o r e v e r y u and

and

In o r d e r t o obtain a relationship between DW and DY, w e shall introduce t h e following p r o p e r t y of Y.

Definition 3.2. (Aubin and Ekeland [2]) Y i s said t o b e u p p e r locally Lipschitz at u^ if t h e r e e x i s t a neighborhood

N

of

C

and a positive constant M such t h a t

Y ( ~ ) C Y ( C ) + M ! U

- C I B

^{f o r w u}

^{E N}

^(3.6)

Remark 3.1. If Y i s u p p e r locally Lipschitz at

C ,

t h e n i t i s u p p e r semicontinu- ous at

6 ,

i.e., f o r any &

>

^0, t h e r e e x i s t s a positive number

6

such t h a t

Y ( u ) c Y ( C ) + & B f o r V U , ! U A

116 .

Depinition 3.3. Let S b e a set in RP and P b e a nonempty closed convex cone in R P . A point

<

^{E S} i s said t o b e a p r o p e r l y P-minimal point of S if

Of c o u r s e , e v e r y p r o p e r l y P-minimal point of S i s P-minimal, since

Proposition 3.1. If y^ is a p r o p e r l y P-minimal point of Y ( 6 ) and if Y is u p p e r locally Lipschitz at f , t h e n

D(Y+P)(f,y^)(u)=DY(f,y^)(u)+P

f o r V u E U

.

(3.8) ( P r o o f ) In view of Proposition 2.1,

D Y ( f , y ^ ) ( u ) + P C D ( Y + P ) ( f , y ^ ) ( u ) for V u E U

.

Hence w e p r o v e t h e c o n v e r s e inclusion. Let y E D ( Y

+

P )

(6

,y^ ) ( u ). From t h e definition t h e r e e x i s t sequences lhk j C

d,,

^{f u k}

1 c

U and f y k

1 c

^{R p} such t h a t hk + O + , uk + u , y k + y .and

5

+ h k y E Y ( C + h k u k ) + P f o r V k

.

T h e r e f o r e t h e r e e x i s t s a sequence f d k

1

^{E P} such t h a t

37

+

h k y k - d k ~ y ( f + h k u k ) f o r V k

,

t i

y ^ + h k ( y k - - ) c Y ( f + h k u k ) f o r V k

.

h k

d k

Suppose t h a t t h e sequence

1-1

^{h a s a} convergent subsequence. In t h i s c a s e , w e

h k

may assume without loss of generality t h a t

-

a ⁺ d f o r some ( 1 . Since P is a closed

h k

d k

s e t , d E P . Moreover, t h e convergence y k

--

^{+ y}

-

^{d implies t h a t}

h k

y

-

^{d E} DY(c

,g

) ( u ), namely t h a t y E D Y ( 6 ,

<

^{) ( u )}

⁺

^{P. Hence w e} have t h e con- clusion of t h e proposition. T h e r e f o r e i t completes t h e proof of t h e proposition t o

d k

show t h a t

1-1

necessarily h a s a convergent subsequence. If t h i s were not t h e

h k

Idkl

c a s e , then

-

⁺

+

^a. Since Y i s u p p e r locally Lipschitz at

6 ,

t h e r e e x i s t a

h k

neighborhood N of f and a positive number M satisfying (3.6). Since f

+

h k u k +

6 ,

C

+

hk u E N f o r a l l k sufficiently l a r g e . Hence t h e r e e x i s t s a sequence f y ^ k

1

ⁱⁿ Y ( f ) such t h a t

f o r a l l k sufficiently l a r g e . S i n c e u k ^-+ u , t h e right-hand s i d e of t h e a b o v e ine- quality c o n v e r g e s t o

MI

u

I.

T h e r e f o r e , t h e s e q u e n c e

f-6

1

-

^{y k )}

+

y k

- -

h k

d k

I

h k

1

dkl

i s bounded. S i n c e

-

^-,

+

^00, t h e s e q u e n c e

h k

c o n v e r g e s t o t h e z e r o v e c t o r in

RP.

S i n c e y k -, y , t h e second t e r m c o n v e r g e s t o d k

t h e z e r o v e c t o r . We may assume t h a t

-

^-, f o r some

2

^{E P with}

[zl =

1. Hence

Bdkl -k

-, d . However, t h i s implies t h a t d E [ d

y

a ( Y ( O )

-

y^)] n ( - P ) , which

l d k ! a>O

c o n t r a d i c t s t h e assumption of t h e p r o p e r P-minimality of y^. This completes t h e

proof of t h e proposition. I

CoroLLary 3.1. If y^ i s a p r o p e r l y P-minimal point of Y ( f ) a n d if Y i s u p p e r lo- cally Lipschitz at 6 , th e n

M i n p ~ y ( f , y ^ ) ( u )

=

MinpD(Y

+

P ) ( 6 , y ^ ) ( u ) f o r Vu E

U .

^(3.9)

( P r o o f ) In view of P r o p o s i t i o n 3.1, by using P r o p o s i t i o n 3.1.2 of Sawaragi et al. [6], we c a n p r o v e t h a t

MinpDY(O,y^) ( u ) = M z n p ( M ( G , y ^ ) ( u ) + P ) =MinpD(Y + P ) (^u,y^) ( u )

.

By combining Theorem 2 . 1 a n d Corollary 3.1, w e h a v e t h e following t h e o r e m . h e o r e m 3.2. If Y i s P-minicomplete n e a r

f

a n d u p p e r locally Lipschitz at

6 ,

a n d if y^ i s a p r o p e r l y P-minimal point of Y ( f ), t h e n

MinpDY(6,y^) ( u ) c D W ( f , c ) ( u ) f o r V u E U

.

Example 3.1 shows t h a t t h e minicompleteness of Y i s e s s e n t i a l f o r t h e a b o v e theorem. The following two examples i l l u s t r a t e t h e importance of t h e o t h e r two conditions in Theorem 3.1, namely t h e Lipschitz p r o p e r t y of Y a n d t h e p r o p e r minimality of y^

.

Example 3.3. (Y i s not u p p e r locally Lipschitz at

ii).

Let

U =

R , p = 1 , P

=

R + a n d Y b e d e f i n e d by

Then

Let

=

0 and

< ⁼

^{0 . Then}

10j if u # O MinpDY(O.0) ( u )

=

_{(b if u = O}

Hence

lo] =

M i n p M ( O , O ) (u )

k

DW(O.0) ( u )

= 4

f o r u

>

0

.

Example 3.4. (y^ i s not p r o p e r l y P-minimal). Let

U =

R, p

=

2 , P

=

R: and Y b e defined by

~ ( u ) = l y l ~ ~ + y ~ = O , Y ~ j U j u ~ y l y l + ~ 2 + l = 0 . ~ l > O j

.

Then

Let

C =

0 and y^

=

(0,O). Then

DW(C ,y^)(u)

= IY

^I Y + y 2

=

0 , Y

<=

^{min(0,u ) j}

.

Hence

(1, -1) % DW(C , c ) ( l ) , while (1, -1) E

M ~ ~ ~ D Y ( C

, < ) ( l )

.

4. SENSITIVITY ANALYSIS

IET

GENERAL MULTIOBJECTJYE OPTIMIZATION In t h i s s e c t i o n w e d e a l with a g e n e r a l multiobjective optimization problem in which t h e f e a s i b l e s e t Y ( u ) is given by t h e composition of t h e f e a s i b l e decision set X ( u ) and t h e o b j e c t i v e function J'

( z

, u ). Namely, l e t

X

b e a set-valued map f r o m

R m

t o

R n

, J' b e a continuously d i f f e r e n t i a b l e function from

R n

x Rm i n t o

RP

a n d Y b e defined by

Y ( u ) = J ' ( X ( u ) , u ) = t y

l y

= J ' ( z , u ) , z ~ X ( u ) j f o r V U € R m

.

^(4.1)

F i r s t , w e investigate a r e l a t i o n s h i p between t h e contingent d e r i v a t i v e s of X and Y.

L e t 6 € R m , z^ E X ( G ) a n d y ^ = J ' ( z ^ , C ) c Y ( 6 ) . R o p o s i t i o n 4.1. F o r a n y u E

R m .

where V,p (2 , G ) ( o r V,J ( 2 , s )) i s t h e p x n ( o r p x m ) matrix whose (i

,

j ) com-

a ~ ' ,

( z ^ , G

a ~ ' ,

^{(2 ,C}

1

ponent i s ( o r ). Moreover, l e t

If

f

i s u p p e r locally Lipschitz at ( C ,y^) a n d T ( C , y ^ )

= lz^

1, t h e n t h e c o n v e r s e inclu- sion of (4.2) i s a l s o valid, i.e.,

V , J ' ( z ^ , C ) . D Y ( G , ? ) ( u ) + V U J ' ( z ^ , C ) . u = D Y ( C , y ^ ) ( u ) f o r V u € R m ( 4 . 4 ) ( P r o o f ) . F i r s t w e p r o v e (4.2). Let

z

^{E DX(G ,z^)} ( u ) ; Then t h e r e e x i s t s e q u e n c e s

lhk ( c

k+,luk I

^c

^{R m}

^{a n d lzk ( c}

^{R n}

s u c h t h a t hk ⁺ O+, u k ^-+ u . z k -, z and 2 + h k z k EX(^ + h k ) f o r V k

.

Then

f ( 2

+

h k z k , f

+

h k u k ) E Y ( f

+

h k u k ) f o r V k

f ( 2

+

h k Z k , f

+

h k u k )

-

f ( f , f )

$

+ h k E Y ( C + h k u k ) f o r V k

.

h k

S i n c e hk + 0+, u k -, u a n d z k -, z ,

Hence

V,f(z^.G)

.

z

+

V u f ( 2 , . i i ) . u E

m(.ii

,y') ( u )

.

Thus (4.2) h a s been e s t a b l i s h e d . Next w e p r o v e (4.4). Let y € DY(% , y ' ) ( u ) along with s e q u e n c e s !Ak

I

^C

d,.

^{luk C}

^{R m}

^{a n d}

l y k

^C

RP

s u c h t h a t hk +O+, u -, u , y ⁺ y a n d

y' +

hk y ^E ~ (

+

6^hku ). Then t h e r e e x i s t s anoth- er s e q u e n c e lzk j C

R~

s u c h t h a t

S i n c e

2

i s u p p e r locally Lipszhitz at ( 6 , y^ ) a n d f ( 6 , y' )

=

[G (, t h e r e e x i s t s a posi- t i v e number M s u c h t h a t

h +

^{h k z k}

-z^! 5

^{~ l ( 6}

+

h k u k , y '

+

h k y k )

-

( 6 , c ) I

f o r all k sufficiently l a r g e . S i n c e t h e right-hand s i d e of t h e a b o v e inequality con- v e r g e s t o Ml(u ,y )( as k ^-,

-,

we may assume without l o s s of g e n e r a l i t y t h a t z k con- v e r g e s t o some z

.

Then c l e a r l y z € D X ( ~ , Z ) ( u ). Moreover,

f ( z ^

+

h k z k , c

+

h k u k ) - f ( z . e ) y

=

lim y k

=

lim

k ^+- k +- h k

T h e r e f o r e

Y E V , J ( Z 1 , & ) . L X ( U ^ , Z ) ( u )

+

V , j ( z ^ , & )

- u

and t h e proof of t h e proposition is completed. ^I

The following two examples show t h a t t h e additional conditions in Proposition 4.1 are essential f o r (4.4).

E z a m p l e 4.1.

( f ( 6

,y^ )

+ iz^ 1 ) .

Let

X ( u ) = i z

E R I O <=z

s m a x ( l . 1 + u ) j f o r u E R ,

Hence, by taking hk

= -

1 u k

=

1 and y k

=

1 , w e can p r o v e t h a t k '

On t h e o t h e r hand, D X ( 6 , C ) ( 1 )

=

R+, V , f ( z ^ , c )

=

-1 and V , f ( z ^ , 6 )

=

0 . There- f o r e

and ( 4 . 4 ) i s not t r u e .

E z a m p l e 4.2.

(2

i s not u p p e r locally Lipschitz at ( 6 , y^ )). Replace X ( u ) by

in Example 4.1. In t h i s case

X ( 6

,y^ )

=

IOj, but

2

i s not u p p e r locally Lipschitz at

(6

, y^ ). W e can analogously p r o v e t h a t

1 E DY(6 , y ^ ) ( l ) but

E z a m p l e 4.3.

(2

i s not u p p e r locally Lipschitz at

(6

_,y^)). Let X ( u )

=

[ 0 , 1 ] c R f o r e v e r y u E R , j ( z , u )

= z 2 , f

= 0 ,

z^

= O and

y^

= O . Then Y ( u )

=

[ O , l ] and

A ' ( ~ , ~ ^ ) ( u ) = R + f o r V u E R

.

However, V , f ( z ^ , u ^ ) DX(C , S ) ( u )

+

V , f ( z ^ , i i ) . u

= to].

^{In t h i s case}

R(u

, y )

=

d y f o r y

2

0 and any u , which i s not u p p e r locally Lipschitz at (0,O).

Finally w e should note sufficient conditions f o r t h e Lipschitz continuity of Y at

c .

Lemma 4.1. If

X

is u p p e r locally Lipschitz at

6

and if X ( 6 ) i s bounded, t h e n Y i s u p p e r locally Lipschitz at

6 .

(Proof). Since X is u p p e r locally Lipschitz at 6 , t h e r e e x i s t some neighbor- hood N of

6

and a positive number M l such t h a t

X ( U ) C X ( ~ )

+ M 1 I u - - u ^ b

f o r V u E N Since f is continuously differentiable,

For any u E N and y E Y ( u ), t h e r e e x i s t s

z

^E .Y(u ) such t h a t f

( z

,u )

=

y

.

^Then

t h e r e e x i s t s

f

E X ( 6 ) such t h a t

kc z'! <= M l h ^1? 1.

Hence

Putting M

=

( 1

+

M 1 ) M 2 , w e h a v e

Namely Y is u p p e r locally Lipschitz at

6 .

This completes t h e proof of t h e

lemma. I

The following example shows t h a t t h e condition of t h e boundedness of X ( 6 ) i s essential in Lemma 4.1.

Ezample 4.4. ( X ( 6 ) i s not bounded). Let

X ( u ) = l z E R ~ Z ~ = U ] f o r u E R and f

( z

,u )

= z l z 2 .

Then

Y ( u ) = t y E R ! ~

= u z 2 j

for u E R

.

Clearly Y is not u p p e r locally Lipschitz at

6 =

0 .

Finally, w e h a v e t h e following theorem. Note t h a t Y is P-minicomplete n e a r f if X ( u ) is compact f o r e a c h u n e a r f

.

7'heotem 4.1. Assume t h e following conditions:

(i)

X

i s u p p e r locally Lipschitz at

c

^;

(ii) X is compact f o r e a c h u n e a r

6

;

(iii) y^ i s a p r o p e r l y P-minimal point of Y ( c ) ; (iv) f ( C ,yl)

= IS{;

(v)

2

i s u p p e r locally Lipschitz at ( f ,$).

Then, f o r a n y u E R m ,

5. SENSITIVITY ANALYSIS IN MULTIOBJECTIVE PROGRAIBUNG

In t h i s s e c t i o n w e a p p l y t h e r e s u l t s obtained in t h e p r e c e d i n g s e c t i o n t o a usu- a l multiobjective programming problem:

P-minimize 'j( z )

=

U 1 ( z ) ,...,fp ( 2 ) ) (5.1) s u b j e c t t o g ( z )

=

( g l ( z )

,...,

g m ( z ) )

<=

^{0 , z E R n}

a n d d i s c u s s t h e sensitivity in connection with t h e L a g r a n g e multipliers. Recall t h a t in usual nonlinear programming, t h e sensitivity of t h e p e r t u r b a t i o n function with r e s p e c t t o t h e p a r a m e t e r o n t h e right-hand s i d e of e a c h inequality c o n s t r a i n t i s given by -Aj(j

=

1 ,

...,

m ) , where A j i s t h e c o r r e s p o n d i n g L a g r a n g e multiplier. Our final r e s u l t will b e a n extension of t h i s f a c t . Throughout t h i s section, e a c h func- tion

pi

(i

=

I, ...,p) a n d g j ( j

=

I,

...,

m ) i s assumed t o b e continuously d i f f e r e n t i a b l e .

L e t X b e t h e set-valued map from R m t o R n defined by

Hence, in t h i s c a s e , t h e f e a s i b l e set-map Y from R m t o RP i s defined by

Of c o u r s e , t h e nominal value of t h e p a r a m e t e r v e c t o r u is 0 in R m . Take a point

2

E X ( 0 ) and d e n o t e t h e index set of t h e a c t i v e c o n s t r a i n t s at

2

by J ( Z ) , i.e.

F i r s t , we c o n s i d e r t h e contingent d e r i v a t i v e of t h e set-valued map X . Lemma 5.1. The contingent d e r i v a t i v e of X at ( 0 , s ) i s given as follows:

D X ( O , Z ) ( u )

=

t x I ^!

<

V g j ( 2 ) , x

>

< = u j f o r V j ^E J ( 2 ) { ( 5 . 5 ) w h e r e

<

^{a ,}

>

d e n o t e s t h e i n n e r p r o d u c t in t h e Euclidean s p a c e .

( P r o o f ) Note t h a t

i s specified by m inequality c o n s t r a i n t s . The g r a d i e n t v e c t o r of t h e j t h c o n s t r a i n t at ( 0 . 2 ) with r e s p e c t t o ( u .x ) i s ( - e l . Vgj

( 2 ) ) .

w h e r e e f i s t h e j t h b a s i c unit vec- t o r in Rm

.

^{i.e. e l}

⁼

^{0 if k}

⁺

^{j a n d e l}

⁼

^1. Hence t h e s e g r a d i e n t v e c t o r s are l i n e a r l y independent a n d s o t h e t a n g e n t c o n e t o g r a p h X i s given b y

= ~ ( u , x ) ! < V g j ( 8 ) , x

>

<uj f o r j E

J ( B ) {

This completes t h e proof of t h e lemma. ^I

In t h i s case X ( u ) i s a c l o s e d set f o r e v e r y u , s i n c e g is continuous. The n e x t lemma p r o v i d e s s u f f i c i e n t conditions f o r t h e Lipschitz continuity of X a r o u n d u

= o .

Lemma 5.2. If t h e r e e x i s t s a v e c t o r

>

0 s u c h t h a t X ( c ) i s bounded, X ( 0 )

+

^(b a n d if t h e Cottle c o n s t r a i n t qualification i s s a t i s f i e d at e v e r y

5

^E X(O), i.e.,

A j V g j ( z ) = 0 and X j

2 0

^{f o r j E J ( z )}

f

imply t h a t X j

=

0 f o r V j E J ( z )

,

t h e n X i s compact-valued a n d Lipschitz in a neighborhood of

6 =

0.

( P r o o f ) This lemma i s d u e t o R o c k a f e l l a r [ 5 ] (combine Theorem 2.1 a n d Corol- l a r y 3.3 in [ 5 ] ) .

Of c o u r s e , if X i s Lipschitz in a neighborhood of

6 ,

t h e n it is u p p e r locally Lipschitz at

c .

Analogously w e h a v e t h e following lemma concerning t h e set-valued map

Lemma 5.3. If

2

i s locally bounded a r o u n d (O,y^), 2(0,y^)

+

^$ a n d t h e Mangasarian-Fromovitz c o n s t r a i n t qualification i s s a t i s f i e d at e v e r y

9

E f ( 0 , 3 ) , i.e.

f & v f , ( i )

+

XjVgj(9)=0 a n d

j =I J

X j

2

0 f o r j E J ( 9 ) imply t h a t

& = O f o r all i =1, . . . , p a n d X j = O f o r all j ^E J ( 3 ) , (5.8) t h e n

f

i s compact-valued a n d Lipschitz in a n e i g h b o r h o o d of (O,y^).

W e will p r o c e e d with t h e discussion u n d e r t h e following assumptions:

Assumption 5.1.

(i) T h e r e e x i s t s

>

0 s u c h t h a t X ( u ) i s bounded.

(ii) The Cottle c o n s t r a i n t qualification (5.6) i s s a t i s f i e d at e a c h ^E X(0).

(iii) Z ( O , < )

=

Iz I f ( z )

=

<, g ( z )

S O ] = 13j.

(iv) The Mangasarian-Fromovitz c o n s t r a i n t qualification (5.8) i s s a t i s f i e d at

9 .

In addition to Assumption 5.1, w e also assume t h a t y^ i s a p r o p e r l y P-minimal point of ~ ( 0 ) ~ . Then w e c a n a p p l y Theorem 4.1 t o o b t a i n t h e r e l a t i o n s h i p

MinpVf ( 9 ) DX(O,Z)(u)

c

DW(O,y^)(u) f o r V u E Rm

.

(5.9) In view of (5.5)

Vf(9) .DX(O,Z)(u)

= ty l y t = <

Vf,(f),z

>

^{f o r i}

⁼

¹

^,...,

^{p ;}

< V g j ( z ) , z > s u j f o r V j € J ( i ) j

.

Hence t h e left-hand s i d e of (5.9) c o n s i s t s of all t h e P-minimal v a l u e s of t h e l i n e a r multiobjective programming problem:

P -minimize

<

V f I ( 3 ) J ^{> s}

I

ⁱ

⁼

^1, . . . , p

s u b j e c t to

<

Vgj(9),z

>

u j , j E J ( 2 )

.

'ln t h i s c a s e w e c a l l

3

a properly P - m i n i m a l s o l u t i o n t o t h e problem (5.1).

The necessary and sufficient P-minimality conditions f o r t h e above problems are t h a t t h e r e e x i s t a multiplier v e c t o r ( p , A ) E RP X R m such t h a t

f:

& v f i ( l ^ ) f

z

A j V g j ( 5 ) = 0 ( 5 . 1 0 )

i = I j

p € i n t ~ + = ~ v € ~ ~ I < v , d

> > o

f o r ~d # O E P ] ( 5 . 1 1 ) Aj

2

0 f o r j E J ( % ) ( 5 . 1 2 ) A j ( < Vgj (%), z

> -

^{u j )}

=

0 f o r j E J ( % )

.

( 5 . 1 3 ) Since % is a p r o p e r l y P-minimal solution t o t h e problem ( 5 . 1 ) , t h e r e e x i s t s a vec- t o r ( p , A) E R m X RP satisfying ( 5 . 1 0 )

-

( 5 . 1 2 ) . Hence, if z E R n satisfies

I ^<

V g j ( % ) . z

> 5

^{u j f o r Vj E J ( % )} such t h a t Aj

=

0

<

V g j ( % ) , z

> =

u j f o r

bj

^E J ( % ) such t h a t Aj

>

0 , ( 5 . 1 4 )

then Vf

(2) .

z E F d i n p D Y ( O , c ) ( u ) . Moreover

Thus we have proved t h e following theorem.

Theorem 5.1. Suppose t h a t 5 is a p r o p e r l y P-minimal solution t o t h e multiob- jective programming problem ( 5 . 1 ) and Assumption 5 . 1 is satisfied. Let @ , A ) b e t h e corresponding multiplier v e c t o r . Then, f o r e a c h z E R n satisfying ( 5 . 1 4 ) ,

Moreover,

6 . CONCLUSION

In this p a p e r w e have studied sensitivity analysis in multiobjective optimiza- tion. The essential r e s u l t w e have proved is t h a t e v e r y cone minimal v e c t o r of t h e contingent derivative of t h e feasible set map in a direction i s a l s o t h e element of t h e contingent derivative of t h e p e r t u r b a t i o n map in t h a t direction under some

conditions (Theorem 3.2). W e have also obtained t h e relationship between t h e con- tingent derivative of t h e p e r t u r b a t i o n map and t h e Lagrange multipliers f o r mul- tiobjective programming problems (Theorem 5.1).

However, t h e r e remain s e v e r a l open problems which should b e solved in t h e f u t u r e . Some of them a r e t h e following. F i r s t , t h e contingent derivative of t h e p e r t u r b a t i o n map is not completely c h a r a c t e r i z e d . In o t h e r words, sufficient con- ditions f o r t h e c o n v e r s e inclusion of Theorem 3.2 have not been obtained yet.

Secondly, t h e Lipschitz continuity of t h e p e r t u r b a t i o n map is not studied h e r e . Thirdly, some more refined r e s u l t s may b e obtained in t h e c a s e of multiobjective programming. Finally, w e should clarify e f f e c t s of t h e convexity o r linearity as- sumption.

REFERENCES

[I] J.P. Aubin, "Contingent derivatives of set-valued maps and existence of solu- tions t o nonlinear inclusions and differential inclusions", in Advances i n Mathematics S u p p l e m e n t a r y S t u d i e s , L. Nachbin (ed.), Academic P r e s s , N e w York, pp. 160-232 (1981).

[2] J.P. Aubin and I. Ekeland, Applied Nonlinear A n a l y s i s , Wiley, N e w York (1984).

[3] A.V. Fiacco, I n t r o d u c t i o n to S e n s i t i v i t y a n d S t a b i l i t y A n a l y s i s i n Non- l i n e a r Programming, Academic P r e s s , N e w York (1983).

[4] R.T. Rockafellar, ' Z a g r a n g e multipliers and subderivatives of optimal value functions in nonlinear programming", Mathematical Programming S t u d y 1 7 , pp. 28-66 (1982).

[5] R.T. Rockafellar, 'Zipschitzian p r o p e r t i e s of multifunctions", Nonlinear A n a l y s i s , TMA, Vol. 9 , No. 8 , pp. 867-885 (1985).

[6] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective m t i m i z a - t i o n , Academic P r e s s , N e w York (1985).

[7] .T. Tanino and Y. Sawaragi, "Stability of nondominated solutions in multicri- t e r i a decision-making", Journal of @ t i m i z a t i o n 77zeory a n d A p p l i c a t i o n s , Vol. 30, No. 2, pp. 229-253 (1980).