Epsilon Solutions and Duality in Vector Optimization

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NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

EPSILON SOLUTIONS AND DUALITY IN VECTOR OPTIMIZATION

I s t v d n VdLyi

May 1987 WP-87-43

Working 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 a n d 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 its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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PREFACE

This p a p e r is a continuation of t h e a u t h o r ' s p r e v i o u s investigations in t h e t h e o r y of epsilon-solutions in convex v e c t o r optimization a n d s e r v e s as a t h e o r e t i c a l background f o r t h e r e s e a r c h of SDS in t h e field of m u l t i c r i t e r i a optimization. With t h e stress laid on duality t h e o r y , t h e r e s u l t s p r e s e n t e d h e r e g i v e some insight into t h e problems a r i s i n g when e x a c t solutions h a v e t o b e s u b s t i t u t e d by a p p r o x i m a t e ones. J u s t like in t h e s c a l a r c a s e , t h e available computational techniques f r e - quently l e a d to s u c h a situation in m u l t i c r i t e r i a optimization.

Alexander B. Kurzhanski Chairman

Systems and Decision S c i e n c e s Area

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CONTENTS

1. Introduction

2. Epsilon Optimal Elements 3 . P e r t u r b a t i o n Map a n d Duality

4 . Conical S u p p o r t s 5. Conclusion

6. R e f e r e n c e s

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EPSILON SOLUTIONS AND

DUALITY

IN VECTOR OF'TIMIZATION Istv&n V&lyi

The study of epsilon solutions in v e c t o r optimization problems w a s s t a r t e d in 1979 by S. S. Kutateladze [I]. These t y p e s of solutions a r e i n t e r e s t i n g b e c a u s e of t h e i r relation t o nondifferentiable optimization a n d t h e v e c t o r valued e x t e n s i o n s of Ekeland's v a r i a t i o n a l p r i n c i p l e as c o n s i d e r e d by P. Loridan [2] a n d I . Vdlyi [3], b u t computational a s p e c t s a r e p e r h a p s e v e n more important. In p r a c t i c a l s i t u a t i o n s , namely, we o f t e n s t o p t h e calculations at values t h a t we c o n s i d e r sufficiently close to t h e optimal solution, or use algorithms t h a t r e s u l t i n some a p p r o x i m a t e s of t h e P a r e t o s e t . Such p r o c e d u r e s c a n r e s u l t in epsilon solutions t h a t a r e u n d e r s t u d y in this p a p e r . A p a p e r by D. J. White [4] d e a l s with t h i s i s s u e a n d i n v e s t i g a t e s how well t h e s e solutions a p p r o x i m a t e t h e e x a c t solutions.

Motivated by t h e above, i n t h e p r e s e n t p a p e r w e s t u d y t h e implications in duality t h e o r y of substituting e x a c t solutions with epsilon solutions. Although t h e well known r e s u l t s h a v e t h e i r c o u n t e r p a r t s , o u r findings show t h a t i n some c a s e s s p e - cial caution i s r e q u i r e d .

For t h e s a k e of simplicity i n formulation w e s h a l l r e s t r i c t o u r c o n s i d e r a t i o n t o finite dimensional s p a c e s , a l t h o u g h a l l t h e r e s u l t s h a v e a c o r r e s p o n d i n g v e r s i o n i n in- finite dimensions. Our m a j o r tool i s t h e s a d d l e point t h e o r e m f o r epsilon solutions and t h e techniques used in s t a n d a r d v e c t o r duality t h e o r y . For d e t a i l s s e e I. Vdlyi [5] and t h e book by Y. S a w a r a g i , H Nakayama a n d T. Tanino [6]. A s a c o n s e q u e n c e of t h e f a c t t h a t t h e notion of a p p r o x i m a t e solution c o i n c i d e s with t h a t of e x a c t solution in t h e c a s e when t h e approximation e r r o r i s z e r o , o u r r e s u l t s r e d u c e to those r e l a t e d t o e x a c t v e c t o r optimization. From a n o t h e r point of view t h e y a r e parallel t o t h e t h e o r y of epsilon solutions in t h e s c a l a r valued c a s e as expounded

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e . g , by J . J . S t r o d i o t , V. H. Nguyen a n d N. Heukernes [7], or in t h e v e c t o r i a l c a s e for a b s o l u t e o p t i m a l i t y by I. VAlyi

[a].

In t h i s p a p e r w e g i v e t h e p r o o f s of t h e r e s u l t s p r e s e n t e d a t t h e VII-th I n t e r n a t i o n - a l C o n f e r e n c e o n Multiple C r i t e r i a Decision Making, h e l d b e t w e e n 18-22 August, 1986, in Kyoto, J a p a n .

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- 3 - 2. EPSILON OPTIMAL ELEMENTS

In t h i s s e c t i o n w e r e c a l l s o m e b a s i c n o t i o n s a n d known f a c t s ( ~ ~ i t h o u t p r o o f s ) r e l a t - e d t o &-optimal s o l u t i o n s i n v e c t o r optimization. All t h e v e c t o r s p a c e s t h r o u g h o u t t h e p a p e r a r e r e a l , f i n i t e dimensional a n d o r d e r i n g c o n e s a r e s u p p o s e d t o b e c o n - v e x , p o i n t e d , c l o s e d a n d to h a v e a nonempty i n t e r i o r . X , Y a n d 2 d e n o t e v e c t o r s p a c e s while C a n d

K

a r e t h e p o s i t i v e c o n e s of Y a n d Z r e s p e c t i v e l y . The d u a l s p a c e of Y i s

I"

a n d t h e c o n e of p o s i t i v e f u n c t i o n a l s with r e s p e c t t o CCY, or t h e d u a l of C , i s C + a n d L+(Z,Y)CL(Z,Y) s t a n d s f o r t h e c o n e of p o s i t i v e l i n e a r maps f r o m Z to Y.

F o r t h e v a r i o u s o r d e r i n g r e l a t i o n s h i p s b e t w e e n t w o e I e m e n t s of a n o r d e r e d v e c t o r s p a c e we s h a l l u s e t h e following n o t a t i o n s , f o r e x a m p l e i n Y:

Y 2 2 Y 1 iff y 2 - y 1 E C Y z Z Y 1 i f f Y ~ - Y ~ E C \ t O j y ,

>

y 1 iff y ,

-

y , E i n t ( C )

a n d y z y l will r e f e r t o t h e f a c t t h a t y l € Y d o e s n o t d o m i n a t e y 2 E Y f r o m below.

Now f o r t h e r e a d e r s c o n v e n i e n c e we q u o t e t h e following d e f i n i t i o n f r o m e. g . D. T.

Luc [9].

Definition 2.1.

T h e set H c Y i s C - c o n v e x if H + C c Y i s c o n v e x .

T h e f u n c t i o n f :X+Y i s C - c o n v e x if t h e set

f

f (x _{) E Y}: x _Edomf j i s C - c o n v e x . The set H c Y i s C - c o m p a c t if t h e r e e x i s t s a b o u n d e d set HoCY with t h e p r o p e r t y H c H o +C a n d if H +C CY i s c l o s e d .

Now t u r n to t h e c o n s i d e r a t i o n of E-optimality. T h r o u g h o u t t h e p a p e r t h e v e c t o r s

E,E, ECCY will r e p r e s e n t t h e a p p r o x i m a t i o n e r r o r a n d t h e i r v a l u e will b e f i x e d . Definition 2.2.

T h e v e c t o r y € H i s a n &-minimal e l e m e n t of H c Y , i n n o t a t i o n y ^EE-min(H), if ( y - E - C ) n H

c

f y - ~ j ,

i t i s weakly &-minimal, i n n o t a t i o n y E E - - w m i n ( H ) , if ( y

-

^E- i n t ( C ) ) n H

c

f y

-

c j

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a n d p r o p e r l y &-minimal, i n n o t a t i o n y E E - p m i n ( H ) , if t h e r e e x i s t s a y * € i n t ( C t ) s u c h t h a t < y * , y - E > S < y * , h > V h E H .

The a p p r o x i m a t e l y maximal e l e m e n t s a r e t o b e d e f i n e d in a c o r r e s p o n d i n g m a n n e r

The following s t a t e m e n t s a r e e a s y c o n s e q u e n c e s of t h e d e f i n i t i o n s a n d c l a r i f y t h e r e l a t i o n s h i p s b e t w e e n t h e d i f f e r e n t n o t i o n s of minimal e l e m e n t .

P r o p o s i t i o n 2.1.

S u p p o s e t h a t Then we h a v e

P r o p o s i t i o n 2 . 2 .

& - p m i n ( H ) c e - m i n ( H ) c e w m i n (H) P r o p o s i t i o n 2 . 3 .

C o n s i d e r a s e q u e n c e t r , EC : n E N

1

d e c r e a s i n g to a E C . Then

c - m i n ( H )

c n I&,

- m i n ( H ) : n E N

c

e w m i n (H) a n d

Now t h e d e f i n i t i o n of t h e c o n v e x v e c t o r v a l u e d minimization p r o b l e m a n d t h e c o r r e s p o n d i n g v e c t o r v a l u e d L a g r a n g i a n follows. Then w e r e c a l l t h e r e l a t i o n s h i p s b e t w e e n &-solutions of t h e minimization p r o b l e m a n d e-saddle p o i n t s of t h e L,agran- g i a n .

Definition 2 . 3 . L e t

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be p r o p e r C-convex (and K-convex, r e s p e c t i v e l y ) functions with A = d o m f n d o m g

+ #.

We define t h e minimization problem (MP) a s follows

&-minimize

JO

( F ) (MP)

where F c X is t h e feasibility s e t of t h e problem ( h P ) defined by t h e equality

A s we a l r e a d y pointed i t o u t , t h e c a s e E=O r e p r e s e n t s t h e solutions in t h e usual ( e x a c t ) s e n s e .

The Lagrangian of t h e minimization problem (MP)

@ : A XL

+(z,Y)

-+ I', is defined by t h e equality

The element ( x o , R o ) E X X L (Z,Y) i s a n &-saddle point f o r t h e Lagrangian 9 if t h e following is met:

( a ) @(xo,Ro) E E - m i n [ 9 ( x , R o ) ^GI' : X E A j

(b) @(x o,Ro) E E-maa: [ @ (I o , R ) E Y : R E L ' ( 2 , ~ ) j

.

Definition 2.4.

We s a y t h a t t h e S l a t e r condition holds f o r t h e problem

(IMP)

if t h e r e e x i s t s a n xlEA with g ( x i ) € 0

.

Theorem 2.1.

The element ( z o , R o ) E X X L (Z,Y) i s a &-saddle point of t h e Lagrangian @, iff (a) (P(xo,Ro) E E-min [ @ ( x , R o ) E Y : a: E A{

(b) x o E F

( c ) - E

$

R o

.

!J ( z O )

5

0.

The p r o p e r t y s t a t e d in Theorem 2.1. i s as much n e g a t i v e as positive. Point (c), namely, t u r n s i n t o t h e well-known complementarity condition R o g (I,)

=

0 in t h e case of e x a c t saddle points. When E + O , i t only means t h a t

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w h e r e t h e r i g h t h a n d s i d e is a n unbounded s e t .

T h e o r e m 2.2.

S u p p o s e t h a t t h e p o i n t ( x o , R o ) ~ x L ( Z , Y ) i s a &-saddle p o i n t of t h e L a g r a n g i a n @.

Then xoGY i s a n (&-Ro.g ( x o ) ) -solution of t h e minimization p r o b l e m (PVIP).

F o r t h e a p p r o x i m a t i o n e r r o r e-Ro.g ( z o ) e C w e h a v e 0 5 &-Ro.g ( s o ) 2 2 . E as a c o n s e q u e n c e of t h e p o i n t ( c ) in T h e o r e m 2.1. However, unlike t h e s c a l a r i z e d c a s e , t r a n s i t i v i t y f o r t h e r e l a t i o n of non-domination d o e s n o t h o l d , a n d s o w e c a n n o t claim t h a t xo&Y is a ( 2 ~ ) - s o l u t i o n .

T h e o r e m 2.3.

S u p p o s e t h a t f o r t h e p r o b l e m (MP) t h e S l a t e r condition h o l d s . If z o a i s a p r o p e r

&-solution of t h e p r o b l e m (MP) t h e n t h e r e e x i s t a n o p e r a t o r R o E L + (Z,Y) s u c h t h a t ( x o , R o ) ~ A ~ L '(2,Y) i s a n &-saddle p o i n t of t h e L a g r a n g i a n Q.

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- 7 - 3. PERTURBATION

MAP

A M ) DUALITY

The p r o c e d u r e t h a t w e s h a l l follnw i s s t a n d a r d . We start by defining a p a r a m e t r i z e d family of p r o b l e m s ( t h e family of p e r t u r b e d p r o b l e m s ) t h a t includes o u r o r i g i n a l minimization problem. The primal map will t h e n b e defined as a function taking t h e optimal e l e m e n t s of t h e p e r t u r b e d p r o b l e m s a s v a l u e s , while for t h e dual map t h i s will h a p p e n via t h e Lagrangian. V e c t o r minimization p r o b l e m s usual- ly d o n o t h a v e unique s o l u t i o n s in t h e e x a c t c a s e a n d t h i s i s e v e n m o r e so now.

T h e r e f o r e t h e s e f u n c t i o n s will b e s e t valued maps.

We s h a l l not r e i t e r a t e t h e a n a l o g i e s to t h e known r e s u l t s in e x a c t v e c t o r optimization o r s c a l a r e-optimization b u t w e should l i k e t o c a l l f o r s p e c i a l a t t e n t i o n t o t h i s issue.

In t h i s s e c t i o n w e s h a l l assume t h a t

(i) f i s C-continuous a n d g i s K-continuous, (ii) A c Y i s c o m p a c t ,

(iii) int (C ^t,+q5.

We d e f i n e f o r e a c h u €2

and

A s i s w e l l known, u n d e r o u r assumptions, F ( u ) c X i s a c o n v e x s e t a n d Y ( u ) c Y i s C- convex. F u r t h e r m o r e in t h e c a s e when u =O, F ( u ) a n d Y ( u ) c o i n c i d e s with F a n d f ( F ) r e s p e c t i v e l y . Hence t h e following definition of t h e p e r t u r b e d p r o b l e m s (P,) r e a l l y means embedding ( W ) in a p a r a m e t r i z e d set of p r o b l e m s :

Definition 3.1.

We d e f i n e t h e p e r t u r b e d p r o b l e m s as follows

a n d we c a l l t h e s e t valued map defined by t h e e q u a l i t y

t h e primal ( o r &-primal) map.

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Now w e s t a t e t h e b a s i c p r o p e r t i e s of t h e p r i m a l mapping.

P r o p o s i t i o n 3 . 1 . The e q u a l i t y

h o l d s a n d s o ,

W,

i s a C-convex set v a l u e d C-convex map.

P r o o f .

The e q u a l i t y i s a c o n s e q u e n c e of P r o p o s i t i o n 5 . 2 . 1 . of Y. S a w a r a g i , I-I. Nakayama a n d T. Tanino [GI. From t h i s a n d t h e C - c o n v e x i t y of t h e mapping Y(.) t h e whole s t a t e m e n t follows.

P r o p o s i t i o n 3 . 2 .

S u p p o s e t h a t z 1 S f 2 a n d u

tsu2.

Then w e h a v e

a n d

i . e . t h e p r i m a l map i s monotonous i n E a n d C-monotonous i n t h e v a r i a b l e u . P r o o f .

The f i r s t inclusion follows f r o m P r o p o s i t i o n 2.1. On t h e o t h e r h a n d t h e monotonicity of t h e mapping Y(.) t o g e t h e r with P r o p o s i t i o n 3 . 1 . imply

Now t u r n to t h e d u a l map a n d s h a l l u s e t h e c o n s t r u c t i o n o r i g i n a t i n g f r o m T. Tanino a n d Y . S a w a r a g i .

Definition 3 . 2 .

L e t f o r e a c h REL ⁺( 2 , ~ ) b e

t h e n w e d e f i n e t h e d u a l ( o r &-dual) map b y t h e e q u a l i t y

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a n d t h e d u a l p r o b l e m as

P r o p o s i t i o n 3 . 3 .

The following r e l a t i o n h o l d s f o r e a c h r €[O,l]:

a n d s o , t h e f u n c t i o n D , i s a c o m p a c t , C - c o n c a v e set v a l u e d , C - c o n c a v e map.

P r o o f .

The map

a(.

, R ) i s C - c o n v e x b e c a u s e i t i s a sum of t w o C - c o n v e x f u n c t i o n s . In o t h e r words t h i s means t h a t n ( R ) i s a C - c o n v e x set. By t h e C - c o n t i n u i t y of t h e f u n c t i o n f , t h e K-continuity of g a n d t h e c o m p a c t n e s s of A c X , R ( R ) c X w e c a n a p p l y Lemmas 2.5. a n d 2.4. of D. T. Luc [9] implying f i r s t t h a t R ( R ) c Y i s C - c o m p a c t a n d t h e n t h a t

H e n c e w e c a n c o n c l u d e t h a t D , ( R ) c Y i s C - c o n v e x .

T h e C - c o n c a v i t y of t h e mapping D , is implied b y t h e following s e q u e n c e of rela- t i o n s :

P r o p o s i t i o n 3 . 4 .

D , ( R ) = E-min

u f

W , ( u ) + R , u : u €2' ]

v

R E L '(2,~)

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P r o o f .

The s t a t e m e n t follows f r o m t h e e q u a l i t y

R ( R ) + C = ~ T N , ( U ) + R ~ U : U E Z ~

V R E L ' ( Z , Y )

t h a t w e s h a l l p r o v e .

L e t f i r s t b e y € R ( R ) . Then w e h a v e

Y

= P

^{( ~ 1 )}+ R . 2 ~ 1 w h e r e w e u s e d t h e n o t a t i o n ul =g ( z l ) This implies

b e c a u s e f ( x i ) EY(U H e n c e

y E W e ( u l )

+

^C

+

^{R . u l}

c

f W , ( u ) + R . u : uEZ

1.

On t h e o t h e r h a n d , if y E W c ( u l ) + R ~ u l f o r some U ~ E Z t h e n , b y t h e d e f i n i t i o n of t h e p r i m a l map,

a n d c o n s e q u e n t l y t h e r e e x i s t s a x l E A s u c h t h a t

H e n c e w e h a v e

Y

2 P

( 1 1 )

+

R . g ( X I )

o r t h a t y EIZ(R)+C.

W e are a b l e now to f o r m u l a t e t h e weak a n d t h e t h e s t r o n g d u a l i t y t h e o r e m s f o r o u r

&-solutions.

T h e o r e m 3.1.

The r e l a t i o n

Y ^-&

4 P ( d

h o l d s f o r e a c h z E F a n d y E D , ( R ) with REL ' ( Z , Y ) .

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P r o o f .

By t h e definition of t h e d u a l map,

holds a n d t h e r e s t of t h e conditions e n s u r e R . g ( x l ) s O .

Theorem 3.2.

( a ) S u p p o s e t h a t f o r some x E F a n d

R

EL ⁺( 2 , ~ ) t h e r e l a t i o n j' ( x ) E D c ( R ) holds.

Then x EX is a n &-solution f o r t h e problem (MP) a n d j ' ( x ) ~ Y i s a n &-maximal element of t h e &-dual problem (D,).

( b ) S u p p o s e t h a t f o r t h e problem (MP) t h e S l a t e r condition h o l d s a n d x EX i s a p r o p e r &-solution of (MP). Then j'(x)EY i s a n &-maximal element of t h e &-dual problem (D ,).

P r o o f .

We p r o v e ( a ) by c o n t r a d i c t i o n . If t h e r e e x i s t s a n x lEF s u c h t h a t j' ( x )

-

^E²^j^'^{( x}

t h e n we a l s o h a v e

c o n t r a d i c t i n g t o t h e assumptions. To s e e t h e r e s t , l e t us s u p p o s e t h a t t h e r e e x i s t s a n RIEL ⁺( 2 , Y ) a n d a y EDc(R1) with t h e p r o p e r t y t h a t

P ( x ) + & ~ Y . This implies a g a i n

t h a t c o n t r a d i c t s t o t h e assumption o n y EY.

We start t h e p r o o f of ( b ) by using Theorem 2.3. to e s t a b l i s h t h e e x i s t e n c e of a n REL ' ( 2 , ~ ) with t h e p r o p e r t y t h a t j'(x)ED,(R). Now ( a ) c a n b e a p p l i e d a n d t h i s c o m p l e t e s t h e p r o o f .

Notice t h a t t h e problem a r i s i n g in Theorem 2.1. d o e s n o t a p p e a r h e r e , o n l y b e c a u s e we use assumptions t h a t a r e s t r o n g e r t h a n t h e &-saddle point p r o p e r t y in t h e c a s e when E # O .

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4. CONICAL SUPPORTS

In t h i s s e c t i o n w e g i v e a g e o m e t r i c i n t e r p r e t a t i o n of t h e d u a l i t y a n d s a d d l e p o i n t t h e o r e m s a n d s u m m a r i z e o u r r e s u l t s in o n e s e q u e n c e of e q u i v a l e n t s t a t e m e n t . This will c l e a r l y show w h e r e d o w e h a v e t h e a n a l o g i e s as e x p e c t e d a n d w h e r e d o p e c u - l i a r i t i e s a r i s e .

The f a c t t h a t t h e f u n c t i o n W , t a k e s v a l u e s among s u b s e t s of a n o r d e r e d v e c t o r s p a c e implies t h a t t h e a p p r o p r i a t e notion of e p i g r a p h i s to b e d e f i n e d in t h e following way:

epi W , =

1

( u , ~ ) E Z X Y : y E W , ( U ) + C , u E Z

1

By P r o p o s i t i o n 3.2., of c o u r s e , h e r e we h a v e t h e e q u a l i t y epi W ,

=

e p i Y ( . ) . We e x p e c t t h a t p a s s i n g f r o m e x a c t s o l u t i o n s t o E-solutions a n d f r o m s c a l a r v a l u e s to v e c t o r v a l u e s means t h e c h a n g e f r o m a s u p p o r t i n g h y p e r p l a n e to a n ' & - s u p p o r t i n g ' t r a n s l a t e of a c o n e . This i s i n d e e d s o . Given a n o p e r a t o r R EL + ( 2 , ~ ) let u s , namely, d e f i n e a c o n e MR i n t h e p r o d u c t s p a c e Z X Y as follows:

a n d l e t u s d e n o t e i t s l i n e a r i t y s p a c e of by 1 (rVR), t h a t is l e t b e

This c o n e is c l o s e l y r e l a t e d to t h e s t r u c t u r e of Y a n d 2 , a n d h a s t h e r e g u l a r i t y p r o p e r t i e s f o r m u l a t e d in t h e following p r o p o s i t i o n .

P r o p o s i t i o n 4 .l.

F o r e a c h R E L + ( z , Y ) , M R c Z X Y i s a c l o s e d c o n v e x c o n e s u c h t h a t

w h e r e t h e l i n e a l i t y s p a c e of MR, 1 ( M R ) c Z X Y , i s i s o m o r p h i c to t h e s p a c e Z . P r o o f .

B e c a u s e of t h e p o i n t e d n e s s of t h e c o n e C C Y we h a v e t h a t ( u , y ) f L ( M R ) if a n d only if y =-R.U . NOW t a k e a n y p a i r ( u ,y ) f Z X Y a n d c o n s i d e r t h e r e p r e s e n t a t i o n

E a s y c a l c u l a t i o n s show t h a t

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o n o n e hand a n d t h a t t h e mapping

1 : Z ⁺L(?dR) defined by

i s a n isomorphism.

Definition 4.1.

We s a y t h a t t h e c o n e M c Z X Y &-supports t h e set valued map h : Z Z Y

at t h e point ( u , y ) E g r a p h h if

( - M ) + I ( ~ , Y - E ) ~

n

^{e p i h} c ~ ( u , Y - E ) { .

Analogously, t h e h y p e r p l a n e H c Z X Y & - s u p p o r t s t h e map h at ( u , y ) E g r a p h h if H

+

f O l x ( - C )

+

I ( ~ , Y - E ) ]

n

^{e p i h} ^c I ( U , Y - - E ) ] .

Let us t a k e now z * €2' a n d y * EY a n d r ER, defining a h y p e r p l a n e in Z X Y , a s fol- lows:

In p a r t i c u l a r , w e use t h e following notation f o r t h e &-supporting h y p e r p l a n e s of e p i W,:

w h e r e

r o

=

s u p

I

^r : H ( Z * , Y * , T - < ~ * , & > ) + [ O ~ X C 3 e p i W ,

1

Now w e f o r m u l a t e t h e r e l a t i o n s h i p s between &-supporting c o n e s a n d h y p e r p l a n e s of t h e p e r t u r b a t i o n function W , .

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Proposition 4 . 2 .

(a) Suppose t h a t H ( z * , y * ) c Z x Y is a n &-supporting h y p e r p l a n e of e p i

W,

_att h e point ( u , y ) E g r a p h W,, y* € i n t ( C +) and t h a t f o r t h e c o n e MR C Z xY t h e r e l a - tion

L

(rUR)cH(z*, y* , 0 ) holds. Then MRcZxY & - s u p p o r t s e p i

W,

_{a t}t h e point (u. , y ) E g r a p h W,.

(b) Suppose t h a t MRcZXY &-supports e p i

W,

at t h e point ( u , y ) E g r a p h

W,.

Then t h e r e e x i s t s a n o n z e r o v e c t o r y* EC' and a h y p e r p l a n e H(z* , y * , r ) c Z x Y t h a t &-supports e p i

W,

at t h e point ( u , y ) E g r a p h

W,

arid f o r which t h e r e l a - tion ^!^I (MR)CH(z*, y* , 0 ) holds.

P r o o f .

To p r o v e (a) w e r e a s o n by contradict,ion. If 1URcZxY d o e s not &-support e p i

W,

a t t h e point ( u

,

y ) E g r a p h W , t h e n t h e r e e x i s t s a n o t h e r point ( u l , y l ) E g r a p h

W,

s u c h t h a t

F o r a n y y* Eint (C +), t h i s implies

On t h e o t h e r hand, relying on t h e formula r e p r e s e n t i n g t h e elements of l ( M R ) c Z x Y , and t h e assumption

L

( M R ) c H ( z * , y* , 0 ) we o b t a i n

T o g e t h e r with t h e p r e v i o u s inequality, t h i s implies t h e r e l a t i o n

c o n t r a d i c t i n g t o t h e &-supporting p r o p e r t y of t h e h y p e r p l a n e H(z* , y * ) c Z x Y . Let u s p r o v e now (b). By t h e assumptions we have t h a t

Consider now t h e mapping

defined by t h e equality

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By t h e l i n e a r i t y of t h e map J a n d P r o p o s i t i o n 3 . 1 . we h a v e t h a t J ( e p i W,)cY i s a c o n v e x set, a n d t h e r e f o r e i t c a n b e s e p a r a t e d b y a p o s i t i v e f u n c t i o n a l f r o m t h e c o n e J ( - ( M R ) + ( u , y - E ) ) c Y . If we d e n o t e t h i s f u n c t i o n a l b y ~ * E c + a n d t h e n , b y a r e a s o n i n g s i m i l a r to t h e a b o v e , w e o b t a i n t h a t t h e h y p e r p l a n e H ( R * . ~ * , ~ * , < ~ * , ~ + R . U > ) C Z X Y & - s u p p o r t s e p i W , . T h e i n c l u s i o n L ( M ~ ) C H ( R ' . ~ * , 0 ) follows f r o m t h e c o n s t r u c t i o n .

Now we s u m m a r i z e t h e m a j o r r e s u l t s of t h e p r e v i o u s s e c t i o n s i n t h e f o r m of a set of e q u i v a l e n t s t a t e m e n t s .

T h e o r e m 4.1.

C o n s i d e r a n e l e m e n t ( z o , R o ) E A X L + (z,Y). T h e n t h e following s t a t e m e n t s a r e e q u i v a l e n t :

( a ) T h e p a i r ( z o , R o ) ~ A X L + ( z , Y ) i s a n &-saddle p o i n t of t h e L a g r a n g i a n a . ( b ) (P(X,,R~) E E - m i n I@(X,R,) E Y : x E A ]

x, f F

- & + ~ o . g ( z o ) S 0 .

( c ) x o € A i s a n (&-Ro.g (xo))-solution of t h e p r o b l e m (MP) a n d f ( z , ) ~ Y i s a n (E-RO-g ( x O))-maximal p o i n t f o r t h e d u a l p r o b l e m (D, -Ro.g (=,)).

(d) x o € A i s a n ( E - R o . g ( z o ) ) - s o l u t i o n of t h e p r o b l e m (MP) a n d t h e c o n e iURo ^E- s u p p o r t s e p i W, at t h e p o i n t ( O , f ( x o ) ) .

P r o o f .

The e q u i v a l e n c e of ( a ) a n d (b) i s s t a t e d in T h e o r e m 2.1.

L e t u s c o n s i d e r now t h e i m p l i c a t i o n f r o m ( b ) to (c). T h a t X,EA i s a n (e-RO.g(xo))- s o l u t i o n of t h e p r o b l e m (MP) follows f r o m T h e o r e m 2 . 2 . , a n d b y t h e d e f i n i t i o n of t h e d u a l map t h i s also m e a n s t h a t ~ ( X ~ ) E D , - ~ , . ~ ( ~ ~ . H e n c e T h e o r e m 3 . 2 . c a n b e a p p l i e d a n d t h i s y i e l d s t h e rest. To see t h e r e v e r s e i m p l i c a t i o n w e n o t e f i r s t t h a t f ( x ~ ) E D ~ - ~ o - g ( ~ d i s a r e f o r m u l a t i o n of t h e f i r s t r e l a t i o n of ( b ) . T h e s e c o n d r e l a -

t i o n follows f r o m t h e f a c t t h a t z o € A i s a n ( e - R o - g ( x o ) ) - s o l u t i o n of t h e p r o b l e m ( W ) . To p r o v e t h e l a s t r e l a t i o n we n o t e f i r s t t h a t b y t h e p o s i t i v i t y of R o € L (Z,Y), t h e i n e q u a l i t y R o , g (xo)dO h o l d s . W e a l s o know t h a t

b e c a u s e f ( x o) EY i s a n (&-Ro.g ( x o%maximal e l e m e n t of t h e d u a l p r o b l e m a n d so we

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p r o v e d t h e t h i r d r e l a t i o n , as well.

( c ) implies (d) b e c a u s e t h e i r f i r s t p a r t s a r e i d e n t i c a l a n d t h e s e c o n d p a r t of (d) i s just a r e f o r m u l a t i o n of t h e f i r s t r e l a t i o n i n (b). T h e l a s t p a r t of t h e a r g u m e n t used to p r o v e t h e implication f r o m (b) t o ( c ) shows t h a t (d) also implies ( c ) .

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5. CONCLUSION

In t h i s p a p e r w e d e v e l o p e d t h e a n a l o g u e of t h e d u a l i t y t h e o r y in v e c t o r optimization, on o n e h a n d , a n d of t h e s c a l a r valued &-duality r e s u l t s , on t h e o t h e r , f o r E -

solutions in v e c t o r optimization. The significance of &-solutions in v e c t o r optimization a r i s e s from t h e f a c t t h a t optimization a l g o r i t h m s o f t e n p r o d u c e s u c h r e s u l t s i n s t e a d of t h e e x a c t solutions. Using t h e a b o v e t h e o r y we o b t a i n g u i d a n c e in s i t u a - tions when we want t o use d u a l i t y without knowing t h e solutions e x a c t l y . I t a p p e a r s t h a t t h e duality r e l a t i o n s r e m a i n t r u e , in g e n e r a l , b u t we h a v e t o c o p e with s l ~ c h problems as t h e i n c r e a s e of t h e a p p r o x i m a t i o n e r r o r when we start e . g . f r o m s a d - d l e points.

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WHITE D. J . , E p s i l o n E m c i e n c y , J o u r n a l of Optimization T h e o r y a n d Applica- t i o n s , Vol. 4 9 , No. 2 , p p . 319-337, 1 9 8 6 .

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NAKAYAMA H., Geometric Consideration of D u a l i t y in Vector O p t i m i z a t i o n , J o u r n a l of Optimization T h e o r y a n d Applications, Vol. 44, pp. 625-655, 1 9 8 4 .