pareto efficiency in locally convex spaces i

This article was downloaded by: [UZH Hauptbibliothek / Zentralbibliothek Zürich]On: 12 July 2014, At: 21:22Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Numerical Functional Analysis and OptimizationPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lnfa20

Pareto efficiency in locally convex spaces iNikolaos S. Papageorgiou aa Department of Mathematics , University of Illinois , Urbana, Illinois, 61801Published online: 09 Jun 2010.

To cite this article: Nikolaos S. Papageorgiou (1985) Pareto efficiency in locally convex spaces i, Numerical FunctionalAnalysis and Optimization, 8:1-2, 83-116, DOI: 10.1080/01630568508816205

To link to this article: http://dx.doi.org/10.1080/01630568508816205

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/lnfa20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/01630568508816205

http://dx.doi.org/10.1080/01630568508816205

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

NUMER. FUNCT. ANAL. AND OPTIMIZ., 8(1 ) , 83-116 (1985)

PARETO EFFICIENCY IN LOCALLY C O N V E X SPACES I

Nikolaos S. Papageorgiou

University of I l l i no i s Department of Mathematics Urbana, I l l ino is 61801

ABSTRACT

A general notion of Pareto efficiency i s introduced and we study the algebraic and topological properties of the se t of a1 1 e f f ic ien t points. We also define a notion of weak Pareto e f f i - ciency and compare i t with the i n i t i a l one. Then we pass t o the study of Pareto efficiency in the case where the se t s under con- sideration are random. We obtain characterizations of the elements of the efficiency s e t and using the theory of s e t valued integration we characterize the aggregate efficiency s e t .

1 ) INTRODUCTION

In recent years there has been an increasing interest in opti-

mization problems with several objectives conflicting with one

another. The subject has i t s origins in economics and in particular in welfare theory. From there, i t found i t s way into equilibrium theory, production theory, game theory, decision theory and vectori a1

optimization. This i s exemplified by the works of Arrow-Barankin Blackwell [ I ] , Debreu [I21 [I31 and Smale [35] in mathematical economics, the works of Bl ackwell-Girschi k [61 and Keeney-Raifa [221 in decision theory and f ina l ly the works of Benson [4], Bitran-Magnanti [5], Borwein [71, Cesari-Suryanarayana [81 [9], Geoffrion [ I 71, Nacchache [241, Yu [36] and others in multiobjective (vectorial

optimization). For t h i s type of problem there does not ex is t a

Copyright @ 1985 by Marcel Dekker, Inc. 0l63~563/85/08014083$3.50/0

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

8 4 PAPAGEORGZOU

universally accepted solution concept. However we can say that a "good" choice i s a decision which cannot be dominated by other a l te r -

natives, in the sense that there are no other alternatives that give

t o the optimizer a greater sat isfact ion (or less dissat isfact ion i f

he i s a minimizer). This solution concept was f i r s t introduced by

the I ta l ian economist V . Pareto in his pioneering work on welfare

economics. Since then i s the most widely used solution concept f o r

mu1 tiobjective optimization problems. Here we are going t o intro-

duce a general concept of such multiobjective optimality, valid for

any locally convex space and study i t s properties. In the l a s t

section we deal with stochastic Pareto efficiency. We deal with

generally i n f i n i t e dimensional spaces, because t h i s i s the natural context t o study several applications. For example, in mathematical

economics the assumption tha t the commodities are not in f i n i t e number agrees with many classical s i tuat ions for economic theory: dif-

ferent iat ion of commodi t i e s , intertemporal equil ibrium with an

i n f in i t e horizon and a world of uncertainty where there are in f in i te ly many s ta tes .

In the continuation of t h i s paper (see [28]) we will study the

s t ab i l i t y of the s e t of e f f ic ien t points under perturbations of the

data.

2 ) DEFINITIONS A N D PROPERTIES OF THE EFFICIENCY SET

Let Y be a locally convex space with a par t ia l ordering

induced by a closed, convex cone Y, of positive elements. This cone specifies the domination structure of the decision maker. For

y1 ,yZ E Y we will write y2 5 yl i f and only i f yl - y2 E Y+,

we will write y2 < yl i f and only i f y2 2 y1 and yl # y2

( i .e. y1 - y2 E Y+\ 101) and f ina l ly i f int' Y, # $ we will

write y2 << yl i f and only i f yl - y2 E i n t Y+.

Now we are ready t o introduce the concept of Pareto efficiency.

Definition 2.1: Given a nonempty S 5 Y a point y E Y i s said to be Pareto e f f ic ien t (or Pareto optimal for the s e t S i f and only i f y E w - cl S and there i s no s E S s . t . s < y.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY 1 8 5

The c o l l e c t i o n of a l l Pareto e f f i c i e n t po in t s o f S w i l l be

denoted by Eff (S) . Convert ing D e f i n i t i o n 2.1 above i n t o mathe-

matic81 symbols, we can w r i t e t h a t E f f ( S ) = {g E w - c l S :

( q - !+In S = $1 where + = Y 0 I n economics, t h i s se t i s

a l so known as the "Pareto f r o n t i e r o f S."

Fo l lowing Cesari-Suryanarayana [81, we w i l l say t h a t Y, has

p rope r t y i f and on ly i f there e x i s t s y* E Y: s e t . f o r a l l

6 > 0 v ~ ( ~ * ) = I y E Y+ : (y*,y) 5 6) i s w-compact. S l i g h t l y

genera l iz ing the proof o f Lemma 4.1 o f [8 ] Ne can show t h a t i f Y, has proper ty (n) and S 5 Y i s minorized, then E f f ( S ) # 4 . I n t h i s paper we w i l l encounter other s i t u a t i o n s where E f f ( S ) # I$.

I n t h i s sec t ion we w i l l study i n d e t a i l t h e p rope r t i es o f the

se t E f f ( S ) . Recal l (see f o r example Yu [36]) t h a t S i s sa id

t o be Y+-convex i f and on ly i f S + Y+ convex. Observe t h a t a

convex se t i s always Y+-convex, bu t the converse i s no t t r u e as 2 2 the f o l l o w i n g simple exmaple shows. Le t Y = R , 'I', = IR, and

2 2 S = {(xl,x2) E IR': x1 t xZ = A x , 5 0, x2 5 0). C lea r l y S t Y+

i s convex, bu t S i s not . I t i s easy t o check tha t S 5 Y i s

Y,-convex i f and on ly i f , when s, ,s2 E S and h E ( 0 , l ) then we

can f i n d s E S

I n what f o

e f f i c i e n c y sets

Propos i t ion 2.1.

Proof. D i r e c t l y

s. t . s 2 Asl + (1 - A)s2.

lows, t o avoid t r i v i a l i t i e s we w i l l assume t h a t t he

nvol ved are nonempty.

For any S c Y E f f ( S ) = E f f ( S + Y + ) . -

from D e f i n i t i o n 2.1, we can say t h a t

Eff(S + Y+) _c E f f (S) . On the o ther hand, f o r any u E S + Y+, there i s an s E S s. t . s ( u and so E f f ( S ) 5 E f f ( ~ + Y+).

Hence we conclude t h a t Eff(S) = E f f ( S t Y + ) . Q.E.D.

From t h i s r e s u l t , we can show t h a t under m i l d hypotheses

convex i f i ca t i on o f the i n i t i a l se t does no t e f f e c t Pareto e f f i c i e n t

po in t s .

Propos i t ion 2.2. I f S c Y i s Y+-convex then Ef f (conv S) = Ef f (S1. -

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

8 6 PAPAGEORGIOU

Proo f . From P r o p o s i t i o n 1 . I , we have t h a t -

But by hypo thes is , S i s Y,-convex, so E f f ( c o n v ( S + Y,)) =

= E f f ( S + Y,). R e c a l l t h a t conv(S t Y,) = conv S + Y+. So we g e t

t h a t

A new a p p l i c a t i o n o f P r o p o s i t i o n 1.1 t e l l s us t h a t

Combining ( 1 ) , ( 2 ) and ( 3 ) above we conclude t h a t

E f f ( S ) = E f f ( c o n v S) .

Q.E.D.

Remark: I t i s easy t o see f r o m D e f i n i t i o n 2.1 t h a t E f f ( w - c l S) =

= E f f ( S ) . So we can complete t h e p r e v i o u s p r o p o s i t i o n by s a y i n g

t h a t i f S i s Y,-convex t h e n E f f (w - c l conv S ) = E f f ( S ) .

It i s w e l l known t h a t one o f t h e ma jo r advantages o f t h e use

o f c o n v e x i t y i n o p t i m i z a t i o n i s t h e g l o b a l i z a t i o n o f o p t i m a l i t y

r e s u l t s . Th is i m p o r t a n t f a c t i s r e f l e c t e d i n t h e n e x t r e u s l t ,

which says t h a t f o r c l o s e d and convex se ts , l o c a l Pare to e f f i c i e n c y

imp1 i e s g l o b a l Pare to e f f i c i e n c y .

P r o p o s i t i o n 2.3. I f S 5 Y i s c l o s e d and convex, U i s a ne igh-

borhood o f q € S and q E E f f ( S n U ) then q E E f f ( S ) .

P roo f . Suppose t h a t q 4 E f f ( S ) . T h i s means t h a t t h e r e e x i s t s

s E S s . t . s < q. So f o r some y + E Y, we w i l l have t h a t

q - s = y,. Because Y i s l o c a l l y convex, w i t h o u t l o s s o f gener-

a l i t y , we can t a k e U t o be convex. Hence we can see t h a t f o r

some h E ( 0 , l ) we w i l l have t h a t h ( s - q ) + q E U and

As + ( 1 - h ) q E S. There fo re h ( s - q ) + q E U n S and t h e n

q ' = q - Xy+ E U n S which c o n t r a d i c t s t h e assumption t h a t

q E E f f ( U n S ) . T h e r e f o r e w e m u s t h a v e t h a t q E E f f ( S ) . Q.E.D.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY I 87

It i s use fu l f r o m b o t h a t h e o r e t i c a l and p r a c t i c a l v i e w p o i n t

t o c h a r a c t e r i z e Pare to e f f i c i e n t p o i n t s i n a dual way, u s i n g t h e

s o - c a l l e d " p r i c e v e c t o r s " . By a p r i c e v e c t o r we mean an element

i n Y: and t h e name i s t a k e n f r o m economics, where when Y = IRn , a p r i c e v e c t o r i s j u s t an element p = (pl . . . pn) E IR: where

pi E IR, r e p r e s e n t s t h e p r i c e o f t h e i t h good i n t h e n-dirnen-

s i o n a l commodity space I R ~ . The n e x t r e s u l t p r o v i d e s t h e d e s i r a b l e

dual d e s c r i p t i o n o f t h e Pare to e f f i c i e n t p o i n t s and t e l l s us t h a t

every e f f i c i e n t p o i n t i s suppor ted b y a p r i c e system, a l s o known as

t h e e f f i c i e n c y p r i c e v e c t o r .

P r o p o s i t i o n 2.4. I f S 5 Y i s convex w i t h i n t S +' $ and

q E E f f ( S ) t h e n t h e r e e x i s t s y* E i: s . t . (y*,q) = i n f (y*,s). sE S

Proof . From t h e d e f i n i t i o n o f Pare to e f f i c i e n c y we know t h a t

( q - Y,) S = 4 . Since i n t S # + t h e f i r s t s e p a r a t i o n theorem * '* o f convex s e t s t e l l s us t h a t t h e r e e x i s t s y E Y s . t .

(y*.q ; Y,) 5 (y*,S). Hence (y*,q - Y+) 5 (y*,q) which shows t h a t

y* € Y+. Fur thermore i t i s c l e a r t h a t (y*,q) = i n f ( y X , s ) . Q.E.D. se S

Remark: I f Y i s f i n i t e d imensional t h e r e i s no need f o r a nonempty i n t e r i o r assumption, i f Y, i s p o i n t e d .

I n f a c t we can g e n e r a l i z e t h e above r e s u l t

P r o p o s i t i o n 2.5. I f SLY i s Y+-convex w i t h

q E E f f ( S 1 t h e n t h e r e e x i s t s y* E Y: s . t . ( y *

Proof . From P r o p o s i t i o n 2.1 we know t h a t E f f (S -

as f o l l o w s

i n t S f 4 and

q ) = i n f (y* ,s) . SE S

So q E E f f ( S + Y,). S ince S t Y, i s convex and i n t ( S + Y+) # 0, * f rom Proposo t ion 2.4 we know t h a t we can f i n d y* E Y+ s . t .

(y*,q) = i n f (y* ,z) . But (y*,Y,) 5 0. So i n f (y*,z) = 7 5 sty, SStY,

= i n f (y*,z). There fo re we conclude t h a t (y*,q) = i n f (y* ,s) . E S sES Q.E.D.

* M o t i v a t e d f r o m t h e above r e s u l t s f o r y* E Y,, we s e t S(y*)=

= Iq E w - c1 S :(y*,q) = in f (y * ,S) ) . We can see t h a t E f f ( S ) 5 C U * S(y* ) . I n f a c t , i t i s easy t o see t h a t i f f o r a l l boundary -

y*Y+

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

8 8 PAPAGEORGIOU

* p o i n t s o f Y+, S(y*) i s e i t h e r empty o r a s i n g l e t o n , t h e n e q u a l i t y

ho lds . I n t h e n e x t p r o p o s i t i o n , we w i l l determine ano ther s e t t h a t

bounds E f f (S) f r o m below. F i r s t we need t h e f o l l o w i n g lemma. F o r

a p r o o f o f i t see f o r example Borwein [ 7 ] .

Lemma a. Suppose t h a t i n t Y: # 41. Then y * t i n t Y: if and o n l y

F o r f i n i t e d imensional Y, every c losed, convex cone K f o r

which a f f ( K ) i s t h e whole space, we have t h a t i n t K f $. I n

i n f i n i t e d imensional spaces, we cannot guarantee so e a s i l y t h a t

i n t K # 4 . However, i f K has a compact base (wh ich i s e q u i v a l e n t * t o s a y i n g t h a t K i s l o c a l l y compact) then we can f i n d y* E Y + s . t . * (y*,Y+) > 0 u s i n g a r e s u l t o f K lee [23]. O f course, i f Y i s

normed, t h e n we know t h a t t h e requ i rement o f compact base i s

e q u i v a l e n t t o say ing t h a t i n t Y: # @ (see Asimow-El l is [Z] ) and

so we go back t o t h e s i t u a t i o n o f Lemma 2.

Using t h a t Lernma, we can have a r e s u l t t h a t complements

P r o p o s i t i o n 2.5.

* * P r o p o s i t i o n 2.6. I f q E w - c l S and f o r some y E i n t Y+

(y*,q) = i n f ( y * , ~ ) t h e n q E E f f ( S ) ) .

P roo f . I f not , we can f i n d s E S s . t . q - s = y + E Y+. Us ing

t h e lemma (y*,q) (y* ,s) i s a c o n t r a d i c t i o n . Q.E.D.

We w i l l c l o s e t h i s s e c t i o n w i t h a r e s u l t t h a t t e l l s us t h a t

cone c o n v e x i t y o f t h e o r i g i n a l s e t i m p l i e s t h e same f o r t h e s e t o f

S i s e f f i c i e n t p o i n t s . Assume t h a t Y+ has p r o p e r t y (n) and

mi n o r i zed.

P r o p o s i t i o n 2.7. I f S i s w-closed t h e n S + Y+ = E f f ( S

So i f S i s Y+-convex t h e n so i s E f f ( S ) .

P roo f . S ince by hypo thes is , S i s w-c losed we have t h a t

E f f ( S ) 5 S. So E f f ( S ) + Y+ 5 S + .Y+. Next, l e t u E S t Y,. Then

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY I

u = s + y ' f o r some s E S and some y; E Y,. If s E Eff (S) + then we are done. So suppose t h a t s $: E f f (S) and take

q~ E f f ( ( s - ? + I n S) # $. C lea r l y q~ s - t,. So q = s - y t w i t h y, E Y+. Then u = q + y; + y+ E f f ( S ) t Y+. Therefore

S + Y, = E f f ( S ) + Y+.

The l a s t p a r t of the p ropos i t i on fo l lows f rom the d e f i n i t i o n

o f Y+-convexi t y .

Remark: I n f a c t a ca re fu l look a t the proof shows t h a t S + Y+ =

= E f f ( S ) + Y,.

2 ) TOPOLOGICAL PROPERTIES OF THE EFFICIENCY SET-WEAK PARETO

EFFICIENCY

I n t h i s sec t ion we pass t o a systematic examination o f t h e

topo log i ca l p rope r t i es o f the s e t o f e f f i c i e n t and weakly e f f i c i e n t

po in t s ( t o be def ined).

We s t a r t . w i t h a d e f i n i t i o n

D e f i n i t i o n 3,1. We say t h a t a se t S c Y+ i s Y+-w-compact i f f o r - a l l s E S, ( S - Y,) n S i s w-compact,

C lea r l y every w-compact se t i s Y,-w-compact, wh i l e t h e con-

verse i s no t t r u e i n general, as simple two dimensional examples

manifest,

The next r e s u l t es tab l ishes another s i t u a t i o n where t h e f a m i l y

o f Pareto e f f i c i e n t po in t s i s nonempty. Recal l t h a t a cone Y+ i s acute i f and on ly i f there e x i s t s y * E Y* s . t . iy*,<) > 0. It

i s easy t o check t h a t a cone possessing proper ty ( 1 i s acute.

Furthermore, t o say t h a t Y+ is acute i s equ iva lent t o saying t h a t * i n t Y+ # $.

P ropos i t i on 3.1. If Y+ i s acute and S 5 Y i s Y+-w-compact then

E f f (S ) i s nonempty.

* * Proof. Because Y, i s acute, we know t h a t there e x i s t s y E Y . s . t .

( y * , ~ + ) > 0. Also, s ince by hypothesis S i s Y+-w-compact, f o r

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

90 PAPAGEORGIOU

a l l s E S, ( S - Y+) n S i s w-compact and so bounded. I n t h e * sequence f i x s E S. Then f r o m t h e c o n t i n u i t y o f y and t h e

boundedness o f ( s - Y+) n S we have t h a t (y*, ( s - Y+) n S) i s

a bounded subse t o f IR. L e t m = i n f ( y * , ( s - Y,) n S) > - and f o r m a n e t fs,},,, s . t . (yX,s6) + m. From t h e w-compact-

ness o f ( s - Y,) n S we know t h a t we can f i n d {spIBEn' 5

S. fs,3,,, be ing a subnet s . t . s &-+ E S. We c l a i m t h a t B ; E E f f ( S ) . Suppose n o t . Then we can f i n d p~ S s . t . p < g. Then p E ( s - Y+) n S and (yk,p) < m a c o n t r a d i c t i o n . So

S E E f f ( S ) . Q.E.D.

Simple two d imensional examples can convince t h e reader t h a t

E f f ( S ) need n o t be connected. The n e x t r e s u l t g i v e s us s u f f i -

c i e n t c o n d i t i o n s f o r t h e connectedness o f E f f (S) . F o r t h a t p u r -

pose assume t h a t Y i s separab le Banach space o rdered by Y, and

t h e dual cone Y: has nonempty i n t e r i o r .

P r o p o s i t i o n 3.2. I f S i s w-compact and Y+-convex t h e n E f f ( S 1

i s connected.

* Proo f . F o r y * E Y, c o n s i d e r t h e f o l l o w i n g s e t -

* C l e a r l y f o r each y* E i n t Y , t h i s i s a r~onempty, w-closed,

convex ( t h e r e f o r e conencted) subset o f S. So we can d e f i n e t h e

m u l t i f u n c t i o n y* -+ A (yk ) . We c l a i m t h a t t h i s m u l t i f u n c t i o n i s

weakly upper semicont inuous. F o r t h a t purpose, observe t h a t s i n c e

S i s a w-compact set , f r o m Theorem 3, p. 434 o f [15], we know t h a t

t h e weak t o p o l o g y on S i s rne t r i zab le . So w-upper s e m i c o n t i n u i t y * S W

i s e q u i v a l e n t t o s a y i n g t h a t " i f yn - y, sn - s and sn€ A(yG1

f o r a l l n 2 1 t h e n s E A ( y * ) " i .e. w-upper s e m i c o n t i n u i t y i s

e q u i v a l e n t t o t h e c losedness o f t h e m u l t i f u n c t i o n .

I n f a c t , f o r every 2 E S we have t h a t (y;,sn) 5 (y;,E).

Tak ing t h e l i m i t as n + m we g e t t h a t (y*,s) 5 (y*,?). So

(y*,s) = in f (y * ,S) wh ich means t h a t s E A(y*) . So A ( * ) i s a

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY I ' 9 1

* w-U.S.C. mu l t i f unc t i on . But i n t Y, being convex, i s connected.

So A ( i n t Y:) i s connected too (see f o r example [18]1. Furthermore * from Propos i t ions 2.5 and 2.6 we know t h a t A ( i n t Y,) =

= u * A(y*) c Eff (S) C c1 A l i n t Y:) = u * A(Y*). Therefore - - y % i n t Y, Y*Y+

we conclude t h a t E f f (S ) i s connected. Q.E.D.

Now we w i l l in t roduce a weaker no t i on o f Pareto e f f i c i e n c y

t h a t corresponds t o a l a r g e r e f f i c i e n c y set . So we assume t h a t

i n t Y+ f +. D e f i n i t i o n 3.2: The se t o f weak Pareto e f f i c i e n t po in t s o f S c Y - i s def ined by

wEff(S) = {q E w - c lS: ( q - i n t Y,) n S = $1.

Obviously E f f (S) wEff (S).

For t he next r e s u l t , which gives a t opo log i ca l p roper ty o f

wEff(S) assume t h a t Y = IRn.

Propos i t ion 3.3. I f Y, i s a polyhedral cone then wEff(S1 i s

closed.

* Proof. Since Y, i s polyhedral , Y+ i s polyhedral t oo and so by

Minkowski 's theorem (see Rockafe l la r [30]) we know t h a t i t i s f i - * n i t e l y generated. Le t G(Y,) = {y?,.. .,y$ be the se t o f i t s

generators.

Le t ~q,~nLl L w E f f l S ) and q n + q . Suppose t h a t q e wEf f lS) .

This means t h a t we can f i n d s 5 S s. t . s << q and so . * (y*,s) < (y*,q) f o r a l l y * E Y, (Lemma a i n Sect ion 2 ) . Con-

s i de r the func t i on y* : y + (y*,y). Since i t i s continuous, we

can f i n d a neighborhood U o f q s. t . (y*,s) < (y*,qi) f o r

a l l q ' E U Le t Y* rn

Y*' U = n U where IY;IT=~ i s t he se t

* i = l Y$ o f generators o f Y,. For n 2 max n ( y r ) we have t h a t qn E U

l< i<m - and so ( ~ 7 , s ) < (y7,q) f o r i = 1- ... m which impl ies t h a t .* (y*,s) < (y*,qi 1 f o r a l l y * E Y+. The l a s t i n e q u a l i t y means t h a t

s << q a c o n t r a d i c t i o n s ince f o r a l l n L 1 q n E w E f f ( S ) . So we n conclude t h a t q E wEff(S1 and so wEff(S1 i s closed. Q.E.D.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

92 PAPAGEORGIOU

As t h e above p r o o f shows, o u r r e s u l t depends h e a v i l y on t h e

f i n i t e d i m e n s i o n a l i t y o f Y . F o r genera l i n f i n i t e d imensional

spaces w E f f ( S ) may n o t be c losed i n any t o p o l o g y compat ib le w i t h

t h e dual p a i r ( y a y * ) . It w i l l be n i c e t o know what a re t h e m i n i -

mal hypotheses t h a t guarantee c losedness i n any such topo logy . As

a f i r s t s tep toward o b t a i n i n g such a genera l theorem we have t h e

f o l l o w i n g r e s u l t . Assume t h a t Y i s a r e f l e x i v e Banach space.

P r o p o s i t i o n 3.4. If S i s c losed, convex and has nonempty i n t e r -

i o r t h e n s - c l E f f ( S ) g w E f f ( S ) .

Proof . L e t iq,InLl 5 E f f ( S ) and suppose t h a t q q. From e*"

P r o p o s i t i o n 2.5 we know t h a t t h e r e e x i s t s y; E Y, n 2 1 s . t .

(y*,q ) = inf(y;,S). C l e a r l y , we can t a k e lly;ll 5 1 f o r a l l n n -;I. From A l a o g l u ' s theorem we know t h a t t h e u n i t b a l l i n Y

*

i s w-compact. So we can f i n d i m l 5 i n 1 s . t . y; y*. We

c l a i m t h a t (y*,q) = i n f (y*,S) . Again we proceed by c o n t r a d i c t i o n .

Suppose t h a t we can f i n d s E S s . t . (y*,s) < (y*,q). L e t

0 c E = (y*,q-s). S ince (y;qn) + (y*,q) t h e r e e x i s t s an no s . t .

f o r a l l n 2 no we have t h a t

and a

So we

1 < € /3

nl s . t . f o r n 2 nl we have t h a t

< €/3.

can see t h a t f o r n "n max(no,nl ) we wi 11 have

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


and so f o r n - > i, (y;,s) < (y i ,qn . However, f o r a l l n 2 1 we

know t h a t (y;,qn) = inf(y;,S) and so we have a c o n t r a d i c t i o n .

There fo re (y;,q) = i n f ( y * , S ) . Now if q $ wEff (S) we c o u l d f i n d

s E S s . t . s = q - y+ where y+ E i n t Y,. Hence (y*,s) < ( Y * , ~ )

a c o n t r a d i c t i o n . Q.E.D.

Th is p r o p o s i t i o n t o g e t h e r w i t h e a r l i e r r e s u l t s , emphasizes t h e

i n t u i t i v e f a c t , t h a t when i n t Y+ # 4 , then we can o b t a i n s t r o n g e r

r e s u l t s about t h e e f f i c i e n c y s e t . So we would 1 i k e t o know when

i t i s p o s s i b l e t o r e l a t e , i n some sense, E f f ( S ) t o t h e e f f i c i e n t

s e t o f S w i t h r e s p e c t t o a c losed, convex cone t h a t has a non-

empty i n t e r i o r . To achieve t h a t we need t o i n t r o d u c e t h e n o t i o n o f

a "Y+-approximating" fami l y o f closed, convex cones ~Y!I~,~. T h i s n o t i o n was f i r s t i n t r o d u c e d i n dua l f o r m by Bi t ran-Magnant i

151 and t h e n used i n p r i m a l f o r m by Nieuwenhius [25]. I n b o t h

papers Y = Rn. Here we ex tend t h i s n o t i o n t o genera l Banach

spaces.

S D e f i n i t i o n 3.3. The decreas ing f a m i l y {Y+36,0 o f closed, convex

cones w i t h nonempty i n t e r i o r i s s a i d t o be "Y+-approximating" if

and o n l y i f 6

( i ) Y , s i n t Y+ f o r a l l B E (0,S ] 0 6 X

( i i L e t y6 E Y, n B1 ( 0 ) . Then a1 1 convergent subsequences

of {y6) have t h e i r l i m i t i n Y+.

It i s easy t o check t h a t when Y+ i s p o i n t e d i .e.

Y+ n (-Y+) = (01 (which i s always t h e case i n t h i s paper ) , then

such an approx imat ing f a m i l y e x i s t s and v i c e versa.

If Y i s r e f l e x i v e , t h e f o l l o w i n g remarkable r e s u l t r e 1 a tes

t h e s e t s Effy (S) and E f f y g ( S ) 6 > 0 ( t h e s u b s c r i p t s i n d i c a t e + +

t h e cone w i t h r e s p e c t t o which t h e g f f i c i e n c y s e t i s t a k e n ) .

6o Theorem 3.1. I f S 5 Y i s w-closed and bounded and Y, i s

acute t h e n

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

9 4 PAPAGEORGIOU

u ~ f f , , a ( s ) 5 E f f y ( S ) 5 w-cl u Effy6(S) = r + 610 +

Proo f . L e t q E

1.5 we know t h a t

and t h i s i m p l i e s

Now l e t q

u Eff * ( S ) . Then f o r some > 0 we have t h a t 6>0 + A '0 ch means t h a t ( q - Yt) n S = 4 . From D e f i n i t i o n

; i '? Hence we have t h a t ( q - i+) 0 S = 4 + - +' t h a t q € Eff,, ( S ) . So U E f f y & ( S ) 5 E f f y (S) .

+ S>O t + E f f (S ) and'suppose q - $ - r . Then we can f i n d

y+ a weak neighborhood o f t h e o r i g i n s . t . ( q + U) n E f f V 6 ( S ) = 4 f o r

' +

a l l 5 > 0. Since b y h y p o t h e s i s YfO i s acute, we know f r o m

e a r l i e r remarks t h a t i n t Y fO* C 4 . A1 so Y+ 5 Y: 2 Y ~ O f o r a1 1

assumed t h a t q $ r , t h e supremum i s over a nonempty s e t and

u ( 6 ) > 0. Because S i s w-closed and bounded and Y i s a

r e f l e x i v e Banach space, we know f r o m A laog l u ' s theorem t h a t S i s - 6

W-compact. SO t h e r e i s a e Y+ w i t h q - iS E S s . t .

u ( 6 ) = (Y* , i6 ) . L e t S = l / n n 2 no where 60 , l / n O . Then

i n s ( q - S ) n ? : c q - S f o r a l l n ' n 0 and t h e l a t t e r s e t i s

w-compact s i n c e S i s . From t h e Eber le in -Smul ian theorem (see

[15]) we know t h a t i t i s w - s e q u e n t i a l l y compact and so we can

f i n d a subsequence {Sm! 5 {$,,I s. t . im . We c l a i m t h a t

$ # 0. Suppose n o t . Then $, % 0. So f o r m 1 arge enough we

w i l l have t h a t -Em. U which means t h a t q - Bm q + U and so

q - $, $ r f o r m l a r g e enough. T h i s l a s t f a c t means t h a t t h e r e .m

a r e s,ES and y m € Y + s . t . q - $ m + y m = s m t S . Since

9m + ym E Y: from t h e d e f i n i t i o n o f u(m) we have t h a t

* But (y*,?,) = u(m) and 0 < (yX,ym) s i n c e y * E i n t Y+ and

y . So we have a c o n t r a d i c t i o n which shows t h a t f 0.

A lso Y E q - S.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


Next we w i l l show t h a t i E Y+. F o r t h a t we w i l l r e f e r t o

D e f i n i t i o n 3.3. Consider t h e normal i zed sequence {im/l ;ml lmzn0

( i t i s usefu l t o r e c a l l a t t h i s p o i n t t h a t ym # 0 f o r m > n o ) . X

- By r e f l e x i v i t y B1 (0) i s w-compact and so w - s e q u e n t i a l l y com-

p a c t (Eber le in-Smul ian theorem [15] ) . So w i t h o u t any l o s s o f gen-

e r a l i t y we may assume t h a t / z e 0 From D e f i n i -

t i o n 3.3 we g e t t h a t z E Y+.

From t h e weak lower s e m i c o n t i n u i t y o f t h e norm f u n c t i o n a l we

know t h a t l i m i n f II;,II 2 II;~I. Hence l i m i n f ( l / l l i m l l ) . m f m llw'

(y*,;,) - i l/ll$ll (T*,;) f o r a l l i* t Y:. Bu t n o t e t h a t I h h -

l i m i n f ( 1 / 1 l ? ~ l l )(Y*,;~) = l i m (y*,ym/llymll) = (y* ,z) . So f i n a l l y we P ntrm

get t h a t ( i * , z ) - < (Y*,$II;II) f o r a l l i* E Y;. T h i s means t h a t

~ / I I ~ I I E ?+ and so E 'i+. R e c a p i t u l a t i n g we see t h a t we have ; E q - S and E 'i,.

Hence f o r some s E S s = q - $ which i m p l i e s t h a t q $ E f f y (S) ,

a c o n t r a d i c t i o n . ' + Q.E.D.

We have seen t h a t i f S i s convex and i n t S # @, t h e n . * Pare to e f f i c i e n t p o i n t s a r e suppor ted by e f f i c i e n c y p r i c e s i n Y,. AS p o i n t e d o u t e a r l i e r t h e converse i s t r u e i f f o r a l l y* E bdi:

t h e s e t s S(y*) = { q E w - c l S : (y*,q) = i n f ( y x , S ) 1 a r e e i t h e r

empty o r s i n g l e t o n s . I n t h e genera l case we can have t h e f o l l o w i n g

"theorem o f t h e a1 t e r n a t i v e " f o r any S 5 Y and i n t Y+ # $.

.* Theorem 3.2. I f f o r some y* E Y+ we have t h a t (y*,q) = i n f ( y k , S )

then q E E f f ( S ) o r f o r some y+€bdY+ we have q - y+E S.

Proof. Suppose q 4 E f f ( S ) . Then by d e f i n i t i o n , we have t h a t - ( q - f + ) n S $ 4 . L e t s E ( q - ;+ ins. Then s = q - y + where

y+ E Y+. Suppose t h a t y+ E i n t Y+. Then we have t h a t (y*,s) =

= (y*,q - y+) = i n f (y*,S) - (y*,y+). S ince yt E i n t Y+ and * .* y E Y+, f r o m Lemma a of S e c t i o n 2 we know t h a t (yX,y+) > 0.

Hence f o r s E S we have t h a t (y*,s) < in f (y * ,S) wh ich i s absurd.

So y+ E bdY+.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

96 PAPAGEORGIOU

Now assume t h a t f o r a l l y, E bdY+, q - y, $ S. Then exact ly as above, we can show t h a t t h i s ac tua l ly holds f o r a l l yt E Y,, which proves t h a t (q - ?+) n S = 4 i . e . q c Eff (S) . Q . E . D .

Closing t h i s sec t ion we would l i k e t o mention t h a t when S i s

closed, convex and minorized and Ef f (S ) # 3 , then it n (-As S) # @

where As S denotes the asymptotic ( r ecess ion) cone of S ( see

[28]). To prove t h i s necessary condit ion f o r nonemptiness of t h e

e f f i c i ency s e t we use t h e f a c t t h a t f o r a l l p E As S p + S - c S.

4 ) MULTIOBJECTIVE OPTIMIZATION

In [26] and [27] we studied extensions of convex analys is and

of Clarke ' s theory f o r vector valued funct ions , whose range was an

ordered topological vector space. We will b r i e f l y r eca l l some

basic d e f i n i t i o n s f ro~n those papers, t h a t we wi l l need i n t h e

sequence. So l e t Y be an ordered topological vector-space. An

opera tor f : X + Y i s s a id t o be convex i f and only i f f o r a l l

A € [0,1] and a l l x l , x 2 E X we have t h a t

1 f ( x l ) + ( 1 - X)f (x2) - f(Axl t (1 - 1 ) ~ ~ ) E Y,.

For such an operator we defined the subd i f f e ren t i a l of a

point xo E X t o be the following s e t

ac f (x0) = (A E l ( X , Y ) : A(x - xol 5 f ( x ) - f i x o ) f o r a l l x E xi.

The study of convex operators brought i n t o t h e p ic tu re , i n a

very natural way the loca l ly o-Lipschitz opera tors . So assume

t h a t X i s a normed space and Y a normed l a t t i c e . An operator

f : X - Y i s s a id t o be loca l ly o-Lipschitz i f f o r every bounded open s e t U of X t he re i s a y E Y, s . t . l f ( z ) - f ( x ) \ 2 < yllz - xll f o r a l l x,z E U . ( I n f a c t t h i s d e f i n i t i o n makes - per fec t ly good sense i f X i s any l o c a l l y convex space and Y

any loca l ly convex l a t t i c e . In t h a t case we use seminorms instead

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


o f norms.) For such operators we were able t o def ine a general ized

o-di r e c t i o n a l d e r i v a t i v e as f o l lows.

Observe t h a t f o r any wn = (zn,hn) (x,O) i n X x R we t

have t h a t

So t he quo t i en ts e x i s t s . Since

we conclude t h a t fo(x;d) i s we l l de f ined and fur thermore we have

obtained a bound f o r it, namely

I n a d d i t i o n i t i s easy t o check t h a t d -. fo(x ;d) i s sub-

l i n e a r f o r a l l x € U. Having in t roduced the above n o t i o n o f

general ized o - d i r e c t i o n a l d e r i v a t i v e we were abl e t o proceed and

de f i ne a new subd i f f e ren t i a1 , the so c a l l e d general i zed s u b d i f f e r -

e n t i a l . So the general ized s u b d i f f e r e n t i a l o f f ( * a t x i s the

se t

I t i s no t d i f f i c u l t t o see t h a t

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

98 PAPAGEORGIOU

We w i l l say t h a t an o p e r a t o r f : X -+ Y i s 1 s . . ( o r * * Y+-closed) if and o n l y i f f o r a l l y* E Y+ we have t h a t t h e r e a l

va lued f u n c t i o n x -+ ( y * , f ( x ) ) i s 1 .s .c . * F o r t h e n e x t r e s u l t assume t h a t t h e dual p o s i t i v e once Y+ i s * * *

g e n e r a t i n g i . e . Y = Y+ - Y+. Th is i s t h e case f o r example when

Y i s a normal l o c a l l y convex o rdered v e c t o r space ( K r e i n ' s

Theorem see [ 2 7 1 ) .

P r o p o s i t i o n 4.1. I f SEX i s w-closed and f : X - t Y i s .* Y+-1.s.c. and t h e r e i s a G* E Y+ s e t . f i s w - i n f -

compact t h e n q E ~ f f [ f ( ~ ) ] i m p l i e s t h a t t h e r e i s an so E S s . t .

f ( s o ) = q.

P roo f . S ince by hypo thes is q € E f f [ f ( S ) ] , t h e n q E w-c l f ( S ) . - W

So we can f i n d a n e t {s616Ea 5 f ( S ) s . t . f ( s 6 ) - q. Hence

f o r a l l y * E Y* (y* , f ( s6 ) ) - (y*.q). Th is means t h a t f o r some

a0 E n and f o r a l l 6 2 ?iO we w i l l have t h a t ( y * , f ( S ) ) 5 * * < (y*,q) + 1. But b y hypo thes is f o r i* E Y+ ( y , f ( * ) ) i s

w-inf-compact. So we can f i n d !sglgEnl 5 is?i)6Ep subnet s . t .

so E S. S ince f ( * ) i s Y+-1.s.c. we have t h a t f o r every S~ -* y; Y E l i m (y;,f(s 1 ) 2 ( y ~ , f ( s o ) ) . Observe t h a t

+ 6 l i m (y;,f(s6)) = (y:,q). Hence we g e t t h a t (y;,q) 2 (y:,f(s0)) -

fi

f o r a l l y; E Y:. I f f o r some y: E Y+ s t r i c t i n e q u a l i t y h o l d s .. *

we g e t t h a t q > f ( s o ) which c o n t r a d i c t s t h e f a c t t h a t

q E E f f i f i S ) ) . So f o r a l l y * E Y: we have t h a t (y:,q)*= +*

= (y:,f(so)). Now l e t y * E Y . Since b y hypo thes is , Y+ i s

g e n e r a t i n g we can f i n d y i + and yZ+ b o t h i n Y: s . t .

y* = y?+ - y;+. So (y*,q) = ( y * , f ( s o ) ) and s i n c e t h i s i s t r u e

f o r a l l y * E Y * we conclude t h a t f ( s o ) = q. Q.E.D.

Assuming, i n a d d i t i o n c o n v e x i t y o f t h e o p e r a t o r f : X -+ Y

we can have t h e f o l l o w i n g s t r o n g e r v e r s i o n o f P r o p o s i t i o n 4.1.

Again we assume t h a t Y: i s g e n e r a t i n g and by f ( 0 ) denote t h e Y*

IR-valued f u n c t i o n x - ( y X , f ( x ) 1.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY I

* P r o p o s i t i o n 4.2. I f f : X - Y i s Y+-1.s.c. and convex,

i n t f ( X ) #c$ q € E f f ( f ( X ) ) and t h e r e i s y * t t :s . t .

( y * , f ( * ) ) = f * ( * I i s w-inf-compact t h e n t h e r e e x i s t so E X and Y

y* E Y: s . t . f ( s o ) = q and 0 E a f * ( s o ) . Y

Proof . The e x i s t e n c e o f so E X s . t . f ( s o ) = q f o l l o w s f r o m - P r o p o s i t i o n 4.1. Fur thermore i t i s easy t o check t h a t f f X ) i s ' * Y,-convex. So f r o m P r o p o s i t i o n 2.5 we know t h a t t h e r e i s y* E Y+

s . t . (y*,q) = ( y * , f ( s o ) ) = i n f ( y * , f ( X ) ) . Bu t n o t e t h a t t h e func -

t i o n f i s convex. So from a w e l l known o p t i m a l i t y c o n d i t i o n Y

o f convex a n a l y s i s (see [30]) we have t h a t OE a f ( s o ) . Q.E.D. Y*

We can have t h e f o l l o w i n g p a r t i a l converse o f t h e p r e v i o u s

p r o p o s i t i o n . Assume t h a t i n t Y: # @.

p r o p o s i t i o n 4.3. If f o r some y* € i n t Y: 0 6 a f y * ( j ) t h e n

f ( 2 ) E E f f ( f ( X ) ) .

Proof . L e t y * E i n t Y: s . t . 0 E a f *( ; I . Then f r o m t h e d e f i n i - - Y t i o n o f t h e s u b d i f f e r e n t i a l o f a convex f u n c t i o n , we can see t h a t

x + f ( X I = (y* , f ( X I a t t a i n s i t s in f imum a t ;. So we have Y*

t h a t (y*,f(;)) = i n f ( y * , f ( X ) ) which b y P r o p o s i t i o n 2.6 t e l l s us

t h a t f ( i ) E E f f ( f ( X 1 ) . O.E.D.

U n t i l now we concen t ra ted o u r a t t e n t i o n on unconstra ined,

convex m u l t i o b j e c t i v e o p t i m i z a t i o n problems. I n t h e sequel, we

wi 11 c o n s i d e r c o n s t r a i n e d convex problems and even tua l l y pass t o

nonconvex ones. So l e t f : X + Y be a con t inuous convex o p e r a t o r

and S - c X convex w i t h i n t S # 4 . We w i l l s tudy t h e f o l l o w i n g

ext remal problem.

Such a p o i n t o f S i s c a l l e d a Pare to s o l u t i o n of (PI. We - a d j o i n t o Y a g r e a t e s t element +a, and ex tend t h e v e c t o r space

o p e r a t i o n s i n a n a t u r a l way. There fo re we have a l s o a d j o i n e d a

s m a l l e s t element -m. We w i l l denote t h e augmented space

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

+= E hU) U t Y + = n

Y.

Proposi t

100 PAPAGEORGIOU

Y u {+a1 by ?, w h i l e Y = Y u { t m l . The t o p o l o g y o f Y i s

then extended t o ? as f o l l o w s , A s e t w i l l be c a l l e d open i n

9 i f t h e t r a c e o f ^U on Y i s o p e n and i f - W E hU ( resp .

U - Y+ = U ( resp .

d e f i n e s a t o p o l o g y on

t h e n f o r U = n Y we have t h a t

U ) . I t i s easy t o check t h a t t h i s

i o n 4.4. If ; E S i s a Pare to s o l u t i o n f o r (PI - then

t h e r e e x i s t s y* B Y: and x* t a f s . t . (x*,I*) = i n f (x*,S). Y

Proo f . S ince by hypo thes is , x E S i s a Pareto s o l u t i o n o f (P_) - -- '* we know t h a t t h e r e i s a y* E Y+ s . t . f (.) achieves i t s

Y* minimum on S a t 2. Now c o n s i d e r t h e o p e r a t o r ? ' ( - ) = f ( * ) +

S S 6 ( 0 ) where 6 ( a ) i s t h e i n d i c a t o r o p e r a t o r o f S i . e .

+a i f X E S 6 s ( ~ ) = . The p o i n t x E S i s an uncons t ra ined

0 o t h e r w i s e

minimum i f i , ( * ) . Hence 0 E a? ( i ) = a ( f t as )(I*). But Y Y* Y* Y*

s i n c e f ( 0 ) i s assumed t o be c o n t ~ n u o u s on S f r o m t h e Moreau-

Rockafe l l a r theorem (see [30]) we have t h a t

D i r e c t l y from t h e d e f i n i t i o n o f t h e convex s u b d i f f e r e n t i a l we

have t h a t 36',(̂ x) = C ; c * E X* : (x*,S - i ) < 0). Y -

L e t x* E af *(j) and z* E as:,(;) s . t . x* + z* = 0. We Y

have f o r a l l z E S (x*,;) 5 (x*,z) which shows t h a t

(x*,;) = in f (x * ,S) . Q.E.D.

Be fo re pass ing i n t h e n e x t r e s u l t , we need t o i n t r o d u c e t h e

n o t i o n o f weak Pare to o p t i m a l i t y f o r m u l t i o b j e c t i v e ext remal prob-

lems. So l e t X be a l o c a l l y convex space and Y a l o c a l l y con-

vex o rdered space, which i s normal, w-sequent ia l l y complete, w i t h

Y, c l o s e d and w - i n t Y+ # 4 . L e t f : x -+ Y be an o p e r a t o r and l e t S 5 X be a w-closed

s e t .

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


D e f i n i t i o n 4.1. We say t h a t x E S i s a weak Pare to minimum o f f

on S i f and o n l y i f

{ x ' E X : f ( x ) - f ( x 1 ) E w - i n t Y + l n S = $.

We have t h e f o l l o w i n g r e s u l t f o r weak Pare to minima o f convex

opera to rs . By T ~ ( x ~ ) we denote t h e r a d i a l C l a r k e ' s tangen t cone

t o S a t xo (see [211) .

P r o p o s i t i o n 4.5. I f f : X + Y i s convex and xo i s a weak Pare to

m i n i m u m o f f ( * ) on S t h e n

r { h E X : f t ( x 0 , h ) E -w - i n t Y + l n T ~ ( x ~ ) = 4 .

Remark: The o p e r a t o r f ' ( x ; h ) i s t h e o - d i r e c t i o n a l d e r i v a t i v e of

f ( - 1 a t x i n t h e d i r e c t i o n h and i s d e f i n e d b y

Because o f t h e c o n v e x i t y of f ( * ) t h e l i m i t e x i s t s (see 1261) .

P roo f . Suppose no t . Then we can f i n d h e r:(xoi s . t .

f 1 ( x 0 ; h ) E -W - i n t Y,. S ince h r:(xo), we know f r o m [ Z l ] t h a t

t h e r e e x i s t s a sequence C$.,ln2, 5 IR+ s . t . An i 0 and

xO + X n h c S f o r a l l n > 1. A l s o -

f ( x o + Ah) - f ( x o ) f ( x 0 + Anh) - f ( x 0 ) f 1 ( x 0 ; h ) = i n f

X = 0 - l i m A>O kMo A n

and u s i n g Lemma 8 o f V a l a d i e r 1331

Since f t (xo;h) E -w - i n t Y+ we deduce t h a t t h e r e i s an

no s . t . f o r n 2 no

f ( x o + Anh) - f ( x o ) E -W - i n t Y,.

A n

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

102 PAPAGEORGIOU

A lso xo t Anh E S and f ( x o t Anh) << f ( x o ) , which c o n t r a -

d i c t s t h e assumption t h a t xo i s a weak Pare to minimum o f f ( . )

on S. Q.E.D.

We w i l l c l o s e o u r s tudy o f convex Pare to extrernal problems

w i t h a s u f f i c i e n t c o n d i t i o n f o r an uncons t ra ined Pare to in f imum of

an o p e r a t o r f ( * ) t o e x i s t .

So l e t X, Y be as b e f o r e .

P r o p o s i t i o n 4.6. I f f : X -t Y i s convex and f o r some y * E i n t Y;

fl,(x,d) 2 0 f o r a l l d E X Y

t h e n x i s a P a r e t o m i n i m u m o f f ( * ) on X.

P roo f . By d e f i n i t i o n we have t h a t

f * ( x t Ad) - f * ( x ) Y = l i m ( ~ * , f ( x t Ad) - f ( X I

f ' (x;d) = l i m A A Y* A+O A+O

From [35] we know t h a t

f ( x t Ad) - f ( x ) = o-lim f ( x + Ad) - f ( x ) f 1 ( x ; d ) = o - l i r n A A A40 X+O

So we g e t t h a t

S ince A -t f ( x ' Ad) - f ( x ) i s i n c r e a s i n g w i t h A we have A t h a t f t ( x ; d ) f ( x t d) - f ( x ) f o r a l l d E X and so f ' Y* (x ;d) =

= (y* , f1 ( x ; d ) ) 5 (y* , f (x + d ) - f ( x ) ) . But b y h y p o t h e s i s

f ' (x;d) 2 0 f o r a l l d E X. SO ( y * , f ( x + d) - f ( x ) ) 2 0 f o r Y*

a l l d E X which t e l l s us t h a t ( y * , f ( x ) ) = i n f ( y * , f ( X ) ) . S ince

y * E i n t Y:, u s i n g P r o p o s i t i o n 2.6 we conclude t h a t

f ( x ) t E f f ( f ( X ) ) . So indeed x i s a Pare tomin imum of f ( . ) on X. Q.E.D.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


As we a l r e a d y ment ioned i n [27] we developed an analog o f

C l a r k e ' s t h e o r y [ l o ] f o r l o c a l l y o - L i p s c h i t z o p e r a t o r s .

I f we have Y = Bn, wh ich i s a Banach l a t t i c e f o r t h e c o o r d i -

na te o r d e r i n g ( i .e. t h e o r d e r i n g induced by t h e p o s i t i v e o r t h a n t )

then we can e a s i l y check t h a t each component f u n c t i o n o f

f = f 1 . . . f n : X + R n i s a r e a l va lued l o c a l l y L i p s c h i t z func -

t i o n and t h e n t h e g e n e r a l i z e d o - d i r e c t i o n a l d e r i v a t i v e o f f ( * )

t akes t h e f o r m

Fur thermore, s i n c e t h e o r d e r i n g o f lRn i s coord ina tew ise we

conclude t h a t

where a f . i = 1, .. .,n i s t h e usual C l a r k e ' s s u b d i f f e r e n t i a l f o r 1

t h e R - va lued l a c a l l L i p s c h i t z fi ( * ) .

So f o r t h e n e x t r e s u l t , l e t X be a normed space.

Theorem 4.1. I f f : X -+ I R ~ i s l o c a l l y o - L i p s c h i t z , f ( X ) i s

IRP-convex w i t h i n t f ( X ) # 4 and i s a Pare to minimum o f f ( . )

on X t h e n f o r some y* E IR: and some A aft:) we have t h a t

(y*,Au) = 0 f o r a l l u E X.

Proof. Because ? i s a Pare to minimum o f f ( 0 ) on X, t h e n - f(G) E E f f ( f ( X ) ) . So we know f r o m P r o p o s i t i o n 2.6 t h a t t h e r e

e x i s t s y* E &: s . t . (y*,f(;)) = i n f ( y * , f ( X ) ) . Hence i e X i s a minimum of t h e IR- va lued l o c a l l y L i p s c h i t z f u n c t i o n

x + f ( x ) = ( y * , f ( x ) 1 . Then we know t h a t 0 E af ( ; I . Y* n Y* Observe t h a t a f ( 1 = a Y * f = a 1 (Y;,fi ( i ) ) and

Y* i = l s i n c e t h e g e n e r a l i z e d s u b d i f f e r e n t i a l i s s u b a d d i t i v e , we have t h a t

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

104 PAPAGEORGIOU

n n Because ?f(;) = a f i i x ) we have t h a t I iy?,af i t j ) ) =

i = l i = l , . = (y*,ai( ;) j . Hence we conclude t h a t 0 E ( ~ * , a f ( 2 ) ) which shows

t h a t t h e r e i s an A E 3f(:) s . t . (y*,Au) = 0 f o r a l l u E X.

Q . E . D .

F i n a l l y we are go ing t o see how t h e e f f i c i e n c y s e t i s e f f e c t e d

by t h e a c t i o n o f a l i n e a r o p e r a t o r . But b e f o r e go ing i n t o t h a t we

need t o develop some aux i 1 i a r y m a t e r i a1 . A f i r s t remark t h a t we would l i k e t o make i s t h e f o l l o i w n g .

I f q E Ef f (S) t h e n q E T - bd S. Suppose n o t . Then q E -r - i n t S

So we can f i n d U E N ( 0) = { f i l t e r o f - -ne ighborhoods o f t h e o r i g i n )

s . t . q + U - c S. We can take U t o be symmetr ic. So i f

y + E U n Y, t h e n -y+ E U 0 (-Y,) and q - y, E S. Bu t t h e n

t h i s c o n t r a d i c t s t h e hypo thes is t h a t q E E f f ( S ) . So q E T - b d S.

A second u s e f u l remark i s t h a t i f X i s a l s o ordered, by a c losed,

convex cone and A E L'(x,Y) ( i .e. A(X+) 5 Y,) then * * * * A* E L' ( y ,X ) where X , Y are endowed w i t h t h e dual o r d e r i n g .

F i n a l l y we remind t h e reader t h a t f o r a convex se t , t h e i r c l o s u r e s i n

t h e o r i g i n a l and weak t o p o l o g i e s c o i n c i d e ( i .e. c losedness i s d u a l i t y

i n v a r i a n t f o r convex s e t s [33]).

For t h e n e x t r e s u l t assume t h a t X, Y are l o c a l l y convex * * * * F r e c h e t o rdered space and t h a t Y, i s g e n e r a t i n g i . e . Y = Y, = Y+.

* * Theorem 4.2. If A E L' ( x ,Y) and A* E L'(Y ,X ) a re b o t h s u r j e c -

t i v e and S 5 X i s open and convex t h e n E f f ( A ( S ) ) = A ( E f f ( S ) ) .

Proof. F i r s t l e t q s E f f ( S ) . Then qs w - cl S. From t h e

w - c o n t i n u i t y o f A we have t h a t A(w - c l S) 5 w - c l A (S) . Hence

A ( q ) E w - c l A(S) . Suppose t h a t A ( q ) $ E f f ( A ( S ) ) . Then t h e r e i s ' * s E S s . t . A (s ) < A ( q ) . So we can f i n d y * E Y-, s . t . ( y * ,A(s ) )

< (y * ,A(q ) ) which i m p l i e s t h a t ( A * ~ * , S ) ' ( A * ~ * , ~ ) . Using t h e * * we have t h a t A y* E X+. Then t h e l a s t

q which c o n t r a d i c t s t h e f a c t t h a t

g e t t h a t

f a c t t h a t

i nequal i t y

q E E f f ( S )

A ( E f f

* * A* E L+(Y ,X

says t h a t s <

. There fo r0 we

' ( S ) ) 5 E f f (A (S)

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


S A

and

Now l e t p E E f f (A(S) 1 . Consider t h e s e t A - ' ( p ) . Suppose

t h a t f o r some u E A - ' ( ~ ) we have t h a t u 9 E f f ( S ) . Th is means

t h a t t h e r e e x i s t s s f S s . t . s < u. Note t h a t ~ ( u ) = p E bd A(S)

w h i l e A(s ) E A(S) and by t h e "Open Map Theorem" A(S) i s open.

So A ( u ) # A ( s ) and s i n c e by hypo thes is A i s monotone, we conclude

t h a t A ( s ) < p. But t h e n we c o n t r a d i c t t h e f a c t p E E f f ( A ( S ) ) .

So f o r a l l o E E f f ( A ( S ) ) we have t h a t A - ' ( ~ ) c E f f ( S ) and so - f ( S ) ) which means t h a t

( 2 ) we conclude t h a t E f f ( A ( S 1 ) = A ( E f f ( S ) ) .

Q.E.D.

Remark. From t h e above p r o o f we see t h a t i n genera l i f A f L + ( x , Y )

A* i s s u r j e c t i v e and S 5 X i s an a r b i t r a r y s e t t h e n

5 ) STOCHASTIC PARETO EFFICIENCY

I n t h i s s e c t i o n we s t u d y P a r e t o e f f i c i e n c y , i n t h e case where

t h e s e t s S depends measurably ( i n a sense t o be d e f i n e d l a t e r )

on a parameter w E $2.

I n a p p l i c a t i o n s , t h i s i s t h e s i t u a t i o n w h e n w e have ameasurespace

(R,C,u) o f agents, each one o f them hav ing a f e a s i b l e s e t o f

d e c i s i o n F ( o ) i n t h e a c t i o n space X. Then we are l o o k i n g f o r

t h e Pare to e f f i c i e n t d e c i s i o n s o f each i n d i v i d u a l agent, b u t we

may a l s o want t o c o n s i d e r t h e aggregat? e f f i c i e n c y s e t i n which

case we need t o d e f i n e i n some sense F(w)du(w) . The use o f i a a measure space of agents i s a common p r a c t i c e among mathemat ica l

economists (see [19]) and i t i s b e l i e v e d t h a t such a model cap-

t u r e s b e t t e r t h e s p i r i t o f p e r f e c t c o m p e t i t i o n .

Another i n t e r e s t i n g i n t e r p r e t a t i o n t h a t we can have, i s t o

assume t h a t t h e s e t o f f e a s i b l e d e c i s i o n s i s t h e outcome o f a random

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

106 PAPAGEORCIOU

event which belongs t o a p r o b a b i l i t y space . I f we a re

i n t e r e s t e d i n t h e mean e f f i c i e n c y , we need t o s tudy t h e s e t i

S = I F(w)dv(w). 12

So we w i l l s t a r t w i t h a b r i e f o u t l i n e o f some b a s i c f a c t s

about measurable m u l t i f u n c t i o n s .

F o r t h e moment, l e t R, Y be two a r b i t r a r y s e t s and c o n s i d e r Y

t h e m u l t i f u n c t i o n ( s e t va lued f u n c t i o n ) F : R - t 2 . We c a l l t h e

graph o f F t h e s e t GrF = i (w ,y ) E R x Y : y E F ( w ) l . A l s o i f

U 5 Y, t h e weak i n v e r s e image o f U under F ( * ) i s d e f i n e d t o be

, t h e s e t F - ( u ) = { w E n : F(w) U # 43. The n e x t r e s u l t c o l l e c t s

and i n t e r r e l a t e s t h e v a r i o u s d e f i n i t i o n s o f measurabi 1 i t y o f m u l t i -

f u n c t i o n s t h a t e x i s t i n t h e l i t e r a t u r e ( f o r d e t a i l s see ( H i l d e r n b r a n d

[19], Himmelberg [20], R o c k a f e l l a r [31] 1321)

Theorem 5.1. L e t ( R Y E ) be a measurable space and Y a separab le

m e t r i c space.

L e t F

c l o s e d f o r a

( 1 ) F-

( 2 ) F-

( 3 ) F -

Sl -t 2' be a m u l t i f u n c t i o n s . t . F ( w ) i s nonempty and

1 w E R. Consider t h e f o l l o w i n g s tatements.

B ) E C f o r a l l B E B(Y) = Bore1 o - f i e l d o f Y .

C) E C f o r each c l o s e d s e t C.

U) E C f o r each open s e t U.

(4) LO -+ d (y ,F(w)) i s a measurable f u n c t i o n f o r a l l y E Y.

( 5 ) F = c 1 f l where fn : Q - Y are measurable se lec -

t o r s o f F ( * ) ( "Cas ta ing r e p r e s e n t a t i o n ) .

( 6 ) Gr F E 1 x B(X) .

Then we have t h e f o l l o w i n g r e l a t i o n s

(a) ( 1 ) * ( 2 ) * ( 3 ) * ( 4 ) * ( 6 ) .

(4) I f Y i s P o l i s h t h e n ( 3 ) * ( 5 ) .

(y) I f Y i s P o l i s h and t h e r e i s a complete a - f i n i t e measure

on a. then a1 1 t h e above statements are e q u i v a l e n t .

I n t h e sequel, we w i l l assume t h a t ( R , C , u ) i s a complete

o - f i n i t e measure space ( f o r example a complete p r o b a b i l i t y space)

and Y a separable Banach space o rdered by a c losed, convex cone * Y+. A l s o we w i l l assume t h a t Y+ i s genera t ing . The v e c t o r

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


valued i n t e g r a l s t h a t w i l l be cons idered i n t h i s s e c i t o n a re

assumed t o be taken i n t h e sense o f Bochner. Furthermore, r e c a l l

t h a t s i n c e Y i s separable, a l l t h r e e k i n d s o f v e c t o r i a l measur-

a b i l i t y (Bochner, P e t t i s and Bore1 c o i n c i d e (see [14 ] ) .

We s t a r t w i t h two u s e f u l lemmas.

Lemma a. I f f : R + Y i s measurable then @ ( w ) = f ( w ) - Y+

has a graph i n C x B(Y) .

Proof . By d e f i n i t i o n Gr $ = {(w,z) R x Y : f ( w ) - z € Y+}.

Def ine t h e f u n c t i o n g : R x Y -t Y by g(w,z) = f ( o ) - z. C l e a r l y

t h i s i s a Caratheodory f u n c t i o n and so i s j o i n t l y measurable.

S ince Y i s separab le we have t h a t I ( w , z ) E R x Y : g(w,z) E Y + l E C x B(Y) Hence Gr 4 E C x B(Y) . Q.E.D.

Ex tend ing t h e n o t a t i o n i n t r o d u c e d i n S e c t i o n 2, t o f u n c t i o n s ,

we s e t f, ( f2 i f f l (w) ( f 2 ( w ) u-a.e. and fl < f2 i f

fl ( f 2 and v h E R : f 2 h ) - f l (w) E ?+} > 0.

Proof. ( i ) Since by hypo thes is fl(w) 5 f 2 ( w ) p-a.e. then f o r

a11 y* E ~f we have t h a t (y*, f l ( a ) ) - < i y * , f 2 ( w l l p-a.e. Hence

which means t h a t

( i i ) L e t A = { w e S3 : f 2 ( w ) - f , lw) E ?+I . Since r r r

Jn f i (w)du(w) = fi (w)dp(w) +

J A fi(w)dv(w) i = 1,2, i t J, A

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

108 PAPAGEORGIOU

su f f i ces t o show t h a t 1 f l (w)dP(w) r f Z ( w i d u ( o ) . Suppose t h a t J A 1 A

r I

J A f l (

* f2 ( t i )du (w) . Then f o r a l l y* r Y, we have t h a t

r (y*, I Q ( f 2 ( ~ ) - f l ( u ) ) d u ( u ) ) = 0 and so (y*,f2(w) -

A - f l ( L o ) ) d i ~ ( ~ ) = 0. Hence ( y * , f Z ( o ) - f l (cu) i = 0 p-a.e. f o r a l l

y* E Y:. Since Y: i s generating we concl ude t h a t

( y * , f 2 ( u ) - f l ( w ) ) = 0 p-a.e. f o r a l l y* t Y* which means t h a t f l ( d ) = f 2 ( w ) u-a.e. a contradic t ion. Q . E . D .

* Remark: The assumption t h a t Y + i s generating i s not a t a l l * r e s t r i c t i v e . We know t h a t i f Y, i s a normal cone, then Y + i s

generating (Kre in ' s theorem). So f o r a l l Banach l a t t i c e s , Y: i s

generating. * * Another case where Y+ i s generating, i s when i n t Y + f @.

This i s t he case with M ( K ) = Regular Bore1 measures on a compact

Hausdorff space K . Recall t h a t M ( K ) = [ c ( K ) I * . Simi lar ly f o r 1 * ~ ~ ( n ) = [ L ( 2 ) ] .

We say tha t a measurable multifunction F : P + 2' i s

in tegrably bounded i f I F ( ~ ) i = sup{liyll : y E F ( w ) } ( u(o ) u-a.e. 1 with u ( . ) E L (2). By P f ( Y ) we wi l l denote the nonempty closed

subsets of Y, by P f c ( Y ) the elements of Pf ( Y ) , t h a t are a l so

convex and f ina l ly by P k c ( Y ) , the nonempty, compact, convex sub-

s e t s of Y . When the l e t t e r w appears in f ron t of f ( r e s p . k ) we wi l l mean weakly closed ( r e sp . weakly compact). Recall t h a t in

a loca l ly convex space a convex s e t i s w-closed i f and only i f i t i s 1 closed. By SF we will denote the s e t of a l l i n t eg rab le se l ec to r s

1 1 of F ( . ) i . e . SF = I f E ~~(l i) : f (w) E F(w) u-a.e.1. Using t h a t

s e t we can define an in tegra l f o r F ( * )

The in tegra l was f i r s t introduced by Aumann [3] f o r Y = IRn , as a genera l iza t ion of t h e in t eg ra l of a s i n g l e valued function and

of the sum of s e t s . Clearly i f S: = 4 then la F(w)d~iw! = @ . How-

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


ever i f F ( * ) i s i n t e g r a b l e bounded and has a measurable graph

t h e n by Aumann's s e l e c t i o n theorem (see [30]) we conclude t h a t 1 SF f 4 and so In F(w)dv(w) f $I. Assume t h a t Y* has t h e Radon-

Nikodym p r o p e r t y (see [14 ] ) .

Theorem 5.2. If F : R -. Pwkc(Y) i s an i n t e g r a b l y bounded m u l t i - 1 f u n c t i o n then f o r every q E E f f ( S ) we can f i n d f t SF s . t .

q = I, f ( w ) d u ( w ) and f (w) E E f f ( F ( w ) ) p-a.e.

P roo f . We w i l l s t a r t by showing t h a t S = F ( w l d v ( w ) i s w-com- 1 1 * p a c t . Note t h a t SF i s w-closed. L e t g E L ~ * ( R ) = [ L y ( n ) l and

c o n s i d e r t h e mu1 t i f u n c t i o n

Since F( . ) i s PwkC(Y) -va lued A(w) # @ f o r a l l w E R. ~ l s o

i t i s convex, c l o s e d and by R o c k a f e l l a r [321 we g e t t h a t w -t A(w)

i s measurable. Hence we can app ly t h e Kura towsk i -Ry l l Nardzewski

s e l e c t i o n theorem t o f i n d f" : R -. Y measurable s . t . ?(w) E A ( i i i )

1 w Q. C l e a r l y ? E SF. Now f r o m Theorem 2.2 o f 1181 we g e t t h a t

r sup ( g , f ) = sup ( g ( w i , f ( w ) d y ( o ) = j Sup ( g ( ~ ) , x ) d u i w ) . fa: fa: Jn R x=F(w)

1 where ( . , a ) denotes t h e d u a l i t y b r a c k e t s between Ly ( i l l and 1 * $ ( i l ) . S ince g ( - ) t L ~ * ( Q ) = [ L y ( R ) l was a r b i t r a r y we conclude

from James1 theorem (see [16]) t h a t 5: i s w-compact. S ince t h e i n t e g r a l i s a w-continuous l i n e a r o p e r a t o r we conclude t h a t S i s

w-compact. So q E S. Hence we know t h a t t h e r e e x i s t s f E S: s . t .

q = f ( w ) d u ( w ) . Suppose t h a t t h e r e e x i s t s A € Z s . t . I J ( A ) > 0

and f o r w E A f ( w ) + E f f ( F ( o 1 ) . Th is means t h a t f o r a l l w E A

- Y,) n F ( a ) f 4 . Working on t h e measure space (AYE n A,yA) we de f ine @ l ( ~ ) = ( f ( w ) - % + I n F ( w ) . We know f r o m Lemma a i n

t h i s s e c t i o n t h a t G r ( f ( * ) - P,) E E x B(Y) and f r o m Theorem 5.1

we have t h a t G r F E C x B(Y) . Hence Gr E (Z n A) x B(Y) .

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

110 PAPAGEORGIOU

Using Aumann's s e l e c t i o n theorem [201 we g e t s : A + Y measurable

s . t . s (w) E $l(w) pA-a.e.

Going back t o t h e f u l l measure space (R,c,~) cons ider t h e

f u n c t i o n

A

C l e a r l y t h i s i s measurable. A lso s < f. Using Lemma 4 we

deduce t h a t ;(w)du(w) < j f ( w ) d u ( ~ u ) . But ; = In ;(w)du(w) E S J R R

F ( w ) d p ( w ) . Hence we c o n t r a d i c t t h e hypo thes is q E E f f ( S ) . So

f (w) E E f f ( F ( w ) ) p-a.e. Q.E.D.

Remark. Roughly speaking t h e above r e s u l t says t h a t ~ f f (Il? F ( w ) d u ( o ) 1

c / Ef f iF( ;u i )dU(w). I n [28] we have a r i g o r o u s p r o o f and under - R some m i l d a d d i t i o n a l hypo thes is we prove t h a t e q u a l i t y h o l d s .

Wi th a d d i t i o n a l assumptions on ( R , c , ~ ) we can have t h e f o l -

l o w i n g remarkable r e s u l t about E f f (S) .

Theorem 5.3 . I f i n a d d i t i o n (il,C,u) i s atomless then E f f ( S ) i s

connected.

Proof . From Theorem 4.2 o f [ I 8 1 we g e t t h a t c l S i s convex. But

as we saw i n t h e p r o o f o f Theorem 5.2 S i s w-compact. Hence S

i s convex and w-compact. Then app ly P r o p o s i t i o n 3.2. Q.E.D.

For t h e Pare to e f f i c i e n t p o i n t s o f S, we have t h e f o l l o w i n g

u s e f u l " p r i c e c h a r a c t e r i z a t i o n " .

Theorem 5.4, I f F : n + Pfc(Y) i s i n t e g r a b l y bounded, ' * i n t F(w) f + u-a.e. and q E E f f ( S ) then t h e r e e x i s t s y* E Y+ s . t .

(y*,q) = i n f ( y * ,y )du(w) . L ,F(u,

P roo f . F i r s t n o t e t h a t f r o m [ll! we have t h a t i n t S = -- = I O i n t F ( w ) d p ( w ) # I$. So f r o m P r o p o s i t i o n 2.4 we know t h a t t h e r e

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PARETO EFFICIENCY I

. * e x i s t s y* E Y+ (y*,q) = i n f (y*,S) and so

(y*,q) = i n f ( y * , = i n f ( y * , j f ( w i d u ( w i ) . Using

'=F R Theorem 2.2 o f [18] we deduce t h a t

I n s e q u e n t i a l d e c i s i o n problems, t h e a - f i e l d i s n o t f i x e d ,

b u t i n s t e a d we have a sequence o f s u b - n - f i e l d s {CnInLl o f L

t h a t i nc reases as we accumulate i n f o r m a t i o n about t h e process.

Lo So i n cases l i k e t h i s , we need t o s t u d y E F ( * ) , where co 5 C .

I n [18], t h e au thors i n t r o d u c e d a s e t va lued c o n d i t i o n a l expecta-

t i o n u s i n g Aumann's s e t va lued i n t e g r a l , Th is c o n d i t i o n a l expecta- - Lo

t i o n i s a m u l t i f u n c t i o n E F ( * ) : R + Pf(Y) which i s Co-measur- - 1

a b l e and S' = c l { E f : f E SF! t h e c l o s u r e b e i n g taken i n t h e

?F L '-norm.

F o r t h e n e x t r e s u l t l e t Y = lRn and suppose t h a t Y i s

p o l y h e d r a l . Then Y * i s p o l y h e d r a l t o o and so f i n i t e l y generated +* * n

(see [301) . L e t G(Y+) = iykIkzl be t h e s e t o f t h e generators o f

y+.

Theorem 5.5. I f F : -+ P f ( Y ) i s an i n t e g r a b l y bounded m u l t i -

f u n c t i o n and f ( w ) E w E f f ( F ( w ) ) p-a.e. then

Co I 0 E f ( o ) E w E f f ( E F ( w ) ) p-a.e.

I 0 1 Proof , L e t g(w) = E f ( w ) E Ly(R,CO). Suppose t h a t t h e conc lu -

s i o n o f t h e theorem i s f a l s e . T h i s means t h a t t h e r e i s an

Lo A E LO s . t . v ( A ) > 0 and f o r w E A g(w) w E f f ( E F ( w ) ) . The

" Lo l a t t e r i m p l i e s t h a t (g (w) - i n t Y,) n E F ( w ) # + f o r w E A.

Consider t h e m u l t i f u n c t i o n d : (AYEA = C n A.uA) + 2' def ined P

by $ ( a ) = ( g ( w ) - i n t Yt) n E F h ) . Using Lemma 4 i n [ l g ] and

Theorem 5.1 we can see t h a t Gr 4 E Co x B ( Y ) . So by Aumann's

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

1 1 2 PAPAGEORGIOU

s e l e c t i o n theorem we can f i n d s : A -t Z IoA-measurable s .t.

s(w) E 4 (w) uA-a.e.

Now c o n s i d e r t h e f o l l w o i n g f u n c t i o n d e f i n e d on ( R , C , L I )

- 1 C l e a r l y t h i s i s Co-measurable and f u r t h e r m o r e s E S T A

1 E-"F Co L y ( " , C o ) ,,

So t h e r e e x i s t s { s 5 . t . E P,, -- + s. By pass-

i n g t o an a p p r o p r i a t e subsequence {m} - c I n 1 we can have t h a t n

I\

E P,(w) - S ( O ) p-a lmost u n i f o r m l y Hence f o r any E > 0

0 < E < LI (A) we can f i n d n o ( € ) s a t . f o r n 2 n 0 ( c ) we have t h a t

Lo f o r a l l U E 2 = A \ A E w h e r e O < v ( A ) E < E and so 0 < E ( f - p n ) ( u )

f o r u E A, n 2 n o ( & ) . A

NOW c o n s i d e r t h e m u l t i f u n c t i o n : (A.zO ,q) - Y: d e f i n e d H

by $(u) = {y* E GIY;) : ( y * , ( f - q n ) (w) 5 01 . Suppose t h a t

+(w) # 4 f o r w E A ' E_ A w i t h ~ J ( A ' ) > 0. Viewed as a m u l t i -

f u n c t i o n on A ' , + ( a ) has nonempty, c l o s e d va lues and s i n c e

G(Y:) i s f i n i t e i t i s ZgnA1-measurable. Hence f r o m t h e Kuratowski -

R y l l Nardzewski s e l e c t i o n theorem [20 ] we can f i n d a xf lA'-measur-

a b l e s e l e c t o r ^y* : A ' 4 G(Y:) s , t . * € ( a ) w ' A ' . So we n

have t h a t ;*(w) = (wly; Ai ' i 0 n A 1 y; E G(Y:). Observe i = l *I io

t h a t f o r v E A ' 0 < (y * (w) ,E f - p n ( 1 and so

Lo We c l a i m t h a t E xA = XAi. To prove t h a t we proceed as

i f o l l o w s

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


Lo Hence we have t h a t E xA (w) < 1 - pAl-a.e. Le t h E ( 0 , l ) . i,*

Then consider Di = {U E Ai . . C lea r l y Dig z 0 n A '

So O < (A - l)u(Di). Since by hypothesis A € ( 0 , l ) we get t h a t

0" "Di) = 0. So E x ( w ) f o r LLI Ai. L e t t i n g X + l we Ai ,*

f i n a l l y ge t t h a t U E x (w) = 1 on A. p

A i , A,-a.e. Since

00 C * 0 E L In) we conclude t h a t E x (u) = x (w) pA1-a.e.

A i Ai ,* Ai

Using t h i s f a c t i t i s easy t o see t h a t ;O;*(LLI) = j * ( u ) uA,-a.e. So we have

But from the d e f i n i t i o n o f ;(.) we have t h a t

fo r n 2 n0(s) a con t rad i c t i on . So $(w) = $ pi-a.e. Hence

0 ( f - pn)(w) u p e . n 2 no(&). Next de f i ne f o r n 2 n o ( r ) .

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

PAPAGEORGIOU

A 1 Clearly G n i f and u n € SF. So f ( a ) $ Eff(F(w)) f o r

w E LI(?) > 0, a contradic t ion, Therefore

Q . E . D .

In [28] we complete our study on i n f i n i t e dimensional Pareto

e f f i c i ency by studying t h e s t a b i l i t y of the e f f i c i ency s e t under

perturbations of the data determining i t .

ACKNOWLEDGMENTS

I wish t o express my graditude t o Professor Gi lbe r t Strang

( M . I .T . ) and Professor Roger Brockett (Harvard) f o r continuous

support and encouragement during t h e preparation of t h i s work.

REFERENCES

[ I ] K . J . Arrow - E . W . Barankin - D. Blackwell, Admissible points of convex s e t s , i n Contributions t o t h e Theory of Games, Vol. 2 eds. H . Kuhn , A . W . Tucker, Princeton Univ. Press, Princeton (1953).

[2] L . Asimow - A . J . E l l i s , Convexity Theory and i t s Applications t o Functional Analysis, Academic Press, London (1980).

[3] R . Aumann, " In teg ra l s of s e t valued funct ions , J . Math. Anal. Appl. E ( 1 9 6 5 ) , 1-12.

[4] H . P. Benson, Existence of e f f i c i e n t so lu t ions f o r vector maxi- mization problems, J . Optimization Theory Appl . 6( 19781, 569-580.

[ 5 ] G. R . Bitran - T. Magnanti, The s t r u c t u r e of admissible points with respect t o cone dominance, J . Optimization Theory Appl. 29( 1979), 573-614. -

[6] D. Blackwell - M. A . Girshick, Theory of Games and S t a t i s - t i c a l Decisions, Wiley, New York (1954).

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014


J. Borwein, Proper e f f i c i e n t p o i n t s f o r max im iza t ion w i t h r e s p e c t t o cones, SIAM J. C o n t r o l O p t i m i z a t i o n l!j( 19771, 57-63.

L. Cesar i - M. B. Suryanarayana, Ex is tence theorems f o r Pare to o p t i m i z a t i o n mu1 t i valued and Banach space va lued f u n c t i o n a l s, Trans. Amer. Math. Soc. %(1978), 37-65.

, An e x i s t e n c e theorem f o r Pareto problems, N o n l i n e a r Anal . 2( l 9 7 8 ) , 225-233.

F. C la rke , Genera l i zed g r a d i e n t s o f L i p s c h i t z f u n c t i o n a l s, Adv. Hath. 40(1981) , 52-67. - R. Cornwal l , Cond i t i ons f o r t h e graph and t h e i n t e g r a l o f a correspondence t o be open, J. Math. Anal . Appl . 39( 19721, - 771 -792.

G. Debreu, Represen ta t ion o f a p r e f e r e n c e o r d e r i n g by a numer ica l f u n c t i o n , i n D e c i s i o n Processes, eds. R. M. T h r a l l , C. H. Coornbs, R. L. Davis, Wiley, New York (1954) .

, Theory o f Value, Yale Uni v. Press, New Haven ( 1959).

J. D i e s t e l - J. J. Uhl, Vec to r Measures, Math. Surveys Vol. 15, A.M.S. Prov idence (1977) .

N. Dunford - J. Schwartz, L i n e a r Operators, Vol . I , Wiley, New York (1958) .

K. F l o r e t , Weakly compact se ts , L e c t u r e Notes i n Math. Vol . 801 Spr inger , B e r l i n (1980) .

A. M. G e o f f r i o n , Proper e f f i c i e n c y and t h e t h e o r y of v e c t o r maximizat ion, J. Math. Anal. Appl . 22( l 9 6 8 ) , 618-30. - F. H i a i - H. Umegaki, I n t e g r a l s , c o n d i t i o n a l e x p e c t a t i o n s and M a r t i n g a l e s o f M u l t i v a l u e d Func t ions , J. M u l t . Anal. 7(1977) , 149-182.

-

W. H i ldenbrand, Core and E q u i l i b r i a o f a Large Economy, P r i n c e t o n Uni v. Press, P r i n c e t o n (1974) .

C . Himmelberg, Measurable r e 1 a t i o n s , Fund. Math. LXXXVII (19751, 53-72.

J. 0 . H i r i a r t - U r r u t y , New concepts i n n o n d i f f e r e n t i a b l e p r o - gramming, B u l l . French Math. Soc. g( 1979), 57-85.

R. Keeney - H. Ra i fa , Dec is ions w i t h M u l t i p l e O b j e c t i v e s : P re fe rences and Value Trade-o f f s , W i l e y , New York (1976) .

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

116 PAPAGEORGIOU

[23] V . Klee, Separa t ion p r o p e r t i e s o f convex cones, Proc. Amer. Math. Soc. 6 (1955) , 313-318.

[24] P . H. Nacchache, Connectedness o f t h e s e t o f nondominated outcomes i n m u l t i c r i t e r i a n o p t i m i z a t i o n , J. O p t i m i z a t i o n Theory Appl . - 25( 1978), 459-467.

[25] J . W. Nieuwenhui s, Some r e s u l t s about nondomi na ted s o l u t i o n s , J. O p t i m i z a t i o n Theory Appl . 36-( 19821, 289-301

[26] N. S. Papageorgiou, "Nonsmooth a n a l y s i s on p a r t i a l l y o rdered v e c t o r spaces, P a r t 1: Convex case, P a c i f i c J. Math 107(2) ( 19831, 403-459.

[271 , Nonsmooth a n a l y s i s on p a r t i a l l y o rdered v e c t o r spaces,Part 2: Nonconvex case, C l a r k e ' s theory , P a c i f i c J. Math. l O g ( 2 ) ( l 9 8 3 ) , 463-495.

[ 2 8 I , Pare to e f f i c i e n c y i n l o c a l l y convex spaces 11: S t a b i l i t y r e s u l t s , submi t ted .

[29] A. P e r e s s i n i , Ordered T o p o l o g i c a l Vec to r Spaces, Harper-Row, New York ( 1967).

[30] R. T. Rockafe l l a r , Convex Ana lys is , P r i n c e t o n Univ . Press, P r i n c e t o n (1976) .

131 1 , Measurable dependence o f convex s e t s and f u n c t i o n s on parameters, J. Math. Anal. Appl. 28(1969), 4-25.

1321 , I n t e g r a l f u n c t i o n a l s, normal i n t e g r a n d s and measurable s e l e c t o r s , i n N o n l i n e a r Operators and t h e Ca lcu lus o f V a r i a t i o n s , eds. Gossez e t a1 . , L e c t u r e Notes i n Math. Vol. 543, B e r l i n (1976) .

[33] H. Schaefer, Topo log ica l v e c t o r spaces, Graduate Texts i n Math. Vol . 3, Spr inger , New York (1971) .

[34] S. Smale, Global a n a l y s i s and economics 111: P a r e t o opt ima and p r i c e e q u i l i b r i a , J. Math. Econ. 1 (1974) , 107-117.

[35] M. Va lad ie r , Sous d i f f e r e n t i a b i l i t 6 de f o n c t i o n s convexes a v a l e u r s dans une espace v e c t o r i e l l e ordonn6, Math. Scand. 3O( l 9 7 2 ) , 65-74. -

[36] P. L. Yu, Cone convex i t y , cone extreme p o i n t s and nondominates s o l u t i o n s , i n Dec is ions Problems w i t h M u l t i p l e Ob jec t i ves , J. O p t i m i z a t i o n Theory Appl . E i 1 9 7 4 1 , 319-337.

Dow

nloa

ded

by [

UZ

H H

aupt

bibl

ioth

ek /

Zen

tral

bibl

ioth

ek Z

üric

h] a

t 21:

22 1

2 Ju

ly 2

014

pareto efficiency in locally convex spaces i

Documents