On Approximate Vector Optimization

Working Paper

ON APPROXIMATE VECTOR OPTIMIZATION

Istvtin V t i l y i

January 1986 W - 8 6 - 7

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOE QUOTATION WITHOUT THE PERMISSIOK OF THE AUTHOR

ON

APPROXIMATE VECTOR OPTIMIZATION

Istv&n ^V&lyi

January

1986

WP-86-7

Working Papers are interim r e p o r t s o n 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 Anaiysis a n d h a v e r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o n o t 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 I n s t i t u t e o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 L a x e n b u r g , Austria

PREFACE

The r o o t s of c u r r e n t i n t e r e s t in t h e t h e o r y of approximate soiutions of optimiza- tion problems lie in approximation t h e o r y and nondifferentiable optimization. In this p a p e r a n approximate saddle point t h e o r y is p r e s e n t e d f o r v e c t o r vaiuea con- vex optimization problems. The considerations c o v e r different possible t y p e s of approximate optimality, including both t h e efficient, o r Pareto-type, which is more frequently used in p r a c t i c a l decision making applications, and t h e absolute, o r s t r i c t type, which is more of t h e o r e t i c a l i n t e r e s t . The saddle point theorems a r e used t o study duality in t h e context of approximate solutions. The a p p r o a c h of t h e p a p e r a i s o provides f o r a unified, view of a number of r e s u l t s achieved e i t h e r in approximate s c a l a r optimization o r e x a c t v e c t o r optimization.

Alexander B. Kurzhanski Chairman

Systems and Decision Sciences Area

CONTENTS

2. S t r i c t Optima

2.1. Approximate Extremai Points and Approximate Soiutions 2.2. T'ne Sacdie Point Tneorems

2.3. Primai and Dual Problems

3.1. Approximate Non-dominated Eiements 3.2. Saddle Point Tneorems

3.3. Primal and Dual Functions

4. References

ON APPROXIMATE VECTOR OPTIMIZATION Istvdn Vdlyi

1. INTRODUCTION

The aim of t h e p r e s e n t p a p e r i s t o give t h e p r o o f s of t h e t h e o r y p r e s e n t e d at t h e IIASA Workshop on Nondifferentiable Optimization held between 1'7 and 22 Sep- tember, 1984, Sopron, Hungary. (See Vdlyi (1985a)),but some m o r e r e c e n t r e l a t e d r e s u l t s are a l s o included.

The c e n t r a l r e s u l t s are Hurwitz-type saddle point theorems corresponding t o ap- proximate solutions extending t h e t h e o r y developed by Zowe (19'7'7) f o r one t y p e of optima, o r by Tanino and Sawaragi (1980) f o r a n o t h e r . By t h e s e theorems then w e investigate t h e r e s p e c t i v e duality t y p e problems. The study of this s u b j e c t was s t a r t e d by Hiriart-Urruty (1982) and S t r o d i o t et a l . (1983) in t h e s c a l a r c a s e , and by Kutateladze (19'78) and Loridan (1984) in t h e v e c t o r valued c a s e .

The p a p e r is divided into two p a r t s according t o t h e t y p e of optimality considered.

Chapter 2. c o v e r s t h e c a s e of s t r i c t , o r non-Pareto optimality. This type of optim- ization in o r d e r e d s p a c e s i s r e g a r d e d by many as having little p r a c t i c a l use. This criticism, however is of l e s s f o r c e in t h e approximate case, since f o r some vaiue of t h e approximation e r r o r w e may find soiutions even if e x a c t solutions d o not e x i s t (like t h e s o called utopia point s o often used in t h e P a r e t o case). Anyway, i n t e r e s t in i t a p p e a r s t o be lasting as is shown e. g. by t h e r e c e n t p a p e r of Azimov (1982).

Section 2.1. is devoted t o some basic p r o p e r t i e s of approximate extremal elements in o r d e r e d v e c t o r s p a c e s and in Section 2.2. t h e main r e s u l t s are proved. Appiica- tions expounded in Section 2.3. c l a r i f y t h e relationships between approximate sad- dle points and approximate solutions of t h e primal and dual problems associated t o t h e original problem. In this w e a l s o show t h e connections t o analogous r e s u l t s , namely t h e corresponding Kuhn-Tucker theorems based on t h e c-subgradient cal- culus, obtained by Kutateladze (19'78). In such a way t h e analogy will be complete with t h e t h e o r y developed f o r t h e scaiar case in t h e p a p e r by Strodiot e t al.

(1983). Finally w e give a p a r t i a l generalization t o t h e v e c t o r valued c a s e of

Golshtein's duality t h e o r e m dealing with generalized solutions of convex optimiza- tion problems and of Tuy's r e s u l t c h a r a c t e r i z i n g well posed problems, i. e. t h o s e w h e r e t h e primai a n d d u a l vaiues coincide. (See 14H in Holmes (1975)).

C h a p t e r 3. d e a l s with P a r e t o , o r nondominated optimality. H e r e we define dif- f e r e n t t y p e s of a p p r o x i m a t e efficient solutions t o v e c t o r optimization problems and deveiop t h e c o r r e s p o n d i n g saddle point t h e o r e m s along t h e logics of Tanino and Sawaragi (1980) o r Luc (1984).

Section 3.1. i s devoted t o definitions a n d some b a s i c p r o p e r t i e s of a p p r o x i m a t e e x - t r e m a i eiements in o r d e r e d v e c t o r s p a c e s a n d in S e c t i o n 3.2. t h e s a d d i e point t h e o r e m s are p r o v e d . As f o r appiications, in Section 3.3. we show t h e equivalence between a p p r o x i m a t e s a d d l e points a n d t h e c o r r e s p o n d i n g primal-dual p a i r s of solutions.

As a consequence 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 coincides with t h a i of ( e x a c t ) soiution in t n e case 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 t o t h o s e obtained in t h e a b o v e mentioned p a p e r s . Throughout t h e p a p e r we r e l y on a knowledge of convex analysis and t h e t h e o r y of o r d e r e d v e c t o r s p a c e s , a n d t h e r e f o r e b a s i c notions a n d f a c t s are used without s p e c i a l expianation.

If needed see e. g. P e r e s s i n i (1967), Holmes (1975) o r Akilov a n d Kutateladze (1970).

All t h e v e c t o r s p a c e s a p p e a r i n g in t h e p a p e r are r e a l a n d o r d e r i n g c o n e s are sup- posed t o b e convex, pointed a n d a l g e b r a i c a l l y closed. In t h e p r e s e n c e of a topolog- ical s t r u c t u r e w e s u p p o s e compatibility, i. e. t h a t t h e o r d e r i n g c o n e i s closed. W e .denote by X a n d V v e c t o r s p a c e s and by ( Z , K ) a n o r d e r e d v e c t o r s p a c e with

core (K) +d, w h e r e core r e f e r s t o t h e a l g e b r a i c i n t e r i o r . Similarly, r c o r e d e n o t e s t h e r e l a t i v e a l g e b r a i c i n t e r i o r . (Y,C) i s a n o r d e r compiete s p a c e , i. e. a v e c t o r l a t t i c e w n e r e e v e r y nonvoid set with a lower bound p o s s e s s e s a n infimum. In o r d e r t o e n s u r e t h e e x i s t e n c e of infima, r e s p , suprema f o r e v e r y (i.e. nonbounded) sets, we supplement t h e s p a c e (Y,C) with t h e elements = a n d -= using t h e notation

Y=Yu!

^{-=, = j ,} a n d s u p p o s e t h a t t h e usual a i g e b r a i c a n d o r d e r i n g p r o p e r t i e s hold.

Hence f o r t h e set H c Y , which i s n o t bounded from below, we h a v e inf (H)=-- and inf (@)=a. The dual s p a c e of Y i s Y while t h e topological dual i s rC

.

The c o n e of positive functionais with r e s p e c t t o t h e c o n e C cY, o r t h e dual of C i s C S . The func- tional y* EY d e n o t e s a n eiement of C + . L +(Z,Y) cL (Z,Y), o r A+(Z,Y)cA(Z,Y) s t a n d s f o r t h e c o n e of positive l i n e a r , o r continuous positive l i n e a r maps from Z t o Y, r e s p e c t i v e l y .

W e r e c a l l now that f o r t h e v a r i o u s o r d e r i n g r e i a t i o n s h i p s between two elements of a n o r d e r e d v e c t o r s p a c e w e s h a l l use t h e foilowing notations f o r exampie in ( Y , C ) :

Yz L Y 1 iff Yz - Y 1 E C Y ~ iff ~Y ~ - Y ~ E C \ Y ~ 0 Y Z

>

y 1 iff y z - Y ? E c o r e ( C )

To d e n o t e o p p o s i t e r e l a t i o n s we u s e symbols Like ;): a n d $. Accordingly

r e f e r t o t h e f a c t t h a t y l ~ Y dominates o r d o e s not dominate y 2 E Y from below, r e s p e c t i v e l y .

The v e c t o r s e , en , e 7 E Y a n d t h e s c a l a r s E , E , dR r e p r e s e n t t h e approximation e r - r o r , of them w e s u p p o s e t h a t e 5 0 , e , 2 0 and e,r_O holds a n d similarly t h a t E , E , a r e nonnegative.

Now t h e usuai definition of t h e main s u b j e c t of s t u d y in t h i s p a p e r foliows, i. e . t h a t of t n e convex minimization p r o b l e n a n d of t h e c o r r e s p o n d i n g v e c t o r valued Lagrangian ( s e e e . g. Zowe (1976)).

Definition 1.1.

Let

p r o p e r convex functions with A

=

d o m j' n d o m h , a n d L ^E L ( X , V ) . We define t h e minimization problem ( M P ) by way of t h e set of solutions:

where

F = l z EX:^ ~ A , h ( z ) s O , L ( z ) = O j

i s called t h e feasibility s e t of t h e problem ( M P ) .

The a l g e b r a i c Lagrangian of t h e convex minimization problem ( M P )

i s defined by t h e equaiity

I

^eo i f x e ' A

@ L ( ~ , R , S )

= 4

f ( x ) + R . h ( x ) + S - L ( x ) i f x ~ A a n d ~ a + ( Z , Y )

I I --

if X E A and R ~ ' L + ( Z , Y )

with t h e set

cailed t h e domain of @L

.

The element ( z o , R o , S o ) E dom @L i s a s a d d l e point of t h e Lagrangian

aL

^if t h e fol- lowing i s met

(i) @ L ( x o , R o , S o ) E M l i Y [ @ L ( ~ , R o , S o ) : z

EX^

(ii) 5,L(zo,Ro,So) ci M k X j 9 L ( x o , R , S ) : ( R , S ) E L (Z,Y) X L(V,Y){

.

Instead of t h e symbol MliY o r MAX, o n e h a s t o s u b s t i t u t e o n e of t h e a p p r o x i m z t e ( o r e x a c t ) notions of minimality or maximality f r o m t h e l a t e r foilowing r e s p e c t i v e de- finitions. Depending on t h i s c h o i c e , w e cali t h e eiements of MIN(W) solutions of tine probiem (MP) of t h e c o r r e s p o n d i n g a p p r o x i m a t e ( o r e x a c t ) t y p e .

The continuous L a g r a n g i a n i s defined as t h e r e s t r i c t i o n of 4L t o X x A(Z,Y) x A(V,Y) and t h e notion of s a d d l e point of t h e continuous Lagrangian @,, i s defined in a c o r r e s p o n d i n g manner.

The a b o v e n o t a t i o n s and conditions are supposed to b e valid t h r o u g h o u t t h e p a p e r a n d will not b e mentioned again.

2. STRICT OPTKMA

2.1. A p p r o x i m a t e E x t r e m a l P o i n t s and A p p r o x i m a t e Solutions

Now we start with t h e definition of s t r i c t extremal and s t r i c t approximate extremal points, and t h e n w e formulate some simple relationships between approximate ex- tremal points corresponding t o d i f f e r e n t values of t h e approximation p a r a m e t e r . Definition 2.1.1.

Suppose t h a t H C

Y.

Then a n element y EH i s called a s t r i c t minimal element of H, o r

The set

S(e)-MIN(H)

=

^jy EH : H c y - e - C j

is called t h e set of s t r i c t e-approximate minimal o r S(e)-minimal points of H.

By convention w e say t h a t

S - M ( $ ) = S ( e ) - M ( 4 )

=

and if H c Y is not bounded from below

S-MIN(H)

=

S ( e ) - M ( H )

=

f - a j Remark 2.1.1.

By t h e pointedness of t h e cone C

cY

t h e set S -MIN(H) cannot have more t h a n one element. If i t h a s one, t h i s obviously means t h a t

in4

(H)

=

S - M ( H ) .

The notions of S(e)-maximal and S-maximal elements are t o b e defined in a corresponding manner.

The statements in t h e following proposition are straightforward consequences of t h e definitions.

Proposition 2.1.1.

(a) S -Mn\l(H)

=

S ( 0 ) -Mn\l(H)

(b) If 0

5

e l 5 e 2 , t h e n S ( e l ) - M l N ( H )

c

S ( e 2 ) - M I N ( H ) .

( c ) If H c Y i s bounded from below t h e n S ( e ) -MIN(H)=(inf ( H ) + e - C ) n H . Corollary 2.1.1.

Let ( Y , C ) b e equipped with a topological s t r u c t u r e and H

c

Y closed. Suppose t h a t a n e t l e y E C : y E

rj

d e c r e a s i n g t o e E Y e x i s t s with

( a ) S ( e y ) - M l N ( H ) n Y # d V y ~ and r

(b) S ( e y J -MIN(H)

c

Y i s compact f o r some y o d ' . Then S ( e )-MlN ( H ) #

@.

P r o o f .

A s a consequence of t h e closedness of C

c

Y , w e h a v e t h a t e E C , and s o S ( e ) -Mn\l(H) i s well defined. S ( e y ) -Mn\l(H)#@ obviously implies i n f ( H ) # w , and s o w e c a n apply (b) and ( c ) in Proposition 2.1.1. t o conclude t h a t S ( e ) -Mn\l(H) i s t h e i n t e r s e c t i o n of nonvoid compact sets.

Proposition 2.1.2.

Let ( Y , C ) b e equipped with a topological s t r u c t u r e and l e y E C : y E

rj

a d e c r e a s - ing n e t t h a t c o n v e r g e s t o e EY.

Then

P r o o f .

By Corollary 3.2., Chap. 2 . in P e r e s s i n i ( 1 9 6 7 ) w e h a v e e

=

i n f le E C : y E

rj.

Hence by Proposition 2.1.1. t h e l e f t hand s i a e in ( 2 . 1 ) i s a s u b s e t of t h e r i g h t hand s i a e .

F o r t h e r e v e r s e inclusion l e t y E Y b e a n element of S ( e ?) -Mn\l(H) f o r e a c h y E

r.

This means t h a t y E H and f o r e a c h fixed h E H , t h e n e t fh --y + e Y € Y : y € r j i s contained in t h e closed c o n e C cY, h e n c e h --y +e EC a i s o holds.

8

Corollary 2.1.2.

Let (Y,C) b e equipped with a weakly sequential complete topology, t h e o r d e r i n g cone C c Y normal and suppose t h a t f e n EC : n E N ! i s a decreasing sequence.

Then

e x i s t s and

Proof.

The statement i s a consequence of o u r Proposition 2.1.2. and t h e Corollary 3.5.

Chap. 2. in P e r e s s i n i (1967).

Remark 2.1.2.

A s a consequence of Proposition 2.1.1. t h e case with e =O provides f o r conditions ensuring t h e existence of e x a c t extremal points based on information about ap- proximate ones in t h e previous Proposition and Corollaries.

Now, using Propositions 2.1.1. and 2.1.2. w e formulate a f e w simple p r o p e r t i e s of t h e approximate solutions.

Corollary 2.1.3.

Suppose, t h a t (Y,C) i s equipped with a topoiogical s t r u c t u r e . Then t h e following hold:

(a) S -M.ZiY (MP)

=

S (0) -MhV ( W )

(b) O ~ e l ~ e 2 i m p l i e s S ( e l ) - M I N ( M P ) c S ( e 2 ) - M ( M P ) .

(c) If f e y E C : y E

rj

^{i s} a decreasing net t h a t converges t o e EY, t h e n n IS(e,.)-MnV(MP) : y E

rj

= S ( e )

-

^MIN(MP).

(d) If t h e topology of (Y,C) is weakly sequentially complete, t h e cone CCY i s nor- mal, {en E C : n E

N ]

is a decreasing sequence with t h e infimum e EY, t h e n

niS(en)-MIN(MP) : n EN]

=

S(e)-IWN(MP).

(e) Suppose t h a t t h e set f (F) E Y i s closed, ten E C : n E

N ]

is a decreasing se- quence t h a t converges t o e EY, zyEX is a n S(ey)-solution of (MP) f o r e a c h

7 ~ r

and t h e r e is a y o E

r

such t h a t t h e set s ( e Y J - M { f (z) E Y : z EFj C Y

- 8 -

i s compact. Then ( M P ) h a s a n S(e)-minimal solution.

2.2. The Saddle Point Theorems

The p r e s e n t s e c t i o n i s closely r e l a t e d t o Zowe's r e s u l t s both as f a r as p r o o f s a n d notions are c o n c e r n e d (Zowe (1976) a n d Zowe (1977)). As t h e r e , in t h e case of ex- act solutions, o n e implication between t h e e x i s t e n c e of solutions a n d s a d d l e points i s valid u n d e r f a i r l y g e n e r a l conditions while w e need additional assumptions in t h e case of t h e o t h e r .

Proposition 2.2.1.

If ( z o , R o , S o ) Edom cPL i s a S ( e )-saddie point of t h e Lagrangian cPL , t h e n

P r o o f .

Follows from zo@ a n d (ii) of Definition 1.1. if o n e c o n s i d e r s t h e case (R,S)=(O,O) in S(e)-MAX.

Theorem 2.2.1.

If ( z o , R o , S o ) E dom cPL i s a n S ( e )-saddle point of t h e Lagrangian cPL, t h e n z o ^E X i s a n S (2.e )-solution of ( M P )

.

P r o o f .

F i r s t w e p r o v e t h a t t h i s implies t h a t zo@.

z o ED follows from t h e r e l a t i o n ( z o , R o , S o ) E dom cPL, Using t h e c h o i c e ( z , R , S ) = ( z o , R , S o ) in (ii) of Definition 1.1. w e obtain t h a t

a n d using ( z , R , S ) = ( z o , R o , S ) t h a t

If h ( z o) !Z -K, then t h e s t r i c t s e p a r a t i o n theorem applied t o t h e singleton set i h ( z o ) j c Z and t h e algebraically closed convex cone KcZ with a nonempty c o r e (Kothe S e c t . 17, 5, (2)) yields a z* EK+ with

sup

I

< z 4 , - k > : ~ E K

1 =

0

<

< z 4 , h ( z 0 ) > .

Let c EC\ I O j b e a fixed v e c t o r , t o b e specified l a t e r and l e t us define

REL+(z,Y)

with t h e equation

By inequality (2.2) now w e have

Selecting f i r s t a n y c EC\ I0 j, w e see t h a t Ro.h ( z o ) +e +O holds, t h e r e f o r e w e are allowed t o set at a second s t e p

leading t o a contradiction with (2.4).

A similar argument shows t h e impossibility of 1 ( z o ) 2 0 , and s o w e can conclude t h a t z o U .

Again, by t h e definition of @L and t h e S(e)-saddle point, w e have f o r e a c h ( z ,R , S ) E dom GL :

f ( z o ) + R e h ( z o ) + S . L ( z o ) - e $ f ( x ) + R o . h ( z ) + S O . L(z)

+

e

A s a consequence of (zo,Ro,So) E dom @L t h e relation RoEL+(2,Y) hoids and t h e r e - f o r e a substitution ( z , R , S ) = ( z ,O,O) completes t h e proof.

Using t h e topological version of t h e s t r i c t s e p a r a t i o n theorem in t h e a b o v e proof, w e readily obtain t h e following f o r t h e continuous Lagrangian QA.

Theorem 2.2.2.

Suppose t h a t (Y,C), (Z,K) and V a r e equipped with a topological s t r u c t u r e .

If ( z o , R o , S 0 ) E dom is an S(e)-saadle point of t h e Lagrangian Q A then zo€X is a S (2.e )-solution of (MP).

Definition 2.2.1.

W e say t h a t t h e problem (MP) meets t h e a l g e b r a i c Slater-Uzawa c o n s t r a i n t qualifi- cation if e i t h e r

( i ) t h e r e e x i s t s a n z l € r c o r e (A) with h (zl)€-rcore (K) and 1 ( z l ) =0, or

(ii) no l i n e a r c o n s t r a i n t is p r e s e n t and t h e r e e x i s t s a n z l € A with

Definition 2.2.2.

The problem (MP), where (Y,C), (Z,K) and V are topological s p a c e s meets t h e topo- logical Slater-Uzawa c o n s t r a i n t qualification if t h e r e e x i s t s an zl€int (A) with h ( z l ) € i n t ( K ) and l(zl)=O.

Now f o r t h e convenience of t h e r e a d e r w e quote from Zowe (1976) t h e a l g e b r a i c and topological v e c t o r valued versions of t h e Farkas-Minkowski lemma.

Theorem 2.2.3.

Suppose t h a t t h e minimization problem (MP) meets t h e a l g e b r a i c Slater-Uzawa con- s t r a i n t qualification.

Then t h e following statements a r e equivalent:

( a ) f ( z ) S O vz*

(b) t h e r e e x i s t o p e r a t o r s R EL+(Z,Y), SEL O/,Y) such t h a t f ( z ) + R . h ( z ) + S . I ( z ) & O V Z E A .

Theorem 2.2.4.

Let (Y,C) and ( Z , K ) b e equipped with a topological s t r u c t u r e , X a completely metrizable topological v e c t o r s p a c e and t h e cone C c Y normal. Let f u r t h e r V b e a Hilbert s p a c e with 1 (X)CV a closed s u b s p a c e and suppose t h a t t h e minimization problem (MP) meets t h e topological Slater-Uzawa c o n s t r a i n t qualification.

Then t h e following statements are equivalent:

( a ) f ( z ) S O vz*

(b) t h e r e e x i s t o p e r a t o r s REA+(z,Y), S€AO/,Y) such t h a t f ( z ) + R . h ( z ) + S . L ( z ) z O V Z E A .

Now w e are a b l e t o formulate and p r o v e t h e c o n v e r s e statements t o Theorems 2.2.1. and 2.2.2.

Theorem 2.2.5.

Under t h e assumptions of Theorem 2.2.3. t h e following holds:

If zo€X is a n S(e)-solution of t h e problem

(MP),

then t h e r e e x i s t o p e r a t o r s R o EL '(2 ,Y) and So EL (V, Y), such t h a t ( z o , Ro,So) E dom Q is a n S ( e )-saddle point of t h e Lagrangian

%.

Proof.

A s z o is a n S(e)-solution, w e c a n apply Theorem 2.2.3. f o r t h e function fl, where

instead of t h e original ^j'

.

T h e r e f o r e t h e r e e x i s t o p e r a t o r s such t h a t

From Proposition 2.2.1. and (2.5) now w e have

on one hand, and by

on t h e o t h e r , completing t h e proof.

Repeating t h e above p r o c e d u r e with t h e topological Theorem 2.2.4. instead of Theorem 2.2.3. w e obtain:

Theorem 2.2.6.

Under t h e assumptions of Theorem 2.2.4. t h e following holds:

If zo€X is a n S(e)-solution of t h e minimization problem ( M P ) , t h e n t h e r e e x i s t o p e r a t o r s R ~ E A + ( Z , Y ) and S0€A(V,Y), such t h a t (zo,Ro,So) E dom IpA i s a n S ( e ) - saddle point of t h e continuous Lagrangian IpA.

Remark 2.2.1.

A s a consequence of Proposition 2.1.1. (a) o u r r e s u i t s r e d u c e t o those of Zowe (1977) and Zowe (1976) in t h e c a s e when e =O.

2.3. Primal and Dual Problems

In t h i s section w e place t h e r e s u l t s of Section 3. in t h e context of some r e l a t e d r e s u l t s and apply them t o analyze t h e primal slid dual problems associated with t h e problem (MP).

Definition 2.3.1.

Consider t h e following functions:

and

which w e call t h e (algebraic) s t r i c t primal and dual functions of t h e minimization problem (MP), respectively. The v e c t o r s defined as

and

are t h e (algebraic) s t r i c t value and dual value, respectively.

The a l g e b r a i c s t r i c t primal and dual problems are formulated by way of t h e sets of solutions:

and

The relationship between t h e original minimization problem ( M P ) and i t s primal problem ( P ) is shown by t h e following proposition, namely t h a t t h e l a t t e r i s just t h e reformulation of a constrained problem into a nonconstrained one.

Proposition 2 . 3 . 1 .

If t h e s p a c e ( Y , C ) is Archimedean then t h e problem ( P ) is equivalent t o ( W ) , i.e.

Proof.

If z E F , t h e n w e h a v e

and t h e r e f o r e

but t h e equality i s valid in t h e c a s e ( R , S )

=

( 0 , O ) .

If z g F because of z g A, t h e n @L ( z , R , S ) = - , and hence P ( x )=-. If x $Z F because of h ( x

)a,

t h e n by t h e s e p a r a t i o n argument in t h e proof of Theorem 2 . 2 . 1 . e n s u r e s t h e existence of a z*

EK+

with <z* ,h ( x ) >

>

0 . This enables u s t o c o n s t r u c t a se- quence of o p e r a t o r s fRn E L + ( Z , Y ) : n E N j with

Let, namely b e c EC\ f 0 { a fixed v e c t o r and define Rn E L + ( Z , Y ) by t h e equation

From t h e Archimedean p r o p e r t y of ( Y , C ) now ( 2 . 6 ) follows, and a similar argument shows P ( z )== in t h e case of 1 ( z ) $ 0 .

P r o p o s i t i o n 2.3.2.

The primal function P i s convex and t h e duai function D i s c o n c a v e . P r o o f .

The f i r s t s t a t e m e n t d i r e c t l y follows from P r o p o s i t i o n 2.3.1. The domain of t h e d u a l function i s c l e a r l y convex, a n d concavity follows f r o m t h e s u p e r a d d i t i v i t y of t h e

in4

o p e r a t i o n .

P r o p o s i t i o n 2.3.3.

( a ) Tine primai v a i u e i s g r e a t e r o r e q u a l t h a n t h e d u a l value.

(b) If

+

EX i s a n S ( e ) - s o i u t i o n of t h e primai problem ( P ) a n d ( R o , S o ) U (Z,Y)xL (V,Y) i s t h a t of t h e dual problem (D) t h e n P ( z 0 )

r,

D ( R O l S 0 ) .

(c) L e t u s n a v e f o r some x o m , (Ro,So)EL(Z,Y)XL (V,Y)

Then s o i s a n S ( e )-solution of t h e primal problem ( P ) a n d ( R o t s o ) i s a n S(e)-solution of t h e dual problem (D).

P r o o f :

The s t a t e m e n t i s a n obvious c o n s e q u e n c e of t h e definitions.

Riow w e t u r n to t h e c o n s i d e r a t i o n of t h e connection between o u r Hurwitz-type r e s u l t s a n d t h o s e o b t a i n e d b y Kutateladze (1978). In t h i s w e s h a l l r e l y on t h e no- tion of p e r t u r b a t i o n function a n d a p p r o x i m a t e s u b g r a d i e n t s .

Definition 2.3.2.

The function

w h e r e

i s called t h e p e r t u r b a t i o n function a s s o c i a t e d with t n e problem ( M P ) . Proposition 2.3.4.

Suppose t h a t I? EL ⁺ ( z , Y ) , t h e n

inf j p ( z , v )

+

^R

.

z + S . v : ( z , v ) € 2 x V

1

= D ( R , S )

P r o o f .

The foilowing equation is a d i r e c t consequence of t h e definitions;

inj' i p ( z , v ) + R . z + S . v : ( z , v ) € 2 x V ! =

T h e r e f o r e we only h a v e t o p r o v e t h a t t h e r i g h t hand s i d e e q u a l s with D ( R , S ) . To d o t h i s c o n s i d e r t h e inclusion

By t h i s a n d t h e definition of t h e dual function D t h e r e l a t i o n 5 always holds. On t h e o t h e r hand R d ' ( 2 , ~ ) implies

a n d h e n c e we a l s o h a v e t h e o p p o s i t e r e i a t i o n .

The definition of a p p r o x i m a t e , o r e-subgradient a n d t h e foliowing t h e o r e m i s t a k e n from Kutateladze (1978).

Definition 2.3.3.

The set

i s called t h e a p p r o x i m a t e , o r e-subdifferential of f at

zom.

Remark 2.3.1.

The statement t h a t O € a e f ( z o ) i s obviously equivalent t o t h e r e l a t i o n z o ES (e ) - M I N ( W ) if t h e r e are no feasibility c o n s t r a i n t s .

Theorem 2.3.1.

Suppose t h a t t h e problem ( W ) meets t h e a l g e b r a i c Slater-Uzawa c o n s t r a i n t quaiif- ication, t h e n

if and only if

t h e r e e x i s t RoEL +(z,Y), S O E L (V,Y) and e l g o ,

ez.O

^with

s u c h t h a t

Theorem 2.3.2.

Suppose t h a t S(e)-MlX(MP)f d and f o r (Ro,So) EL+(z,Y)xL (V,Y)

holds.

Then

P r o o f .

A s t h e conaitions e n s u r e t h e r e i s a xoES(e)-MlN(MP).

Now using t h i s a n d t h e definition of t h e e-subgradient w e o b t a i n

P ( ~ O ) - ~ . ~ S P ( ~ , V ) + ( R ~ . S ~ > - ( ~ , V ) V ( z , v ) E Z X I / : P r o p o s i t i o n 2.3.4. yields

and by feasibility t h e p r o o f i s complete.

Theorem 2.3.3.

Suppose t h a t t h e problem ( W ) meets t h e a l g e b r a i c Slater-Uzawa c o n s t r a i n t qualif- ication, a n d s u p p o s e t h a t S ( e ) -MlX(MP) f 0.

holds t h e n

P r o o f .

Sy t h e conEitions w e h a v e ar. zoES(e)-MIN(M.P) and h e n c e

Tneorem 2.2.5. e n s u r e s t h e e x i s t e n c e of a p a i r ( R ~ , s ~ ) E L + ( Z , Y ) ~ L (V,Y) such t h a t (zo,R1,S1) ^E dom @L i s a S(e2)-saddie point f o r t h e Lagrangian

aL,

t h a t i s by t h e definition of t h e probiem (D) and (i) of Definition 1.1. th i s means t n a t

Now P r o p o s i t i o n 2.3.1. implies

f

( z o >

-

²

.

^e S D(R1,S1) and as ( R o , S o ) ~ S ( e ) -Mn\l(D), w e a l s o h a v e

From h e r e by Proposition 2.3.4. t h e s t a t e m e n t follows.

Theorem 2.3.4.

Suppose t h a t t h e probiem ( M P ) meets t h e Siater-Uzawa c o n s t r a i n t qualification and c o n s i d e r t h e foiiowing s t a t e m e n t s .

( a ) ( z o , R o , S o ) ^{E dom} 9L i s a n S ( e l ) - s a d d l e point f o r

aL.

(b) F o r ( z o,Ro,So) E dom

aL

^{w e h a v e}

with

e' 2 0 , e"

s:

⁰ and 0

g

e'

+

e" g R o h ( z O )

+

e 2

Then (a) implies (b) with e2=2.el, a n d (b) implies (a) with el=2.e2.

P r o o f .

If ( a ) holds t h e n a c c o r d i n g to Theorem 2.2.1., xot;S(2.e ) -IUIN(IUP) a n d so Theorem 2.3.1. e n s u r e s (2.11) and. (2.12). On t h e o t h e r hand, by P r o p o s i t i o n 2.2.1. a n d t h e s a d d l e point p r o p e r t y we h a v e

a n d h e n c e

From h e r e by P r o p o s i t i o n 2.3.4. (b) follows.

S u p p o s e now t h a t (b) holds, t h a t i s a g a i n by P r o p o s i t i o n 2.3.4. o n o n e hand we h a v e

As by Theorem 2.3.1. xo€S(e2)-IWN(MP), implying

On t h e o t h e r , b y f e a s i b i l i t y a n d (2.12):

In view of P r o p o s i t i o n 2.3.1. t h e a b o v e c a n b e r e f o r m u l a t e d as:

C o r o i l a r y 2.3.1.

S u p p o s e t h a t t h e problem ( W ) m e e t s t h e Slater-Uzawa c o n s t r a i n t qualification a n d c o n s i d e r t h e following s t a t e m e n t s :

(a) ( x o , R o , S o ) ^E dom

aL

i s a n S ( e l ) - s a d d l e point f o r t h e L a g r a n g i a n

$

a s s o c i a t - e d with t h e problem ( W ) .

( b ) F o r ( x o , R o , S o ) E dom

eL

w e h a v e

(i) x o € X i s a S(e2)-solution of t h e primal problem (P) a n d

(ii) (Ro,So) EL ⁺ (Z,Y) >(L ( V , Y ) is a S ( e 2)-solution of t h e d u a l problem (D).

Then ( a ) implies (b) with e2=4.el, a n d ( b ) implies (a) with el=6.e2.

Now w e t u r n t o t h e c o n s i d e r a t i o n of generalized solutions.

Definition 2.3.4.

Suppose t h a t (Y,C) i s eqiupped with a topological s t r u c t u r e , a n d Ie7€C : 7 € r j i s a d e c r e a s i n g n e t t h a t c o n v e r g e s t o OEY. Let f u r t h e r z7€X b e a n S(e7)-solution of (MP) f o r e a c h 7 €

r.

Then we c a l l t h e n e t Iz7€X : y E r j a generalized s t r i c t solu- tion of t h e minimization problem (MP).

Suppose, in addition t h a t t h e r e e x i s t s a n e t f(R7,S7) EL (Z,Y)xL (V,Y) : y € r j with t h e p r o p e r t y t h a t (z7,R7,S7) is a n S(e7)-saddle point f o r t h e Lagrangian

aL.

^Then

w e c a l l t h e n e t f(z7,R7,S7) EXxL (Z,Y) >(L (V,Y) : y ~ r j a g e n e r a l i z e d s t r i c t saddle point of t h e i a g r a n g i a n

aL.

Proposition 2.3.5.

F o r t h e s t r i c t vaiue of (MP), v

EY

w e h a v e

v

= in9

[ f (z7) E Y : !z7 E X : 7 E

r{

a generaiized solution, 7 E

r

^{{ .}

P r o o f .

The equaiity i s a d i r e c t consequence of t h e definitions.

Definition 2.3.5.

Suppose t h a t (Y,C) i s equipped with a topological s t r u c t u r e . W e c a l l t h e problem ( M P ) weli posed if t h e r e e x i s t s a n e t i ( z 7,R7,S7) : y ~j such t h a t r

Remark 2.3.2.

By t h e definition of infimum a n d supremum, o u r definition coincides with t h e single r e q u i r e m e n t of v =v* in t h e s c a l a r vaiued case.

Theorem 2.3.5.

Suppose t h a t (Y, C ) i s equipped with a topological s t r u c t u r e a n d t h a t t h e c o n e C c Y i s normal. If t n e Lagrangian Q,L h a s a generalized s a d d l e point, t h e n t h e problem (MP) i s well posed.

P r o o f .

A s a consequence of P r o p o s i t i o n 2.3.3. (a) w e only h a v e t o p r o v e v 5 v * . By t h e definition of t h e g e n e r a l i z e d s a d d l e point, t h e r e e x i s t a d e c r e a s i n g n e t

!e 7 ~ C :

~ ~ r i ,

t h a t c o n v e r g e s t o OEY such t h a t

Hence f o r e v e r y fixed 6 E w e have

and, by t h e normality of t h e cone C c Y , from h e r e t h e statement follows.

m

Corollary 2.3.1.

Under t h e conaitions of T'neorems 2.2.5. and 2.3.5. t h e e x i s t e n c e of a generalized.

s t r i c t solution t o t h e problem ( M P ) impiies t h a t t h e probiem is well posed.

Proof.

Easiiy follows from t h e combination of t h e quoted theorems.

Remark 2.3.3.

I t i s worth noting t h a t t h e r e v e r s e implication seems not t o hold in t h e vectorial c a s e while i t is trivial f o r s c a l a r s .

Similariy t o t h e preceding, notions and statements of Section 2.3. c a n a l s o b e f o r - mulated in a purely topological way. P r o o f s a r e analogous, but of c o u r s e relying on Theorem 2.2.6. instead of Theorem 2.2.5.

3. NON-DOMINATED OPTIMA

3.1. A p p r o x i m a t e N o n - d o m i n a t e d E l e m e n t s

Definition 3.1.1.

The v e c t o r y EH is a P(e)-minimal eiement of

HCY

o r a p p r o x i m a t e l y P a r e t o minimal, in notation

WP(e )-minimal, in notation

( y

-

e

-

c o r e (C)) n H

=

@.

H e r e , of c o u r s e , we need t h e condition t h a t c o r e ( C ) + @ a n d s p e a k i n g a b o u t WP- minimality, we always s u p p o s e i t .

a n d P ( y * , &)-minimal, in notation

By convention, we s a y t h a t a l l kinds of minima of t h e void set consist of t h e single element

~ E Y .

The approximately maximal elements are t o b e defined in a c o r r e s p o n d i n g manner.

Remark 3.1.1.

Our definitions, in t h e case of e =0, o r &=O, r e p r o d u c e t h e usual e x a c t notions of minimality. Weak a p p r o x i m a t e minimality means t h e c o r r e s p o n d i n g a p p r o x i m a t e minimality with r e s p e c t t o t h e (algebraically non-closed) c o n e C'

=

I0 j ucore (C).

The notion of y EY being P ( y * ,&)-minimal means t h a t <y* , y > d R i s a P(&)-minimal element of t h e set y * ( H ) = { < y * , h > E R : h ~ H i .

Remark 3.1.2.

In t h e scalar case t h e d i f f e r e n t notions of a p p r o x i m a t e solutions f o r t h e minimiza- tion problem (MP) coincide a n d t h e r e w e simply s p e a k of &-solutions o r &-saddle points.

Let us formulate some simple f a c t s t h a t are e a s y consequences of t h e definitions b u t are s t i l l i n t e r e s t i n g b e c a u s e t h e y c l a r i f y t h e r e l a t i o n s h i p s between t h e dif- f e r e n t notions of minimal solution. Omitted p r o o f s are t r i v i a l .

Proposition 3.1.1.

Suppose t h a t e l s e 2 and

E ~ B ~ ~ .

Then we h a v e

W P (e ,)

-

^{MIN (MP)}

^c

^{W P (e 2)}

-

^{MIN (MP)}

Proposition 3.1.2.

(a) S u p p o s e t h a t we h a v e < y * , e

> >

0. Then

with

e'

=

^E . e

<y* , e

>

(b) WP(e)-MRY(MP)

=

u I P ( y * , < y * , e >)-MIN(MP) : y * E

c+\

f O j j.

Proposition 3.1.3.

Suppose t h a t (Y,C) i s equipped with s u c h a weakly sequentially complete topology t h a t t h e o r d e r i n g c o n e C c Y i s normal. Consider a s e q u e n c e !en EC : n € R j de- c r e a s i n g t o e EC.

Then

P ( e ) - M I N ( M P ) c ntP(e,)-MIIY(MP) : n~ Nj c WP(e)-MIN(MP)

and

ntWP (en)-MIIY(MP) : n~ Nj

=

WP(e)-MIN(MP)

Proof.

The f i r s t inclusion i s obvious.

For t h e second l e t us r e a s o n by contradiction and suppose t h a t t h e element x o W i s not WP(e)-minimal. This means t h a t w e can find a n o t h e r x l W with

By normality int ( C ) # $ and t h e r e f o r e int ( C ) = c o r e ( C ) . Hence t h e formula under ( 3 . 1 ) is equivalent t o

A s a consequence of Corollary 3 . 5 . Chap. 2. in Peressini (1967) f o r t h e sequence w e have

Lim f f ( x O ) - e n - P ( x l ) : n ~ N j = f ' ( x ~ ) - e - P ( x l ) E i n t ( C ) , and so, t h e r e exists a n m EN with

This means t h a t J' ( x l ) dominates t h e element f ( x o ) -em EY from below.

The proof of t h e second statement i s analogous.

Proposition 3 . 1 . 4 .

Suppose t h a t t h e sequence

I

^{E ,} E R + : n EN j d e c r e a s e s t o E

a+.

Then

3.2. Saddle Point Theorems

Proposition 3.2.1.

The element (zo,Ro,So) ~ d o m cPL is a P(e)-saddle point of t h e Lagrangian cPL, iff (a) @L(zO,RO,SO) €P(e)-MIN !cPL(z,Ro,So) E

Y :

^{z E X {}

(b) z o E F

(c)

-

^e

+

^{R o}

^.

h ( z o )

s

0.

Proof.

Condition (a) is identical with t h e f i r s t p a r t of t h e definition. Suppose now t h a t ( z o , R o , S o ) ~ d o m

aL

i s a P ( e )-sacidle point. The definition of dom @L immediately yields (b), and w e have

From t h e definition of t h e P ( e )-saddle point w e a l s o know t h a t

f o r e a c h (R,S)EL (Z,Y)xL (V,Y). Selecting S=So and R =Ro w e obtain

and

respectively. Suppose now t h a t h ( z o ) s O does not hold. Then by t h e s t r i c t alge- b r a i c s e p a r a t i o n theorem ( s e e Kothe (1966) Section 17.5. (2)) applied f o r t h e sets

f h ( z o ) j

cZ

and - K c Z , t h e existence of a functional z*

EK'

is guaranteed with

Let c 2 0 , c EY b e a n a r b i t r a r y , fixed element, and define t h e map REL (Z,Y) as

F o r this o p e r a t o r R w e obviously have REL +(Z,Y) and

in contradiction with (3.3). A similar argument leads t o contradiction with (3.4), if w e suppose 1 ( z o) +O. H e r e w e define a n o p e r a t o r S EL (T/,Y) as

The last inequality in (c) is a consequence of z and RoEL +(Z,Y), while t h e f i r s t follows from (3.2) if w e choose (R ,S)=(O,O).

To p r o v e t h e r e v e r s e implication, suppose t h a t (a), (b) and (c) are valid. From t h e l a s t two w e have t h e following relations:

f o r e a c h (R , S ) EL +(Z,Y)

xL

(T/,Y) implying t h e missing relationship f o r (zo,Ro,So) ~ d o m

aL

^{t o b e} a P ( e )-saddle point.

Remark 3.2.1.

The p r o p e r t y s t a t e d in Proposition 3.2.1. is as much negative as positive, and t h e r e f o r e i s a f i r s t sign of t h e problems t o b e s e e n in t h e sequel. Point (c), name- ly, t u r n s into t h e well-known complementarity condition

in t h e case of e x a c t saddle points. In g e n e r a l , however i t only means t h a t

and t h e r i g h t hand side h e r e i s a n unbounded set.

The proof of t h e following two statements i s analogous.

Proposition 3.2.2.

The element (zo,Ro,So) Edom @L is a WP(e )-saddle point of t h e Lagrangian @L iff (a) @L ( z o,Ro,So) E WP(e ) -MY @L ( z , R o , s o ) E

Y

: z E X j

(b) z o E F

(c) - e $ R o e h ( z o ) S 0.

Proposition 3.2.3.

The element ( z ,, Ro.So) E d o m

$

is a P ( y

*

, &)-saddle point of t h e Lagrangian @L iff

(a) @ L ( z o , R o , S o ) E P(y*,&)-MlIY [ @ L ( z , R o , S o ) E

p :

z E X !

(b) zo E F

(c)

-

^E

⁵ <

Y* , R o h (20)

>

^{5_ 0,}

Theorem 3.2.1.

Suppose t h a t t h e point (zo,Ro,So) Edom @L i s a P(e)-saddle point

/

WP(e )-saddle point

/

P ( y + ,&)-saddle point of t h e Lagrangian @ L .

Then zo€X i s a n approximate solution of t h e minimization problem

( M P )

in t h e r e s p e c t i v e s e n s e where t h e approximation e r r o r i s

in t h e f i r s t and second, and

in t h e l a s t case.

Proof.

By Proposition 3.2.1. z o € X i s a feasible point. If z EF is a n o t h e r , then f o r t h e s a m e r e a s o n w e have

and this means

By feasibility I (zo)=O, and s o t h e f i r s t case i s proved.

The proof of t h e r e s t is analogous, with t h e additional use in t h e l a s t case of t h e transitivity of t h e relation on R.

Remark 3.'2.2.

Instead of t h e r e l a t i o n (3.5) f o r t h e approximation e r r o r ~ ' E Y w e have 0 8 e 1 ) 2 . e and O p e ' t 2 . e .

as a consequence of t h e points (c) in Proposition 3.2.1. and 3.2.2., respectiveiy.

However, unlike t h e scalarized case, transitivity f o r t h e relation of non- domination or weak non-domination does not hold, and s o we cannot claim in Theorem 3.2.1. t h a t zo€X is a P(2.e)-solution o r WP(2.e)-solution.

Theorem 3.2.2.

Suppose t h a t t h e problem (MP) meets t h e a l g e b r a i c Slater-Uzawa c o n s t r a i n t qualif- ication. If zo€X i s a P ( y * ,&)-approximate solution of t h e problem, t h e n t h e r e ex- i s t o p e r a t o r s R o G ' ( 2 , ~ ) a n d SOEL (V,Y) such t h a t ( z o , R o , S o ) ~ dom @L i s a P ( y * ,&)-saddle point of t h e Lagrangian IPL.

P r o o f .

I t i s supposed t h a t z o € X i s a n &-solution of t h e s c a l a r valued optimization problem m i n

I

< y * , f ( z ) > E R : z E A, h ( z ) 5 0 , l ( z ) = 0 j

By Theorem 2.2.5. in t h e s c a l a r valued c a s e , t h e r e e x i s t functionals T * ~ U + a n d S*~EV' e n s u r i n g t h a t ( ~ ~ , r * ~ , s * ~ ) i s a n &-saddle point f o r t h e Lagrangian c o r r e s p o n d i n g t o t h e a b o v e s c a l a r problem, i.e.

If c EC i s a n element with <y* , c >=I, t h e n defining R o EL ' ( 2 , ~ ) a n d S O E L (V,Y) with t h e following c o r r e s p o n d e n c e s ,

t h e t h e o r e m i s p r o v e d .

Theorem 3.2.3.

S u p p o s e t h a t t h e problem (MP) meets t h e Slater-Uzawa c o n s t r a i n t qualification, and c o r e ( C ) # @ . If zo€X i s a WP(e)-solution of t h e problem ( M P ) t h e n t h e r e e x i s t o p e r a t o r s RoEL '(2,Y) a n d S O E L (V,Y) s u c h t h a t ( z o , R o , S o ) ~ d o m

aL

^{i s a WP(e)-}

s a d d l e point of t h e Lagrangian @ L . P r o o f .

By point ( c ) in Proposition 3.1.2. t h e r e e x i s t s a n y * E C + such t h a t z o € X i s a P ( y * , < y * , e >)-solution of ( M P ) a n d s o Theorem 3.2.2. implies t h a t t h e r e e x i s t a P ( y * , <y* , e >)-saddle point f o r Now, obviously y * EC' i s s t r i c t l y positive f o r t h e c o n e C1=core ( C ) u f O j . From a n a r g u m e n t similar t o t h e o n e used in t h e proof of

(b) in P r o p o s i t i o n 3.1.2. w e c a n conclude t h a t t h i s P ( y * , <y* ,e >)-saddle point i s a WP(e )-sacidle point as well.

w

Remark 3.2.3.

A r e s p e c t i v e t h e o r e m c o n c e r n i n g P(e)-solutions cannot. b e s t a t e d as a y * EC+, which i s s t r i c t l y positive f o r t h e whole c o n e C CY, d o e s not always e x i s t .

3.3. Primal and Dual Functions

In t h i s final s e c t i o n w e only d e a l with t h e s c a l a r i z e d case, i. e. P ( ~ * , E ) - t y p e minimality, as o t h e r w i s e being t h e solution of t h e r e s p e c t i v e a p p r o x i m a t e primal problem c a r r i e s l i t t l e information, as i s indicated in Remark 3.3.1.

Definition 3.3.1.

W e c a l i t h e following set valued maps t h e a p p r o x i m a t e primal a n d d u a l functions of t h e minimization problem ( M P ) :

and

The a p p r o x i m a t e primal a n d d u a l problems ( P ( y * , E)) a n d (D(y* , E ) ) are defined in terms of t h e functions P ( y * , E ) a n d D ( y * ,E). Accordingly z o E X o r (Ro,So)EL +(z,Y)%L (V,Y) i s a solution of t h e a p p r o x i m a t e primal or d u a l problems, if

respectively.

Proposition 3.3.1.

Proof.

For z E F w e have

@ , ~ ( z , R , S ) E Y u ^!-,a{ : R € L f ( Z , Y ) , S €L(V,Y) j

=

Remark 3.3.1.

If w e define e.g. t h e approximate primal problem ( P ( e ) ) in a corresponding manner t o Definition 3.3.1. then t h e analogue of Proposition 3.3.1. i s valid, and in such a way t h a t t h e set P ( e ) ( z ) is not bounded from below if z E F and h ( z ) # O . A s a consequence, i t would have only -a a s a solution. A s w e know from e.g. Luc (1984) t h i s i r r e g u l a r i t y d i s a p p e a r s if e =O.

Proposition 3.3.2.

(a) If zo€X i s a P ( y * ,&)-solution of t h e problem ( W ) t h e n i t is a solution of t h e problem ( P ( y * ,&)).

(b) If zo€X i s a solution of t h e problem ( P ( y * ,&)) t h e n i t i s a P ( y * ,4&) solution of t h e problem ( W ) . Proof.

(a) By Proposition 3.3.1. w e have f o r all z E F t h a t

T h e r e f o r e i t i s sufficient t o p r o v e t h a t

Again by t h e last proposition:

Hence by t h e definition of P ( y * ,3&) -MAV, t h e validity of (3.6) follows from t h e ine- quality:

And t h i s is a consequence of t h e relation w e supposed.

(b) Let us suppose now t h a t zo€X solves ( P ( y * ,&)), i.e. t h e r e e x i s t s a n

Belonging t o t h e f i r s t set means t h a t

where c o € C a n d OS<y* ,c O > S & . AS w e have f o r all z EX\F t h a t P ( y * , ~ ) ( z ) = t = j , i t is enough t o consider z EF, impiying

Hence belonging t o t h e second set implies:

and by (3.7)

< y * , f ( z o ) > - 4 c S < y * , f ( z ) >

vz ^EX.

Definition 3.3.2.

The element (zo,Ro,So) EXXL (Z,Y)

xL

(V,Y) i s called a P ( y * ,&)-dual p a i r of solu- tions if

(i)

z o E

X

i s a solution of t h e problem ( P ( y * , E ) )

and (ii)

Remark 3.3.2.

The definition could equivalently b e formulated as: x o € X and (RolSo)EL(Z,Y)XL (V,Y) i s a solution of t h e primal and t h e dual problem r e s p e c - tively, where t h e latter i s valid by way of j' (xo)€Y.

Theorem 3.3.1.

(a) If (xo,Ro.So) Edom cPL i s a P ( y * ,&)-saddle point of t h e Lagrangian IPL, t h e n i t is a P ( y * ,&)-dual p a i r of solutions.

(b) If ( z o l R o , S o ) EXXL ( Z , Y ) U (V,Y) i s a P ( y * , &)-dual p a i r of solutions t h e n i t is a P ( y * ,2&)-saddle point of t h e Lagrangian cPL

.

Proof.

(a) On one hand by Proposition 3.3.1. w e h a v e

On t h e o t h e r , by Theorem 3.2.1. w e know t h a t x o € X i s a P ( y * ,2e)-solution of t h e problem

(W).

Together with Proposition 3.3.1. t h i s yields t h e r e l a t i o n

This p r o v e s t h e f i r s t requirement of (xo,Ro,So) E$ being a P ( y * ,&)-dual p a i r of solutions. If (xo,Ro,So) E dom

aL

i s a P ( y * ,&)-saddle point t h e n by (c) in Proposi- tion 3.2.3. w e h a v e

and a l s o by t h e definition of t h e saddle point

If w e combine t h e s e two r e i a t i o n s then w e obtain

a n d a s a consequence

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

If t h i s i s n o t s o t h e n t h e r e e x i s t R EL ( Z , Y ) , S EL ( V , Y ) a n d y l ~ ( y * , . z ) ( R , S ) such t h a t

H e r e i t i s n e c e s s a r y t h a t R E L + ( Z , Y ) b e valid b e c a u s e o t h e r w i s e D(y* , & ) ( R , S )

= I --I

a n d consequently <y* , y >=-w. T h e r e f o r e

f o r some

z l W .

Using ( c ) in P r o p o s i t i o n 3.2.3 a n d t h e formula u n d e r (3.8), w e ob- t a i n

This, a n d y € D ( y * , & ) ( R , S ) , however, c o n t r a d i c t to (3.9). S o t h e s e c o n d r e q u i r e - ment i s p r o v e d .

( b ) By t h e f i r s t p a r t of t h e definition of t h e P ( y * ,&)-dual p a i r of solutions, t h e conditions imply t h a t w e P ( y * , .z)(z0), and, t h e r e f o r e z o W . By t h e s e c o n d w e know t h a t ^{- 0 4} D ( y * , 2 & ) ( R o , S o ) a n d t h e r e f o r e R o e + ( Z , Y ) . Hence, ( z o , R 0 , S o ) € d o m GL holds. A s a c o n s e q u e n c e of

z

w e h a v e

a n d so

implies

From (3.10) i t also follows t h a t

B y (3.11) (3.12) and t h e relation

sow,

Proposition 3.2.3. holds and t h e r e f o r e

sow

^{is a P ( y *} ,2e)-saddle point of

aL.

8

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