D E R I V A T I V E C H A R A C T E R I Z A T I O N

O F

C O N S T R A I N E D EXTREMA O F F U N C T I O N A L S

A SURVEY

Mohamed E l - H o d i r i

University of K a n s a s

R e s e a r c h Paper N o . 34

AUGUST 1970

RESEARCH P A P E R S I N T H E O R E T I C A L

AND

A P P L I E D ECONOMICS

DEPARTMENT O F E C O N O M I C S THE U N I V E R S I T Y OF KANSAS

LAURENCE, KANSAS

INTRODUCTION

One of t h e b a s i c axioms of economic a n a l y s i s is t h e axiom of ration-.:

a l i t y , i d e a , of p o s t u l a t i n g t h a t economic behavior r e s u l t s from a process

of opt imiza t ion . A s it i s , a t b e s t , d i f f i c u l t t o d i r e c t l y t e s t t h e axiom,

economists have been i n t e r e s t e d i n c h a r a c t e r i z a t i o n theorems of optirniza-

t i o n theory. In t h i s paper we review the theorems t h a t c h a r a c t e r i z e o p t i -

mal i ty by way of d e r i v a t i v e s . F i r s t we formulate a very gene ra l optimiza-

t i o n problem. Then we p resen t c h a r a c t e r i z a t i o n theorems f o r t h r e e types

of problems: F i n i t e dimensional, v a r i a t i o n a l and problems i n l i n e a r topo-

l o g i c a l spaces. I n each case we present theorems f o r e q u a l i t y - inequal-

i t y c o n s t r a i n t s . The theorems i n each case a r e : first o rde r necessary

cond i t ions , first o rde r s u f f i c i e n t cond i t ions , second o rde r necessary

condi t ions and second or~er:,sufficienTy&osditions. The scheme of repre-

s e n t a t i o n is a s fol lows: Statements of theorems are followed by remarks.

r e f e r r i n g t h e r eade r t o t h e e a r l i e s t , known t o us , proofs of t h e theorems.

I n some ins t ances , s l i g h t gene ra l i za t ions of some theorems appear he re f o r

t he first t ime, an i n d i c a t i o n of necessary modif icat ions t o e x i s t i n g

proofs a r e indica ted . A case is "solved" if proofs f o r a l l t h e fou r types

of c h a r a c t e r i z a t i o n theorems e x i s t . The only "unsolved" case i s t h a t of

problems i n l i n e a r t opo log ica l spaces with i n e q u a l i t y and wi th e q u a l i t y - i n e q u a l i t y c o n s t r a i n t s . For t h i s case , we p resen t two con jec tu res about

second o rde r condi t ions t h a t a r e analcgousto t h e e q u a l i t y c o n s t r a i n t case.

2

We now s t a t e t h e "general" op t imiza t ion problem:

Let A , B , C , D be r e a l l i n e a r t o p o l o g i c a l spaces. Consider t h e func-

t i o n s :

Let P be a p a r t i a l o rder ing of B y l e t > - be def ined on C - a s u s u a l - by

way of a convex cone and l e t e l e2 e 3 and e 4 be t h e n e u t r a l element o f 3 3 '

a d d i t i o n - ze ro elements - of A,B,c and D r e spec t ive ly . The problem may

( 1 ) - be s t a t e d a s fo l lows . A a

Find E A such t h a t f ( x f is P-maximal ( 2 ) sub jec t t o :

I n case B is t h e r e a l l i n e , P is t h e r e l a t i o n ">" - def ined for r e a l numbers

and w e have a problem of s c a l e r op t imiza t ion . I n ca se B is f i n i t e dimen-

s i o n a l we have..a f i n i t e - v e c t o r maximization problem. I n gene ra l , charac-

t e r i z a t i o n s of s o l u t i o n s t o f i n i t e - v e c t o r maximization problems may be

derived from c h a r a c t e r i z a t i o n s of s o l u t i o n s of scaler maximization prob-

l e m s . We s h a l l restrict ou r p re sen ta t ion t o s c a l e r maximization prob3ems.

As an a p p l i c a t i o n of s c a l e r maximization theorems we may "solve" a p a r t -

t i c u l a r f i n i t e - v e c t o r maximization problem, namely f o r t h e ca se where P

is taken t o be a aret to'^) order ing of B. FQ' i n f i n i t e - v e c t o r maximization

problems, a s c a l e r i z a t i o n theorem appears t o be the most app ropr i a t e i n t e r -

3

mediate s t e p t o de r ive c h a r a c t e r i z a t i o n theorems. Such a theorem w a s

proved by Hurwicz [20]. The method, f o r f i n i t e - v e c t o r m3ximization, con-

s i s t s ( ' ) of simply observing t h a t t h e problem is equiva lent t o an in-

f i n i t e number of s c a l e r maximization problems. However, w e s h a l l only be

concerned with s c a l e r maximization problems.

2. FINITE DIMENSIONAL PROBLEMS

In t h i s p a r t we l e t A = E", B = E', C = E~ and D = E', where E", E ~ , Em I I . E Eucl id ian spaces of dimensions n , one, m and R r e spec t ive ly . For

the is p a r t we reformulate t h e s c a l e r maximization problem a s follows:

Problem 1. Find x E E~ such t h a t f (x ) > - f f o r a l l s a t i s f y i n g

B ga(x ) = 0, a = 1 ,..., m , h (x) 2 0 , - B = 1 , ..., R . .-

We f u r t h e r s t a t e some p rope r t i e s of t h e c o n s t r a i n t s t h a t w i l l be used

i n d i scuss ing t h e theorems i n t h i s s ec t ion . Some of t h e nomenclature,

designated by " " is Karush's C231.

0 D.1) Def in i t ion . E f f e c t i v e c o n s t r a i n t s . 'Le t x b e a po in t t h a t s a t i s f i e s h B 0 (x) > 0 , = 1,. . . , m. Let r ( R ) be the s e t of i nd ices B such t h a t h B -

($1 = 0 . The c o n s t r a i n t s with ind ices B E r i:) ' k i l l be r e f e r r e d t o as t h e

0 e f f e c t i v e cons t r a in t6 a t x .

D.2) Def in i t ion . "Admissible Diyectiop'! Let h be d i f f e r e n t i a b l e . We say t h a t

X is an admissible d i r e c t i o n i f X is a non-tridal. s o l u t i o n t o t h e i n e q u a l i t i e s :

0 D.3) D e f i n i t i o n . "A curve i s s u i n g from x i n t h e d i r e c t i o n X." By t h a t w e

mean,. an n-vector va lued func t i on , §(t), o f a r e a l v a r i a b l e t such t h a t

d E ( 0 ) = $ and 5 ' ( 0 ) = - = A'.

d t S ( t ) I t z o 0

D.4) D e f i n i t i o n . admissible a r c i s s u i n g from x i n t h e d i r e c t i o n A " is

0 a n arc i s s u i n g from x i n t h e d i r e c t i o n X such t h a t h ( S ( t ) ) > - 0.

0 D . 5 ) D e f i n i t i o n . "Proper ty Q" - f o r i n e q u a l i t y c o n s t r a i n t s , is s a t i s f i e d a t x.

i f f : For each admis s ib l e d i r e c t i o n A , t h e r e e x i s t s a n admis s ib l e a r c

i s s u i n g from 8.

D.6) D e f i n i t i o n . The rank c o n d i t i o n f o r i n e q u a l i t y c o n s t r a i n t s . We s a y t h a t

t h e rank condi t ion h is s a t i s f i e d a t $ i f f 1.) is d i f f e r e n t i a b l e and 2 ) t h e

B rank of t h e m a t r i x c R . I , f3 E r ($) , e q u a l s t h e number o f e f f e c t i v e con- 1

s t r a i n t s .

D.7) D e f i n i t i o n . The rank c o n d i t i o n f o r e q u a l i t y _ - i n e q u a l i t y c o n s t r a i n t s is -- - --

0 s a t i s f i e d a t x i f f 1) t h e f u n c t i o n s g and h are d i f f e r e n t i a b l e and 2 ) t h e

0 ag" 9 c o n s t r a i n t s a t x, where EOL .i = - a x . I x =x:

L

. > ,.: . . . )

D.8) D e f i n i t i o n . The rank c o n d i t i o n f o r e q u a l i t y c o n s t r a i n t s is s a t i s -

f i e d a t i f t h e rank of t h e m a t r i x [E?I is a . 1

F i n a l l y wa d e f i n e l o c a l and g l o b a l c o n s t r a i n t maxima:

D . 9 ) D e f i n i t i o n . x is s a i d t o be a l o c a l s o l u t i o n t o problem 1. i f f : A ,. ,.

There e x i s t s a neighborhood of x, N(,x), such t h a t f ( x ) > f ( x ) for a l l -

,. ? r N(;) satisfying g ( * ) = 0 and h(x) 2 0. ,. ,.

D.lO) Definition. x is said to -. be a global solution to.problem 1. iff f(x) 3

f(x) for all x satisfying g(x) = 0 and h(x) > 0.

2. 1. First order necessary conditions. +.

Theorem - 1. If f, g - and h - are contjnuously differentiable and if x -- is a

global solution to problem 1, then there exists a vector (A 0' v,p) = (Ao, 1 m L

v , . . . , v ,e , . . . , p a ) 0 such that, 1 > 0 and ; 0 -

1, e B > 0 , - ,OhB (;) =o . . - 0 0 0 2) Fx = 0, where F = X f + w. g + p. h, Fx is the vector of partial

0 A

derivatives of F with respect to the component's of x evaluated at k = x.

n For the case of equality constraints, i .e. the set {x I h (x) 0 )=E ,

( 6 ) Theorem 1 was proved by ~aratheodor~(~) C93 and Bliss C71. For the case

n of inequality constraints, i . e . the set {x I g (y) = 0) = E , theorem 1 -

except for the non-negativity of p in condition 1 of the conclusion - was proved by ~arush'~) [231. The non-negativity of p may be proved by using

( 8 a separation theorem . This was, essentially, the crux of Fritz John's

[22] proof who has treated a more general problem of an infinite number of

inequality constraints. For the equality-inequality constraints the proof

involves writing equality constraints as inequality constraints (g l o and - g (x) > 0) and applying Karush's or Fritz Jone's theorem. -

(9 ) Theorem 2. If, in addition to the assumptions of theorem 1 we have either. - . .

a) Property Q for inequality constraints, D. 4, and the rank condition for ..

equality constraints, D. 7, are satisfied at x. I

a ) -- The rank condition ~- for .. equality -inequali ty constraints is satisfied A

at x. - Then the conclusions of theorem 1 follow with X 0 .

---L--I)-

For equality constraints the theorem was proved as a corollary of

theorem 1 by ~liss(lO' [ 7 ] . For the case of inequality constraints the

theorem was proved by Karush (I1) C231. The proof, for equality-inequality

constraints, may be accomplished by converting inequality constraints to

equality constraints and obtaining the theorem as a corollary to theorem 1.

This was presented by Pennisi (13) C271. A direct proof was presenied by

Hestenes (l" [l9].

2.2. First Order Sufficient Conditions.

Theorem 3. If: 1) f. E and h are differentiable 2 ) The conclusions of A ,. A

theorem 1 hold with X _ > J at a point x with g(x) = 0 , h (x) > 0 . 3 ) "

Either (3.a) FO, of theorem 1, is concave or (3.b) G! is linear and h is ,.

concave. Then x is a rloh31 solution to ~roblem 1.

The theorem follows from the fact that a concave function lies above

its tangent plane. The implications of this fact was utilized by Kuhn-

Tucker C241 in the proof of their equivalence theorem. The present the-

orem may be proved by applying Lemma 3 of Kuhn and Tucker C241 to FO.

2 .3 Second Order Necessary Conditicns.

Theorem 4 . If: 1) f , g and h have second order continuous p a r t i a l de- -- A

r i v a t i v e s . 2: x is a solutLon t o problem 1. 3) The rank cond i t i on f o r --

e q u a l i t y - i n e q u a l i t y c c a s t r a i n t s , 9.5. . I .... 5 .I . ..-!:. -- ---

1 m 1 R Thenthere exis-? m u l t i p l i e r s ( v , p ) = ( v , . . :, v ; LI , . . . , LI ) such t h a t :

For t h e e q u a l i t y c o n s t r a i n t s , thc! theorem w a s proved by Caratheodory (15)

[9] and ~ L i s s ( ~ ~ ) [ ' 7 ] . For i n e q u a l i t y c o n s t r a i n t s t h e theorem was proved

by Karush (17) [231. For t h e equa l i t y - inequa l i t y ca se , t h e theorem was

proved by Pennis i (19 ) ( l R ) [27] , under t h e , d i spens ib l e , s t i p u l a t i o n t h a t

t he number 02 n o ~ l - ~ s r o m u l t i p l i e r s a t t ached t o e f f e c t i v e i n e q u a l i t y con-

s t r a i n t s is a t most one.

2.4. Second order s x f f i , . e n t cond i t i ons .

Theorem 5,. I f : l? f , g c 7 d h have cont inuoas second o rde r d e r i v a t i v e s . A A A

2 ) The c o n c l u s i o ~ ~ s G! theorem 2 holds at :; with g (x) = 0 , h (x) > 0. - -

3 ) ~ h e conclusion o f theoren. 4 hc lds with s t r i c t i n e q u a l i t y f o r r, f 0.

A

Then x is a l o c a l s o l u t i o n t o p r o b l ~ n 1.

For e q u a l i t y c o n s t r a ~ n t ; f;ic ti !?orex tris proved by B l i s s ' 20 )[ 7 I. Caratheodory

( 2 1 ) [9 ] assumes, c ~ r i ; o ~ c l y , t h a t t h e r a ~ k cond i t i on holds . For in-

e q u a l i t y c o n s t r i i n t s t he theo rec w z s proved by ~ a r u s h ( ~ ~ ) [ 2 3 ] . A very

c l o s e l y r e l a t e d theorem was proved by ~ e n n i s i ' 2 3 ) ~ 27 I , where the res t r i c t - ^' > 0 for - ind iceg 4 with ions on n a r e augmented by r e q u i r i n g t h a t C.h. 1 1 " i -

p0 > 0 . Theorem 5 may be proved by applying t h e s u f f i c i e n c y theorem f o r

e q u a l i t y c o n s t r a i n t s by us ing Karush I s device ' 2 4 ) ~ 2 3 I . A d i r e c t proof was

provided by H e ~ t e n e s ( ~ ~ ) [ 1 9 1 .

3 . A VARIATIONAL PROBLEM

I n view of t h e v a s t l i t e r a t u r e on t h e ca l cu lus of v a r i a t i o n s and

t h e wide a c c e s s a b i l i t y of such l i t e r a t u r e , t h i s p a r t w i l l be very b r i e f .

Many i n t e r e s t i n g t o p i c s w i l l be l e f t o u t , e .g . problems with r e t a r d e d a r -

(26 guments . For a survey of v a r i a t i o n a l problems with e q u a l i t y c o n s t r a i n t s ,

t h e r eade r is r e f e r r e d t o B l i s s [ 6 ] , which we s h a l l t ake a s a p o i n t of de-

p a r t u r e f o r t h i s p a r t . We formulate t he Bolza-Hestenes ( 2 7 ) problem i n t h e

Calculus of v a r i a t i o n s a s fo l lows: L e t T be a subse t of t he non-negative

h a l f of t h e r e a l l i n e . Consider t h e c l a s s o f piecewise smooth func t ions

x ( t ) def ined on t and having va lues i n E" t oge the r with t h e i r d e r i v a t i v e s k.

The problem is : ^ A A

A A A A

-ern 2.Find t t l , k ( t ) , ~(t), k(?), x ( t 3 = 3 t h a t maximizes J0 [XI = 0 0

L1 0 It f ( t , x, A) d t + rQ (to, tl, x ( t o ) , x ( t ) ) sub jec t t o 0

1 +

B ( 3 ) ip ( t , X, A ) = 0 , a = 1 ,...,. 21,

B - ( 4 ) 4 ( t , x , 2 ) > - 0 , B = R + 1,. .. , L , 1

with t < t 0 1'

A s was noted, by Berkovitz C51 and Hestenes C171, t he above

problem is equiva len t t o t h e problem of opt imal c o n t r o l ( 2 8 ) . The r e -

s u l t s of opt imal c o n t r o l theory a r e de r ivab le from t h e r e s u l t s t h a t

we s h a l l p re sen t . ( 2 9 )

3.1. F i r s t o rde r necessary cond i t i ons .

Theorem 6 . I f : 1 ) 1 f e r e n t i a b l e . a s func t ions of r e a l v a r i a b l e s . 2) Z is a s o l u t i o n t o Dro-

- and where t h e index desig-

0 A na te s i nd ices B with 4 (2 ) = 0, i '= 1,. . . , n.

1 m 1 Then t h e r e e x i s t a cons tan t vec to r A , q,o) = ( A - ; q ,. . . , q ; o , . . , us)

1 R A A

and a vec to r func t ion p = (p ,..., p ) def ined on Ct , t 1 such t h a t :

1 ) There does no t e x i s t t E Ct t l l with L A n , q , W , PI = 0 , a l s o (ho ,q ,o) 0.. A'.

3) P ( t ) is piecewise cont inuous , cont inuous a t p o i n t s of c o n t i n u i t y of - A

k , p t > - 0 , pB ( t ) +OL i31 = 0

5 ) - d Fk = I , i = 1 ,..., n , where F = A o f 0 t T CY qCY fa t pB 0' , X

d t i i

a' A A

7 ) d G +[ !F-fxiFx. ) d t t Z FA d x i ( t ) lY' l = 0 i s an i d e n t i t y & dxi Y 1 y y=o i i

.-. A ,. A A .-. ,. 8) E ( t , X , X , k ) = F ( t , X , X) - F ( t , X , k ) - C ( X - k.) FA 2 ru E $

1 i 1 i B

A

whenever (t, x , A ) s a t i s f y t h e . c o n s t r a i n t s ( 1 ) - ( 6 ) .

For f i x e d end p o i n t s ( i . e . t 0 ,

x ( t end x ( t a r e c o n s l a n t s ) , 0 1

(30-31) Valent ine [29] proved t h a t conclusion: 5 E 8) a r e necessary with conclu- -

B s i o n 3 ) holding f o r p except f o r t h e non-nega t iv i ty of p . The non-neg- - a t i v i t y of pB was proved by apply ing a Clebsch ( 3 2 ) type second o rde r nec-

e s s a r y condi t ion ' 33) . Valent ine I s method c o n s i s t e d of conver t ing d i f f e r -

e n t i a l i n e q u a l i t y c o n s t r a i n t s t o equa t ions by s u b t r a c t i n g from each , t h e

square o f a d e r i v a t i v e of an added v a r i a b l e . Then he der ived h i s r e s u l t s

a s a p p l i c a t i o n s o f c h a r a c t e r i z a t i o n theorems f o r t h e problem o f Balza with

e q u a l i t y c o n s t r a i n t s (34). For t h e g e n e r a l problem we may conver t t h e i n -

e q u a l i t y c o n s t r a i n t s ( 2 ) and ( 6 ) t o e q u a l i t y c o n s t r a i n t s by i n t roduc ing

- - - -

t h r e e new sets of v a r i a b l e s ya 1 ( t ) , Y: ( t ) and y; ( t as fol lows : - - - - -

ci y, ( t l ) f r e e , G: 2 y 3 3 1

B ( t ) = 0, y ( t ) f r e e and $Y= 0 , y Y ( t ) f r e e , and cons ider

0 . . equ iva l en~ prcblen of maximizing J s u b j e c t t o c o n s t r a i n t s (11, (31, ( 5 )

. - - - - - - 1 -a 1 -.G 1

and: (2 ) j0 = J~ - (yl ( t l ) 1 2 = 0 , ( 4 ) = 6' - = 0 and ( 6 ) =

- - ci 2

qa - (y3 ( t l ) ) = O . We then g e t , i n a d d i t i o n t o Va len t ine ' s cond i t i ons ,

cond i t i on 7 ) of theorem 6 . Noting t h a t condi t ion 6 ) may be obtained from

5 ) and 7 ) w e would have a l l of t he cond i t i ons of t h e theorem. A d i r e c t . .

e l egan t proof of theorem 6 was provided by Hestenes [18].

We now d i s c u s s a condi t ion t h a t guarantees t h a t A i n theorem 6 is 0

non-zero. The case where t h e m u l t i p l i e r s a r e unique, choosing X = 1, is 0

what is known i n t h e l i t e r a t u r e a s t h e normal case . Although cond i t i ons

f o r normali ty a r e very hard t o v e r i f y i n a p p l i c a t i o n s , we s h a l l p re sen t one

o f t hese cond i t i ons i n t h i s s e c t i o n (35)

*

D.11) Def in i t i on . Normality. Z i s s a i d t o be normal i f f conclusions 1 ) - 5) and

7) of theorem 6 hold with ho = 1 and t h e m u l t i p l i e r s q,p,w a r e unique.

D.12) Def in i t i on . Admissible v a r i a t i o n s . Consider a po in t 2 and a vec to r valued

u b u - - - func t ions E ( t ) = (C1 ( t ) , . . . . Sn ( t ) ) ; 0 1,. .. , m t s , where m = m t

1 -

- t he number of c o n s t r a i n t s J~ t h a t hol4 a s equat ions a t 2 , e f f e c t i v e at t h e

- - end p o i n t s of Z , and where 5 = S , + t h e number of c o n s t r a i n t s G Y t h a t a r e

- - 1 m t s e f f e c t i v e a t g . E ( t ) = (5 ( t ) , ..., 5 ( t ) ) i s s a i d t o be admissible

v a r i a t i o n s i f f :

1) 5 . ( t ) a r e d i f f e r e n t i a b l e on C t 1 0 ' i l l ,

- - - 3 s a L? *' = c f o r a wi th 4 p ( Z ) = 0, where m0 = -

2.2) 4 - (SO) = ? 4x Si + l 4x Si ' i l i 1

- - and where , a 4, and 4; are de f ined s i m i l a r l y .

i i i

D,13) De f in i t i on . The rank c o n d i t i o n . The f irst rank c o n d i t i o n is s a i d t o be

0 s a t i s f i e d a t 2 i f t h e r e e x i s t s a set of admis s ib l e v a r i a t i o n s 5 , and a r b i -

u 0 - - t r a r y c o n s t a n t s T 0 ' -ri . 0.1,. . . , m + s , such t h ~ t t h s ma t r i x

L . I

- - - a m t s , where c1 = x ( t o ) - fa ( I o , X ( t o ) , x ( t o ) ) ) T, u '! 1 o i ( t )

0

- - - ' 0 -

S? (tl) t 1:' (, 5; ( t ) + r: ;.(t)) d t l , 6 denotes indices of t Xi 1

0 i

- - e f f e c t i v e c o n s t r a i n t s J [XI a t Z , " = 1, ..., m + s , - 3

- - - - - - u -

( t l ) l , 7 denotes i nd ices

- of $Y t h a t a r e e f f e c t i v e a t 5 , 0 = 1,. . . , m + s , and where s u b s c r i p t s

denote ~ a r t i a l der ivatLves with r e s ~ e c t t o i nd ica t ed v a r i a b l e s and " - I f

- above an express ion i n d i c a t e s t h a t it is evaluated a t X.

Theorem 7. If f , g , 4 and 1) a r e d i f f e r e n t i a b l e t hen t h e rank cond i t i on , - -

d e f i n i t i o n D.12., a t S is necessary and s u f f i c i e n t f o r t h e normali ty of 3.

I n t h e absence o f i n e q u a l i t y c o n s t r a i n t s , t h e theorem was proved

( 3 6 by B l i s s [ 6 ] . Reformulating t h e problem with added i n e q u a l i t y c o n s t r a i n t s ,

a s i nd ica t ed above, t h e theorem i s obtained a s a s t r a i g h t forward a ~ p l i c a t i o n

o f B l i s s ' s theorem.

3 . 2 . F i r s t Order S u f f i c i e n t Condit ions.

Theorem 8. If 1) f , g , b and $ are d i f f e r e n t i a b l e , and concave. 2 ) Con-

c l u s i o n s 1 ) - 5 ) and 7 : of theorem 6 a r e s a t i s f i e d wi th : 2 : i )

h > o , 2. i i ) qY > a, :iY > o , y = I,.. . , ml, y = l,.. . , sl, a t a po in t 2 0

~~- -- ,. t h a t s a t i s f i e s t h e c o n s t r a i n t s (1) - ( 6 ) . Then Z is a g l o b a l s o l u t i o n t o

The theorem was moved by Mangasarian C261 f o r t h e canon ica l , opt imal

c o n t r o l , problem with f i x e d to and t 1 ' Theorem 8 may be proved by

r epea t ing t h e s t e p s of Mangasarianls proof f o r problem 2. A s Man-

gasa r i an n o t e s , i n t h e absence of e q u a l i t y c o n s t r a i n t s cond i t i on 2. ii)

of theorem 8 i s no t needed, no t i s it req-uired i f t h e e q u a l i t y c o n s t r a i n t s

a r e l i n e a r .

3.3. Second Order Necessary Condit ions. -

Theorem 9. ( J acob i - Myer - Bliss) I f : lj f , g , h and + have cont inuous A A

second o rde r d e r i v a t i v e s . 2 ) 3 is a s o l u t i o n t o ~ r o b l e m 2 and 3 ) Z is

normal. Then t h e r e e x i s t m u l t i p l i e r s a s i n theorem 6 with X = 1 such t h a t : 0

't, - A * *

+ I , (.I. F 5 . 5 . + 2 . I . F 5 . 5 . + .Z. F. t . t . ) d t < , f o r 1 , X.X 1 ] I,] X . ? 1 ] l,] X.? 1 J -

t n 1 j 1 j 1 j

( T , 5 ) = ( T o , T ~ , El ( t ) ;... , En ( t ) ) 0 s a t i s f y i n g

^ B ( i ) @ ( 5 ) = O , - B Q ( 5 ) = 0

.I

with t h e terms i n ( i ) and (ii) def ined a s i n ( 2 ) of D.12 and as i n D . 1 3

5 5 with (T, < ) a s a m a t r i x with - one rov! (T , 5 ) = ( r , <) ) , and where:

2 0 A

d G ( 2 ; ' r , 5 ) i s t h e s x o n d d i f f e r e n t i a l o f G a t 2 with ( r , 5) a s in-

crements, a s i n g l e s u b s c r i p t denotes a f i r s t d e r i v a t i v e and a double

s u b s c r i p t denote second (mixed p a r t i a l ) d e r i v a t i v e s with " - s i g n i f y i n g

eva lua t ion a t Z .

For e q u a l i t y c o n s t r a i n t s t h e theorem was proved by ~ l i s s ( ~ ~ ) [ 6 1

and [ 8 1 ! ~ ~ ) 1 . 1 ad ied i : lequal i ty c o n s t r a i n t t h e theorem may be proved by

applying Bliss's theorem t o our problem, a f t e r conver t ing it t o a problem

with e q u a l i t y c o n s t r a i n t s as we ind ica t ed i n t h e d iscuss ion of theorem 6 .

Theorem 10. (The Clebsch cond i t i on ) . If t h e hypotheses of theorem 9 a r e

s a t i s f i e d then 1. F. . II.II. < 0 f o r II = (II ,,... , nn) f 0 s a t i s f y i n g i,] x,x, 1 1 -

- ? iB n . = o , B = ~ ,..., i=l x, 1

L I iB x, II. 1 = 0 , B a r e i n d i c e s o f i n e q u a l i t y

c o n s t r a i n t s t h a t a r e e f f e c t i v e a t Z -

-a A

C ,a 11. = 0 , = . m l , i ~ l l i ; fk: 1 = 0 a i n d i c a t e s c o n s t r a i n t s

A

J~ e f f e c t i v e a t Z.

Theorem (10) w a s proved by Valent ine ' 3 9 ' ~ ~ 9 ~ f o r f i x e d end po in t s .

Valent ine ' s method o f proof can , e a s i l y , be app l i ed t o prove ou r theorem.

3 .4 . Second Order S u f f i c i e n t Conditions

For t h e purposes of t h i s s e c t i o n , we have t o d e f i n e a weak l o c a l

s o l u t i o n of problcm 2 . A

D.13. Def in i t ion . Weak Local Solu t ion . We say t h a t Z is a weak l o c a l s o l u t i o n A

if JO [ a ] > JO [a] f o r a l l 3 s a t i s f y i n g t h e c o n s t r a i n t s (1 ) - ( 6 ) with -

1 ) (X-X, - 1 I I < E -or sone E > 0 , where 1 1 * I I denotes t h e Eucl id ian norm.

Theorem (Penn i s i ) I f 1) f , g , 4 and + have continuous second o r d e r

16.

P a r t i a l d e r i v a t i v e s . 2) Conclusions 1 - 5 and 7 of theorem 6 a r e s a t i s f i e d a t a

po in t Z t h a t s a t i s f i e s t h e c o n s t r a i n t s 1 - 6. 3 ) The ma t r ix A

of theorem 6 , has f u l l rank . 4 ) The form Q (Z; r , 5 ) of

theorem 9 is negat ive d e f i n i t e under c o n s t r a i n t s ( i ) and ( i i ) @A

i

A

'\

' ( i n t h e s ta tement of theorem 9) f o r (T, 5) -) 0. Then t h e r e "exis ts an

E > 0 such t h a t 2 is a weak l o c a l s o l u t i o n of ~ r o b l e m 2, i n t h e sense of

d e f i n i t i o n D. 13.

Assuming t h a t X = 1 and t h a t a t most one i n e q u a l i t y c o n s t r a i n t 0

A

is e f f e c t i v e a t Z, Valent ine (40 ) proved an analogous theorem i n C291.

P e n n i ~ i ( ~ ~ ) [ 2 7 ] proved a s u f f i c i e n c y theorem without t h e assumptions of

Valent ine. P e n n i s i ' s theorem i s s t ronge r t han theorem 11 i n t h e sense

t h a t Q is negat ive on t h e subse t of v a r i a t i o n s (-r, 0 ) of theorem 11

which i n a + d i t i o n ( t o ( i ) and ( i i ) ) of theorem 9 ) s a t i s f y some inequal-

i t i e s f o r t hose c o n s t r a i n t s t h a t a r e e f f e c t i v e but have zero m u l t i p l i e r s .

One way t o prove theorem 11 is t o conver t t h e problem i n t o one with

e q u a l i t y c o n s t r a i n t s and apnly P e n n i s i ' s theorem. Another way is t o

no te t h a t theorem 11 is c o r o l l a r y o f P e n n i s i ' s theorem, a f t e r modifying

t h e l a t t e r t o t a k e c a r e o f t h e a d d i t i o n a l c o n s t r a i n t s .

I

4. A PROBLEM I N LINEAR TOPOLOGICAL SPACES.

Let A , C, D be r e a l Banach spaces and l e t R denote t h e r e a l l i n e .

Consider f : A - R : A -t C and h : A -t D. The problem we s t u d y h e r e i s :

Problem 3. t:r:Lr-!ize f(::j s u b j e c t t o g(x) = 0 and h(::) L €14 3

where t h e

i n e q u a l i t y 0 and 8 a r e as de f i ned i n t h e i n t r o d u c t i o n . 3 4

Dealing w i th d i z f e r e n t i a b l e f u n c t i o n s we n o t e t h a t t h e r e are num-

e rous e q u i v a l e n t (42) vr2.y~ o f d e f i n i n g d e r i v a t i v e s i n l i n e a r space s . We

s h a l l u s e Frechr:tls d e f i n i t i o n and mean ":'rechc.t d i f f e r e n t i a b l e " when

we s a y tha- t a f unc t i on is d i f f e r e n t h b l c . ('13

4.1. F i r s t Order Necessa~%y Condi t ions A

Theorem 1 2 . If f , g and h are d i f f c r c n t i a b l e and i f x i s a s o l u t i o n t o

problem 3 then t h e r e e x i s t s a c o n s t a n t X > 0 and l i n e a r f u n c t i o n a l s 0 -

2 2 C + R , x2 : D + R such t h a t 1 ) > 0 , [ a 2 , h(;)l = 0 , where [e , h l -

deno t e s t h e va lue of -. t h e f u n c t i o n a l k2 a t h ( r ) . 2 ) For any

1 i 2 2 y E C , y2 E D , t h e t r i p l e ( A , , [PA , yl!, [ a , y ] / 0 3) F 1 (x) = 0 ,

1 2 . A

where F = X .f t [ R , g l + [R , h l and F' (x) = d F (x, 5) with 5 as t h e n

"increment" i n t h e d e f i n i t i o n o f t h e d i f f e r e n t i a l .

~ 1 , ~ 1; 1.2.: -?.J,TJ ;--,? 1 JL,;, 6.;- - . . . -1-. ,. *.-:-I theorem 2 . 1 i n Duboviski i and . .

M i l y u t i n ( " 4 ) [ l ~ ~ .

Condi t ions t h a t gua ran t ee t h a t X > 0 and t h a t t h e f u n c t i o n a l s 0

1 2 R and R a r e unique a r e r e f e r r e d t o i n t h e l i t e r a t u r e , p e r t a i n i n g t o

problem 3 , a s r e g u l a r i t y cond i t i ons and a s c o n s t r a i n t q -ua l i f i ca t ions .

We s h a l l now l i s t t h e s e cond i t i ons and p re sen t some s u f f i c i e n t condi-

t i o n s f o r them t o hold.

(4 .1.1) Regular i ty Condit ions:

(R . l ) (Gapushkin [14]): For e q u a l i t y c o n s t r a i n t s , x is s a i d t o (R.1) r e g u l a r

if f o r every 5 E A with g 1 (X, 5 ) = e3, 5 we have: There e x i s t s a

func t ion of a r e a l v a r i a b l e t , V : [O ,l] + A such t h a t ~ ( 0 ) = 2, g(T~(t)) = 8 3

f o r t E [0, 11, V ' ( t , T) e x i s t s f o r t E C0, l l and V ' ( 0 , -r) = 5.

(R. 2) (Hurwicz [20]) : For t h e i n e q u a l i t y c o n s t r a i n t , x is ( R . 2 ) r e g u l a r i f f :

For any 5 s A wi th 5 0 such t h a t x = x + 6 impl ies h ' (x, 5 ) + h ( x ) 1 e 4 , 1

we have: There e x i s t s a func t ion of a r e a l v a r i a b l e t , V: [O, 11 + A

such t h a t :

( i ) V ' ( t , T ) e x i s t s f o r t E C O , 11 -

( i i ) x = V ( 0 )

( i i i ) h (V ( t ) ) > 0 4 , t E C O , l l - ( i v ) V' ( 0 , r ) = 5 , r > 0.

(R.3) (Gapushkin C141): 2 is sa id t o be ( R . 3 ) - r e g u l a r i f f : For any 5 E A

with g 1 (;, 5 ) = 0 and h + h ' (x, 0 2 O 4 we have: There e x i s t s a 3

func t ion of a r e a l v a r i a b l e t; V: [ O , .11 + A such t h a t

( a ) v (0 ) = x ( b ) g (V ( t ) ) = 0 3 , h (V ( t ) ) > e 4 , t E [ O , l l

(c) V' ( t , T ) e x i s t s f o r t E C0,lI

( d ) v ' ( 0 , T) = 5 , T > 0 -

Remark 1 : (R.3) i s a s p e c i a l i z a t i o n of Gapushkin's r e g u l a r i t y condi t ion

which is a uniform r e g u l a r i t y condi t ion(46) . (R.1) i s a f u r t h e r s p e c i a l -

i z a t i o n f o r t h e ca se of e q u a l i t y c o n s t r a i n t s .

Remark 2: Reca l l t h a t t h e i n e q u a l i t y i n c o n s t r a i n t 2) of problem 3 is

def ined i n terms of a c losed convex cone, s ay , K2. Let K = { e I&, K,, 3

where ( 0 1 is a cone t h a t con ta ins only 0 Let 8 = 0 3 Q 0 4 and l e t 3 3 '

G : A -t C D be t h e "pa i r" valued func t ion < g , h >. Then we may w r i t e

c o n s t r a i n t s 1) and 2) i n t h e form G(x) > 8 where ">" is i n t h e sense o f K. - -

With t h a t formula t ion , (R.2) becomes a r e g u l a r i t y cond i t i on f o r equa l i t y -

i n e q u a l i t y c o n s t r a i n t s .

4.1.2. S u f f i c i e n t Conditions f o r R e ~ u l a r i t v :

We now present some condi t ions t h a t imply r e g u l a r i t y . We present

some s u f f i c i e n c y lemmas f o r e q u a l i t y c o n s t r a i n t s and some s u f f i c i e n c y lemmas

f o r e q u a l i t y - i n e q u a l i t y c o n s t r a i n t s . These l a s t lemmas a r e , of course ,

s u f f i c i e n c y lemmas, f o r R . 1 and R . l r e g u l a r i t y . However, they may be

s t reng thened by s ~ s c i a l i z i n g t h e c o n d i t i o n s when we a r e concerned wi th

'(R..l) - r e g u l a r i t y o r ( R . 2 ) - r e g u l a r i t y . Before s t a t i n g t h e s e condi-

t i o n s we i n t roduce some n o t a t i o n s .

0 . 1 ) The c o n s t r a i n t s e t N = {x E A: g ( x ) = O 3 and h ( x ) > - e4] .

0.2) Let I I 1 1 , t h e norm of t h e space A , a sohe re i n A w i th c e n t w a t ; and

- r a d i u s 6 w i l l b e denoted by y (x, 6 ) and y ( x , 6 ) = { x : I I x - x I In 5 t i ) -

0.3) The s e t D; = 1 F E A : F' (x, 5 ) = 03} i s a subspace of A . Let P- b e t h e X

p r o j e c t i o n o p e r a t o r w i th P- A = D-. X X

... .. 0 .4 ) Let 2 be a l i n e a r space , w e denote by 2 t h e space of l i n e a r e f u n c t i o n a l s

de f i ned on Z.

0 .5 ) Denote by N t h e s e t N = U . y ( x 6 6 X E N O '

E ) - 0

ncw l',st t h e c o n d i t i o n s which w e u s e i n t h e s t a t emen t s o f s u f f i -

c iency theorems f o r r e g u l a r i t y .

(S.1) The f u n c t i o n h i ( x , 5 ) and g ' (x ,S) a r e con t inuous and bounded on N . (S .2 ) g' (x, 5 ) maps A on to C. Fur thermore, t h e snace A may be w r i t t e n as t h e

d i r e c t sum of D- and ano the r s u b s ~ a c e E- i . e . , A = D- 4l E- where t h e pro- X x ' X X

j e c t i o n o p e r a t o r ( s e e 0 .3 ) P- is bounded, i . e . , t h e r e e x i s t s a p o s i t i v e X

cons t an t M such t h a t I I P- I I < M. x A - .'. .I. .'.

(S.2) 'The s;,ace A" may be w r i t t e n as t h e d i r e c t sum o f two s u b s ~ a c e s R: and S" X X

.'. .5 ... .'. .I. .L .'. ... -.. ,. - .. .. i.e., A" = R- W S-, where R- = { g ' (a, F1) = 0, 5" E c " } and \,,here g 1 is X X X 1

(48 ) ... *.. t h e con juga te o p e r a t o r o f g ' and where S- is a subanace of A". Fur ther -

X ... ... ,. more, t h e p r o j e c t i o n o p e r a t o r Q- w i t h Q- A = R- i s bounded.

X X X

(5 .3) Given 6 > 0 . For any x E y ( x , 6 ) ,q M, t h e snace A can be w r i t t e n a s .#.

t h e d i r e c t sum of A = D x E J x and t h e p r o j e c t i o n o p e r a t o r s P x wi th P x A" = Rx

a r e bounded and s a t i s f y t h e L i n s c h l i t z c o n d i t i o n , i. e . , I / Px ( I < - M and

X1' X2 6 y (x, 6 ) n N , where

P,: and M a r c p o s i t i v e c o n s t a n t s . 1 .'. .'. .I.

(5 .3) ' For any x E A(;, 6 ) n N we have A" = R ; @ S ~ where t h e p r o j e c t i o n I.. -7-

o p e r a t o r s Q wi th Cx A = R a r e bounded and L i n s c h i t z i a n . x ' x ' , -9. -1. -1. -9.

( S . 4 ) 1 1 g '' (X, 5:) I / ) M 1 1 51 1 1 , f o r any F ; E c* ' , where M > 0.

( S . 4 ) ' 1 1 g t (X, n ) ( 1 M1 I I 4 I l n , f o r any q E E- x ' where M 1 0 .

t - (S.4)" For any y E C , t h e equa t i on g ( x , b ) = y h a s a s o l u t i o n b ( y ) w i th

1 1 b (y) I I A M2 / I y I I C where M i s a p o s i t i v e c o n s t a n t . 2

( 5 . 5 ) There e x i s t s E. r A wi th / 1 5 / I A K such t h a t : ( i ) g r ( x , 5 ) = e3. 1

( i i ) [L, ( h (x) + h (x, ?))I 2 P , where L is a non-negative l i n e a r -

f u n c t i o n a l wi th I I L I I = 1 and where P and K a r e p o s i t i v e c o n s t a n t s .

We now p r e s e n t s u f f i c i e n c y c o n d i t i o n s f o r r e g u l a r i t y . These

1el11111cte fo l l.ow from Gapushkin's theorems L-141 on uniform r e g u l a r i t y o f N.

Lemma 1. For e q u a l i t y cons t ra in t s - , !S,3) => ( R . 1 ) . -.

The lemma fo l l ows from theorem 2 o f Gapushkin [ 2 8 ] .

1

Lemma 2. If A i s r e f l e x i v e t hen (S .3 ) => (R.1) f o r e q u a l i t y c o n s t r a i n t s .

Th i s f o l l ows from lemma 1 ( a s c o r o l l a r y t o theorem 2 o f Gapushkin [14]. I I t

Lemma 3b. S . l , S.2 and e i t h e r S . 4 , S . 4 o r S . 4 2 (R.1) f o r e q u a l i t y

c o n s t r a i n t s .

T h i s f o l l ows from theorem 3 of Gapushkin C141 and i t s c o r o l l a r i e s .

Lemma 4 : In t h e p resence o f e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s - any o f t h e - fo l lowing c o n d i t i o n s is s u f f i c i e n t f o r ( R . 3 ) :

( i ) The e q u a l i t y c o n s t r a i n t s a t i s f i e s (R. l ) , and (s. 1 ) and (s . 5) a r e

s a t i s f i e d . ( i i ) ( S . l ) , (S .3) and (S .5 )

( i i i ) A i s r e f l e x i v e - ( S . 1 ) , (S .3 ) and (S .5 )

1

( i v ) S . , 2 4 and (S .5 )

1

(v ) ( S . l ) , (S .21 , (S .4 ) and ( S . 5 ) 1 r

( v i ) S l y 2 4 and ( S . 5 ) .

The lemma fo l l ows from theorem 4 o f Gapushkin C141 and from h i s

remark a t t h e end o f s e c t i o n 4 of [141.

4 .1 .3 . F i r s t Order nece s sa ry c o n d i t i o n s f o r t h e r e g u l a r c a s e A

Theorem 13 . I f , i n a d d i t i o n t o t h e assumptions of theorem 1 2 , x is

(R.3) - r e g u l a r t hen t h e conc lu s ions of theorem 12 f o l l o w wi th X A > 0 "

and k1 and e2 a r e unique ( t a k i n g h = 1 ) . 0

For t h e ca se o f e q u a l i t y c o n s t r a i n t s , t h e theorem w a s proved d i r e c t l y (49)

by Go lds t i ne C151, ? ~ t i l i z l a g ( S . 2 ) without assuming t h a t A i s t h e d i r e c t

sum o f S.2. ( 5 0 ) The theorem was proved d i r e c t l y by ~ u r w i c z ( ~ ~ ) [ 2 0 ] , and

it fo l l ows from theorem 5 of Gapushkin, who r e s t r i c t s A and C t o be

r e f l e x i v e .

4.2. F i r s t Order Suf f i c i en t Conditions.

Theorem 14) If 1) The funct ions f , g and h a r e d i f f e r e n t i a b l e , 2) The A

conclusions of theorem 12 a r e s a t i s f i e d , with A n > 0 , a t a poin t x t h a t

s a t i s f i e s t h e c o n s t r a i n t s of problem 3 , and i f e i t h e r : 3 . n ) The func t iona l

F of theorem 12 i s concave, 3.b) The e q u a l i t y c o n s t r a i n t g is l i n e a r and A

f and h a r e concave. Then x is a g loba l so lu t ion of problem 3.

For an o u t l i n e of t h e proof of t h i s theorem see t h e proof of theorem'

v.3.3. of Hurwicz [20] where he u t i l i z e s t h e f a c t t h a t t h e d i f f e rence be-

tween the values of a concave func'cional, say J ( X I , a t two d i f f e r e n t po in t s

is l e s s o r equal t o t h e d i f f e r e n t i a l , i . e . , J ( x l ' ) - J ( x f ) c J 1 ( x " - x')).

Guignard [16], using cl c o n s t r a i n t q u a l i f i c a t i o n , proves theorem 14 with

pseudo-concavity of f and h r ep lac ing assumption 3 of t h e theorem ( i n

t h e absence of e q u a l i t y c o n s t r a i n t s ) .

4.3 Second O ~ d e r Necessary Conditions. Conjecture 1. If 1) t h e ,.

funct ions f , g , and h have second order d i f f e r e n t i a l s , 2) x is a so lu t ion

t o problem 3 and 3) x is regu la r . Then t h e upper bound of F" (x, 5 ) is

non-posi t ive, f o r 5 with 1 1 5 [ I A = 1 t h a t s a t i s f y a) g' (x, 5) = 0, b ) If ,. ,.

h i s e f f e c t i v e i . e . , i f h ( x ) = 0 then h ' (x, 5 ) = 0 , where F is as defined

i n theorem 12.

FOP e q u a l i t y c o n s t r a i n t s t h e conjec ture was proved by ~ o l d s t e i n ( ~ ~ ) [ l ~ ] .

4.4; Second Order S u f f i c i e n t condi t ions .

24.

Conjecture 2. If 1 ) f , g and h have second d i f f e r e n t i a l , 2) The con- ,.

c l u s i o n s of theorem 12 a r e s a t i s f i e d a t a p o i n t x t h a t s a t i s f i e s c o n s t r a i n t s A

1 ) and 2 ) of problem 3 , 3 ) The po in t x i s r e g u l a r and 4) The upper bound 6

of F" (x, 5) is negat ive f o r I: with I I 6 I I = 1 s a t i s f y i n g a) g 1 (x, 6 ) = 0 A A

and b ) i f h is e f f e c t i v e a t x then h ' (x, 5 ) = 0 , where F i s a s def ined

i n theorem 12. Then t h e r e e x i s t s a neighborhood N i n A such t h a t *

f (x) > f ( x ) f o r x E fin N. -

This con jec tu re was proved by ~ o l d s t e i n ( ~ ~ ) [ 5 1 f o r t h e case of

e q u a l i t y c o n s t r a i n t s .

25. FOOTNOTES

This gene ra l s ta tement of t h e problem i s due t o Hurwicz C201.

W e s h a l l r e s t r i c t our a t t e n t i o n t o maximization. Cha rac t e r i za t ions of s o l u t i o n s t o minimization problems fo l low t r i v i a l l y from maximization t h e o ~ e n s .

i I n t h a t c a s e , x i s s a i d t o Se Pare to s u p e r i o r t o y i f f (x ) 2

A

fL ( y ) f o r i = 1,. . . , r (r is t h e f i n i t e dimension o f B ) , and x is Pa re to opt imal (p - maximal) i f t h e r e does not e x i s t a p o i n t y E A s a t i s f y i n g g ( y ) 2 02, h (y ) = 0 which i s Pa re to supe r io r t o x

3

Suggested t o t h e a u t h o ~ by Hurwicz.

Theorem 2 , sect!-on 187 p a t 11.

Theorem 1.1.

Theorem 3.1.

Separa t ing t h e l i n e a r s e t s : +. B f i Si < 0 and fii 1 > 0. See Dubovskii t Milyut in 1101 f o r an i - extens ive d i scuss ion .

T1.- condi.i-jons -:.:lxt kcllci.r -.?e aLternat i i re f o r x s o f t S c c o n s t r a i n t q u a l i f i c a t i o n , see Ar~ow-I-furwicz, Uzawa C11 f o r o t h e r forms and f o r r e l a t i o n s among va r ious forms of t h e c o n s t r a i n t q u a l i f i c a t i o n .

In remarks fol lowing theorem 1.1.

Theorem 3 . 2 .

B D Following Karush C231, by w r i t i n g h ( x ) > 0 as h (x) - ( z8 l2 = 0 - and so lv ing t h e problem i n t h e space of n + m - v e c t o r s ( x , g).

Theorem 3.1.

Theorem '0.1, Chapter 1.

Theorem 3, s e c t i o n 212, Pa r t 11.

Theorem 1 . 2 .

Theorem 5.1.

Corol la ry t o theorem 3.2.

See my no te s [Ill, where P e n n i s i ' s theorem is proved by d i r e c t l y apply ing t h e second o r d e r necessary cond i t i ons f o r e q u a l i t y c o n s t r a i n t s us ing Karush's dev ice ( f o o t n o t e 1 2 ) .

Theorem 1 .3 .

Theorem 4 , s e c t i o n 213, P a r t 11.

Theorem 6.1.

Theorem 3.3.

See [121.

Theorem 10 .3 , chap te r 1.

See E l ' s g o l ' c [121, Halanay C171 and Ewing [13].

See Hestenes [18].

A s s t a t e d i n t h e s e two papers .

See Berkovi tz C51 and Hestenes El81 f o r t h e necessary t r ans fo rma t ions .

F i r s t Necessary Condit ion I , Page (412) .

Second Necessary Condit ion 11, Page ( 4 1 4 ) .

See s e c t i o n 3.2 i n t h i s paper .

Coro l l a ry 3.4.

These .theorems may be found i n Bliss's paper [ 6 ] .

See Berkovi tz [5], s e c t i o n V I I I , theorem 3 , f o r a l t e r n a t i v e s u f f i c i e n t cond i t i on f o r normal i ty .

Sec t ion 9 , Page 693.

Sec t ions (24) - ( 2 6 ) .

Theorem 80, P. 228, t h e s ta tement and proof he re a r e more complete than they a r e i n [61.

Coro l l a ry 3 :4 P . 9.

Theorem 10 .2 , s e c t i o n 10.

Theorem 2 -1.

See Averbukh and Smolyanov [2] and [3].

See Vainberg [281 and L i u s t e r n i k and Sobolev [ 2 5 ] f o r an e x p o s i t i o n of c a l c u l u s i n l i n e a r sDaces.

Duboskii & Milyu t i n [lo] u t i l i z e t h e f a c t t h a t t h e s e t o f "va r i a t i ons1 ' t h a t g i v e t h e maximand va lue g r e a t e r t h a n t h e m a cou ld n o t i n t e r s e c t w i th t h e s e t s o f " v a r i a t i o n s " t h a t s a t i s f y t h e c o n s t r a i n t s . By v a r i a t i o n s t h e y mean d i f f e r e n t i a l s a t x. S ince t h e s e s e t s are de- f i n e d by l i n e a r i n e q u a l i t i e s and equa t i ons t h e y a r e convex. Using a s e p a r a t i o n theorem t h e y d e r i v e what t h e y ca l l t h e E u l e r equa t ion . Wri t ing t h e Eu l e r equa t i on i n t e rms of d i f f e r e n t i a l s of t h e maximand and c o n s t r a i n t s w e o b t a i n conc lus ion 3 of theorem 12.

For f i n i t e d imens iona l space s , t h i s is e q u i v a l e n t t o t h e r ank cond i t i on .

S e c t i o n 2 , page 592, i n t h e s ense t h a t t h e c o n d i t i o n ho ld s f o r a l l p o i n t s o f t h e c o n s t r a i n t s e t .

Th is i s Gapushkin's [14] n o t a t i o n .

See Kantorovich [211 Chapter XII, nage 476.

Without u s i n g theorem 1 2 , theorem 2.1.

See L i u s t e r n i k and.7Sobolev C251, (page-204) fcr a proof t h a t t h i s p a r t of (S.2) is d i spensab l e and f o r a n e l e g a n t proof o f theorem 1 3 f o r e q u a l i t y c o n s t r a i n t s .

Theorem V.3.3.2 (page 971, s e e remark 1 i n 4 . 1 . 1 o f t h i s paper .

Theorem 2 . 3 (page 147 ) .

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