On the Anti-Monotonicity of Differential Mappings Connected with General Equilibrium Problem

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

ON THE ANTI-HONOTONICITY OF DIFFERENTLAL MAPPINGS CONNEClXD WITH GENERAL

EQUJLIBRIUM PROBLM

S.P. Urias'ev

Working P a p e r s a r e interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions expressed h e r e i n do not necessarily r e p r e s e n t those of t h e Institute o r of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

FOREWORD

This p a p e r i s concerned with t h e anti-monotonicity of differential mappings connected with g e n e r a l equilibrium problems. These r e s u l t s c a n be used f o r the in- vestigation of different game t h e o r y problems, f o r example Nash equilibria f o r noncooperative n - p e r s o n games. Such an approach gives possibility t o c o n s t r u c t r e c u r r e n t algorithms f o r finding t h e equilibria point.

This r e s e a r c h w a s conducted within the framework of t h e Adaptation and Op- timization P r o j e c t in t h e System and Decision Sciences Program.

Alexander B. K u n h a n s k i Chairman System and Decision Sciences Program

AUTHOR

Senior research scientist Stanislav Urias'ev from the Institute of Cybernetics Academy of Sciences of the Ukr. SSR (252207 Kiev, USSR) worked on stochastic optimization and game theory and wrote this paper during his t w o week visit a t IIA- SA.

CONTENTS

1 Introduction

2 Weakly Convex Functions

3 On the Anti-Monotonicity of the Dierential Maps for the Weakly Convex Functions 4 The Monotonicity of Dierential Map f o r

Quasi Convex-Concave Functions References

ABSTRACT

L e t X be a s u b s e t o f a Hilbert s p a c e H a n d ck:X X X

- + R ,

(k(z, z ) = O f o r a l l z E X . Let G(z)

= ay

ck(x,

v)ly ==

denote generalized d i e r e n t i a l with respect to t h e second argument a t t h e point ( z , z ) . W e s h a l l b e concerned with t h e p r o p e r t i e s of t h e function ck sucient to e n s u r e t h e anti-monotonicity of t h e map G(x). I t will b e shown t h a t f o r t h e anti-monotonicity of t h e map G(x) i t i s sucient to assume convexity-concavity of t h e function ck. In t h e case of t h e weakly convex-concave function ck t h e map G(x) i s anti-monotone under some conditions o n t h e remainder terms. In t h e case of t h e quasi convex-concave function ck, t h e condition similar to t h e anti-monotonicity condition hold.

Some p r o p e r t i e s of t h e weakly convex functions used in a r t i c l e will b e proved.

-

vii

-

ON THE ANTI-MONOTONICITY OF DFFEFWMTIAL MF'PINGS CONNECTED WITH GENERAL

EQUILIBRIUM PROBLEM

SP. _Urias'ev

The mathematical problems discussed in this a r t i c l e were stimulated by t h e in- vestigations of simulation model f o r international oil t r a d e (SMIOT) developed at t h e International Institute f o r Applied Systems Analysis [ I ] .

Briey t h e main idea of t h i s model is t h e following. T h e r e i s a market of a single homogeneous p r o d u c t , which consists of some sellers ( e x p o r t e r s ) and a single b u y e r (importer). Let i

=

1,

. . . ,

n be t h e e x p o r t e r s , f i (2) b e t h e marginal c o s t of which any e x p o r t e r i p r o d u c e s t h e amount z of t h e product f o r marketing and

r ( z ) b e t h e p r i c e at which t h e i m p o r t e r would agree t o buy t h e amount z of t h e product. If z ; denotes t h e amount of t h e p r o d u c t sold by e x p o r t e r i , then t h e r e v e n u e qi ( z ) of t h e e x p o r t e r i , can be e x p r e s s e d as follows:

Let s b e a number of a time point, zf b e t h e amount of t h e product sold by ex- p o r t e r i at time s . The dynamics of t h e model i s given by t h e relation

xt

^+l

⁼

^{max 0,}

zt -

^p,

t

^{a q i}

^a

^{zi ( z S )}

I ^,

^S

⁼

^{0 , 1 ,}

^...

where ps , s

=

0 , 1 , .

. .

are t h e positive s c a l a r values.

In more g e n e r a l form

where nX(-) denotes t h e o p e r a t i o n of projection on feasible set

is a d i e r e n t i a l map of p r e f e r e n c e .

The aim of t h e study is to formulate assumptions on t h e functions r ( 2 ) . f i ( z ) , i

=

1 ,

. . .

, n s u c h t h a t t h e p r o c e s s (1) h a s a s t a b l e cycle, converges to a Nash equilibrium point o r converges to some point. The r e s u l t s of t h i s a r t i c l e allow us to formulate conditions o n t h e functions r ( z ) , f i ( z ) , i

=

1,

. . .

^{, n} insuring the con- v e r g e n c e of p r o c e s s (1) to a Nash equilibrium point. These questions were dis- cussed in p a p e r s [3]

- 151

a n d i n more g e n e r a l situations in 161, P I , etc.

We study some conditions on t h e payo functions sucient f o r t h e anti- monotonicity of mapping g ( z ) o r o t h e r s d i e r e n t i a l mappings. Hence we c a n formu- l a t e t h e conditions under which many i t e r a t i v e algorithms f o r s e a r c h equilibrium points are applicable (see, f o r example, [8]

-

^1141).

STATEMENT OF THE

PROBLEM

Let X be a s u b s e t of a Hilbert s p a c e H and

<., .>

denote t h e i n n e r product. W e s a y t h e multivalued mapping G i s anti-monotone o n X i f < g ( z ) - g ( y ) , y - z > Z O f o r a l l z , y E X , g ( z ) E G ( z ) , g ( y ) € G ( y ) .

Let Y:X XX

-

^{R b e a} quasi o r weakly

1151

convex-concave function, Y ( 2 , z )

=

⁰ f o r a l l z E X. Let G ( 2 )

=

8Y Y ( z . y ) &

.,

denote generalized d i e r e n t i a l of t h e function Y with r e s p e c t t o t h e second argument at the point

( 2 8 2 ) .

W e shall be concerned with t h e p r o p e r t i e s of t h e function Y sucient t o e n s u r e t h e anti-monotonicity of t h e map G ( z ) . I t will be shown t h a t f o r t h e anti- monotonicity of t h e map G ( z ) , i t i s sucient t o assume convexity-concavity of t h e function Y. In the c a s e of t h e weakly convex-concave function

*

the map G ( z ) i s anti-monotone under some conditions on the remainder terms. Some p r o p e r t i e s of t h e weakly convex functions used in a r t i c l e will b e proved. Investigation of such questions c a n b e motivated by t h e problem of nding t h e points

z*

E X dened by variational inequality

where g ( z * ) € ~ ( z * ) .

If t h e multivalued map G ( z ) i s anti-monotone as s t r i c t l y anti-monotone w e c a n use r e s u l t s s t a t e d in [ 8 ]

-

[14], etc. f o r solving t h e problem (2).

In t h e case of t h e quasi convex-concave function 9 ( 2 , y ) a n d 9 ( z , z )

=

0 f o r z E X t h e weaker condition t h e n t h e anti-monotonicity holds:

<gr(z),z* - z > r O f o r a l l z ~ X , g ( z ) E G ( ~ ) ,

where G ( z ) i s a d i e r e n t i a l of quasi concave function 9 ( z , z ) with r e s p e c t to second argument.

Let us c o n s i d e r some problems which c a n b e r e d u c e d to t h e variational ine- quality (2).

EXAMPLE 1 N a s h e q u i l i b r i a f o r n o n c o o p e r a t i v e n - p e r s o n games. Let X b e a convex closed bounded s u b s e t of t h e production H I X XHn of a Hilbert s p a c e s H i , i

=

1,

. . . ,

n . A point zi E Hi i s a s t r a t e g y of i -th p l a y e r i

=

1,

. . . ,

n and cpi(z)

=

cpi ( z l ,

. . .

^,

z,)

i s his payo function. The element (z1,

. . .

, zi

* * *

yi

,

zi ⁺

l , . . ,

zi ) i s denoted by (yi / z ). The point z

=

( z l

, . . .

, zn ) E X i s re- f e r r e d to ^as t h e Nash equilibrium of n - p e r s o n game if f o r i

=

1,

. . .

^, n

Let u s introduce t h e function 9 ( z

,

y ) :

I t i s not d i c u l t to see t h a t '3) ( z , z )

=

0, z E X . W e suppose t h a t t h e functions

vi

^{( z ),}

i

=

1,

. . . ,

n are continuous on X. The point z* E X i s dened as t h e normalized equilibrium point if

LEMMA 1 (See f o r example [16]). The normalized equilibrium point i s t h e equilibrium point, t h e r e v e r s e is t r u e if X

=

X l x

- . X X , , &

^C Hi.

The condition ( 2 ) i s a n e c e s s a r y optimality condition f o r t h e problem (3), f o r t h i s r e a s o n t h e problem of nding Nash equilibrium is reduced to t h e problem ( Z ) ,

EXAMPLE 2 An equilibrium point is dened in [17] as follows. Let X be a con- vex closed subset of a n Euclidean space R n , the functions

be concave with r e s p e c t to y E X f o r each z E X and continuous with r e s p e c t t o

z ,

y on X X X . Let us denote

The problem consists in nding the point z * such t h a t

* *

z * E x ( z * ) ; max ( Q , ( z * , y ) : y c ~ ( z * ) j

=

Q,(z . I

.

If the condition X ( z ) = X is t r u e f o r all z f X I then t h e problem i s reduced t o problem (3), where Q ( z , y )

=

Q o ( z , y )

-

Q , ( z , z ) . It is easy to s e e 9 ( z , z ) = O for all z ^f X. The necessary condition ( 2 ) can be used in this case too.

EXAMPLE 3 Let us consider one more problem which can be reduced to (3).

We assume t h e X is a subset of a Hilbert space

H

(or m o r e general space), and 9 : X X X ^-4 R i s a function satisfying sup Q ( y , y ) S 0. The problem consists in

Y EX nding t h e point z

*

E X such t h a t

s u p Q ( z * . y )

s o .

Y E X

We suppose 9 ( z

,

y )

=

4 ( z , y )

-

^Q ( z , z ) . If the point z

*

E

x

^satises (3), then i t satises (4) too. For this reason the necessary condition (2) c a n be used in this case as w e l l .

Theorems concerning t h e existence of problem (4) solutions were formulated in [18]. In t h e same book t h e r e a r e r e f e r e n c e s on the original p a p e r s related to this problem.

2.

WEAKLY

CONVM FUNCTIONS

In this section basic p r o p e r t i e s of weakly convex functions [15] a r e investi- gated. The family of weakly convex functions includes smooth and convex functions and i s closed with r e s p e c t to t h e summation and pointwise maximum. We give new denition of this family useful f o r applications. It will be shown t h a t this denition and t h e denition given by E. Nurminski [I51 a r e equivalent.

DEFINITION 1 Let X be a convez subset of

a

Hilbert space H. A c o n t i n u o u s f u n c t i o n f :X ⁴

R

is called w e a k l y convex o n X i f for all

z

E X, y E X,

0 S a S 1 t h e following i n e q u a l i t y holds.

where t h e r e m a i n d e r

r

: X x X ⁴

R

s a t i s e s

for all 2 EX.

The set Bf

( z )

i s c a l l e d a d i e r e n t i a l of a weakly c o n v e x function f

( z )

at a p o i n t z ~ X o n X i f f o r a l l g ( z ) E B f ( z )

f o r all

z ,

y E X

W e s a y t h a t a function f

( z )

i s weakly c o n c a v e o n X, if

-

^f

( z )

i s weakly con- v e x on X. A d i e r e n t i a l of t h e weakly c o n c a v e function f

( z )

i s d e n e d

as

a d i e r e n t i a l of t h e weakly c o n v e x function

-

^f

^{( z )}

t a k e n with s i g n minus.

THEOREM 1 Let X be

an

open convez subset of

a

filbert space H a n d the f i n c t i o n f ( z )

i s

w e a k l y convez o n X. Then thR set B f ( z ) ,

z

E X

i s

non-empty,

convez, closed bounded a n d

where f l ( z , p ) is

a

d e r i v a t i v e of t h e

fimtwn

f ( z )

at

a p o i n t

z

along

a

direc- t i o n p .

PROOF W e start with t h e following lemma.

LEMMA 2 For a n y

z

E X, p E

H

a d e r i v a t i v e

z +

A p ) - f ( z )

/ ' ( z .

p )

=

1imJ"

A 4 0 A

e x i s t s a n d i s nite.

PROOF F i r s t of all we p r o v e a n e x i s t e n c e of t h e nite or innite d e r i v a t i v e f l ( z , p ) f o r a n y

z

^E X, p E H . F o r 0 S A2

<

A, inequality ( 5 ) implies

consequently

The last inequality implies t h e existence of t h e derivative fl(z, p ) because (Z +.'P*

z,

-P 0 _{f o r A} r 0

.

The derivative fl(z, p ) c a n not t a k e t h e value

+ -.

Let us p r o v e t h a t t h e derfva- Live f'(z, p ) bounded below. For E

>

0 , A

>

0 t h e inequality (5) implies

After t h e equivalent transformation

If E and A are suciently small. then from (6) obtain - + P s A + & ⁺

> 6 =

const

.

consequently

LEMMA

3 The d e r i v a t i v e fl(z, p ) i s a c o n v e z positive-homogeneous * n o

twn

c o n t i n u o u s

at

t h e p o i n t 0 w i t h respect t o p.

PROOF Let us p r o v e a positive-homogeneity of t h e function fl(z, p). Accord- ing to t h e denition of a d e r i v a t i v e f o r a

>

⁰

fl(z, u p )

=

lim f(z +Asp) -f(z)

A 4 0 h

For A,, A, r 0, A ,

+

X2

=

1 t h e inequality (5) implies

Passing to limit A r 0

Prove t h a t t h e function f

' ( z

^{, p )} i s continuous with r e s p e c t t o p at t h e point 0 f o r any

z

E X . To check t h i s i t is sucient t o prove boundedness of the function f ' ( 2 , p ) in

a

neighborhood of t h e point 0 (see f o r example [19]).

Passing t o limit A2 r 0 in ( 9 ) obtain

The condition (6) implies t h a t t h e r e exists a neighborhood U ( 0 ) of t h e point 0

r ( z +

Alp,

z )

such t h a t

1 I <

6

=

const f o r a l l p E U(0). Since f

( z )

is continuous, A1

then w e can suppose that t h e function f

( z +

Alp ) is bounded on U ( 0 ) with r e s p e c t

to

p

.

Therefore t h e boundedness above of f ' ( 2 , p ) with r e s p e c t to p in the neigh- borhood U ( 0 ) follows from t h e last inequality.

LEMMA

4 A dierential Bpf ' ( 2 , 0 ) of the convez &nction f ' ( 2 , p ) w t t h respect to argument p at t h e point 0 coincides w i t h 8 f ( 2 ) .

PROOF Dierential Bpf ' ( 2 , 0 ) i s dened as followed

The inequality (5) implies

z

+ a ( y - 2 ) )

- j ( z )

f ( 2 4 )

- f ( z )

²

1( +

⁽¹

- a ) r ( z ,

y ) ,

a >

⁰

.

a Passing to limit

a

& 0

Let z belong t o $ f f ( z , 0 ) , then from t h e last inequality w e have

consequently Bpf ' ( 2 , 0 )

c

d f

( z

^).

Let u s prove t h e r e v e r s e conclusion. If

z

E

8 f ( z ) ,

then

Hence

J ( z

+

X P ) - f

( 2 )

r ( z + X p , z ) A

^Z

< z , p > +

X

and f

' ( z , p )

2 <z,

p >,

consequently

8 1 ( z ) c 8 p f ' ( z , o

). According t o Minkovski duality, since f l ( z ,

p )

i s a convex positive homogeneous function with r e s p e c t to

p

and continuous a t t h e point

0

then t h e set

B p f l ( z , 0 )

is non-empty bounded convex closed and

Consequently t h e

set

a f

( z )

i s non-empty bounded convex close and relation (8) is t r u e . The theorem h a s been proved.

W e shall give equivalent denition of t h e weakly convex functions.

DEFINITION 2

Let

X

be a convez subset of a Hilbat space

H .

We s a y thut a f i n c t i o n c o n t i n u o u s o n

X

i s weakly convez o n

X I

i f f o r a n y z

^E X

the set

G

( z

)

consisting of the vectors

g ,

s u c h thut

is empty, and remainder t a m <(z , y

)

in each compact subset K c

X is

uniform- l y small relatively to llz - ^y 11, i.e. for a n y

&

> 0 there e z i s t s 6 > 0 t h a t

f i r l l z - y I I < & z , u fK.

E. Nurminski 1151 h a s introduced this denition in case X

=

H

= R n

where

R n

i s an Euclidean n-dimenional space.

THEOREM 2

Let

X be

a convez open subset of a Hilbert space

H ,

t h e n t h e d e n i t w n s

1

a n d

2

are equivalent in the fillowing sense:

a ) ^{i f}

a N n c t i o n i s weakly convez in the sense of t h e d e n i t w n

2,

t h e n i t i s weakly convez in the sense of t h e d e n i t i o n

2

and <(z

,

y

)

= r ( z

,

y

);

b) ^{i f}

a N n d i o n i s weakly convez in the sense of the d e n i t i o n

2,

then i t i s

weakly convez in t h e sense of the d e n i t i o n

1

and

PROOF

It i s not dicult to see t h a t (6) implies the uniform convergence on a compact s u b s e t K . Hence the statement a ) of t h e theorem follows from t h e denition of a weakly convex function and theorem 1.

Let u s prove the statement b). In the view of

( l o ) ,

^{w e have}

f o r a l , a 2 2 0 , a 1

+

a 2

=

1.

From last t w o inequalities w e obtain

A s < ( z , y ) / l l z - y l ( - * O f o r z + z , y - + z , then

n'r

( ( ( 2 , Z

+

a I + a E = l cri.%+"

a 2 ( v - z ) ) / a $ l y - z ( (

+

< ( y . y + a l ( z - y ) ) / a l l l y -211) - + O f o r z - 2 ,

y ^{--+ 2 .}

The theorem h a s been proved.

W e shall note some c a s e s , where the remainder t e r m s in t h e denitions 1, 2 coincide.

COROLLARY 1 Let X be a convex open subset o f a Hilbert space H . u a weakly convez function

p

( z ) s a t i s e s

f o r all z , y E X , then

Conversely the second inequality implies the rst o n e f o r a n y g E 8f ( z ).

This c o r o l l a r y i s w e l l known, if p

>

^{0, in this} case the function f ( z ) i s strong- ly convex.

Let u s consider t h e case when function f ( z ) is twice continuously dierentiable at e a c h z ^E

X,

where

X

i s an open s u b s e t of a Hilbert s p a c e

H,

i.e. f o r all z , y

E X

Vf ( z ) i s a g r a d i e n t at a point z ; A ( z )

:X

+

H

i s a l i n e a r o p e r a t o r generating the symmetric bilinear function

The function i s called twice continuously dierentiable on X

c

_H, if i t i s dierentiable at e a c h point z ^E

X

and

COROLLARY 2

V

a f u n c t i o n f ( z ) is twice c o n t i n u o u s l y d i e r e n t i a b l e o n an o p e n s u b s e t

X

of a Hilbert s p a c e H , i.e. t h e c o n d i t i o n s @I), &?) a r e t r u e , t h e n

w h e r e

C o n v e r s e l y , t h e i n e q u a l i t y (23) i m p l i e s @I).

PROOF

The c o n v e r s e statement follows from t h e a ) of t h e theorem 2. Let us p r o v e t h e d i r e c t statment using t h e b ) of t h e theorem 2. In t h i s case w e c a n denote

consequently

Thus, t h e inequality (13) i s t r u e .

3. ON THE ANTI-HONOTONICITY OF THE DIFFERENTIAL

MAPS

FOE

THE WEAKLY

CONVEX FUNCTIONS

Let

*

^:

^X

^X

^X

^-.,

^R

^{b e a} function dened on a p r o d u c t

X

X

X,

where

X

i s a convex open s u b s e t of a Hilbert s p a c e H. The function 9 ( z , y ) i s weakly convex on

X

with r e s p e c t t o t h e rst argument, i.e.

f o r a l l z , y , z

E X ;

a l + a 2 = l ; a l , a z 2 0 a n d r , ( z , 24)

- 0 if llz

-

^y((-.,O f o r a l l z

E X .

llz

- u I1

W e suppose t h a t t h e function *(z, y ) i s weakly concave with r e s p e c t t o t h e second argument on

X,

i.e.

~ ( z S u )

- 0 if 112

-

y ( 1 - 0 f o r all z

E X .

llz

- u II

We say t h e function *(z, y ) i s weakly convex-concave, if i t s a t i s e s (14), (15).

Let

Q(z, x ) i O f o r a l l z

E X .

(16)

Denote G ( z )

= By

*(z , y )

1 ^==,

i.e. G ( z ) i s a d i e r e n t i a l of the function *(z, z ) with r e s p e c t to t h e second argument at a point ( z , z ) .

We shall formulate t h e sucient conditions of the anti-monotonicity of the mul- tivalued map

G ( z ) ,

i.e. for all

g ( z )

E

G ( z ) ,

g ( y ) E G ( y )

< g ( z ) - g ( y ) ,

y

- z > r O

f o r a l l

z ,

y E X

.

THEOREM 3

Let

X be

an open convez subset of a Hilbert space H, a mnction

9 : X x X ^--, R be

weakly convez-concave, the remainder r, ( z ,

y ) be

continuous with respect to

z ,

the function

9

satis& condition

(323). T h e n

jbr all z , y

^E X ;

g ( z )

^E

G(z),

g ( y ) E

G ( y )

PROOF We c a n assume y

=

0. I t also can be assumed t h a t g ( 0 )

=

0

as

a n anti- monotonicity of the map

G ( z )

does not depend upon a linear t e r m of t h e function

9(z

y ) with r e s p e c t to t h e second argument. It is necessary to prove that

In t h e view of

(14)

w e g e t

a Q ( z , a z ) + (1 - a)9(0, a z )

2

9 ( a z , a z ) + a(1 - a ) r a , ( z ,

0 )

.

Taking into account the p r o p e r t i e s of t h e weakly concave functions f r o m the last inequality obtain

a [ 9 ( z , a z ) - 9 ( z , z ) ] r (1 - ^a)[9(0,

^{0 )}

-

^{9 ( 0 ,}

^{a z ) ] + a(1} - a ) r a , ( z ,

0 )

Since

&(a z , 0 ) / a

^-+ 0 if

a

4 0 , then t h e last inequality implies

9 ( z ,

0 )

-

^{9 ( z ,}

^{z )} =

lim

[ 9 ( z , a z ) - 9 ( z , z ) ] r

a40

Consequently, taking into account the weakly concavity of the function

9 ( z ,

y ) with r e s p e c t to t h e second argument, we obtain

Theorem has been proved.

REMARK W e c a n see f r o m t h e p r o o f o f t h e t h e o r e m t h a t c o n d i t i o n ( 1 4 ) c a n be r e p l a c e d b y

* ( z ,

y )

- * ( z , z ) S

< g ( z ) , y

- z >

+ @ , ( y ,

z )

f o r a l l z , y

E X

COROLLARY 1 Let a l l c o n d i t i o n s of t h e theorem 9 be fullled a n d

r g ( z ,

y )

- & ( u s

^{Z )} 2 O f o r a l l z . y

EX,

t h e n t h e m a p G ( z ) i s a n t i - m o n o t o n e o n

X.

COROLLARY 2 Let

X

be a n open subset of a filbert space H , a f u n c t i o n

@ ( z ,

-

y ) be c o n t i n u o u s a n d convex-concave o n

X

X

X,

t h e f?unction

@ ( z , z )

^be

concave o n

X, then

t h e m u l t i v a l u e d m a p G

( z ) =

B y

@(z

^{, y} )

l y =,

i s a n t i m o n o t i n e o n

X.

PROOF I t i s easy to get t h i s s t a t e m e n t i f a s s u m e

COROLLARY 3 Let all c o n d i t i o n s o f t h e theorem 9 b e f u l l l e d a n d

jbr a l l z , y

EX,

w h e r e

0 ' 1 ~

- Y ' " ) - 0 _if

2 - 2 , y - 2

llz -

^Y

112

un.igormly

with

respect t o z

EX, then

jbr a l l z , y

E X

a n d j b r a l l g ( z ) E G ( z ) , g ( y ) E G ( y ) .

PROOF L e t

z ,

^y

E X ; z #

y . In t h e v i e w o f ( 1 7 ) , ( 1 8 ) w e get

< g ( z ) - g ( y ) , v - z > = lim

n + -

( 2

Corollary has been proved.

Let u s suppose t h a t function

O ( z , y )

i s twice dierentiable on e a c h argument.

Denote by A

( z , y

)

=

9=

( z ,

y ) t h e second d e r i v a t i v e with r e s p e c t t o t h e rst argu- ment ( s e e ( l l ) ) , and i t h e same way

B ( z

,

y

)

=

9yy

( z

,

y

) with r e s p e c t t o t h e second argument and

g ( z ) =

Vy

9 ( z , y

)

ly =, .

THEOREM 4

Let

X

be a n open convez subset of a Hilbert space

H ,

function

9 :

X

X

X

^-+

R be twice dierentiable with respect t o each argument, and

J J A ( z , ~ ) - A ( z , z ) J I - + o ,

if

z

- 2 ,

y

4 2 uniformlyfor

z E X

;

J I B ( z , y ) - B ( z , z ) l ( - - 0 ,

^if

z 4 2 , y - + z

uniformly'for

z E X

; t h e o p e r a t o r

Q ( z , z ) = A ( z , z ) - ^{B ( z}

^,

z )

satisfy

f o r a l l z

E X , h ^E

H a n d

Y

do m t d e p e n c i o n z ,

h .

Then

< g ( z ) - g ( y ) , y - z > r - v l l z

1

- ^y

f o r all

z , y E X .

2 (20

PROOF The statement of t h e theorem follows from t h e c o r o l l a r y 2 of t h e theorem

2,

and t h e c o r o l l a r y 3 of t h e theorem 3. According t o the c o r o l l a r y

2

d t h e theorem2

In t h e view of

(19)

Thus, t h e conditions of t h e c o r o l l a r y 3 of t h e theorem 3 are satised.

Let u s consider t h e example which i l l u s t r a t e s the last theorem.

EXAMPLE

₄ Model f o r i n t e r n a t i o n a l o i l t r a d e

[Z].

T h e r e i s a market of a single homogeneous p r o d u c t , which consists of some sellers ( e x p o r t e r s ) and a sin- g l e b u y e r (importer). Let i

=

1,

. . .

^, n be t h e e x p o r t e r s , f i ( z ) be t h e marginal c o s t of which any e x p o r t e r i produces t h e amount z of t h e p r o d u c t f o r marketing and r ( z ) be t h e p r i c e at which t h e i m p o r t e r would a g r e e to buy t h e amount z of t h e product. If zi denotes t h e amount of t h e product sold by e x p o r t e r i , then t h e r e v e n u e p i ( z ) of t h e e x p o r t e r i , c a n be e x p r e s s e d as follows:

N o t e t h a t t o t h e s e n s e o f t h e p r o b l e m z t r 0 , f i ( z ) 2 0 , r ( z ) r O , i = 1 ,

. . . ,

n . W e assume also t h a t e x p o r t e r i , i

=

1,

. . .

, n i s a b l e to s e l l no more t h e n &

amount of t h e product. If w e suppase t h a t e a c h seller i s going to choose t h e amount zi in o r d e r to maximize his r e v e n u e in any market's situation c h a r a c t e r i z e d by a v e c t o r z

=

(zl,

. . .

, z,), t h e n t h e problem w i l l b e as follows: t o nd a n

* * *

equilibrium's situation z

=

( z l ,

. . . ,

zn ) s u c h t h a t

The admissible set X

=

[z E R n :O s zt s p i , i

=

1 ,

. . . ,

n

1

i s convex and compact, The function +(z

,

y ) (see example 1 ) denes as

W e assume t h a t t h e functions f z ) i

=

1

. . .

, n are continuously dierentiable and t h e function r ( z ) i s twice continuously dierentiable on some open s u b s e t [ z E R : z

=

C r = l z t , z E

U c

R~

1,

where

U

i s open s u b s e t such t h a t X

c U

c R n . Denote

It is not dicult t o nd

Let us investigate under what conditions the matrix Q ( z , z ) satises (19). Let e

=

(1,

. . .

^, 1 ) be the ndimensional vector, and

Assume that

r Z ( z ) 4 O r z z ( z ) 2 0 f o r ^{z E X}

.

We can write the following inequality

Hence, [ - 2 d n x : = l p f r , ( z )

+

2p(z)

1 ^.+

^v > O implies (19). The lastinequality is c o r r e c t if

f o r a l l z E X I 1 6 i 6 n . Consequently (21), (22) imply (20) and t h e map g ( z )

=

(V, ,pl,

. . . ,

V,, p, ) i s anti-monotone.

I t should be noted t h a t in t h e view of t h e (21), (22) the function *(z, y ) i s strongly concave with r e s p e c t t o y because t h e matrix B (z, y ) i s negatively dened and t h e Nash point equilibrium exists [20].

4 THE YONOTONICITY OF DIFFERENTIAL

MAP

_FOR

QUASI

CONVEX-CONCAVE F'UNCTIONS

Let X be a closed convex subset of a Hilbert s p a c e H, U be an open subset of H and X

c

U. A function

*

^{:X X X --,}

^R

i s quasiconvex with r e s p e c t t o t h e rst argu- ment, i.e.

for all al, a 2 ² 0; al

+

a 2

=

1 and f o r all z , y

,

z E X and quasiconcave with r e s p e c t t o t h e second argument, i.e.

f o r a l l al, ,a2 2 0 ; al + a 2 = l a n d f o r a l l z , z , y E X .

For the f u r t h e r development we assume t h a t t h e function ck(z, y ) satises re- gularity condition with r e s p e c t to the second argument, i.e. f o r any z € X and any y such t h a t y

#

a r g max, ,X 9 ( z , z )

int f u E U : +(z, u )

=

ck(z, y ) {

=

0

,

where int A denotes the i n t e r i o r of a set A and *(z, z )

=

0 for a l l z E X. The fam- ilies of weakly convex and quasiconvex functions intersect but are not embedded into e a c h other.

A cone Df(z)

=

f g € H : < g , y - z > 6 0 for all y €CL(z)l is called a dierential of a quasiconvex function f (z ) at a point z on a s e t U C

H ,

where M(z )

=

f y : f (y ) 6 f (z ), y E U

1.

A dierential of t h e quasiconcave function p(z ) i s

dened as a d i e r e n t i a l of t h e quasiconvex function

-

^{q ( z )} t a k e n with sign minus.

We s a y a function Q ( z , y ) i s quasi convex-concave if i t i s quasiconvex with r e s p e c t t o t h e rst argument a n d i s quasiconcave with r e s p e c t t o t h e second one.

Denote G ( z )

= ^Dy

Q

( z

, y )

ly =,

where

Dy

i s a dierential of a function Q ( z , y ) with r e s p e c t to y in t h e s e n s e d e s c r i b e d above.

Let a point

z *

^E X be a solution of t h e equation max

* ( z S ,

y )

= o .

Y E X

W e will p r o v e t h a t ( 2 5 ) implies

f o r all

z

E X and f o r all g

( z )

E G ( z ) . Consequently f o r nding a point

z *

r e s u l t s of t h e p a p e r [14] c a n be used, f o r example.

THEOREM 5 Let U be

an

o p e n s u b s e t o f a Hilbert space H , X be

a

closed con- v e z s u b s e t of H a n d X

c

U ,

a

f i n c t i o n

*

^{: X X X 4} R be c o n t i n u o u s o n X X X a n d q u a s i conuez-concave o n X X X,

at

l e a s t one of t h e i n e q u a l i t i e s @), @4) be s t r i c t @oral # O ,

al

# I ) . Y ( z , z ) = O f b r a l l z ~ X . f l a ~ o i n t z * ~ ~ s a t i r a s (23) t h e n t h e v a r i a t i o n a l i n e q u a l i t y (26) h o l d s for all

z

E X a n d for a l l

~ ( 2 ) f G ( z ) .

PROOF W e c a n assume

z* =

0 . Consequently i t i s n e c e s s a r y t o p r o v e t h a t

Assume at rst t h a t t h e inequality ( 2 3 ) i s s t r i c t , t h e n max [ Q ( z , az), Q ( 0 ,

a z ) ] >

Q ( a z , a m )

=

0 , 0

< a <

¹

The equality ( 2 5 ) implies

Q(0, a m ) r Q(0,O)

=

0 f o r 0 r

a s

1 ,

z

f X

.

W e g e t Q ( z , a m )

>

0 f o r 0

< a <

1 taking i n t o account (27). Passing to limit when

a

^.( 0 obtain Q ( z , 0 ) 2 0 and Q ( z , 0 )

-

^{Q ( z ,}

z )

² 0 . In a c c o r d a n c e to t h e denition of t h e d i e r e n t i a l of a quasiconcave function w e obtain statement of t h e theorem from t h e last inequality.

Let u s consider t h e case with t h e s t r i c t inequality (24). In t h i s case Q ( O , ~ U C ) < Q ( O , O ) = O f o r O < a < l , z EX,^ # O

.

The inequality (23) implies

m a x [ Q ( z , a z ) , Q ( O , c l u : ) ] 2 Q ( a ~ , a z ) = O f o r 0 4 a 4 l .

Taking into account t h e last two inequalities w e g e t Q ( z , az) 2 0 f o r 0

<

a

<

1.

F u r t h e r consideration coincides with t h e previous case. The theorem i s proved.

Let us c o n s i d e r t h e case when a function Q ( z , y ) i s dierentiable with r e s p e c t to t h e argument y

.

Denote

COROLLARY Let all conditions of the theorem 5 be satised a n d t h e fiLnction

* ( z ,

y ) i s dierentiable w i t h respect to the second argument. Then the e q u a l i t y (25) implies.

PROOF The theorem statement follows from t h e inclusion q

( z )

^€ G ( z ) , where G ( z ) i s a d i e r e n t i a l of quasiconcave function

Q(z

,

z

) with r e s p e c t to t h e second argument.

Let us consider an example illustrating t h i s corollary.

EXAMPLE Wald's production model [ Z ] . Let n p r o d u c t s are produced and

r

r e s o u r c e s used in a n economy,

a . . .

,

a ,

be t h e amounts of t h e s e r e s o u r c e s . The values a i j ( i

=

1,

. . .

, n ; j

=

1,

. . .

, r ) denote t h e input of j - t h r e s o u r c e neces- sary to produce t h e unit of i - t h product. The p r i c e s of t h e p r o d u c t s depend on t h e amounts of t h e produced products. Let f i

( z ) =

f i

( z l , . . .

^,

z n )

⁽ⁱ

=

1,

. . .

^,

n )

be t h e p r i c e of a unit of i - t h e p r o d u c t if t h e p r o d u c t s are produced in t h e amounts

z . . .

^,

zi ,... z,;

X be a feasible set

n

X

= I z (

^{i = l}

aijzi

4

a,,

j = I , .

. .

, r ; z i 2 0 , i = I , .

. . , n t

Under some conditions on t h e functions f i

( z ) ,

⁽ⁱ

=

1,

. . .

^,

n )

t h e existence of a

* *

non-negative production v e c t o r

z * = ( z

l ,

. . .

^{, 2,)} and non-negative r e s o u r c e p r i c e v e c t o r such t h a t

r

* * *

f:

a i j z ; ~ a j ( j = I . .

. .

^, r ) ;

x

a y U j Z Y i ( z l n . . ..z,,)

i = l j = l

w a s p r o v e d in [Z], [I?]. For a l l non-negative and continuous o n t h e s e t z Z 0 func- tions t h e existence of a non-negative production v e c t o r z

*

E X being the solution of t h e following l i n e a r programming problem

n

max

C ^li

^{( z}

*

^)zi

= C

^{f t (2 *)z;}

E € X i = l i = l

w a s proved in [Zl]. The e x i s t e n c e of v e c t o r s z * , U* satisfying (29) follows from t h e last equality and t h e duality theorem of t h e l i n e a r programming.

Denote

Since the function 9 ( Z , y ) i s l i n e a r with r e s p e c t t o y t h e n i t i s concave in y . If t h e point z

*

i s a solution of t h e problem (30), t h e n z

*

i s a l s o the solution of t h e problem (25). Hence in o r d e r to satisfy inequality (28) it i s s u c i e n t t h a t t h e func- tion 9 ( z , y ) be s t r i c t l y quasiconvex with r e s p e c t t o z on X, i.e.

max [*(z, y ),

W z _,

y ) l

>

9 ( a l z + 4 2 2 I Y

f o r all a l , a 2 0 ,

#

1 , Q l

+

a2

=

1 and f o r a l l z , z , y E X , z

#

2 .

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N-

P e r s o n Games. Econometrics, 1965, 33, No. 3.

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-

Verlag Berlin, 1974.

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