Appeox. Theory & its AppL, 9:3, Sept. 1993 �9 9 �9

A KIND OF APPROXIMATION OPERATORS AND ITS APPLICATION

Zhang Yufei

(Naniing University, China)

Received May 22, 1991 Revised July 16, 1992

Abstract

In this paper we introduce a kind o f kernels called ~-kernels. Based on them linear opera-

tor.~ from the continuous mapping space to R " are constructed. We give an important appli-

cation in Optimization after discussing the properties o f these operators.

1. Introduction

Let F : �9 c R " --+ R be continuous, here �9 is a convex, compact subset of IR" with

nonempty interior. We denote the following index set by I(F,~) :

I(F,tb) --- {i : x l eCP, F(x l) = maxlF(y)l}.

If I (F/b) is finite we define a linear operatoe A (F,q~,v) : x --, R " by

= Y. L f ( x , ) , f x, (1) /It.#}

where R P~ v -- (v I ,..-,vp) r t> 0, ~ v I --- I, p --- [I(F,O)[ is the number o f t h e elements in

I(F,~), and X the continuous mapping space from R " to itself.

The main purpose of the paper is to construct a sequence of linear operators to ap-

proximate the operator in (I) under some restricti6ns, and to analysis their approximation

property. This is what we are going to do in Section 2 and Section 3. In Section 4, as an ap-

plication of these new operators, ah algorithm is presented to solve the following problem:

Finding x ' e r so that F ( x ' ) > ~ F ( y ) for all y in r (2)

The problem in (2) is a basic one in Optimization. So we provide in theory a new

method in th.is area. The auther also believes that the method is a prospective one in

Optimization, especially in Nonsmooth Optimization.

Here are some notations used in this paper:

�9 10 �9 Zhang YufeL" A kindofApproximation Operators

I ( F , $ ) : -- { ie l (F ,q~) :x ,e~} for any ~ = ~ > .

c : - - : = ! } .

C (I) : = {ve Cp : v~ ffi O,i~l} for any index set

I1" I1: t he maximum norm.

(x ) i : the j th coordinate of the vector x

U(x) : a neighborhood of x e Rm restricted on

B ( x , ~ ) : -- { y ~ R " : I l Y - xll ~< ~}.

N(x,8) : = { y ~ R " : I l y - x ~ 2 < t } .

d iam~ : = m a x U x . - YU.

2. Lemmas

.

Definition 1. F is said to have L i proper ty on �9 if for ir I(F,~) there exist a neigh-

borhood U(x~) o f x I, positive constants aj , / /a and a~ so that

P i l l x i - x l l ~ <<.F(x,)-F(x)<<.cttilx~-xll ~ for xr (3)

Let S : q~ ~ S(~) = �9 be an affine transform of the form S(x) = (1 - r)x o + rx, here

x 0 e R " and re(0,1]. Whether can the L i property be kept when F is replaced by F * S?

In general, (3) does not hold for F * S while it doe~ for F. But if we put

G(x) = F . S ( x ) - h , I

h = t /maxF(y) + (1 - ~/)minF(a), ~/e[0,1), (4) r ~ O siS(O]

we have the following result.

/ 1 Lcmma 1. F has L 1 property on #P, There exists ~1 e[0,~) independent o f the mapping

S so that ~I in (4) satisfies tl <<. ~l'. Then for ieI(F,S(~)) ~ ~b there exist positive constants �9 t

a ~ and [ 3 independent o f the mapping S so that

i i I i

/~ IIx, - x II ~ ~< ! - IG(x)l / G ( x , ) < =, IIx i - x II ~ t

for x e S - I ( U ( x ~))Aep, x I = S - I ( x ~).

Proof.

t �9 I i i 1 - I G ( x ) l / G ( x , ) ~ 1 - G(x) / G ( x , ) = (F(a(x,)) - F(S(x))) / G(x )

r c c i r

, IIx, - x 110,, G(x,)

(5)

Approx. Theoo, & it.~ Appi., 9:3, Sept. 1993 �9 11 �9

on the other hand,

- I G ( x ) l / G ( x ' , ) 1

e l

I ~tr , IIx] - xll ~ , >t [ G(x,)

] + G(x) / G(x t) i> ! - ~ / / ( I - q) i> 1 - q//(I - r/'),

Since

t

r*' / G(x t)

if G(x) >>. O,

if G(x) < O.

r (l - r/')/~ r ~ ( d i a m . ) % -lI' ~< G(xl) <~ atr ( d i a m . ) ~ ,

is uniformly bounded for re(0,1]. Then we choose positive constants

s i i t s

a t >t~r / G ( x t ) , re(0,1],

p '__min{ in fp , r o, / G ( x l ) ' l - 2 q ' ,.o.u (1 - P/') (diam~)li '

which satisfy the lemma. The proof is completed.

Next we give two other useful lemmas:

.

(6)

L e m m a 2 . Let e>O be bounded, then I dt >t O| dt for so,he positive con- .1 oNB(xx) a B(x,t)

stant 0 and all x in ~.

Proof int �9 ~:O implies there exist eo > 0 and y0~O so that B(Y0::0)e(l). If el

an upper bound of e, i.e. ~1 ~> e, then

/ d t>~(e . / (d iam~+~, ) ) ' / dt ONB(x,~) 110% ,% )

> ~ ( E 0 / ( d i a m ~ + e j ) ) " j" dt B(x,c)

Set 0 = (e0 / (diam(5 + t~ ) ) ' , then the desired conclusion follows.

L e m m a 3. l f F has L I property on q~, then for ~ > 0 so that B(x t,~) c U(x t) ,

i e I (F ,~)

f ' j " (, - t " , " ' " - ' " (F(x))6dx...(F(x,)) t at, (6~oo). m(x t ,t)NO 0

Proof Let e' > 0 so that N(x t / ) c B(x ,,e) and F(x) >I 0 for x e N ( x t:').

Since B(xt,e) c U(xt) , we can prove by the continuity of F and Lemma 2

iim[ (~(x))'dx / j" (F(x))'d~-- 0 a . ,0 - (#(xj ~) - S(x I ~ * S(x I ~ N O

implying

i s

�9 12 �9 Zhang Yufei: A kindofApproxlmation Operators

(F(x))'ax~f (F(x))'ax, (6-, oo). (7) f J(x~.,)Ae x~,.,~|

But

(F(x))Sdx ~ f ( F ( x , ) - B, IIx, - x ~ ~ )Sdx f N(x I J)NO S(x~ ~)

<~(f )f~(F(x,)-Mg,t~ ~f] t~ "-I ~ ( F ( x , ) ) 0 (I - dt, (6 -. oo),

where ~ is the surface of the ball N(x~,~") and M some positive constant, and

(F(x))S dx ~ f (F(x ~) - ~, IIx~ - xll ~ )S dx f N(xj~)no s(x~Nk~

0 / ~ ~1 m - I >/(f~,)f ( F ( x , ) - M c t t ) t de

If. ~(F(x l ) ) s -- t ~ (6-, oO),

where ~ ' is the intersection of the surface of the ball N(x~,g) and the cone k~ -- {x :x

==xj + t ( y -x~) , ycB0,0,e0), t >~0}, here B(Y0,~0) is defined as in Lemma 2, and M'

some positive constant. By (7), it is clear that the lemma holds.

3. 6 -kerne l s and Approximation Operators

Definition2. S u p p o s e F ( x j ) > 0 for all ir An operator K : Y x R - , Y , Y t h e

space of the continuous functional of R " , satisfies

a) K(F,6)(x) ~ 0 if F(x) >i 0,

b) K(F,6)(x ,) = K(F,6)(x ~) for any i,/r162

c) K(F,6)(x) > K(F,6)(y) and lims_.s ~ K(F,6)(y) / K(F,6)(x) = O, 6 o ~ R or

6 0 = oo, if IF(x)l > IF(y)I. Then

Ko(F,r --- K(F,6)(x) / f K(F,6)(t)dt @

is called 6-kernel with respect to F and �9 whenever f K(F,6)(t)dt ~ O. o

We denote Ko(F,r by -g0 for simplicity. It is not difficult to see the following

properties of the kernel K o by using lemmas in Section 2.

Approx. Theory & its Appl., 9:3, Scot. 1993 �9 | 3 �9

1. I K o = l . e

2.1imlq~3uKo--i a n d l i m f K o - - 0 , U = ~ U(x,). 1~6e I ' J e " O \ U /~F.e)

Definition 3. Operator B(F,q~,6) : x --, R " is defined by

B(F,q~,6)(]) .= ~.Ko(F,q,,~)(t)/(t)dt, f~X .

Theorem 4.

where p ~. II(F,q~)I

Proof. Let

lira inf sup II(A (F,~,v) - B(F,~,6))(I)II = 0, ~l~JI e vsC4, fGH

and H is any equally continuous functional set o f R " .

K o + (I / P ) / Ko, i~l(F, qJ), O\B

B = U B(x,,,). /(.r.e)

T h e n

inf sup [](d (F,~,v) - B(F,~o,~))(])II , iC t fen

" , 0 O~T~ ~< sup I[(A(F,q~,v ~ - B(F,q),6))(/)ll !v o tv~, '",vp) )

f~H

< s u p m a x max II/(t)-/(x,)ll +2supmax[[ ] [ t ) l l k o - , 0 , (~--,oo, ~ - ,0 ) . /'ell le.~F.e) t~.~/(zt.~ ) l i e

The proof is completed by the arbitrariness of ~.

We are more interested in K o when it is of the form

K~174 - - (F(X)) s / / . (F(t)) ~ dt"

In fact, it is a 6 -kerna l for 6 o = ~ with respect to F and q). For this kind of kernel we

have another property.

3 0 . Ko(tF,cb,6) = Ko(F,Cb,6) for any t > 0.

In the rest o f the paper we assume K o to be this kind of kernel.

Theorem 4'. I f F has L I property on r then

lim inf sup II(A(F,q),v) - B(F,~,~))(/)[I -- 0, 6 - ~ ,iCw(/) fell

where I .. {i~l(F,q)) : o" I < max t(~.|162

Proof. It follows directly from Theorem 4 and Lemma 3.

�9 14 �9 Zhang Yufei: A kindofApproximation Operators

Next theorem is the main result of the paper.

, 1 Theorem 5. I f F has L n property on ~, I(F,S(qO) # ~ , and there exists ~1 r in-

dependent o f the mapping S so that ~ <~ ~', then for any d~(O,1) there exist a positive constant

independent o f the mapping S so that

inf ( / , ) s u p U(ACF,SCff~),v) - B(F- h , S (r a - ') l l < d, root. foil

where p, = II(F,S(~b))l, I ,= {i~l(F,(~)): or, < maxjatr,s(| and h, is defined as in (4).

Proof.

IICA ( F , S C ~ ) , v ) - SCF - h , , S ( ~ ) , , S ) ) ( f . S - ' )II

s

where x, -- S-Z(x~), ir G is defined as in (4). From the proof of Theorem 4 we

only need to prove that for small

limf KnCG,t~,6)(t)dtffil and limf Ko(G,q~,6)(t)dt=O (8) i-.| r - I - | O\B

hold uniformly for'the mapping S, here Bffi U ~,~(c.| �9 Note that

suplG(x)l/G(xl)< 1 uniformly for re(0,1]. (9) z I O \ J

In fact, it follows directly from

p " g l #d �9

sup suplG(x)l/G(x,)~!l-#'r ~ / G ( x , ) < l , (G(x)>~O), ,.,,.,,..,, tn' l (i - n') < I. (GCx) < 0).

and t

max sup IG(x)l/G(x,) < I, reiro,ll z 6 e ~

here roe(0 ,1) so that S ( ~ ) c U(xj) if r < r o. But Lemma 1 ensures that we can replace

F i n L e m m a 3 b y G to get

--t ) t at, or-- maxorl, (or--, oo), (I0) ONB /ao(o,o)

since

lim~ ( l - t " ) J t ' - I d t / ( l - t ' l ) J t = - I d t f f i O a-'D J'O 0

Then (8) can be proved by (9) and (10). The proof is completed.

if r < err

Approx. Theory & its Appl., 9:3, Sept. 1993 �9 15 �9

4. Application

In this section we propose a method to solve the problem in (2) in the case of �9 as-

sumed be a rectangle, i.e,

= { x e R " :a t~<(x)~<bt, i= l , . - - ,m}.

Algorithm. Let de (0,~ ), 3, e> 0 begiven.

l . k < l , j < l .

F k so tha tF k=r/max | r/)min | ~/e[0~). 2. Finding

3. (xk) j -- f ,Ko(F- Fk,~,~)(t)( t) idt.

1 1 1 4. lf(xk)~ t> (at + b i ) / 2 then a t ,-- ( ~ + d)a/ + ( ~ - d)b j else b t ,,- ( d)a j

1 +(~ +d)b .

5. If/< m than/.--/+ I goto 3 else ifdiam �9 < e then STOP else j,-- I, k*-- k + I

goto 2.

The idea in the algorithm above is to apply Operator B(F- F~,tD,6) to approximate

Operator d(FJb,v), then act on the identity mapping of IR" to itself. On the convergence

of the algorithm we have the following result.

i property on �9 and there exists ~'~(0,2 I-) so that ~ <<. ~', then Theorem 6. if F has L

for any de (0,~) there exists a positive constant ~ so that the sequence {x k } generated by the

algorithm converges r-linearly to some x " ~, a solution to the problem in (2).

Proof. Let (}kj denotes the jth integral region at the kth iterate in the algorithm, i.e. k - I k - 1 b k i = l , ' " , j - l , and a <<.(x)~ <~bq , ~Pkj = { x e R " :at ~<(x)l< , '

q =/,...,m},

S , ~ ( t ) -- (a ~ k k - 1 ~ - ~ ) r 1 ~ ~ ~ o o l , ' " , a i_~ ,a l , ' " ,a + ( ~ + d ) k- ( t - ( a l , . . . , a / _ 1 , a i , . . . , a m ) r ) ,

G k / ( t ) - - F ( S ~ j ( t ) ) - F k , t~(I)ki , j - - 1,. . . ,m, k = 1,2:-

Then

(xk)1 -- ; ko(F - Fk,t~k/,~)(t)(t)jdt -- (B(F- F~ ,O~j,~)(t))j. @k I

Pkj --It(F,r Let

( l l )

�9 16 �9 Zhang Yufei: A kind of Approximation Operators

I~, = {ieI(F,cb~,):a~ < max o,}. i~;I(F, O I~ I )

Since ~ 1, c ~ i1' for j = 2, . . . ,m, x ~ ,l, x'~ = S ~ l (x ,), i e I ( F , O k,)' let

- 1 ' - I s = S,, (S~(x)), ~ = S~1 (x~).

Then

and

o e

IG ~,(x)l/Gkt(x ~) -- IG kl (~)1/Gkl(s 1 )

t - 1 i �9

I I x - x l l l = IIS,, (S,1(s S~ 1 - ( S , 1 ( s = I 1 - ~ - s

which implies that (9) and (10) in Theorem 5 hold uniformly for Gkj for any k and j. So by

the proof of Theorem 5 we know there exists a positive constant.6 so that

- i (b~ - aj). inf I[(A(F,q~k,,v ) - B ( F - Fk,CY~kl,~))(Ski )][ < d min 0 *t~C t , ' ( l l, l ) i " l , . ' , m

By (1 l)

So

and

1 I o (vi)r Cp~, I ~ v ,Cx , ) , - - (xk ) , l<d(~ +all k- ( b ~ - a , ) for some v = (Ik/). t ( F , o t~ )

k-i 1 d)(b~-] aj . if ( x ~ ) , < a . + b j ) / 2 rain ( x i ) j < a , + ( ~ + k- l ) k-] k-i l (u k t ) - I t~ I !

bk-! 1 ~-l *-l (X*)i k-l bk-1 ~,%'-',max ( x , ) i > i - - (~ +d)(b --a, ) if >~(a t + j ) / 2 . (12)

Next we complete the proof of the theorem by induction. It is clear tha t I ( F , �9 n ) -- I n

#q~. We may assume I(F,cbkl)--Ikl #qb, the by (12) we have that I(F,~k2)--Ika #qb.

Repeating the process we get that I(F, aP k, ) --Ikj ~ qb for j---- 3,.-.,m. Then I(F,ap (k + i)1 )

-- I(k + i)1 # q~ follows from I(F,~ ~ ) -- I ~ ~ qb. The proof is completed if we note that

1 d)~-i diamq~kl ~<(~ + diamq~ n.

Department of Mathematics

Nanjing University

Nanjing, 210008 PRC