BẤT ĐẲNG THỨC BIẾN PHÂN TỰA ĐƠN ĐIỆU VÀ THUẬT TOÁN XẤP XỈ GIÁ TRỊ
NGUYỄN VĂN THÙY
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CHV0NG 2
A '" " ,:?
NGUYENLYBAITOANBOTR0
, .,
2.1 Thu~itoancdso
Trong [2, 3, 4], mQtnh6mcac thu~ttoandtSgiai bai toan (1) dil dtiQc
th6ngnha'ttrongclingmQtkhuonkhBva dtiQcgQiIa nguyen15'bai toanbB
trQ.Y ttidngchinhcuathu~ttoannayd1.,1'aVaGnh~nxetsanday.
Xet hambBtrQM :X ~ 9i Idi m;:tnhvakhavi GateauxtrenX , va Ela
mQtso'dtiongchotrudc.Vdi x E X chotrtidc,xetbaitoanbBtrQ:
(11) mill (M(y) +( EF(x) - M'(x),y) ).
Y EOXad
Gia sa y (x)Ianghi~mcuabaitoan(11).
A.p dl.lllgbB d€ 1.5vdi FI( . ) =M( .) Idi,khavi GateauxtrenX va
F2(-) =< EF(x) - M' (x),.) la hamIdi ( vi tuye'ntinh). Theo(9) , Y (x) thoa
- - -
(M' (y(x», y - y(x» +(E F(x) - M' (x),y) - (EF(x) - M' (x),y(x» 2 0
v Y E Xad,
hoi:icd d;:tngnit gQn
(12) (M'cY(X»+EF(x)-M'(x),y-Y(X»20 VYEXad.
Ne'uy(x)=x thi (12)suyra
(EF(x),y-X)20 VYEXad,
tlicla
(FcY(X»,y-Y(X»20 VYEXad,
lien y (x) Ia nghi~mcuabai toan(1).
D1.,1'atrennh~nxet nay,xay d1.,1'ngthu~toanco sd sanday dtSgiai bai toan
(1).
14
Thu(it tmln 1 ( Thu(it tmln cosO')
Cho tru'deday cae sO'du'ong{8k, k E ~} 0
(i)
(ii)
(13)
(iii)
Chc;mdii!mxucltphat XOEX tily yo
abuack, bierXk, tinh Xk+1 :=y(Xk) b!ing vi?c giai bai loan b5 trq
(11) vai x thaybai Xk, 8bai 8k
mil}cl(M(Y)+(8k P(xk)-M'(xk),y»).
YEX
Neu II Xk+l - Xk II nho hcJn melt giai hC;lncho truac thi dang 0 Nguqc
IC;li,tra v~buac(ii) vai k +- k +1 0
B6d~2.1
T~lim6ibuaccuathu~tloantren,Xk+la nghi?mduynhC[tcuabai loanbien
ph{m:
(14)
vai:
(15)
(pk(Xk+l),X-Xk+l):2: 0 \1XEXad ,
pk(X)=8kp(Xk)+M'(x)-M'(Xk) 0
Chung minh.
Go? ? (14)
/ h
'
hOA k+l, k+l / 1,la su co al ng H~mx va y , We a
(pk(xk+l),yk+l -xk+l):2: 0,
Suy ra
( P k(y k+1), Xk+1- Yk+1) :2: 0 ,
(pk(Xk+l)_pk(yk+l),xk+l - yk+l):::;0,
ho~cvie'td d,;lllgkhac
(16) (M'(xk+l)-M'(yk+l),xk+l_yk+l):::;O,
M~tkhac,doM Iaham16im~nhnentheom~nhd61.3secoh~ngsO'
a>0 saocho
(17) (M'(Xk+l)-M'(yk+l),Xk+l _yk+l):2:a II Xk+l _yk+1112 ,
15
Tli (16)va (17)suyraXk+l=l+l . .
2.2 DjnhIy hQit\1d1!atrenHnht1!adondi~um~nh
Trangphftnnay,chungta se chungminhslj hQiW cuathu~tloan 1
tranghaitnionghQpvoi loantll'F trangbailoan(1)Ia dontr!vadatr!,voi
giii thie'tF Iatl!adcJndi~um~nh.
2.2.1 TruonghQptm!ntll'dontrj
Caegiathie't
1. F Ia tl!adondi~um~nhvoi'hangsO'e lIen xad,
2. F Ia lien WcLipschitzvoi hangsO'A lIen X ,
3. M' Ia dondi~um~nhvoi hangsO'b lIen xad.
B6 d~2.2
V6'igid thilt F la tZ!adondi?um(;mh,nlu bai loan(1)co nghi?m,thl nghi?m
do la duynh{[t.
Chungminh.
* * d
Giii sabailoan(1)cohainghi~mIa Xlva X2E Xa .tucIa
(18)
(19)
* * ad
(F(XI)' X - Xl ) ~ 0 '1/X EX,
* * ad
(F(X2),X-X2)~O 'l/XEX .
Thay X=X; trong(18),taduQc
* * *
(F(XI),X2-Xl) ~0 .
DoF Ia tl!adondi~um~nhnentant~ihangsO'e >0saocho
* * * * * 2
(20) (F(X2)'X2 -Xl) ~ ell X2 -Xl II .
*
M~tkhac,thayX= Xl trong(19)thi thuduQc
(21)
* * *
(F(x2),X2 - Xl)::; 0 .
* *
Tli (20)va (21),suyra xl =X2 . .
16
DjnhIy 2.1
Gidsa riinghili loan(1)conghi~mx* . Ne'uM' la dondi~umqmhvai hiing
sffh trenXld, thi t6ntqziduynhatnghi~mXk+lcila hili loanh8 trei(13).Ne'u
F la tf:Cadondi~umqznhvai hiings6 e tren;:ad( thix*duynhd't) va lien tl;tc
Lipschitzvaihiings6A tren;:adva:
\j k E ~, ex0,~>0,
A +~
thiday{Xk}hQi tl;tmqznhv~x* . Hon mla,ntu M' la lien tl;tCLipschitzvai
hiings6B tren;:ad, thicouaclu(,fngsai s6:
(22)
(23) Ilxk+l-x*II~~(~ +A)lIxk+l-xkll.e c;
Chung minh.
a) Sf:Ct6ntqzivaduynhd'tnghi~m.
Ap dl.mgb6d~1.5va (12),taco Xk+lla nghi<%mcuabai toan(13)ne'u
vachine'u
(24) <M'(Xk+l) - M'(Xk) +c;kF(Xk), x - Xk+l) ~0 \j X E Xad .
Ap dl.mgb6d~1.1vdiA la M', f la M'(Xk),cpla c;kF(Xk)thiA va cpd~
tha'ythoa3giathie'tdftutiencuab6d~.Takit3mtfagiathie'tcu6i.
R5rang0 E domcpvaM' dondi<%um(;lnhvdih~ngso'bnen
< M' (x) - M' (x*), x - x* ) ~b II x - x* 112 ,
thayx*=0 thi tadu<;)c
< M' (x),x ) ~b II X 112 +< M' (0), x ) .
Dodotaduoc
<M'(x),x)+cp(x)~bl!x 11+<c;kp(xk)+M'(O),x)
II x II II x II '
suy fa
17
< M' ex),x) +cp(x)-} +ex) khi IIx II-} +ex).
IIx II
Do v~y,theob6 d~1.1thibai tmln(24)luautant<,tiit nha'tmQtnghi~m,
nghi~mnayduQcgQiIii Xk+l.
Tinh duynha'tcuaXk+lsuyra tub6 d~2.1.
b) Day (XkjhQitl,im~mhv~x*.
*
x langhi~mct'iabaitoan(1)nen
* *
(25) <F(x ), x - x ) ~0
EHit
v X E Xad.
(26)
* *
ct>(x)=M(x ) - M(x) - < M' (x),x - x) .
Vi M' dondi~um<,tnhnentub6d~1.2,taduQc
(27) ct>(Xk)~ b II xk - x* 112 ~0 .
2
X6t s1fbie'nd6i ct'iact>t<,tim6ibuckcuathu~toan1.
6.t+1:=ct>(Xk+l)- ct>(Xk)
=M(Xk) - M(Xk+l) - <M'(Xk) , Xk- Xk+l ) + <M'(Xk) - M'(Xk+l), x*- Xk+l).
Vi M' dondi~um<,tnhnentub6 d~1.2,taduQc
M(Xk+l) - M(Xk) - <M' (Xk), Xk+l- Xk) ~ b II xk+l - xk 112.
2
Dodo
sl:=M(Xk)-M(Xk+l)_<M'(Xk),Xk_Xk+l)::::; - b Ilxk+l-xk 112.
2
Thay x =x*vao(24)thi taduQc
(28) S2 :=< M'(Xk) - M'(Xk+l), x* - Xk+l)
::::;Sk <F(Xk), x* - Xk+l )
::::;Sk <F(Xk) - F(Xk+l) , x* - Xk+l) +Ek <F(Xk+l) , x* - Xk+l).
Do F tl,l'adondi~um<,tnhva
< F(x*) , Xk+l- X* ) ~0 ,
nen
18
(29) (F(Xk+l) ,Xk+l~X*);::: ell xk+l-x* 112.
Do d6
S2:::;- e skllxk+l - x*112+ sk( F(xk) - F(xk+l), x* - xk+l).
V~y
k+l
L'.k
:::;_lll xk+l - xk 112- e Ek II xk+l - x * 112+Ek(F(xk) - F(xk+l), X* - xk+l )2
:::; _lll xk+l - xk 112- e'Ek II xk+l - x * 112+Ek II F(xk) - F(xk+l) IIII x * - xk+l II2
:::;-lll xk+l - xk 112- e Ek II xk+l - x * 112+EkA II xk+l - xk IIII xk+l - x * II2
-
(
vib
II
k+l k
II
EkA
II
k+l *
11)
2
(
A2E2k k
J
II
k+l *
11
2- - - x - x - - x - x + - eE x - x
-Ii J2b 2b
:::;E2k
(A2 - +] II xk+l - x * 112.2b E
2
"1'. k 2eb ~ e A +/3 D d/Y1 a . 0 0
A +/3 E 2b
E2k
(
A 2 - ~
J
<- /3a2.
2b Ek 2b
V~y
(30) L'.~+1:::;- a2/3II xk+l - x* 112.
2b
Tli d6 ta suy ra L'.t+1:::;0 tilc la <D(Xk+l):::;<D(xk).Do v~y,day {<D(Xk)}
giamva bi ch~ndtioibdi 0 lien hQiW. Do d6, L'.t+1-:; 0 va tli (30)suyra
day {Xk}hQiW m(;lnhv6 x* .
c) Changminh(23).
*
Thayx =x vao(24)vado(29),tadtiQc
(M'(xk+l)-M'(xk),X* _Xk+l) +Ek(F(xk)-F(xk+l),x* -xk+l);:::
;:::-Ek(F(xk+l),x* -xk+l)
19
;::::Ek e II Xk+l - X * 112.
M~tkhac,doM' lien WcLipschitzvdi h~ngso'B va F lien WcLipschitzvdi
h~ngso'A nen
:::;B II Xk+l - x * IIII xk+l - xk II,
:::; A II Xk+l - x * IIII Xk+l - x k II .
Dodotaco
Eke!! Xk+l -x* 112:::;II Xk+l -x* IIII Xk+l -xk II (B+EkA),
tlic la
II Xk+l - x' II ,; : (~+A) II Xk+l - Xk II .
.
2.2.2 TruonghQ'ptmintii'datrj
Trongph~nnay,chungtasexettru'onghQpF la loantii'datIt,giatricua
F lucnayla illQtt~pconcuaX. Bai loan(1)lucnaytrdthanh:
* dTIm x EX" SCWcho
* * * * d
::3r E F(x ) : ;::::0'Vx E Xa .
TrongtnronghQpF Ia loantii'da trt,cacdint nghlatti'1.5de'n1.13
tliongling choant X<;ldatri seco dliQcb~ngcachthayF(.) bdi r E F(.).
Chingh<;ln,dint nghlatt,(adondi~uill<;lnhcuaant X<;ldontridliQcthaythe'
bdidintnghlasanday:
::3e > 0, 'V Xl, X2 E xad , 'V rl E F(Xl), 'V r2 E F(X2),
(31)
(32) ;::::0=> ;::::e II Xl - X2 112.
Thu~tloancosdvftnnhlicu,nhlingdoF(Xk)Ia illQtt~phQp,nenla'yba't
ky rk E F(Xk)thayehoF(Xk)trongloan tii'dontIt, va day {Ek}trongtrliong
hQpnaythoa
(33)
-tco . -tco
Ek >0, LEk =+CXJ,L(Ek)2 <+CXJ
k=O k=O
20
Thu~ttmin 2
Bdt dauta dilm xuc{tphatXOE X. Tc;zibuckk, bie'tXk,am Xk+lbangcach
gidi bai loanbdIrq:
(34) mill (M(X)+(ck rk -M'(Xk),X»),
x E Xad
/0 k
F(
k
)VCllrEx.
Caegiathie't
Trangphftnnay,cacgiathie'tv~ngillnguyennhu'trong2.2.1.Riengtint
lient\lCLipschitzcuaF du'Qcd6ithanh
(35) ::3a> 0, ::3~>0 saochoV x E Xad,V r E F(x), II r II ~a II x II +~.
Chti Y 2.1
Ke'tquacuab6 d~2.2v~ndungtrangtru'onghQpF la loantli'da trio
Chungminhdi~unaytu'dngt11nhu'chungminhb6d~2.2,chuy dingluc
* * ~ * *
nayF(x})va F(X2)du'QcthayboiVrEF(x})va VsEF(X2)'
Djnhly 2.2
Gid sa bai loan (31)co nghi?mx*. Ne'uM' la dondi?u mc;znhvdi hangso
b trenxad,thi tbntc;ziduynht/tmiltnghi?m~+1chobai loanb6 trq (34).Ne'u
F la ti!adondi?u mc;znhvdi hangso'e tren)(ld(x*la duynha't) va thoaman
(35),vane'uday{I} thoaman(33),thiday{Xk}hili tl;tmc;znhv~x*.
Chung minh.
a) Si! tbntc;zivaduynha'tnghi?m.
Chung minh hoan loan tu'dngt1fnhu'dinh 1:92.1, ChI thay F(Xk)bdi
k k
)V r E F(x .
b) Day {Xk}hili tl;tmc;znhv~x*.
x* la nghi~mcuabailoan(31)ne'uvaChIne'u
* * * * d
::3r EF(x):(r,x-x ):::::0 VXEXa.
1.5va (12),taco Xk+lla nghi~mcuabai loanb6 trQ(34)
(36)
, ? '
Ap d\lngb6de
ne'uvaChIne'u
21
(37) (M' (Xk+l)- M' (Xk), X - Xk+l) +8k (l, x - Xk+l ) ~0
~. k
F(
k
)VOl rEX.
\j X E xad ,
Xet ham
* *
(x)=M(x ) - M(x)- ( M'(x),x - x) .
VI M' la dondi~um<;lnhnentub6 d~1.2,tadtl-Qc
(Xk)~ b II xk -x* 112~O.
2
llt+l : =(Xk+l)- (Xk)
=M(Xk)- M(Xk+l) - (M'(Xk) , Xk- Xk+l) +(M'(Xk) - M'(Xk+l),x* - Xk+l).
VI M' dondi~um<;lnhentub5d~1.2,taco
M(Xk+l) - M(Xk) - ( M' (Xk), Xk+l- Xk) ~b II xk+l - xk 112.
. 2
Do v~y
Sj :=M(Xk) - M(Xk+l)- (M' (Xk), Xk- Xk+l):::;- b II xk+l - xk 112.
2
Thay x =x* vao(37),taduQc
S2 := (M'(Xk) - M'(Xk+l), x* - Xk+l)
< k (
.k X* - k+l)- 8 I, X
< k (
k * k
)
k k k k+1
-8 r,x-x +8(r,x-x).
Thay x =Xkvao(36)thitaduQc'
* k *
(r ,x -x )~O.
M<.Hkhac, F tl,(adon di~um<;lnhnen
(rk , Xk- x* ) ~ ell Xk - x* 112.
Dodo
S2:::;-e8k Ilxk-x*112+ 8k(rk,xk-xk+l).
V~y
llt+1 :::;- b II Xk+l - xk 112-e 8k II xk - x * 112+8k II rk IIII Xk+l - xk II2
[
k
~
2 k 2
:::;_b Ilxk+l-xkll-~llrkll +lillrkIl2-e8kllxk-x*1I22 b 2b
22
(
k
)2 *
::::;~ IIrk 112-eEk II Xk - X 112.
2b
Tli (35) suy ra
(38) Ilrkll::::;allxkll+~::::;allxk-x*ll+allx*II+~ .
M~t khac ta co
( u +v )2::::;2( U2+V2) '\I u, V E iR .
Do v~y,tli (38) suy ra
IIr~::::; ~2(a211 Xk -x* 112+(a II x* II+~)2)2b 2b
2
a k * 2 l
(
*
)
2
::::;-11x -x II +- a IIx II+~
b b
::::;Yllxk_x*112+8,
trongdo
y =~ 8 =(a IIx*II+~)
b' b
Do do
,0.~+1::::;-eEk IIXk -x* 112+(Ek)2(y II xk -x* 112+8).
Voi bat ky s6 tlf nhien N, ta co
N-l N-l
I,0.~+l::::;I(-eEk IIXk -x* 112+(Ek)2(YII Xk -x* 112+8)),
k=O k=O
ke'thQpvoi (27), ta duQc
b IIxN - x * 112::::;<:D(xN)
2
N-l
[ ](39) ::::;<:D(xo)+ I -eEk IIXk -x* 112+(Ek)2( Y IIXk -x* 112+8)
k=O
N-l
::::;<:D(xo)+I(Ek)2(Yllxk -x* 112+8),
k=O
SHY ra
23
N * 2
II x -X II ~
° o 2 28N-1 N-1 2 *
~ 2cD(x ) + 2(£ ) Y IIxO-X* 112+- L(£k)2 + L ~(£k)211 Xk -X 112
b b b k=0 k=l b
N-I
~l1N + L~Lk Ilxk-x*f,
k=l
ydi
28 N-1
N = 2cD(xO)+ 2y (£°)211XO-X* 112+- L(£k)2,
11 b b b k=O
k - 2y
(
k
)
2
~L - - £ .
b
Ta CO
\i k, II Xk - X * 112~ SUp II Xl - X * 112,
l::;k+1
"k 2y" k 2
LJL =- L./£ ) <+00
kE~ b kE~
bdi vi L(8k)2 <+00.
kE~
Vi I(8k)2 0 saGcho l1N ~11, \iN E ~.
kE~
Luc nay,haiday{II Xk - x* 112}va {l1k}thoagiathi6tcuab6d61.6,suy
ra day {II xk - x* 112}bi ch~nvadododay{Xk}bi ch~n.
D~t f(x) =II X - X* 112,theo dinh 1y gia tri tIling binh
f(x) - fey)=(f(z), x - y)
*
=2(z-x ,x-y),
vdi z =AX +(1- A )y, A E (0, 1) .
Ta co
* *
If(x)-f(y)I~21Iz-x 1IIIx-yll~2supllz-x 1IIIx-yll,
ZEK
vdi K la baa16iciiaday {Xk},va doK bi ch~nnensuyra
(40)
Han Hila,tu (39)taduQc
f lien WcLipschitz.
24
N-l N-l N-l
eL Ek II Xk - X* 112:::;<D(XO)+ L Y(Ek)2 II Xk - X* 112+8L (Ek)2 .
k=O k=O k=O
Vi day {II Xk - x* 112}bi ch<;inen
k * 2
3p >0: II x -x II :::;p \t k,
suyra
N-l N-l
el>k Ilxk -x* 112:::;<D(xO)+(yp+8)L(Ek)2 .
k=O k=O
Vi 2.:(8k)2 <+00nen tli tren suy ra
kE~
(41) 2.:Ek II xk - x* 112 <+00 .
kE~
Thay x =Xkvao (37) ta du<;jc
+Ek 20,
Wcla
-bllxk+l-xk 112+Ek Ilrk 1IIIxk+l-xk 1120.
Ke'th<;jpvoi (35) thi tadu<;jc
II Xk+l - Xk II :::; ~II rk II :::;Ek (exII Xk II +0) (ne'u II Xk+l - xk 11:;t0) .
b b
Vi day {Xk}bi ch<;innend<;it~=exII x:11+0 thi tli trensuyra
(42) II Xk+l -xk II:::; ~Ek .
(42)vftndungtrangtru'ongh<;jpII xk+l - xk II =O.
Tli (40),(41),(42)tasuyra cacgia thie'tcuab5 d@1.7du<;jcthoaman.Do
d6 tadUdc
hm Ilxk_X* 11=0.
k-Hoo
V~y,day {Xk}hQitumanhv@x* . .
25
2.3 DinhIy hQitv dtfatrentlnheMitttfaDunn
Trangph~nnay,chungtasechungmint slfhQit~lcuathu~toan1trang
tnionghQptoan111F trangbai toan(1) Ia dontri va F Ia tlfaDunn.
Caegiathie't
1. F co tint cha't11!aDunnvdi hangso'E trenXad,
2. F lien WcHoldertrenXadngmaIa
:3c>0vaD> 0saGchoV x,y EXad,II F(x) - F(y) II::;D IIx - y Ilc ,
3. M' Ia dondi~umanhvdihangso'b va lient~lCLipschitzvdihangso'B
A
X adtren .
DinhIy 2.3
Oidsabaitoan(1)conghi?mx*.NtuM' la dondi?um{;mhwJi hangsob
tren;:ad, thit5ntqziduynh(Jtnghi?mXk+lchobaitoanb8tre!(13).Hannlla,
ntuF la tf!aDunnvaihangs(/E tren;:advane/u:
(43) \-I k \0..> k+I , I--'> ,
E+~
thi dc7y{ F(Xk) } h()i tl,{v§ F(x*), II Xk+l -Xk II h()itl,{v§ a.va dc7y{ Xk}bi
ch4n.Ntu themgid thilt la M' lien tl,{cLipschitzvaF la lien tl,{CHolder tren
;:ad,thim6iddm tl,{ylu cuadc7y{Xk} la m()tnghi?mcua(1).
Chung minh.
a) Sf!t5ntqzivaduynhatnghi?m.
Slf tan t~iva duynha'tnghi~mciia bai toanb6 trQ(13) da:duQcchung
mint d dint 192.1.
b) Sf! h()i tl,{.
Bat
(44) \P(x, E)=cD(x) +D(x, E),
vdi
(45)
* *
cD(x)=M(x ) - M(x) - ,
* *
D(x, E)=E.
26
Theo (27),taco
cD(xk) ::::b II x k - X * 112.
2
*
Do x Ia nghi~mcuabai roan(1)lien
(\ k k - k
<
* k- *
»~l(X , S ) - S F(x), x x - 0 .
Dodo
(46) '¥(Xk,Sk);::::bllxk-x*112::::0.
2
Ta xet s11'bi€n d6i cuaham'¥ doi vdi m6ibudccuathu~tloan 1.
B~ngcach dungcac ky hi~uva tinh loan tltongt11'nhu trongchungminh
cuadinhIy 2.1,taduc;fC
1:+1:='¥(xk+1,sk+1)- '¥(Xk,sk)
=cD(Xk+1) - cD(xk) +Q(Xk+1, Sk+1) - Q(Xk , Sk)
k+! * k+1 * k * k *
= s] + S2+S < F(x ),x - x ) - S <F(x ), x - x )
=s] +S2 +S3 ,
vdi
S] ~ - b II xk+1 - xk 112,
2
< k<F(
k
)
* k+1
)S2- S x,x - x ,
k+] * k+l * k * k *
S3= S <F(x ), x - x ) - S <F(x ), x - x ).
Ta co
S2~ Sk<F(Xk ), X* - Xk+1 ) =Sk <F(Xk), X* - Xk ) +Sk <F(Xk ), Xk - Xk+l ) .
DoF cotinhcha'tuaDunnva
* k *
<F(x ), x - x );::::0,
lien
<F(Xk),Xk-x*)::::! II F(xk)-F(x*) 112.
E
Dodo
k
S2~-~ II F(Xk) -F(x*) 112+sk<F(Xk),Xk _Xk+1) .
E
27
M ~ kh/ d k+l < k ~';it ac, 0 E - E nen
< k * k+l k )S3- E (F(x ), X - X .
V~y
k
S2+S3 :;;-~IIF(Xk)-F(x*)112 +Ek(F(Xk)-F(x*),Xk_Xk+l).
E
Suy ra
r:+1 :;;-~llxk+l_Xk 112-~ IIF(Xk)-F(x*) 112+
+ Ek II F(Xk) - F(x*) 1111Xk - Xk+l II .
Vi
Ek II F(Xk) - F(x *) 11IIXk - Xk+l II :;; (Ek)2 II F(x k) - F(x *) 112+2~
+~ II Xk - Xk+l 112,
nentadudc
rk+l < ~-b Ilxk -xk+1112-Ek
(
~-~
)
IIF(xk)-F(X*)112.
k - 2 E 2~
V /' '\ b ' k 2~ h' /01I\,< va ex<E <- t 1taco
E+~
rk+l :;;-b-~llxk+l-xk 112- ex~ IIF(xk)-F(x*)112.
k 2 E(E +~)
. Ne'u Xk+l=Xkva F(Xk ) =F(x* ) thi tU(25),suyra Xkla nghi~mcuabai
roan(1).
. NguQc l';ii,ta duQc
r:+1 =\Jf(xk+l,Ek+l)- \Jf(Xk,Ek) <0 ,
do do day {\Jf(Xk, Ek)} giam, bi chi;indudi bdi 0 nen hQi tl,l.Do do, ta co
r:+1 = \Jf(xk+l,Ek+l)- \Jf(Xk, Ek) ~ 0 khi k ~ +00.
Do v~ytaduQc
II xk+1 - X k II ~ 0 khi k ~ +00,
28
IIF(xk)-F(x*)II--+O khik--++oo.
Hon nlia, day {'¥(Xk, 8k)} hQi tl.llien bi ch~nva do do, tU(46) suy fa
'¥(Xk,8k) ~~II Xk -x* 112.
2
V~yday{II xk -x* II} bi ch~nvadododay{Xk}bi ch~n.
Bay giG,gia sii'z la mQtdit5mtl.lyeticila day {Xk},vagia sii'day con {Xkj}
hQi tl.lv~z.
Ta viet l~i (24) la
(M' (Xk+l) - M' (Xk ), X - Xk+l) +8k ( F(xk ), X - Xk+l) ~0 v X E xad .
M' lien WcLipschitzvdi h~ngsO'B lien
II M'(Xk+l)-M'(Xk) II ::=;Bllxk+l-xkII.
M~tkhac,VI 8k>a lien
(F(xk), X - Xk+l) ~- B II xk+l - xk III1 x - xk+l IIa
vX E Xad .
Do do taduoe
(F(xki),x-xki+l)~_Bllxki+l-xki II Ilx-xki+l II VXEXad.
a
VI
F(xki)--+F(x*), xkj --+z, Ilxki+l-xki 11--+0 khi ki --++00,
lien
(47) (F(x*), x - z) ~0 V X E Xad .
Ngoai fa, VI (F(z), x kj - z) --+0, ki --++00lien lieU F(z) =0 thlfa rangz Ia
nghi~mcua(1).
NeuF(z)"*0 , d~t
yki = Xki - (F(z), Xki -z)
II F(z) 112 '
thl
(48) ( F(z) , ykj - z) =(F(z), Xki - Z ) -
29
- 12(F(z),Xki-z).(F(z),F(Z»=0
II F(z) II
D@thfty
Ilykj-xkill::;;llxki-zll~O,
nen
II yki - Xki II ~ 0, ki ~ +00 .
Do F lien tueHoldernen
II F(yki ) - F(xki ) II ::;;D II yki - Xki Ilc (e>O,D>O).
Do d6 tadude
(49) F(yki ) ~ F(x *) .
Mat khae
Ilyki -zll ::;;llyki-Xki 11+llxki-zll,
nen
II yki - z II~ 0 khi ki ~ +00.
Do F tlfaDunnnentil (48)SHYra
(F(yki), yki -z):2:~IIF(yki)-F(z)11 .
E
Cho ki ~ +00thi SHYra
(50)
Til (49) va(50) SHYra
F( yki ) ~ F(z)
*
F(z ) =F(x ) .
Do d6 (47) trdthanh
( F(z), x - z ) :2:0
V~yz langhi~meuabailoan(1).
v X E xad.
.
30