l nghlala ,k E 8Ekf(xk).

+00 1 c. L{Ek - :\«p' (xk, xk+l), xk - xk+l)} < +00 k=l k Tli (2.62) ta co Ek = f(xk) - 'ljJi(k}(xk+l) + lk «p' (xk, xk+l), xk - xk+l) :s; ~[J(xk) - f(xk+l)] + lk «p' (xk, xk+1), xk - xk+l).

39 Nhu v~y n n I) Ek - :k «I>'(xk, xk+l), xk - xk+l)} ~ ~I)f(xk) - f(xk+l)] k=l k=l - ~[J(xl) - f(xn+l )]. Vi f bi ch~n duoi nen +00 1 L {Ek - ~«I>' (xk, xk+l), xk - xkH)} < +00. k=l k d. f(xk) -+ ! = inf{f(x) Ix 2:O}

Day {f(xk)} khong tang, hQi tl;l de'n f. Gia su phan chung ding! > 1* :=

inf f(x), nghla la t6n t<;liy E IRPva 8> 0 thoa f(y) + 8 < f(xk), Vk dli IOn.

xE!R.P

Vi Ek-lk «I>/(xk,xk+l),xk - xk+l) -+ 0 nen t6n t<;liko d~ voi k 2: ko thl

Ek- ~, «I>/ (xk xk+l' ),Xk - xk+l) <-8

/\k 2'

Tu B6 d€ 2.2.2 voi a = xk,b = xkH va c = y, ta co

Ily- xk+1112-Ily - xkl12~ -t(y - xk+l, I (xk,xk+l))

= - t(y - xk, I(xk, xk+ 1)) - t(xk - xk+ 1, I(xk , xk+ 1)) .

Tu (2.64), ta co ngay

-t(xk - xk+l,/(xk,xk+l))< ~k(~ - Ek)

M~t khac tu ph~n b va ryk= -lk '(xk, xk+l) E OEkf(xk) ta du<;5c

-t(y - xk, I(xk, xk+l)) = ~k (ryk,Y - xk)

f(Xk) - 8 > f(y) 2: f(xk) + (-l, y - xk) - Ek'

Ke't h<;5pvoi (2.67) va (2.68) ta du<;5c

1 k I k k+l Ak

- e (y - x , (x , x )) < e [-8 + Ek].

Cu<3icling tu (2.65), (2.66) va (2.69) ta du<;5c

k+ 1 2 k 2 Ak 8 k 2 8

lIy- x II -< lIy- x II +-[-() 2 - Ek- 8+Ek] = lIy - x II - Ak-'2()

(2.64) (2.65) (2.66) (2.67) (2.68) (2.69)

Liy t6ng bit ding thue tren voi mQi k > ko ta du'Qe k-l (y

0 ::; Ilxk - yl12 ::; Ilxko - yl12 - 2() L Ak' k=ko

Cho k --t +00 thl L Ak ::; 2: Ilxko - yl12 < +00, di€u nay mall thuffn voi gia thie't.

k=ko

Bay giG ta gia sil' f co eve ti€u x tren IR~va {Ad bi eh~n. e. {xk} bi eh~n (adsbygoogle = window.adsbygoogle || []).push({});

Dung bit ding thue (2.65) voi y = x ta du'Qe

Ilxk+l- xl12::; Ilxk- xl12 - t(x- xk, <1>'(xk, xk+l)) - t(xk - xk+l, <1>'(xk, xk+l )).

Tli dinh nghla eua ryk= -lk <1>'(xk, xk+l) E OEkf(xk), ta co -b(x - xk, <1>'(xk, xk+l)) = ~k(ryk,x- xk)

::; ~k[f(x)- f(xk) +Ek]::;~k Ek.

Do do

Ilxk+l - x112::;Ilxk- xl12+ ~k[Ek- :k «1>'(xk,xk+l),xk - xk+l)].

Vi {Ak} bi eh~n ( do gia thie't ), ap d1;1ngph~n e ta co

+= A 1

L ()k[Ek- :\«1>'(xk,xk+l),xk - xk+l)] < +00.

k=l k

Dung B6 d€ 2.1 ta suy ra {llxk- xllhhQit1;1.V~y {xk} bi eh~n.

f. MQi di€m gioi h<;lnx* eua {xk} Ia eve ti€u eua f tren IR~ va xk --t x*

Cho xnk --t x*. VI f lien t1;1enen f(xnk) --t f(x*). Ap d1;1ngph~n d, f(xk) --t J=

inf f(x). Do do f(x*) = j. VI x* E IR~nen x* la eve ti€u eua ham f tren IR~.

xER~

Tli (2.70) thay x bdi x* ta du'Qe

Ilxk+l - x*112::;Ilxk- x*112+(yk,

(2.70)

trong do

{y = Ak[E - ~ «1>'(xk xk+l ) xk - xk+l )]

k () k Ak " .

ViL (jk < +00 nen ap d1;1ngmQt ke't qua eua Correa va Lemareehal ([7], M~nh

k=l

2.7 Ktt qua tinh toan 86.

f)~ thu~t gi<li dti d~Hdu'Qc, ta cin gi<li bai toan con sail

{

mill 7/Ji(y)+ 1kd<p(y,xk), Y E JR~+.

V(ji 7/Ji(y)= max {f(yj) + (s(yj), Y -yj) I j = 0,. . ., i-I}, bai toan tren tu'dngdu'dng (adsbygoogle = window.adsbygoogle || []).push({});

v(ji

(SPh,i

mill v + 1kd<p(Y,xk),

v 2: f(yj) + (s(yj), y - yj) j = 0, . . . , i-I,

Y E JR~+.

Ta tha'y dng ne'u (yi, vi) la nghi~m cua bai toan tren thl

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