một số phương pháp proximal điểm trong cho bài toán cực điểm lồi và cho bất đẳng thức biến thân, chương 2

38

Do m < 1, ta thu du'ejc tli ba"t d~ng thuc nay rang f(xk) :s; f(y*). Dieu nay dfin
dSn

1 1

f(xk) + Akd
VI y* Ia nghi~m duy nha"t nen xk = y* va do B6 de 2.6.2 ta suy ra xk Ia cvc tieu

cua f tren JR!.~.

(ii) Ta ky hi~u i(k) la ChI s61~p ung voi xk du'ejc c~p nh~t, nghla la yi(k} = xk+l.

Ta dinh nghla "yk- ,i(k} E 8'ljJi(k}(xk+l). Ta bitt rang

,k = - :k

Tli nhung ky hi~u tren ta chung minh cac kh~ng dinh sau:

a. {f(xk)} kh6ng tang. VI

f(xk) - f(xk+1) ~ m[J(xk) - 'ljJi(k}(xk+1)] (2.62)

nen theo Chti y 2.6, f(xk) - 'ljJi(k}(xk+l) la ham kh6ng am, do do {f(xk)} cling

kh6ng tang. Do do ta co the gia sa {f(xk)} bi ch~n du'oi (nSu kh6ng thl f(xk) ~

-00 va chung minh xong ).

b. ,k E 8Ekf(xk). Voi

Ek = f(xk) - 'ljJi(k}(xk+l) + :k «p'(xk, xk+1),xk - xk+l).

Tli dinh nghla cua ,k, ta thffy ngay du'ejcrang

Ek = f(xk) - 'ljJi(k}(xk+l) - (,k,xk - xk+l).

M~t khac do f ~ 'ljJi(k}va ,k E 8'ljJi(k}(xk+l) nen

Vy, f(y) ~ 'ljJi(k}(y)~ 'ljJi(k}(xk+l)+ (,k,y - xk+l). (2.63 )

Trang tru'ong hejp d~c bi~t, voi y = xk thl Ek~ O. TIT (2.63) ta co

Vy, f(y) ~ f(xk) + 'ljJi(k}(xk+l) - f(xk) + (rk,y - xk) + (,k,xk - xk+l),

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 - k+l )
8

, ' , x <-

/\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>I(xk,xk+l))

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

Tu (2.64), ta co ngay

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

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

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

va

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

(2.64)

(2.65)

(2.66)

(2.67)

(2.68)

(2.69)



40

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

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]::; ~kEk.

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 - xllh hQi t1;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 " .
+=

Vi L (jk < +00 nen ap d1;1ngmQtke't qua eua Correa va Lemareehal ([7], M~nh
k=l

d€ 1.3) thl toan bQ day {xk} hQi t1;1de'n x*. .



41

2.7 Ktt qua tinh toan 86.

f)~ thu~t gi

{

mill 7/Ji(y)+ 1kd
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
v(ji

(SPh,i

mill v + 1kd
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

Vi = max{f(yj) + (s(yj),y - yj)}.

Vi r.p(t) = ~(t - 1)2 + p,(t -logt - 1) nen ham ml,lc tieu cua bai toan (SPk,i) "cvc

ky" phi tuye'n va vi~c Hm nghi~m cua bai toan (SPk,i) co th~ ra't kho khan. Tuy
nhien, ne'u

P

dm=1 m

thl ham ml,lc tieu la tach du'QCva cach gi

ng~u cua no. V(ji mQi m, ta d~t Zm= ;7: va Z = (zm) thl bai toan (SPk,i) co th~Tn

vie't l(,li nhu' sail

(MSPh,i

p

mm v + L amr.p(zm),
m=1

(sj Z) - v < b., - J' j=0,...,i-1,

Z E JR~+,

t d ' - ~ (
k

)2 j - ( j ) k - 1
'

b. - ( ( j ) j ) - f( j ) '-

rang 0 am - Ak xm , sm - S Y xm' m - ,..., p va J - s y ,y y, J -

1,..., i-I. Ham Lagrange tu'dng ling v(ji bai toan (MSP)k,i la
p i-I

L(v, z, p,)= v + L amr.p(Zm) + LP,j [(sj, z) - v - bj]

m=1 j=O

va ham d6i ng~u la

d(p,) = inf{L(v,z,p,) I v E JR,z > O}

!

inf{t amr.p(zm)+ ~p'j[(sj,z) - bj]}- z>o
- m=1 j=O

-00

i-I

ne'u L p,j = 1,
j=1

ngu'Qc l(,li.



42

Do v~y b~titmln d6i ng§:u tu'ong ling Ia

{

max d(~),

L ~j = 1
(D)

~j2::0,j=O"",i-l,

trong do
P i-I

d(~) = L dm(~) - L ~jbj,
m=I j=O

i-I

dm(~) = inf{ G:m<P(Zm)+ L ~jstnzm I Zm > a},
j=O


Hon nua, m6i ham dm la khcl vi va

Vdm(~) = (stnz~ - bj)O~j~i-I'

i-I

trong do z~ = arg min{ G:m a}.
j=O

Vi (D) Ia bai tmln trail nen ham mvc tieu cua no de u'oc 1u'<;Jng,ta co th~ sa

dvng ba't ky phu'ong phap c6 di~n nao d~ gicli no. GQi ~* la nghit%mcua bai

toan (D) thl z* = (z~) trong do

i-I

z:n = arg mill {G:mj=O

va
i-I i-I

* -
(~ * j *) ~ *b '

v - ~ ~j s ,Z - ~ ~j J
j=O j=O

la nghit%mcua bai toan (SPk,i). Th~t v~y, do di€u kit%ndQ It%chbli ta co

i-I

L~;[(sj, z*) - v* - bj] = 0,
j=O

i-I

Vi L~; = 1 nen
j=o

i-I i-I i-I
* - ~ * * -

(~ * j *) ~ *b '
v - ~~jV - ~~jS ,Z - ~~j J'

j=o j=o j=o



43

Thi dl;l: f(x) = max{fj(x) = xTQ(j)x + (cj? x,j = 1,..., 5}
yoi

Q~k = etcos(ik)sinj, i < k

Qii = *Isinjl+ L IQikl
i=/=k

cj = e}sin(ij),i=l,...,n.

ChQn:

xQ = (1 1 1 1 1 1 1 1 1 1 )

A = 0.1000

m = 0.4000

Tieu chuffn dung:

Ilxold - xl12~ 0.001000

Gi
Ke't qua:

2 61.74627860 1
3 16.82191257 4
4 7.51790296 2
5 2.74151481 4
6 1.62981397 3
7 0.32026335 4
8 0.02442314 5
9 -0.06580683 6
10 -0.14407693 10
11 -0.16407434 8
12 -0.17470473 9
13 -0.17881529 8
14 -0.18039020 13
15 -0.18219034 10
16 -0.18263927 14
17 -0.17967192 40
18 -0.18234564 13
19 -0.18315244 11
20 -0.18042678 40

Solution:

0.0000 0.0006 0.0121 0.0363 0.0773 0.0000 0.0721 0.0744 0.0459 0.0189

f = -0.18042678