p chi Khoa ligc Trudng Dgi hgc Cdn Tha Phdn A: Khoa hgc Tv nhien, Cong «gA# va Moi trudng: 35 (2014): 111-118^
Tap chl Khoa hoc Tru'dng Oai hpc Can Thcf website: sj.ctu.edu.vn
T A P B I E N P H A N T I E M C A N C A P H A I V A U K G D U N G Le Thanh Tiing'
' Khoa Khoa hgc Tu nhiin, TrudngDgi hgc Cdn Tha ThSng tin chung:
Ngdy nhdn: 03/09/2014 Ngdy chdp nhdn: 29/12/2014 Title:
Asymptotic second-order variational sets and applications
Tir khda:
Tgp biin phdn, dgo hdm tiim cgn cdp hai, tdp biin phdn tiim can cdp hai, bdi todn toi uu da trj, dieu kiin loi uu cdp hai Keywords:
Variational sets, asymptotic second-order derivatives, variational asymptotic second- order sets, set-valued optimization problems, second-order oplimalily conditions
ABSTRACT
By combining variational sets, proposed by Khanh and Tuan in 2008, with asymptotic second-order derivative defined from asymptotic second-order cone, presented by Penol in 1998, we propose a new definition, asymptotic second-order variational sets, establish some their calculus rules and apply them to establish the optimal conditions for set-valued optimization problems.
TOM T A T
Kit hgp giQa khdi niim tgp biin phdn dugc dinh nghia bdi Khdnh vd Tudn ndm 2008 vd khdi niim dgo hdm tiim c4n xdy dung tir non tiim can, dugc trinh bdy bdi Penot ndm 1998, chung toi dua ra khdi niim mdi la khdi ni^m tgp bien phdn tiem can, khdo sdt mgt so phep todn cua chung vd Ong dung ldp biin phdn tiem can ndy di xet dieu kiin toi uu cua bdi todn toi uu da trj.
1 MO BAU
Ra ddi vio nhfhig nim 30 cda thi ky XX, gidi tieh da frj cd mgt vai frd quan frgng frong toin hgc vd trong cdc dng dyng cda toan hgc nhu ly thuydt tdi uu, 1;^ thuydt dilu khiln, van tru hgc, phuong frinh vi phdn, phuang trinh dgo ham ridng,... vi ed nhieu hdm sd frong thyc te Id hdm sd da fri, nghTa Id nhgn gii fri la mdt tap hgp. Khi khdo sdt hdm da trj, cdc logi dgo ham da tri li mdt cdng cy quan frgng cho vide nghidn eihi nhieu van dd khae nhau nhu; cie didu kidn tdi uu, cac dinh ly ve hdm ngugc, him in, sy dn dinh nghiem, tinh duy nhdt nghidm,... frong nhieu md hinh toin bge khic nhau.
M$c du, cd mgt sd loai dao him da fri dugc sd dyng nhidu nhu ddi dgo hdm Mordukbovich, dgo hdm contingent,... nhung frong mOt sd trudng hgp vin ed nhflng loai dgo hdm khdc cd thd sd dyng thugn Igi hon. Trong (Khdnh vi Tudn, 2008) da
dua ra khdi niem tip biln phdn chda dugc nhilu logi dgo hdm da fri nen dp dung thu§n lgi khi xdy dyng eic dilu kien toi uu cua bii todn tii im da trj so vdi each su dung dgo ham da tri contingent vi mgt so logi dgo ham da fri khdc. Cdc phdp todn cila tip bien phdn vd mdt sd iing dyng cdc ph^p todn cda tip bien phin da dugc chdng tdi khdo sdt frong (Anh et al., 2011). Khi khdo sat ldp hdm khdng lien tyc, tip xip xi, dugc dinh nghTa bdi Jourani vi Thibault nam 1991, da dugc chiing tdi ip dyng hidu qud vdo xet tinh duy nhit n ^ d m cda bdi toin bit dang thuc biln phan frong (Khinh vi Tung, 2012), va xet cic didu kidn toi uu cua cdc dang bdi toin toi uu hda don tri ttong (Khanh vd TCmg, 2013), (Khdnh v i Tung, 2014), va da ttj frong (Tung, 2013). Chung tdi cimg di ndu ra cdc djnh nghTa mdi la dao him radial cip eao trong (Anh et al, 2011), radial-contingent cip cao frong (Diem et al, 2014) v i khio sit cic iing dyng cua chdng
Tgp chi Khoa hgc Trudng Dgi hgc Cdn Tho Phdn A: Khoa hgc Tg nhiin, Cdng nghi vd Mdi trudng: 35 (2014): 111-118
frong vide xet dilu kidn tii uu vi khao sat tinh dn dinh nghidm dgng dinh lugng. Mgt sd dang dgo ham da trj vi iing dung cua chung ed thi tham khio tiiem frong (Gutierrdz et al, 2009), (Li vi Zhai, 2013), (Wang et al, 2011). Trong (Penot, 1998) da dua ra khdi nidm ndn tidm can cip 2 v i dua ra vi dy cho thdy ndn tidm can cdp 2 khac vdi tap contingent cap 2. Dua fren ndn tidm can c ^ 2, (Kalashnikov et al, 2006) di dua ra khdi niem dgo hdm ti$m cin cdp 2 v i ip dung de xdy dyng dieu kien tdi uu eho bii toan tdi uu da fri. Cich xdy dyng dieu kidn tdi uu theo hudi^ tidp can ciia Dubovitskii-Milutin sii dung tren dao him tiem cdn cip 2 da dugc khdo sdt ttong (Khan vd Tammer, 2013). Mgt sd tinh chit cua ndn tidm can cap 2 dugc khao sdt frong (Giorgi et al, 2010).
Mgt dang dgo hdm tiem can dugc diing de khdo sit didu kidn toi uu cua nghiem huu hidu chdt trong (Li et 01,2012).
Trong bdi bdo nay, kit hgp giua khdi nidm tap bidn phan vd khdi nidm dao him tidm can xay dyng td ndn tiera cdn, chung tdi dua ra khai niem mdi la khai nidm t$p bidn phdn tiem can, khdo sat mgt sd phep toin eua chung vd iing dung tip bidn phdn tidm can niy d& xet dieu kien tdi uu cua bii toin toi uu da fri. Mgt sd vi dy cho thdy tip bien phin tidm cdn cd till ip dung hon trong vide xdy dyng dieu kidn tdi uu so vdi vide dung dao ham tiem can frong (Kalashnikov et al, 2006) va (Li el al, 2012).
2 KIEN THU'C C H U A N BI
x„ -> x^,3y„ e II(x„) :y„~^ y},va\ x—^^-^ nghTa Id X —> XoVi H{x) ^ 0 .
Gidi ban dudi theo Painleve-Kuratowski ciia inh xa H khi X hdi ty ve x^ duge xdc dinh bdi Limuif H{x) = {y e y I VJ:„ e domH:
x„ -> x^, 3y^ G H(xJ :>'„->•>'}.
Vdi A^X, int A, clA, BA ki hieu tuong iing cho phan frong, bao ddng, bidn cua A . X* li khdng gian ddi ngau cua X v i B^.,Bj,la hinh ciu don vj ddng frong X, Y. Vdi x^ e X, C/(X(,) la tap cdc lin cin cua a^. Vdi A^X,UQX, cdc ndn sau thudng dugc dCmg coneA = {Aa | .i S 0, a^A}, c(me.^A = {Aa\A>0,a&A},A{u) = cone{A+u).
Trudc hdt, mdt sd khdi niem lien quan dugc nhac Igi nhu sau.
Djnh nghia 2.1 (Aubin vd Frankowska, 1990) Vdi S c:X,x^ eS,ueX, ndn contingent cua S tai x^ xdc dinh bdi 7'(5,j:o) = Limsup ~. S~x
Tip contingent eip 2 eua S tgi (X(„«)li T{S,x^, u) = Limsup'^"^^"^" .
- Cho F : X —*2^, dao him contingent cua F tgi (ig, y^) e GrF ji mdt anh xa da fri tii X vao ¥ Trong bdi bdo nay neu khdng gia thiet gi them,
chung tdi xet X, Y, Z l i eic khdng gian dinh chuin thyc, G QY,D c. Z\a cie ndn ldi ddng cd dinh
vdi phin frong khae ring. Vdi inh xa^da fri^^^^JhQa^g^-^^i^.J/o) = T{GrF,{x^,y^)). Dgo hdm /? : X —> 2^, tip xac dinh, dd thi, frdn dd tin cua contingent eip 2 eua Ftgi (x^, jQ)tuangiing vdi H tuong ling dugc djnh nghTa nhu sau
domH = {XGX : H(x) * 0 } , GrH = {{x,y) eX>cY:ye B{x)}, epiH = {{x, y)eXxY:ye H{x) + C).
Anh xg profile cua H, ki hieu H^, djnli nghia bdi H^{x)==H{x)-{-C.
Gidi hgn tren theo Painleve-Kuratowski cua dnh xg i / khi X hgi tu vl x^ duge xdc djnb bdi Limsup H{x) = {ysY\3x„e domH:
(«j,7)J e XxY Id dnh xg da tti td Xvdo Ythda GrD'F{x„y„u,v) = T\GrF,{x^,yJ.{u^,vX-
- Dinh nghTa dgo ham contingent cip 1 va cdp 2 frong (ii) tuong duong vdi cdc dinh nghia sau
^^(^..VoXw) -Limsup-^^^ i - ^ ,
F{x^ +lu^+t^u')~ya -/v.
= Limsup—
Sinh nghia 2.2
Tgp chi Khoa hgc Trudng Dgi hgc Cdn Tha Phan A: Khoa hgc Tv nhien, Cong nghi vd Mdi trudng: 35 (2014): 111-11^
(Penot, 1998) Vdi S c X,x^ eS,ue X, ndn tidm cdn cdp 2 cua S tai {x^,u) dugc xac dinh
^^r(S,x^,^)= Limsup i z ^ J l f ^ .
- ((Kalashnikov et al, 2006), (Li et al, 2012)) Dgo bdm tidm cin cip 2 cua F tgi
(X(j,>'g)tuongdngvdi ( t i j , u j € X x y Idanhxa da fri tir X vio Y thda
GrDlF{x^,y^,u,,v^) = T^{GrF,(x^,y^),{u^,v^)).
Dinh nghia ndy tuong duong vdi DlF{x^,y^,u^,v^){u) =
F{x^ -\-ti\-\-Hu')-y^- tv^
Limsup rt (i,r)-.(0-.0*),^0
Vi dy sau cho thiy djnh nghTa ndn ti?m can cdp 2 cda Penot frong (Penot, 1998) khac vdi tgp contingent eip 2.
Vi du 2.!. (Penot, 1998) Cho X = R\S = {{x- y)eX\y = x^'^}. Khi do, vdi (aTgij/g) = (0;0),(Uj;Uj) = (Q;l), dl ding kilm fra duge
r'(s,(\,!/.),(«„«,)) = '3, r'(S,(i„,!/„),("„»,)) = {(•a;v) £R'\v>a}.
Djnh nghia 2.3. (Khanh va Tuin, 2008) Cho i? : X - . 2',(!„,!;,) 6 GrFva j,,o,,...,u,. , 6 7 . Tilp biln phSn logi 1 dirge dinh nghia nhu sau:
" ( - f . ^ i
»0
,y
«•) =
, f „ - FW
Limsup
" . - , ) = -y„- i", -
t F{x)-
...-r
y«- t'
-'»
-to,
1
Vuex.
- D ' F ( i j , ! ( , i , j , , O j ) , D V ( 3 : , , j ( , , i i , , t i J t r o n g tniong hpp t6ng quat la khac nhau.
= IlF(i„!,.)(<j),V«eX.
Cac nhan xdt fren the hien rd frong cdc vi dy sau.
Vf du 2.2 Cho X = Y = R,F:X^ 2^ xdc dinh bdi
[x^;+oo),khix > 0, ,khi x<0.
F{x) =
Khi do, vdi
(.\;y„) = (0;0) e Grf,(u,;t),) = {l;0),tac6 D''F(x^,y„,u^,v^){u) = [l;-l-oo),Vu e X, I / ' ( f , i „ , SJ , u,, 0,) = [C;+00).
Dodo,
n'F(x^,y^,u^,«^){tt)CV'(F,x^,y^,u,,v,),WeX.
Vf du 2.3. Cho X = Y = R,F:X^2''xic dinh boi F(x) = {y £ R\ y>x^ \/y = —x^}.
Khi do, voi
(i,;y,) = (0;0)sGrF,(!(,;v,) = (l;0),tac6 D'F(x„,y„u„v,)(u) = {veR\vilvv = -l), VueX, EfF(x^,y^,u^,v,Xu)= {vs«|v>0},Vu€X
Dodo,
D'F(x^,y^,u^,v,),DlF{x„,y^,u^,v^)tmng lru5ng hop t6ng quat la khac nhau.
3 TAP BIEN PHAN TIfM CAN CAP HAI VA tfNG DUNG
Kk hop gifla khai niem tap biSn ph&i trong (Khanh va Tu&, 2008) khai ni^m dao Mm ti^m can xSy dung tii non tiem c|n ttong (Penot, 1998), chiing toi dua ra khai niem mdi la khai nigm tip biln phan tiem c|n tiong dmh nghTa sau.
Djnh Dghia 3.1 Cho
F:X-^2'Xx„,yJsGrFyi v, s K . Tap bien phSn tiem can cip 2 dupe dinh nghia nhu sau:
Tgp chi Khoa hgc TrudngDgi hgc Can Tha Phdn A: Khoa hgc Tu nhiin. Cdng nghi vd Mdi iruang: 35 (2014): 111-118
Fix).
Limsup
t i . ' H ( o - , o - ) . i ^ o j — ^
Tu dinh nghia Limsup, cic tinh chit sau suy ra de ddng.
Menh d l 3.1
I^A^,x„y„v,Xu)^V^{F,x„y„v,),Vu^X, - VXF,x„y„O)=V\F,x„y„0)=V{F,x„y,l
K'(F>x„y„0) =
{y^y\ lUninf d(y^+tv,+rty,F{x)) = 0}' r'{F,x^,y^,v^) = {veY\ 3(i„,r ) - (0^0-),
^ - > a , 3 x „ — ^ x , 3 v „ - > - r^
^(F,^o'y«A) = {" e 5'l 3((.,r) -*{tt,0*), .2--.0,3a; '^-•1,39 e F{x ) ,
J. " n n
Vidu3.1. Cho
A-= 7 = «, f : X - > 2'" xac dinh boi Fixy^iysRlyi-x"}.
Khi do,
voi (i„;j<„) = (0;0)6GrF,(a,;V|) = (l;0), ta co i5,'f (x„y„.!<„v,X") = R..VusX.
y'F(.Xt,yo,v,) = R.
vdd ( i , , s J = (0,(0;0))£GrF,Oj=(l;0), ta cd l'"(J^.\,S.) = «,x{0},I/'(F,i.,s.,.,) = {(0;0)}, V;(F,j:„,j,.,t.,) = {0}xJi^.
Tiong phan tiep theo, mOt s6 phep toan don gian cua t ^ bien phan tiem c|n duoc khao sat.
Minh Al 3.2.
Cho F : X ^ 2 ' ' , ! = !,...,/t, tl, e y,(i„,!/,) e Gri?,. = l,...,/t. Khi do,
(=] 1=1
Chiing minh:
Voi JCsU^ffiJi.^.liitan t?i i„,y^V;(.F^,y^.y„^').
Theo dinh nghia tSp bien phan tiem can, ton tai (i„,r)—»((y',0*),——*0, a;„—'^—i3j, va y„ —+ysao cho % +/„v, +f,;;j', € F^ (xJ c ( J ^ ( x , ) , Vn.
I
Dodo, J ' e ^ i ' d J ' ^ . ^ o . ^ . i ' i ) -
1.1
Nguoclaii, voi 31 e V^^F^,x^,yg,v^),t6ntai
(I.,/-.)->(0*,0*),-!-^0,x, u-?
sao cho 3'o +/„v, +t„r„y„ &\^FXx„\yn. Khi dd, Dinh nghia Up Win phan ti|m can cip hai la \i,at^ i^ sao cho :)'o+'.v,+r,r,7, eFj^(j,),VH.
khac biet so vdi t|p bien phan cap hai duoc minh
hga trong vldu duoi day. O^^,^^ y<^v;{F^.x„y^,v,)^(]V^(F„x„,y„v,).
Vfdu3.2. Cho
X = R,Y = R-,F:X ^ 2' xac dinh boi
F(x) = Kbi do,
{(0,0)},khi 1 = 0,
l ( — , — ) ) , k h i I = i , n = 1,2,...
' V " n' »
Menh de 3.2.
Cho F :JSr->2>',> = ],...,«:,
», € y,(3:,,li,) £ GrF,i = !,..,«:. Khi do,
K'(n^.V!'..»i)ch''.'(^.V!'..'',)-
1=1 1=1
Chihig minh:
Tap chi Khoa hpc TrudngDgi h^K Cdn Tha Phdn A-Khoa hoc Tif nhiin. Cong nghi va Udi Irudng: 35 (2014): liJ-U8*
Vcri ye[yi(F„^,y^v,\-ii ^€(f(i;,^,y„,v,),Vi Khi dd, vdi mgi r, t6n tai (/„);)->(0*,0*),-=.->0, I , — ^ A : „ va
r^
y„ ->>'sao cho yf,+t„Vj +t.f„y„ e^(x„),Vn.
Do do, tdn tai ('.,'-.)->(0*,0*),-^->0, x„—M-^ va 7„ -> y sao cho
yu + '.''i + '.V, <^(\FXx,)yn.
Vi du dudi dSy cho thay bao ham thijrc ngugc Iai trong Menh de 3.2 khong xay la liong tiudng hgp tdng quat.
V[d(i3.3.
Cho X = y = « , F „ F , :A'->2'xac djnh nhu J[-l;l],tWA: = 0, r{0},«nx = 0, F.{x) = < F.(x) = \
[{0},tWi*0, • l[0;l],«rri*0, (x„;y„) = (0;0)6Gr-f;,1=1,2 va v, =1. Khi V:{F,.x„y„v,) = V;(F^,x„y,.v,) = R^
v;(.F„x„y„v,)r-.y,'CF„x„,y,.v,) = R.
Nhung, (F,nF,)(x) = {0}, nln v;(F,nF„x„y„v,) = 0.
De xay dung phep toan cOng cua tip biSn phan - tiSm c4n cip hai, gii thiet tiong dinh nghia sau ' dugc sii dung.
Djnh nghia 3.2 (i) Cho F : A ' - > 2 ' , ( x „ y , ) e G r F v a v , e y . T|p biln phan ke lifim can c3p 2 dugc dinh nghia nhu sau:
Vl(F.x„y.,^',)^
Liminf Z W r A z i . (ii) Anh xa F dugc ggi la cd tap pioto-bi6n phan tiem c|n d p hai t^i ix^.y^) e GrF tuong ling vdi v , € y nlu V;iF.x„y„v,) = !i(.F.x„y„v,).
M|nhae3.3. Cho
F ; : ; f - > 2 ^ r = l,...,t,
v,„...,v, J £ r,(.x„yi) eGrF„i = l,...,t,
domF, cdoniF^.i = 2,...,/t. Nlu F„i = 2 keo cac tap proto-bien phSn tiem c|n cap 2 tuong ung la F„'(F;,x„,;c„v,,),i = 2,...,i,thi
tK'(.F:,x„y,.y,,)^V,\f^F,.x,.-Zy,.f^v„).
Chung minh:
Xet cdc v, e V^{F^,XQ,y^,v^J,i = \,...,k. Vi Vi e K,"(F|,J:O,3',,V, , ) , nen tdn tgi a „ ' O - ^ ( 0 ' > 0 ' ) ' - ^ 0 , x^-^x, vi y,„ e F,{x„) sao cho l i m — ( y i „ - y i - ' n V , „ ) = v,.
"^^ r,t„
Vi i^,i = 2,...,itcd cdc tgp proto-bien phdn tidm can eip 2 tuong dng Id V^{F^,XQ,y,v^,),i=2,...,k v i chmF, czchnF^,i = Z-,k, nen tdn tgi y,„ s F,{x„),i = 2,..., k sao cho
lim iy,„~yi "i,,^,„) = v„i = 2,...,k.
Dodo,
(Ly..»-ily' ''..ll\^)^'L^r
Vdy, t,v.€V^HZF„x„Y.y„j]v,^,).
•.^^^^=^Biy_gid,-~su--dyng t§p bien phan tidm cgn eip hai, chung tdi xay dyng didu kien tdi uu cho m$t sd dgng bii todn tdi uu hda.
Cho X I ' l a cic khdng gian djnh chuin thyc, C c r li eic ndn ldi ddng cd dinh vdi phin frong khic rdng, anh xa da fri F : AT -> 2*^,5 <= X. Xet bii toan tdi uu fren tip:
(P) minF(x), vdi X S ^ " .
Dilm (xc^o) e GrF li nghidm h&u hidu yen dia phuang cda (P) nlu ldn tai mdt lin cdn U cda x^sao cho ( F ( 5 n t / ) - > ' „ ) n ( - u i t C ) = 0 v d i F{SnU)= U Fix).
Tgp chi Khoa hgc Trudng Dgi hgc Cdn Tho Phan A: Khoa hgc Tv nhiin. Cdng nghi vd Mdi trudng: 35 (2014): 111-118 Dd xiy dyng didu kien toi UU cua (P), bd dl sau tai mdt lan can U cua X^sao cbo dugc dung.
B6 d l 3.1. (Khanh vi Tudn, 2008) NIU K cXlk ndn ldi, ddng cd phin frong khac rong, 2 o ^ ~ ^ i ze-intcone(X^+Z(|), —(z„-Zo)->zva r„ -* 0*, thi z„ e -int.^ vdi« du lan.
Dinh Iy 3.1. Nlu (xQ,yfj)la nghiem hiiu hidu ydu dja phuang ciia (P) vdi mgi
V, e V'(/; ,x^,y^)n(-5C), ta cd
^'(^..-o.>'o.v,)n(-intC(v.)) = 0 , V^\F^,x„,y„,v,)n{-mtC{v,)) = 0.
Chung minh: Vdi
V, 6 F' {F^, Xo ,>-(,) n {-dC), ta sd ehiing minh
^;'(^*.>^o=)'o'V,)n(-intC(v,)) = 0 (1) (cdn F^(F,,.v„,:^„v,)n(-intC(v,)) = 0 chiing mmh tuong ty).
Gid thidt phin chung, tdn tai y e V^{F^,-ia,yo,v,)n{-intC(v,)). Khi dd, tdn tgi ( r „ , r J ^ ( 0 \ 0 * X - - > 0 , ; v , - ^ v a y„^F{x^)+Csao
l l
cho — (;'„ -y^ -/„v, )—>-y vdi _v e - int C(v,).
( F ( ^ n t / ) - ; ; o ) n ( - i n t C ) - 0 .
Djnh ly 3.2. Neu {Xi^,yg)\a nghiem hihi hidu ylu dia phuang cua(CP) vi z^ eG{Xfj)n-D, vdi mgi {v„w,)eV*({F,G)^,Xa,yo)n-diCxD(Za)),tac6
V\iF,G)^,x„y„(v„w,))n-mt{C(v,)xD{7,)) = 0, V;HF,G),,x„,y„,{v„w,))r.~mt(C{v,)xD(z^))^0.
Chirng minh: Vdi
{v,,^^0eV{{F,G)^,x„y,)ry-&(OxlX2,)^ta se chung minh vXiF,G),,x^,y^,(v„w,))r^-inl{Civ^)xD(z^y) = 0
(chdng rainh V\(F,G)„x^,ya,^v^,w^))r^-iat^C(v^) y-D{z^)) = 0 xem frong (Khinh va Tuan, 2008)).
G i i thidt phan chung, tdn tai y,z)BVl{{F,G)„x„y^,{v„wJ)r^-MM<:{v,)xD{z,)).
x„—^x^\a {y„,z„)B{F,G){x„) + Cy.D sao cho—iiy„,z„)-{ya,Za)-h(yx>'^\))^{y^^)
r„r„
vdi>'e-intC(v,)va zg-intZ)(Zo).
Do dd, —(•^""•^"-v.) -> y. Theo Bd dl 3.1, Dodo, J-C^" -''°-v,)-»:v. Theo B d d l 3.1, vdiwduldn, ^" •*'° e - i n t C ^ > - „ - ; ' o e - i n t C .
miu thudn vdi gid thilt, (Xo,yQ)ld nghiem hihi hi|u ylu dia phuong cua (P).
Cho j r , y , Z la cdc khdng gian dinh chuin thyc, C c K, Z) c Z la cae ndn ldi ddng ed duih vdi phin frong kliie rdng, a&ib xa da tri F:X-i'2^,G:X^2^,S(zX .Xet bdi todn tdi uu cd ring huge :
(CP) minf(x), vdi j : e 5 , G ( x ) n - Z > ? i 0 . Ggi A={xeS:G{x)r\-D7^0)l^ tip chip nb|n dugc cda (CP). Dilm (x^,ya)eGrFla nghidm hiiu hidu y l u dja phuong cua (CP) nlu tdn
Vi w^ e-int D{Zf,); tdn tgi d^sDva a t > 0 sao cho w, = '-a(d^ + ZQ ) . Khi do, tu
( z . - ^ Q - / „ " ' , ) = {z -'Z„+t„a{d,+z„)) r^t„ rj„
= (z„ +t„ad, -(1-/„Q:)ZO)
r„l„ 1 - t„a
i p dung Bd dd 3.1 (dat — = —, thi h^ -> 0*), vdi n du ldn,
Tap chi Khoa hoc Trudng Dgi hgc Cdn Tha
z„-htad, . „ . -^—-—-e-inlD:^z„-i-t„aa. e-
=> r. e - D - i n t D c - i n t f l .
Mat khac, vi ( ; ; „ z , ) E ( F , G ) ( i . ) + C x f l , nen tdn tji ( 7 „ r . ) £ ( F , G X i J , (c„d,)eCxD sao cho(y,„z,) = (J.,J,)+(c.,rf.).
Do do, vdi« dii ldn, ta cd
>'.-j4e-intC,G{x:.)r-.-fl*0 (vi z,e-irtD), mau thuin vdi gia diilt, (X{),7o)Ia nghiem hUu hi^u yeu dia phuong cua (CP).
Cdc vi du sau cho thiy cic dieu ki$n tdi uu dimg tip bidn phan tiem can ap dung tdt hon so vdi cdc dieu kien toi im diing dao ham ti^m c ^ .
V(dv3.4.
Cho X=r=R,S=X,C=R„ix„,y„)=(.0,Ol anh xa F:jr->2' xac djnh bdiF(x)={;;sR| j'>-j^}.Khi dd, V, sIXF„x„,y„Xn)r^~eC={0}.Vn eXMv, =0 vatacd
D'(F„x„y„u„v,Xu) = I>'(F„;t„ >•„«,,v,)(u) = R^,VueX, C'(F„i.,>.„«„v,)(!;)n(-lntC-{v,» = OiW.*o..>'»."i.i',X'')'->(-intC-{v,))*0,VueX
Dinh ly 3.1 trong (Kalashnikov et al., 2006), klidng thi dimg dl bac bd ket luan (x^jy^) khong phai la nghidm hiiu hieu ylu cua (P). Nhung vdi V, s >"(F.,A:„,j,.)n(-a(C><fl(2.)) = {0>,thi
>''('\,:to,y.,v,) = y^(.F„x„y„.v,) = R ,nto y\K,x„y,.v,)i^iaC=v;(F,.x„,y^v,)ry-]atC*e!.
Ap dung Dinh ly 3.1, (x„,j'„)khdng phai la nghiem hflu hieu ylu ciia (P).
Vldu 3.4.
Cho
Jr=J'=Z=«S=jr, C = B = .R.,(x„,;;J = (0,0),z„ = 0, anh X9 F : X -> 2*' xac dinh bdi
F{A:) = {;;e*|,)>a-jt'}, anh xa G : A - ^ 2 ^ xac dinh bdi G(jr) = {0}. Khi do. Vr^ €T(S,x,)=R, thi 7'(S,V^)=?W^.r^)=^, vdi 0eZia:ji,0Xi^)={0}
va vdi V, sfl(F„x„,j;„X"i)r-i-5C = {0>, thi dl
Phdn A: Khoa hoc Tv nhiin. Cdng nghi vd Mdi Irudng: 35 (2014): lll-lis'
dang kilra tia dugc cac tinh chit cua Dinh 1;^ 3.1 tiong (Li errr/., 2012) diu thda vdi
Z?(F,>4,;4,ir„v,X«)= 4«.^.>'..",.ViX")=«„
Vir e X. tiong do ky hieu DfFlx„y„u,.v,)^
F(.x„+tu^-l--t-u')-y„-tv^
,t:'o'vi'.
DliF(.x„y„a^
L i m i n f
<',0-»<o*,o*).--»o,«
t
. V i ) =
F(x,
• * . 2
+ '", +ir/rr')- rl
-y^- -tv^
Do do, khdng thd ip dung Dinh ly 3.1 frong (Li el ai, 2012) dd bac bd kdt luin (j:o,_y(,) khdng phai li nghidm hflu hidu ylu cda (CP) (vdi ring budc dang 0 e G{x), la trudng hgp die bidt cua G ( ; c ) n - / ) ^ 0 ) .
Vi A = {xeS:O^G{x)) = S. Ndn bii todn (CP) vdi ring budc dgng 0 e G{x) trung vdi (P).
Tuong ty nhu Vi dy 3.3, diing Djnh ly 3.1 cd till ket luin (xo,;vo)khdng phdi Id nghi?m hiru hieu ylu cda (CP).
4 KET LUAN
Trong bdi bao niy, khai ni?m tap bien phan tidm can cip hai da dugc dinh nghTa, khio sit mdt sd phep toan v i dng dyng vio xet didu kidn tdi uu eua mgt sd dgng bii todn tdi uu da fri. Cic vi du cho thiy tip bien phdn tidm c|n cdp hai khdc vdi t^p bien phdn cip hai va cd till ap dyng tdt hon khi^
xdt cic dilu kidn tdi uu so vdi diing dao him tidm can cap hai. Cac didu kidn tdi uu cho cdc dang nghiem hihi hidu khdc cd thd xdy dyng tuong ty.
Tgp bien phdn tidm can cd thd ung dyng vdo xet sy dn dinh nghiem dang dinh lugng tuong ty nhu t4p bien phin. Ngoai ra, cd thi dya vio tip biln phin cdp m, vdi m > 2 , dd md rgng tuong ty cho tgp bidn phdn tidm cin cdp m, vdi m > 2 .
TAI LI J u THAM KHAO
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