TRUCiNG DAI HOC DONG THAP Tgp chi Khoa hpc so 41 (12-2019)
DIEU KIEN CAN HUtJ HIEU CHO NGHIEM SIEU HU*U HIEU DIA PHU^OfNG CUA BAI TOAN CAN BANG VECTO CO RANG BUOC BAT DANG THlTC
T 6 N G QUAT VA AP DUNG '
• Tran Van Sti'"', Nguyen Thanh Phong", Tran Ngpc Qu6c'"\ Nguyen Thi Bich Laf^
Tdm tat
Trong bdi bdo ndy chung toi sie dung khdi niem dao ham Studniarski trong khdng gian Banach vdi ldp ham khdng tron de ihiet lap dieu kien cdn hieu hieu cho nghiem sieu hieu hieu dia phuong cua bai loan cdn bdng veclo co rang bupc tap vd bat ddng thuc long qudt. Ket qud thu duoc se dp dung true liip vdo bdi loan bdt ddng Ihiec bien phdn veclo vd tdi uu vecto cd chung rang buoc tap vd bdt ddng thiee tdng qudt.
Tir khda: Diiu kien cdn huu hieu; Bdi todn cdn bdng vecta; Bdi todn bdt ddng thiee biin phdn vecto; Bdi todn tdi uu vecto; Nghiem sieu him hiiu dia phuong; Dao ham Studniarski.
eho bai toan can bing vecto cd rang bupc, Ve dieu ki$n toi uu cho bai toan can bang vecto vdi mpt rang bupc tap va mpt rang bupc bit dang 1. Mff dau
Nam 1994, Blum va Oettii [2] da xay dyng ldp cae bai toan can bing vd hudng fren co sd tong quat hda cac bai toan ly thuyet trd choi khong hpp tac kieu Nash va bai toan bit dang thirc bien phan kilu vo hudng Nam 1997, Bianchi-Hadjisawas- Schaible [5] de xuat xay dung ldp cae bai toan can bang vecto tdng quat tren co sd md rpng bai toan can bang vd hudng va sau dd eae tae gia thu duoe cac ket qua ve dieu kien ton tai nghiem cho Idp bai toan can bang nay. Nam 2000, Ansari [1] nghien cuu tinh on dinh nghiem va i e u kien hiiu hieu cho bai toan can bang vecto va sau dd ap dung tryc tiep cac kit qua eho bai toan bat ding thiic bien phan vecto Nam 2010, Gong [5] da su dyng cdng cu eua giai tich khong fran va giai tich loi de thiet lap cac dieu kien can va dii toi uu eho nghiem hiJu hieu ylu, hiiu hieu Henig, huu hieu toan cue va sieu hiru hieu cho bai toan can bang vecto dya tren tinh loi tong quat ciia cae ham myc tieu va
thire tong quat theo ngdn ngii dao ham theo hudng (eu the dao ham Studniarski) la chua dupe xem xet tdi [6]
Nam 1986, Studniarski [8] de xuat khai niem dao ham Studniarski cap cao va ap dung cdng cy nay de thiet lap cac dieu kien cin va du hiiu hieu cho cyc tieu ch^t dia phuong vdi ldp cac ham khong fron frong cac bai toan toi uu hoa vecto va bat dang thirc bien phan vecto. Ket hpp nhan dinh nay vdi nhan dinh ben fren chung toi nhan thay dieu kien can hiru hieu cho cac lo?i nghiem hiiu hieu dia phuong eua bai toan can bang vecto tong quat vdi rang bupc tap va bat dang thue tong quat theo ngdn ngii dao ham Studniarski vdi Idp ham khdng frpn la ehua dupe nghien ciru frong khong gian vd han ehilu ciing nhu mdt so ap dung eiia chiing.
Muc dich ciia bai bao nay la sir dung cong ham rang bupc cung vdi mpt so ap dung ket qua cy cua dao ham Studniarski d l xay dyng dieu thu dupe eho bai toan bat dang thire bien phan kien can hiru hieu cho nghiem sieu hiru hieu dia vecto va bai toan toi uu vecto. Nam 2011, Long- phuong ciia bai toan can bing vecto cd rang Huang-Peng [8] md rong kit qua cua Gong 2010 bupc tap va bit ding thuc tong quat eung vdi tir gia thiet tinh loi tong quat sang tinh loi suy mpt so ap dung ciia chiing. Kit qua nhan dupe rpng tong quat va nhan duoc eae dilu kien cin va ciia chimg tdi trong bai bao nay la mdi va ehua dii hiiu hieu cho nghiem hiru hieu Henig va sieu timg duoc nghien euu trude day. Trong tupng hihi hieu cua bai toan can bang vecto cimg vdi ap lai kit qua dat dupe nay ed the ap dyng de dung cho bai toan bit dang thiic bien phan vecto nghien ciru tinh on dinh nghiem cua bai toan va bai toan toi uu vecto Nam 2015, Khanh-Tung can bang tham so va xay dung cae thuat toan so [7] nghien ciiu dilu kien toi uu va tinh doi ngau tim nghiem toi uu eho Idp bai toan can bing vecto ndi ehung va ldp bai toan toi uu hda vecto ''' Tnrong D^i hoc Quang Binh ndi rieng.
TRUCfNG DAI HOC DONG THAP Tap chi Khoa hoc so 41 (12-2019) 2. Kien thirc chuan bi
Cho X, Y va Z la cae khdng gian Banach thuc va C la mdt tap khae rong eiia X, trong do Y va Z dupe sap thu tu bdi cac ndn loi, ddng va cd phan frong khae rdng Q va S tuong img. Phan frong, bao ddng va bao ndn eua mdt tap eon A trongXdupe ky hieu tuong img bdi int4, cL4 vd cone(A), d day cone (A) = [ta. a&A, t > O}. Ky hieu I, II thay cho mpt "chuan" tiong mpi khdng gian Banach thyc va ky hieu x„ —> x nghia la
Iim x,^-x hay Iim x^ - x = 0, Tap hpp eae so tu nhien, so nguyen va so thyc duoc ky hieu lan lupt bdi N, Z va K. Ky hieu H - « thay cho n la sd ty nhien du ldn, va ky hieu /„ —> 0"^ thay cho mpt day sd thue duong (/„ )^^^ vdi giai han bang 0. Vdi mdi x^^X va S>0, hinh eau md tam XQ ban kinh 5 duoe ky hieu bdi
B(XQ, S) = ixeX: ||^-^o|| <^]^ va chiing la mpt tap md trong X. Khdng gian doi ngau topd cua X, Y va Z theo thir ty dupe ky hieu bdi X', Y* va Z' tuong img; cac ndn ddi ngau eiia QvaS duoc xae dinh Ian lupt bdi
Q'={^^Y':<^,q>>0 yq^Q}
va S^ ={r}&Z':<r},s>>Q V-se^}, Chii y ring cae ndn Q^ va S* la loi va ddng yeu*. Tua phan tiong cua ndn doi ngau Q" ky hieu Q^ va duoc xac dinh bdi
G ^ = { ^ e Q ^ < ^ , ^ > > 0 yq^Q\{0)}. Theo Gong [5] va Long et al. [8], mpt tap eon loi B khae rdng cua mpt ndn loi Q duoe gpi la mpt ea sd cua ndn Q neu 0 ^ cl{B) va Q = cone(5) Ta de dang kiem tia tya phan trong Q" ^(fi neu va chi neu ndn loi Q cd mot co sd loi B. Nhu vay neu B la co sd loi eua ndn g thi 0 ^ cl{B). Theo mot dinh li tach cae tap loi rdi nhau {O} va cl(fl) (xem Rockarfeller [10]), ton tai philm ham tuyen tinh lien tuc / g ' y —?^M vdi fg^O sao eho rg=M{f^{b): b&B}>0. Cac ky hieu
^ ' /fl> ' s se dupe ed dinh xuyen suot bai bao.
Ky hieu L{X, Y) la khdng gian cae anh xa tuyIn tinh bi chan tir X vao Y. Xet mpt anh xa vectp
T.X^L(X, Y). Khi do vdi moi XGK,TX la mdt phiem ham tuyen tinh bi ehan tir A'vao Y.
Trong bai bao nay chimg tdi nghien ciiu bai toan can bang vecto ed rang buoc tap va bat dang thiic tong quat (GVEP)' Tim XGK sao cho
F(x, x)^-mtQ (VXGK). Trong do F:XxX ^Y la mot song ham vecto thda man dieu kien F(x,x)-OyxGX; ham rang bupc g- X ->Z Tap chap nhan dupe eiia (GVEP) dupe ky hieu bdi K—{XGC g(x)e-S} De don gian, vdi mdi xeK, ta viet
F(1, K)=[JF{X, X) = |F(X, xy. XGK\.
xsK
Dinh nghia 2.1 ([5, 8]). Vecto XGK duoc gpi la mpt nghiem sieu hiiu hieu cua bai toan (GVEP) neu vdi mdi lan can V cua 0 frong Y, ton tai mdt lan can Uciia 0 trong X thda man
e o n e ( F ( x , . ^ ) ) n ( ; 7 - G ) c l > ^ (1) Neu thay K bdi K r\Byx, S) vdi so thye
^ > 0 thi XGK thda man (1) duoc gpi la nghiem sieu huu hieu dia phuong eua bai toan (GVEP) Nhu vay, neu xeK la nghiem sieu hiiu hieu cua bai toan (GVEP) thi xeK ciing la nghiem sieu hiiu hieu dia phuong ciia bai toan (GVEP), Do do, cac ket qua nghien cim v l dieu kien can hiiu hieu nlu dimg cho nghiem sieu hiiu hieu dia phuong thi ciing dimg eho nghiem sieu hiiu hieu eiia bai toan (GVEP).
Hai frudng hop dae biet cho bai toan (GVEP) bao gdm bai toan tdi uu vecto cd rang bupc bat ding thuc tdng quat (GVOP) va bai toan bat dang thirc bien phan vectp cd rang bude bat dang thiic tong quat ( G W I ) dupe dinh nghTa nhu sau.
Dinh nghia 2.2 ([5, 8]). Cho trude mdt anh xa f:X->-Y. Neu song ham
F(x,y)=fiy)-f(x) 'v'x,y&X va neu xeK la nghiem sieu hiiu hieu dia phupng eua (GVEP) thi x&K dupe gpi la nghiem sieu hiiu bieu dia
TRUcnsiG DAI HOC DONG THAP Tgp chi Khoa hpc so 41 (12-2019) phuong cua (GVOP) Trong trudng hpp nay ta
gpi K la tap chap nhan duoe eua (GVOP) Dinh nghia 2.3 ([5, 8]). N l u F{^,y) -<Tx, y-x>\fx,y&X va neu x^K la nghiem sieu hiiu hieu dia phucmg cua bai toan (GVEP) thi ji: e A^ duoe goi la nghiem sieu hiiu hi?u dia phuong eua ( G W I ) . Tap K cdn goi la chap nhan dupe cua ( G W I ) .
Tiep theo chimg tdi nhic lai cac dinh nghTa quan frpng sau:
Dinh nghia 2.4 ([9, U]). Cho anh xa / ' A" —> 7 va cac X,VGX, m > 1 (m e N ) Dao ham Studniarski cap m cua / tai dilm (x, vl duoc ky hieu nhu dgf{x;v) va duoe xae dinh bdi
frx + lu)-f(x) d7f(x;v) = Hm —^^ ^ ^ ^
neu gidi ban tdn tai. Trong trudng hop m ^ 1, ta Viet dsf(x,v) thay eho dlf(x;v). Ngoai ra, nlu
/ la Lipscbitz dia phuong thi f{x + tv]~fG:)
dsfix,v)^lim—^ ^ ^ ^ Day chinh la dao ham theo hudng cd diin eua / tai x theo hudng V, Neu dao ham Studniarski cap 1 ton tai, nd keo theo d&o ham theo hudng cd diln tdn tai,
Khi dinh nghTa dao ham Studniarski cap 1 theo nghTa "limsup" va "liminf trimg nhau thi dao ham nay chinh la dao ham theo hudng Dim fren va dong thdi eiing la dao ham theo hudng Dmi dudi, Tuy nhien d day chiing toi dinh nghTa dao ham Studniarski cip 1 theo nghTa "Iim" cd mpt khd khan do la sy hpi tu eua day thuong
f[x + lu)-f(x
Dinh nghia 2.6 ([4, 9]). Ndn tiep lien phan frong ciia tap A^X tai diem x e clA la
inA.x)^[v^X:3l„^0- saochoVv.,^v. x+(.v„ s.< (V«-i)j K y h i e u
[T(A,x) = ^v&X-3t„^0^ sao cho x + t.,veA (V«-«>)}, Ta cd su bao ham
IT(A, x) c IT(A, x) <= T(A, x),
Mdt dae tnmg tupng duong cua non tiep lien do Giorgi va Guerraggio [4] eung cap dupe phat bieu nhu sau.
Menh de 2.1 ([4]). Ndn tilp lien eiia tip Ad X tai diem xecLi la
T(,A.x) = \v^X 3x„^A\[x], x„->x sao cho / ' ^ ^ n [ U (o).
3. Ket qua mdi cua bai bao
Sii dung cdng cy chinh la dao ham Studniarski tiong khong gian Banach va cac non tiep hen, ndn tiep lien phan frong doi vdi mpt tap tai mpt diem cho trude, frong tieu muc nay chiing tdi eung cap cae dieu kien can hiiu hieu co ban va ddi ngau cho nghiem sieu hiiu hieu va sieu huu hieu dia phuong ciia bai toan (GVEP) Kit qua thu dupe se ap dung true tiep cho hai bai toan neng ciia (GVEP) do la (GVOP) va ( G W I ) .
Dinh Ii 3.1. Cho x^K thda man dieu kien can bing Fix, x\ = 0 va B \a. eo sd loi, ddng va bi ehan cua ndn Q. Gia su cac dao ham Studniarski d^F{x, x, v) va d^g{x\ v) ton tai theo mpi phuong v&X. Khi do, neu x^K M mpt nghiem sieu hiiu hieu dia phuong eua bai toan (GVEP) thi ton tai mpt lan can loi va can doi f/ ciia 0 vdi f; c {>' e 7 • \fs{y)\ < ^ \ thoa man khi W->V k b d n g p b a i luc d,F[x,x.nc,x)r^\u^X d,glx.«)^-mtS}]r\(-intconeiU+B))=^
Chimg minh. Cho XGK la mot nghi?m sieu hiiu hieu dia phuong eua bai toan (GVEP).
Theo Long et al ([8], Remark 2,1), ton tai mpt lan can loi va can doi U eua 0 vdi nao ciing xay ra bdi vi khdng gian Y khdng hiiu
ban va ham / la tiiy y. Vi vay, trong mpi phat bieu cua bai toan chimg toi luon gia thuyet cac d&o ham Stiidniarski cap 1 tai dilm dudi su xem xet luon ton tai theo moi phuong
Djnh nghia 2.5 ([4, 9]). Ndn tiep Hen cua tap AczX tai dilm x G clA la
T(A,x) = ivsX3l„>0,3x,^A. ):„-»iiaocAo/,(i„ - x)-» vj
i 7 c | j ' £ 7 ; | / , M | < | j thoa [F(x,Kn,B(x,S)))f]{-\mcone(U + B)) = ^. Djt Q*=clcom(B + U). Khi do Q* la non loi (Jong
TRUCJNG DAI HOC DONG THAP Tap chi Khoa hoc sd 41 (12-2019)
va nhpn trong Y thda man Q\{0} a intQ* (xem [4, 6]), De dang c6 dupe
coneI^F(x, K n B(x, S))) f] ( - i n t Q * ) ^ ^ . Dieu nay dan den
F(x-,Kr^B{x,S))r\{-'mtQ'')^t^ (2) Ta kiem tra
d,F[x,x,nC,~x)r,{u<^X rf,g(x,«)e-int5})n(-mtO*) = ,* (3) That vay, neu (3) khong diing thi ta tim dupe mpt hudng VGT(C,X) sao eho d^g{x;v)G-\ntS va dgF(x, x;v)G-\ntQ*. Bdi vi intS" khdng ehira 0 nen v^O Ap dung Menh dl 2 1, tdn tai mpt day (x^) c C \ | x | , x„ -> x thda man
lim
™ P „ - J : | | HI"
Vdi moi so ty nhien
, Jk-4
(4)
IIHI
Ap dung (4) de ed ket
qua ^„ —> 0^ va v„ -> v. Theo each thiet lap tren, bang mpt tinh toan don gian cho ta
X ^ = X + / „ V „ G C ( V « > 1 ) . (5) Sii dung khai niem dao ham Studiniarski {T)inh nghTa 2.4), ta ed
FIX, X + I„U)-FIX, X\
dsF{x, x; v) = lim -
dsg{x;v)=\im-
F{X.X,)~F{X.X) t
v)-s(^)
g ( ^ , ) - g ( ^ )
(6)
(7)
Do intS la tap ma, ket hop (5) va (7), ton tai so thuc dirong^ > 0 sao cho
hay la.
-int5 iyn>A),
g ( x + ( „ v . ) G - S ( V n > ^ ) , (8)
dp ndn S loi, g{x)G-S, S + \niS = mxS<dS va / int 5 = int 5 Mat khae, ta eiing cd
^'„ -> X G Bix, S\, tdn tai mpt sp thyc B vdi B >
A sao eho x„ e BIX, 3) va cae quan he phu thude trong cac dieu kien rang bupc (5) va (8) dimg vdi mpi « > B Vay,
^ - ^ = x + / „ v „ G i : n S ( x , <5) ( V n > S ) . (9) Tuong tu nhu lap luan tien, ap dung Q*
thay cho K, va khong mat tinh tong quat
Fllc,x + t„v„)e-mtQ* {Vn>B) (10) Ket hop (2) va (9) dan den mpt sy mau thuan vdi (10) Vay (3) thda man keo theo CI,F{X, X; 7'(C, x)r~,[i, e X d^gix, ii) e-imS^'jC\(-mtcone(U-^-B)) = (*
Dinh li dupe chiing mmh diy du n Mdt he qua true tilp tii Dinh li 3.1 la kit qua sau.
He qua 3.2. Cho xeK thda man dieu kien can bing F(X, XI - 0 va 5 la eo sd loi, ddng va bi ehan ciia ndn Q. Gia su cac dao ham Studniarski d^F(x,x,v) va d^g(x;v) ton tai theo mpi phuong VGX. Khi dd, neu xeK la mot nghiem sieu hiiu hieu cua bai toan (GVEP) thi ton tai mdt lan can loi va can doi U eua 0 vdi UdlyGY: \f,(y)\ < ^ \ thda man
d,F[x.x,nC,~x)r.{usX d,g(x.„)e-,ntS}]n{-imcmL-{U + B))^^
Chieng minh Bdi vi mot nghiem sieu hiiu hieu eua bai toan (GVEP) ciing la mot nghiem sieu hiiu hieu dia phuong ciia bai toan do Tir Dinh li 3.1 ta ed dieu phai chiing mmh. n
Mot dilu kien hiru bieu phat bieu d dang ddi ngau cho nghiem sieu hiru hieu (dia phuong) eiia bai toan (GVEP) dya tren eae Dinh li 3,1 va He qua 3,2 nhu sau,
Dinh li 3.3. Cho XGK thda man dilu kien can bing fix, xj = 0 va B la co sd loi, dong va bi ehan cua ndn Q. Gia sir cac dao ham Studniarski d^F(x, x; v) va d^g(x; v) ton tai theo moi phuong v&X. Khi dd, neu XGK \a mpt nghiem sieu hiiu hieu dia phuong (hay
TRUCJNG DAI HOC DONG THAP Tap chi Khoa hocsd41 (12-2019) nghiem sieu hiiu hieu) eua bai toan (GVEP) thi
vdi mpi V e TiC, x) sao eho d^gix, vj G - i n t 5 , ton tai / G m t ( Q ^ ) va keS* thda man
f[d,F{x,x;v)Yk[d,g(x-v)]>Q. (11) Chieng minh. Ap dung kit qua thu dupe frong Dinh li 3.1 va He qua 3 2 tren, ton tai mpt lan can loi va can ddi U eiia 0 vdi
Ud[yeY:\f,(y)\<'AxUaman
rf,F(x,r.r{C,x)r.{„6X </,g(r,«)e-int5))n(-intawje([/ + 5)) = fll Vdi tiiy y V e T{C, x) sao cho d^gix, v\ G - i n t 5 , tacd
dsF{x,x, v) ^-'mXconeiU + B).
He qua la
(dsF(x,^;v), dsg(x\v)^{~'m\.com{U + B))y.{-intS) Ap dung djnb li tach trong Rockerfellar [10],ton tai / G T * va keZ', khdng ddng thdi bang khdng sao cho
\/qG'mtcone{U+B) Vz-Gint^
f[d,F(x, X. v))+f(q)+k[d,g(x, v)) + k{r) > 0> 0 02) Bdi •vi non cone(U + B) va 5 chiia goc nen bat ding thiic (11) dimg. De thay k^S^. Be kiem tra / G int(Q'), ta chimg minh
f(q)>0 yqGCone(U + BX / ^ O . (**) Dieu nay ed duoc la do (**) dimg keo theo / G [ c o n e ( U + B ) r \ { 0 } c i n t ( Q " ) (xem Bo dl 2 1 (i) va (iii) tiong [8]). De cd (**), vdi moi / >
0, ta cd iq G cone(U + B). Su dyng bat dang thirc (12), ta cd
f(dsF{x, x; v)) + //(^) > 0 V^ e coneiU + B) Mt > 0, hay tuong duong
l-'f{d,F{x,x,v))+nq)>0 Vq€Cone{U + B),'^t>0.in) Cho / ->+co trong (13) din den (**) thda man.
Chii y ring / ^ 0 la do ^^^(x, v) G - i n t 5 . Vay
dinh Ii dupe chimg minh. D Tilp theo chiing toi cung cap dilu kien can
hiru hieu eho nghiem sieu him hieu dia phuong cua bai toan (GVOP) va ( G W I ) .
Dinh h' 3.4. Cho B la mpt co sd loi, ddng va bi chan eua ndn Q, xeK va gia sii
Fix, y) ^ f(y) -fix) yx,y eX vdi / X ^>Y la mpt anh xa. Gia sir cac dao ham Studniarski d^f^x; v) va d^gix; v) ton tai theo moi phuong v G ^ . Neu xeK la mpt nghiem sieu hiiu hieu dia phuong (hay nghiem sieu huu hieu) ciia (GVEP) thi vdi moi V G r ( C , x) sao cho dgg(x,v\e-intS, ton tai / G i n t ( 2 * ) va keS'^ thda man
f[dj{x,v))+k(d,g{~x,v))>0^ (14) Chung minh Ta co F(x,y)=f(y)-f(x) \fx,ye.K vdi song ham F: KxK ^- Y Khi do, vdi moi x G .^ thoa man dieu kien can bang F ( x , xl - 0 va hon nua, F(x, x) = / ( x ) - / ( x ) Vx G AT Do vay, dao ham Studniarski d^f(x;v) tdn tai theo mpi phuong veX khi va ehi khi d&o ham Studniarski d^F(x, X, v) ton tai theo moi phuong veX.
Ngoai ra, d^f(x; v) - d^F{x, x; v) vdi mpi V G X , Ap dung Dinh li 3,2, ton tai cac phiem ham tuyen tinh lien tyc / G int(g^) va keS*
thda man (14), D Dinh li 3.5. Cho B la ca sd loi, ddng va bj
chan cua non Q, xeK va gia su
^(-^. JV)=<7x, > ' - x > Vx,>'GXvdi anh xa T.K-^L{X,Y). Gia sir dao ham Studniarski
^sSi^'^ ^) ton tai theo moi phuong VG JC Khi dd, neu XG A^ la mdt nghiem sieu hiiu hieu cBa phuong (hay nghiem sieu hiiu hieu) eiia bai toan (GVEP) thi vdi mpi veT(C,^) sap cho dsg[x,v)e-mtS, tdn tai / G i n t ( 2 " ) va keS* thda man
f{Tx(v)) + k(d,g[x,vyj>0 (15) Chieng minh. Ta dinh nghTa F{x, y) ^<Tx, y-x>'^x,yeK vdi mpt song ham
TRUCJNG DAI HOC DONG THAP Tap chf Khoa hoc so 41 (12-2019) F:KxK—> Y. Lue nay vdi mpi xeK thda
man dieu kien can bing F(X, XI = 0 va ngoai ra, F{x, x) =<Tx, x-x> VxeK Do dd dao ham Studniarski d^F{x, x; v) luon ton tai theo moi phucmg V G X va hon nua Tx {y) = d^F{x, x, v) vdi mpi V G X Thep Dinh Ii 3.3, tdn tai cae phiem ham tuyIn tinh Hen tue / G \nt{Q^) va k€.S* thda man dieu kien (15) va chiing ta kit
thiic chimg minh dinh li D Chii y 3.6. Kit qua thu duoc ciia Dinh li 3.1,
3.3, 3.4, 3.5 va He qua 3.2 van cdn diing frong
la mpt Ian can loi can doi ciia 0, Theo Long et al.
[8], X = (0,0) la mot nghiem sieu hiiu hieu dia phuong ciia bai toan (GVEP), Mat khae vdi mpi
(v, w) G R^, Tinh toan tryc tiep ta dupe dsF{x,x,{v,w) = {v,w) va <s-^(x,(v,w)) = (-v,-w) Do do tat ca cac gia thiet eua Dinh li 3,3 duoe thda man. D l y ring T(C, x) = M^ Theo Dinh li 3.3, vdi moi (v,w) G R ^ thda man d^gix, v) - -(v, w) e -int R^, ton tai phiem ham tuyen tinh lien tye /={./;,/j)eintM^ va A = (O,0)eS"
sao eho / I t / ^ F l x , x; V M > 0 . That vay, bang trudng hgp ndn tilp lien T(C, x) bi buy bd va phuong phap thu tinic tiep, ta ludn chpn dupe cac chimg duoc thay the bdi ndn tilp lien phan tiong so / j > O, / j > O thi f ^ (0,0) va bon niia
/ ( C ? 5 F ( X , x;v)) = / j . v + / 2 , w > 0 . Vay Djnh H 3.3 dupe kiem tia day du.
4. Ket luan
Bai bao da chimg minh dupe ket qua ve dilu kien can hiiu hieu eho nghiem sieu hiiu hieu va nghiem sieu hiiu hieu dia phuong ciia bai toan can bing vecto cd rang bupc tap va bat dang thirc tong quat, Ket qua nay duoc ap dyng tryc tiep vao hai bai toan rieng do la bai toan toi uu hda
^ vecto va bai toan bat dang thirc bien phan vecto Khi do, tap chap nhan dupe cua (GVEP) la theo ngon ngii dao ham Studniarski vdi ldp ham IT{C, x) hay IT(C, x). Cudi cimg chiing tdi cung
cip mpt vi dy dl mo ta cho Dinh 11 3,3 tiong trudng hpp nghiem sieu hiiu hieu dia phuong.
Vi du 3.7. Xet bai toan can bang vecto cd rang buoc tap va bit dang thirc tong quat (GVEP) trong dd X-y = Z = M'. Q - 5 - K - , C^i\
],(,,.),(,,,)}.[-i,i]'
va J: = (0,0). Ta xet cac anh xa F(*, ), g-.X^S.' duoc dinh nghta tuong img boi f (J, <.',r)) = (Jt, y] {Vx.y e K), g<,x,y) = {-x.-y) {Vx.yeR).
X = . | ( l , 0 ) , ( l , l ) , | i , l Hien nhien mpt eo sd loi ddng va bi chan eiia ndn Q ludn ed dgng B = {b = (b^,b^)GRl: b^+b.^=\] Chimg tdi djnh ng iTa f/^Jx = (xi,x2)Ei
khong fron trong khdng gian Banach. Ket qua dat dupe la hoan toan mdi va duoc sii dyng cho viec nghien ciiu tinh on dinh nghiem eua bai toan can bang tham so va dimg trong viec thiet ke thuat toan so tim nghiem sieu hiiu hieu dia phuong cho bai toan can bang vecto ed rang bude bit ding thuc tdng quat trong tuong lai /
Tai lieu tham khao
[1], Q. H Ansari (2000), "Vector Equilibrium Problems and Vector Vanational Inequalities, in Vector Vanational Inequalities and Vector EquilibnaMathematical Theories", Edited by Prof F.
Giannessi, Kluwer Academic Publishers, Dordrecht-Boston-London, pp. 1-16,
[2]. M. Bianchi, N, Hadjisawas, S, Schaible (1997), "Vector equilibnum problems with generalized monotone biflmctions", y Optim. Theory Appl, (92), pp. 527-542
[3], E, Blum, W Oettii (1994), "From optimization and variational inequalities to equilibrium problems", Maf/j Stud, (63), pp. 127-149
[4], G Giorgi, A. Guerraggio (1992), On the notion of tangent cone in mathematical programming. Optimization, (25), pp 11-23
[5]. X. H. Gong (2010), "Sealarization and optimality conditions for vector equilibrium problems". Nonlinear Analysis, (73), pp. 3598-3612.
TRUCJNG DAI HOC DONG THAP Tap chi Khoa hpc so 41 (12-2019) [6]. A, D. loffe (1984), "Calculus of diru subdifferentials of functions and contingent coderivatives of set-valued maps". Nonlinear Analysis: Theory , Methods & Applications, 3 (5), pp.
517-539.
[7], P, Q. Khanh, N, M. Tung (2015), "Optimization conditions and duality for nonsmooth vector equilibnum problems with constiaints". Optimization, (64), pp, 1547-1575.
[8], X, J, Long, Y. Q Huang, Z, Y Peng (2011), "Optimality conditions for the Henig efficient solution of vector equilibnum problems with constraints", Optim. Lett., (5), pp. 717-728.
[9], D, V. Luu (2008), "Higher-order necessary and sufficient conditipns for strict Ipcal Pareto minima in terms of Studniarski's derivatives". Optimization, (57), pp, 593-605.
[10]. R. T, Rockafellar (1970), Convex Analysis, Princeton Univ. Press, Princeton [11], M. Studniarski (1986), Necessary and sufficient conditions for isolated local minima of nonsmootii functions, 5Z4My Optim., (24), pp 1044-1049.
NECESSARY EFFICIENCY CONDITIONS FOR THE LOCAL SUPEREFFICIENT SOLUTIONS OF VECTOR EQUILIBRIUM PROBLEMS
WITH GENERAL INEQUALITY CONSTRAINTS AND APPLICATIONS Summary
In this article, we use the concept of Studniarski's derivatives in Banach spaces with a class of non-smooth functions to establish necessary efficiency conditions for the local superefficient solution of vector equilibrium problem with a set constraint and a general inequality eonsfraint. The obtained results are directly applied to the vector variational inequality problem and the vector optimization problem with their common set and general inequality constiaints.
Keywords: Necessary efficiency conditions, vector equilibnum problems, vector optimizahon problems, vector variational inequality problems, local superefficient solutions, studniarski's derivatives,
Ngdynhdnbdi 2& 82019; Ngdy nhdn Igi: 01/11/2019; Ngdy duyet dang. 05/12/2019.
