fCJNG DAI Hex; DONG T H A P Tap chl Khoa hoc so 13 (6-2015)
pc
6 N P H A P T U Y E N T H E O Hl/dNG VA DIEU KIEN T 6 I tfU• ThS. Vo Dijfc Thinh'*' Tom tat
Trong bdi bdo ndy, chiing toi nghiin ciiu mgt sd tinh chdt cua non phdp tuyin Frechet theo ng vd ndn phdp tuyin qua gidi hgn theo hUdng. DSng thdi. chiing toi cdi bien dinh nghia cua ndn p tuyen qua gidi fign theo hudng vd nghien ctiu mgt sd tinh cfidt cua non cdi bien ndy. Sau dd, ng tdi dua ra nhiing vi du minh hga cho su khdc nhau giQa cdc ndn phdp tuyin theo hudng. Cudi g, chdng tdi xdy dung cdc kftdi ni$m dudi vi pfidn theo hUdng thdng qua cdc ndn pfidp tuyen theo ng. Bang cdch svc dtmg dudi vi phdn theo hudng, chiing toi chi ra diiu kien can cho nghiem tdi heo hudng c&a mgt bdi todn tdi Uu.
TU khda: Non phdp tuyin theo hudng, dudi vi phdn theo hudng, dieu kiin tdi Uu.
1. Gidi thi€u va tdng quan
L^ thuyet toi ifu la m6t Hnh vifc quan Nam 1994, B. S. Mordukhovich [5] da gidi lg cfla to^n hoc. Trong \^ thuydt t6'i ufu, cdc thieu cdc non phap tuy6'n nhif sau.
gid quan tam den bai todn sau (xem [2], gj^jj ^^^ 1 2 , Cho kh6ng gian Banach , [51, [61 va cdc tai M u tham khdo). ,^ '„ - , - . > - , ^^ r^--? • v A,M^
•^ ^'"^ ^ ^ X,s>0 va tan Q.<^X. Diem x GX aifdc iTun/(x), (P) ' -^
lg d6 f:X->R la ham m3ra liSn tuc difdi D lk tap con cfla khdng gian X.
Trong pham vi bai bao nky, chflng toi dac lg dilu kidn can cho nghiem cfla b&i toan
thfing qua dtfdi vi phSn theo htfdng. Chung lu6n gia thiet X la khSng gian Banach;
x,r) la hinh c l u tam x bdn Mnh r; B lk hinh 1 ddn vi cfla X va n(;(o'"):=arginin||x-j:J| la cdc dnh cua phep ehieu vuSng gdc tCf x^, 16n M6t tap n c . ^ dtfdc gpi lk dong dia tang xung quanh x n^u t6n tai 13n cSn U I X sao cho QniC/ la tSp dong. Ngoai ra, : k i h i e u C ^ . x — ^ x ^ vk x "'^ >x^ dtfcfc
gpi la mot s -pimp tuyen cfla O tai Xg G Q neu ( x ' , X - X ( i )
limsup-
T$p ta't ca cac s-phdp tuye'n cfla Q tai X(jei2 dtfdc ki hieu la N^(XQ,Q). Theo dinh nghia ta co
(x*,x-X(,) N^{x„n):= x* &X' |hmsup-
h nghia nhtf trong [3].
B§y giS, chflng toi nh^c lai mot so khai m sau.
Dinh nghTa 1.1. Cho khdng gian Banach vk lap DcX. Biim x^eD dtfcfc goi la liim tdi uu djapfiuang cfla bai todn (P) neu I tai sd thtfc r > 0 sao cho
/ ( X o ) < / ( x ) V x G 5 ( X o , r ) n D . (1.1)
h-4
Neu x^ifi thi chflng ta qui tfdc JV^(X(,,n):=0. N6'u s=0 thi chflng ta viet 7V'(Xo,n) thay iV(,(X(„Q) v^ gpi la ndn phdp tuyen Frechet.
Non pfidp tuyen qua gidi hgn cfla Q tai Xg dtfpc kl hieu 1^ N{Xf^,^) va xdc dinh nhtf sau:
N{x^, Q) := Lim sup N^ (x, H), trong do n^u F:X-
X vao Y thi
• 2 Id mot anh xa da tri tuf fveriadayj:
.ineNj
:hoa Sir pham ToiSn-Tin, Tnfdng Dai hpc Dong TMp.
LimsupF(..,. .
NhSn xet 1.3. V&i mgi Xf, e H c X ta cd Nix„Q)^N(x,,Q).
63
TRUONG Bid HOC DONG THAP
Ngoai ra, mot sS tinh cliat c i a ndn pliap tuygn Fr6chet va non phap qua gidi lian cung dUdc tic gii ctil ra [5]. Cu tli^ cliung ta cd cac menh de sau.
Menh a^ 1.4. Cho .\;..X, Id cdc khdng gian Banach, O, Id cde lap eon etia X,, n. Id cdc tdpconcia X,, i,^Q. va x,_eQ,. Khidotaco
(i)iV(j,n)=JV(x„n,)xJV():;,n;), (1.2)
(2)JV(l,n) = iV(x„n,)x«(j:;,£l,), (1.3) M6nh a^ 1.5. Neu n Id tap loi, x„ 6 Q vd U la mpt ldn can eUa x^ thl
A'K,n)={«-e,VK«'.«-^.>=so.v:«s"^c;}. d-'*'
Trong [5], B. S. Mordukhovich da dac tnmg tinh chft cua non phip tuye'n qua gidi han u-ong ichdng gian hilu han chieu. Tinh chft n^y giiip chung ta thuan ldi hdn trong viec m6 ti cSc ndn phap niy^n qua gidi han trong Ichdng gian hOu han chieu. Cu t h l chdng ta cd dinh li sau.
Dinh li 1.6. Gid su- x, e n c R" vdi Q. Id cdc tap ddng dia phuang quanh Xg. Khi dd ta cd cdc khdng dinh sau:
( l ) W ( i „ , n ) = LimsupW(A:,n), (2) JV(x„,n) =Limsup[cone(x-n(3:,n))], trong dd n(x,Q) Id tap cdc dnh cua phep chieu vudng goc tU X lin Q.
Tit c5c non phdp tuygn nay, B. S.
Morduldiovich [5] da gidi thieu khai niem doi dao hSm Frechet, dgi dao ham qua gidi han va dudi vi phSn qua gidi han,... Trf do, tdc gia dac tning cdc tinh chft nhu Lipschitz, gia Lipscliitz, m6tric chinh qui,... ciia dnh xa da tn thong qua cdc do'i dao h&m cung nhd xay dung cdc dieu ki6n cin va dieu kien dij cho nghiem toi tfu cua bai todn (P).
Ndm 2011-2012, F. Ginchev vd B. S.
Mordukhovich [21, [3] da gidi tiii6u non phdp luyen qua gidi han ttfdng d:ng vdi m6t tap thong
Tap chi Khoa hoc so 13 (6-;o)»f
qua Idiai niem non phap tuye'n Frechet tiidL:
d-ng vdi mot tap nhil sau.
Dinh nghla 1.7. Cho X la mot khong jii, Banach, £ > 0, Q, g la cdc tap con ciia X it X ^X. Khi dd vdi moi (5>0, Q^:=Q + SB. f5iem . v ' e X' dtfdc goi la mj
£ -phdp luyen lUang I'Olg rdi lap 0 cQa Jl
tx',x-x„)
x.eCl neu limsup -^j j — £ s
Tap ta't ca cdc s -phdp tuyen luong HIIJIM tap Q cua n tai -V, dtfdc ki hieu la .V.„(.i,.ai Theo dinh nghla ta co
7V^o(x„.n);=A',(-v„,n^^(x,+C„)). (IS:
Ne'u £ = 0 thi chiing ta vie't W„(i,,ai thay cho N^g(x„,n) va goi la non plidplm Frecliel luang ttng vdi tap Q ciia n tai i N6'u x„ i Cl thi chung ta dat 7V„(x„,n):=0 Non phdp luyen qua gidi hqn lumg rllij vi tdp Q cua n lai ,r„ e fJ ditdc xdc dinli nhllsal
N°(x,„a):= Limsup N,gXx,ai (LS
trong do limt/
Xp nghia la x - l e Q i i
•I'tf do. cdc tdc gia xay dtfng Idiai niei dtfdi vi phan theo htfdng va xay dtfng moi qui tac tlnh cho dtfdi vi phan theo hifdng q»
gidi han trong mot so trtfdng hdp cij 'h^-J'"!
[3], cac tdc gia da thiet lap cac dieu Idenl^
va dli cho mot diem x„ cho trtfdc la nghieml tfu dia phtfdng cho mfit trtfSng hdp cu llie •
bai todn (P). J Trong bai bao nay, tren cd sd cac kliJiB!^
non phap myen trong [2], [3], chiing toi i<&\
mot khai niem non phap tuye'n khac (cu llie^
(S > 0 bfli S>0). Sau do, chung loi xay m cac menh de. cdc vi du de so sanh non!»
tuyen mdi niiy vifi cac non phap luyen Irong Pl- 6 4
' t ^ G D A I HOC DONG THAP Tap chi Khoa hpc so 13 (6-2015) 2. K^'t qua chinh
Trong phan nay, chflng toi gidi thi6u mot i niem khdc ve non phap tuy^n qua gidi theo htfdng. Sau dd, chung toi trinh bay s^ tinh chlft cfla ndn phap tuyen do va dtfa i du chd'ng minh r^ng cdc tinh chat nay ng dflng cho ndn phdp tuy^n qua gidi han g [3], Trtfdc het, ta c6 dinh nghia sau.
Djnh nglua 2.1. Cho X la mot khdng gian ach, 0.,Q \k cac tap con cfla X vk
\X. K h i d d vdi moi 5>Q dat Qg-Q-^SB, ndn pfidp tuyen qua gidi hgn tucmg ihig vdi Q cfla n tai Xo G a dtfdc xdc dinh nhtf sau:
(2.1) NQ{X(^,0):= Limsup ^j.g {x,0).
Chfl ^ rang trong Dinh nghia 2.1, so 5 cd bkng 0 khdc vdi dieu kien ^ > 0 trong Dinh la 1.7. Vi du sau chi ra rang hai ndn phap
;'n trong Dinh ngMa 1.7 vd Dinh nghia 2.1 la c nhau,
V i d u 2.2. Xdt Q=EM{(xi,Xj)|;i^ <-|j(^ | } ,
= {(;Ci,X2)eQ|x,>0}, x = (0,0). Khi dd (x,f^) = {(x,,X2)|x,<X2<0}. Bay gid vdi ki (5>0 thi Cg^ =IR\ Do do nn(x+Cg^)=ii : h o b S ' t l d x — ^ ^ ^ ^ x t a c d :
{(Xj.Xj)IXj = -x^ +2x|',Xj >Xj}
n e u x - ( x ° , x ; ' ) ^(0,0);
(x;n)-^{(x^,Xj)lXj=X2 + 2x°,Xj<-Xj|
neu x - ( x ° , - x ° ) ^ ( 0 , 0 ) ;
|o} neu ngtfdc lai.
id
( x , n ) = {(x„x,)|x, =X2 <Ohay;c, = - x , <0}.
lg tinh todn trtfc tiep ta cd
Ng{x,a) = {{x„x^)\x^<Qr.x.,<^x,}.
Dtfdi day, chflng tdi xay diftig mdt sd' tinh : cfla ndn phdp tuyen theo Dinh nghia 2.1 lg tfng vdi tinh chSft (1.2) vd (1.3) cfla ndn p tuy^n qua gidi han. Chfl ^ r?lng, mdt s6 chkx se khdng dung cho ndn phdp hiy^'n I Dinh nghia 1.7 va khi dd chung tdi se xay g nhifng vi du minh hoa de lam rd stf khdc J niiy.
Dinh li 2.3. Gid sii Xj, J^, la hat khdng gian Banach, X=X^xX^,Q=QxQ^, n^n^xS\ vd x„ =(x^,xl')eQ. Khi dd ta co cdc khdng dinh sau.
(1) N^{x,,n) = N^(x,\a,)xNQ,ix°,a,).
(2) N^ix„n) = NQ^ix,\a,)xN^(xla,).
Chtfng minh. Vi ndn ti^n phdp tuyen khong phu thuoc vdo cac chuin ttfdng dtfdng trdn X^ va X^, chflng ta cd the co' dinh mdt chuin bat Id trdn cdc khdng gian nay va dinh nghla chuan trdn khdng gian tich X bdi
ll(^.^.)||=W+lhll-
v m bft ki £>0,<5>0,x = ( x „ X ; ) e f 2 ta lay x*=(n*,)^)eiV^g_(xi„n) vdi x^nX^.x^eXl, ta tun dtfdc lan can U =U^xU2 cua x sao cho X £ f i n ( x + C g ) r ^ f / t a c d
< x ' , x - i f > < 2 £ | | x - i f | | . D o d d
<x,*,x;-^>+<Xj,Xj-^><2s(||x,-^|| + | | x i - ^ | | ) vdi X, E n , (^ (x, + Cg^ ){^U„ vdi moi i = 1,2.
Chpn Xj = Xj. Khi dd vdi ba't ki e > 0 , x , eEn,r-,(:^+Cg^)r^Lf„ tacd
< x ' , x , - ^ > < 2 £ | | x , - ^ II.
Suy ra x' G N,^,. (i^, QJ ).
Chon X| = x^. Khi dd cho
^ > 0 , X j e Q 2 n i ( x , + C g ^ ^ ) n t / 2 t a c d
<xJ,Xj-Xj><2e||xj-Xj||.
Suy ra X^GTV^^Q ( X J , ^ ! ^ ) . x*GiV^^„ ( ^ i , ^ ) ^ ^ 2 £ 0 , (^2'^2). Bi^u nay suy ra N,^(x,atizN„^^(x„n,)xN^^(^,n,).
Ngtfcfc lai, liy
X* =(xi*,x3)eiV,^g^JJ^,Q,)xiV^^g_^(x5,a,), talim dtfdc cdc 13n cdn U^,U, cua x,,!^ sao cho (2.2) va (2.3) dtfdc thoa man. Bat (/:= f/, x f/j, khi dd vdi bft Id x = (X|,X2)Gnf-i(3c+Co )r^U thi X[ eQ,oGq+Cg^)nC/|,XjGn2n(x,+Cg^pr^C/2.
T a c d
<x,',xi-^>£3ellx,-^il; (x^,Xj-^) <3si| x , - x , 11.
D o d d
<x*,x-x)<3£(||xi-x, || + ||xj-Xj||) = 3 £ | | x - x | | . (2.2)
(2.3) Do dd
TB.UCSNG DAI H O C D O N G THAP Tap chi Khoa hoc so 13 (6-2o«
Suy ra x'GA'„g^(3c,n). Khi do vdi bat ki e>0, ta cd
*.,j,(i.l2)c*„^,t5i.Q,)x*„,j,(S„Q,)c*„,j,(i,n). ( 2 . 4 ) Cho £ = 0, X =X|, trong (2.4) ta dtfdc (1).
hiy gidi han trong (2.4) khi £ - > 0 , x - > x „ ta dtfdc (2). Do dd dinh li dtfdc chiing minh. D
Sau day, chiing toi se xdy dtfng mpt tinh chft ttfdng ttf cho ndn phdp tuyen qua gidi han theo htfdng thdng qua dinh li sau.
Djnh If 2.4. Gid si x„ G n c R" va
^ c I R " sao cho Q. vd (x^+Cp) Id cdc tap ddng dja phirtmg xung quanh Xg vd C^+CgizCQ. Khi do ta cd cdc khdng dinh sau.
(1) iVg(x„,n) = Limsup./Vg(x,n), ' ^ ' JVe(jr„Q)-Linisup[conc(A:-n(»,Qn(»,+Cg)))].
Chiihg minh. Trtfdc tien chdng ta chd'ng minh (1), nghia la chung ta cd thg Hy s,S = 0 u-ong Dinh nghia 2.1 ve ndn phdp tuygn theo htfdng cho tap ddng dia phtfdng trong khdng gian hifu ban chilu. Chd f rang
Afjtxi.Si)" Limsup W,^_(x,n)3LimsupiVg(x,n), Chung ta chi can chflng minh bao ham thflc ngtfdc lai. Cg dinh x' eA'j,(x„n) ta tun dtfdc cdc day S^ \0,s^\0, x^ "-^ >x„ va x ; ^ x * sao cho x;G.W,^g^(xj,n) vdi mpi
* s N . - V i X = r = K " vd cdc tap n , x „ + C e Id ddng dia phtfdng xung quanh x„ nen vdi moi
* e N cho bft ki a > 0 ton tai i f , e n ( x , + a x ; , n n ( x „ + C „ ) ) . Khi dd ta cd bd't ding thflc sau
h^axl^^.Ua^Wxll
Khdng mft tinh tgng qudt, gid sft chua'n dang xdt la chuan Euclid, ta cd
| | x . + « ; - w , f = | | x . - w , f + 2<.<x;,x,-„.>+a^|j;,;||\
Do dd ta cd
||x. -w,f <. 2a(x\w^ - x . ) ,Va > 0. (2.5)
6 6
Vi u', -> X, khi Q- \ 0 va xr G Af u n H ' t G n n ( x „ + C o ) c n n ( x „ + C g ^ ^ ) , suy
< ^ i . n - *t> ^ 2ffj j|w, - x\l. Ke't hdp vdi Q.
ta cd ||xt-w,j|<4a,.ffj, trong dd a,~a.i ra w, —^—>x„ khi jt -> +«.
Hdn ntfa. dat Uj" := xj + — (x, ~w.) latj a^
| K - J ^ i ' | s + = . *'^ ";->-v' khi t-^+=c. Taseei ra rang if," G W^ (ir,, £2) vdi mpi kefi. M viy, CO dinh A- G N va vdi mpi diem ciio insSi x e n r ^ ( n ' ^ + C p ) idii d d -ve-r^+Co+CpCjCj+f, t a c d
= (ai,xl +JCj -x,a^xl +x^-w^)-(x^ +^k^l-x,M\-ii
~{a^xl + x^ -w^,x-'W^)-{a^xl + x,^ -w,^.a^x\ + x^-,t|
= -2a,{wlx-w,) + \\x-i.,f.
Do dd ( w ' . . Y - x , ) < ||x-w,f \6\m
X G n n ( W j + C g ) . I2,(ii Vdi bd't ki £• > 0 la lim dtfdc lan cii
U{w^,s) cua w^ siio cho vdi moi x €[/(ii',.,f|
ta cd |x-w^||<2aj.£'. Vi vay yvl ^NQ{<.\,^\
vdi moi k&^.
SiJ dung dinh nghia cua ndn phap luyci qua gidi han ttfdng iJng vdi tap Q ta o x' e NQ (xp, n ) . Do dd (1) dtfdc chiJng minh,
D^ chd'ng minh (2), la can chiTng minh Limsup,??g(x,Q) = Limsiip[cone(x-n(*,an(x+Cj))t Quan sdt rSng A'p(5:.n) = A'(x,Qn{i+Cp Sfl dung chfl"ng minh trong Dinh li 1.6 [3] tac(
LimsupJV(jr,Qn(A:+Cg)) = Limsup[cone(.r-n(.v,Qn(.ttr D o d d t a c d
Limsup7^g(x,£J)=LimsLip[cone(.r-n(x,f2n(i-i-C;)!
Vay dinh li dtfdc chtfng minh.
Chfl J r^ng Dinh li 2.4 khong diing ^ ndn phdp tuyg'n theo htfdng qua gidi hanW Dinh nghia 1.1. That vay, ta xel vidu sau
LrcaNGDAI HOC DONG THAP Tgp chl Khoa hoc so 13 (6-2015) Vi du 2 ^ . Cho a va e nhtf trong Vi du
• KM dd vdi x = (0,0) thi theo ket qua trong du 2.2 ta thsTy rang Dinh li 2.4 khdng thda n cho ndn phdp tuyen theo htfdng qua gidi 1 trong Dinh nghla 1.7.
Dinh If 2.6. Gid sU H va Q Id cde tap con I X, U la mot ldn can cua diim x^ e Q sao
^ n r-i (XQ + Cg ) o C/ Id tap ldi. Khi dd vdi mgi iO tacd
,g(x„,n) = { x - G X ! < x ' . x - X o ) < e | | x - x , | |
• e n n ( X o + C g ) o C / } .
Chdhg minh. Chfl y rang bao ham thfl'c )" trong b i l u thfl'c tren dat dtfdc ttf dinh iia vdi hS't ki tap €l,Q vk bat ki ISn cdn U 1 Xj,. Chflng ta chi can chd'ng minh chieu Jdc lai khi nr-i(Xo+Cg)}nC/ la tdp l6i.
y gid, llfy bd't ki x* e 7 ^ ^ g ( x , n ) vd co dinh 'm x e f i i n ( X o + C g ) n f / . Khi dd ta cd : = X o + Q r ( x - X o ) e n n ( x o + C g ) r i t / vdi mpi : a < l do tinh l6i cfla tap £3r>(Xo4-Cg)ni7.
n ntfa, day x^ -> x^, khi a -> 0*. Lay bat ki
>0, ta cd {x',x^-XQ)<{e + y)\x^~x^\ khi or > 0. Dinh H dtfdc chflng minh. D
Sau ddy, chflng toi gidi thidu khai nidm 3i vi phdn theo htfdng thdng qua khdi ni6m a phap tuyd'n theo htfdng va dp dung dtfdi vi in theo htfdng d^ tim dieu kidn c i n cho lidm t^i tfu cua bdi todn (P). Trtfdc h^t, ing toi se gidi thidu khai nidm ctfc tiiu ttg dttg vdi mdt tap nhtf sau.
Dinh nghia 2.8. Cho / Id mgt ham nuTa 1 tuc dtfdi tCf khong gian Banach X vdo R.
\m Xf^sX dtfdc gpi la cue tiiu dia phuang ng ling vdi Q(zX\{0} cua / nd'u t6n tai
0,(5 > 0 sao cho
/ ( x ) > / ( x j V x e Z ) g ( x „ ^ , r ) , igdd£>e(Xo,^,/-):=(j:o+Cg^)n5(Xo,r).
Tid'p theo, chung tdi se gidi thidu khdi m dtfdi vi phan theo htfdng thdng qua khdi
nidm ndn phdp tuyd'n theo htfdng. Sau dd chflng tdi sfl dung dtfdi vi phdn theo htfdng de dac tnmg dieu kidn cin cho nghidm ctfc tieu theo htfdng cfla mdt ham sd / .
Cho f:X->lx la mdt hdm nufa Udn tuc dtfdi. Khi dd ta cd cac ki hidu sau
domf := {x 6 ^ I / ( x ) < +00} va
^P'f '•= {(.X.'') EXX^\X^ domf, r > / ( x ) } . Dinh nghia 2.9. Cho n,Q la cac tdp con ddng cfla X va X(, e Q . Dudi vi phdn qua gidi ftgn tuang iing rdi Q cua / dtfdc xdc dinh bdi V K ) •= {^' ^ ^ I (^'.-1) e A'e((x„,/(x,)).e/«/)}.
Nd'u Q-[u] vdi « ^ 0 thi ta ki hieu
^u/(^o) *hay 5, . / ( X Q ) vd gpi Id dudi vi phdn qua gidi fign theo hudng u cfla / tai x^.
Djnh ll 2.10. Niu x^, Id cue tiiu dia phuang tuang ling vdi QcX\{0} eUa f thi 0 6 5 „ / ( x , ) .
Chtfng minh. Ldy x„:=x^ '^'^'^ >x^, X* := 0 — ^ ^ ^ 0 va la'y b^t ki £„ -> 0,^„ -> 0^
thi vdi m6i ^ > 0 chpn r^ = r^ > 0 bat ki ta cd {K^x-x^)-ir- p{x„)) = -ir - ^(Xo))
<-(p(x)-9>(:^))<(£„+^)(||x-x„||+|r-K=c„)|) vdi mpi xGDQ{x^,r„,SJr\dom<p,r>g){x). Do
d d O s a ^ ^ X o ) . n Sau day, chflng tdi dtfa ra dieu kien c i n
cho ctfc tieu dia phtfdng cfla bdi loan (P). Trtfdc tien, chflng ta cd h6 di sau.
B ^ d^ 2.11. Niu Xj, la cue tiiu dia phuang cua f tbi Xf^ Id cue tiiu dia phuang theo mgi hudng M e X \ {0} cua f.
C h i ^ g minh. Suy ra trtfc tie'p ttf Dinh
nghia 2.8 vd Dinh nghia 2.9. n Dinh Ii 2.12. Niu x^ Id cue tiiu dia phuang
cua f thl vdi mgi u e X \ { 0 } tacd 0 e dj{x^).
Chtfhg minh. Suy ra ttf Dinh li 2.10 vd Bo
d e 2 . 1 L n
TuaONG SAI HOC DONG THAT Tap chi Khoa hpc 50 13 (6-;ow Tai li^u tham khao
[1]. N. L. H. Anh and P. Q. Khanh (2013), "Higher-order optimality conditions in set-valutj optimization using radial sets and radial derivaUves", J. Glob. Optim., (56), p. 519-536.
[2]. I. Gmchev and B. S. Jvlordukhovich (2011), "On directionally depend™
subdifferentials", C. R. Acad. Bulgare Sci., (64), p. 497-508.
[3]. I. Ginchev and B. S. Mordukhovich (2012), "Directional subdifferentials and optimaliiy conditions", Positivity, (16), p. IQH-l'il.
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THE DIRECTIONALLY NORIMAL CONE AND OPTEVIAL COlNDinON Summary
This paper is devoted to smdy some properties of the directionally Ff chet normal cones ad the directionally limiting normal cones. Moreover, we also modify those directionally iimitijij normal cones and estabhsh some properties of the modified normal cones. Then we provide some examples to illustrate the differences between these normal cones. Finally, we generate li concepts of the directionally sub-differential via the directionally normal cones. By using tin dkectionally sub-differentials, we come up with a necessary condition for the directional) optimal solution of an optimization problem.
Keywords: Directionally normal cone, directionally sub-differential, optimality condition.
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