Luận văn Vài đạo hàm suy rộng và tối ưu hóa không trơn

ChUO1lg3 UNG DUNG. TrangchuangmlYtasedungHessianx~pxi dethi~tl~pdi~uki~ncAnvadlit6iuu c~phaichobaitmlnquihoC;lchp ituy~nv6irangbuQcd~ngthucvab~td~ngthuc. Saildota sirdl,mgJacobix~pxi Frechetdetimdi~uki~nduynh~tnghi~mdia phuangvatoanC\lCd6iv6ibaitminb~td~ngthucbi~nphfmmarQngv6idfrki~n lient\lCkhongLipschitzdiaphuang. 3.1BAI ToAN QUI HO~CH PHI TUYEN X6tbaitmin(P) san mm lex) , g(x)~0, hex)=o. 39 TrongdoI: 9{n~ 9{,g: 9{n~ 9{P,h: 9{n~ 9{q lacachamkhflvi, lient\lCdp m('>t. D~tg=(gp...,gp),h=(hp...,hq). lea)={i:gi(a)=O}. C :={xE 9{n:g(x) ~0,hex)=O}Iat~pch~pnh~nduQ'Cvam6idi~mtrongt~pnay duQ'cgQiIadi~mch~pnh~nduQ'cuabaitoan(P). D~t C(.<)={xe c: t.<;g;(X) ~ 0}. Non cuahuangch~pnh~nduQ'cd6ivai t~pcon S c 9{nt~iXES duQ'Cxacdinhb6i F(S,x) ={uE9{n:38>O,VO~a~8,x+auES}. Binb ngbia3.1Di~mch~pnh~nduQ'caEC duQ'cgQileidi~mchinhquicuabai toan(P)n~ucacvectaVgi,ViE lea),Vhj(a),j =1,...,q d('>cI~ptuy~ntinh. HamLagrangeduQ'cxacdinhb6i L(x,1,Jl):=lex) +(1,g(x))+(Jl,hex)). B8d~3.1( DinhIi Karush-Kuhn-Tucker)Choa lil cvctiiudiaphlfO'ngcuahili loan (P). Niu a lil didmchinh qui cua hili loan (P) thi t{mtc;zi1 E 9{P,JlE 9{qsao cho [ VL(a,1,Jl) =0 1Tg(a)=O,1~O g(a)~O,h(a)=O. Di~uki~ntrenduQ'cgQiIadi~uki~nKarush-Kuhn-Tucker. Di~uki~n1Tg(a)=0 duQ'cgQileidi~uki~nd('>l~chbu. Nb~nxet3.1 N~ubaitoan(P) labaitoanC1JCd~ithidi~uki~n1~0 duQ'cthay bing di~uki~n1 ~0 ( max lex) =-mill (-lex))).Cacdi~uki~nkhacvftngill nguyen. 3.1.1ei~uki~ncan t6i U'ucAphai Binh Ii 3.1 Choa fa cvc tiJu iliaphuangcuahai toim(P). Gid sir vaim6i 1E 9F"uE 9)Q,J2L(a,1"u) faHessianxdpxihichi}.ncuaL(.,l"u)tqia.Niu a fa iliJm chinhqui thi t6n tqi 1 E9)P"uE9)QsaDcho (a,l"u) thod maniliJu ki?n Karush-Kuhn-Tucker.Han nua Vu E F(C(1),a),3M E J2 L(a,l"u) saDcho (u,Mu);:::O. ChungminhTheogiathuy€tthiB6d~3.1thoamannentirdi~uki~nKarush- Kuhn-Tuckertaco (uV'Lt (a,l"u),u)= 1imsup(uV'L)(a+tu,l"u) - (uV'L)(a,l"u) lio t 1. (u,V'L(a+tu,l"u))-(u,VL(a,l"u))= Imsuplio t 1. (u,V'L(a+tu,l"u))= Imsup . lio t Cho uEF(C(l),a), theodinhnghlat6nt~i(J >0 saochov6i mQi0S;as; (J, a+auEC(l). Khido O<~«J, k 1 1 L(a+-U,A"u)=f(a +-,u);:::lea) =L(a,A"u).k k Hay 1 L(a+-u,l"u)-L(a,l"u);::: O. k Theo£)inhIi giatritrungbinht6nt~is6nguyenN >0 va 0N saocho (u,V'L(a +tkU,A"u));:::0,Vk;:::N. Dodo . (u,VL(a+tu,A,j.i)) hmsup 2 0 . 11,0 t Vi J2L(a,A,j.i) laHessianx~pxicuaLCA,j.i) t~ia nentaco VuEF(C(A),a), sup (Nu,u)2 (uVLY ((a,A,j.i),U) NEJ'L(a,A,fl) 1. (u,VL(a+tU,A,j.i))= Imsup 20. Ita t Theogiathuy~tJ2L(a,A,j.i) bi ch~nnen3M E J2L(a,A,j.i) saGcho (Mu,u)2 sup (NU,U)20. NEJ'L(a,A.fl) 0 H~ qua 3.1 Cho a fa qrc tidu aia ph14angcua hai loan (P). Gia s14 J2lea),J2g(a),J2h(a) facacHessianxdpxi hi ch(mcuaf,g,h t14angungt(lia. Niu a faaiimchinhquithi t6nt(liAE9iP,f1E9iqsaGcho(a,A,j.i)thoamanai~u ki?nKarush-Kuhn-Tucker.Han mfa, Vu E F(C(A),a),3N E J2 f(a),P E J2g(a),Q E J2h(a) saGcho ((N +ATP +j.iTQ~,u) 2 O. Chtfngminh TuTinhch~t(iii)taco J2 L(a, A,j.i) = J2 lea) +ATJ2g(a) +j.iTJ2h(a) litHessianx~pxi cuaLC A,j.i) t~ia. V~ytheoDinhIi 3.1 3M E J2L(a,A,j.i) saGcho Vu E F(C(A),a) ,(u,Mu) 2 O. Vi M =N +ATP+ j.iTQ, trongdo N E J2 f(a),P E J2g(a),Q E J2h(a) nen ((N+J! P +j.iTQ~,u)20, VuEF(C(A),a). 0 42 3.1.2£)i~uki(!ndu tOiU'ucAp hai Trangphc1nnaytasetrinhbaydi~uki~ndud.phaichobaitmln(P). f)~t J ={iE I(a) :A;>O}. Q ={yE Bn :yTVg;(a) =O,iE J, yTVhj(a)=O,j =1,...,q}. Cho c:>0,0 >0, 2 (c:,0) = {dE B n :3y E Y,O <g(d) <0, lid - yll < c:,a +g(d)d E C} . Djnh Ii 3.2 Choa la ilidmchapnhqnillf9'Ccuabaitoim(P).GiGsir (a,A"u)thoG manili~uki?nKarush-Kuhn-Tucker.GiGSltthemrlingvaimQix trongIancqncua a,J2L(x,A"u)laHessianxapxi hich(mcuaL(-,A"u)tgia. Niu t6ntgic:>0,0>0 saGchovaim9i dE 2(8,0) vavaimQi0<a <1, \:1ME J2 L(a +ad,A"u),(Md,d) ~0, thi a la cvc tiduilia phlfO'ngcua bai toim(P). Chungminh Gicisu a khonglaqlc ti~udiaphuangcuabaitoim(P). Khi dot6nt(;lidaych~pnh~nduQ'c{xk}cuabaitmln(P)saocho Xk ~ a khik ~ +00va f(Xk) <f(a), \:Ik. Gicisu Xk=a+okdk. Trongdolldkll=l,ok >0,Ok~O khik ~+oo. Vi Ildkll=1 nenday{dk}codayconhQitl,1. Khongm~tinht6ngquatavin ki hi~uladay{dk }saocho dk ~ y khik ~ +00 v6i Ilyll =1. TheoTinh ch~t(iv) taco 0> f(Xk) - f(a) =okd[Vf(a +17okokdk),O<17ok<1, 43 0 ~ gj(Xk) - gj(a) =gkdJVgj(a +17jkgkdk),O<17jk<1,Vi E lea), 0=h/Xk) - h/a) =gkdJVhj(a +;jkgkdk),O<;jk <1,Vj =1,...,q . Chiacacphuongtrinhvab:ltphuangtrinhtrencho gkvaI:lygi6ih(;1nkhi k ~ +00. Ta duQ'c yTVf(a)~O,yTVgi(a)~0,Vi E l(a),yTVh/a) =0,Vj =1,...,q. Giasutfmt(;1iitnh:lti E J saochoyTVgj(a)<O.Taco q 0~yTVf(a) =-LAjyTVgj(a)- LPjyTVh/a) >0, jEJ j=l voly. V?y yTVg;{a)=O,ViEJ hoi;icJ=0. V?y YEQ. Theogiathuy~t(a,A,p) th6amandi~uki~nKarush-Kuhn-Tucker.Ta co Aj~0,Ajgj(a)=O,i=1,...,p, q VL(a,A,p) =Vf(a) + LAiVgi(a) +LPjVhj(a) =O. jE/(a) j=l Mi;itkhacvi f(Xk) <lea),Vk vasudt,mgTinh ch:lt(ix) chohamL(x,A,J1)t(;1ia. Taco lea) >f(Xk) q ~f(Xk)+ LAjgi(Xk) +LPjh/Xk) jE/(a) j=1 ;, f(a) +,~)A,g,(a)+ tJljh/a) +Okdi(Vf(a) +,~)A,vg,(a) +tJljVhj (a)J +! - 2min (Mkgkdk,gkdk) 2 MkECOJ L(a+8kokdk,A,/l) =lea) +!g; mill (Mkdk,dk) 2 MkEJ2L(a+Okokdk,A,/l) 1 J ( 0 }=lea)+2g; Mkdk,dk. Trongdo O<Bk<1va MZ EJ2L(a+Bkgkdk,A,p). Dodov6imQik, (MZdk,dk}<O. 44 Mauthufinv6i giathuy~tsauIldkll=l,dk~ Y E Y,5k~ 0 khi k ~ +00,0<B/ik <1 khi k du Ian va a +5kdk la huang ch<1pnh~nduQ'cv6i 1119ik . Dodov6ik dulan,dkEZ(&,5)va(M~dk,dk)?o. V~ya IaClJCti~udiaphuOTIgcuabaitmin(P). 0 Binh Ii 3.3 Cho a lit aiim chdpnhijnalt(lccuabatloan(P). GiGsir (a,A"u) thoa manaiJu ki?nKarush-Kuhn-Tucker.GiGsirthemding vaimQix trongfancijncua a,J2L(x,A"u) laHessianxdpxibichij.ncua L(.,A"u) t(1ia.Niutfmtgi &>0,5>0 saDchovaimQid E Z(&,5) vavaimQi0<a <1, \/M E J2 L(a +ad,A,J-L),(Md,d) >0 thi a la C1!Ctiiu chij.taiaphltang cua bat loan (P). Chungminh (TuO'ngtlJ chungminhDinhIi 3.2) 0 3.2BM ToAN BAT DANG THUC BIEN PHAN ChoK Iat~p16i,dong,khacr6ngtrongkh6nggianEuclidean~n. Anhx~f,g: ~n~ ~nIatoantuphituy~nlientl,lC,g xacdinhtrenK. Bai toanb<1td~ngthucbi~nphanmafQngV(f, g,K) duQ'cxacdinhnhusau. Tim a E ~n saDcho (f(a),g(x) - g(a))? 0,\/x E ~n. Nontiip xuccuat~pK t~ix EK duQ'cxacdinhbat T(K,x):= ~imti(xi -x): Xi E K'Xi ~ x,ti >OJ.->oo Nonsinhb6'it~pK duQ'cxacdinhbai cone(K):={tx:x EK,t? O}. 45 Cho A ~9{nhi t~pkhaer6ng.Nonlidxaeuat~pA duQ'exaedinhb6i A", :=~limtiXj:Xi E A,tj >0,~imti=Of~'" ,~"" Nonet6iC1!Ceuat~pK duQ'cxacdinhb6i K' :={UE9{n:(u,x)~O,VxEK}. Non t&ihc;zncua I, g t~iX duQ'cxacdinhb6i CU,g)(K,x):={vET(K,g(x)):(/(x),v)=O}. M(>t~pduQ'cgQiIaetadi?nn€u no Iagiaohlhlh~ncaelllIakh6nggiandong. Tinheh~t?Pdadi~nA~ 9{n: (a) T(A,x)=cone(A- x),VxE A; (b)V6'imQia E A t6nt~iIfmc~nU cuaa saocho cone(A- a)~ cone(A- x),VxE An U . D~tJ tea)=JI(a) u ((JI(a))",\ {o}). B6d~3.2 GiGsu I: 9{n -+ 291mla hametatrj nua lien t1j.ctren tc;zia E 9{n,Cho ti >0 h6it1j.etin 0, qj Ecol(a+tjBn) saocho limllq,II=00 valim ll qi .ll =q.,Khieto ,~'" ,~",q, q.E (col(a)L. Han nfra,niu co(/(a))oocoetinhthi q.Eco(/(a)L =(col(a))",. ChungminhTheogiathuy€tI lahamdatftlllIalient\Jctrent~iaE9{n.Thea dinhnghla,v6'imQiF::>0,t6nt~iiodu1611saocho I(a +tjBJ c tea) +F::Bm'Vi ~io' Dodo qi E co(/(a +F::Bm))C co(/(a +F::Bm))+F::Bm'Vi ~io' Vi v~y 46 q. E [co(f(a)+t:BJ+ t:B(O,l)L,c [co(f(a)+t:BJL, c (cof(a)L. M~tkhacta1uanco co(f(a)L c (cof(a)L b6'ivi lea) c cof(a)va co(f(a)L 1anon 16i,dong. Ta c~nchungminhchi~ungugcl(;li. m+l Cho P E (cof(a)L,p:I: o. Theo dinh 1yCaratheodory,t6n t(;lit6 hgp 16iPi =LAijPij' ~ /=1 m+1 v6i Aij2 O'Pij E lea) va LAij =1 saocho /=1 II~II=~~~II~:IIva~~~llpill=oo. Khangm~tinht6ngquattagicistr m+1 limAij =A/2O,j=l,...,m+1vaLA/ =1.1-->00 /=1 V6i m6i j, xetday {AijPijIllpilltl . Ta dn chungminhdaynaybi ch~n. Th~tv~y,gicistrday {AijPijIllpilltl khangbi ch~n. Cho aij =AijPijIllpJ N~uc~ntal~ydaycon,gicistrr~ng Ilaijoll=max~laijll:j=l,...,m+l},Vi. Do do ~~~llaijoII=00 . m+l Vi PiIllp,ll=L aulientaco j=l m+1a.. 0=hm Pi =hmL !!- . HOO Ilpililiaijoll HOO J=l Ilau",1 Ta gicistrday {auIllauollL,oh0itl,1d~naOjE (cof(a))oo,j=1,"',m+1.B6'ivi daynaybi m+1 ch~nlienkhi aD/a:I:0 phuangtrinh0=Lao/ chirar~ngco(f(a)L khangcodinh. /=1 Mfmthuftnv6i gicithuy~t. 47 V?y day {AijPijflip,lit!bich~n.Tagiilsirday{AijPij/Ilpilitlh(>it\l d~nPO}E (f(a)L . Khi do m+l P =LPo) E co(f(a)L. }=I 0 3.2.1 Tinh duy nhatdia phlJ'O'ng Blnh Ii 3.4 Cho K ~innla t()p16i,ilong,khacr6ng,f, g :inn---+innla toimtIrphi tuyin lien t7,lC,g xac ilinh trenK va Jf(a),Jg(a) la cac Jacobi xap xi cua f,g tuangungtgia. Niu a langhi?mcuaV(f,g,K) thim(jttrangbailiJu ki?nsaula iluili choa langhi?miliaphuangduynhat. (i) K la t()p16iiladi?nva vaim9iME J lea), N E J g(a)taco (M(v),N(v)) >0,\Iv E inn\ {O}, trongilo N(v) E CU,g)(K,a),M(v) E [CU,g)(K,a))*,. (ii) K la t(zp16i ila di?nvava-im9iME J f(a),N E J g(a)taco (M(v),N(v)) >0,\Iv E inn\ {O}, trongilo N(v) E CU,g)(K,a),lea) +M(v) E [T(K,g(a)))*,. - - (iii) Vaim9iME J lea),N E J g(a) taco (M(v),N(v))>0,\IvE inn\ {O}, trongilo N(v) E CU,g)(K,a). Chung minh TruactientachUngminh(ii) suyra(i). - - Th?tV?y,cho vEinn\ {O},ME J f(a),N E J g(a) taco N(v)E CU,g)(K,a),lea) +M(v) E [T(K,g(a)))*. Ta c~nchungto M(v) E [CU,g)(K,a))*. 48 Th~tv~y,cho UE CU,g)(K,a), theodinh nghlata c6 U E T(K,g(a)) va(f(a),u)=o. Khi d6 0~(f(a) +M(v),u)=(f(a),u) +(M(v),u)=(M(v),u). V~yM(v) E[CU,g)(K,a)]*. Giasu(ii)thoaman,tac~nchungminha Ianghi~mduynhfttdiaphuongcua V(f,g,K) . GiasunguQ'cl~ia kh6nglanghi~mduynhfttdiaphuangeuaV(f, g,K) . Tuc la tfmt~i{x,}langhi~mcuaV(f, g,K) saGcho {x,}hQit\1d€n a. Tagiasu {cxi-a)/llxi-all}hQit\1d€nv;tO. Vi Xivaa langhi~mcuaV(f,g,K) nentac6 lea) E [T(K,g(a))]*,f(xJ E [T(K,g(x,))]*, (f (a),g(Xi)- g(a));:::0,(f (Xi),g(Xi)- g(a))~0. TheoTinhchftt(b)cuat~pdadi~n,t6nt~iio;:::1saGeho [T(K,g(Xi))]*s;;;; [T(K,g(a))]*, Vi;:::io' Do d6 f(xJ - lea) E [T(K, g(a))]*- lea), Vi;:::io . HannuaJf(a),Jg(a)lacaeJacobixftpxi Frechetcuaf,g tuangungt~ia. Ta co Mi E Jf(a), Ni E Jg(a) saGcho f(xJ - lea) =M;(Xi - a) +rJxi - a), g(x,) - g(a) =Ni(Xi - a) +r2(Xi- a). Trang d6 rj(Xi - a) /IIXi- all~ 0 va r2(Xi - a) /IIXi- all~ 0 khi i ~ r:f). ThayvaGbi€u thuc(3.2)va(3.3)taduQ'c Mi(Xi-a) +lj(Xi-a) E [T(K,g(a))]*- f(a), (f(a),Ni(xi -a)+r2(xi-a));:::0, (f(x,),Ni(Xi -a)+r2(xi -a))~O,Vi;:::io' 49 (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) NSu{Mj}bich~nthitagiaSlrnohQitvdSnME Jf(a). Chia(3.4)choIlxi- allva choquagiai h,;mkhi i ~ 00, taduqc M(v) Econe([T(K,g(a))]*- lea)). Do dotfmt~it >0 saccho lea) +M(tv) E [T(K, g(a))]*. NSu {M,}kh6ngbi ch~nthi ta gia Slr (3.7) ~~~IIMill= 00 va~~~II~:II=ME (Jf(a))" \ {o}. Chia (3.4) cho IIM,llllXj- all va cho quagiai h~nkhi i ~ 00 taclingthuduqc(3.7). NSu{Ni}bich~nthitagiaSlrnohQitvdSnN E Jg(a). Chia(3.5)cho Ilxi- allvachoquagiai h~nkhi i ~ 00, taduqc (f(a),N(v))~o. TuangW chia(3.6)cho Ilxi- allvachoquagiai h~nkhi i ~ 00, taduqc (f(a), N(v))s:;o. Dodo (f(a),N(v)) =0. (3.8) Hannlia,tir(3.5)va(3.6)taco (f(xJ- f(a),g(xJ-g(a))S:;O. (3.9) Hay (M(v),N(v))s:;O. (3.10) Lienh~gilia(3.7),(3.8)va(3.10)tath~ynomfmthuftnvaigiathuySt(ii). NSu{Nj}kh6ngbi ch~nthitagiaSlr ~~~IINfII = 00 va~~~II~:II= N E (Jg( a))" \ {O}. Chia(3.5)va(3.6)choIIN,llllxi- allvachia(3.9)ho~ccho IINjlllixi- al12 khi {M,}bi ch~nho~ccho IIMj1IIINi1IIIxi- al12khi {Mi}kh6ngbi ch~nva cho quagiai h~nkhi 50 i ~ 00,taclingthuduQ'c(8),(10) cungvai (7) ta th~yno mfmthu~nvai giathuy~t (ii). Cho(iii) thoaman.Giasua kh6ng1anghi~mdiaphuangduynh~tcuaV(f, g,K) . Tilc lat6nt(;li{x,}langhi~mcuaV(f, g,K) saDcho {Xi}hQit1,1d~na thoaman(3.1) va(3.2).Tird6suyra(3.5),(3.6),b~ngeachIi lu~nnhutrentaclingthuduQ'c(3.8) va(3.10).M6i lienh~gifra(8)va(10)mfmthu~nvaigiathuy~t(iii). 0 Nh~nxet3.2 Khi glatocintud6ngnh~tthib~td~ngthilcbi~nphfmV(f, g,K) tra v~b~td~ngthilcbi~nphanc6di~n.Khi d6di~uki~n(i) a DinhIi 3.4seduQ'cthay b~ng:vai m6i VECU,g)(K,a)va M(V)E[CU,g)(K,a)f,(M(v),v)=O=>v=O.Khi f,g 1acachamLipschitzdiaphuangthiJacobimarQngClarkeduQ'cdungnhula Jacobix~pxi Frechet. 3.2.2 Slf duy nhAttoimcl;IC Di;it Ko :=co{xE 9{n:g(x) E K}. Binh Ii 3.5 Cho f, g :9{n~ 9{nfa toimfirphi tuyin lien t1;lC,g xac atnhtrenK. Jf, Jg fa cac anh xq Jacobi xdp xi cua f, g tU'O'ngung. Gia sir vai m6i MEUXEKOCO(Jf(x))u((coJf(x)L\{o})va NEUXEKoCO(Jg(x))u((coJg(x)L\ {Onma tr(mNTM xacatnhdU'O'ng.KhiaobailoanV(f, g,K) coduynhdtnghi~m. Ch.rngminh Gia subaitoanV(f, g,K) c6hainghi~mphanbi~tXova Yo'Khi d6 [xo,Yo]c Ko va (f(xo) - f(Yo),g(xo) - g(yo))~o. ' (3.11) 51 Taxethamvahuangx H (f(x),g(xo) - g(yo)). Khi d6baad6ngcuat~p F(x):={M(g(xo)- g(yo)):ME Jf(x)} laJacobix~pxi cua (fO,g(xo) - g(yo)) t~i x. Apd\mgTinhch~t(iv)vaihamvahuangnaytren[xo,Yo],tac6 3c E (xo'YO),qiE coF(c) saocho (f(xo) - f(Yo),g(xo) - g(yo))=~im(qi,g(XO)- g(yo)).1-'>00 (3.12) Vi coF(c)=[coJf(c)](xo- Yo) ta c6 th€ tim Mi E coJf(c) saocho qi =Mi(xo - Yo). Neu {MJ bi chi[Ln,tagiasirn6h<)it1,1den Mo EcoJf(c). Khi d6tir (3.11)va(3.12) suyra (f(xo) - f(yo), g(xo)- g(yo)) =(Mo(xo- Yo)'g(xo)- g(yo))~o. Neu {MJ khongbi chi[Ln,tagiasir (3.13) limllM;11=r:fJ va ~im 11 M; !! =Mo E (coJf(c)L \ {O}. 1-'>00 1-'>00 M; Tir phuangtrinh(3.12)tac6 (Mo(xo- Yo)'g(xo)- g(yo))=lim([M,/IIM,II](xo- Yo)'g(xo)- g(yo)) ~o.1-'>00 (3.14) Xethamvahuangx H (Mo(xo- yo),g(x)),trongd6 Mo lamatr~nthuduQ'ctir co(Jf(c))u ((coJf(c)L \ {on. TaIy lu~ntuang1\1nhudillamvaihamxH (f(x),g(xo)- g(yo)), 3d E (xo,yo),N; E coJg(d) saocho (Mo(xo- yo),g(xo)- g(yo))=lim(Mo(xo- yo)'Ni(xo - Yo)).1-'>00 KethQ'Pvai (3.13)va(3.14)tac6 3NoEco(Jg(d))u ((coJg(d)L\ {onsaocho (Mo(xo- Yo)'No(xo- Yo))~o. 52 Di€u m'iymfmthu~nv6i giathuy~tleimatr~nN/ M 0 xacdinhduang. 0 Nh~nxet3.3Khi g leitoantud6ngnh~t,aco K =Ko'Khi dob~td~ngthucbi~n phanV(f,g,K) trav€ b~td~ngthucbi~nphanc6di~n. 53