Thuật toán tìm không điểm cho tổng hai toán tử đơn điệu cực đại

<;1cgQila mi€n anh cuaT vakihi~ula ImT.V~y: 1mT={y EH 13 x E H: (x,y) E G(T)}. NghichdaocuaroantitT la roantitcod6thila {(y,x)/(x,y)E G(T)}. Dinh nghTa1.1 a. MQlloan III T dUrfcgQiLaddndi~uneu: 20 V(x,y),(x',y') EG(T). b. MQlloan lit T dUrfcgQiLaddndi~uchc;itntu : >0 V(x,y),(x',i) EG(T) va(x,y)7(x',y'). c. MQIloanta T dUrfCgQiLaddndi~umc;mhWYimoduloyneu : 2 y /x' - x P V(x,y) , (x',y') E G(T). Trang7 Vidu: ChoE la mQtkhonggianEuclide, E1va Z la hait~pconkhongtr6ng cuaE. X6tham16if :E ~ E1.Giasirr~ngZ c damfeE la ta'tcacacdi~mz maai(z)*~(doh~qua2.2chu'dng1cua[l]), Z * 0. X6ttoantii'T :E ~ 2E xacdjnhbdi T(Z)={:(Z) neUZEZ, neuzEE \ Z. Hi~nnhiendamT =Z. Chungtachirar~ngT la ddndi~u. Chungminh: Th~tv~y, giasux vax'lahaidi~mtUyy trongZ vay ,y' lahaiph~ntu tuyy trong T(x) va T(x'). B~ngdinhnghIadu'divi phan(Subdifferential) (chu'dng1ph~n2cua [1])chungtaco : rex')~ rex)+, rex) ~ f(x') + . CQnghaibfftdiingthuctrentadu'Qc f(x')+rex)~rex)+f(x')++. Dodo ~o, nghTala T ddndi~u. ChuYIII daychungla kh6ngphanhi?l girlaloantll vad6thicuachung. Dinh nghia1.2 MQI loan Iii ddndi?u ia qtc dgi neunokh6ngnlimlrongmQlloan111ddn di?ukhackh6nglrungwJi no. Trang2 NguyenThe'Uy Chu'o'ng1 TminTii'Do'nDi~u Saudayzacactinh chattijpanh cualoan titddndi?uc1/cd(li .. DtnhIj 1.1.. Ne'uT : E -+ 2Ela tmln tti ddn di<%ucljc d<;tithl cha XOEdam T, t~p T(xo) la 16iva dong, Chungminh: . TachungminhT(xo)ladong. Go') ') ° I ' k? d" k T( 0 ) (k 12 ) T ~ h hK d d'" ?1a Sl( Y =1m y , d ay y E X =, , III C at dn IyU cuak~oo T chI fa ~0 vdi (x,y) E G(T). Quagidi h<;tnkhi k -+ r:fJ taduQc: ° ° ~O, Vdi ( x,y ) E G(T) tuyy,Vi T la ddndi<%uqic d<;li, ta thuduQcyOE T(xo)co nghIala T(xo)la dong . TachungminhT(xo)la 16i: , Gia sll'y' , y"E T(xO)va 0 < a < 1. Vdi mQix Edam T va y E T(x) chungtaco , o 0 ~ , " ° 0 ~ . E)~t ya=ay' + y" , tadliQc a o ~O, Trang3 VI T laddndi~uqtcd~ikeotheoya E T(xo).NenT(xo)la t~p16i. TrongtHronghQprieng, n€u XOE int ( domT), taco dinhIy sail Djnh lj 1.2 N€u XOE int ( domT) va T :E ~ 2Ela toclntii' ddndi~ucvcd~i,thlT(xo) la t~p16icompact. Chungminh : (Xemchu'dng4 ph~nI cua[l]) 1.2 Tminto-khongdaD: Dinhnghia1.3 MQl loanlit Q: E ~ 2£dl1rjcgQiLakh6ngdiinntu domQ ~ fjJva /q' - q" /::; /z.'- z" j, Vdi (z', q' ), ( z" , q" ) E G(Q.). Dinhnghia1.4 MQlloanlit T:E ~ 2£dl1rjcgQiLamiJrQngcila loanla T: E ~ 2£ ntu G(T) c G( T) . Khi d6 loanla T dl1rjCgQi Lalhuh~p cila loanlaT. Djnhlj 1.3 Cho mQttoantii'khongclanQ :E ~ E thl t6n t~imQttoantii'khong clan Q marQngmadom Q= E Chungminh: (Chu'dng4 cua[1]) Trang4 DufJi dayzalienh~giila loan li'tddndi~uvaloanli'tkhongdiin Xet bi€n d6i tuy€n tint <D: Ex E ---+E x E dintnghlabai ( (z+t) (Z-t) )(u,q}=<D(z,t)= -Ii' -J2 . Do (u+q) z- h - '\!2 va . (u - q) t= -Ii chung taco <D=<D-1. T~pQ(u) ={q E E : (u ,q) E G(Q) =<D(G (T))},xac dint toantitQ : E ---+2Emad6thi la <D(G (T)).Tacolienh~giG'aT va<D(T)nhu'sau Djnhly1.4 (i) MQt toantti'T : E ---+2Ela ddndi~un€u va chin€u Q =<D(T)la kh6ngdan. (ii) MQttoantitT :E ---+2Elatoantitddndi~uC\tcd~in€u vachin€u Q =<D(T)la kh6ngdanvadamQ =E. H~qua1.1 M6i toantitddndi~uco mQ~toantti'marQngIa ddndi~uqtc d~i. 1.3 Tinhddndi~ucl,icd~icuat6nghaitmlntd'ddndi~ucl,icd~i Tru'dckhi chungmint dint 19cdban(dinh191.8), chungtanh~cI(~dmQt sf)dint 19dadltQCchungmint trong[1]. Trang5 DtnhIf 1.5 MQttoclnti'tT : E ~ 2Ela ddndit?un€u va chi n€u phu'dngtrlnhT(z) +z =h co nghit?mchom6ih E Eo DtnhIf 1.6 GiitsU'T:E ~ 2Ela loantii'ddndit?uqtc d~i,ZOE ri (domT) va S c E la mQtt~pcompact. Giit si'tVOlm6iZ E S n domT t6nt~imQtvectdtE T(z)VOl<t , z - ZO> 2::0, Khi do phltdngtrlnhT(z) co nghit?mtrongconv S n domT. Dinh If 1.7 Giit si'tT :E ~ 2Ela loantU'ddndit?uct;t'cd~i.N€u domT la dong hayT ]addndit?um~nhyoimoduJoy thlphu'dngtrlnhT(z)=0conghit?m. j Djnh Iy CC1bitn. Dinh If 1.8 N'" T 1 E 2E, T2 E 2E I' , ? d d'" d '"eu : ~ va : ~ a loan tli dn lyU ct;t'c~lman (domT1) n ri (domT2)* ~thl loan tU'T =T1+T2clinghI loantU'ddndit?uqtc d~i. Chlingminh: Tntoch€t t~mgiit sitr~ngint (domT2)* 0 vadomT2la dong.GiitsU'ZO N - E ri (dom T1) n int (domT2). B~ngcachthay T1va T2bdi Tl va Tzyoi - TI(Z)=T1(z+ZO)-t, - I;(z)=T2 (z+ZO). Trang6 Chungtaco th~h~ncht slfnghiencuuvaotru'onghQpZO=0 va 0 E T\O). Tli dinh190.5), d~T la dondi~uclfcd~ithldi€u ki~nau la, vdi tEE, phu'ongtrlnh Tl(Z)+T\z) +z - t=0 co nghi~m.Ntu cgn, thayT\z) b~ngT2(z)- t,chungtaco th~gidi h~ntrong tru'ongh9P t =O. V~yd6chungminhT la tocintli'dondi~uclfcd~ichungtaphaichira mOtnghi~mcuaphltdngtrlnh . T1(z)+T2(z)+z =O. (1) 0) co nghi~mntu va chi ntu t6nt~icac vectoz* , t* , t1E T1(z*) , t2E T\z) ma I I * * t +-z =-t 2 ' (2) 2 1 * *t +-z =-t . 2 GiaSltT la toantltdinhnghTaboi - i . 1 T (z) =T' (z)+- (z), 2 i Ti vz E E vaQI la nghichdaocua T (i=1,2). VI Ti la toantitdondi~uclfcd~i,apdvngdinh191.5(di€u ki~nchi ntu)tathayphltongtrlnh - i T (z)+z-t=O laconghi~mvdim6it E T. Trang7 - i Do aiSuki~naua ainh1y5 , tatha'yT 1atmintU'adnai~uva Qi(i =1,2) cGngla tmintii'ddndi~uqic d~i. Hdnnlia,doHnhddndi~um~nhcua ri k60theophu'dngtrinh - i T (z) -t=O coduynha'tnghi~mt E T. Vi v~yQila anhx~datri madomQi=E (i=1,2). X6t loan tli Q dinhnghIabai congthuc. . Q(t) =Q2(t)- Ql(-t) , tEE. B~ngh~qua1.1trong[1],loantii'ddndi~uqic d~iQl va Q21alien tvctrenE. Voi Q 1addnai~uapd~ngh~qua"1.1trong[1]1ffnnlia chungtakSt 1u~nr~ng Q la ddndi~uqic a~i.DungQi chungtaco th€ viSt (2)du'oid~ngtu'dngdu'dng 1a . z*=Ql (~t*), z* =Q2(t*). hay Q(t*)=Q2(t*)- Ql( -t*) =O. VI v~y,d6tlmnghi~mcuaphu'dngtrinh(1)thl tachi cffnchi ra nghi~mcua phu'dngtrinh . Q(t)=0, (3) . Chungminht6ntginghi~mcua(3) Chungtadungdinh1y1.6,nhu'ngtru'octieDchungta chi ra r~ngt6nt~i mQtt~pcompactS baoquanhtoE E va >0 vdi m6i tE S va va z=Q(t).Gia sl(ta1a'yS 1am~tcffu C(O,a)={t:.Itl=a, tE E} Trang8 Voi a> 0vatoladie'mxua'tph,Hthl > 0 la'ytn ;;::0 , t E C(O,a) . (4) tnQl(O)=0, tntinhcha'tddndi~ucuaQI keo theo ;;::0 , cho tE E. Hdnnua(4)chIraba'tdgngthuc ;;::0 tE C (O,a) . (5) . Changminhdungkhia au[an. Chungminh(5), chungta la'ydiSm t'E E va dungtinhcha'tddndi~ucua Q2 chungtavi6t ;;::0 , hay ;;::+ . Tn Q2(t), Q\t') Edam T2va damT21adongtheogia thi6t,chungtaco ;;::-All t'l +. (6) Voi h~ngsoAl >0vat, t' tuyytrongE. K6 ti6pchungtachIrar~ngcothSch9nt' ph\)thuQctdSv6phaicua(6) ladlfdngtrenC(O,cx)voia duIOn. Tn 0 E int (dam f2) =int(damT2),t~pint (dam f2) chuaquac~u S(O,s)={ZE:Izi<s}, Trang9 vdi E >0 dli be. Ap d\lngb6 de 1.6,T2la tmintU'ddndi~uqtc d'.liva t~p - 2 sea,c;)c int(domT ) la compact.Chungta ke'tlu~nr~ngt6nt'.limQth~ngs6 A2>Omalt'1 ::;A2va IQ2(t')I::;c.GiaSll' Itl>O,va t' ET2 (~ ItI ), Q2 (t)=~ ItI . Tli (6)suyraba'td~ngthuc \Q2(t) , t)?: -AI A 2 +c;ltl' (7) dung cho tEE. Bay gio giLlsU'r~ng a?: AlA2 c; , chungtased'.ltdlt<}C(5)tli (4). . Chungminhd§ydu,labodigiLlthie'tT21adongvaint(domT2)"# 0.D~t T2=T2+Pa a d dayPala toantitdlt<}Cdinhnghia 0 ne'ulzl<a ne'ulzl=a ne-ulzl>a a Pa(z)=~Aa:a>O Trang70 NguyenThe'Uy Chu'o'ng1 TminTii'Do'nDi~u tmlntil'Pexla roantU'ddndi~uqic d<;li,mi€n Pexla dong va ZO= 0 E int (dom T2)n int (domPex).Do 6 tren, ta co T: la ddn di~uqic d<;li. Chungta co domT: c domPexva ZO=0 E int (domT:) n ri (domT1) chungtaket lu~nroantit r] +r2=r1+r2+P =T+Pa a a la ddndi~uqic d O.NhlingtrongtruonghQpnay,b6i b6 d€ (3.2), T cfingla ddndi~uqic d<;li. . Cu6i clIngchungtagiitiquyetgia thietint (domT2)* 0.Gia sU'r~ng ZO=0 E ri (domTl) n ri (domT2). Gia sU'L1va L2la haikhonggianconcuaE thuduQccfinggiongnhu cuadom T1va domT2 T1 +r2 =T1+T2+P =T+Pa a a B6d~: MQtroantl'iT :E~2EmadomTc M, voiM =zo+L(6dayZoEL va L lakhonggianconcuaE) la ddndi~uqic d<;lin€u vachi n€u t6nt<;limQtroan tl'iddndi~uc\icd<;li - T:M~2M va domT=domT - T (z)=T(z)+L~,voizEdomT 6dayLdimE =dimL+dimL~. NguyenThe'Uy Chtidng1 TminTd DdnDi~u Vdi b6 d€ tren, loan tti'ddndi~uqtc d(;liTi co thebieu di~nthanh d(;lng Ti(z)=Ti(z) + (Li)"\ Z E damT1 aday - i T :L. ~2 L,I la loantti'ddndi~uqtc d(;li(i =1,2...).Do N(Li , z)=(Lil, z E Li . (i=1,2,...) £)u'<;$cchungminhtrong(chu'dng1[1]) .taco - j Tj =T +N(Lj), i =1,2,... (7') Chungtad~t L=C nL2 va '. I T~ =T +NI (L) d dayky hi~uN1(L)conghiala loantitN(L) xettrenIqIonggianEclideL1c E, VIv~y N1(L)(z)=N(L)(z) n L1 Vdi Z E E.Hdn nna - I T :C ~ 2L[ la ddndi~uQtcd(;li, va Nj (L):C ~2Ll ciingla ddndi~uQtcd(;liva ZO=0 thuQcv€ ph~nchungdam7'1va mi€n trong cuadamN1(L)=L . Chungtachungminhdu'<;$c TOI: LI ---72 LJ la loantitddndi~uC\tcd(;li, ma Trang72 Nguy~nThe'Uy Chu'Ong1 TminTii'DOnDi~u domT~cL Ap dl,lllgb6as 1.4[1]vdi ~ I T~=To +N}(L), dday ~ I To:L~2L la ddndi~uqtc d(;li. DungdinhnghIacua Talva biSudi€n cuaTa}, chungtathodtiQc N1(L)+N1(L)=N1(L), ,~ ~ I T l(z)=To (z)+N}(L)(z),zEdomT2 nL, (8) d day ~ I To :L~2L la loantli'ddndi~uqtc d(;li,tudngtl!cho rzchungtaco : ,~ ~ z T 2(z)=To(z)+Nz(L)(z),zEdomTz nL, dday ~ ZI To :L~2L la loantitddndi~uqic d(;liva N2(L)(z)=N(L) (z) n L. Ke'thQp(7')(8) (9)va chuyWi (LI)1-+(L2l =L1- va L 1-+L 1-n L1 + L 1-n L2 =L 1- chung ta tho duQc r =r1+rz =~l+~z+N(L) Trang73 (10) d day: la loantitadndi~uclfcd~i(i =1,2). ~I :L ~ 21 Tli - 1 - 2 Zn Eint (domTo) n int(domTo), vatUnhungdi€u dachungminhd tren.Tacotoclntti' ~ ~l ~2 70 =To +To tuL taiL la ddndi~uqtcd~i V~ydung(10)va b6d€ tre~chungtak€t lu~nT la toclntti'ddndi~uqtc d~i. Trang74 ._.