on methods for discrete topology optimization of continuum ...13221/fulltext01.pdfe1>a> !...
TRANSCRIPT
"!#$%&'!()*+%,-.%/1023%445* 6344)7
89:<;#=?>@A8B>
;DCFEHGJIJKLENMPOQR;HSUTVWTJX5Y[Z[Z[\
!!#"#$&%'()+*,-.'/0$1 12345"6#"768696#!:;
+<=?>A@B>DC:E=F>DGHIEHJK1LM=?N@OM$4PNGQ?L0+N<EQ=F@RNSH=$G#TUVE=?PN@OE=F>AWSM
$G8LEXYHM=?>D=?Z=?N'GTU[NSW\PHGXAG]L12F:^O=FGW\_PGXA@I`a1bc2d0+2d
/_EJNS@B>AM?_BE:e1PEHJXA>DH]fM?G@g@BNSJO=?>AXDXhM=Fi#HaJjE:eB%ZH]XklNS_H>AM?_Em($n]M?_GXhE#HRToQFE@BXAp]]N:M=F>DXAXaGTqToNH=FXD>A]f]Q\E#HaM_1H>AH])TonQ,E8eXhp#]]E#HJN$E:e^=?NS_HGXAG]>AN$JG_=FGQ\M?N rE#@RNSHs`#ToQ?N:JE#]fJNH@OE#Q\M!#f_1XDGW\_EHVSt f>uQFZ@v0)w`xl>DHaJM=FNSJ1=\Mep]NHKy`%ZH]XslNS_1H>AM?_ER($n]M?_GXhE#Hs`=FGWF_1PGXD@Kt
z E#=FM4bNQF@BN`E#Q\WFP|!#
*QF>AH=S~d$H>AeNQ\M?>=FN =FMFM?NQFe>hW N)+3
"!# $%"'&!# )(*+!,-./+#,&0/+21!2(34$1 4!#56&#(78:9;<.##=#5?>A@?#(B(CD' &E2GFIH
JKMLONPLON0QOR QTS?U)V#WX4R QONPQ
@?;8P +80
e1>A>
! #"!$%&('()!*"+,'-./*"+001'2'30,"+01&45 60)&01!6! ,%&7'()*98!.:"!*./&.7;<! #>=?'@! #01!?8!A*!0B*C2"+!D! E'(F&C2'*?/"!'2"+&&A(83%7))BG&)F)&4HIJ6"+)B*2'$,"J:'8!K'!BD"!D"!JL1'&C'L%AM@=N*?B*%3B*7'O"!'P,"3D'8!Q.!"+&"!L?./!&3$"RL,"+&AP./!&3(S'2"+8T0(&+&@"+L+ UBD"+F&V"+?JWD.!"+&!)3=ND?B*%4TXJB*B*!OT"+??0,"!01F&>6Y!&01!&Z"![! ./![\]U&U]Y0&L?B^ _+&B$)?"+4
HI<R=&0,"!01F&T'8!`."&"!L?]"&ML,"+&AaL)(K]"+)B*2'b,"+D[./!V'&))(&U2"!"!)"!??NA*Z"+c/UB*U?/"'#4d J2e%).!"!?%d-"!''(8-"fBD"!??,01N./:.!"!?)!gεg7$"!??E'8!].!"&V"+L?gL1!D:,">"+&Y.'K"!'D:,"+>"&h=,??2'D;>NKBD"+&"!?i4j`NBD"!??#01N./Y?2;F&UL#!)'*hNk#6BD"+&NlDL1B*01N./f'(=,N: _+&:"+??'84hHm ?NAR3?B*%h:F"!82'Wn_ o&BpBD"+&"!?ED./'R!&: o&Bp./']DBD"+&"+?oq:F;`8?L"!?#k#BD"+&Nl*rs)>"9?;t&Z"!c@B*7'N=2"++ u:?'Dh"!,'@7)dv F&BD"!Ss\]+&&!SwjO%7'L)(&AR o!&B$)?"2"!PL1$)'TKB*0)3@'0?V"+B*%YM87L1!)&8h'(8x*%?NA%4 x@%5N.(NIA3"!?)?"!52"+39L1"!00?2'KY!%l7>! 5"-8!7L1)&%7']"+&ZRB*7'y4 v hYB*0)(2''0?V"+B*%>"&l"!9MKzSm87L1)(&%7'an_;6P*'(8O.!"+&"!L?*fF"!8!2'q:$87L1)(&%7'2"+&ZRB*('*;>??y=,,'K"-?%2"+?u0(B$)B;6K&01>$$zSm8!7L1)&%('#4
[,'a0,"+01&D0&7*8?!L,"!??NA<0(BD"+?UC&!Sm[?)(!DO!B*RBD"!??U2"+?01?8+AW0(B*C2"Q0(&L?B*-'(=,2'PQ'&FC2'Q%7))B|'8!'!BD"!4R V'(2"h,"h-?)(!f"!ML1-)2'M"+YL1ZBD"&c7U;>T8K;~"+?8!!&NB*h o!&=,,'(8$0)&YC&!Smf?)>$!0#!?8!A]!0B*C2"+!R0&!L?B*4
HI$N&Z'-0,"+01&5&)?N5 o&B=&0"!01&d"+&6l7,'(2'3Y?)'>"!?Y6"!;>&>&UJ
ε > 045HI@J"!UUNk#JBD"+&Nl$;>??19?8&L1U01N./h'(=,N
;>ZaB*2"+-"+$ v &BD"+Ss\]!&&(SwjO77'(L)(&AO _!&B-)?"T2"!bM?8&*L1R"+00?'y4 *Z,"!88R! 6*+&3s;J[L,"&AO'8W.!"+&"+L?9]N&3001NDL"+&AP.!"!?)9;>????&)?NYP"D?2;&Z"+cRZ"!8!gyL):3C$! 9&2'()2'[Nk#YBD"+&Nl];>??5Z,"!8;>-6'(8E4Hm)(&)2g!2;./&2g!,"5N501L?6h!B*0)(dk#+ ,?2;&Z"!cZ,"!8Ux*%?NAR"!?*;>N!)Uf01N./9?2;&hL1),'#4 9x*%:.7NsA2"!?)?V"d2"+$$L16)2'$-J o&Z"!B*F;+&c-! #"h8!(L1!)&77'-2"+&Z@B*7'#4HI-2"!@$B*0?F]zFSI87L1)(&%7'P"!,'P"K)Lf! J*Sm8!7L1)&%7'P97./8"+2'Kf"+&Z] _!&:"$?%2"!??NAR!0BD"!?y!?)y4
HI]Y _!)&R0,"+01&>fN.7NsAK2"!?)?"+Y'./?012'K]YN&Z'K0,"!01F&h"&9)2'*8&Z"+f=&h"+,']0,"&"!?u'R!&Z'(&h"!00&2l(B">! f!?2"&: _)U))S"!??NAP0(&%fM01?!8!AP!0B*C2"+P0&!L?B*4$ *"!00(&2l(BD"!f"&$M)'T8&Z"+>)L0(&L?BD-s;J9'(k#F&%2e%)%"!?%8!&d0&!8!&Z"+B*BD8hB*7'6n v( HP"+,'v Hug7&0#./?NAqF4J!hB*7'(8!F&Z"+f"-2e%)h+ N&Z"!R01!%,">"!L1[0(&./bO7.&8RW"P?%2"+?h!0B$)B;>N<&01DPOzFSm87L1!)&%7'#4 B*7'(>"+&Y']!R!B*9'(Nk#&%601?!8!A]0(B*C"+]0&L?B*4
f= VR0,"+01&h'(B*&Z"+6"+> v( HbB*('K'.?012'K]f0&.7)U0"!01&2"!]L1f"+00?2']"+?$-hB*Z,"+B'(8]0&L?B;>NR&U&Z"+74HI]+&Z'&68&Z"+$$)L0&!L?B*hO"D i"+9;"AMBD"+??5'0?"+B*%f"+&@"!)B*2'yg#;>ZPB*0?,"+@]x*%D.7NsAb2"+?)?"+]'(&./2'bbKN&Z'b0,"+01&D2"!aL1R)'y4` 7)B*&2"+?#&)?>,'("+Y,">fB*F7'2"+RL1Y)2'@?2;J&UY&:"!'K??E8"- _)"!?EB*F"!BR4
HI[3Nl7T0,"!01F&h v HtB*7'['./?012'[[9 _)(&M0"!01&hh)2'T"+f"D010&%+&>-!LZ"!K?%2"+??NAK!0BD"!?1C&!SmY!?)>Z"&8- V&B^"3&!)'2'*?)
e1>A>D>
$!&&01,'(8R%7))f0&!L?BR4* @7)B*&2"+?&)?N9'2"+$,"f$B*('2"+R01& o!&B^;??u"!U"$01U0&%!&24
./%-0"!01&5"U+&2"!?0"!01&"+57./8"+J."+?V'(NsA3! B*B*?NA)'@01N./U?2;&L1!),'
ε@U'8*.!"+&"!L?d;>DZ"83"!'@?.78f01!?!8!A*!0S
B*C"+0(&L?B*J'='*K'(&C2'?"'2"&&A(8-%7))B&))&4 :BD"!&)?:0(&%2']&9>,"+:"!R0(B"+?
εSzY?)(![*"!
εSm01&)&L1'h'&F3B*S
B$)B ;8!%d0&!L?B ;>N@!&Z"+%!@B*0?"!g/.*\]&J"+,'-&Z"+@&8!A'Ng1U0BD"+?wg,"+ V&:&),'(8$*CF&!Smg@f+&&01'8I)01&)&L12'h'(S&60&!L?BR4 d?'5N 16&Z"+%d$601&)(&L#'-0&!L?B~"+&62"+& o)??NA*'(=,2'"!'
ε > 0>)x@%?NAKBD"!??i4
E01?!8!AW!0B*C2"+!Eg v &9!&Z"!%g0, 1
0(&L?B*g v N.7NsAM2"+?S)?V"gh8!7L1)&%7'b2"+&ZaB*7'(g v 2e%)%V"+?6%8&*0&8+&Z"!B*B*8(gXJB*0?"!%B*Z,"+B'8!Eg,!:0(&78
4PNQFN EQ?N EXDG=^G#T$<kNG<XDNPGVPE8eN M=?GGJL @fL M?>hJNj=?PQFGZ]PGZ=m=?PN LN:E#Q\MmEMfEJGW7=FGQ\E=?N'M=?ZJNSH=:t,bGQ\JM$WE#HKHG=$QFNSE#XAXALjN r<QFNSMFM =FPNm]Q\E=?>D=?ZaJN,ToNSNXs=FG EQFJMMG@RNGTLGZst
|L JNSN<kNSM=B]Q\E=F>D=?ZJN]G1NSMf=?G*QFG#ToNSMFM?GQR%QF>AM=?NSQB1e#E#HkNSQ?]VTqGQkNSXD>ANe1>AH] >AH @BNNSHGZ]P)=?G=\E#_N@BNGHmEMsP>AM[JGW7=FGQ\E=FNM=FZJNSH=Stbc>=FPGZ=[P>hMlM?Z<<kGQ=:`:NHW GZQ\E#]NS@RNSH=EHJ_1>AHJHNSMFM =FP>hM4=?PN:M>hM4@OE8LjHG=PE8eNkNNSHQF>=?=?NHlt
|L=FPE#H_M']G|E#XhMGI=?G|*QFG#ToNSMFM?GQ'$HaJNQ\M'xl>DHaJ Z>AM='ToGQmN@B<XAG8L1>AH]K@BNjEM'E|JGW7=FGQ\E=FNIM=?ZJNSH=jE#=R=FPN|0/>Ae1>AM?>AGH G#T'+<=?>A@B>DC:E=?>AGH EHJ 1LM=FN@OMB4PNGQFLt 4PNKG#=FPNQT EW ZXD=LI@RNS@!kNQ\M/E#HJ|JGW7=FGQ\E=?NmM=?ZJNSH=\M+E=$=FPN^0/>Ae1>hM>AGH|G#T,+<=?>A@R>ACSE#=?>AGHE#HaJ1LM=FN@OM4PNGQ?LjJNSM?NQFeN/=?PaE#H_M,ToGQ4<Q?Ge1>AJ>DH]RM?ZW\PIE#HNH#"G8LE$XDN%,GQ?_1>AH]^NHe1>AQ?GH@BNH=:tH <E#Q?=?>hW ZXhE#Q:`Y=FPNOWG@B<E#H1LG#T @fLQFG1G@B@OE=FNB=FN T EH $XA@'&Q:`[Et _kt EtR0+Qm($LWGHs`lPEMkNSNHj]Q?N:E=FXDLRE<<QFNSW >hE=FNSJR=FPQ?GZ]PGZ==FPN+LNSEQFMSt4PN/PGM?<>=\E#XA>=LBEHJO<kGM?>=F>DeN$=FP>DH_>AH])G#Tk0+E8e1>AJ^$H>hM?>1PEM(kNNSH^E+Q?N:E#X>AHM<>DQ\E=F>DGH'=FG+@BNt*)QFNSJQF>A_!+ E#QFXhM?M?GHs`8@fL'XAGH]/=F>D@BN<aE#Q?=?HNSQ9>DH9G<=?>A@R>ACSE#=?>AGHs~+b N,>DXAXE#X- E8LM+PE8eNB$GQ?QF_n<>DH]atf/NNSJ )M?E8L|@BGQFN/.0)GQP>AM<EMFM>AGH>AHKM<kGQ=\M`1P>hWFPI2kNXA>ANeN>hM4NeNSHI]QFNSE=FNQ,=FPE#HI@B>AHN`*UNQ2dH3e>hM=St[GeN4 ZaM=\E:e1> `u5>AXAXUHG#=)ToGQ?]N =)=?PE#=LGZ6kN:E=@BN'LGHNBMN:W GHaJ>AH87EM?EXDG<<kN =:t!)GQ'E#XAX=FPNToZHjJ>AMFW ZMFM>AGHMU[=FPE#H_^VE9"Em%'E#Q\EM?E#XAG%PGmMNSN@OMU=FGmM?PE#QFN4@fLRM?NHM?NG#TYP1Z@BGZQ:t:GPaE#HI%'E#QFXhM?M?GHs`ToGQLGZQ ToZH XAGe>AH]OE#HJN:EM?L9]G>AH]^<kNQ\MGHE#XA>D=Lt2>AM?PKE#XAXY=FPN;kNSM==FG|=?PN Q?NSXAE#=?>AeNXALVHN< ]ZLMS`|>A_E#NSX=)aE#XAXA]QFNH EHJ /HJNSQFMf|nXDXANQ:tI$XAX,G#TLGZ @OE#_NWG@B>AH]^=FG',GQ?_j>AH=FPN'@RGQ?H>DH]OERQ?N:E#X>"G8Lt
$=?PEH_|*QFG#ToN:M?M?GQE=FP>AEM1=?GXA<kN^ToGQ>AHe1>D=?>AH]I@BN^=?GK0) ToGQ'EIMPGQ=)QFNSM?NSEQFW\Pe1>hM>D=St4PaE#H_M =?GOP>A@v24EME#XAN)=FGOM?NNm> NSJNH>AH=FPN^+XAL1@R<>AW)]GXhJ >AHI>hW N'PGW\_NL>AH!#;t
dE@>AHJN<=FNSJ =?GBi#Q?=FNHjToGQE#X- E8LMdQFN@B>DHJ>DH]R@BN)G#Tl=?PN)=?P>AH]M,=FPE=$E#QFN/GTlQFNSEX>A@B<kGQ?=FEHW N)>AHKXD>DToNtd|LkNXAG8eN:JM?>AM=?NSQ%'EQ?>AHIJNSM?NQFeNSM,=FPE#H_1M4ToGQ_NNS<>AH]B@RNmM=?NSEJL>AH=?>A@BNSM+GTdJGZ =St'$XhMGa`a=?PNRW G@R<aE#HLKG#TdVE==F>AEM`uM?E>DXA>AH]jGQ="ZM=PEH]>AH] GZ=:`YPEMkNSNHKQ?N? E#Q\J>AH]t
xlEM=@Z=/HG#=$XDN:EM=S`,=?PEH_ @fLjT E=?PNQ$EHJA7/>AQF]>AH>hEfToGQ4=FPN>AQ$_>AHJKMZ<<kGQ?=St
1=?GW\_1PGXA@K` )N?Q?ZaE#QFL !#
VE=FMb NSQ?@BN
>r
A
tA $<<XD>hWE#=?>AGHMG#T[=FG<kGXAG]LOG<=F>D@B>ACSE#=?>AGH t'tt'tt'tt'tt'tt'ttt'tt't t ! EM?>hW)EM?M?Z@B<=F>DGH tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't t w +,GH=?>AHZZ@ =FG<kGXAG]LOG<=F>D@B>ACSE=F>AGHKtt'tt'tt'tt'tt'tt'tt'ttt'tt't !t )>DH>=FN'NXAN@BNH=/J>hM?WQ?N=?>ACSE=F>AGHt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't !t y +<=F>D@OE#XA>D=LW GHJ>D=?>AGHM tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't ;t ; E#r1>A@fZ@ M=?>kHNSMFMJN:M>A]HaMftt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't 6t 6 H=FNQF<kGXhE=F>DGH@RGJNXhM t'tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't 8!t =FQFE>DHINHNQF]LjJNHM?>=F>DN:M tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't St " =FQ?N:M?MW GHM=?Q\E#>AH=FMIt'tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't 8ytAS E>DHKWGH=FQ?>-Z=F>AGHM4G#T[=FP>hM4=?PN:M>hM tt'tt'tt'tt'tt'tt'tt'ttt'tt't :tA 1Z@B@OE#QFLjGT[=?PN'<E<kNQ\M t'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't :"tA:! NS@BEQ?_jGHI/G#=\E=F>DGH t'tt'tt'tt'tt'tt'tt'tt'tt'tt'tt'ttt'tt't !
"!#%$ &&
r1>
8
4P>AM+=FPNSM?>hMW GHaM>hM=\M+G#T E#H >DH=?QFGJZW =?>AGHE#HJ MNSeNH M?N<aE#Q\E=?NRE#<<kNSHJNSJV<E<kNQ\Mtm4PN<ZQ?<kGMNIG#T/=FPNK>DH=?QFGJZW =?>AGH >hM^=?G ]>AeNKE#H G8eNSQ?e1>AN< G#T/=?PNNXhJ GT/=?G<kGXAG]L G<=?>D@B>ACSE#=?>AGHst^H<EQ=F>AWZXhE#Q:`Y<QFG$XAN@OMJNHN:J GHJ>hMFW QFN =?>ACN:JVXDGEJ WEQ?QFL1>DH]IW GH=?>AHZZ@M=?QFZW =?ZQFNSME#QFNWGHM?>AJNQ?N:Jut
!"#$%&')(*+,-./0
4PN)ZaMNG#T[=?G<kGXDG]LBG<=?>A@B>DC:E=F>DGHIWE#HaN'M?NNSH>AHIEReEQ?>AN =LOG#TJ> YNQFNH=1NXhJMSt+ME#HNrE#@B<XAN+G#Ts=FG<kGXAG]LRG<=F>D@B>ACSE#=?>AGH G#T[XDGEJ WE#QFQ?L1>AH]fM=?QFZW7=FZQFNSMS`=?PN%,N>A]P= QFNSJZaW7=?>AGHGT=FPNXANSEJ>AH]VNSJ]N QF> aMf>AH E#>AQFWQFE#Tq=!>AH]MS`32 549`WSE#H kN@BNH=?>AGHN:Jut $XhMGa`>DH62 wy54 `E >DHaJ =?ZQ >AHNXhEJN @OE#>AH M?<E#Qf@OEJNjGZ=^G#T$WG@B<kGM?>D=?NSMm>hMfG<=F>D@B>ACNSJstVlG<kGXAG]LG<=?>A@B>DC:E=F>DGHRWEHRE#XhM?G kN4ZM?NSJf=?G'JN:M>A]H^NSXDN:W7=FQ?G=?PNSQ?@OE#XEW =?ZE#=?GQFM[>AHR@B>AWQ?GM?WSE#XAN`72 y#84 `EHJm>AHm=FPN4JN:M>A]HfGTaW G@R<XD>hE#H=@BNSW\PE#H>hM?@BMS`#M?NN:9<;>=?;2 "84ToGQUM?G@BN,N rE#@B<XANSMSt[4PN ZM?NGTa=FG<kGXAG]L'G<=F>D@B>ACSE#=?>AGHR>AHRGQFJNQ=FG]N =@OEHZT EW7=?ZQFE$XDN4<PG#=?GH>hW4W QFLM=\E#XM=FQ?ZaW7=?ZQFNSM>hMJNSMFW QF> kN:J^>DH@9<;>=?;A2ASw54EHJ2A849t[HB2 ;!C49`=?G<kGXDG]L'G<=?>A@B>DC:E=F>DGH^>hMZM?NSJ^>AHR=?PN$JN:M>A]HGTE#HfEW GZM=?>hWUPGQ?Hlt/=FPNQ[N rE#@B<XANSMsGTE#<<XD>hWE#=?>AGHMlWEH;kNToGZHJ'>AH9D;>=7;E2D:54EHJF2A:!C4EHJ =?PN'Q?NToNQFNHW N:M,=FPNQFN>AHst
)QFG@ HG9 GHs`YE#XAXl=?G<kGXDG]L G<=?>A@B>DC:E=F>DGH|<QFG$XAN@OM$>AH|=?P>hM$=FPNSM?>hM+E#QFNmJNHNSJ|ToGQXAGEJWSE#QFQ?L1>AH]RM=FQFZW7=?ZQ?N:M`NeNSHI>DT[=FP>AM>hMHG#=$N r<XA>AW>=FXDLM=FE#=?N:Jut
HG JIK EL) E EM,N+&
+,G@B@RGH^=FGmE#XAX<QFG$XDNS@BMW GHM?>AJNSQ?N:J^>DHB=?P>hM=?PNSM?>AM>AM=?PE#==FPN$@OE=FNQF>AEX>hMEMFM?Z@BNSJf=FGkN4>hMG=?QFG<>hW#`8=?PE#=>AMS`#>D=PEM[=FPNM?E@BN,@OE#=?NQF>hE#X1<Q?G<kNQ?=?>ANSM[>AHBE#XAXJ>AQ?N:W7=?>AGHaMt )ZQ=FPNQ:`#>=>hMEMFM?Z@BNSJf=?PE#==?PN$JN ToGQ?@OE#=?>AGHME#QFNM@OE#XAXE#HJ^=?PE#=XD>AHN:E#QNXhEM=F>AW>=LfPGXhJM`POQ;R9<;=FPE==FPNQFN>hMlEXA>DHNSE#QlQFNXhE=?>AGHkN= NNH'=?PNdM=?QFNSMFMsEHJ)=?PNdM=?Q\E#>AHst[4P>hMs>A@B<XD>ANSMs=FPNToGXAXDG9>DH]QFNXhE=F>DGHMS`uEXAM?G_1HG9HEM(/G1G_N?S M)XAE'`
ε = FσEHJ
σ = Eε` PNQFN
ε>AM)=?PNjM=FQFE>DH
eN:W7=FGQ:`σ
>hM=?PN^M=FQ?N:M?M$eNSW7=FGQ:`F
E#HaJE
EQ?N=FPNfM=?QFNSMFM =FG#M=?Q\E>AHKE#HJ|M=?Q\E#>AH =?G#M=?Q?N:M?M@OE#=?QF>AWNSMS`QFNSM?<aN:W7=F>DeNSXDLt[|GQ?NSG8eNSQS`8<XhE#HN4M=?QFNSMFM[>hMUZM?ZE#XAXALmEMFM?Z@BNSJu`#P>hWFPR@BNSEHM[=FPE=
! "!$#%'&()+*,
GHKEO!9J>D@BNHaM>AGHEXuJG@OE>DH=?PNSM?NQFNXhE=?>AGHaM kNSWG@BN
ε =
εx
εy
γxy
= Fσ =1
E
1 −ν 0−ν 1 00 0 2(1 + ν)
σx
σy
τxy
,
σ =
σx
σy
τxy
= Eε =E
1 − ν2
1 ν 0ν 1 00 0 1−ν
2
εx
εy
γxy
,
- tD/.
PNSQ?NE
>AM4=FPN.GZH]AS M@RGJZXAZM$E#HJν
>hM4=?PNm*G>AMFM?GH S M4QFE#=?>AGt(/NQ?Nεi
-σi
.JNHG=?N:M=FPNM=?Q\E#>AH - M=FQ?NSMFM0.l>AH^=FPN
i=?PB<QF>DHW>D<aE#XJ>AQ?NSW =?>AGHs`
γxy
>hM=FPNM?PNSEQE#HaJτxy
>AMU=?PN$M?PNSEQM=?QFNSMFMt)4PN^Q?N:MZX=\M/>DH=FP>hM/=?PNSM>hM)E#QFN`kPG9,NeNSQS`kHG#=)XA>A@R>D=?N:J|=FG <XhE#HNRM=?QFNSMFM Z=WSE#H>AHI<Q?>AHW >A<XANkNmE#<<XA>ANSJ>DTU<XAEHN'M=?Q\E#>AH>hMEM?M?Z@BNSJ>DHaM=FNSEJIG#T<XhE#HNmM=FQ?N:M?MSt
)21 3 003MM, E0%0 ')( 0+ ,-&.+M?M?Z@BN=FPE=E rN:J'JN:M>A]HmJG@BE>DH
Ω>AH
IR2GQ
IR3>hMl]>AeNHst[4PN,W GH=?>AHZZ@-=FG<kGXDG]L
G<=?>A@B>DC:E=?>AGHB<QFG$XDNS@ >AM=?G%HJOE'M?Z YJG@OE>DHΩ 46587 ⊂ Ω
P>hWFPO>hMKXAXANSJ'>=FPO@BE#=?NSQ?>hE#XM?ZWFP =FPE=$Ef]>DeNSH G$1"NSW =?>AeN+ToZHW7=F>AGH>hM @B>AH>D@B>ACN:JP>AXANE= =FPNMFE#@BN/=F>D@BN)=?PN]>AeNHW GHM=?Q\E#>AH=FMjEQ?N|HG= e1>DGXAE#=?N:Jut M?GXAZ=?>AGH =FG =?P>hMj>AH$H>=FN J>D@BNHaM>AGHEX$<Q?G XDNS@ >AME ToZHW7=F>AGH
χPE:e1>AH] =FPN <Q?G<kNQ?=L =FPE=
χ(y) = 1>DT
y ∈ Ω 46597 E#HJχ(y) = 0
>DTy ∈ Ω\Ω 46597 t 4PNmG$1"NSW7=F>DeNfWEHkN9<;>=?;a=?Gj@B>DH>A@B>DCSN'=?PNmWG@B<XA>AEHW N - @BE#r>D@B>ACN=FPN]XAG E#XM=F> YHNSMFM0.UG#Tk=FPN/M=?QFZW7=FZQ?N$EHJ^=FPN/W GHM=?Q\E#>AH=dWE#HRToGQ>AHM=\E#HWN$JN:M?WQ?>-kN$EXA>D@B>D=GH=FPNmE#@BGZH=GT@OE=?NSQ?>hE#XY=?PaE=/WEHAkN'ZM?NSJ>AHI=?PNmJN:M>A]Hst
9=l>hMs_1HG9H)=FPE=l=?PNdG<=F>A@BEXMGXDZ=?>AGH)=FG=?PNd@B>DH>D@fZ@5W G@B<XD>hE#HaW N<Q?G XDNS@->DH'=?P>hMM?N ==F>AH]f>AM,EfJNSM?>D]H,>D=?P EHO>AH$H>D=?N+H1Z@!kNQ,GTsM?@OE#XAXPGXANSMS`P2 w!54 tU4PZMd=?PN+<Q?G XAN@g>AMHG==,NSXDXa<kGM?NSJutHjGQ\JNQ=FGmG$=\E#>AHjE;,NXAX<kGMN:JR<QFG$XDNS@I`WGHM=?Q\E#>AH=\MW GH=FQ?GXDXA>DH])=FPN]NSG@BN =FQ?LBG#Ts=?PNJG@OE#>AH
Ω 46597 @OE:L,kN)EJJNSJO=FGf=?PN+<Q?G XDNS@gToGQF@fZXhE=F>DGHst+HN+<kGMM?> >AXA>=L^>hMU=FG'>AH=FQ?GJZW N<kNSQ?>A@BN =FNQdW GHM=?Q\E#>AH=FMS`M?NN 9D;>=7; 2 !6<4aE#HaJ2 549t4PNSM?N<kNSQ?>A@BN =FNQW GHM=?Q\E#>AH=FMfEQ?N ZMN:J =?G6kGZHJ =?PN H1Z@!kNQRG#T/WFPaE#H]NSM'ToQFG@ @OE=FNQF>hE#X=FGVeG>hJ EHJToQFG@veG>hJj=FGB@OE=FNQF>AEX t
HGQFJNSQ,=FGBM?GXAeN/=FPN)=FG<kGXAG]LBG<=F>A@R>ACSE#=?>AGH<QFG$XDNS@HZ@RNSQ?>hWEXDXALO>D=>hM4HNSWNSM?MFE#QFL=FGJ>AMFW QFN =F>DCSNm=?PNR<QFG$XAN@Kt4PN^@BGM=W G@R@BGHVE#<<Q?GEWFP|>hM/=FGJ>AMFW QFN =F>DCSNm=?PNRJNSM?>A]HJG@BE>DHIZM?>AH] H>D=?N'NSXDNS@BNH=\M$E#HJ =FP>hM>AM4=?PN=FG<>hW)GT[=?PNToGXDXAG>AH]OMN:W7=F>DGHst
): ; &,<=< >< ,?< 0@ EBAC</&./0H <Q\EW =?>hW NR=?PNjJNSM?>A]H JG@OE#>AH>AM'ZaMZEXDXAL J>AMFW QFN =F>DCSNSJ >AH=FGEI]Q?GZHJ M=?QFZW =?ZQFN'>D=?PnH>D=?NNXAN@BNH=FMSt 4PNKToZHW7=F>DGH
χ>hMBQFN?Z>AQFNSJ =FGkNW GHM=FEH=OGeNQONSEW\P NXAN@BNH=
P>hWFP @BNSEHM)=FPE=m>D=fWSE#H kNOQ?NS<XAEW N:J6L=FPNn9J>D@BNHaM>AGHEXJNSM?>D]HeE#QF>hE#XANReNSW7=FGQ
x = (x1, x2, . . . , xn)T`8>AHJ>hWE#=?>AH]/<QFNSM?NHWNGQE#M?NHaW NGT@OE=?NSQ?>hE#X>DHm=?PN,e#E#QF>DGZMaH>=FN
NXAN@BNSH=FMSt)1>DHaW Nm=FPN ZN:M=F>DGH|G#T<QFNSM?NHWNmGQ+E$MNSHW Nf>AM+Ej@OE==FNQ+G#TDLNSM>E'GQFDHG+E7`kEHE#=?ZQ\E#XUWFPG>AWNf>hM/=FGIJN aHNf=FPNBJNSM?>D]He#E#QF>AE$XANSM/EM
xj = 1PNSH=?PN
j=?P NXAN@BNH=)>hM
0 w
aXDXANSJ >D=?P @OE=FNQF>AEXEHJxj = 0
PNSH =FPNj=?P NXAN@BNSH=m>hMeG>AJstO4P1ZM
x ∈ 0, 1n`
PNQFN0, 1n
JNHG=?NSM4=?PNmM?N =$G#Tn9J>A@BNHM?>AGHEX >DHEQ?LjeN:W7=FGQ\Mt
bc>=FP|=?PN%aH>=FNfNXAN@BNSH=)J>hM?WQ?N=?>ACSE=F>AGHs`=?PNfJ>AM?<XhEW NS@BNH=NXhJKGTU=?PNf>AH$H>D=?NRJ>D@BNSHM>AGHaE#X4<Q?G XDNS@ >AMjE#<<QFG8r1>A@OE=FNSJ 1L EJ>hM?<XAEW NS@RNSH=OeN:W7=FGQ
u ∈ IRd`2PNQ?N
dJNHG#=FNSM=?PNmHZ@!kNSQ/G#TJN]Q?NSNSMG#TToQFNN:JG@ - JG#T M0.4>AHK=?PNfJ>AMFW QFN =F>DCSNSJKJN:M>A]H|JG@OE#>AHst)GQdM?>D@B<XA>hW >D=L`E#XAXHZ@BNSQ?>hWEXN rE#@B<XANSM>DHB=?P>AM=FPNSM?>AMdE#QFN$!8J>A@BNHM?>DGHE#XEHJ^=?PNSQ?NToGQFN=FPN$J>hM?W ZaM?M?>DGH^>AH^=?P>AMM?NSW7=F>DGH^>hM=FG'E)XAEQ?]N,N r1=FNH=QFNSM=?QF>AW =?N:Jm=?G)=FPN/!#0 WEM?Nt($G9,NSeNQ:`=FPN'@BN =FPGJM$JNSMFW QF> kN:JWSE#HI>AHI]NHNSQFEX kN'E#<<XD>ANSJ=?GOw#9J>D@BNHaM>AGHEXY<QFG$XAN@OMSt
4PN = GVNXAN@BNSH=^=L1<kNSM^ZMN:J >AH =FPNH1Z@RNSQ?>hWEX,N rE@R<XDN:Mf>DH =?P>AMf=FPNSM?>hM^EQ?Nj=?PNZEJQ\E=?>hWmToGZQHGJN - ,.$NSXDNS@BNH=mE#HaJ|=FPN,ZaEJQ\E=F>AWfH>AHNBHGJN - " ./NXAN@BNH=S`lM?NN)>A]ZQFNtAE#HJRt !t[4PN $E#HJ "4NXAN@BNSH=FM[E#QFN=?PN@RGM=[W G@R@BGHXDL+ZMN:J)>AH=?PN)aNXhJGT=FG<kGXDG]LjG<=?>A@B>DC:E=F>DGHKG#TXDGEJKWEQ?QFL1>AH]BM=?QFZW7=FZQFNSM@Z=$=FPNm@BN=?PGJM/JNSMFW QF> kN:JI>AH=FP>hMl=?PNSM?>AMEQ?NdHG#=QFNSM=?QF>hW7=?N:J=?G/=?PN:MN,NSXDNS@RNSH==L1<kNSMSt )GQ[@BGQFNd>DHToGQF@BE#=?>AGHmQFN]EQFJ>AH]=FPNSM?N'NXAN@BNH=\M$E#HJG=?PNSQFMM?NN#9<;>=?; 2 !#549t
(−1,1)
1 2
34
(−1,−1) (1,−1)
(1,1)
ξ
η
)>A]ZQFN tA~ 4PN H>D=?NNSXDNS@BNH= >AHXAGWSE#X[W G1GQFJ>AHE#=?NSM
(ξ, η)t$4PNfXAGWE#XlHGJN
H1Z@!kNQF>DH]B>hME#XhM?GBMPGHst
(−1,1)
1 2
34
(−1,−1) (1,−1)
(1,1)
5
698
7
(−1,0)(1,0)
(0,0)
(0,1)
(0,−1)
ξ
η
)>A]ZQFN t !~ 4PN H>D=?NNSXDNS@BNH= " >AHXAGWSE#X[W G1GQFJ>AHE#=?NSM
(ξ, η)t$4PNfXAGWEXsHGJN
H1Z@!kNQF>AH]B>hME#XhM?GBMPGHst
[G >DXAXAZM=FQFE#=?NV=FPN J>hMFW QFN =?>ACSE#=?>AGH <Q?GW NSJZQ?NE - eNSQ?L W GE#Q\MN.FaH>=FNNSXDNS@RNSH=|J>AMWQ?N=?>ACSE#=?>AGHKGTE JNSM?>D]HJG@OE>DH|>hM+MPGH|>AH6)>D]ZQ?Njt wt/($NQFN`a=?PNRJG@OE#>AH|PEM kNNSHJ>AMFW QFN =F>DCSNSJI>DH=?G
n = 3ZaEJQ\E=F>AW+ToGZQ$HGJN:H>D=?NmNSXDNS@BNH=\M$]>Ae>AH]O>=+ER=FG#=\E#XsGTUN>A]P=
HGJNSM,E#HJd = 12
JG#T MStU/G#=FN$=?PE#=d=?PN+J>hM<XAEW N@BNSH=FM,WGQFQ?N:M<kGHJ>AH]=?GfHGJNm+E#HJO
2
3
5
6
7
8
1 4
3
1
2
u1
u2
u3
u4
u5
u6
u7
u8
u9
u10
u11
u12
)>D]ZQ?NBt w~ 5M?@OE#XAXuN rE@R<XDNGTUE H>D=?N'NXAN@BNSH=/J>hM?WQ?N=?>ACSE#=?>AGHst
EQ?N`EMdE'QFNSM?ZXD=dG#TY=FPN+rE=F>DGHM`HG#= DTqQFNN E$JN]Q?NSNSMGTYToQ?NSNSJG@ E#HJB=?PNSLBE#QFN4=?PNSQ?NToGQFN
"!$#%'&()+*,
HG=4<EQ=G#TutUbc>D=?PI=?P>hMJ>hM?WQ?N=?>ACSE#=?>AGHs`=FPN'W GHM=FQ?ZaW7=?>AGHG#TEHN r1=?NSQ?HaE#XYXDGEJ eNSW =?GQ
]>AeNSHA1Lp ∈ IRd
>hM$M=FQFE>D]P=ToGQ 4E#QFJut/MFMGW>AE#=?N:J >D=?P'N:EW\P H>D=?NdNXAN@BNH=l>hMlE#HNSXDNS@BNH=M=F> YHN:M?Ms@OE#=?QF>r
Ke ∈ IR2m×2m`
PNSQ?Nm
>AM =?PNHZ@!aNSQG#T[HGJN:M >DH=?PNNSXDNS@RNSH= - =?PN)T EW7=?GQ!f>hM,ToGQEB!89J>D@BNHaM>AGHEXNXAN@BNSH=0. t )GQ$E#HINSXDNS@RNSH= >D=?PKZH>D=/E#QFNSERE#HJZH>==FP>hWF_1HN:M?MS`1=?PN@OE#=?QF>r
Ke
WEHAkNN r<QFNSMFM?NSJEM
Ke =
∫∫
Ωe
BT(ξ, η)EB(ξ, η)JξJη =
g∑
i=1
ωiBT(ξi, ηi)EB(ξi, ηi),
- t !.
PNSQ?Ng
>AM =FPN=?G#=\E#XuH1Z@!kNQ$G#T 4 E#ZMFM <kG>DH=FM4ZaMN:J >AH=FPNH1Z@BNQF>AWSE#Xu>DH=?NS]Q\E=?>AGHE#HaJωi
>hM^=?PN,N>A]P=RGT$=FPNi=?P 4 EZMFM^<kG>AH=
(ξi, ηi)t |GQ?N>AHToGQF@OE=?>AGH GH HZ@RNSQ?>hWEX
>AH=FN]Q\E=F>DGH|WSE#H kNmToGZHaJ|>AH 9<; =7;)2 1!C4[GQ 2 y#w549t/4PNf@OE#=?QF>rE
>hM+JN aHNSJ>DH - tA.$EHJ=FPN
3 × 2m@BE#=?QF>r
B(ξ, η)>hM4]>AeNSHAL
B(ξ, η) =
∂∂ξ
0
0 ∂∂η
∂∂η
∂∂ξ
[
φ1(ξ, η) 0 . . . φm(ξ, η) 00 φ1(ξ, η) . . . 0 φm(ξ, η)
]
,- t w .
PNSQ?Nφi(ξ, η), i = 1, . . . ,m
E#QFNB=FPNEM>hM'ToZHW =?>AGHMfToGQf=?PN ]>DeNSH NXAN@BNH=^>AH =?PNXAGWEXkW G1GQFJ>AHE=FNSM
(ξ, η)t4PN EM>hMToZHW7=F>AGHM PE:eN$=FPN+<QFG<kNSQ=L^=?PE#=
φi
>hM,N?ZE#Xa=?G1>AH^=?PN
i=?PBHGJNE#HJ^N?ZEX=?G
0>AHBE#XAXG#=FPNQHGJNSMSt )GQ=FPN H>D=?NNXAN@BNH=U=FPN EM>hM
ToZHW =?>AGHM'EQ?N >AXA>DHN:E#Q'E#HJToGQ)=?PNBH>AHNRHGJN "#H>D=?NBNSXDNS@RNSH==?PN'EM>hM/ToZHW7=F>DGHME#QFN%> ZEJQ\E=F>AWt
4PN/NSXDNS@RNSH=4M=?>kHN:M?M,@OE=FQ?>DrB>AHjXAGWEXaW GGQ\J>DHaE=?N:M`Ke
`>AXAX kN)EfML1@B@BN =FQ?>hW$<kGM?>=F>DeN$M?N@B>hJN H>=FN
2m×2m@BE#=?QF>r >D=?POQFEH_
2m−3>DTuE'!9J>D@BNHM?>AGHEX7aH>=FNNSXDNS@BNH=
>hM/ZaMN:Jut)4PNfQFNSEMGHK>hM/=?PE#=+ToGQE M>AH]XDN H>=FN^NXAN@BNSH=S`sE=)XDN:EM=/=?PQFNNRJG#T M+@fZM=%kNrNSJ|>AH|GQFJNQ=FGj]N=+EjM=FE#=?>hWEXDXALIJN=?NQF@B>DHNSJ|NXAN@BNH= - = ,GJG#T M$ToGQ/=?Q\E#HM?XhE=?>AGH|EHJGHNToGQ/Q?G=FE=F>DGH .7t )GQ=?PN:MNmNXAN@BNSH=FMS`=FPNmQ\E#H_ GT=?PNmNXAN@BNH=+M=?>kHN:M?M$@OE=FQF>r>hM+yToGQ=?PN BNXAN@BNH=+E#HJ 8ymToGQ4=FPN "^NSXDNS@BNH=:t
HGQ\JNQ$=FG ]N=)Ej@BE#=?QF>Dr=FPE=JN:M?WQ?>-kNSM$=?PNRM=?>kHN:M?M+GTU=FPN^NH=F>AQ?N^M=?QFZW7=FZQFN'=FPN]XAG E#XM=?>kHN:M?Mm@OE=FQ?>Dr PEM=?G kN W G@B<Z=?NSJutI4P>AM EM?>hWE#XAXALQ?N/Z>DQFNSM'E|MZ@B@OE=F>AGHGeNQ+=?PNBNXAN@BNH='M=?>kHN:M?M@BE#=?QF>hW NSMS` Z=mE=?=?NSH=F>DGHVPEM/=FGAkNR<aE#>hJ|=?G=FPNBJ>YNQ?NSHW N>AH XDGWSE#X$E#HJ ]XAG$EXWG1GQ\J>AHE=FNSMR>DH =FPNBH>D=?NNXAN@BNH= J>hM?WQ?N=?>ACSE#=?>AGHstc4P>AMO<QFGW NJZQFN^>hM)WEXDXANSJ|=?PNBEM?M?N@!XA>AH] G#T=?PN^]XDG E#XM=?>YHNSMFM+@OE=FQ?>Drut+M)EHNrE@B<XDN^GT=FPN=FQFEHMToGQ?@OE=F>AGH kN = ,NNSHXDGWE#XYEHJ ]XAG$EXkWG1GQ\J>AHE=FNSMS`WGHM?>AJNQ,=?PN)]XAG$aE#XkHGJN;^>DH)>A]ZQFN t wO=FPE=>AMW GHHNSW7=FNSJ|=FGAkG#=?PVNSXDNS@BNH=m! E#HJVwt;)QFG@)>A]ZQFN tDf>D=>AMW XANSE#Q=FPE=4=?PNXDGWSE#XYHGJN'!^G#T[NXAN@BNSH=/!j=?PNXAGWEXkHGJNR)G#TNXAN@BNH=/wRW GQFQFNSM?<kGHJB=FG=FPN^]XAG$EXlHGJN^;t)4P>hM/@BN:E#HM/=?PE#=+=?PN^XAGWSE#XJG#T M)w EHJ| - =?PNSQ?NREQ?Nm= GJGT M$ToGQNSEWFPOHGJN.UToGQdNSXDNS@RNSH=4!R=?PN/XAGWSE#XaJG#T M//E#HJj!)ToGQdNXAN@BNSH= wmW GQ?QFNSM?<kGHJ^=?G'=FPN]XAG E#XuJG#T M/6^EHJIt
xsN =Kj ∈ IRd×d
kN'=?PNm]XAG$EXuN?Z>AeEXDNSH=$GT=?PNmNSXDNS@RNSH=+M=?>YHNSMFM$@BE#=?QF>rKe
ToGQ=FPN
j=?PKNXAN@BNH=G =FE>DHNSJIETq=?NSQ4=?PN=?Q\E#HMToGQF@OE=F>DGHOToQFG@ XAGWSE#Xk=FGB]XAG$EXYWG1GQ\J>AHE=FNSM
0 y
P>AW\P EM QF>DNLfJNSMFW QF>-kNSJ^E$kG8eNtl2eNSH^=?PGZ]P^=?PNM?>DCSN GT=FPN4@OE#=?QF>rKj
>hMU]QFNSE#=?NSQ=FPE#HI=?PNmM?>ACNG#T=?PN'@OE=FQ?>Dr
Ke
`1=FPN'Q\E#H_ >DXAXlM=?>AXDX kN2m − 3
tbc>=FPK=FPNSM?N'NXAN@BNSH=+M=F>kHN:M?M$@OE=FQ?>hW N:M4>AHK]XDG E#XsWG1GQ\J>AHE=FNSMS`=FPNm]XAG$EXsM=?>kHN:M?M
@OE#=?QF>rK(x) ∈ IRd×d
>hM4]>DeNSHA1L
K(x) =n∑
j=1
xjKj ,- t .
PNQFN`EM!kN ToGQ?N`n
>AMf=?PNHZ@!kNSQ^GT$NXAN@BNH=FM^>DH =?PNH>D=?NNSXDNS@RNSH=R@BGJNSX tV9=R>hMEM?M?Z@BNSJ =?PE#=
K(e)>hM<kGM?>=F>DeN'JNH>D=?N` PNSQ?N
e = (1, . . . , 1)T ∈ IRnt
4 >AeNH =?PN ]XDG E#XM=F> YHNSMFM@OE#=?QF>r =FPN J>AM?<XhEW NS@BNH=KeNSW7=FGQ:`u ∈ IRd
`+WSE#H kNWG@B<Z=FNSJKEMEBM?GXAZ=?>AGH=?GB=?PN'NXhEM=F>AW)N?Z>AXA> QF>AZ@vN?ZE=F>DGHM
K(x)u = p,- t y .
<Q?Ge1>AJN:J|=?PaE=)=FPNSM?NRN/ZE=F>DGHM+PE8eNREIM?GXAZ=?>AGHltfgHNSWNSM?MFE#QFL|W GHaJ>=F>DGHVToGQ - t y .$=FGPaE:eNIE M?GXAZ=F>DGH >hM^=?PE#=
p ∈ R(K(x))`OQ;R9<;
p@fZM='kNXAGH]V=FGV=FPNIQFEH]N GT
K(x)t
(/G NeNQ:`EM2>AXAX kN'J>AMFW ZaM?M?NSJ XAE#=?NQ:`=?P>hMW GHJ>D=?>AGH>AM4HG#=M?ZjW >ANH= ToGQ - t y .d=FG^PaE:eNERZH>-ZNmM?GXAZ=?>AGHI>DT
x ∈ 0, 1nt
XAG#=mGT,=FPNO<Q?G XDNS@OM'W GHM>hJNSQ?N:JV>AH=?P>AM'=?PN:M>hMmJN:E#X >D=?P =?PNj>DHWGQF<kGQ\E=?>AGH G#TeGHj|>AM?NSM,M=FQ?N:M?M?NSM>AHO=?PN/<QFG$XDNS@ ToGQF@fZXhE=?>AGHst4PN/M ZE#QFNSJReGHj|>hM?NSMdM=?QFNSMFM>AHO=?PNNSXDNS@BNH= - >TU>D=$>hM3XAXANSJ>=FPI@OE#=?NQF>hE#X'.d>hM$W G@B<Z=FNSJKEM
σ2e = uT
e Seue,- t ;.
PNQFNue ∈ IR2m
>hM=FPN^J>AM?<XhEWN@BNH=/eNSW =?GQ/W GQ?QFNSM?<kGHJ>AH]R=FGO=?PNfNSXDNS@RNSH=)E#HaJI=?PN@OE#=?QF>r
Se ∈ IR2m×2m>AM]>AeNHL
Se =
∫∫
Ωe
BT(ξ, η)EMEB(ξ, η)JξJη =
g∑
i=1
ωiBT(ξi, ηi)EMEB(ξi, ηi),
- t 6 .
PNQFNE`B
EHJωi
EQ?N)=?PNmMFE#@BN'EME#kGeNEHJAPNQFNM
>AM$JN aHNSJKEM
M =
1 − 1
20
− 1
21 0
0 0 3
.- t .
4P>hM)@BN:E#HM+=FPE=σ2
e
>AM)=?PN',N>A]P=?NSJE:eNSQFE]NfG#T,=?PNOM ZE#QFNSJeGH >AM?NSM'M=FQ?N:M?M?NSM>AH =?PN 4 EZM?M^<kG>DH=FMBE#HJ EXAM?G=FPNKE:eNSQFE]N M ZaE#QFNSJ eGH |>hMN:MRM=?QFNSMFMf>AH =FPNPGXDNNSXDNS@BNH=:td9=/PGXhJM4=FPE=/QFEH_
(Se) =QFEH_
(Ke) = r` OQ;R9D;aToGQ4=FPN'= GjJ> YNQFNH==L1<kNSMGT
NSXDNS@BNH=\M - OE#HJ ".d=?PN'@OE=FQF>rSe
>AMS`XD>A_NKe
`G#TQ\E#H_yREHJ :y`1QFNSM?<kNSW =?>AeNXALtxsN=
Sj ∈ IRd×dkN=?PNI]XAG$EX,N?Z>AeE#XANH=^GT=FPNI@BE#=?QF>r
Se
G =FE>DHNSJ ETq=?NSQ^=?PN=FQFEHMToGQF@BE#=?>AGHTqQFG@ XAGWEXY=FGO]XDG E#XlW G1GQFJ>AHE#=?NSMStd+M4ToGQ
Kj
`>D=/PGXhJM=?PE#=$=?PNmQ\E#H_GT
Sj
>DXAX kNrt
; "!$#%'&()+*,
) $,NK 8( 0 @&&0 4PNdJ>hMFW Q?N=?>ACSE#=?>AGH%1L/=?PN)H>D=?NNSXDNS@BNH=[@RN=?PGJ)@OE_NSMu>D=l<kGMFM>-XANU=?GToGQF@fZXAE#=?NU=FG<kGXG]LBG<=?>A@B>DC:E=F>DGH <Q?G XDNS@BM4JNHNSJGHH>D=?N'J>A@BNHM?>DGHE#XuM?<EW NSM >DHaM=FNSEJ GTlToZHW7=F>DGHM?<EWNSMSt($G9,NSeNQ:`$kN ToGQ?N/J>AMFW ZMFM?>DH]fM?GXAZ=?>AGHO@BN =FPGJM=?PN/>AMFM?ZN$G#T DM?GXAZ=?>AGH,EEHJR=FPNZNSM=?>AGHIGTUPG9-=?GjW\PEQFEW7=?NSQ?>ACN'ED]G1GJ EM?GXAZ=?>AGHK@fZM=5kNfEJJQFNSMFMN:Jut,4P>hM$M?NSW =?>AGH>hM4JNeG#=FNSJj=FG^ToZHW =?>AGHM$JN HN:JGHIWGH=F>DH1ZGZaM4M?N =FME#HJ =?PZaM4HG#=4ToGQ4=?PN'J>hMFW QFN =?NM?N =0, 1n
GHP>AW\P =?PN+=FG<kGXDG]LRG<=F>D@B>ACSE#=?>AGH <QFG$XAN@>AM ZM?ZE#XAXALB<kGM?NSJut$GHN =FPNXANSMFMS`>D='>hMeNQFLWG@B@BGH >AH<Q\EW =?>hW N^=FGKWGHM?>hJNQ)=FPNOWGH=?>AHZGZM)QFNXhErE#=?>AGHV<kGM?NSJVGHV=FPNW GH=F>DH1ZGZM[MN=
[0, 1]n- GQ[E$W GH=F>DH1ZGZM[E#<<QFG:r>A@OE=F>DGH)<kGM?NSJGH
[ε, 1]nToGQ[M?G@BN
ε > 0`
M?NN'1N:W7=F>DGH t ;tA .7t|GQ?NSG8eNSQS`1W GH=?>AHZGZM QFNXhErE#=?>AGHM E#QFN+G#Tq=FNH=?QFNSE#=?N:Ju`ToGQ4N rE#@B<XAN>AH E6Q\E#HaWFPE#HaJ1 kGZHJ WGH=FN r1=S`,EMR>AH 9D;>=7;+2 !C4$E#HJ62 4 `>AH =?PNK<QFGW N:M?M^G#T)MGXDe1>AH]=FG<kGXAG]LjG<=?>A@B>DC:E=?>AGH|<Q?G XAN@OM$JNHNSJ|GH|J>AMFW QFN =FNmM?N =FM/M?ZW\P|EM
0, 1nt44P1ZM=FPN
W GHW NS<=/G#TM?GXAZ=F>DGHI=?GO=FP>hM$QFNXhErNSJI<QFG$XDNS@ >AM$>A@B<kGQ?=FEH=St|GQ?N'>AHToGQF@OE=F>DGH|E#kGZ=Q\E#HW\PE#HJ1 kGZHJj@BN=?PGJMWSE#HkN)ToGZHJI>DH 9<;>=?; 2 w6<49t
HOGQFJNQ=FGmToGQF@BEXD>ACN=FPN+W GHW NS<=,G#TlMGXDZ=F>DGHjW GHM>hJNQ=FPN$ToGXDXAG9>DH]m]NHNSQFEXHGHXA>DHNSE#QG<=?>A@B>DC:E=?>AGHI<QFG$XAN@
minx
f0(x)Mt =St
fi(x) ≤ 0, i = 1, 2, . . . ,m,
- t " .
PNSQ?Nfi : IRn → IR, i = 0, 1, . . . ,m
E#QFNWGH=F>DH1ZGZM?XAL J>kNSQ?NSH=F>AE$XDNToZHW =?>AGHMSt4PNR]GE#X PNH MGXDe1>AH]IE#H G<=F>D@B>ACSE#=?>AGHV<Q?G XDNS@ GHV=?PNRToGQF@ - t " .+>AM)=?GHJE
ToNSEM>-XAN/<kG>AH=x∗
M?ZWFPO=FPE=f0(x
∗) ≤ f0(x)TqGQ E#XAX
x ∈ F`PNQFN
F>hM=FPN$ToN:EM?> XDN+M?N =
G#T - t " .4]>DeNSHLF = x ∈ IRn | fi(x) ≤ 0, i = 1, . . . ,m
t1ZWFPEB<kG>DH=/>AM/WEXDXANSJKE]XAG E#Xk@B>AH>A@R>ACNSQ - GQ4]XDG E#XYG<=F>D@fZ@ . G#T - t ". tNSXAE#=?NSJ =?G^=FPN]XDG E#XkG<=?>A@fZ@v>AMEXAGWEXsG<=F>D@fZ@P>hWFP|>hM4=FPN kN:M=/<kG>AH=/>AHEBHNS>D]PkGZQFPG1GJM?ZQ?QFGZHJ>AH]R=?P>hM$<kG>DH=S`OQ;R9<;EfToNSEM>-XAN+<kG>DH=
x∗>hM4E^XDGWSE#XkG<=?>A@fZ@ >DT
f0(x∗) ≤ f0(x)
ToGQ4EXDXx ∈ F
MZW\P =?PE#=‖x∗ − x‖ ≤ κ
ToGQ$MG@BNκ > 0
` PNSQ?N‖ · ‖
>AM$EBM?Z>=\E#XANHGQ?@KtlG]N =)<Q\EW =?>hWE#XAXALZaMN ToZXG<=?>A@OE#XA>=L|WGHJ>D=?>AGHaM+M?G@BNf_1>DHaJ|GTdWGHM=?Q\E#>AH=%ZE#XA>
WE#=?>AGHK>hM$HNSNSJN:Jut44PNfW GHM=FQFE>DH=5ZaE#XA> aWSE=F>DGHI]ZE#Q\E#H=?NSNSM =FPE=+EjJNSMFW QF>A<=?>AGHKGT=FPNToNSEM>-XANfM?N =+GT - t ".@aEM?NSJKGH|XA>DHN:E#QF>DC:E=F>DGHMG#T
fi, i = 1, . . . ,m>AM+M?ZjW >ANH=?XALKEWWZ
Q\E=FNt4PN'EM?M?Z@B<=F>DGHGTXD>AHNSEQ?XALj>DHJN<kNHJNH=/]Q\EJ>DNSH=FM4G#T[=?PN'EW7=F>DeN'W GHM=?Q\E#>AH=\M >AMW G@B@RGH^EHJf>=>A@R<XD>ANSM - E@RGH]+G=?PNSQ[=FP>DH]M0.s=?PaE=U=?PNMGWEXDXANSJ Q\M=UGQ\JNSQWGHM=?Q\E#>AH=ZE#XA> aWE#=?>AGHM$PGXhJut )GQ+EO@BGQFNmJN =FE>DXANSJJ>hM?WZM?M?>AGHKGHW GHM=FQFE>DH=5ZEXD> aWE#=?>AGHaM$MNSN9<;>=?;)2 ;84 t+4PNfG<=?>A@OE#XA>=LKWGHJ>=F>DGHM+E#QFNfM=\E=FNSJ|>DH=?NQF@OM/G#T=?PN^x[E#]Q\E#H]>hE#HIToZHW7=F>DGHL(x,λ)
PNQFNλ
>hMm=?PNeNSW =?GQfG#T/xlE]Q\E#H]NO@fZXD=?>A<XA>DNSQFMSt 4PNxlE]Q\E#H]>hE#HTqZHW7=F>DGHW GQ?QFNSM?<kGHJ>AH]f=FG - t " . >hMJNHNSJKEM
L(x,λ) = f0(x) +m∑
i=1
λifi(x).- tAS.
4PN,MG/WEXDXANSJ:QFM=lGQFJNQsHNSW N:M?MFE#QFL+W GHJ>D=?>AGHM[M=\E=FNSMu=FPE=[>DTx∗
>AM[EXAGWEX@B>DH>A@B>DCSNQG#T - t ".4E#HJ=?PN%QFM=$GQ\JNQWGHM=?Q\E#>AH= ZEXD> aWE#=?>AGHM4PGXAJKE=
x∗`=FPNHI=FPNQFNNr>AM=FM$E
0 6
eN:W7=FGQλ∗ ∈ IRm
MZW\P=FPE=
∇xL(x∗,λ∗) = 0,- tA.
λ∗ ≥ 0,
- tA:!.
fi(x∗) ≤ 0, i = 1, . . . ,m,
- tD:w.
λ∗
i fi(x∗) = 0, i = 1, . . . ,m,
- tDS .
PNQFN∇xL(x∗,λ∗)
JNSHG#=FNSM,=?PN)]QFEJ>ANH= G#Tl=?PN'xlE#]QFEH]>AE#H >=FPQ?N:M<kN:W7=,=?Gx`Ne#E#XD
ZaE=?N:JK>DHK=FPNm<kG>DH=(x∗,λ∗) ∈ IRn × IRm
t4PN%Q\M=/GQ\JNSQG<=?>A@OE#XA>=LIW GHJ>D=?>AGHM/E#QFNEXAM?GR_1HG9HKEM =?PNm%'EQ?ZaMP%ZPHlZaWF_NQ - %'%' . W GHJ>D=?>AGHMS`/2 w54lE#HaJ 2 ww549t
|GM=f<Q\EW7=F>AWSE#X@BN =FPGJM'ToGQfM?GXAe1>DH]KG<=?>A@B>DC:E=F>DGH <QFG$XAN@OM=FQ?LV=FG HJ EK<kG>AH==FPE=+MFE=?>hM NSM4=?PN:MN%QFM=$GQ\JNSQ$W GHJ>=F>DGHMt,(/G,NSeNQ:`M?>AHW N=FPNSM?NmWGHJ>D=?>AGHaMEQ?N'GHXALHNSW N:M?MFE#QFL`=?PNSQ?N^>AM+HG ]ZaE#Q\E#H=FNN'=?PE#=Ej<kG>AH=M?E#=?>hMToL1>AH] - tD .9 - tDS .$>hM+EjXAGWSE#X[G<=F>D@fZ@ ToGQER]NSHNQ\E#XYHGHXD>AHNSEQ4G<=F>A@R>ACSE#=?>AGHI<Q?G XDNS@KtUHT EW =S`ToGQ]NSHNQ\E#XYHGHXD>AHNSEQG<=?>A@B>DC:E=F>DGHI<QFG$XDNS@BM4=?PNQFNmE#QFNHGO<Q\EW7=F>hWE#Xs@RN=?PGJM4=FPE=+E#QFN]ZEQFEH=FNNSJ =?GjWGHeNSQ?]NINSeNH =?G E XDGWSE#XG<=?>A@fZ@Kt0)GQjM?G@BN|M?<kNSW>AEX<QFG$XAN@OMS`,PG9,NSeNQ:`,@BGQFN|WE#HkNVM?E>AJ E#kGZ=jE <kG>AH=M?E#=?>hMTqL1>AH]=FPN QFM=jGQ\JNQOHN:W N:M?MFE#QFL WGHJ>D=?>AGHaMt +HNMZW\P>A@B<kGQ?=FEH=$W XhEMFM4G#T<QFG$XAN@OM4>hM =?PNmW GH1eN r <QFG$XAN@OMSt
! "$#
MN=C ⊂ IRn
>AMdEmW GH1eN rBMN =d>DTαy1 +(1−α)y2 ∈ C
ToGQ,E#XAX<kG>DH=FMy1,y2 ∈ C
EHJRToGQEXDX
α ∈ [0, 1]t|GQ?NSG8eNSQS`EBToZHW =?>AGH
f : C → IR>hM/WEXDXANSJ|W GH1eN rI>TTqGQ+E#XAX
y1,y2 ∈ C>D= PGXAJMd=?PE#=f(αy1 +(1−α)y2) ≤ αf(y1)+ (1−α)f(y2)
tU9Tu=FPN)W GHeNr^ToZHaW7=?>AGHf>hM Em= >AWN)WGH=F>DH1ZGZaMXALOJ>kNSQ?NSH=?>hE#XDN+TqZHW7=F>DGHj=FPN)@OE=?QF>DrOG#T[MN:W GHaJjJNSQ?>AeE#=?>AeNSM - =FPN
(/NSMFM>hE#H|@OE#=?QF>r.H(y)
>DXAX(kNf<kGM>D=?>AeNfM?N@B>hJN H>=FNmToGQ+EXDXy ∈ C
t/$HVG<=F>D@B>ACSE#=?>AGH<Q?G XDNS@ GH=?PNToGQ?@ - t " . >hMWEXDXANSJIWGHeNrj>DT[=FPNToNSEM?>-XANM?N =
F>hMEBW GHeNr M?N =$EHJ
f0
>hMEBWGHeNrjToZHW7=F>DGHKGHF
t WGHM?N?ZNSHW NfGT,W GH1eN r>D=LK>hM/=?PaE='E#HLKXAGWSE#X[@B>AH>D@fZ@ GT,E WGH1eN r|<Q?G XAN@ >hM
EXAM?GREf]XDG E#Xk@B>AH>D@fZ@Itd|GQFNGeNQ:`EHLO<kG>AH=4=?PE#=MFE=?>hM NSMd=FPNaQFM=4GQ\JNQ HNSWNSMFM?EQ?LWGHJ>D=F>DGHMm>AM^E|XDGWSE#Xd@B>DH>A@fZ@ E#HaJ=?PZaM^E#XhMGE|]XDG E#X@B>AH>A@fZ@Kt )GQ^EK=?PGQ?GZ]P=FQ?N:E=F@RNSH=$G#TUWGHeNr>=L MNSNF2 1yC4sGQ%2 !#549t
% & '1, M, ?$)( < 7 @ < E '0
4PN@RGM=@,NSXDXsM=?ZaJ>DN:J =?G<kGXAG]LBG<=F>D@B>ACSE#=?>AGH<QFG$XDNS@>AH=?PN'XD>D=?NSQFE#=?ZQFN)>hM,=FPN@OE#r>A@fZ@ M=F> YHN:M?M$<QFG$XAN@K`aMNSN9D;>=7; 2A:!54 ` 2 !#w84 ` 2 "549` 2 y!C49` 2 ;54[E#HaJ=?PNmQFN ToNQFNHWNSM4=?PNSQ?NS>DHst4PNB@OEr>D@fZ@ M=?>kHN:M?M'<QFG$XAN@ >hM@BGQFNBW G@B@BGHXDL_1HGH EM)=?PNO@R>AH>A@fZ@ W G@R<XD>hE#HWN<QFG$XAN@K`1PNSQ?N)=FPNmW G@R<XD>hE#HWN)>hM$JN HN:JKEM
pTut
4 >AeNH|EHIZ<<kNQ5kGZHJV > 0
GH=FPNfE#@BGZH=$GTU@BE#=?NSQ?>hE#Xu=?PE#=/WSE#HkN'ZM?NSJs`=FPN
"!$#%'&()+*,
@B>AH>D@fZ@ W G@B<XA>hE#HWN)<QFG$XDNS@vWEHAkN;QF>=?=?NH|EM
minx,u
pTuMt =St
K(x)u = p,
eTx ≤ V,
x ∈ 0, 1n,
- tA:y .
P>hWFP >hMmEK@R>DrNSJ>AH=?NS]NQmHGHXD>AHNSEQ'<Q?G]Q\E#@ M?>DHWNB=?PNjJNSM?>D]HeE#QF>hE#XANSMEQ?N'>AHE#QFL- O ;R9<;
0, 1.E#HaJ=?PN^J>AM?<XhEWN@BNH=/eN:W7=?GQ>hM/JN HNSJKGH
IRdt4|GJNXAXA>DH]O=?PNm<QFG$XDNS@
>D=?P =?PNIJ>AM?<XhEWN@BNH=^eN:W7=?GQ^EM^EeEQ?>hE#XANj>DH =?PNG<=F>D@B>ACSE=F>AGH <QFG$XAN@ >hMRWE#XAXANSJ=FPN^/+0 - 1>A@fZX=\E#HNSGZM/+/E#XALM>hM$E#HJK0+NSM?>A]H .E#<<QFGEWFPltd1>AHWN - tA:y .4>AM/EB@B>rNSJ>AH=FN]NSQ[HGHXA>AHNSEQl<QFG]QFE#@K`DHJ>AH]/=?PN,]XAG$EXG<=?>A@fZ@ >DXAX `>AHf]NHNQ\E#X `/kNdNr1=?QFN@BNXALJ> jW ZXD=/EHJ =?PNSQ?NToGQFN)PNZQ?>hM=F>AWSME#QFN)G#Tq=FNHKN@B<XDGLNSJ =FGOMGXDeN - tD8y+. tHT EW7=:`GHXAL EXA>D@B>D=?N:J HZ@!kNSQ'G#T,]XAG$aE#XAXDL|G<=F>D@OE#XMGXDZ=?>AGHM+=?G|J>hM?WQ?N=?>ACNSJ=FG<kGXAG]LKG<=F>D@B>ACSE#=?>AGH<QFG$XAN@OME#QFN)_1HG9Hs`M?NN#9<;>=7; 2 y 4 `/2 w54lEHJ 2 yy54 t
HE=FZQ\E#XaM=?Q\E=FN]L PNSHB=FQ?L1>AH]'=FGfM?GXAeN/E'PEQFJBG<=?>A@B>DC:E=F>DGHO<Q?G XAN@g>AM=FGmQFNXhEr=FPNFDPaE#Q\J EWGHM=?Q\E#>AH=\Mt9TY=FPN5>AHEQ?LRWGHM=?Q\E#>AH=\MEQ?N$Q?NSXAE#r1N:J^=?PN$ToGXAXDG9>AH]'HGHXA>DHNSE#Q<QFG]QFE#@>hM4G =FE>DHN:J
minx,u
pTuMSt =:t
K(x)u = p,
eTx ≤ V,
0 ≤ xi ≤ 1, i = 1, 2, . . . , n.
- tAS;.
9==FZQFHMGZ==FPE==FP>hM<QFG$XAN@ >hMNSMFMNSH=?>hE#XAXAL^W GHeNrY`M>AHW N=?PN/M?N =dG#TYToNSEM?>-XDN/JN:M>A]HMS`OQ;R9<;
x ∈ [0, 1]n | eTx ≤ VEHJ =?PNSQ?N'>AM$E=XANSEM=GHN
u ∈ IRdM?ZWFPI=?PE#=
K(x)u = p `
>hMWGHeNru`M?NN%2A:y54ToGQdE<Q?GG#Tt4P>hM@BN:E#HMU=FPE=:`>DHO<QF>DHW>D<XDN` - tAS; .WEHkN/MGXDeN:Jf=FG]XAG E#X1G<=?>A@OE#XA>=LfZM?>DH]'M?G@BNMGXDeNSQToGQHGHXA>DHN:E#QG<=F>D@B>ACSE=F>AGHBMZW\PBEM19<;>=7;#=FPN]XAG$E#XAXALVW GHeNSQ?]NH='eNQ\M>AGH G#T4Vm` - 4 + Vm`02 y#"54.+GQf1<EQFM?NB/GHXA>AHNSEQf+<=F>A@R>ACNSQ- )+*d` 2 !#;54.7t
4PNQFN>hMlEXhMG/E#H'N/Z>De#E#XANH=W GHeN rQ?NToGQF@fZXAE#=?>AGHGT - tD:; .sEM[EXA>DHNSE#QMN@B>hJN aH>=FN<QFG]QFE#@ - 0/* . taHT EW =S`=FPNmW G@R<XA>hE#HWN+ToZHW7=?>AGH|WE#HkN;Q?>D==FNH|EM
c(x) = supw∈IRd
2pTw − wTK(x)w
. - tA86 .
9=$>hMM?PG9H>AH 2 6C4u=?PE#=c(x) ≤ θ
>DTUEHJGHXDLj>DT=?PN'@OE#=?QF>r
[
θ pT
p K(x)
] - tAS.
>hM/<kGM?>D=?>AeN^MNS@R>hJN aH>=FNt)+M>AH]j=?P>AM+QFNSM?ZXD=/@BN:E#HM$=FPE= - tD:; ./WE#H kN!QF>D==FNHVEM=FPN
0 "
ToGXDXAG9>DH]BXA>DHN:E#Q$MNS@B>AJNH>D=?N'<QFG]QFE#@
minx,θ
θ
Mt =St [
θ pT
p K(x)
]
0,
eTx ≤ V,
0 ≤ xi ≤ 1, i = 1, 2, . . . , n,
- tAS" .
P>AW\PWEHkNfM?GXAeNSJ=FGj]XAG$EXuG<=?>A@OE#XA>D=L`9<;>=?; LIZM>AH]O=?PNf>AH=?NSQT EW N'.d/x[|* 2 w#84=FG|WEXDXE|0+*MGXDeNSQXA>D_Nj1NS0+Z|>+2 y6C49t')GQ@BGQFNR>AHToGQF@OE=F>DGH Q?NS]E#Q\J>AH]K0/*,MmM?NN9<;>=7; 2 ;4 t
4PNBE$kG8eNR@RNSH=F>DGHNSJ /+0 ToGQF@fZXAE#=?>AGH - tAS; .+PEM%kNNHZMN:J M?ZWWNSMFMToZXAXAL|GH=FQ?ZaM?MUJNSM?>D]Hf<QFG$XAN@OMS`#MNSN:9<;>=?;2AS;84E#HJ2 !#"549`#E#HJmToGQJ>AMFW QFN =F>DCSNSJfWGH=?>AHZZ@&<QFG$XDNS@BM>AH 9D;>=7;12D6<49`*Z=^=FPNP>A]P H1Z@!kNQRG#T+JG#T M^@BE8LkNSW G@RN<Q?GP> >=F>DeNjToGQRJ>hM?WQ?N=?>ACNSJWGH=?>AHZZ@ <Q?G XDNS@OMB>T'P>D]PNQjGQ\JNQONXAN@BNH=FM E#QFNIZaMN:Jut 4PN|QFNSEMGH >AMB=FPE=
d>hM
=L1<>hWE#XAXAL W GHaM>hJNQ\E#XAL XAEQ?]NQO=?PaE#Hnt /MIE#HcNrE#@B<XANVW GHM?>AJNSQE#]E>AH)>D]ZQ?N t w
PNQFNd = 12
E#HaJn = 3
t9TY=?PN "H>=FN$NXAN@BNH==,GZXAJBPE8eN kNSNHOZM?NSJ^=FPN$HZ@!kNQGTJG#T M GZXhJKPE:eN kNNSH
d = 36Z=/=?PNmH1Z@!kNQ+G#TKH>D=?NfNXAN@BNSH=FM5,GZXhJKM=?>AXAXlGHXAL
kNn = 3
t+HN@ E8Lm=FG'QFN@BGeN =?PN/J>AM?<XhEWN@BNH=FMEMeEQ?>hE#XANSMU>AHR=FPN$<QFG$XAN@ ToGQF@fZXhE=?>AGHB>hM
=FG^>DH=?QFGJZWN+Em<kGM>D=?>AeN/XAG9,NQ2kGZHJjGHj=?PNJNSM?>D]HjeEQ?>hE#XANSMSt4P>hM,@BE_NSMd>D=4<aGM?M?>-XDN=FGBNr1<Q?NSMFM,=FPNmJ>hM<XhEW N@BNH=FM$EME^ToZHW =?>AGHKG#T[=FPNmJN:M>A]HIeEQ?>hE#XANSMSt
" " " "
9T - t y . PEMERZH>-ZNmM?GXAZ=?>AGHToGQ$NeNQFLx
>D=$>hM4<kGM?M?> XDN)=?GOQ?NS@RGeN+=FPNmJ>hM<XhEWN@BNH=e#E#QF>AE$XANSMsToQ?G@ =?PN4<QFG$XAN@ ToGQF@fZXhE=?>AGH^EHJf>DHM=?N:EJfN r<QFNSMFMs=FPN4J>hM?<XAEW NS@RNSH=UeN:W7=?GQEM)E ToZHW =?>AGHG#Td=?PNOJN:M>A]HVe#E#QF>AE$XANSMSt'+M)@BNH=?>AGHN:JVNSEQ?XA>DNSQS` - t y .+@OE:L|HG=PE8eNREZH>-ZNIM?GXAZ=F>DGH ToGQRE#XAX
x ∈ 0, 1nt4PNQFNSEM?GH >hMm=FPE=^=FPNIM=F>kHN:M?M^@OE=FQ?>Dr @OE8L
kNRM?>DH]ZXhE#Q/>DT,MG@RNxi = 0
M?>AHW Nf=?PN - ML1@B@BN =?QF>hW .$]XAG$EX[M=?>kHN:M?M+@OE=FQ?>DrI=FPNHV@OE8LWGH=FE#>AH^M?G@BN,Q?G9$ME#HJfWGXAZ@BHM PNQFN E#XAX@OE#=?QF>rmNXAN@BNH=FMUEQ?N,CNQFGt[4PNSM?N,CNSQ?G+Q?G9$MEHJKW GXAZ@BHM$E#QFN'EBWGHM?N?ZNSHW NG#TUHGJNSMPE#H]>DH]R>AH D@B>AJ1E#>AQ8E+>AHI=?PN:H>D=FN'NSXDNS@RNSH=J>AMFW QFN =F>DC:E=?>AGHl`M?NN;)>A]ZQFNRt t
)[>A]ZQ?NBt ~ /GJNSMPE#H]>DH]R>DH D@R>hJ1E#>AQ>E7t
S "!$#%'&()+*,
)GQNSeNQFLx ∈ 0, 1n
>=$>hM4<kGM?M?>-XDN)=?GO<E#Q?=?>D=?>AGH=?PN']XAG$EXuM=F> YHN:M?M@OE=FQ?>DrEM
K(x) =
[
K O
O O
]
,- t !#.
PNSQ?NK ∈ IRd×d
>hMdEQFNSJZWNSJBM=?>YHNSMFM@OE=?QF>Drf=?PaE=,W GQ?QFNSM?<kGHJM=?G'=?PN DEW7=F>DeN$JGToM>E7t+M?M?Z@B>DH] =FPE=j=?PNN r1=?NSQ?HEXXAGEJ eNSW7=FGQ
pJG1NSMOHG#=jPE8eN|E HGH CSNQFG W G@B<kGHNH=
E#@BGH]^=?PN(D<EM?M?>AeN)JG#T M>E\`>A@B<XA>DN:M =FPE= - t y .4WSE#HkNQF>D==?NSHKEM[
K O
O O
](
u
u0
)
=
(
p
0
)
.- t !.
)Q?G@ - t ! .>D=/>hM/W XANSEQ=?PaE=u0
WEHkNfW\PGM?NHE#Q >=FQFEQ?>AXALt )ZQ?=?PNSQS`a>DTU=?PNm@OE=FQ?>DrK>hM <kGM?>D=?>AeN)JN H>D=?N)=FPN'J>hM<XhEWN@BNH=FM >AH =FPN'EW7=F>DeNJGT M`
u`WSE#HAkNW G@B<Z=?N:JEMd=FPN
ZH> ZNmMGXDZ=?>AGH=?G
Ku = p.- t !! .
+HN2 E:L)=?G+@BE_N,E#XAXJG#T MBDEW7=?>AeN EdEHJ=?P1ZM[NSHMZQ?N=FPE=K(x) = K 0
ToGQE#XAX1JNSM?>D]HM>hM/=?G>AH=FQ?GJZW NREj<kGM>D=?>AeN^XDG9,NQ%kGZHJ
εGH=?PNBJNSM?>D]He#E#QF>AE$XDN:M` PNQFN
ε > 0>hM)E
M?@BEXAXUH1Z@!kNQ - 9<;>=?;ε = 10−3
.7tBbc>D=?P=FP>hMXAG NQkGZHJV=?PNB]XDG E#XM=?>kHNSM?M'@OE=FQ?>DrWEHkNQF>D==?NSH
K(x) =n∑
j=1
xjKj =n∑
j=1
εKj +n∑
j=1
(xj − ε)Kj = εK(e) +n∑
j=1
(xj − ε)Kj .- t !#w.
1>AHW N>=R>AM^EM?M?Z@BNSJ=FPE=K(e) 0
>D=fToGXDXAG9$M'=?PaE=εK(e) 0
>Tε > 0
t8)ZQ?=?PNQ:`M?>DHWN
xj − ε ≥ 0ToGQfE#XAX
ε ≤ xj ≤ 1EHJ
Kj 0>D='ToGXAXAG$M)=FPE=
K(x) 0ToGQmEXDX
x ∈ [ε, 1]nt4PZM
K(x)−1N r>hM=\M'ToGQ^EXDX
x ∈ [ε, 1]nP>AW\P @BN:E#HM'=FPE=f=?PN M?LM=?N@
- t y .fPEMRE ZH> ZNIM?GXAZ=F>DGH ToGQBEXDXx ∈ [ε, 1]n
t 4P>hMBM?GXAZ=?>AGH >hM`d>DH =FPNKM?GWE#XAXANSJDHNSM=?N:J|E#<<Q?GEW\P,E\`]>AeNH 1L
u(x) = K(x)−1pt/4PNf@R>AH>A@fZ@ WG@B<XA>AEHW Nm<QFG$XDNS@
WEHIHG0kN;Q?>D==FNH
minx
pTu(x)MSt =St
eTx ≤ V,
ε ≤ xj ≤ 1, j = 1, 2, . . . , n.
- t ! .
4PNf@B>AH>D@fZ@ W G@B<XA>hE#HWNm<QFG$XDNS@ - t !# .$PaEM$= ,GE=?=?Q\EW =?>AeNm<QFG<kNSQ=F>DN:Mt5)>AQFM=S`=FPN']Q\EJ>ANH=4G#T[=FPN'G$1"NSW =?>AeN
pTu(x)>hM4N:EM?>DXAL W G@R<Z=?N:JutHT EW7=:`M?>AHW N
K(x)K(x)−1 = I ⇒∂
(
K(x)K(x)−1)
∂xj
=∂I
∂xj
= O ∈ IRd×d- t !y .
>D=ToGXAXDG9$M =?PE#=
∂(
K(x)−1)
∂xj
= −K(x)−1KjK(x)−1.- t !#;.
0
4PN'JNSQ?>AeE#=?>AeN)G#T[=FPNmW G@R<XD>hE#HWN)TqZHW7=F>DGH>=FPIQFNSM?<kNSW =4=?Gxj
>hM4=?PNSHK]>AeNHL
∂(
pTu(x))
∂xj
=∂
(
pTK(x)−1p)
∂xj
= pT∂
(
K(x)−1)
∂xj
p
= −pTK(x)−1KjK(x)−1p = −u(x)TKju(x),
- t !6 .
P>AW\P WE#H,kN+WG@B<Z=FNSJjE= E'eNQFL^XAGcWGM=d>Tu=FPN+J>hM<XAEW N@BNSH=,eNSW7=FGQu(x)
PEM kNNSHWG@B<Z=FNSJut4P>hM,@OE#_N:M - t !.=,NSXDXuM?Z>=FNSJjToGQ ]QFEJ>DNSH= EM?NSJjG<=F>D@B>ACSE#=?>AGH @BN =?PGJMSt
)ZQ=FPNQ:`>D=^WSE#H kN M?PGHl`UMNSN 2 y#84 `[=?PaE=f=?PN($N:M?M?>hE#H GT4=?PN G$1"NSW =?>AeNOToZHW =?>AGH>hM)<kGM>D=F>DeNBM?N@B>AJNH>D=?NRToGQ'E#XAX
x ∈ [ε, 1]nt^4P1ZM - t !.+>hMEIWGHeNr<Q?G XAN@P>hW\P
@BN:E#HM =?PE#=$E^XAGWEXY@B>DH>A@fZ@ >AME#XhM?GRER]XAG$EXk@B>AH>D@fZ@It2eNSH=?PGZ]P =?P>hM4>hME^eNSQ?LE#==FQFEW7=?>AeNR<QFG<kNSQ=L`u=?PNOG$=\E#>AHNSJ ]XAG$aE#XAXDL|G<=?>A@OE#XMGXAZ=?>AGH >hM)=L1<>AWSE#XAXDLHG#=
ε, 1t
HaM=FNSEJ E XAG#=OGT+>AH=FNQF@RN:J>hE=?NKe#E#XAZNSMRGH =FPN|JN:M>A]H eE#QF>AE$XANSMREQ?NIG$=\E#>AHNSJ EMR=?PNM?GXAZ=F>DGHstU2rE#@B<XANOtA)>AXDXAZM=?Q\E=FNSM =?P>hMSt
! P9 O ?99 9 O - t !# . )O"!#!"9 ?9$ O = % 8O &' (9 =8O")59 !O *0O =+,9.- ;"/7;*012!#EOF93 14 95!#9 ?9$ O = % 8O 6 ( 5O 9! O'7D9 O !8-9:<;>=@?BA PO"!9#9 9F9 !CDE!"P96F1+ E=0G 5. %+,H+I5 15 9!4!8-KJJ(D!#95LM5O G B %!HO 1J?;ON P!#9' (D
pDQ ( 8O!R O+,! 9 P+ PO"!M' ;S9.)11H+, 9 O" O"!E ( 9!M!8
/KJ,TUWV5!"P9 1"HXY K [Z\$!C$+ !C+,9<;2] 9$ O = 6) 1 O K 9& 1$9C 8O 7=^!_ 9 9 9 !C` %)DO E=.19a!# b 9 9 Wc9 !9 6de!"P9PO S099`Af3E9 g!_g!#9a) %H+P9- e!#9a'1091+
ε 1h9!!8
ε = 10−3;i95YK 9j (h1H)59 6+I O 7=lknmpoaohqr)O"!#s!#9
$!_[K O 7=a O"!9 Og!" %!t!"P9:X 1$UWV`!"P99$ O 1+ %0)C9 !_1 YVEuvuaw [5O"! O - tA . - tA.Y1+, %!193I 9 9
5 · 10−3;* 9 1 O O" O#!Rdh)O"!"9R9[ !Q!_ - tA:! . - tASw .
O O"E O O#!x O 9kvmpoPohqy)O""z9!R+, QV 9 1 O 9`%H+,!HO O Vv 9 9[3?O$!;59 ! 5O 9 %H+,!HO O99 ! 9 #O a*0O =%+,95- ;#| g 9- ;"-7;
3
3
6
6
p
)>A]ZQFN$t y~O4PN,JNSM?>D]HmJG@OE#>AHmE#HJXAGEJWSEM?Nt
)>A]ZQFN+t ;~I4PN,G =FE>AHNSJfM?GXAZ=?>AGH!PNSHM?GXAe1>AH] - t !.cGH =?PN JNSM?>A]H JG@BE>DH]>DeNH->DH )>D]ZQFN t y1t /G#=?N=FPN - ZHJN M?>AQ?N:J.,]QFNLjQ?NS]>AGHst
:! "!$#%'&()+*,
x eTx p
Tu(x)
e x
E#XANKtA~ !"#$!$" %'&$(*)+,#-&" !$./&0" 1!$2(3&4)5%+&" )5 &/0%+-)+.6 7 85 9 4:5 !$";#!$" %!#-4<=>&?!$"1 @.A%+&2)5B!C)D)+4!$" " EF<"1!$HG.%+ 8>B!$%I)5J#!$" ><=&$)+K&" % L!!$" M%5 %B!$%<=N.&L%5 8?OP2QB )54" -)+%
SR T " " # " " !
9Tk=FPNG<=?>A@B>DC:E=F>DGHR<QFG$XAN@ PEMUGHXDLfGHNWGHM=?Q\E#>AH= - N rW NS<=ToGQ=FPN$M?>D@B<[email protected]`XA>D_NI=?PNK@B>AH>A@fZ@ W G@R<XA>hE#HWN<Q?G XDNS@ - t !#. `,M?GWSE#XAXANSJ G<=F>D@OEXD>D=L WQ?>D=?NSQ?>hE - ;+ .@BN =FPGJMS`[WSE#H kNjE#<<XD>ANSJ =?GM?GXAeN^=FPN@Kt 4PN'aEM?>AWR>hJNSEKG#T$;+5@BN =FPGJM'>hM=?G|ZM?N=FPN%Q\M=$GQFJNQ$HNSWNSMFM?EQ?LW GHaJ>=F>DGHM$G#TG<=F>D@OEXD>D=LKJ>AQ?N:W7=?XAL>AHK=?PNfM?NSE#Q\W\PToGQ+EO<kG>AH=MFE=F>AMToL1>DH]I=FPNSM?NOWGHJ>D=F>DGHMtj4PN:MNO@RN=?PGJM>D=?NSQFE#=?>AeNXALVE#<<Q?G8r>D@OE#=?N^=?PN xlE#]QFEH]N@fZXD=?>A<XA>DNSQFM/E#HaJ=?PNfJN:M>A]HKeEQ?>hE#XANSME#QFN'W\PE#H]NSJIEWWGQ\J>DH]^=?G MG@RNmMFWFPNS@BN'ZH=F>DXE]>AeNSH W GHeNQF]NSHW NBWQ?>D=?NSQ?>AGH>hM'@BN =:t|GQFNB>AHToGQF@OE=F>DGH E#kGZ=^;+&@RN=?PGJMmWEH kNToGZHJI>DH 9D;>=7; 2D8!C4sGQ%2 !yC4lE#HaJ =?PN'QFN ToNSQ?NHaW NSM =?PNSQ?NS>DHlt
)HU V70<$AP%0&& ,N @ < 4PNJ>AMFW QFN =F>DCSNSJ WGH=F>DH1ZZ@=FG<kGXAG]LRG<=?>A@B>DC:E=F>DGH <QFG$XAN@>hM,=?GBJNSW>AJN%P>hWFPaH>=FNNXAN@BNSH=FM=?G:aXDX3>D=?PO@OE#=?NQF>hE#XE#HJ'P>AW\PFH>D=?N$NSXDNS@RNSH=\M=?PE#=,MPGZXhJkN_NS<=,EMeG>AJst4PZaMf=?PN]Q?NSL9MFWE#XDN M?GXAZ=F>DGH G$=\E#>AHNSJ >AH )>D]ZQ?N t ;>hM^HG#=RQFNSEXDXAL =FPNKE#HM ,NQ^GHNM?NN_MS` OQ;R9<;k=FPNf]XDG E#XAXAL G<=?>A@OE#XM?GXAZ=F>DGHK=?Gj=?PN^WGH=F>DH1ZGZM/HNSM=?NSJ|eNSQFM?>AGHKGT - tA:y .JG1N:M4HG#=$]>AeN'E D]G1GJ E)MGXDZ=?>AGH=?GR=FPN'<QFG$XAN@ GHN EH=\M =?GjMGXDeNt
4PN ZNSM=?>AGHR>hMU=?PNHBPG9 =FGG =FE>DH'XAEWF_E#HJ P>D=?N -0, 1
.JN:M>A]HaMt+ME#XAQFNSEJL@BNH=?>AGHN:JB>AH 1NSW =?>AGH|t ;`EmW G@B@BGHjE#<<Q?GEW\PR>AM=?G%Q\M=dQFNXhEr^=?PN5>AHE#QFL^WGHM=?Q\E#>AH==FG
x ∈ [0, 1]nt$N r1=:`l=FPNOXA>AHNSEQmJNS<kNHJNSHW NjGH
x>AH=?PNO]XDG E#XM=?>kHNSMFM'@OE=FQ?>Dr >AM
QFN<XhEWNSJ1LERHGHXA>AHNSEQ ToZHW =?>AGHh : [0, 1] → [0, 1]
>=FPI=?PN'<QFG<kNSQ=LB=FPE=
h(x)
= 0,>DT
x = 0,
≤ x,>DT
0 < x < 1E#HJ
= 1,>DT
x = 1.
- t !#.
4P>hM]>AeNSM =?PNToGXAXAG>AH]B]XAG$EXuM=?>kHN:M?M@OE=FQ?>Dr
K(x) =n∑
j=1
h(xj)Kj ,- t !#".
PNSQ?N)=FPNToZHW7=F>DGHh
WSE#HkN'>DH=?NQF<QFN =?N:JIEME D?JNHM?>D=LjToZHW =?>AGH,E7t|GJNXAXA>DH]R=?PN']XDG E#XsM=?>YHNSMFM@OE=FQ?>Dr>DHK=?P>hMT EM?P>DGHI@BNSEHM4=?PE#=1aH>=FNmNXAN@BNH=FM
>D=?P >AH=?NSQ?@BNSJ>AE#=?N`(aN= NNH KE#HaJ `leE#XAZN:M)GH =?PNjJN:M>A]HeEQ?>hE#XANSME#QFNR]>DeNSH XANSMFMM=?>kHNSM?M'>AH W G@R<aE#QF>AM?GH=FGI=FPN,,NS>D]P=mG#T,=?PNFH>D=?NONSXDNS@BNH=:t 4P>AM'QFNSM?ZXD=FM>AH E>AEM
0 :w
=FG4E#Q\JM0, 1
JNSM?>A]HMS`$Z=,<ZQ?N XhEW\_9E#HaJ1 P>=FNJNSM?>A]HMEQ?N]NSHNQ\E#XAXDLfHG#=,EW\P>DNSeNSJ>D=?PI=FP>AM$E<<QFGEWFPst
4PN@BGM=lWG@B@BGH>AH=FNQF<kGXhE=?>AGH@BN =FPGJ>AM[|* - 1GXA>hJ)M?G#=FQ?G<>hWU|>hW QFGM=?QFZW7=FZQ?N>D=?P*UNHEXD>ACSE#=?>AGH . `sM?NN 9<;>=?;02 549`K2D 4E#HJ 2 16C49tmHV=FPNj1|* @BGJNSX `u=FPNRToZHW7=F>DGH
h>hM+JN HN:JEMh(x) = xp
ToGQ)M?G@BNp ≥ 1
t+4PN^P>A]PNSQ$=FPNf<kNSHE#XA>ACSE=F>DGH|<E#Q\E#@BN=?NQp`
=FPN D@RGQ?N EXhEW\_9EHJ1 P>=FN=?PN^JNSM?>D]H>AXAX kNt$($G9,NeNQS`>DT=?PNfHNSM=?NSJE#<<Q?GEW\PI>hMZaMN:Ju`aEBP>A]P|eE#XAZN'GH
p@OE:LXANSEJ=FGOH1Z@BNQF>AWSE#XlJ> jW ZXD=?>ANSM+M>AHW N'=?PNf]XAG$EXsM=?>kHN:M?M
@OE#=?QF>r8kNSWG@BNSM'>DXAXW GHJ>D=?>AGHN:JJZNO=?G|NH=?QF>DN:M'WXDGMNR=FGεp
>DH =?PNj@BE#=?QF>rut $HG=?PNSQ>hMFMZNK>hMR=FPE=O=?PNW\PG>hW N
p > 1@OE_NSMR=?PN|<QFG$XAN@ HGHW GHeN r P>hWFP @RN:E#HMR=FPE=
aHJ>AH]=FPN$]XAG$EXG<=?>A@fZ@>AXDX `>DHO]NSHNQ\E#X `#kNeNSQ?LfPEQFJutU($G9,NeNSQS`#=?PN$ZM?NGTuMG@RNG<=?>A@B>DC:E=F>DGH M?G#Tq= E#QFNIM?ZW\P EM 9<;>=7; 4 + >AXDX$NHM?ZQ?N|W GHeNQF]NSHW N =FG E %'%' <kG>DH= - O ;R9<;lE <kG>AH=+=FPE='M?E#=?>hM NSM$=FPNBW GHaJ>=F>DGHM - tA . - tDS .5>=FP>AH M?G@BN^M?<kNSW> aNSJ=FGXANQ\E#HWN. t
4PN^ToGXAXAG>AH]N rE#@B<XANRJNS@BGHM=?Q\E=FNSM/=?PNBN YNSW7='G#Td=?PNO|*-@BN=?PGJ6PNSHE#<<XD>ANSJ=?GB=?PN'@B>AH>D@fZ@ W G@B<XA>hE#HWN)<QFG$XDNS@ - t !#.7t
! & 29M 9!R!HO E= O!#9 1 9 (3O 3 14 9- ;"-+,! 1e!#9ohL> 9!# O+I9 2)O"!#
h(xj) = x3j
;59tYK 9 (41H)59 !C 2! O" 9$D,Ax$!@)O"!"g= ?nA PO"!9:9 9X 9 !C !#9 )O"!"S=EA PO"!9 9 9 9 !;92K! 5O 9 .%H+,! O fP9 ^!"P9xV1+, ?9A PO"!9@9 9 9 !B= ?OP+I9 !O =8O")59 LO s*0O =%+,9^-?;R9 K 9g- ;O: e!#9^! 5O 9 %H+,!HO^)O"!#a!#9&=9 9 9 !O99 ! 9 O \*O =+,95- ;; g K 95-?;NE;
)>A]ZQFN+t 61~I4PN,G =FE>DHNSJfM?GXAZ=?>AGH!PNSHM?GXAe1>DH] - t !.cGH =?PN JNSM?>A]H JG@BE>DH]>DeNSH >DH )>D]ZQ?N-t y ZM?>DH]51|* >D=?Ph(xj) = x3
j
EHJ 1H>D=?N,NXAN@BNH=FMSt[/G#=FN=FPNmE#<<kN:E#Q\E#HWN+GTUWFPN:WF_NSQ kGE#QFJMSt
)>A]ZQFN+t ~I4PN,G =FE>AHNSJfM?GXAZ=?>AGH!PNSHM?GXAe1>AH] - t !.cGH =?PN JNSM?>A]H JG@BE>DH]>DeNH >DH )>D]ZQFN t y ZM?>DH]51|* >D=?Ph(xj) = x3
j
EHJ " H>D=?N,NXAN@BNH=FMSt[/G#=FN=FPN'QFNSJZWNSJHZ@!aNSQ$G#TUW\PNSW\_NQ kGEQFJMt
$G=?>hW NSE$XAN^ToQFG@ 2rE@R<XDNKt ! >hM)=FPE==FPNjMGXDZ=F>DGH >AH)>D]ZQ?NKt 6j>AM@fZaWFP W XAGM?NQ=FGXhEW\_9EHJ1 P>=FN/=?PEH =FPN'MGXDZ=F>DGH G =FE>DHNSJ >DH)>D]ZQ?NRt ;t )ZQ=FPNQ:`=FPN'MGBWEXDXANSJW\PN:WF_NQ kGEQFJ <QFG$XDNS@ kNSW G@BNSMBE#<<aE#QFNH=:t 4PNKQ?N:EM?GH TqGQR=?PN|E<<kNSE#Q\E#HaW N GT/=?PNW\PN:WF_NQ kGEQFJ<E#==FNQFHM>AM'<kG1GQH1Z@BNQF>AWSE#X@BGJNXAXA>DH]IGT,=?PNjM=?>kHN:M?M'>AH =FPNSM?NOEQ?NSEM`M?NN 2 !4 t +MmWSE#HaN MNSNH >AH )>D]ZQ?N|t =FPNjM?>D=?ZE#=?>AGH >AMfMG@RN?PE=mQ?N:W7=F> N:J LV=?PN
"!$#%'&()+*,
x eTx p
Tu(x)
e x J P
E#XAN t !~ 2: !$" #!$" %&$( )+'#-&" E!$.A&0" 1!$( & )5J%+&" )5 &A05%+-)+. 7 85 DL:5 !$" #!$" % B!#-L<=&?!$" @.?%+&')5B!C)D)54!$" " E <"1!$HG,.%+ 8A!$%I)5#!$" ><;&$)5N&" % D'!$B!$" M%+ %B!$%<=?.& %5 8,O PLQB )+ " -)5%I!$.?&$)I%E 8
x eTx p
Tu(x)
e x J 9
E#XAN t w~ D2: !$" #!$" %&$( )+'#-&" E!$.A&0" 1!$( & )5J%+&" )5 &A0%+-)+. 7 8 DL:5 !$" #!$" % B!#-L<=&?!$" @.?%+&')5B!C)D)54!$" " E <"1!$+G.%+ 8A!$%I)5#!$" ><;&$)5N&" % D'!$B!$" M%+ %B!$%<=?.& %5 82O4QB )+ " -)5%I!$.?&$)I%E 8
ZM?N^G#TP>D]PNQ+GQFJNQ+NXAN@BNH=\M)M?ZW\PVEM$=?PN^H>AHN^HGJN "jNSXDNS@RNSH=:` Z=)=?P>AMJG1NSM+HG#=QFNSM?GXAeN/=FPNmJ> jW ZXD=LG#TUW\PN:WF_NSQ kGEQFJ>AH]RW G@B<XDN=?NSXDLt
4PN/W\PNSW\_NQ kGEQFJ^<E=?=?NSQ?HMdE<<kNSEQ?>AH]>AH )>A]ZQFNmt 6)E#HJ,)>A]ZQFN't mE#QFNHE=FZQ\E#XAXDLZHJNSM?>DQ\E#XANO>DH <Q\EW7=F>AWNOM?>AHW NB=FPNSM?NOQFN]>DGHMPE8eNBeNSQ?LVXAG M=?>kHN:M?MfEHJ@OE:LWEZMN=FPNKM=FQ?ZW =?ZQFNj=FGW GXDXhE#<aMNtH =?PNIXA>=FNQ\E=FZQ?N =?PNSQ?NIEQ?NIMG@RNIJ>kNSQ?NSH=B@BN =?PGJMfToGQJN:E#XA>DH] >=FP-W\PNSW\_NQ kGEQFJMSt 4PN @BGM=KW G@R@BGH >AM =FPN ZaMN GTfJ>YNQ?NSH=XD=?NQF>AH]=FNSWFPH>-ZN:M`,M?NN 2 y4E#HJ =FPNIQFN ToNSQ?NHaW NSMf=?PNQFN>AH ToGQOE#H GeNQFe1>DN?'t 4PNI>hJNSE G#T$=FPNXD=?NSQ?>AH]O=FNSW\PH>-ZN:M/>AM$=?GDM?@BNSE#Q$GZ=>E'=?PN^J> YNQFNHaW NSM5kN = ,NNSHJ>YNQFNH=+<aE#Q?=FM$G#TU=FPNJN:M>A]Hst4P>AM@RN:E#HM =?PaE=4=FPN'M=?>YHNSMFM4>AHKERWNQ?=FE>DH<kG>DH=
yJNS<kNHJMGH=?PN'<kG>AH=FM>DH
EfW NQ?=FE>DHOHN>A]PkGZQ?PG1GJu`$PGMN+M>ACN/>AM,JN =?NSQ?@B>AHNSJ,1L^=FPN+XD=?NSQ,Q\EJ>AZMS`M?ZQFQ?GZHJ>AH]=FPN<kG>AH=
yt /HG#=FPNQ|E#<<Q?GEWFP >hM =FG >AH=?QFGJZaW NW GHM=?Q\E#>AH=FMGH =FPN D]QFEJ>DNSH=GT
=FPN)JNSM?>D]H,E\`1MNSN 2 a 49tU($NQFN$=?PN+]QFEJ>DNSH=,G#Tu=?PN)JNSHM?>=LRToZHW7=F>DGHh
>hMdQFNSM=?QF>AW =?N:J'P>hWFP@BNSEHM[=FPE=U=FPNJN:M>A]HRWEH^HG=Ue#E#QFL D=?G1G)@fZW\P,E >AHBE)M@OEXDXQ?NS]>AGHst[/HG#=FPNQE#<<QFGEWFP>hM =FGOEJJ<kNQF>D@BN =FNQ$WGHM=?Q\E#>AH=\M4EM4@BNH=?>AGHN:J>AH1NSW7=F>DGHt wt
9Tu=FPN+GQF>D]>DHaE#X0, 1
<Q?G XDNS@g>hM=FQ?N:E=FNSJO>DHM=FNSEJOGTu=?PN)W GH=?>AHZGZMdQFNXhErE=F>DGHR=FPNW\PNSW\_NQ kGEQFJB<Q?G XDNS@ WE#HNSEM?>AXDL,aNQ?NS@RGeNSJ L EJJ>AH]BE#<<QFG<Q?>hE=FN$XA>AHNSEQ >DHN?ZE#XD>D=?>ANSM4=?GB=?PN'<QFG$XDNS@ToGQF@fZXhE=?>AGHlt
) 3A) < <$AP')( @ < E><
bcPNSH JN:M>A]H>DH] E XAGEJcWSE#QFQ?L1>AH] M=?QFZW =?ZQFN>D=>hM HG=GHXAL =?PNM=F> YHN:M?Mj=FPE=I>AM GT>A@B<aGQ=\E#HWNtl9T=FPN M=FQ?ZaW7=?ZQ?Nd>AMMZ3>"N:W7=?N:J'=?G+M?ZjW >ANH=FXDL'XAEQ?]NJNToGQF@BE#=?>AGHMl>DH^W NQ?=FE>DH<XhEWNSM[<kNSQ?@OE#HNH=JNToGQF@BE#=?>AGHf@OE:L'QFNSM?ZXD=UEME+WGHM?N?ZNSHW Nt[4P>hMPE#<<kNSHM PNHf=FPNM?GWSE#XAXANSJ M=FQFE>DH NHNSQ?]L JNSHM?>=LQFNSEW\PNSMfEW NQ?=FE>AH @OE=FNQF>AEXdJNS<kNHJNSH=RXD>A@B>=:tV4PZM>D=O@OE:L kNKG#T+>AH=FNQFNSM=R=FG W GH=FQ?GX,=FPN|M=?Q\E#>AH NSHNQF]L JNHaM>D=L >DH =FPN|JNSM?>A]H <QFGW N:M?MSt4PNRM=?Q\E#>AHNHNQ?]LKJNSHM>D=LK>AH=?PN
j=?P H>D=?NRNXAN@BNH= - >T,>=>hMXDXANSJ6>=FPV@OE=FNQF>AEX .$>AM
]>AeNSH'LuTKju
` PNSQ?Nu
>AM,E'M?GXAZ=F>DGHR=?G - t y .7td+HN$<kGMFM?> XANG<=?>A@B>DC:E=F>DGHB<QFG$XDNS@>AHeGXDe1>AH]^M=?Q\E#>AH NHNSQ?]LBJNSHM?>=F>DN:M >AM,=?PN)<Q?G XAN@G#Tl@B>AH>D@B>AC>AH]^=?PN)@OEr>D@fZ@ M=FQFE>DH
0 8y
NSHNQF]L JNHaM>D=L` >=FP|EReGXAZ@BN'W GHM=FQFE>DH=St4P>hM4<Q?G XDNS@vWEHkNQF>=?=?NSHKEM
minx,u
maxj:xj=1
uTKju
MSt =StK(x)u = p,
eTx ≤ V,
x ∈ 0, 1n.
- t w .
/E#=?ZQ\E#XAXAL`a=FPNRM=?Q\E#>AHNHNQF]LKJNSHM>D=F>DN:M)WEH kN kGZHJN:J>T W GHaM=FQFE>DH=FM+E#QFN^EJJNSJNr<XD>hW >D=?XAL|XA>D@B>D=?>AH]=FPNR@OEr>D@fZ@ M=FQFE>DHVNHNSQ?]L|JNHM?>=Ltm4P>hM)>AMW G@B<XDN=?NXALM>A@B>AXAEQ=FG=?PN<QFG$XAN@ G#T$PE8e1>DH]VE kGZHJ GH =?PN@OEr>D@fZ@ E#XAXAG NSJ M=FQ?N:M?Mm>AH =FPNJN:M>A]HP>AW\PI>hM$J>AMFW ZaM?M?NSJ>DHI=?PNToGXAXDG9>DH]BM?NSW7=F>DGHst
AC< E 0 A)0 /HG#=FPNQ,>A@B<kGQ?=FEH=d>hMFMZN/>hM=?PN+>D@B<kGM?>AH]mG#TlM=?QFNSMFMdW GHaM=FQFE>DH=FM>DHO=FG<kGXDG]LfG<=?>A@B>ACSE=F>DGH M?>DHWNIE:eG>hJ>DH]VP>A]P M=FQ?NSMFM?NSMf>hM^ZM?ZE#XAXDL E >D]]NQRWGHWNQFH =?PEH EW\P>ANe1>DH]VP>A]PM=?>YHNSMFMt(/G,NSeNQ:`u=FP>hM'>hM?M?ZNO>hM'HG#=fEM,NSXDX,M=FZJ>ANSJ EM)=FPN D?M=FE#HaJE#Q\J EBW G@B<XA>hE#HW N<Q?G XDNS@OM - tAS; .EHJ - t !#.7t +HN QFNSEMGHP1L M=FQ?N:M?MIW GHaM=FQFE>DHN:Jc<Q?G XAN@OM PaE:eNQFNSWN>AeNSJ @fZaWFP XANSMFMBE==FNH=?>AGH =FPE#H =?PNK@B>AH>D@^Z@ W G@B<XA>hE#HWN<QFG$XDNS@ >AMR=?PE#=B=?PNH1Z@!kNQ)G#TdW GHaM=FQFE>DH=FM$>AH|=?PN^G<=F>D@B>ACSE#=?>AGH|<QFG$XDNS@ kN:W G@BN:M$XhE#QF]Nt$4PN^Q?NSEMGHIToGQ=FP>hMd>hM=FPE=,=?PN)M=FQ?N:M?MdW GHM=?Q\E#>AH=FMZM?ZE#XAXALRM=FE#=?N$=?PE#=d=?PN)M=FQ?N:M?M>AHjNSEWFPFH>=FN/NSXDNS@RNSH=M?PGZXAJ'kNXANSMFMU=FPE#HjM?G@BN<QFNSMFW Q?>-kNSJRZ<<kNQ=kGZHJ
σmax > 0tU4PZaMGHNHNNSJMU=FGmEJJ
M=?QFNSMFMW GHM=?Q\E#>AH=FM GH=?PNToGQF@
σ2j ≤ σmax, j = 1, 2, . . . , n,
- t w/.
PNQFNσj
>hM$W G@B<Z=?NSJKEM4>AH - t ; . t$HG=?PNSQ4Q?N:EM?GHj>hM,=?PN DM?>AH]ZXhE#QF>D=LE$<Q?G XDNS@-QFM=J>hM?WZM?M?NSJ >DH 2 ;84 PNSQ?NEm=FQ?ZMFM
NrE#@B<XAN>hM4]>DeNSHI>DHP>AW\P|EBW NSQ=\E#>AHKJNSM?>D]HKWEHIHG#= kNmEW\P>ANeN:J =?PQFGZ]PIEOW GH=?>AHZGZMJN:W QFNSEM?N4G#TsEE#QdEQ?N:E/=FG4E#Q\JMUCNSQ?GmE#QFNSE%>D=?PGZ=de1>DGXAE#=?>AH])=FPN/M=FQFNSM?MW GHM=?Q\E#>AH=:t4P>AM'>hM'M?NNH E#XhMGI>AH2 w 4 PNQFNR=?PNjM=FQ?N:M?M'W GHM=FQFE>DH=FMEQ?NBQ?NSXAE#rNSJ6P>hWFP E#XAXAG$M)ToGQaH>=FN'HGH CNSQ?GReEXDZN:M GTUM=FQ?N:M?M?NSM PNSHKEBJNSM?>A]HIeE#QF>hE#XAN+]G1NSM,=FGBCNSQ?GatU4PNmWEZMNG#T=FP>hM)<PNHG@RNSHE>AM)=?PNB<kGMFM?> XANRJ>AMFW GH=?>AHZ>=LG#T M=?QFNSMFM)W GHM=FQFE>DH=FM PNSHV=FPNBWGQFQ?NM?<kGHaJ>DH]OJN:M>A]HIe#E#QF>AE$XDN:M,]G^=?GOCNSQ?Ga`2AS"549t
2eNSH =FPGZ]PIM=FQ?N:M?M Q?NSXAE#=?NSJ <Q?G XAN@OM EQ?N)HG#=4=?PE#= NXAXuM=?ZJ>DN:J >AH =FPNXD>D=?NSQFE#=?ZQFN`=FPNQFN E#QFNjMG@BNjJ>kNSQ?NSH=^@BN =FPGJM'=FPE=f=?QFLV=?G>AHWGQF<kGQ\E=?NOM=?QFNSMFMN:M>AH =FPN <Q?G XAN@ToGQ?@fZXAE#=?>AGHMSt,H 2 !!54 `=?PNm@B>AH>D@fZ@ ,NS>D]P=/<QFG$XAN@ >D=?PVM=FQ?N:M?M$WGHM=?Q\E#>AH=\M>AM/WGHM?>hJNQFNSJut($NQFN`=?PN'e#E#QF>AE$XANSM EQ?N_N<=/WGH=F>DH1ZGZaM Z==FPNmM=?QFNSMFM4WQ?>D=?NSQ?>AGH>hMEJE#<=FNSJ>AHjGQFJNSQ=?GfG$=FE#>AHjQ?NXANe#E#H=,M=FQ?N:M?MXA>A@B>=\MToGQd@BGJNXhMdEXDXAG9>DH]fEfJNHaM>D=LOJN:M?WQ?>A<=F>DGHOG#T=FPNmM=?QFZW7=FZQFNt4PN'ZaMN'G#TEBW NSQ=\E#>AHT E#>AXDZQFN'WQ?>D=?NSQ?>AGHI>AM$M?NNSHI>AH!2 w"549`1PNQFN>D=$>hMZM?NSJ=FG]N =?PNQ%>D=?P E#H
ε9Q?NSXAE#rE#=?>AGHs` 2AS84 tf($NSQ?N`Y=?PNOM?NHM?>=F>De1>D=LVE#HEXDLM?>hM/>hM)<kNQ?ToGQF@RN:JGH
=FPN WGH=?>AHZZ@5<QFG$XDNS@ EHJ=FPN J>hMFW QFN =?>ACSE#=?>AGHm>AMJGHN,E#Tq=?NSQ EQFJMtsH2 yC49`E/@R>AH>A@fZ@M=?QFNSMFM)<QFG$XAN@ >hM)=?QFNSE#=?NSJ8L=?PNOZMNBGTd=FPNBPG@BG]NSH>DC:E=F>DGHV@RN=?PGJs`)2 849tB$HG=?PNSQE<<QFGEWFP >hMMNSNH >DH 2A86C4 PNQFN|=FPN
0, 1Q?N/Z>DQFN@BNH= >hM Q?NSXAE#r1N:J EHJcE W GHWE8eN
S; "!$#%'&()+*,
ZEJQFE#=?>hWjToZHW =?>AGH >hMBEJJNSJ =FG =FPNKG$1"NSW7=F>DeNToZHW =?>AGH >AH GQFJNSQ^=?G <kNHEXD>ACNI>AH=FNQ?@BNSJ>hE=FNmeE#XAZN:M4GHI=?PNfJNSM?>A]HKeEQ?>hE#XANSMStd4PNmQFNSM?ZXD=?>AH]jM=FQ?N:M?M$W GHM=?Q\E#>AHNSJ@B>AH>A@fZ@,NS>D]P=)<QFG$XAN@ >hM/=?PNSH M?GXAeNSJ >D=?PE#H>AH=FNQF>DGQ+<kG>AH=)@BN=?PGJu`sM?NN9<;>=7;)2 !84UGQ 2 w549tH 2 y#;84 `u=?PNB@B>DH>D@fZ@ ,NS>D]P='<QFG$XAN@>D=?P M=FQ?N:M?MEHJJ>AM?<XhEW NS@BNH=mWGHM=?Q\E#>AH=\M)>AM=FQ?NSE#=?N:Jutj($NSQ?NB=?PN/+0vE#<<QFGEWFP >hM'ZMN:JV=FG]N=?PNSQ;>=FP>AHE#QFLVJNSM?>A]HeEQ?>hE$XDN:Mtbc>D=?P^=FPN4ZMN GTkEJJ>D=?>AGHE#XToGQ\W N XA>D_N,e#E#QF>AE$XDN:M`:=FPN4HGHXA>AHNSEQ>DH=?NS]NQU<QFG]QFE#@5WEH!kNQFNSWEM=+EM/EOXA>AHNSE#Q+@R>DrNSJ|>AH=FN]NSQ/<QFG]QFE#@ EHJK=?PNSHVM?GXAeNSJI=?G ]XAG$EXlG<=F>D@OE#XA>D=Lt)/TW GZQ\MN`>DHR<Q\EW =?>hW N`#GHXALm<Q?G XAN@OM >=FPBE+QFNXhE=F>DeNSXDLmM?@OE#XAX1HZ@!kNQG#TYJNSM?>D]H^eEQ?>hE#XANSMWEH'kN+MGXDeN:J^=?Gm]XAG$aE#XG<=?>A@fZ@KtU4PN/MFE#@BN<QFG$XDNS@I` >D=?PjEHOEJJ>D=?>AGHEXWGHM=?Q\E#>AH=GH =FPNjW G@R<XA>hE#HWN`s>hM'M=?ZaJ>DN:JEHJMGXAeNSJ=FGKXDGWE#XG<=F>D@OEXD>D=L>AH EKW NSQ=\E#>AHM?NHM?NB>DH=FPN'P>DNSQFEQFW\P>hWEXYE<<QFGEW\P >AH!2 w;549t
/M+@BNH=F>DGHNSJVE#kGeN`kGHNfGT=?PNRJ> jW ZX=F>DN:M >D=?P M=FQ?N:M?M+W GHM=FQFE>AH=FM/>AM/=?PE#=+=?PNSLM?PGZXhJOeE#H>hM?PO>Ts=?PN)JNSM?>D]HOeEQ?>hE#XDN>hMdCNSQ?Gat4P>hMd@BNSEHM=?PE#=d=?PN)M ZEQ?N:JReGHj|>hMN:MM=?QFNSMFM >AH=FPN
j=?PKNXAN@BNH=/MPGZXhJAkN
σ2j =
uTSju,>DT
xj = 1
0,>T
xj = 0.
- t w! .
+HNO<kGMFM>-XAN' E8L|=FGEW\P>DNSeNR=?P>hMm>AM=FG|JN HNO=?PN M ZEQ?N:JVeGH |>AM?NSMmM=?QFNSMFM>DH=FPNj=?PKNXAN@BNSH=/EM
σ2j = xju
TSju.- t ww.
9T =FPN JNSM?>A]He#E#QF>AE$XANSMEQ?NB_NS<= >AHEQ?L` - t w! .'E#HJ - t ww .EQ?NBN/Z>De#E#XANH=mEHJ =?P1ZM=FPNmM=FQ?N:M?MW GHM=FQFE>DH=WEHkNQF>D==?NSHIGH=?PNToGQF@
xjuTSju ≤ σmax.
- t w# .4PNQFN^E#QFNmG=?PNSQ E8L1M$=FG NSHMZQ?Nm=FPE=+=FPNRM=?QFNSMFM/W GHM=?Q\E#>AH=+@OE#_NSM/HGQ?N:M=FQ?>hW7=F>AGH
PNSH =FPN'W GQFQFNSM?<kGHJ>AH]^JNSM?>A]HeE#QF>hE#XAN/>hM4CNQFGtUH 2A 49`=FPN)=?G<>AW)G#T[PE8e1>DH]RERJNSM?>A]HJNS<kNHJNSH= ToNSEM>-XAN)MN =4>AM EJJQFNSM?M?NSJst+M E#H N rE#@B<XAN`E'=FQ?ZMFMd<Q?G XDNS@ >AM,]>DeNSH PNQFN=FPN/ToGQ\W NSM - E#HJO=?PZaM E#XhMGm=?PN)M=?QFNSMFMN:M .>DHj=FPN E#Q\MdEQ?N+E#XAXAG,N:JR=FG!kN+HGH9CNSQ?GmGHXALB>T=FPN2E#Q[>hM[<QFNSM?NH=>DHm=FPN JNSM?>A]HstNXhE=FNSJ=FG$=FP>hM[>AM2 w84 PNQFN=?PN M=FQ?NSMFM[W GHaM=FQFE>DH=FMl>DHE+J>hM?WQ?N=?>ACNSJfW GH=?>AHZZ@&MN==F>DH])EQ?Nd@fZX=F>D<XA>ANSJ!1L)=FPN WGQFQ?N:M<kGHJ>DH]/JN:M>A]HfeEQ?>hE#XANSM=FGB]>AeN De#E#H>hMP>AH] E)W GHM=FQFE>DH=FM GH=?PNToGQ?@
xj(uTSju − σmax) ≤ 0, j = 1, 2, . . . , n.
- t wy . e1>AGZaMXALj=FPN^M=?QFNSMFM$W GHM=?Q\E#>AH=$>AH|=?PN
j=?PNSXDNS@BNH= >AXAX[HNeNSQ5kNfEH|>AMFMZNm>T
xj = 0>D=?P =?P>hM'=L<kNjG#TW GHM=FQ\E#>AH=St4P>AMm<E#<kNSQmEXAM?G|J>hM?WZM?M?NSM=?PNOZM?NOGT Nr1=?Q\EIeEQ?>hE#XANSMs ≥ 0 ∈ IRn
EHJIM=?QFNSMFM4WGHM=?Q\E#>AH=\M GH=FPNToGQF@
uTSju − sj − σmax ≤ 0, j = 1, 2, . . . , n,- t w;.
=FG]N=?PNSQ >D=?P|EBWG@B<XAN@BNH=\E#QF>D=L WGHM=?Q\E#>AH=G#T[=FPN=L1<kN
xjsj = 0, j = 1, . . . , n.- t w6 .
0 6
4P>AMU>A@B<XD>ANSMU=?PE#=U=?PN$M=FQ?N:M?MUW GHM=?Q\E#>AH=U>DH^=FPNj=?PRNSXDNS@BNH=dWE#HkN4HN]XANSW =?N:Jf>DT
xj = 0t
-J>YNQFNHW N+>AHj=?PNJN H>=F>DGHM,G#Ts=?PNM=FQ?N:M?M,WGHM=?Q\E#>AH=\M - t w#. ` - t wy .,E#HJ - t w; .7`>hMd=FPE= =?PNQ\M= >AM4JN HN:J M?Gm=FPE= =FPN)M=?QFNSMFMd>AH NSXDNS@RNSH=
j>AM CNQFGf>DT
xj = 0`>P>DXAN+=?PN
G=?PNSQFMd>D@B<XALR=?PE#=d=FPNj=FP M=?QFNSMFMdW GHM=?Q\E#>AH=,J>hM?E<<kNSEQFM>T
xj = 0t/HG#=FPNQ J>YNQFNHWN
>hMB=?PaE=j>DT)=?PNeE#XAZNKGTE JNSM?>D]H eEQ?>hE#XANI>hMBeNQ?L M@OE#XAX Z=> 0
`,=FPNH - t wy .BM=?>AXDXQFN?Z>AQFNSMm=?PaE=
uTSju ≤ σmax` P>AXAN - t w#.f>hMf=L<>AWSE#XAXDL M?E#=?>hM&aNSJ NeNH >DT
uTSju >
σmaxt/M?ZE#XAXDL`EHNXAN@BNH=>AMW GHM>hJNSQ?N:JDeG>AJ E+>DTl=FPNeEXDZN+GTl=?PN'JN:M>A]He#E#QF>AE$XDN+>hM
M?ZjW >ANH=?XALOM?@BEXAX `1EHJB=?PNQ?NToGQFN=FPN+XhE=?=?NQ <Q?G<kNQ?=LR@OE:L'kN)JNSM?>DQ\E#XDNt[Hj=?P>AMd=FPNSM?>hM=FPN4M=?QFNSMFMJN aH>=F>DGH - t ww .s=FG]N=?PNSQ >=FPf=?PN HNSM=FNSJfeNQ\M>AGH
σ2j (x) = xju(x)TSju(x)>AXAX kN)=FPN'@RGM=/W G@B@BGHXDLjZaMNSJst
JJ>DH]OM=?QFNSMFM4W GHM=FQFE>DH=FM,=FGR=?PN'@B>DH>A@fZ@ W G@R<XD>hE#HWN)<QFG$XDNS@ - t !. ]>AeNSM =?PNToGXDXAG9>DH]B<QFG$XAN@
minx
pTu(x)Mt =St
σ2j (x) ≤ σmax, j = 1, 2, . . . , n
eTx ≤ V,
ε ≤ xj ≤ 1, j = 1, 2, . . . , n,
- t w .
PNQFNu(x)
>hM4]>DeNSHA1Lu(x) = K(x)−1p
t0/ZN4=?G=?PN$M=FQ?N:M?MW GHM=FQFE>AH=FMS` - t w .*>DXAX kN$E)HGH9WGHeNrm<QFG$XDNS@ E#HaJf=?P1ZMeNSQ?L
PaE#Q\J =?GM?GXAeNj=FG ]XAG$EX,G<=F>A@BEXD>D=Lt )ZQ=FPNQ:`>T/=?P>hMR<QFG$XAN@ ToGQ?@fZXhE=F>AGH >hM^ZM?NSJu`=FPNG$=FE#>AHN:J MGXDZ=?>AGH >DXAX @BGM=^XA>D_NXAL kN]QFNLM?WSE#XDNjEMf>AH 2rE@B<XDN tD#tH GQ\JNQ=FG EWFP>ANeNK@BGQFNXAEWF_E#HJ1 P>D=?N|JNSM?>D]HM`,GHN|WEH E#<<XAL 9D;>=7;d=?PNV|* @BN =FPGJut4P>AM>AXDXPG,NSeNQQ?N/Z>DQFNf=FPE==FPNOM=FQFNSM?MWGHM=?Q\E#>AH=\M)E#QFNBEJE#<=?NSJV=?G|E#XAXDG95ToGQ=?PNJNHM?>=LRJNSMFW QF>A<=?>AGHB>DH 1|*U`M?NN 9<;>=7;A2 !!C49t/HG#=FPNQ<kGM?M?>->DXA>D=L^>AM=?G`EM>AHB2A86<49`EJJBEWGHWSE:eN ZEJQ\E=F>AW)ToZHW =?>AGH|<kNSHE#XA>DCS>DH]O>DH=?NSQ?@BNSJ>hE#=?NmeE#XAZN:MGHI=FPNfJN:M>A]HKe#E#QF>AE$XANSMSt4P>AM]>AeNSM =?PNToGXDXAG9>DH]B<QFG$XAN@
minx
pTu(x) + µ(x − εe)T(e − x)MSt =St
σ2j (x) ≤ σmax, j = 1, 2, . . . , n
eTx ≤ V,
ε ≤ xj ≤ 1, j = 1, 2, . . . , n,
- t w" .
ToGQ$MG@BNµ ≥ 0
t.N =E#HG=?PNSQO<kGMFM>->AXA>=L >AMB=FG >A@B<kGM?N|E XD>A@B>= GH =?PN DE@BGZH=OG#T]QFNLE=?PE#=j>hM
EXDXAG9,NSJ|>DH=?PNRJNSM?>D]Hst)4P>hM/@BN:E#HM$=FPE=EW GHM=?Q\E#>AH=/QFNSM=?QF>AW =?>AH]O=FPNRJNe1>hE#HWNmToQ?G@EH
ε, 1JNSM?>A]HI>AM$EJJNSJu`]>Ae1>DH]^=?PN)ToGXDXAG9>DH]R<QFG$XAN@
minx
pTu(x)Mt =:t
σ2j (x) ≤ σmax, j = 1, 2, . . . , n
(x − εe)T(e − x) ≤ b
eTx ≤ V,
ε ≤ xj ≤ 1, j = 1, 2, . . . , n,
- t .
S "!$#%'&()+*,
ToGQ/MG@RNb ≥ 0
t9='MPGZXhJ kN^HG#=?N:J|=FPE=:`sEM+@BNH=?>AGHN:JVE$kG8eN`k=?PNBM=?QFNSMFM
σ2j (x), j = 1, . . . , n
>hMW GQ?QFNSW =>DT
x ∈ 0, 1n`Z= >Tl>DH=?NQF@BNSJ>hE=FN$e#E#XAZNSMdGHB=FPN+JN:M>A]HjeE#QF>AE$XANSME#QFN/E#XAXDG9,N:J
=FP>AM@BGJNSX@OE:Lf]>AeN >AHW GQ?QFNSW =UM=?QFNSMFM?NSMSt[($G9,NeNSQS`#@BGM=G#Ta=?PN4<QFG$XDNS@BMU>AH^=?P>AMU=?PNSM?>AME#QFN$JN HN:JBGHR=FPN/J>hM?WQ?N=?N$M?<EWN`EHJ^=?PN$ZMN$G#Tk=FP>hMdJN aH>=F>AGHjWE#HB=?PNSH,kN*"ZM=?> N:JToQFG@ E^=?PNSGQFN =?>hWEXY<kG>DH=GTe1>DN?'t
) & ) 0AP IM+$& !"# < 7
4PN'@BN =?PGJM$JNeNSXDG<kNSJ>DHI=?P>hM4=?PNSM?>AM$E#QFN)@OE#>AHXAL JNeG#=FNSJ =?GB=?PNmJ>AMFW QFN =FN0, 1
- GQε, 1+ε
`ToGQ,M?G@BN$M?@BEXAXε > 0
.MN==F>DH]atU @BE>DHjWGH=FQ?>-Z=F>DGHB>hM=?PN/JNSeNXAG<@BNH=dG#T@BN =FPGJMUToGQN jW >ANH=?XAL^WG@B<Z=F>DH])=FPNNrEW7=J>AM?<XhEW NS@BNH=\M`GQU=?PN$M?G'WEXDXANSJBMNSHM>D=?>Ae>D=?>ANSMGTuJ>hM?<XAEW NS@RNSH=FMS`>AHBHN>A]PkGZQF>DH]mJNSM?>A]HMS`POQ;R9<;1JN:M>A]HM PNQFNGHNGQ= ,GfJNSM?>A]HeEQ?>hE#XDN:ME#QFN4WFPEH]N:Jm=?G)=?PN>AQG<<kGM>D=?N >DHEQ?LmeEXDZN - M?GWSE#XAXDN:J sE#HJB! HNS>D]PkGZQFM0.7t9T=FPN'XAG NQ@kGZHaJ
ε > 0>AMZM?NSJ=FPNf1PNSQ?@OE#H9|GQ?QF>AM?GH bG1G1J1ZQ?LBToGQ?@fZXhEBWEHkN
E#<<XD>ANSJI=FG W G@R<Z=?Nm=?PN^W GQ?QFNSM?<kGHJ>AH]OJ>hM<XhEWN@BNH=/eNSW =?GQ PNHVEOJNSM?>D]HKeEQ?>hE#XDN>hMW\PEH]N:Jj=FGB>D=FMG<<kGM?>=FN%>DHEQ?Lje#E#XAZNt9T`PG9,NSeNQ:`=FPN'<ZQFN
0, 1MN==F>DH]O>hM4ZM?NSJ
=FPNmM=F> YHNSMFM@OE=?QF>Dr>DXAXsHGBXAGH]NQ kN'<kGM?>=F>DeN'JNH>D=?N`aE#HJ =FPNf1PNSQ?@OE#H9|GQ?QF>AM?G#HbG1GJ1ZQ?L^ToGQ?@fZXhEfWE#HjHG^XDGH]NSQ=kN)E#<<XA>ANSJut|GQFNGeNQ:`>DHj=FP>AM WSEM?N`=FPN)M?>DCSN/G#Tu=FPNQFNSJZWNSJ M=?>kHNSMFMm@OE=FQF>r >AXAX,W\PEH]N >=FP =?PNJN:M>A]H W G@R<XD>hWE#=?>AH]K=FP>AH]MmToZQ?=?PNSQSt9=/=?ZQFHM/GZ=$=FPE=/=?PNfM?NHaM>D=?>Ae1>=LIWSE#XhW ZXhE=F>AGHM$JNSQ?>AeNSJIToGQ$=?PN
ε, 1 + εWEMNfWE#H kN
]NSHNQ\E#XA>ACNSJj=?GB=?PNmWSEM?N%PNSHKER<ZQFN0, 1
MN==F>DH]O>AMZM?NSJut4PZMS`>DT=?PN Q?NSXDNSeE#H=[ToZHW7=F>DGHMUEQ?N:9D;>=7;W G@B<XD>hE#HaW N`#M=?Q\E#>AHmNSHNQF]L'JNSHM>D=?>ANSMGQeGH
|>hMN:MM=?QFNSMFMN:M`=?PN$N rEW =WFPEH]N4>AH^ToZHaW7=?>AGHOeE#XAZNWEHkN$W G@B<Z=FNSJBE=dE)XAG W GM=UToGQE#XAXY=?PNHNS>D]PkGZQ?>AH]RJN:M>A]HMSt4P>AM4>A@B<XD>ANSM =FPE=4=FPNN jW >ANH=$MNSHM?>=F>De1>D=L WEXAWZXhE=?>AGHaMWEH8kNOE#<<XA>ANSJV>AHHN>A]P>kGZQFPG1GJVMN:E#Q\WFP M?N =?=?>AH]MStB1>AHW NR=FPNBN rEW7=)ToZHW =?>AGHe#E#XAZNSMToGQ4EXDX 9HN>A]P>kGZQ\MdE#QFN+W G@B<Z=?NSJs`=?PN)G$=\E#>AHNSJ MGXDZ=F>DGHM=>AXAX kN+XAGWEXDXALRG<=F>A@BEXk>DH=FPNO 9HN>A]P>aGZQFPG1G1Jut
$HG=?PNSQ@OE#>AH^WGH=?QF> Z=?>AGHf>hMl=FPN4JNSeNXAG<@BNSH=E#HJf>D@B<XAN@BNH=FE#=?>AGH^G#TaM?N?ZNSH=F>AEX>AH=FN]NSQ'<Q?G]Q\E#@R@B>AH]I@BN =FPGJMSt H=FPNSM?NO@BN =FPGJM'=?PNjN jW >ANH=^MNSHM>D=?>Ae1>=LWE#XhW ZXAE#=F>DGHMjEQ?NKZaMN:J =?G JNQF>DeNBQ\M=E#HJ <EQ=F>AEX$M?NSWGHJ GQ\JNQ E#<<QFG:r>A@OE=F>DH]ToZHW7=F>AGHMSt4PNM?N?ZNSH=F>AEXY>DH=?N]NQ4<Q?G]Q\E#@B@R>AH]f@BN =FPGJM4]NHNSQFEXD>ACN/=?PNHN>A]P>kGZQFPG1GJOM?NSEQFW\P@BN =FPGJM)M?>AHW N^XhE#QF]NQ/HNS>D]PkGZQ?PG1GJM/WEH kN^ZaMN:J|>AH=FPNRM?NSEQFW\PKToGQ)E XAGWEXDXALIG<=F>D@OE#XJNSM?>D]Hstf3 L|ZaM>AH]KEM?<kNSW>AEXD EWF_ =?Q\EWF_1>AH]+EfM=FQ\E=?NS]LK>D='WSE#H8kNBM?PGHV=?PE#=)=FPN@BN =FPGJM1HaJ|EBXAGWEXsG<=F>D@fZ@ >AHK=FPNj 9HN>A]PkGZQFPG1GJ >AHE H>=FNmH1Z@!kNQ/GTM=?N<aMt9=>hMJN@BGHM=?Q\E=FNSJ LfHZ@RNSQ?>hWEXN rE@R<XDN:M[=FPE==FPN@BN =FPGJBWE#HkN$E#<<XD>ANSJ^=?G'QFE#=?PNSQ]NSHNQ\E#X
0, 1=?G<kGXAG]LBG<=?>A@B>DC:E=F>DGH<Q?G XDNS@OMt
|GQFNGeNQS`1>D=$>hMM?PG9H=?PE#=/E#HIG<=?>A@OE#Xε, 1n
M?GXAZ=F>DGH =FGOEH Dε9<kNQ?=?ZQ kNSJ EJ>hM
W QFN =FN)@B>AH>D@fZ@ ,N>A]P=4<Q?G XDNS@ >D=?PKWGHM=?Q\E#>AH=\M,GHIW G@R<XD>hE#HWN`eGHI|>hMN:M4M=?QFNSMFMN:ME#HaJ^M=?Q\E#>AH^NSHNQF]LmJNHM?>=F>DN:M`>hMUG<=?>A@BEX `ETq=FNQQFGZHJ>DH]+=FG
0, 1n`#=?G)=?PNWGQFQ?N:M<kGHJ1
>AH]DZH<kNQ?=?ZQ kNSJ,EJ>hM?WQ?N=?N<QFG$XAN@K`>DTε > 0
>AM$MZ OW>DNSH=?XALM?@OE#XAX t
0 :"
M,-, A$( "# (< K <$AA 4P>AM =FPNSM?>hMW GHaM>hM=\M GTM?NeNHI>DHaJN<kNHaJNH=$<E#<kNSQFM4EHJj=FP>hMM?NSW =?>AGHGkNSQFM4EQF>DNTEHJJNSMFW Q?>A<=F>DeN'M?Z@B@OE#QFLOGT[=?PNS@Kt # " # %f #% #% " # # ! $ !C
d`W GEZ=?PGQFNSJ>=FPK%QF>hM=FNQ$1eEHkNSQ?]!C+ !R+, 1 .o\+,H!HO 8O O O %$d f! O" O'7 1! O D : nN:</KN ?J1D:<JKJ/H=?P>AM+<E<kNQ)EjP>ANQ\E#Q\W\P>AWSE#XlHN>A]P>kGZQFPG1GJ|MN:E#Q\WFP|@BN =FPGJ|ToGQ)M?GXAe1>DH]O=?G<kGXG]LG<=F>D@B>ACSE#=?>AGH<QFG$XAN@OM'JN aHNSJGH J>hMFW QFN =?>ACN:J XD>AHNSEQ?XALNXhEM=?>hWBW GH=?>AHZZ@M=FQ?ZW =?ZQFNSM>hM$<Q?N:MNSH=?N:Jut 4PNmJNSM?>D]HKG#T=FPNfM=?QFZW7=FZQ?N'>hM$Q?NS<QFNSM?NH=FNSJLA>AHEQ?LJNSM?>D]He#E#QF>AE$XDN:M,>AHJ>hWE#=?>AH]B@OE=FNQF>hE#XYGQ4eG>hJ >AH=FPN'eEQ?>AGZaM0H>D=?NmNXAN@BNH=FMSt4PNM=\E#Q?=?>AH]^<kG>AH=4G#TJ>hM?WZMFM>AGH >AM,=FPN)@B>AH>A@fZ@ ,NS>D]P=4<QFG$XAN@ >D=?PIERW G@R<XD>hE#HWN/W GHM=FQ\E#>AH=StU4PN/WG@B<XA>AEHW N/W GHM=?Q\E#>AH=>hMXhE=FNQdQFN<XhEWNSJ1LRWGHM=?Q\E#>AH=\MGHK=?PN^M=?Q\E#>AH|NHNQF]LIJNSHM>D=?>ANSM/>AHK=FPNfNXAN@BNH=FM+EHJ|E#XhM?GB=?G WGHM=?Q\E#>AH=\MGHK=?PNeGH |>hMN:M)M=?QFNSMFMN:M/>AH=FPNRNSXDNS@RNSH=\Mtm9=>hM)M?PG9H=?PE#=S`uZHJNQ'WNQ?=FE#>AH EMFMZ@R<=F>AGHMS`#>D=>AM<kGMFM>-XAN,=FG>AHW XAZJN@BGM=G#Ta=FPN$W GHaM=FQFE>DH=FM>AH^=FPNG$1"NSW =?>AeN ToZHW =?>AGHEHJ M=F>DXAXG =FE>DH E#H N?Z>AeE#XANH='<QFG$XAN@Kt 4PZaM`l=FPNSM?NO<Q?G XAN@OM'WEH kNOQFN ToGQ@^ZXAE#=?N:J=?GO<QFG$XAN@OMNSMFMNSH=?>hE#XAXAL >D=?PGZ=+W GHM=FQFE>DH=FMSt/N r1=S`=FPN'<QFG$XDNS@BM$E#QFN<kNSQ=FZQ kNSJ 1LjERM@OE#XAX
ε > 0P>AW\P@OE#_NSM,=?PN'M=F> YHN:M?M @OE#=?QF>rj<kGM?>=F>AeN)JN aH>=FN
ToGQE#XAXYJNSM?>D]HMtU9T[GHXDLOGHN+NSXDNS@BNH=>hM4WFPEH]N:J - ToQFG@ @BE#=?NSQ?>hE#Xa=FGReG>hJjGQ,ToQ?G@eG>hJ =FG@OE=?NSQ?>hE#X'.+=FPN HN<]XAG$EXdM=F> YHNSMFMf@OE=?QF>Dr>AM "ZM=RE|XAG9gQ\E#H_ @BGJ> WSE=F>DGH|G#T=?PN^GXhJKGHNfEHJI=FPZM/=FPNfG$1"NSW =?>AeNmToZHW =?>AGHVeE#XAZNSM/>DHHNS>D]PkGZQ?>AH]JNSM?>D]HMWE#H'kN/W G@R<Z=?N:JR>AHOE)T EM== E8Lt1>DHaW N=?PN$<QFG$XDNS@BMEQ?NNSMFMNH=?>hE#XAXALmTqGQ@^ZXAE#=?N:J >D=?PGZ=WGHM=?Q\E#>AH=\M`ToNSEM>->AXD>D=LOHNN:JjHG#= kN/=\E#_NSH >DH=?GBEWSW GZH= PNSHWG@B<EQ?>AH]m= ,GRHN>A]PkGZQF>DH]^JNSM>A]HMtU4P1ZM,=?PN)<QFG$XAN@ MN==?>AH]R>AM ,NXAXYM?Z>D=?N:JToGQBEVHN>A]P>kGZQFPG1GJ MN:E#Q\WFP @BN =FPGJut [GVToZQ?=?PNSQBM?<kNN:J Z< =?PNI<QFGWNSM?MS`U=?PN@BN=?PGJ >hM'>A@B<XAN@BNH=FNSJ >DH EKP>ANQ\E#Q\WFP>AWSE#X E8LtI1=FE#Q?=?>AH]KToQFG@ E|W GE#Q\MN H>D=?NNSXDNS@BNH=f@BNSM?Ps`[=?PNjHNS>D]PkGZQFPG1GJ MN:E#Q\WFP>hM'QFN<kNSE#=?NSJXDL E#<<XD>ANSJGH aHNQfEHJaHNQ$@RN:MPNSMStGQFNGeNQS`4>=I>AMM?PG9H =?PaE=>DT'=?PN <kNQ?=?ZQ >AH]
ε > 0>hMM?ZjW >ANH=FXDLcM?@BEXDX)E
]XAG$E#XAXAL G<=F>A@BEX M?GXAZ=?>AGH =?GV=?PN<kNQ?=?ZQ kN:J <Q?G XDNS@ >AXDX kNG<=F>D@OE#X4E#XhMG|=FG=FPN'<QFG$XAN@ >=FPGZ==FPN'<kNQ?=?ZQ E#=?>AGHst
$ !C " #f C ( %%f ! $ !C " ! I!C+ !R+, 1 .o\+,H!HO 8O O O %$d f! O" O'7 1! O DtN::I/ :,|(:D:<JKJK|4P>AM)<E#<kNSQ+<Q?N:MNSH=FM+]XDG E#XAXALIG<=F>A@BEX
0, 1nM?GXAZ=?>AGHM/=?GKMG@RNf=FG<kGXAG]LIG<
=F>A@R>ACSE#=?>AGH<QFG$XDNS@BM'JNHNSJGH J>hMFW QFN =?>ACN:J WGH=?>AHZZ@ JNSM?>A]H JG@OE#>AHMStB4PN]XAG$E#XAXAL'G<=?>A@BEXM?GXAZ=?>AGHaMUEQ?N ToGZHJLfNSHZ@BNQ\E=F>DGHREHJf=?PN$W GHM?>hJNQFNSJf<QFG$XAN@OMfE#QFNR=FPNj@R>AH>A@fZ@ W G@B<XD>hE#HaW NO<Q?G XAN@K`l=?PNj<QFG$XAN@ G#T4@B>AH>D@B>AC>AH]|=?PN@OE#r>D@fZ@gM=FQFE>DHRNHNQF]LfJNHaM>D=LRE#HJ^=?PN$<Q?G XAN@ G#TY@R>AH>A@B>DCS>DH]=FPN@OE#r1>A@fZ@eGH |>AM?NSMmM=FQ?N:M?MStB4PNB<QFG$XAN@OM'E#QFNBM?GXAeNSJVToGQmEIeEQ?>AN =LG#T4J>kNSQ?NSH=meGXDZ@BNWGHM=?Q\E#>AH=\MSt
!# "!$#%'&()+*,
! $ !C " %$ p #% # " # > # l C $ " `WGEZ=?PGQ?N:J >D=?P|%QF>AM=?NSQ$1eEHkNQF] !;,1+, 1V1`+, 9 O 1 o 9!# 1 O E=8O 9 9 O 7=%DQ|,9v-<|I9(J1XY-K| %Dt:IJJ<|H =?P>AMR<aE#<kNQ^=FPNI>hJNSEM^G#T+*UE<aNSQB EQ?NN r1=FNHJN:J =?G>DHWXDZaJN=?PNKWEMNAPNH=FPNQFN'>AMHG
ε > 0@OE#_1>AH]R=FPNmM=?>YHNSMFM@OE=FQ?>Dr <kGM>D=?>AeN'JN H>D=?N'ToGQ/E#XAXlJNSM?>A]HMSt
9TGHXALGHNmGQ$= GONSXDNS@BNH=\M/EQ?NmW\PEH]NSJ - ToQFG@ @OE#=?NQF>hE#Xu=FGjeG>hJIGQToQ?G@ eG>hJ=FG^@OE=?NSQ?>hE#X'.=?PN)HN< ]XAG$EXkM=F> YHNSMFM @OE=?QF>DrO>AM("ZM=$EmXAG9cQ\E#H_B@BGJ> aWE#=?>AGHjGT=FPNGXhJ^GHN`$Z=dM?>DHaW N4=?PNSQ?N>AMHG'XAGH]NQ=FPN
εXAG NQ kGZHaJu`#=FPN]XAG$aE#XM=F> YHNSMFM
@OE=FQ?>hW N:M >AXAXHG XAGH]NQ5kN^<kGM?>=F>DeN^JN H>D=?Nt%)ZQ?=?PNSQS`a=FPNRM?>DCSNfG#T=FPN^Q?N:JZW N:JM=?>kHNSMFM/@BE#=?QF>Dr>AXDXW\PE#H]N >=FP|=?PN^JNSM?>A]H P>hWFP@BNSEHM=FPE=+=?PNRPNQF@OE#H|GQ?QF>hMGH bG1G1J1ZQ?LBToGQF@fZXhE^ZaMN:J>AHI*E#<kNSQ$5WEHIHGBXAGH]NQ@aNmE<<XA>DN:Jut4PN@OE#>AHW GH=?QF>-Z=?>AGH|G#TU=?P>AM/<E#<kNSQ$>hM=?PNfJNSeNXAG<@BNH=/G#TN jW>DNSH=+@RN=?PGJM=?PaE=>AH EKM?NHM?NB]NHNSQFEXD>ACN^=FPN PNQF@BEH|GQFQ?>hMGH bG1GJ1ZQFLIToGQ?@fZXAE=FGE#XAXDG9&ToGQZaW7=?ZaE=?>AGHaM4>AHKM?>DCSN'WEZMN:J1Lj=?PN'XAG-Q\E#H_ W\PE#H]NSMSt4PNdNjW >ANH=UM?NHM?>=F>De1>D=LWEXAWZXAE#=?>AGHaMsEQ?N=FPNHfZMN:J'>DH'=FPNdToQFE@RN?,GQF_/GTE$HNS>D]PkGZQ?PGGJ M?NSEQFW\P@BN =?PGJIEHJIE#<<XD>ANSJ=?GOM?G@BN'J>kNSQ?NSH=$<QFG$XAN@OMSt
%f % % ! 2 #% 1 f ! $ ! ",`W GEZ=?PGQFNSJ>=FPK%QF>hM=FNQ$1eEHkNSQ?]!C$+ !C+, % o\+,H! O 5O O O 1$d 0!HO" O'7 1! O DtN? :9K9X_: %Dt:IJJ,94PNd@BE>AHmW GH=?QF> Z=?>AGH'G#T1=?P>AMl<E<kNQl>AMs=?PNd]NSHNQ\E#XA>ACSE=F>DGH)GT1=?PNdHN>A]PkGZQFPG1GJM?NSEQFW\P@BN =?PGJ ZMN:J>AH *UE<kNQ'++t[($NQFNB=FPN
0, 1QFN?Z>AQ?NS@RNSH=mGH=?PNJNSM?>D]H
e#E#QF>AE$XDN:M>hM_N<=,EHJ^=?PN$HGHXA>AHNSEQ@B>DrNSJR>AH=FN]NQ=FG<kGXAG]LmG<=F>D@B>ACSE#=?>AGHB<QFG$XAN@OME#QFN^M?GXAeNSJ|=?GI<QFG8e#E#XANfXDGWSE#XG<=?>A@BEXD>D=L LK=FPNRZM?N^GT,MN/ZNH=F>hE#X>AH=FN]NQ<QFG]QFE#@B@B>AH]B@BN =FPGJMSt$4PNf_NLI>AJN:EO>AHK=FPN^M?N?ZNH=?>hE#Xl>AH=FN]NQ/<QFG]QFE#@B@B>DH]@BN =FPGJM >AMd=FG^]NHNSQFE#=?N+E#HJ M?GXAeN+EfM?N?ZNSHW N+G#TDN:EM?>DNSQ>E$M?Z <Q?G XDNS@OM=,NQFN$=FPNM?N?ZNSHW NRG#T M?GXAZ=?>AGHaM >AXAXWGH1eNQF]Nm=FGKEXAGWE#XUG<=?>A@fZ@ GTd=?PNBGQF>A]>AHE#X<QFG$XAN@KtH=FP>hM<E<aNSQ4=?PNfMZ <Q?G XAN@OME#QFN)]NHNSQFE#=?NSJAL QFN<XhEW>DH]R=?PN'HGHXA>DHNSE#QToZHaW7=?>AGHaM5LQ\M=)E#HJ|<EQ=F>AEX[MN:W GHaJKGQ\JNQ+E#<<Q?G8r>D@OE=F>DGHMSt 9=)>AM+M?PG9HK=?PaE=>D=?P=FPNSM?NOE<<QFG:r>A@OE=?>AGHaM`Y=FPNjM?Z <QFG$XAN@OM'WEH kNOM?GXAeNSJEMXA>AHNSE#Q'>AH=FN]NQ<QFG]QFE#@OMSt3 LBZM?>AH]m=FPNMNSHM>D=?>Ae1>D=LjWE#XhW ZXAE#=?>AGHM,JNSQ?>AeNSJj>DH*UE#<kNSQ + =?PNMZ <Q?G XAN@OM,WSE#HkNB]NHNQ\E=?N:J|NjW >ANH=FXDLVEHJ=?PNH MGXDeN:J >D=?P MG@RNBM?G#Tq= E#QFNfToGQ>AH=?NS]NSQ)<QFG#]QFE@R@B>AH]t^9='>hM'JN@BGHM=FQFE#=?N:J|=FPE='=?PNO@RN=?PGJWE#H kNOE#<<XA>ANSJV=?G|Ee#E#QF>DN=LGTJ>YNQFNH==?G<kGXAG]LOG<=F>D@B>ACSE#=?>AGH<QFG$XDNS@BMJN HNSJGHIJ>hMFW Q?N=?>ACN:JJNSM?>A]HIJG#@OE#>AHMSt
P ! %P # # 1 %p p % ! qi1$!9)C9 O 6YV`!#EOQ 9 (Q9$9 !9 1!Q!"P99)59 !#51$ ^mx7=9$ !R$+ !R+, 1 .o\+,H! O 5O O O %$d f! O" O'7 %!HOD /9[%+, Dx:IJJ,94P>hMV<E<kNQ JNS@BGHM=?Q\E=FNSM|=?PE#=V=?PN 1N?ZNSH=F>AEX^XD>AHN:E#QV>AH=FN]NSQV<QFG]QFE#@B@B>DH]@BN =FPGJ JNSeNXAG<kNSJ >DH *E#<kNQj0 WEH kNKZM?NSJ E#XhMG PNSH JNSM?>D]H>AH] W G@R<XD>hE#H=
0 !
@BN:WFPEH>hM@OMStU4PN)<QFG$XAN@G#T[M=?QFNSMFM,WGHWNH=FQFE#=?>AGHM,>DH@BNSW\PE#H>hM?@JN:M>A]HaM WE#HkNmPEHJXANSJ1L =?PNm>AH=?QFGJZaW7=?>AGHKGTM=FQ?N:M?M$W GHaM=FQFE>DH=FM4>AHI=FPNm<QFG$XAN@ ToGQ?@fZXAE#=F>AGHst'4PNRM=?QFNSMFM+W GHaM=FQFE>DH=FM+E#QFNm=?PNSHVPEHJXANSJN r<XA>hW >D=?XALK>AH=?PN^XA>DHNSE#Q)>DH=?N]NQM?Z <Q?G XDNS@OMtHGQ\JNSQ+=?GK]NSHNQ\E=FNf=?PNOM?Z <QFG$XDNS@BM)NOW>DNSH=FXDL M@OE#XAXJ>AM?<XhEW NS@BNH=\MEQ?NBEMM?Z@BNSJAP>hWFPI>A@B<XD>ANSM =FPE=4=FPN'M?NHM?>=F>De1>D=LjWSE#XhW ZXhE=F>DGHM JNQF>DeN:Jj>AHI*UE<kNQ5+ WE#HkNmE#<<XD>ANSJPNH|W G@B<Z=?>AH]R=FPN'XD>AHN:E#Q$E#<<QFG8r1>A@OE=F>DH]^ToZHW =?>AGHMSt3 LcHZ@RNSQ?>hWEX)NrE@B<XDN:M>=|>AMKJNS@BGHM=?Q\E=FNSJc=?PE#=I=?PN @RN=?PGJ >hMKWE<E#XAN G#TXAG9,NQF>AH]^=?PNmM=?QFNSMFMN:M4E#HJKM=F>DXAXs]>AeNE^ToZHW7=F>DGHE#Xs@BNSW\PE#H>hM?@Kt
#f %p p " % ! g # ! ! 1 f ! $ !C " ! I+ O"!R! 9 !_ ! ;I%+, %%V%+, 9 O 1zo 9!" 1 O E=8O 9 9 O 7=Hc=FP>hM <E<kNQj=FPN @R>AH>A@fZ@ WG@B<XA>AEHW NV<Q?G XAN@ >=FP M=?QFNSMFM W GHM=FQFE>DH=FMj>hMWGHM?>hJNQFNSJut$H>D=?>hE#XAXAL=FPN
0, 1nQFN?Z>AQ?NS@RNSH=+>AM+Q?NSXAE#r1N:JKEHJK=?PN^XAG,NSQ5kGZHJ
ε > 0>hM,>AH=FQ?GJZW N:JOToGQ =?PNJNSM?>D]H eE#QF>hE#XANSMSt[HGQ\JNQ,=?GRG$=\E#>AHMGXDZ=?>AGHMd=FPE=
EQ?N+WXDGMN$=FGε, 1n
EfM?N?ZNSHW N+G#Tl<Q?G XDNS@OMdG#Ts=?PN/ToGQF@ - t .>hM M?GXAeNSJ,>=FPEM?N?ZNSH=F>AEXDXALOXDG9,NSQ eEXDZN)G#T
btUbcPNSHIERM?Z>D=FE#XDNJN:M>A]HPEM kNNSHG$=\E#>AHNSJs`=?P>AM
M?GXAZ=F>DGH>AM4Q?GZHJN:Jj=?GOE^<ZQFN0, 1n
JNSM?>D]HP>AW\PI>AM4ZM?NSJIEM4ERM=FEQ=F>DH]^<kG>AH=ToGQ,=?PN'1N?ZNSH=F>AEXkXA>DHN:E#Q >DH=?N]NQ <QFG]Q\E#@B@B>DH]m@BN=?PGJ JNeNXAG<kN:JO>AH *E#<kNQ40^t4P>AM$M=?N<|WE#HkN'e1>DN?,NSJIEMER<kGM=<QFGW N:M?M?>DH]RM=FN<st9=>hM)JN@BGHM=FQFE#=?N:J L|HZ@BNQF>hWE#XNrE@B<XDN:M$=FPE=)=?PNO1N/ZNH=F>AEXXA>DHN:E#Q)>DH=?N]NQ<Q?G]Q\E#@R@B>AH]^@BN =FPGJ,GQF_M ,NXAXlEMER<kGM=<QFGW N:M?M?GQSt
#p " $ ! p " C ( ! $ ! ",`W GEZ=?PGQFNSJA>=FP%Q?>hM=FNQ/1e#E#HkNSQ?]q [ 98! 9 4V1Q+ O 1! O O !C$+ !C+, % o\+,H! O 5O O O 1$d 0!HO" O'7 1! O H|=?P>AM/<E<kNQ$=?PNfe#E#XA>AJ>D=LGT=FPN^W G@R@BGHXDLIZMN:J|E#Q?=?> aW>AEXsXAG,NSQ kGZHJ
ε > 0GHB=?PN+JNSM?>D]HBe#E#QF>AE$XDN:MU>hM>DH1eNSM=?>A]E#=?NSJst4PN/@BE>DHOQFNSM?ZXD=>hM=FPE= E#HOG<=F>D@OEXaMGXAZ=F>DGH=FGOEBJ>AMFW QFN =FN D
ε <kNSQ=FZQ kN:J E)@R>AH>A@fZ@ ,NS>D]P=$<Q?G XAN@ >D=FPKWGHM=?Q\E#>AH=\M
GHKW G@B<XD>hE#HaW N`eGHK|>hMN:MM=?QFNSMFMN:M4E#HaJIM=?Q\E#>AHNSHNQF]L JNHaM>D=?>ANSMS`>hM4G<=?>A@OE#X `ETq=FNQQ?GZHJ>AH]`=FG^=?PN'WGQFQ?N:M<kGHJ>AH]^J>AMFW QFN =FN DZH<kNSQ=FZQ kN:J E+<QFG$XAN@K`1<Q?Ge1>AJN:J=FPE=)=FPNBW GHM=FQFE>DH=FM+>DHV=?PNR<kNQ?=FZQ kN:J<Q?G XDNS@ EQ?N^WSE#QFN ToZXAXAL|JN HNSJ E#HJ
ε>hM
M?ZjW >ANH=?XALM?@BEXDX t)ZQ?=?PNSQS`8>D=>AMMPG9H'=?PE#=l=FPNQFNdN r>hM=\M[EGHN =?G GHNW GQ?QFNSM?<kGHJNSHW N >D=?PfQFNSM?<kNSW7==FGm=?PN+GQFJNSQ?>AH]'GTY=FPN+G$1"NSW7=F>DeN/TqZHW7=F>DGHje#E#XAZNSMToGQd=?PN/ToNSEM>-XAN/<kG>AH=FMd=?G!kG#=FP<Q?G XDNS@OMt
G < ,N"A 0 )$&0HI=?PNmMNSeNHI>DHaJN<kNHaJNH=$<E<kNQ\M,=FPE=ToGXDXAG'`=?PN'HG=FE#=?>AGHI>AM>AH=?QFGJZWNSJIM?N<EQFE#=?NSXDL>AHKNSEW\PI<E#<kNSQSt4PN'QFNSEJNSQ4>AMZQF]NSJ =?GO@R>AHJIHG=FE#=?>AGHEXuW GXDXA>hM>AGHlt
!! "!$#%'&()+*,
I &' AK (2A 4 b5t84W\P=FC>A]NQ:t)<=?>A@B>DC:E=?>AGH >=FPmeEQ?>hE#XANM?N =\MsGTWGHM=?Q\E#>AH=\MsEHJ'EHmE<<XA>AWSE=?>AGH
=?G=?QFZMFMfJNSM?>D]Hst mx14+,! 1! O 1 0!HO" O'7 1! O aq O %!HO7`$8y - . ~ ;"1";`!#t
2 !C4 b5tWFP=?CS>D]NQ,EHJO t=?GXD<kNtu[Q?ZaM?M=?G<kGXAG]LfG<=F>D@B>ACSE=F>DGH,>=FP J>hM?WQ?N=?N/JNSM?>A]HeEQ?>hE#XANSM 4 ZE#Q\E#H=FNN:JO]XAG$aE#XaG<=?>A@OE#XA>=LjE#HJ kNSHW\P@OE#QF_BN rE#@B<XANSMSt !C$+ !R+, 1 o\+,H! O 5O O O 1$d 0!HO" O'7 1! O `aw - . ~D!#`!#6t
2 w54 b5tWFP=?CS>D]NQ,EHJO t=?GXD<kNtu[Q?ZaM?M=?G<kGXAG]LfG<=F>D@B>ACSE=F>DGH,>=FP J>hM?WQ?N=?N/JNSM?>A]HeEQ?>hE#XANSM 4 ZEQFEH=?NSNSJ^]XDG E#XG<=?>A@OE#XA>=LRE#HJkNHW\P@OE#QF_fN rE@R<XDN:M4t !C$+ !R+, 1 o\+,H! O 5O O O 1$d 0!HO" O'7 1! O `aw - . ~D!#`!#6t
2 84 4 t$XAXhE#>AQ?Nt, [9 f! O" O'7 %!HOSd.!"P921.=79 AO'7 1!HOgo 9!# #t 1<QF>DH]NQ@7NQ?XAE]`$N? .GQ?_Y`!#!t
2 yC4 4 t:$XAXAE>AQ?N`) t/:GZeN`:EHJ'(ft:E#>AXDXAG#=:tlG<kGXAG]L$G<=?>A@R>ACSE#=?>AGHToGQl@B>AH>A@fZ@5M=FQ?N:M?MJNSM?>A]H,>D=?Pj=?PN+PG@BG]NH>DC:E=?>AGHB@BN =FPGJut !R$+ !R+, 12o\+,H!HO 8O O O %$d fX!HO" O'7 1! O a`k!# - ! . ~ 161"`!##at
2 ;54 t Yt834E#CSEQFEE`:(ft 0^tPNQ\E#XA> `E#HJ ++t t1PN==Lt O 9 1xL =% %2 O 7= P9[%$d % =<1 O"!#I2\tdbc>AXDNSL$N<-.GQF_Y`Y:""wt
2 6<4 mt3 NH[EXYE#HJmt$N@B>AQ?GeM_1> t ZhWV /9 O 9CA AO"! 9L =% %2 O 7=.9X1$d(Dvq =<1 O"!#I2[D q O# 1! O7` W\PE<=?NSQO=?QFZW =?ZQ\E#X40+NSM?>A]Hst %XAZ,NSQRWSEJN@B>hW*Z XA>AM?PNSQFMS`!t
2 54 t *Ut83 NHJMNt1+<=?>A@OE#XM?PE<kNdJN:M>A]HfEMlE$@OE=FNQF>AEXJ>hM=?QF> Z=?>AGHm<QFG$XDNS@It !R$+ X!C+, % f! O" O'7 %!HO`l - . ~AS"w!!`k:""t
2 "54 t *Ut^3 NHJMNt f! O" O'7 %!HO WVw!R$+ !C+, 1!8%'=d(D Y [P9$D %! 9 O 1t1<QF>DH]NQ:`$N<-.GQF_Y`Y:""yt
2AS84' t *Ut3 NHJMN`2$txsZHaJu`mtk+XDPG`E#HJ|^t>D]@fZHJutlG<kGXAG]L +<=?>A@R>ACSE#=?>AGHQFGEJNH>AH] =FPN E#QFNSEMIG#TBE#<<XA>hWE=F>DGHstjmx !C% m@d 9 9! O $7`mw - /.7~ 6wy1`!#yt
2A4' t *UtU3 NSHJM#NE#HJ RtU1>A]@fZHaJutVE=?NSQ?>hE#X,>AH=?NSQ?<kGXAE#=?>AGH M?W\PN@BNSM^>AH =FG<kGXDG]LG<=F>D@B>ACSE#=?>AGHst q EO")595YVqt OQ9 &o@9 AO $7`k;"~ ;wy1;y`YS"""t
2A:!54' t *Ut3 NSHJMNEHJ|^ta1>D]@fZHJut5[1' =d f! O" O'7 %!HO B91$d(Dt 9!" ( [K O %!HOC 7t,1<Q?>AH]NSQS`$N? .GQ?_Y`!#wt
2ASw84*Ut 7t3 GQFNX `1xdt (ft )Q\E#HaJMNSHs`ft(/EQ?< #=?Ps` t%Q?>hM=?NHaMNSHs`1t/>DNS@B> `*Ut+>DH]a` :t Yt:NSHM?NHs`EHJK^t1>A]@fZHJstU0/N:M>A]HIE#HaJOT E$Q?>hWE#=?>AGH G#TM?G>D aEM?NSJO<PG=?GH>AW)WQ?LM=FEXW G@R<kGHNH=FMStBH>L <;EWVh|(!# !9 1! O %Em V 99 9a , R 19 ! f! O 1 9!C 1Z1 1m4 :<JKJ?`a<E]NSM$!6!6#y`!t
0 !w
2A 4*Ut 3 GQFNX `$mt(/EQ?< #=?Ps`+xt (mt )QFEHJM?NHs`/ t%Q?>hM=?NHaMNSHs`*Ut$1P> ` :at ut :NSHMNSHs`EHJ ^t[1>A]@fZHJutI[G<kGXAG]L|G<=?>A@R>ACSE#=?>AGH E#HJ T E$Q?>hWE#=?>AGHGT <PG#=FGH>hWOW QFL1M=FEXM=?QFZW7=FZQ?N:Mt f! O x3$9$7`s:! - " .7~AS""#;!#`!#t
2A:y54'ft#3 QFpHHXAZHJ^E#HaJ^%Bt#eE#H>kNQF]t+<=F>D@OE#X1=?QFZMFM[=FG<kGXAG]L`M?N@B>hJN aH>=FN4<QFG]QFE#@R@B>AH]EHJEj@BN =FPGJ EM?NSJ|GHWGHM?NQFeE#=?>AeNmE#<<Q?G8r>D@OE#=?>AGHMSt/lN:WFPH>AWSE#X[QFN<kGQ=:`+<=F>D@B>ACSE#=?>AGHKE#HJKL1M=FN@OM4PNSGQFL`E=FPN@OE=F>hWM`%'(m`sS""61t
2AS;84'ftY3,Q\p#HHXDZHJ|EHJ%BtY1eEHkNQF]t)+<=?>A@B>ACSE=F>DGH|GTU=FQFZM?M$=FG<kGXAG]L1LKEj<QF>A@BEXJZEX@BN =FPGJuta[NSW\PH>hWEXQ?NS<kGQ?=S`+<=?>A@B>DC:E=F>DGHREHJB1LM=FN@OM4PNSGQFL`E=FPN@OE#=>hWMS`%'(f`u:""6t
2A86C4' t83 ZQF]NQ[E#HaJ''t=\E#>AH_Gt*dPEM?N NXhJ'Q?NSXAE#rE#=?>AGHGT1=FG<kGXDG]L+G<=F>D@B>ACSE=F>DGH;>D=?PXAGWEXYM=?QFNSMFM WGHM=?Q\E#>AH=\Mt [qno I%+, % mx !C1 0!HO" O'7 1! O a`y~DS161`!;t
2AS84 4 t+,PNH]'EHJ ft 4 ZGtε QFNXhErN:JfE#<<Q?GEW\Pf>DHBM=?QFZW =?ZQ\E#X=FG<kGXDG]LG<=?>A@B>ACSE=F>DGHst
!C$+ !C+, % f! O" O'7 1! O a`lSw - ,.7~ !y#!#;;`YS""61t2AS"84 4 t#+,PNH]+EHJUt/:>hE#H]t=FZJLGH=FG<kGXAG]L/G<=?>A@B>DC:E=?>AGH >=FPfM=?QFNSMFMlW GHM=FQFE>DH=FMSt
E=8O 9 9 O 7=a[! O" O'7 %!HO`Y! - !.7~A:!" k`k:""!t2 !#84''t 0^t9+,GG_Y`80ft Yt8E#XA_1ZMS`8E#HaJ' t 2$t:*XANSM?PEt0mx 98! Eq O 1! OEWV *0O PO"!9
9 9 ! q %Hd( Otbc>DXANL`$N? .GQ?_Y`!#!t2 !4'ft80AEC E#HJR^t1>D]@fZHJut3+,PNSW\_NQ kGE#Q\J)<aE==FNQFHM[>DHfXhE:LGZ=[G<=?>A@B>DC:E=?>AGHst !R$+ X
!R+, 1 .o\+,H! O 5O O O %$d f! O" O'7 %!HO`l: - /.7~ y`YS""yt2 !!54*Ut0+ZLM?>DHrBEHJB t *Ut3 NHaJMNtklG<kGXAG]L'G<=?>A@R>ACSE#=?>AGHBG#TuWGH=F>DH1ZZ@ M=FQ?ZaW7=?ZQFNSM
>D=?P|XAGWEXsM=?QFNSMFMWGHM=?Q\E#>AH=\Mt ! 9 %!HO% I%+, %V%+,F9 O % o 9!" ( O E=8O 9 9 O 7=`w - ! . ~Ay#wk 16`k:""t
2 !#w84'(ft ftl2dMFW\PNHEZNQfEHJ ftU+XAPG Ut[G<kGXDG]LG<=F>A@R>ACSE#=?>AGH G#T$W GH=?>AHZZ@ M=?QFZW7=FZQ?N:M~ Q?NSe>AN<'t q OQ9 Bo 9 [ PO 9)<OQ9@\`kya~ ww`!#t
2 ! 4'ft )GQ\M?]Q?NSHs`*Ut 2$t 4 >DXAX `EHJI t (ftb QF>A]P=StUH=FNQF>DGQ4@BN =?PGJM ToGQHGHXA>DHN:E#Q4G<=?>D@B>ACSE=F>DGHstqvo9)<OQ9`~ y!y y#"61`!#!t
2 !y54''t (mt 4 EXDXhE#]PNQRE#HJ ^t ++tk>DNSH_1>DN?>AWCt f! O"2+, !R$+ !C+, 1@] 9 O = l9[%$d5qt O 1!HO7tbc>AXDNSL`s:"6#wt
2 !#;84*Ut 2$t 4 >AXDX `Ob5tf|ZQFQFE8L`BEHJ t mt^E#ZHJNQ\Mt )+*d~fE#H * EXD]GQF>=FP@ToGQ XAEQ?]N M?WE#XAN W GHaM=FQFE>DHN:J&G<=F>D@B>ACSE=F>DGHst qvo ,1+, 1. f! O" O'7 %!HO`8! - ,.7~ "6#" kS;`!#!t
2 !6C4''t 3)t(+E#kNQ:` ++t Yt#:G]`#E#HJ^ t *Ut#3 NHJM#Nt HN? E#<<QFGEWFP'=FG)eEQ?>hE#XAN =?G<kG#XAG]LM?PE#<kN/JN:M>A]HOZM?>AH]fEmW GHM=FQFE>AH=GHO<kNQF>D@BN =FNQ:t !C$+ !R+, 1 o\+,H! O 5O O O 1$d f! O" O'7 %!HO`l~Ak:!`k:"";t
! "!$#%'&()+*,
2 !#84;:t 3)t(/>DQF>hE#Q?=$QFQ?Z=L E#HaJ++tdxsNS@BEQ &:WFPEX t *z+ 1 9 ! % WVSmx )C9[3Sq %Hd( O7t1<QF>DH]NQ:`!#t
2 !#"84;) t :E#QFQFN` t%/GW e#E#QFE`E#HJ :at kG9,Nt )<=?>A@OE#X =?QFZMFMRJNSM?>D]H L >AH=?NSQ?>AGQ? <kG>DH=@BN =?PGJMStqno I%+, %0 f! O" O'7 1! O a`a~AS#k#S61`k:""t
2 w84b5tk%'EQ?ZM?Pst4>DH>A@OEOGTUToZHW =?>AGHM/GTMNSeNQ\E#XseEQ?>hE$XDN:M >D=?P>AHN?ZEXD>D=F>DN:M/EM/M?>AJNW GHJ>D=?>AGHMStEM=?NSQDS M,=FPNSM?>hM`$H>DeNSQFM?>=LjGT +,P>hWE#]G`Y:"w"t
2 w4'mtU%>DQ\MFWFPst +H M>AH]ZXhEQ'=FG<kGXAG]>DN:M'>DH G<=?>A@fZ@ M=?QFZW7=FZQFEX,JNSM?>A]Hst !C$+ !R+, 1 o\+,H! O 5O O O 1$d 0!HO" O'7 1! O `k! - w .7~ASwwk!`YS""t
2 w!54''t 7ft%/GPHE#HaJ 4 t=?Q\E#H]tU+<=F>A@BEXuJNSM?>A]HEHJjQFNXhErE#=?>AGHjG#T[eEQ?>hE=F>DGHE#Xa<QFG$XDNS@OMt mx%25+ PO 1! O > L+,9 wqt OQ9 bo 1!#9 %!HO 7`Rw" - w. ~ wy#w1w66`S";t
2 ww84'(mt b5t%ZPH EHJ mt b5t[ZW\_NQ:tI/GHXA>DHNSE#Qf<Q?G]Q\E#@B@R>AH]at HsL <;4YV /9[ 9Z?9 9d d14( O"+,j wo %!"9 1! O % ! 1! O$!HO \LQYK K O" O"!Rd`,<E]NSMf"!`Y:"yt
2 w# 4;:tlxln#T kNQF]tR.dEXA@R>A<s~fE=?GGX-kG:rToGQm@BGJNXA>DH]|E#HJG<=F>D@B>ACSE#=?>AGH>AH@OE#=?XhE#stOHL <;xYV ! 9 %!HO% d141 O"+, sm %E+,! 9qO ?9 ^mx !C% d($! 92] 9 O =a`a!##t
2 wy542$t#xlZHJRE#HJ :at#=FN]@OE#HHsta+H^M=?QFZW =?ZQ\E#X1G<=F>D@B>ACSE=F>DGHfGTkW G@B<kGM>D=?N4MPNSXDXM=FQ?ZaW7=?ZQFNSMmZM?>AH]VE|J>hMFW QFN =?N W GHM=F>D=?Z=F>DeNj<aE#Q\E#@BN =FQ?>ACSE=F>DGHst O 9H=d` - .7~AS":!#`!yt
2 w;84' t=FGXA<kN)E#HJjt1=?>hJMNSHstl-P>DNSQFEQFW\P>hWEX@BN=?PGJOToGQ4J>AMFW QFN =FN+M=FQ?ZaW7=?ZQFEX=FG<kGXG]LKJNSM?>D]H<Q?G XAN@OM5>D=?PVXAGWEXM=FQ?N:M?M+E#HaJJ>hM<XAEW N@BNH=W GHaM=FQFE>DH=FMSt ! 9 %X!HO% ,1+, 1(V1 `+, 9 O 1o 9!" (:O 7= O 9 9 O E=#`;" - y . ~AS;kS##`!161t
2 w6C4 4 t xdt[$N@BPEZMNSQmEHJ xt mtlbGXhMNSLt !9 =79 6mx%O %!8% O 1 f! O" O'7 1! O atbc>DXANL`sS"""t
2 w84;:t/GW N:JE#XEHJ Yt :atUbQF>D]P=:t `+, 9 O 14!HO" O'7 1! O at 1<QF>AH]NSQS`,$N? .GQ?_Y`!#;t
2 w"84;:t t4*NQFN>AQ\E`$2/t ft@)aE#HaW NXAXAG`+E#HJ ++t Yt34E#Q\W NSXDXAGMStg[G<kGXDG]L G<=?>A@B>DC:E=F>DGH GTW GH=?>AHZZ@ M=FQ?ZW =?ZQFNSM >D=?PV@BE#=?NSQ?>hE#XsT E#>AXAZQ?N^W GHaM=FQFE>DH=FMSt !R$+ !R+, 1 &o\+,H! O"X5O O O %$d f! O" O'7 %!HO`k!; - /.7~ y#;;`!#t
2 84;:t*UN =FNQ\M?M?GHst,1G@RNmW GHeNSQ?]NHWN+QFNSM?ZXD=FM>DHK<kNSQ?>A@BN =FNQ?9WGH=FQ?GXDXANSJj=?G<kGXAG]LjG<=F>D@B>DC:E=?>AGHlttmx14+,!9Mo 9!" (O BqtK OQ9 o 9 [ AO $ 7= O 9 9 O E=#`k61~D8!w`Y:"""t
2 4;:t*N=?NQ\MFMGHOE#HaJ ^t1>A]@fZHJsts1XAG<kN/WGHM=?Q\E#>AHN:Jf=FG<kGXAG]LfG<=F>D@B>ACSE#=?>AGHst ! 9$X%!HO1 ,1+, 1V1+,F9 O 1zo 9!" ( O E=8O 9 9 O 7=`~Aa86 kw#`kS""t
0 !y
2 !54'xt/iJN EHJ 3)t[bNSM=?NQF]QFNHlt Kq o %!"P9 %!HO ZV1 z OQ9 9 E=8O 9 9 O 7=t,=?ZaJNHXA>D==?NSQFE#=?ZQ/+3)`u:""t
2 w84'Kxt (mt8$EM?@fZMFMNSH'EHJm t81=?GXA<kNtlG<kGXAG]L/G<=?>A@B>DC:E=F>DGH'<QFG$XAN@OMuToGQ?@fZXAE#=?NSJEM@OE#=?PNS@BE#=?>hWEX<QFG]Q\E#@OM*>D=?PjN?Z>AXA> QF>AZ@ W GHaM=FQFE>DH=FMStaHPL <;pWV 9!# 51$ mx E=%95 !R$+ !R+, 1 Bo\+, !HO 5O O O 1$d f!HO" O'7 %!HO`Y<E#]NSM'S!6 aSww`!6t
2 4'ft tk>AH]NSQ=FCt$+H|@RN=?PGJMToGQ+J>hM?WQ?N=?NfM=FQ?ZaW7=?ZQ\E#XlG<=F>D@B>ACSE#=?>AGHst E=8O 9 9 O 7= f! O" O'7 %!HO`lSw - /.7~ 16 ;#`YS"t
2 y54''t tGW\_E#ToNXAXAEQStnmx )59[3.q 1Hd( O7t*Q?>AHWN =?GHK$H>DeNSQFM?>=Lj*dQ?N:M?MS`uS"16t
2 ;84 4 t 7t ftYGCSeE#H1L`k t *tu3,NSHJM?G1N`kEHJ|ftk%>AQ\M?W\Pst$xlE8LGZ=/G<=F>D@B>ACSE=F>AGH|GTM=?QFZW7=FZQ?N:Mttqt OQ9 &o 9[ PO 9)DOQ9@7` - ! .7~ ak`YS""yt
2 16C4 4 t 7t ft$GCe#E#HL` t kPGZs`E#HJ^t3,>AQF_NQ:t 4 NSHNQ\E#XA>DCSNSJfMPE<kN,G<=F>D@B>ACSE#=?>AGH!>D=?PGZ=4PG@RG]NSH>DC:E=F>DGHst !R$+ !R+, 1 o\+,H!HO 8O O O %$d f! O" O'7 1! O a` - w. ~ !y#!y!`Y:""!t
2 84'ftdW\PQFE@B@ EHJ t kPGZst N:W NH=jJNeNSXDG<@BNH=FMB>DH W G@B@RNSQ?>hWEX >A@B<XDNS@BNH=\E=?>AGH G#T$=?G<kGXAG]LG<=F>D@BCSE#=?>AGHlt H EKqno d14( O"+,j [1' =d] 9 O = f! O" O'7 %!HO^WV !C$+ !R+, 9D o EO 9 2o 1!9 O % 7`<E#]NSM,!#w"!t<Q?>AH]NQ:`!yt
2 "84f^t1>A]@fZHJutu+HR=?PN$JN:M>A]HBG#TuW G@B<XD>hE#H=@RN:WFPaE#H>hM@OMZM?>DH])=FG<kGXDG]L'G<=?>A@B>ACSE=F>DGHst o 9[ PO 19 6] 9 O = WV !R$+ !C+,9 >o K EO 97`!y - ,.7~ "w y!`:""6t
2 y#84f^t1>D]@fZHJut 0/N:M>A]H GT$@fZXD=?>A<PLM?>AWSMREW7=FZE=FGQ\MmZM>AH]=FG<kGXDG]L G<=F>A@R>ACSE#=?>AGH*E#Q?=,7~+HN @OE=FNQF>AEXM=?QFZW7=FZQ?N:MtQmx%E+,! 94o 9!# ( O &qtK OQ9 5o 9[ PO E=8O 9 9 O 7=`lS" - ".7~ ;y66 1;;#`!#t
2 y4f^t1>A]@fZHJutU|GQ?<PGXAG]L EMN:J,XAEWF_jEHJP>D=?N XD=?NQ\M,ToGQ =FG<kGXAG]LRG<=?>A@B>ACSE=F>DGHst!C+ !R+, 1 .o\+,H!HO 8O O O %$d f! O" O'7 1! O a`aww~ !8`!#161t
2 y!54f^t#1>A]@fZHJ^EHJ!:at8*N=?NQ\MFMGHst1/Z@BNQF>AWSE#X>AHM=FE$>DXA>D=?>ANSM>AHf=?G<kGXDG]LG<=F>D@B>ACSE=F>DGHs~vM?ZQ?eNSL GH <QFGW NSJZQ?N:MmJNSEXD>AH] >=FP WFPN:WF_NSQ kGE#QFJMS`l@BN:MPJNS<kNHJNSHW >ANSMfEHJXAGWEXY@B>AH>A@BEt !R$+ !R+, 1 f!HO" O'7 %!HO`l:; - .7~ ;6y`YS""t
2 y#w84;:atY=FG1NQ)EHJ'tY3 ZXA>DQ\MFWFPst !R 1+ ! O#S!8 `+, 9 O % q %Hd( O7t'1<QF>DH]NQ:`k(/N>DJNSX kNSQ?]a`Y:""wt
2 y 4' ta=?GXD<kNt . 9 . 9!# (xV1=%'K %p[!HO" O'7 1! O#6WVn$!C+ !R+, 1!_[%HX=dt*P05=FPNSM?>hM`GLE#XuHM=F>=FZ=?NmG#TU[NSW\PHGXDG]L - %'( . `a!#wt
!#; "!$#%'&()+*,
2 yy54' tl=FGXA<kN`sb5tsWFP=?CS>D]NQ:`sE#HaJ mts%'E E#@BG=?GtfgWGHWN<=ToGQ]XAG$aE#XG<=?>A@B>DC:E=?>AGH|G#T=?G<kGXAG]L JN:M>A]HK<QFG$XAN@OMStH EKqno d14( O"+, [%' =%d.] 9 O = f! O" O'7 1! O ^YV !C$+ !C+,9D o EO 9$ o %! 9 O 1 7`<E]N:M:ykS"#t1<Q?>AH]NSQS`!#;t
2 y#;84' t4=?GXD<kN E#HJc%Bt eE#H>kNQF]t |GJNXAXD>AH] =?G<kGXAG]L G<=?>A@R>ACSE#=?>AGHc<QFG$XAN@OM EMXD>AHN:E#Qm@B>rN:J #FO<Q?G]Q\E#@OMt !9 1! O % ,%+, %V1 `+, 9 O 1xo 9!" (O 7= O 9 9 O 7=`ky61~ 6#!w6w"`!wt
2 y6C4;:t ) tk=?ZQF@Kt /M?>DH] MNSJZ@B>dt t rY`EB@OE=FXAE$=FG1GX-kG8rjToGQ$G<=?>A@B>ACSE=F>DGHIGeNQ/ML1@R@BN =?QF>hWW GHNSMSt f! O" O'7 1! O go@9!# 1 zWV!R 19 `l~ ;!y1;y#w`k:"""t
2 y#84'%Bt1eEHkNQF]tY+HRXAGWEXE#HJR]XAG$aE#X@B>DH>A@OE>AHOM=FQ?ZaW7=?ZQFEXG<=F>D@B>ACSE#=?>AGHstHO =FQ?N_Y`4 E#XAXhE#]PNSQS`$$E]MFJNXAX `$EHJ Y>DNSH_1>DN?>AWC`$N:J>=FGQ\M` 9y] O" 9 !HO O f!HO"5+,!R$+ !R+, 1] 9 O =atbc>DXANL`sS"#t
2 y#"84'%Bt[1e#E#HkNSQ?]at W XhEMFMGT4]XAG$EXDXALVWGHeNSQ?]NH=G<=?>A@B>ACSE=F>DGH @RN=?PGJM;EMN:JGHW GHMNSQ?e#E=?>AeN W GH1eN r M?N<EQFE$XDN E<<QFG:r>A@BE#=?>AGHMSt [qno I%+, %Q f!HO" O'7 1X!HO`s8! - ! .7~ yyy 1y6w`!!t
2 ;84 4 ta1eNSJKE#HJ t 4 >AHGMSt=?QFZW =?ZQ\E#XuG<=?>A@B>DC:E=?>AGHZHaJNQ$@fZX=F>D<XDNmXDGEJ>AH]t ! 9$X%!HO1 ,%+, 1fWVvo 9[ PO % OQ9 9$7`s:~ w1y`kS";t
2 ;4'xt=7E#HJNHkNSQ?]PN|E#HaJ utd3 GLJut N@B>AJN H>D=?NV*Q?G]Q\E#@B@B>DH]at qno 9)<OQ9`w - . ~ "1"y`k:"";t
2 ;!542$t4b EJ Q?G E#HJ t43 NSQ?]]Q?NHltglG<kGXAG]L G<=F>A@R>ACSE#=?>AGH G#T^E#H-EW GZaM=F>AWVPGQFHstmx14+,!9 o 9!" 1 O ^qtK OQ9 Po 9 [ PO E=8O 9 9 O 7=`S"; - w . ~ !#w;`!#;t