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;E'P?B7:;Î=ÌO:;>=@?<HKJPýEO¼ENC ENJPO:=ÌAKJX?<7:EAV?<7:E9SAK]G ˆEdýo?<=$ýAKJP=@?B9HV;>Jq?<=AKþ?<7:E+W61032 4SHKJPOXH

9BEdO¼U:JPO:HKJPýÿ .›AKJ:E4ýAKJP=@?B9HV;>Jq?…ýNHVJð] E49<EGIA$TKEdO

L

E'

G

;>=@?<HKJPýE>.

E, F

44YSJPOæ;[Jð?B7P;>=

ýHK=BE

L

?B7:;Î==BU:] W610I2 8ADU:ZÎO…7PHhTKE§ADJ:Z[ÿ:B:.

= 4+3+3+3+3−2−2−2−1−1−1−1−1

⁴

GH?

=);[J =ˆ?<ENHKOAKþ?<7:E%9<ERQqU:;>9BEdO

d(d+1)

2 = 6

;[JÓO¼;>GIEJP=B;>AKJ

d = 3

^{P\7ŒUP= L} ;[?-AKU:ZÎO] E O¼E?<ENýo?<ENOA$TKEN9m=

9B;K;ÎOI] EdýHKUP=@E…AKþ?<7:E%9BEdO¼U:JPOPHVJPýÿñHVJPOX=B;[JDU:ZÎHV9<;•?ˆÿ4;[?SýADJD?HV;>JP=R

G

E?BENý?B;>JÓ9<ENO¼UPJPO:HVJ ý;>EN=0HKJPOì=B;[JDU:ZÎHV9<;•?<;[Ed=%;>=+HÓO¼;-7ñýU:Z[?+C:9<AK]PZ[ENG U;•?+;>=)KEJPE9

HVZ>Z[ÿæERQqU:;>ThHKZ[ENJq?Y?<AÓO¼ENý;>O:;[J )7PE?B7PE9AD9%J:AV?%?<7:E9<EñHV9<E4HVZKE]P9<HK;>ýO¼EC^EJ O¼EJPý;[Ed=;>J

HVJìERQqUPHV?B;>AKJÖ=Bÿ¼=ˆ?<EG

L

HKJPO·7:AÉG4HVJŒÿðU:JWŒJ:A)JP=+ýHVJî] E P6¼ENO·]Œÿ·ENHKý7ì=BU:]P=BE?AVþ

ERQqUPH$?<;[ADJP=R

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9<;-D;>O¼;[?ˆÿîýAKJP=B;>O:E9=

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KEADGIE?B9<;ÎýÌ=BCPHKýE'

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AK9Ì;[J =ˆ?HVJPýE

L

?B7:EI=@ERKGIEJq?

CD

^{;[J ?} ;-DU:9BEQ&x]Ó;>=YHñ9<;K;ÎOÓ=@UP] W61032 ;>JYV

G

U^?B7PE

C AD;[Jq?<=ýNHVJ:J:AK?]^E8GIA$TDENO=BECPHK9<HV?BENZ[ÿ

L

]:U:?ýNHVJ]^E9BAK?<H$?<ENOHKJPO…?<9<HKJP=@ZÎH$?<ENOHKZ•?<A'DE?B7PE9

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A-ETKEN9

L

?B7:;Î=Ì=BU:] W610I2 7PHK=SAKJ:Z>ÿ B .

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d(d+1)

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;•?

L

)7:;Îý79<EGIA$TDEN=8AKJ:E…;>JPO¼EC^EJ O¼EJq?S9<AV?H$?B;>AKJRþä9BADG ;[?<=)C^AD=<=@;>]:Z>E%O¼;>=BC:ZÎHKýEGIEJq?<=R

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=@;>JPýE

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L

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d

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L

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L

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L

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L

?B7:;Î==BU:] W61032ë7PHD=…AKJ:Z>ÿ B .

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?B7:ENJÖA$TKEN9 m=

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⁴

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9B;K7q?JŒU:G0]^E9AVþ2O¼;Î=@C:ZÎHKýEGIEJq?=§AVþ^?B7:EÌ=@U:]^W61032

S ⁰

;>JPO¼U ýENOI]Œÿ

O ⁰

^L ;•?58ADU:Z>OGIENHKJ

?B7PHV?

R O 0

;>Jq?B9<AŒO:UPýEd=4HæýAKJq?B9HKO¼;Îýo?<AK9<ÿ KEADGIE?B9<;ÎýRC:9<AKC^E9B?ˆÿì;>Jî?<7:EÊ=@ÿ¼=@?BEG

L

;"E§;>=

J:AV?1

ýAKJP=B;>=@?BENJq?5);[?B7?B7:E%ýADJP=@?B9HV;>JD?=AKþ?B7PE…W61032 ENG0]^ENO:O¼;>J

O ⁰

UðE…9BEþäE9-?B7PE 9BEdHKO¼EN98?<A 9FDE9K=RMkþäAD95HIGIAK9<ESþäAD9BG4HVZkCP9BAŒAVþAVþ?B7:Ed=@E%ýZ>HK;[G4=R

024 A7:9 Xo<>EG<H?7B 7I < 6 S

\7:E0C:9<A¼ýEd=B=SþäAK9YýAKGIC:U:?B;>JR?<7:E

G

L AKþHR=BU:]P=BE?…AVþAD]G ˆENý?<=

O ⁰

;>JæHW61032

S = (O, C)

ýAKU:ZÎO] EO¼Ed=Bý9B;>]^ENOXHK=-þäADZ[Z>A5=U

Q'S68AKGIC:U¼?<E…?B7:E=BE?

R

AVþHKZ[Z2?B7:E…C^AD=<=@;>]:Z>E%9B;K;ÎO¼;-P ýHV?B;>AKJP=-AVþ

O ⁰

=I 6Œ?B9HKý?SH+9<;K;ÎO¼;P^ýH$?<;[ADJ

R O ⁰ ∈ R

^{)7:;Îý7;Î=}

ýAKJ =@;Î=ˆ?<EJq?1);•?<7

C

V

G L).

O ⁰ , R O ⁰ , S

^{4 GH? .}

(O, R O ⁰ )

^4&

¦ ˆºqŠ<“ˆºŒ|‰%ˆºqŠ²d‰DŠ‡Ž´‰Œ–K†q³Šo–…}$Æ̐ˆºqŠ‰q²d‰K­›íd|”•†DŠo–Y³²N‰q‡xm“@|Ž•‰$ˆ‡Ž´‰%ˆºqŠ§“ˆŽ´ÅdŽ•–KŽ´Ôq³o|ˆŽ´²N‰q‡~K‡mŠŠ

? L

?B7:E

G

LëýADGCPU¼?<HV?B;>AKJXC:9<AŒýEN=<=)ýHKJ]^E=B;[GIC:Z>;P ENO?B7 HVJW¼=?<AI?B7:E…þäADZ[Z>A);[JCP9BADC A'

=@;[?B;>AKJ

K=7 7C>?<>?7:B 4

$

O ⁰

O ⁰⁰

^g +&&$'+l[H ` bXc+d

S

^{Z %Z}

O ⁰⁰ ⊂ O ⁰

^{m %Z f}

O ⁰⁰ , R O ⁰⁰ , S

^h

≤

^f

O ⁰ , R O ⁰ , S

^hZn

M

HD=@;ÎýHKZ[Z>ÿ

L

?B7:;Î=0GIEdHVJP=?B7PHV?0?B7PE

G

L ;[J ý9<ENHK=BEN=);•?<7ì?B7:EJŒU:G0]^E9AVþ)AK]+ ˆENýo?=

OmJPO¼EEdO

L

WDEENC:;[J;>JÊGI;[J O?B7PHV?)?B7:E

G

Lë9<EC:9<EN=BEJq?<=-?<7:E%JŒU:G0]^E9SAVþO¼;Î=@C:ZÎHKýEGIEJq?=

HVZ>Z[A-ENO;>JRH=BU:] W61032

L

HVJPO?<7PH$? 8ES9BEd=ˆ?<9B;Îýo?<ENO?B7PEÌýADJP=@?B9HV;>JD?=?BA0] EÌAKJ:Z>ÿ+9<EZÎH$?<;[TDE

?BAì?<7:EÕAD]G ˆENý?<=

L

;•?;>=ýZ>ENHK94?B7PHV?HKOPO¼;[JÖAK]G ˆEdýo?=4?BAHì=@U:]^W61032 ýHKJ:J:AK?R9BENGIA$TKE

O¼;>=BC:ZÎHKýEGIEJq?<=R

2

\7:EJ

L

-EO¼EPPJ:EY?B7:EýAKJPýEC¼?5AKþ 8 ,*

+ T$ '+l[U

S 6UTOB?<>?7B S

9;?BX?9PE+NOCHoYMOCH6/<p7I 7 MH68@<>C

_

+

O ⁰

^{$+l Y `bXcGd}

S

^{# S} 9P?BX?9;E+N

$H 5 \*( (

+&&1T$)+l

g

%Z %Z

n

Ox?9<EN=BU:Z[?<=-þä9<AKG 29BADC Aq=@;[?B;>AKJ QY?B7PHV?RU

K=7 7C>?<>?7:BL1

f

O ⁰ , R O 0 , S

^h

max O ⁰⁰ ∈DM(O ⁰ ,S)

f

O ⁰⁰ , R O 00 , S

^{h m g Z}

f

O ⁰ , S

^{h #: Z:1$ ,*} +&&: $)'+l&:

O ⁰

ⁿ

OmJPO¼EEdO

L

=@;>JPýE?B7PE

G

L ADJ:Z[ÿÊ;>JPý9BEdHK=BEN=)7PEJÕAD]G ˆENý?<=YHV9<E+HDO:O¼ENO

L

?<7:E

G L AVþ

HVJŒÿì=@U:] =@E?+AVþÌAK]G ˆEdýo?<=+;Î=0J:EdýEd=B=<HV9<;[Z>ÿðERQqUPHVZ8?BAÕ?B7:E K9<ENHV?BEd=ˆ?0AKþ5;•?=

G

L1xGI;[JP;[G4HVZ

AKJ:Ed=

2

\7qU =

L

ýAKGIC:U:?B;>J?<7:E

G

L AVþHKJŒÿ=BE?-AVþkAK]G ˆEdýo?=;>JRH0W61032«HVGIAKUPJD?=§?<A+ýADG

C:U¼?B;>J…?<7:E

G

L«AKþ^;[?<=

G

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] EdýHVU =@E…AVþ?B7PE…þäAKZ>Z[A);>JC:9BADC Aq=@;[?B;>AKJEU

K=7 7C>?<>?7:B

d

^{mH 8 ,A} +&&T$ +l 8

d(d−1)

2 + 1

^> &"'+l&

n

OmJPO¼EEdO

L

H=B;>JKZ>E DEADGE?B9<;>ý+AK]+ ˆENýo?…;>JðO¼;>GIEJP=B;>AKJ

d

HKZ[Z>A5=YH$?…Z[EdHK=ˆ?

d

O¼;Î=@CPZ>HDýE&

GENJq?<=R

D

AK9<EA$TDE9

L

;[JÊH

G

L1¤GI;>J:;>GIHKZW61032

L

EdHKý7AK]G ˆEdýo?5G+UP=ˆ?YHKO:OXH$?5Z>ENHD=ˆ?Q…?<A

?B7:E

G L L

AV?B7PE9[);>=BE?<7:EIAK]G ˆEdýo?…;>=YU:J:JPENýEd=B=<HV9<ÿ

L

HVJPOÓ?B7:E4=BU:] W61032 ;Î=ÌJPAV?

G L1

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n

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L

K9<ENH$?<E9§AD9ERQqUPHKZ¼?BA

d + n −1

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d

þäAD9?B7PE PP9=ˆ?AD]G ˆENý?

L

n −1

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L

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d(d+1) 2

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d

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L

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n

AK]G ˆEdýo?=

L

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^{L ;"E}

n ≤ ^d(d−1) ₂ + 1

2

OmJæþ›HDýo?

L

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V G

4…ýAKGIC^AD=BENOæAKJ:Z>ÿÓAKþ-C^AK;>Jq?<=

L

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L

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KZ>E

L

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L

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S

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O

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C

AVþ?ˆÿqC^EN=SAVþ§ýADJP=ˆ?<9<HK;[Jq?=

L

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?B7:E·=@E?AKþ

G

L ¤GI;>J:;[G4HKZ…=BU:]P=BE?= .¡HVJPO ?B7:EN;[9C^AD=<=@;>]:Z>EÕ9<;-D;>O¼;-P ýNH$?B;>AKJ =[44?B7PHV?ýHKJ

HVC:C^ENHK9);[JXHVJŒÿW61032 ;>J

S

^U

Q'S68AKGIC:U¼?<E…?B7:E=BE?

DORmin n

AKþHVZ>Zk?<7:E=@UP]P=@E?<=AKþ

n

AK]G ˆEdýo?<=AKþ?ˆÿŒC^EN=;>J

O

=I

?

AD98EdHKý7

O ∈ DORmin n

L

O¼E?<E9<G;>J:EÌ?B7:E…=BE?

R O

AKþHVZ>Z^?B7:EÌC Aq=B=B;[]PZ[EÌ9B;K;ÎO¼;-P ýH

?B;>AKJP=5AVþ

O

]PHD=@EdORADJXýAKJP=@?B9HV;>Jq?<=-AVþ?ˆÿŒC Ed=);[J

C

V

?

AD9)ENHKý7CPHK;[9

R ∈ R O

U

k

;[þ

∃O ⁰ ( O

^{);[?B7 G L.}

O ⁰ , R ⁰ , S

^{4 G L.}

O, R, S

^{4 L} ?B7:ENJX?B7:EC HV;>9

(O, R)

^;Î=

J:AK?SH+C^AV?<EJq?B;ÎHVZ

G

L1xGI;[J:;>G4HVZ

k

AV?<7:E9[);Î=@E

L

(O, R)

;Î=5HKO:O:ENOR?BAI?B7PE%Z[;Î=ˆ?)AKþC^AV?<EJq?B;ÎHVZ G

L1xG;>J:;>G4HVZ"

\7:;>=-C:9<A¼ýEd=B=?BE9<GI;[J H$?BEd=-] EdýHVU =@EÌAVþ2?<7:E%=@;EÌZ[;>GI;•? K;>TKENJñ;>J 29<AKC^AD=B;•?<;[ADJ V L HKJPO

] EdýHVU =@EI?<7:E4?ˆÿŒC Ed=AKþAD]G ˆENý?<=HKJPOìýADJP=@?B9HV;>JD?=%HV9<E P6¼ENOS ;•?<7ì?B7:;Î=CP9BA¼ýEd=B=

L

AKJPE

ýHVJ CP9BEdýAKGIC:U:?BEÓADJPýEÊþäAD9HKZ[ZYH·?<HK]:Z[EÕAVþHVZ>Z5?<7:EÕC^AD=<=@;>]:Z>EÓCPHK;[9=Y.¡=@E?RAVþ%AK]+ ˆENýo?=

L

9B;K;ÎO¼;-P ýHV?B;>AKJ845)7:;Îý7ÊýHVJ]^E

G

L ¤GI;>J:;[G4HKZk;[JÊHVJŒÿXW61032 AVþ?B7:EýAKJP=B;>O:E9<ENOýZÎHK=<=

AVþW61032= æE57 HhTKE5O¼ADJ:E?B7:;Î=8C:9BEdýADGCPU¼?<HV?B;>AKJ+þäAK9?<7:ESýZ>HD=B=BEN=AKþ2W610I2=8O:;>=<ýUP=<=BENO

HV]^A$TKE

L

HVJPOÓ;•?…9BEd=@UPZ•?<ENOÊ;>JÕHR=BE?YAVþ5V .%Q/Q;[J V G

45=BU:]P=BE?=SAKþAK]G ˆEdýo?<=

L

HK=<=@A¼ý;ÎH$?<ENO?<A

H$?)GIAD=@?V+9<;K;ÎO¼;P^ýH$?<;[ADJP=-ENHDý7

A 68@ 0?B < 6 5 p@Z7BCH?C><>68BX@

\Añý7:EdýW4?B7PE

ýAKJP=B;>=@?BENJPýÿ

L

AKJ:E%JPEENOXHKJ;>J¼þäEN9BENJPýEADC EN9<HV?BAD9HK]:Z>E…?BA4C:9<A¼O¼UPýEHKZ[Z

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] EN9AVþ2O¼;Î=BC:Z>HDýENGENJq?<=§AKþkHK;>TKEJ4=BE?AKþkAK]G ˆEdýo?<=;[JñHW610I2!>Ox?8;Î= WŒJ:A)J .¡=@ENE9FDE9K=RM

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?

AD9+;>JP=@?<HVJ ýE

L

?T8AICPHK9<HKZ[Z>EZvZ>;[J:Ed=);>J = G

HVZ>Z>A AKJ:Z>ÿ =IO¼;Î=BC:Z>HDýENGENJq?<=

L

)7:;[Z>EY?T8AIJ:ADJIxCPHV9HVZ>Z[ENZvZ[;>J:EN=

HVZ>Z[A V

M

EdýHVU =@EAVþ

@

ÿqC^AV?<7:EN=B;Î= = L

ETDE9<ÿ DEAKGIE?B9<;>ýC:9<AKC^E9B?ˆÿ·ýNHVJÖ]^EÊ=ˆ?H$?<ENOÖ]Œÿ

HÓJ:ADJI¤T$HKZ[U:EdOÖýADJP=@?B9HV;>JD? .%)7:;Îý7Ö9<EC:9<EN=BEJq?+Hæ=@;>JKUPZ>HK909<EZÎH$?B;>TKERC:Z>HDýENGENJq?4%AD9+;[?<=

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\7qU =

L

AKJ:E+J:EEdOX?BA]^EHK]:Z[E0?BA;[J¼þäEN9…J:E ýAKJP=@?B9HV;>Jq?<=)þä9<AKG H K;>TKEJæ=@E?ÌAVþ-ýAKJ

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I g

.›=BEE

G

EPPJ:;[?B;>AKJ J40;>=

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?

AD9…?B7:;Î=%C:U:9<C^AD=BE

L

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L

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G L L

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L

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