Computing the Minkowski sum of prisms

(1)

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?B<LÇÆ><D?BA[TbQª?BUTV<>=@?O_<]LNhcKT`\VA[_H9=@»]L\`T5WÉÈ]_HLÇcKT`\VA[_H

a ∈ R ⁿ

¼?BA@S»_5=@?BA@?BcKT

\`_UÊ»_<>TV<KA[=9¼T¼H@?BA[T

¯ a = ( _¯ _a ¹

1 , . . . , _¯ _a ¹

n )

LN<>QQDTV<>_A[TbËK¾

Π = Π ^a ^¯ = {x ∈ R ⁿ | h¯ a, xi ≤ 1 , x ≥ 0 }

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ÑÒÓÔ'ÕÖ[×NØxÙ9Ã9_<7cdTÚ¦LN<]LNhB¾ª=@?O=xÛDR¦?B<DPd_^¼=@P7?0=@CDUÛª»_hB¾7A[_»T`=xW

ÜªÝdÝdÝÊÞß5àá Òâ ß5àGãä[ßånæ]ç>èGé Ò ä`àÇêåBß ØMØ ãë'äMß5àGãÕìÙYí5îïÇðdñ7Û]î5òïÇîdñ7Û]ó5ðdÃð5ô

õ ö{÷øYù9úûüýbø½þ^ú"÷

È]_HA1¼_ÊLNH@ËD?BA@HMLNH@¾"=@CDË>=[TVA[=

A, B ⊂ R ⁿ

^A@S>T ^Þ¦ã^{ìDÿÕÔÉØMÿ} ^ã ^Ø ^ç ^â ?O=QDTÆ><>T`QË7¾A@S>TbE_H@U°CDhÂL

A + B = {z = a + b | a ∈ A, b ∈ B}

LN<>Q.E_H

λ ∈ R

^LN<]Q

A ⊂ R ⁿ

^A@S>TÇâ ^ç7å^àGã ^å^{ãäMß5àGã} ^Õì

?O=JQDTÆ><>T`Q¦Ë7¾

λA = {x = λa | a ∈ A}.

^;E

A

LN<>Q

B

LNH[T°\`_<7cdTÚ¦=[TVA[=bA@S>TV<¦A@S>T.=[TVA[=

A + B

LN<>Q

λA

LNHMTJLNhO=[_°\`_<7cdTÚLN<>Q?E0U_H[T`_^cdTVH

A

LN<>Q

B

LNH[T»_hB¾7A[_»T`=xÛ7A@S>TV<

A + B

LN<>Q

λA

LNH[TLNhO=[_.»_hB¾7A[_»T`=xWSKC>=A@S>TÇR¦?B<DPd_^¼=@P7?n=@CDU _NE0Æ><D?BA[TVhB¾"ULN<7¾»_hB¾7A[_»T`=?O=

A@S>Tg\`_<7cdTÚS7CDhBh{_NEÉA@S>T=@CDUÏ_NEÉ?BA[=TÚªA@H[TVUT»_?B<7A[=xW

;1< LQ>Qª?BA@?O_<ZA[_ÎA@SD?O=H[TV»DH[T`=[TV<7AMLNA@?O_< ?BA"?B=EGH[T`ÐKC>TV<KA@hB¾+S>TVhB»ªEGCDhJA[_S]LxcdTLQªC]LNhJH[TV»DH[T

=[TV<7AMLNA@?O_<"_NE0A@S>TbR¦?B<DPd_^¼=@P7?]=@CDU¿_NEnÆ><D?BA[TVhB¾ÊULN<K¾Ê»_hB¾7A[_»T`='?B<A[TVH@U=9_NEn?B<>T`ÐKC]LNhB?BA@?OT`=xW

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(2)

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_NEÉA@S>TA2¾7»T 1

UL^Ú

hc, xi

CD<>QDTVH

(P ₀ ) x = z ¹ + z ² + . . . + z ^k

¼?BA@S

ha ⁱ , z ⁱ i ≤ α i i ∈ {1, ..., k}, z ⁱ ≥ 0

¼?BA@S

c, a ⁱ , z ⁱ ∈ R ⁿ

LN<>Q

α _i ∈ R .

IKA@H@C>\VA@C>HMLNhÇ»DH[_»TVH@A@?OT`=LNË_CDA"A@S>TR¦?B<DPd_^¼=@P7?J=@CDU _NEg»DH@?O=@U=xÛ=@C>\MS L=A@S>T¦<7CDU.ËTVH

_NEbEL\`T`=ÊLN<>Q=@¾7UÊUTVA@H@?OT`=.E_HA@S>T"dTV<>TVH@?O\t\xL=[TS]LxcdT"ËT`TV<Í=@A@C>Qª?OT`Q?B<23 1WrXT"S]L`cKT

\xLNhBhOT`QL°Æ><D?BA[Tg=@CDU _NEÉ»DH@?O=@U=JL°Á1\`TV»DS>_?OQ7ÄMW

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\xL=[T¦?B< A@S>T¦»DH[T`=[TV<7A»LN»TVHxW ïJ=L\`_<>=[T`ÐKC>TV<>\`T5ÛA@S>T¦H[T`=@CDhBA[=LNH[T¦¼9TxLNPKTVH"ËDCDAU_H[T

dTV<>TVHMLNhW&;1<Z»]LNH@A@?O\VCDhÂLNHxÛÇ¼9T»DH[_xc7?OQDTLÎcKTVH[=@?O_< _NE.A@S>T\`_?B<>\V?OQDTV<>\`TrA@S>T`_H[TVU ¼SD?O\[S

¾7?OTVhOQD='LJ<>T`\`T`=[=MLNH@¾LN<>QÊ=@C4\V?OTV<7A'\`_<>Qª?BA@?O_<ÊE_HÉUL^Úª?BULNhDEL\`T`=65gËDCDAYhÂL\[Pª=YA@S>T=@»T`\V?Æ]\

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SD?O=cdTVH[=@?O_<.A@CDH@<>=_CDAA[_bËT'U°C>\MS.U_H[T9=@CD?BAMLNËDhOT½E_H\`_UÊ»DCDAMLNA@?O_<LNhª»DCDH@»_5=[T`=xW{XZS>TV<

\`_UÊ»DCDA@?B<DA@S>T'UL^Úª?BULNh7EL\`T`=É_NE\`TV»DS>_?OQD=xÛ¼T9QD_b<>_A¼LN<7AA[_bA[T`=@A{E_H<>_<>QDTVdTV<>TVHML\`¾

?B<¦LQªc5LN<>\`T5W8JTV<>\`T5ÛªA@S>TVH[TÇ?O=L.»]LNH@A@?O\VCDhÂLNHbTVUÊ»DS]L=@?O=_<\`_UÊ»DCDAMLNA@?O_<]LNhL=@»T`\VA[=?B<tA@S>T

»DH[T`=[TV<7A»]LN»TVHxW

90TVAC>=QDTV<>_A[TËK¾

0 ∈ R ⁿ

A@S]T"_H@?B5?B<Í_NEÇLN<>QË7¾

e ⁱ

^A@S>T

i

A@SCD<D?BAcKT`\VA[_H_NE

R ⁿ

^E_H

i ∈ {1, ..., n}.

È]_HbL.cKT`\VA[_H

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ ,

^hOTVA

Π ^a =

^\`_<Kc

0, a ¹ , . . . , a ⁿ

¼?BA@S

a ⁱ = a _i e ⁱ , i ∈ {1, ..., n}.

^XT\xLNhBh

Π ^a

^L ^Ö ^ã^Øâ L=[=[_K\V?ÂLNA[T`Q A[_ZA@S>TcKT`\VA[_H

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ .

Ë>=[TVH@cKTA@S]LNA

Π ^a = {x ∈ R ⁿ | h¯ a, xi ≤ α

LN<>Q

x ≥ 0 },

¼S>TVH[T

¯ a =

α a 1 , ..., _a ^α

n

∈ R ⁿ

?O=\xLNhBhOT`QA@S>TÊÕ çªàÒÖÇìÕÖâ ßå;:< ÕNÖ åÒ=Ò å

α > 0

^> ^_NE

Π ^a .

;1<A@SD?O=9»]LN»TVH¼9TJ¼?BhBh0LNhB¼L`¾ª=

\MS>_K_5=[T

α = 1.

XÎT\xLNhBh

∆ ^a =

^\`_<Kc

a ¹ , . . . , a ⁿ

A@S>T@?

ß

Ö[Ò

à Õ

<Vßdä Ò°_NE

Π ^a .

XZS>TV<>TVcKTVH¦¼TrSL`cKTÆ><D?BA[TVhB¾ ULN<K¾ cKT`\AM_H[=xÛ=MLx¾

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

^¼?BA@S

a ^(k) = a ^(k) ₁ , . . . , a ^(k) n

> 0 ∈ R ⁿ ,

¼9T¼?BhBh0¼H@?BA[TÇE_HA@S>T\`_H@HMTV=@»_<>Qª?B<D»DH@?O=@U=

Π ^(k) = Π ^a ^(k) =

^\`_<Kc

n

0, a ^(k),1 , . . . , a ^(k),n o

(3)

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A*BDCEGFIHFG'$J#K$LM(N!)( $IO#"P(Q'P(RK#SF

?

^T

¼?BA@S

a ^(k),i = a ^(k) _i e ⁱ , i ∈ {1, ..., n}, k ∈ {1, . . . , K }.

U VXWDY[Z\YgýJú^]`_Íú^aYþxø½þ^ú"÷cb&ù9þN÷ýJþ _7deY

8JTV<>\`TE_H@A@S¼9T¼H@?BA[T

I = {1, . . . , n}

WÈ]_HÇA@S>T\`_<KcKTÚS7CDhBh½_NEA@S>TÊTÚªA@H[TVUT»_?B<KA[=

a ⁱ ¹ , ..., a ⁱ ^r

_NEÉA@S>TÇ»DH@?O=@U

Π ^a

¼TÇ¼?BhBh0C>=[TÇA@S>T°LNËDËDH[TVc7?ÂLNA@?O_<

\`_<7c

a ⁱ ¹ , . . . , a ⁱ ^r = [a ⁱ ¹ , . . . , a ⁱ ^r ] = ∆ ⁽ _J ^a ⁾ ,

¼?BA@S

J = {i 1 , . . . , i r }.

È>_HL.\`_hBhOT`\VA@?O_<_NE{»DH@?O=@U=

Π ^a ^(k) , k = 1, . . . , K,

¼9TJ¼?BhBh¼H@?BA[T

∆ ^(k) _J _(k) = ∆ ⁽ _J ^a _(k) ⁾ ^(k)

E_HA@S]T\`_<7cdTÚS7CDhBh{_NEÉA@S>TTÚªA@H[TVUT»_?B<KAM=

n a ^(k),l o

l ∈J ^(k) ⊂ Π ^a ^(k)

¼S>TV<>TVcKTVH

J ^(k) ⊆ I

?O==@CDË>=[TVA_NEY\`_7_H[Qª?B<]LNA[T`=xW f½|7ÎsÎoPgGmhg

i{ÕNÖ

ß

=Ò

äxà ÕNÖ

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ

^å

Ò à

Π ^a

^è ^Ò ^àá ^Ò ^ß ^ØMØ`Õ ^äVãß5à^Ò@× ^Ö ^ã^Øâkjml ^á ^Òìon ^< ^ÕÖ

Ò=ÒÖÓÒqp à Ö[Òâ"ÒØxÒ à

S ⊂ ∆ ^a

^àá ^ÒÖ[ÒÒqp ^ã^Ø ^à

J ⊆ I

^Ø çDäáàáªß5à

S =

^\`_<7c

a ⁱ _i _∈J = ∆ ⁽ _J ^a ⁾ .

rgues)s]utv90TVA

S ⊂ Π ^a

ËTtLN<ÍTÚªA@H[TVUTt=[TVAÊ_NEbA@S>TwÉLNH[TVA[_eEL\`T

∆ ^a

^_NE

Π ^a .

S>TV<ÍËK¾ QDTÆ><D?BA@?O_<_NEbLN<TÚªA@H[TVUT=[TVAg¼TS]L`cKTÊA@S]LNAgE_H°TVcKTVH@¾A2¼9_»_?B<7A[=

u, v ∈ Π ^a

¼S>_5=[T hB?B<>T1=[TV5UTV<7A

[u, v]

?B<7A[TVH[=[T`\VA[=

S

^Û½?W^T5W

[u, v] ∩ S 6= ∅

Û½?BA.E_hBhO_^¼=ÊA@S]LNA

u, v ∈ S.

^;E

¼T»TVHE_H@U A@SD?O=°»DH[_7\`T`QªCDH[T¼?BA@SA@S>TTÚªA@H[TVUT»_?B<7A[=._NE

Π ^a

¼TdTVAL\`_hBhOT`\VA@?O_<_NE TÚªA@H[TVUT»_?B<KA[=

a ⁱ _i∈J

^_NE

Π ^a

¼SD?O\[SLNH[TgTVhOTVUTV<KAM=b_NE

S.

8JTV<>\`TÇ?BAE_hBhO_x¼=A@S]LNA

S =

^\`_<Kc

a ⁱ

i ∈J = ∆ ⁽ _J ^a ⁾

S>_hOQD=A@H@C>T¼SD?O\[S»DH[_^cKTV=A@S>TgL=[=[TVH@A@?O_<0W xgB|hgByg

S>T¦PdTV¾E_HLNhBhb?B<7cdT`=@A@?BKLNA@?O_<>=_NEgA@S>T¦=@A@H@C>\VA@CDH[T_NEgA@S>T¦R?<>Pd_^¼=@P7?b=@CDU _NEg»DH@?O=@U=

_Hb\`TV»DS>_?OQD=?O=A@S>TA@S>T`_H[TVU$_<tA@S>T°LQDQª?BA@?O_<_NEEL\`T`=°ÀG=[T`Tzò ÉLN<>Q7ñQGÅÛD¼SD?O\[S¼T¼?BhBh

=@AMLNA[TS>TVH[T?B<tA@S>TÇE_hBhO_x¼?B<D¼Lx¾P1

f½|7Îs oPgÂo|{q}°z{| it|Kºs7~s]w^Â¹^s|s]ns>º5|7wQgkÒ

à

a ⁽¹⁾ , . . . , a ^(K) ∈ R ⁿ

^è

Ò ß

<Vß â ãåÓ"Õ <

ÕNØ ãGàGã =Ò=Ò ä`àÕÖMØ ß ì× åÒ à

Π = X K

k=1

Π ^(k)

Ô

ãGàá

Π ^(k) = Π ^a ^(k)

^< ^ÕÖ

k ∈ {1, ..., K}

^è ^Ò ^àá Ò+KÒìÒÖ ß5à

Ò@×

ä Ò á Õ ã

×GjÒ

à

F ⊂ Π

^è ^Ò ^ß

â ß p ãâ ßå#<Vßdä ÒÕ <

Π,

^ã jÒQjQª?BU

F = (n − 1)

^ß ^ì× ^å^Ò ^à

n _F ∈ R ⁿ

^è ^Ò ^ß ^ìÎÕ ^çªà ^{ÒÖ°ìÕÖâ} ^ßå ^Õ ^<

àá Ò <Vßdä Ò

F,

^ã^jÒQj

(4)

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F = Π ∩ {x ∈ Π | h n _F , xi = d ^∗ }, j = 1, ..., k,

Ô á ÒÖMÒ

d ^∗ = max

x∈ Π h n _F , xi > 0.

l á Òì

àá ÒÖMÒÒqp

ãØ àØ

K ₀ ⊆ {1, . . . , K}

^ß ^ì× ^ãì×KÒqpØxÒ àØ

J ^(k) ⊆ I (k ∈ K ₀ )

^Ø çDäá¦àáªßà

F = X

k∈K 0

∆ ^(k) _J _(k) + p

Ô á ÒÖMÒ

p

^ã^Ø ^ß ^Ø ^ç ^â$Õ ^< ^Òqp ^à^Ö[ÒâÒ ^Õ ^ã^ì ^à^Ø°Õ ^<àá ^ÒgÖ[Òâ ^ßã^ì ^ã^ì#+ ^Ö ^ã^Øâ.Ø

n Π ^(k) o

k / ∈K ₀ .

rgues)s]ut9¾06S>T`_H[TVU5WíA@S>TVH[TgTÚª?O=@A»DH@?O=@U=

Π ⁽ⁱ ¹ ⁾ , ..., Π ⁽ⁱ ^r ⁾

=@C>\MStA@S]LNA

F = S _i ₁ + ... + S _i _r + p

¼?BA@S

S _i _j = {x ∈ Π ⁽ⁱ ^j ⁾ | h n _F , xi = d ⁱ ^j }, j = 1, ..., r,

¼S>TVH[T

d ⁱ ^j = max

x∈ Π ^ij

h n _F , xi > 0

LN<>Q

p

?O=°Lt=@CDU_NETÚªA@H[TVUT»_?B<7A[=g_NEA@S>TÊH[TVULN?B<D?B<D

»DH@?O=@U=xW

wYCDA

K ₀ = {i 1 , . . . , i _r }

WJIK?B<>\`T°TVcKTVH@¾

S _k

?O=JLN<TÚªA@H[TVUT.=@CDË>=[TVAJ_NEYA[S>TwÉLNH[TVA[_EL\`T°_NE

Π ^(k) ,

?BAb\`LN<ËTÇ¼H@?BA@A[TV<L=

S _k = ∆ ^(k) _J _(k)

¼?BA@SLÊ=@CD?BAMLNËDhOTg=@CDË>=[TVA

J ^(k) ⊆ I

^W ^xgB|gByg

8JTV<>\`TTVcdTVH@¾UL^Úª?BULNhEL\`T.\`_<KAMLN?B<>=gLA@HMLN<>=@hÂLNA[T_NE½A@S>TR¦?B<DPd_^¼=@P7?=@CDU _NE'<>_</xTVH[_

Qª?BUTV<>=@?O_<]LNh>TÚªA@H[TVUT=[TVA[=É_NE]»DH@?O=@U=xW Ë>=[TVH@cdT9A@S]LNAA@S>T_CDA[TVHÉ<>_H@ULNhO=Y_NE]A@S>T'UL^Úª?BULNh

EL\`TLN<>QA@S>T"R¦?B<DPd_^¼=@P7?9=@CDU _NE<>_</xTVH[_Qª?BUTV<>=@?O_<]LNhTÚªA@H[TVUT"=[TVA[=._NE»DH[?B=@U=ÊLNH[T

T`ÐKC]LNh0CD»tA[_L.<>_<D<>TVKLNA@?BcKTgU°CDhBA@?B»DhOT5W

8JTV<>\`TE_H@A@S+¼TC>=[T

K = {1, . . . , K}

E_HÊA@S>Tt?B<>QDTÚ+=[TA_NEgL\`_hBhOT`\VA@?O_< _NEÇ»_5=@?BA@?BcdT cKT`\VA[_H[=xWY;1<tc7?OTV¼ _NEÉA@S>TgLNË_xcKT»DH[T`=[TV<KAMLNA@?O_<¦¼9T°L=[=@?B5<tA[_TVcdTVH@¾"UL^Úª?BULNhnEL\`T

F ⊂ Π =

X K k=1

Π ^(k) = X

k ∈K

Π ^(k)

A@S>TÇE_hBhO_^¼?B<D\`_hBhOT`\VA@?O_<_NEÉ»]LN?BH[=_NEÉ?B<>QDTÚ=[TVA[=

I _F =

K ₀ , h J ^(k) i

k ∈K ₀

.

8JTVH[T

K ₀ ⊂ K

?O=ÇA@S>T=[TVA_NE9?B<>Qª?O\`T`=_NE9A@S>_5=[T»DH@?O=@U=

Π ^(k) ,

A@S>T.EL\`T_NE'¼SD?O\[Sr?O=Ç<>_A /xTVH[_e1Qª?BUTV<>=@?O_<LNh{¼?BA@SH[T`=@»T`\VAA[_

n _F

Ûª?WT5WBÛDA@S]LNAb=MLNA@?O=EG¾

Qª?BU

(S _k ) =

^Qª?BU

{x ∈ Π ^(k) | h n _F , xi = d ^k }

≥ 1

¼?BA@S

d ^k = max

x∈ Π ^k

h n _F , xi > 0

WÊïbhO=[_DÛ

J ^(k)

\`_<7AMLN?B<>=gA@S>T?B<>Qª?O\`T`=°_NEA@S>TTÚªA@H[TVUTÊ»_?B<7A[=

_NEÉA@SD?O=EL\`TÇ?B<

Π ^(k) .

XÎT\xLNhBh{=[TVA

I _F

^A@S>T ^äMß ^ìÕì ^ãä[ßå ^Ö[Ò^< ^ÒÖMÒì ^ä ^ÒgØÓNØ ^à ^Òâ$Õ ^<

F.

(5)

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rguesh~s]w^Â¹^sZoPgO)g

Ò à

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

^è ^Ò ^ß<Vß ^â ^ãå^Ó"Õ ^< ^ÕNØ ^ãGàGã ^=Ò=Ò ^ä`à ^ÕÖMØ ^ß ^ì× ^å^Ò ^à

Π =

X K k=1

Π ^(k)

Ô ãGàá

Π ^(k) = Π ^a ^(k)

^< ^ÕÖ

k ∈ K.

^{Ò ^à

F ⊂ Π

^è ^Ò ^ß ^â ^ß ^p ^ã^â ^ßå<Vßdä ^Ò ^ß ^ì×

n _F ∈ R ⁿ

^è ^Ò ^ß ^ì

Õ

çªà ÒÖ°ìÕÖâ

ßå

Õ

<àá

Ò

<ßdä

Ò

F.

l á Òì

n _F

ãØ ß Ø àÖ ãä`àGåÓ

ÕNØ ãGàGã =Ò=Ò ä`àÕÖQj

rgues)s]ut90TVA0C>={L=[=@CDUTYA@S]LNA

F ⊂ Π

?O={LUL^Úª?BULNh5EL\`TY_NE

Π

LN<>QA@S]LNA

n _F = (α ₁ . . . α _n ) ∈ R ⁿ

?O=bLN<¦_CDA[TVH<>_H@ULNh0¼?BA@ShOTVcdTVh

1

^_NE

F.

S>TV<

F = {x ∈ Π | h n _F , xi = 1 }.

9¾"A@S>TgQDT`\`_UÊ»_5=@?BA@?O_<¦»>H@?B<>\V?»>hBTgA@S>TVH[TgTÚª?O=@A»DH@?O=@U=

Π ^(k) (k ∈ K ₀ )

=@C>\MStA@S]LNA

F = X

k ∈K ₀

∆ ^(k) _J _(k) + p

=MLNA@?O=EG¾7?B<D

∆ ^(k) _J _(k) = {x ∈ Π ^(k) | h n _F , xi = d ⁱ ^j }, k ∈ K ₀

¼S>TVH[T

d ^k = max

x∈ Π ^k

h n _F , xi > 0

LN<>Q

p

^?O=ÊL=@CDU _NEbTÚªA@H[TVUT»_?B<7A[=._NEA@S>TH[TVULN?B<D?B<D

»DH@?O=@U=xW½R_H[T`_xcKTVH

h n _F , a ^(k)i i = d ^k (i ∈ J ^(k) ).

IK?B<>\`T

h n _F , a ^(k),i i = α i a ^(k) _i

?BAE_hBhO_x¼=A@S]LNA

α i > 0

LN<>QEGH[_UA@S>T¦L=[=@CDUÊ»DA@?O_< A@S]LNA

F

?O="LUL^Úª?BULNhEL\`T5Û9?WT5W Qª?BU

F = n − 1,

?BA'E_hBhO_x¼='A@S]LNALNhBh\`_UÊ»_<>TV<7A[=_NE

n _F

LNH[T5H[TxLNA[TVH9A@S]LN<^/xTVH[_DWxgB|gByg

}°z{|Gs]uN|7 oPg0g

Ò

à

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

Π =

X K k=1

Π ^(k)

Ô ãGàá

Π ^(k) = Π ^a ^(k)

^< ^ÕÖ

k ∈ K.

^{Ò ^à

F ⊂ Π

^è ^Ò ^ß ^â ^ß ^p ^ã^â ßå)<Vßdä Ò¦Õ <

Π

^Ô ^ãGàá ^Õ ^çªà^ÒÖ

ìÕÖâ ßå

n _F ∈ R ⁿ

^ß ^ì× ^å^Ò ^à ^Ô ^ãGàá ^Ø ç7ãGàßKèåÒ

K ₀ ⊆ K

F = X

k ∈K ₀

∆ ^(k) _J _(k) + p

Ô ãGàá

∆ ^(k) _J _(k) = {x ∈ Π ^(k) | h n _F , xi = d ^k }, k ∈ K ₀

Ô á ÒÖMÒ

d ^k = max

x∈ Π ^k

h n _F , xi > 0

^ß ^ì×

p

^ã^Ø ^ß ^Ø ^ç ^â ^Õ ^< ^Òqp ^à^Ö[ÒâÒ ^Õ ^ã^ì ^à^ØÕ <"àá Ò"Ö[Òâ ßãì ãì#+

Ö ã Øâ.ØQj@l

á

Òìon

<

ÕÖ

ß

ì>Ó×KÒ

ä

Õâ

ÕNØ

ãGàGã ÕìÕ

<

I = {1, . . . , n}

^ã^ì ^à ^Õ ^à Ô'ÕìÕìÒâ à

Ó×

ãØé Õ ãì à

Ø ç>è ØxÒ àØ

I ₀

^ß ^ì×

I ₁

^àá ^{ÒÖ[ÒÊÒqp} ^ã^Ø ^à^Ø

κ ∈ K ₀

^Ø çDäáàáªß5à

J ^(κ)

^áªß ^Ø ^ß ^ìÕìÒâ ^à^Ó ^ã^ì ^à ^ÒÖMØxÒ ^ä`àGã ^Õì

Ô ãGàá¦è Õ àá ØxÒ àØ

I ₀

^ß ^ì×

I ₁ .

(6)

?

TDCEGF$h(Q'P(R%GF#'GFICEGF$ "GFJ

?

rgues)s]uty90TVA'C>=9L=[=@CDUTA@S]LNA½A@S]TbL=[=[TVH@A@?O_<?O=½<>_A½A@H@C]TW;S>TV<A@S>TVH[TbTÚª?O=@A[=9LQDT`\`_UÊ»_e

=@?BA@?O_<_NE

{1, . . . , n}

?B<7A[_A1¼_<>_<>TVUÊ»DA2¾¦Qª?O=_?B<KAÇ=@CDË>=[TVA[=

I ₀

LN<>Q

I ₁

=@C>\MS¦A@S]LNAÇTVcdTVH@¾

=[TVA

J ^(k)

^EGH[_U A@S>Tg\xLN<>_<D?O\xLNh{H[TETVH[TV<>\`Tg=@¾ª=@A[TVU _NE

F

ËTVhO_<Dd=bTV?BA@S>TVHA[_

I ₀

^_HA[_

I ₁

^W

J_^¼gÛªA[_TVcdTVH@¾

κ ∈ K ₀

¼TgL=[=@?B5<¦L.ÆDÚDT`Q?B<>QDTÚ

i _k ∈ J ^(k)

LN<>Q¦L.H[_^¼cdT`\VA[_H

b

a ^(k) = X

i ∈J ^(k)

a ^(k)i − a ^(k)i ^k

IK?B<>\`T

h n _F , a ^(k)i )i = h n _F , a ^(k)i ^k i (i ∈ J ^(k) ,

?BAE_hBhO_x¼=A@S]LNAA@S>TgLNË_xcKTÇH[_^¼cdT`\VA[_Hb?O=_H@A@S>_d_<]LNh{A[_A@S>Tg_CDA[TVH<>_H@ULNh0cKT`\VA[_H

n _F .

J_^¼ ¼TtËDCD?BhOQ LULNA@H@?ÚÍA@S]LNA?B<7cd_hBcKT`=LNhBhA@S>T`=[TtH[_x¼cKT`\VA[_H[=xWIJT`\`T`=[=MLNH@?BhB¾dÛL^EGA[TVH"L

=@CD?BAMLNËDhOT»TVH@U°CDAMLNA@?O_<_NEÉH[_x¼=JLN<>Qt\`_hBCDUÊ<>=xÛDA@S>?B=ULNA@H@?ÚtS]L=A@S>TJE_hBhO_^¼?B<DÊE_H@U1

M =

· · · I ₀ · · · · · · I ₁ · · ·



 



A 0

0 B



 

 .

IK?B<>\`TA@S>T_CDA[TVH<>_H@ULNh0cdT`\VA[_H

n _F

?O=CD<D?OÐKC>TVhB¾QDTVA[TVH@UÊ?B<>T`QtCD»tA[_L°»_5=@?BA@?BcdTÇU.CDhBA@?B»DhOT

LN<>Q¦ËTVhO_<Dd=JA[_A@S>T°PdTVH@<>TVh_NE½A@S>TLNË_^cdTgULNA@H@?Ú

M,

¼T°QDT`QªC>\`T5Û]A@S]LNAJA@S>TgHMLN<DP_NE

M

?O=

(n − 1).

;EbË_A@SÍ=@CDËULNA@H@?O\`T`=

A

LN<>Q

B

QD_<>_AS]LxcdTEGCDhBhHMLN<DPÛYA@S>TV<

M

S]L=ÊLNAU_5=@AHMLN<DP

(n − 2)

¼SD?O\MS?O=J<>_AJ»_5=[=@?Ë>hBT5Û=[_¼T.\xLN<rL=[=@CDUT5ÛA@S]LNA

B

S]L=bEGCDhBhHMLN<DPLN<>Q

A

<>_AxW 8JTV<>\`TÊA@S>TÊT`ÐKC]LNA@?O_<

M n _F = 0

S]L=°L=[_hBCDA@?O_<0Û0¼S>TVH[TLNhBh½\`_UÊ»_<>TV<7A[=°_NE

n _F

¼?BA@S

?B<>Qª?O\`T`=?B<

I ₁

c5LN<D?O=@S¦LN<>QtA@SD?O=?O=bL\`_<7A@HMLQª?O\VA@?O_<¦A[_mwYH[_»_5=@?BA@?O_<îªW3ªW xgB|hgByg

VXWDYÏúþN÷ýJþNûDYg÷ýYVXWDYúù,Y]

?BcKTV<tA2¼9_\`_<KcKTÚt=[TVA[=

A, B ⊂ R ⁿ

¼TQDTV<>_A[TË7¾

A ∨ B =

^\`_<Kc

(A ∪ B )

A@S>Tg\`_<7cdTÚS7CDhBh{_NEÉA@S>TV?BHCD<D?O_<0W½ïbhO=[_¼9T¼H@?BA[T

Π ^a J = ∆ ^a J ∨ {0}

E_HbA@S>T°»DH@?O=@UÏdTV<>TVHMLNA[T`Q¦Ë7¾¦LN<TÚªA@H[TVUT.=[TVAJ_NE

∆ ^a

WS>T°<>_AMLNA@?O_<

Π ^(k) _J _(k)

A@S>TV<¦H[TETVH[=

A[_ÊA@S>TcdT`\VA[_H

a ^(k)

_NE½L°ELNUÊ?BhB¾KW;S>TV<¼TÇS]LxcdT 1

(7)

?

}°z{|Gs]uN|7Ì)gGmhg

Ò à

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

Π = X K k=1

Π ^(k)

Ô ãGàá

Π ^(k) = Π ^a ^(k)

^< ^ÕÖ

k ∈ {1, ..., m}.

^{Ò ^à

F ⊂ Π

^è ^Ò ^ß ^â ^ß ^p ^ã^â ^ßå<Vßdä ^Ò.Õ ^<

Π

^j

l á Òì ß =Ò ä`à ÕÖ

n _F ∈ R ⁿ

^ã^Ø ^ß ^ì ^Õ ^çªà^ÒÖìÕÖâ ^ßå:1à ^Õ ^àá ^Ò ^å^Ò=Ò ^å

1

^> ^Õ ^<àá ^Ò"â ^ß ^p ^ã^â ßåP<Vßdä Ò

F ⊂ Π

^ã^<ß ^ì×"Õì ^å^Ó ^ã^<àá ^{ÒÖ[ÒÊÒqp} ^ã^Ø ^à

K ₀ ⊂ K

^ß ^ì× ^<Vßdä ^ÒMØ

∆ ^(k) _J _(k) ⊂ Π ^(k) , k ∈ K ₀

ß ØgÔ9Ò åÂåß Ø

ÕNØ ãGàGã=Òì ç â è ÒÖMØ

c _k ∈ R

^Ô ^áKãäá ^Ø ^ß5àGã^Ø^< ^Ó ^àá ^Ò ^< ^Õ ^åå^ÕÔ ^ã^ì#+ ^ä ^Õì× ^ãGàGã ^ÕìDØxÙ

ji0ÕÖÒ=ÒÖÓ

k ∈ K ₀

^àá ^Ò ^å^ã^ìÒ ^ß ^Ö ^<Mç ^ì ^ä`àGã ^Õì

x 7→ hn F xi

^ã^Ø ^ä ^ÕìDØ ^àß ^ì ^à ^Òu¡ ^çDßå½à ^Õ

1

^Õì

àá

Ò

<Vßdä

Ò

c k ∆ ^(k) _J _(k)

^j

Ü j ã< n < ÕÖJØ`ÕâÒ

k, l ∈ K ₀

ⁿ ^à^Ô'Õ"× ^ã^¢ ^ÒÖMÒì ^àh<Vßdä ^ÒMØ

c _k ∆ ^(k) _J _(k)

^ß ^ì×

c _l ∆ ^(l) _J _(l)

^áªß ^=ÒÒqp ^à^ÖMÒâÒ

Õ ãì à ØbÔ áKãäáåãÒgÕì àá ÒbØ ß âÒ ä Õ^ÕÖ@× ãì ß5àÒ ß p ãØ£n àá Òì àá ÒMØxÒ°Òqp àÖ[ÒâÒ

Õ ãì àØ ä Õ ãì¤

äVã×KÒQj

¥ jl á Ò Ö ãØâ

Π b = _

k∈K 0

c _k Π ^(k) _J _(k)

ãØ

n

^¤× ^ã^âÒìDØ ^ã^Õì ^ßåß ^ì× ^ãGà ^ØJÕ ^çªà^{ÒÖJìÕÖâ} ^ßå¦:1à ^Õ ^àá ^Ò ^å^Ò=Ò ^å

1

^> ^ã^Ø ^àá ^ÒÇÕ ^çªà ^{ÒÖbìÕÖâ} ^ßå

n _F

Õ <

F.

rgues)s]ut,9{TAC>=L=[=@CDUTJA@S]LNA

F ⊂ Π

?O=L°UL^Úª?BULNhEL\`TÇ_NE

Π

LN<>Q"A@S]LNA

n _F ∈ R ⁿ

^?O=LN<

_CDA[TVH<>_H@ULNh_NE

F.

S>TV<

F = {x ∈ Π | h n _F xi = 1 }.

9¾"A@S>TgQDT`\`_UÊ»_5=@?BA@?O_<¦»>H@?B<>\V?»>hBTgA@S>TVH[TgTÚª?O=@A»DH@?O=@U=

Π ^(k) k ∈ K ₀

=@C>\MStA@S]LNA

F = X

k ∈K ₀

∆ ^(k) _J _(k) + p

¼S>TVH[T

p

?O=°L"=@CDU_NETÚªA@H[TVUT»_?B<KA[=g_NEA@S>TH[TVULN?B<D?B<D»DH@?O=@U=¼?BA@S?<]QDTÚ

k / ∈ K ₀

^W

ïbKLN?B<¼TÇP7<>_^¼ A@S]LNA¼?BA@S

∆ ^(k) _J _(k) = {x ∈ Π ^(k) | h n _F , xi = d ^k }, k ∈ K ₀ ,

¼?BA@S

d ^k = max

x∈ Π ^(k)

h n _F , xi > 0

^W

J_^¼ _Ë>=[TVH@cKTÇA@S]LNAA@S>TÇEL\`T`=

∆ ^(k) _J _(k) (k ∈ K ₀ )

LNH[ThB¾7?B<D?B<t»]LNHMLNhBhOTVh0S7¾7»TH[»DhÂLN<>T`=xÛD<]LNUTVhB¾

H ^k = {z ∈ R ⁿ | h n _F , xi = d ^k }, k ∈ K ₀ ,

(8)

?

^§

¼?BA@S

d ^k > 0 (k ∈ K ₀ ).

^J_x¼ \MS>_K_5=[T

c _k = 1

d ^k , k ∈ K ₀ .

S>TV<A@S>TÇÆ>H[=@A\`_<>Qª?BA@?O_<¦?O==MLNA@?O=Æ]T`QnW

J_^¼ L=[=@CDUTtA@S]LNAA2¼9_rEL\`T`=

c _k ∆ ^(k) _J _(k)

LN<>Q

c _l ∆ ^(l) _J _(l)

S]L`cKT¦TMÚªA[H[TVUT»_?B<7A[=¼SD?O\[SZhB?OT _<A@S>T

j

A@S¦\`_K_H[Qª?B<]LNA[T.L^Úª?O=xÛª?WT

c _k a ^(k) _j e ^j ∈ c _k ∆ ^(k) _J _(k)

LN<>Q

c _l a ^(l) _j e ^j ∈ c _l ∆ ^(l) _J _(l) .

S>TV<

h n _F , a ^(k) _j e ^j i = d ^k

LN<>Q

h n _F , a ^(l) _j e ^j i = d ^l ,

S>TV<>\`T

d ^k a ^(k) _j = d ^l

a ^(l) _j

¼SD?O\MSUTxLN<>=A@S]LNA

c _k a ^(k) _j = c _l a ^(l) _j .

8JTV<>\`T5ÛÉA@S>TA1¼_¦TÚªA@H[TVUT»_?B<KA[=\`_?B<>\V?BQ>TW¦ïÇ=.L\`_<>=[T`ÐKC>TV<>\`T5ÛÉ<>_¦TÚªA@H[TVUT»_?B<7A._NE

LN<7¾EL\`T

c _k ∆ ^(k) _J _(k)

¼?BhBh0ËT`\`_UT°LN<?B<D<>TVH»_?B<7A?B<

Π b = _

k ∈K ₀

c _k Π ^(k) _J _(k)

¼SD?O\MS?BUÊ»DhB?OT`=A@S]LNAbQª?BU

Π b = = n − 1.

R_H[T`_^cdTVHA@S>TJEGCD<>\VA@?O_<

x 7→ h n _F , xi

?O=\`_<>=@AMLN<7AT`ÐKC]LNhA[_

1

^_<

Π b

LN<>Q"A@S>TVH[TE_H[T

n _F ∈ R ⁿ

?O=LN<_CDAMTH<>_H@ULNh½ÀA[_.A@S>TbhOTVcKTVh

1

^Å9_NE

Π b

¼SD?O\MS»DH[_xcKT`=_<>TgQª?BH[T`\VA@?O_<¦_NEÉA@S>TA@S>T`_H[TVUW

È]_H"A@S>T\`_<7cdTVH[=[TQª?BH[T`\VA@?O_< hOTVA"C>=tL=[=@CDUT¦A@S]LNA"A@S>TVH[TTÚª?O=@A[=LÎ=[TVA

K ₀ ⊂ K

^LN<]Q

EL\`T`=

∆ ^(k) _J _(k) ⊂ Π ^(k)

_NEYA@S>T°\`_H@H[T`=@»_<>Qª?B<D"»DH@?O=@U=JL=¼TVhBhÉL=»_5=@?BA@?BcKTg<KCDU.ËTVH[=

c _k ∈ R , k ∈ K ₀

¼?BA@S¦A@S>T E_hBhO_^¼?B<D»DH[_»TVH@A@?OT`=Q1

5WÇÈ]_HbTVcKTVH@¾

k ∈ K ₀

A@S>TghB?B<>TxLNHbEGCD<>\VA@?O_<

x 7→ hn F , xi

?O=b\`_<>=@AMLN<7AJT`ÐKC]LNh{A[_

1

^_<

A@S>TÇEL\`T

c _k ∆ ^(k) _J _(k)

^W

îªWÇ;E½A2¼9_"Qª?©¨nTVH[TV<7AJEL\`TV=

c _k ∆ ^(k) _J _(k)

LN<>Q

c _l ∆ ^(l) _J _(l)

S]L`cKT.TÚªA@H[TVUT°»_?B<KA[=_<A@S>T.=MLNUT

\`_7_H[Qª?B<]LNA[T.L^Úª?O=xÛªA@S>TV<A@S>TV¾\`_?B<>\V?OQDT5W

(9)

?

^ª

3ªW«S>T»DH@?O=@U

À3ªW0^Å

Π b = _

k ∈K ₀

c _k Π ^(k) _J _(k)

?O=

n

1Qª?BUTV<>=@?O_<]LNh>LN<]Q°?BA[=É_CDA[TVHÉ<>_H@ULNh{ÀA[_bA@S>T9hOTVcdTVh

1

Å0\`_?B<>\V?OQDT`=Y¼?BA@S.A@S]T9_CDA[TVH

<>_H@ULNh

n _F

_NE

F.

S>TV<ÊhOTVAÉC>=É\`_<>=@?OQDTVHÉA@S>T_CDAMTHY<>_H@ULNh{ÀA[_bA[S>T9hBTVcKTVh

1

^Å

n _S

_NE]A@S>T9<1Qª?BUTV<>=@?O_<]LNh>»DH@?O=@U

Π b

L=5?BcdTV<Ë7¾+À3ªW0^Å«S>TEGCD<>\VA@?O_<]LNh

f (·) = h n _S , ·i

L\MSD?OTcKT`=g?BA[=UL^Úª?BULNh½c5LNhBC>T`=°_<

A@S>T»DH@?O=@U=

n Π ^(k) o

k∈K 0

_<A@S>TÇEL\`T`=

∆ ^(k) _J _(k) ⊂ Π ^(k) (k ∈ K ₀ ).

IK?B<>\`T.TVcKTVH@¾tUL^Úª?BULNh{EL\`T°_NE

Π

?O=bA@S>T.=@CDU$_NEYEL\`T`=J_NE½A@S>Tg»DH@?O=@U=

Π ^(k) (k ∈ K )

^Û]?BA

E_hBhO_^¼=A@S]LNAJL.A@HMLN<]=@hOLNAMTg_NEÉA@S>T=[TVA

X

k ∈K ₀

c k D ^(k) _J _(k)

?O=\`_<7AMLN?B<>T`Q?B<¦L.UL^Úª?BULNh0EL\VT

F ⊂ Π.

^{J_^¼} ¼TÇ¼?BhBh0»DH[_^cdTÇA@SLNA

Qª?BU



 X

k∈K 0

c _k ∆ ^(k) _J _(k)



 = n − 1

S>_hOQD=.A@H@C>T5W;1<>QDT`T`QnÛË7¾\`_<>Qª?BA@?O_<3dÅg_NEA@S>TL=[=[TVH@A@?O_<ÎA@S>T»DH@?O=@U

W

k ∈K ₀ c _k Π ^(k)

J ^(k)

?O=

EGCDhBh©1Qª?BUTV<>=@?O_<]LNhW;8JTV<>\`T½A@S>TywÉLNH[TVA[_eEL\`T9S]L=0A@S>T'Qª?BUTV<>=@?O_<

(n−1).

IK?B<>\`T'A@S>TywÉLNH[TVA[_e EL\`T_NE

Π b

?O=YA@S>T\`_<KcKTÚS7CDhBh>_NEA@S>TEL\`T`=

c _k ∆ ^(k) _J _(k) (k ∈ K ₀ ),

A@S>TVH[TTÚª?O=@A½T`QªdT`=½_NEA@SD?O=

EL\`TÊ¼SD?O\[SLNH[TÊ_H@A@S>_d_<]LNh'A[_tA@S>TÊcdT`\VA[_H

n _S

LN<>Q=@»]LN<L"hB?B<>TxLNH°=@»]L\`T_NEQª?BUTV<>=@?O_<

(n − 1),

¼SD?O\MSUTxLN<>=A@S]LNAQª?BU

P

k∈K 0 c _k ∆ ^(k) _J _(k) = n − 1.

Ã9_<>=[T`ÐKC>TV<7A@hB¾dÛ

F

LN<>QrA@S>T=[TA

P

k ∈K ₀ c _k ∆ ^(k) _J _(k)

S]LxcdTÊA@S>TÊ=MLNUTÊ_C>A[TVH<>_H@ULNhW"IK?B<>\`T A@S>T=[TVA

P

k∈K 0 c _k ∆ ^(k) _J _(k)

LN<>QÊA@S>T»DH@?O=@U

Π b

SL`cKT\`_UÊU_<_CDA[TVH½<>_H@ULNhO=xÛdA@S>T\`_<KcKTVH[=[T

Qª?BH[T`\VA@?O_<¦?O=»DH[_^cdT`QnW xgB|hgByg

S>T`_H[TVUX3ªW0b5?BcKT`=Lg»_5=[=@?BËD?BhB?BA1¾E_HLN<LNË>=@A@HML\VAQDT`=[\H[?»>A@?O_<t_NE{A@S>TgØ àÖ çDä`àGç Ö[Ò°Õ <6<Vßdä ÒMØ

?B<SD?B5S>TVHbQª?BUTV<>=@?O_<>=xW

rguesh~s]w^Â¹^s )gÂoPg

Ò à

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ

^ß ^ì×

b = (b 1 , . . . , b n ) > 0 ∈ R ⁿ

^è ^Ò ^ÕNØ ^ãGàGã^=Ò=Ò ^ä`à ^ÕÖMØ

ß ì× åÒ à

Π ^a

^ß ^ì×

Π ^b

^è ^Ò ^àá ^Ò ^ß ^ØMØ`Õ ^äVãß5à^Ò[× ^Ö ^ã^Øâ.ØQj

¬

ØMØ ç âÒ àáªß5àÊàá Ò åãìÒ ß Ö <Mç ì ä`àGãÕì

x 7→ ha, xi

^ßdäáKã^Ò=ÒMØ ^ãGà^Ø¦â ^ß ^p ^ã^â ^ç ^â ^Õ=ÒÖ

Π ^b

^ã^ì

àá ÒÒqp àÖMÒâÒ

Õ ã ì à

b ⁱ ⁰ = b _i ₀ e ⁱ ⁰ ∈ Π ^b ,

^àá ^Òì

b _i ₀

a _i ₀ Π ^a ⊇ Π ^b

ß ì×

b _i ₀ e ⁱ ⁰

^ã^Ø ^ß"ä ^{ÕââÕìÒqp} ^à^Ö[ÒâÒ ^Õ ^ã^ì ^à ^Õ ^<°è ^Õ ^àáàá ^Ò ^Ö ^ã^Øâ.Ø

^b _a ⁱ ⁰

i 0 Π ^a

^ß ^ì×

Π ^b

^j

(10)

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^±

rgues)s]utJIK?B<>\`TA@S>TghB?B<>TxLNHEC><>\VA@?O_<

x 7→ ha, xi

L\MSD?OTVcdT`=b?BA[=UL^Úª?BU.CDU$_xcKTVH

Π ^b

?B<A@S>T TÚªA@H[TVUT»_?B<KA

b _i ₀ e ⁱ ⁰ ∈ Π ^b

Ûª¼TÇS]LxcdT

max n

ha, xi | x ∈ Π ^b o

= b _i ₀ a _i ₀

¼SD?O\MS?BUÊ»DhB?OT`=

b i 0

a _i ₀ Π ^a ⊇ Π ^b

LN<>QtA@S]LNA

b i 0

a i 0 Π ^a

LN<>Q

Π ^b

^S]LxcKT

b _i ₀ e ⁱ ⁰

L=bL\`_UÊU_<TÚªA@H[TVUT»_?B<7AxW xgB|hgByg

² ³´ýYaµ´÷û¶b·YJù] üøy´Êø½þ^ú"÷7a

TV<KAMLNA@?BcKTVhB¾¼9T\`_<>=@?OQDTVH½A@S>T\xL=[T_NEA2¼9_»DH@?O=@U=xWÉÈ>_H½\`_<7cdTV<D?OTV<>\`T¼9TC>=[TLÇ=@?BUÊ»DhB?Æ]T`Q

<>_AMLNA@?O_<0WYÈ>_HA2¼_bcdT`\VA[_H[=

a = (a ₁ , . . . , a _n ) > 0 ∈ R ⁿ

LN<>Q

b = (b ₁ , . . . , b _n ) > 0 ∈ R ⁿ

¼TÇ¼H@?BA[T

A = Π ^a =

^\`_<Kc

0, a ¹ , . . . , a ⁿ

LN<>Q

B = Π ^b =

^\`_<7c

0, b ¹ , . . . , b ⁿ

E_HA@S>TL=[=[_K\V?ÂLNA[T`Q»DH@?O=@U=xWY;1<"A@SD?O=\xL=[T5Û7?E{A@S>TJA2¼9_.»DH@?O=@U=

A

LN<>Q

B

S]LxcdTL.\`_UÊU_<

TÚªA@H[TVUT»_?B<KAb_<A@S>TV?BHwÉLNH[TVA[_eEL\`T`=bA@S>TV<A@S>TÇE_hBhO_x¼?B<D=@AMLNA[TVUTV<7AS>_hOQD=Q1

f½|7ÎsDgGmhg

æ]ç

ÕNØxÒ àáªß5à

a ⁱ = b ⁱ

^á ^Õ ^å^×NØ ^à^Ö ^ç ^Ò ^< ^{ÕÖÇØ`ÕâÒ}

i ∈ I.

l á Òì < ÕÖÒ=ÒÖÓ ãì×KÒqp

l 6= i

^á ^Õ ^å^×NØxÙ

j

a ^l ∈ A

^ã^Ø ^ß ^ìÒqp ^à^ÖMÒâÒ ^Õ ^ã^ì ^à ^Õ ^<

A ∨ B

^ã^<gß ^ì×"Õì ^å^Ó ^ã^<

^a _b ^l

l ≥ ^a _b ⁱ

i

Ü j

b ^l ∈ B

^ã^Ø ^ß ^ìÒqp ^àÖ[Òâ"Ò

Õ ãì à Õ <

A ∨ B

^ã^<gß ^ì×"Õì ^å^Ó ^ã^<

^a _b ^l

l ≤ ^a _b ⁱ

i .

rgues)s]ut;1<tc7?OTV¼ _NE

A =

^\`_<7c

0, a ¹ , . . . , a ⁿ

LN<>Q

B =

^\`_<7c

0, b ¹ , . . . , b ⁿ

¼TÇS]LxcKT

A∨B =

^\`_<7c

0, max{a 1 , b ₁ }e ¹ , . . . , max{a i − 1 , b _i − 1 }e ⁱ ⁻ ¹ , a _k e ⁱ , max{a i+1 , b _i+1 }e ⁱ⁺¹ , . . . , max{a _l , b _l }e ^l , . . . , max{a n , b n }e ⁿ o

.

J_^¼

b ^l ∈ B

?O=LN<TÚªA@H[TVUTJ»_?B<KA_NE

A ∨ B

^?EÉLN<>Q"_<DhB¾?E

a _l ≤ b _l

¼SD?O\MS?O=T`ÐKCD?Bc5LNhOTV<KA A[_

a l

b l ≤ 1 = ^a _b ⁱ

i ,

ËT`\xLNC>=[T

a _i = b _i

ËK¾tL=[=@CDUÊ»DA@?O_<0W

S>T»DH[_7_NEY_NEÉA@S>TÇÆ>H[=@A=@AMLNA[TVUTV<7A?O=?OQDTV<7A@?O\`LNh{A[_ÊA@SD?O=_<>T5W xgB|hgByg

J_^¼ ¼9T\`_<>=@?OQDTVHA@S>T\xL=[T_NEÇA@SDH[T`T»DH@?O=@U=xW+XT¦LNCD5UTV<KAA@S>T<>_AMLNA@?O_< ËK¾?B<KA@H[_e

QªC>\V?B<D

C = Π ^c =

^\`_<7c

0, c ¹ , . . . , c ⁿ

(11)

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f½|7ÎsDgÂoPg

Ò à

a, b, c

^è ^Ò ^ÕNØ ^ãGàGã ^{=Ò^=Ò} ^ä`à ^ÕÖMØ ^ß ^ì× ^å^Ò ^à

A, B, C

^è ^Ò ^àá ^Ò ^ß ^ØMØ`Õ ^äVãß5à^Ò[× ^Ö ^ã^Øâ.ØQj

¬

ØMØ ç âÒ

àáªß5àM< ÕÖ ß ì ãì×KÒqp

i ∈ I

^àá ^ÒÒq¡ ^çDß5àGã ^ÕìDØ

a ⁱ = b ⁱ = c ⁱ

^á ^Õ ^å^× ^à ^Ö ^ç ^ÒQj

l á Òìon

<

ÕÖ.Ò=ÒÖÓ

l 6= i

ⁿ ^àá ^ÒÒqp ^àÖ[Òâ"Ò Õ ãì à

c ^l ∈ C

^ã^Ø ^ß ^ìÎÒqp ^à^Ö[ÒâÒ ^Õ ^ã^ì ^à ^Õ ^<

A ∨ B ∨ C

ß ØgÔ9Ò åÂå0ã<gß ì×"Õì åÓ ã<

a _i c _i ≥ a _l

c _l

ß ì×

b _i

c _i ≥ b _l c _l

ãØÇØ ß5àGãØë Ò[×Gj

rgues)s]utS>T.»DH[_7_NE'?O=Ç?OQDTV<KA@?O\xLNhYA[_"A@S>T.»DH[_K_NE_NE'A@S]T°»DHMTc7?O_C>=¸90TVUÊUL^>W05Wg;E

A, B

LN<>Q

C

LNH[T°L=bLNË_^cdT5ÛDA@S>TV<¼9TS]L`cKT

A ∨ B ∨ C =

\`_<7c

0, max{a 1 , b ₁ , c ₁ }e ¹ , . . . , max{a i − 1 , b _i − 1 , c _i − 1 }e ⁱ ⁻ ¹ , a _i e ⁱ , max{a i+1 , b _i+1 , c _i+1 }e ⁱ⁺¹ , . . . , max{a l , b _l , c _l }e ^l , . . . , max{a n , b _n , c _n }e ⁿ o

.

J_^¼

c ^l ∈ C

?O=gLN<rTÚªA@H[TVUT»_?B<7A_NE

A ∨ B ∨ C

^?ELN<>Q_<DhB¾¦?E

a _l ≤ c _l

LN<>Q

b _l ≤ c _l

¼SD?O\MS?O=ÇT`ÐKCD?Bc5LNhOTV<KAJA[_

a l

c l ≤ 1 = ^a _c ⁱ

i

LN<]Q

b l

b l ≤ 1 = ^b _c ⁱ

i

ËT`\xLNC>=[T

a _i = b _i = c _i

S>_hOQD=JËK¾

L=[=@CDUÊ»DA@?O_<0W xgB|hgByg

rguesh~s]w^Â¹^s¹gO)g

Ò

à

a, b

^è ^Ò ^{Õ^Ø} ^ãGàGã^=Ò=Ò ^ä`à ^ÕÖMØ ^ß ^ì× ^å^Ò ^à

A, B

^è ^Ò ^àá ^Ò ^ß ^ØMØVÕ ^äVãß5à ^Ò@× ^Ö ^ãØâ.ØQji0ÕÖ

i ∈ I

^å^Ò ^à

F ⁱ _A =

^\`_<Kc

a ^l | l ∈ {1, ..., n}

^¼?BA@S

a _l b _l ≥ a _i

b _i

ß ì×

F ⁱ B =

^\`_<Kc

b ^l | l ∈ {1, ..., n}

^¼?BA@S

a _l b l ≤ a _i

b i

.

l á Òìon < ÕÖ.Ò=ÒÖÓ ãì×KÒqp

i ∈ I

^àá ^ÒÇØ ^ç ^â

F ⁱ _A + F ⁱ _B

ãØ ä Õì àßã ìÒ@× ãì ß â ß p ãâ ßå<Vßdä Ò.Õ <

A + B.

rgues)s]utïJ=

b i A

LN<>Q

a i B

S]L`cKTL\`_UÊU_<TÚªA@H[TVUT.»_?B<7AxÛA@S>T.H[T`=@CDhBAbE_hBhO_^¼=J?BUÊUT

Qª?ÂLNA[TVhB¾"EGH[_UX90TVUÊUL>W0°LN<>QS>T`_H[TVUº3ªW05W xgB|gByg

XÎTÇULx¾"<>_x¼ H[TE_H@U.CDhÂLNA[TwYH[_»_5=@?BA@?O_<v>W3.?B<A[TVH@U=_NEÉ»TVH@U°CDAMLNA[?B_<]=JL=E_hBhO_^¼=xW

it|»YÂ¹^s°g0g90TVA

a > 0, b > 0

ËTgcKT`\VA[_H[=J_NE

R ⁿ

Wï »TVH@U.CDAMLNA@?O_<

π : I −→ I

?O=\xLNhBhOT`Q¦L

ÕNØ ãGàGã Õì ãì#+

ÒÖâ çªàß5àGãÕì?EÉA@S>Tg=[T`ÐKC>TV<>\`Tg_NEYÐKC>_A@?OTV<KA[=

a _π ₍₁₎

b _π ₍₁₎ ≥ a _π ₍₂₎

b _π ₍₂₎ ≥ . . . ≥ . . . ≥ a _π _(l)

b _π _(l) ≥ . . . . ≥ a _π _(n) b _π _(n)

?O='U_<>_A[_<D?O\xLNhBhB¾"QDT`\VH[TxL=@?<>>WÉXÎT=MLx¾A@S]LNA9A@S>TcdT`\VA[_H[=

a, b

L='¼9TVhBhnL=½A@S>TbA1¼_g»DH@?O=@U=

A = Π ^a

LN<>Q

B = Π ^b

LNH[TÊ?B< ÖMÒ+ ç7åBß Ö

ÕNØ ãGàGã Õì?EA@S>TV?BH°»_5=@?BA@?O_<D?B<D»TVH@U.CDAMLNA@?O_<?O=

CD<D?OÐKC>TVhB¾tQDTÆ><>T`QnWyS>TÇ»_5=@?BA@?O_<D?B<D»TVH@U°CDAMLNA[?B_<?O=A@S>TV<¦QDTV<>_A[T`QtË7¾

π _(A,B) .

(12)

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?

^*

ï »_5=@?BA@?O_<D?B<Dt»TVH@U.CDAMLNA@?O_<H[T¼]T`\AM=ÇA@S>T½ÂÖ[Ò åBß5àGã=Ò

ÕNØ ãGàGã Õì#¾'_NE'A@S>TÊL=[=[_7\V?ÂLNA[T`Q»DH@?O=@U=

A

LN<>Q

B

W°IKC>\[SL»TVH@U°CDAMLNA@?O_<Q>_KT`=LNhB¼L`¾ª=TÚª?O=@AxÛ]A@S]_CD5S?BAJULx¾<>_AÇËT.CD<D?OÐKC>TVhB¾ QDTVA[TVH@UÊ?B<>T`QnW_.L`cK_?OQQDTVdTV<>TVHMLNA[T`Q"\xL=[T`=xÛK¼9T¼?BhBh]S>TV<>\`TE_H@A@StLNhB¼L`¾ª=L=[=@CDUTA@S]LNALNhBh

»DH@?O=@U=JLNH[TÇ»]LN?BH@¼?O=[T?B<H[TV5CDhÂLNH»_5=@?BA@?O_<0W

¿

s]uesvGvsDu^IgÂ¹{qnsDºd|Kw"s>{°rJ|Kuy0¹xsª¹^s{wQg«{Ò à

a = (a ₁ , . . . , a _n ) > 0 ∈ R ⁿ

^ß ^ì×

b = (b ₁ , . . . , b _n ) > 0 ∈ R ⁿ

^è ^ÒB+ ^ã^=ÒìPj{Ò ^à

π _(A,B)

^è ^Ò ^àá ^Ò ^ÕNØ ^ãGàGã ^Õì ^ã^ì#+ ^ÒÖVâ ^{çªàß5àGã}^ÕìÕ ^<

A

ß

ì×

B.

^l ^á ^Òìon ^< ^{ÕÖ.Ò=ÒÖÓ}

i ∈ I

^ß ^ì×

F ⁱ A =

^\`_<Kc

n

a ^l | l ∈ {1, ..., n}

^¼?BA@S

π _(A,B) ⁻ ¹ (l) ≤ π _(A,B) ⁻ ¹ (i) o

ß ì×

F ⁱ B =

^\`_<7c

n

b ^l | l ∈ {1, ..., n}

^¼?BA@S

π ⁻¹ _(A,B) (l) ≥ π _(A,B) ⁻¹ (i) o

àá ÒØ ç â

F ⁱ A + F ⁱ B

A + B.

?BcKTV<tA2¼9_Ê»DH@?O=@U=

A

LN<>Q

B,

¼SD?O\[SLNH[T?B<tH[TV5CDhÂLNHb»_5=@?BA@?O_<0WS>TV<A@S]TÇ<A@CD»DhOT

Π A

B

=

π _(A,B) (1), π _(A,B) (2), . . . , π _(A,B) (n) .

?O=bLÊ=@A@H@?O\VA@hB¾QDT`\VH[TxL=@?B<D"=[TAb_NEÉ?B<>Qª?O\`T`=xÛDA@S>TÕÖ[×KÒÖ.?B<>QªC>\`T`QËK¾

π _(A,B) ,

^?W^T5W

i ≥ j ⇐⇒ π _(A,B) ⁻ ¹ (i) ≤ π _(A,B) ⁻ ¹ (j).

90TVAC>=<>_^¼ ?B<KA@H[_7QªC>\`TA@S>T°LNËDËDH[TVc7?ÂLNA@?O_<

AB (i) = F ⁱ _A + F ⁱ _B ,

E_H'A@S>Tb=@CDU _NE0A1¼_EL\`T`=L='QDTÆ><>T`QLNË_xcKT5WS>TV<E_H9TVcdTVH@¾

i ∈ I

A@S>Tb»]LNH@A9_NE0A@S>TEL\`T

AB (i)

¼SD?O\[SËTVhO_<Dd=9A[_

A

?O=9QDTVA[TVH@UÊ?B<>T`QËK¾A@S>Tb?B<>Qª?O\`T`=9¼SD?O\MStLNH[TJ5H[TxLNA[TVH_HT`ÐKC]LNh A[_

k

?B<A@S>Tg_H[QDTVH?B<>QªC>\`T`QtË7¾

π _(A,B)

LN<>QtA@S>T»]LNH@Ab_NEÉA@S>TJEL\`T

AB(i)

¼SD?O\MSËTVhO_<Dd=

A[_

B

?O=9QDTVA[TVH@UÊ?B<>T`Q"Ë7¾ÊA@S>Tb?B<>Qª?O\`T`=9¼SD?O\MStLNH[TbhOT`=[=9_HT`ÐKC]LNh]AM_

k

?B<"A@S>TJ_H[QDTVH9?B<>QªC>\`T`Q Ë7¾

π _(A,B) .

;1<tA[TVH@U=b_NEÉA@S>T»_5=@?BA@?O_<D?B<D»TVH@U.CDAMLNA@?O_<¦¼TQDT`QªC>\`TÇEGH[_UX90TVUÊUL>Wî1

rguesh~s]w^Â¹^s¹gOl)g

æ]ç

ÕNØxÒÊÔ9Ò ß Ö[Ò+ ã=Òì

3

^=Ò ^ä`à ^ÕÖMØ

a > 0 ∈ R ⁿ , b > 0 ∈ R ⁿ ,

^ß ^ì×

c > 0 ∈ R ⁿ

^ß ^ì×

åÒ à

A, B

^ß ^ì×

C

^è ^Ò ^àá ^Ò ^ß ^ØMØ`Õ ^äVãß5à^Ò[× ^Ö ^ã^Øâ.ØQj ^Þ ÕÖ[Ò@Õ=ÒÖ£n åÒ à

π _(A,B)

^ß ^ì×

π _(A,C)

^è ^Ò ^àá ^Ò

ä

ÕÖÖ[ÒMØ

Õì×

ã

ì#+

ÕNØ

ãGàGã Õì

ã

ì#+

ÒÖâ

çªàß5àGã ÕìDØQj

l á Òì < ÕÖÒ=ÒÖÓ ãì×KÒqp

i ∈ I

^àá ^ÒØ ^ç ^â

ABC(i) = F ⁱ A + F ⁱ B + F ⁱ C

Ô ãGàá

F ⁱ A =

^\`_<7c

n

a ^l | l ∈ I

^¼?BA@S

π ⁻ _(A,B) ¹ (l) ≤ π ⁻ _(A,B) ¹ (i)

LN<>Q

π _(A,C) ⁻ ¹ (l) ≤ π ⁻ _(A,C) ¹ (i) o

(13)

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^T

F ⁱ B =

^\`_<Kc

n

a ^l | l ∈ {1, ..., n}

^¼?BA@S

π _(B,A) ⁻ ¹ (l) ≤ π _(B,A) ⁻ ¹ (i)

LN<>Q

π ⁻ _(B,C) ¹ (l) ≤ π ⁻ _(B,C) ¹ (i) o

ß ì×

F ⁱ _C =

^\`_<7c

n

a ^l | l ∈ {1, ..., n}

^¼?BA@S

π _(C,A) ⁻ ¹ (l) ≤ π _(C,A) ⁻ ¹ (i)

LN<>Q

π _(C,B) ⁻ ¹ (l) ≤ π ⁻ _(C,B) ¹ (i) o

A + B + C .

rgues)s]utYXÎT'P7<>_^¼A@SLNA

b _i c _i A, a _i c _i B

LN<>Q

a _i b _i C

S]L`cKTLJ\`_UÊU_<TÚªA@H[TVUT9»_?B<KAxWJ_^¼ A@S>TH[T`=@CDhBAE_hBhO_x¼=?BUÊUT`Qª?ÂLNA[TVh¾EGH[_UX90TVUÊUL>WîLN<>QS>T`_H[TVUº3ªW05W xgB|hgByg

J_A[TtA@S>TA@S>T¦L=@?BUÊ?BhÂLNH\`_<>Qª?BA@?O_< =@AMLNA[T`QÍ?B<¹wYH[_»_5=@?BA@?O_<I>Wò?O=ÊA@H@C>TE_HLN<K¾ÎÆ><D?BA[T

=@CDUW

rguesh~s]w^Â¹^s¹gkMg

æ]ç

ÕNØxÒÔ9Ò áªß =Ò

3

^=Ò ^ä`à ^ÕÖMØ

a > 0 ∈ R ⁿ , b > 0 ∈ R ⁿ ,

^ß ^ì×

c > 0 ∈ R ⁿ

^ß ^ì× ^å^Ò ^à

A = Π ^a , B = Π ^b ,

^ß ^ì×

C = Π ^c

^è ^Ò ^àá ^Ò ^ß ^ØMØ`Õ ^äVãß5à^Ò[× ^Ö ^ã^Øâ.ØQj ^Þ ÕÖMÒ@Õ=ÒÖ£n åÒ à

π _(A,B)

^ß ^ì×

π _(A,C)

^è ^Ò ^àá ^Ò ^ä ^ÕÖÖ[ÒMØ ^Õì× ^ã^ì#+ ^ÕNØ ^ãGàGã ^Õì ^ã^ì#+ ^ÒÖâ ^{çªàß5àGã}^ÕìDØQj

À

<;<

ÕÖ

à

Ô'Õ

ã

ì×

ãä ÒMØ

i, l ∈ I

^Ô9Ò ^áªß ^=ÒxÙ

Á

i ≥ l

?B<A@S>T_H[QDTVH?B<>QªC>\`T`QËK¾

π _(A,B) ,

Á

l ≥ i

?B<A@S>T_H[QDTVH?B<>QªC>\`T`QËK¾

π _(A,C) ,

àá Òì

AB(l)

^ß ^ì×

AC(i)

^áªß ^=Ò ^ß ^ß ^Ö ^ßåå^Ò ^å ^Ò@×Q+KÒÔ ^áKãäáã^Ø ^ß ^Ö ^ßåÂå^Ò ^å'à^Õ ^àá ^Ò ^å^ãìÒØ`Ò+âÒì à è Ò àÔ9Ò[Òì

a ⁱ

^ß ^ì×

a ^l

^ß ^ì× ^ä ^{Õì=ÒÖMØxÒ} ^å^ÓGj

rgues)s]ut9¾ÂwYH[_»_5=@?BA@?O_<v>W3.¼TÇS]LxcdT 1

F ^l A =

^\`_<Kc

a ^r | r ∈ I

^¼?BA@S

r ≥ l

?B<tA@S>Tg_H[QDTVH?B<>QªC>\`T`QtË7¾

π _(A,B) , F ^l _B =

^\`_<Kc

b ^s | s ∈ I

^¼?BA@S

s ≤ l

?B<tA@S>Tg_H[QDTVH?B<>QªC>\`T`QtË7¾

π _(A,B) , F ⁱ A =

^\`_<7c

a ^s | s ∈ I

^¼?BA@S

s ≥ i

?B<A@S>Tg_H[QDTVH?B<>QªC>\`T`QtË7¾

π _(A,C)

LN<>Q

F ⁱ C =

^\`_<Kc

c ^s | s ∈ I

^¼?BA@S

s ≤ i

?B<A@S>T_HMQDTVH?B<>QªC>\`T`QtË7¾

π _(A,C) .

IK?B<>\`T

i ≥ l

?B<tA@S>Tg_H[QDTVH?B<>QªC>\`T`QËK¾

π _(A,B)

?O=E_hBhO_^¼=xÛxA@S]LNA0A@S>TYhB?B<>TY=[TV5UTV<7A

[a ⁱ , a ^l ] ⊂ F ^l _A .

ïb<]LNhO_d_C>=@hB¾KÛ0=@?B<>\`T

l ≥ i

?B<A@S>T_H[QDTVH?B<>QDC>\`TVQËK¾

π _(A,B)

?BAJE_hBhO_^¼=xÛnA@S]LNAA@S>T hB?B<>Tg=[TV5UTV<7A

[a ⁱ , a ^l ] ⊂ F ⁱ A .

8JTV<>\`T

[a ⁱ , a ^l ] ⊂ AB(l)

LN<>Q

[a ⁱ , a ^l ] ⊂ AC(i).

S>Tg\`_<7cdTVH[=[TgQª?BH[T`\VA@?O_<?O=bLNhO=[_\VhOTxLNHxW xgB|hgByg

wYH[_»_5=@?BA@?O_<Ã>Wñ5?BcKT`=.L»_5=[=@?BËD?BhB?BA2¾rA[_¦\`_<>=@A@H@C>\VAÊLtUL^Úª?BULNh'EL\`TËK¾ÎLQ2C>=@A@?B<DA1¼_

»]LNHMLNhBhOTVhT`QªdT`=xWYXTÇ¼?BhBh0C>=[TÇA@S>T<>_AMLNA@?O_<1

AB (l)AC(i) = F ^l A + F ^l B + F ⁱ C = F ⁱ A + F ^l B + F ⁱ C .

È]_HA@S>TdTV<>TVHMLNh{\xL=[T¼9TS]L`cKT 1

(14)

?

ABDCEGFIÄ%ÅQ&oL !GJ#F# !O#"$ 'GF#LGL;ÆÇCGÈ$ÉÊG¯J#K#SF#L

?

}°z{|Gs]uN|7ËgOp)g

Ò à

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

Π = X K k=1

Π ^(k)

Ô ãGàá

Π ^(k) = Π ^a ^(k)

^< ^ÕÖ

k ∈ K.

^Ò ^à

F ⊂ Π

^è ^Ò ^ß ^â ^ß ^p ^ã^â ^ßå<Vßdä ^Ò9Ô ^{ãGàá.ä[ß} ^ìÕì ^ãä[ßå ^Ö[Ò^< ^ÒÖMÒì ^ä ^Ò

ØÓNØ à Òâ

I _F =

K ₀ , h J ^(k) i

k∈K 0

Ø

çDäá¦àá7ß5à

F = X

k ∈K ₀

∆ ^(k) _J _(k) + p,

Ô ãGàáß Ø ç7ãGàßKèåÒØ ç â

p

^Õ ^< ^Òqp ^à^ÖMÒâÒ ^Õ ^ã^ì ^à^Ø°Õ ^<àá ^ÒgÖ[Òâ ^ßã^ì ^ã^ì#+ ^Ö ^ã^Øâ.Ø ^ã^ØÇØ ^ß5àGã ^Ø^ë ^Ò[×Gj

l á Òìon < ÕÖÇÒ=ÒÖÓ àÔ'Õ× ã¢ ÒÖ[Òì àã ì× ãä ÒMØ

p, q ∈ K ₀

^n{Ô9Ò ^áªß ^=Ò ^< ^ÕÖ ^àá ^Ò ^ãì×KÒqp°ØxÒ à Ø

J ^(p)

^ß ^ì×

J ^(q)

^àáªß5à ^Ô ^ãàá ^ÖMÒMØ ^Ò ^ä`à9à^Õ ^àá ^{Ò.ÕÖ@×KÒÖ} ^ã ^ì× ^çDä ^Ò[× ^è ^Ó

π ( Π ^(p) ,Π ^(q) )

àá Ò ãìÒu¡ çDßåãGà Ó

min n J ^(p) o

≥ max n J ^(q) o

á Õ å×NØ àÖ ç ÒQj

rgues)s]utSD?O=?O=bLN<t?BUÊUT`Qª?ÂLNA[T°\`_<>=[T`ÐKC>TV<>\`Tg_NE;wYH[_»_5=@?BA@?O_<v>W3ªW xgB|hgByg

Ì VXWDY[Í ûÎÉü7aÉøy]`Yg÷øÏb&ù9ú¦ýYaaÐ|VIÑ ú`ÒÓD´k]`_7d Ya

ÔÕ0s>7~Év|PgGmhgXÎT\`_<>=@?OQDTVH

3

»DH@?O=@U=?B<

R ⁴

¼SD?O\MSLNH[T5?BcdTV<ËK¾P1

A

^Ö ^\`_<Kc

{(0, 0, 0, 0), (7, 0, 0, 0), (0, 5, 0, 0), (0, 0, 3, 0), (0, 0, 0, 1)}

B

^Ö ^\`_<Kc

{(0, 0, 0, 0), (2, 0, 0, 0), (0, 1, 0, 0), (0, 0, 10, 0), (0, 0, 0, 1)}

C

^Ö ^\`_<Kc

{(0, 0, 0, 0), (1, 0, 0, 0), (0, 6, 0, 0), (0, 0, 6, 0), (0, 0, 0, 5)}

È]_HA@S>T»TH[U°CDAMLNA@?O_<>=b¼9TS]LxcdT

π _(A,B) =

1 2 3 4

2 1 4 3

π _(A,C) =

1 2 3 4

π _(B,C) =

1 2 3 4

1 3 2 4

8JTV<>\`T¼TÇS]LxcdT 1

Π A

B

= [ 2, 1, 4, 3 ]

(15)

?

ABDCEGFIÄ%ÅQ&oL !GJ#F# !O#"$ 'GF#LGL;ÆÇCGÈ$ÉÊG¯J#K#SF#L

?

Π A

C

= [ 1, 2, 3, 4 ] Π

B C

= [ 1, 3, 2, 4 ]

l

ßKèå Ò.Õ

< i ßdä

ÒMØ

×PØ£ÙÛÚÜ Ý ÞßRàÜGáâÛãäå Þ;ßRàÜGáâÛãäå ÞßRàÜGáâÛãäå ÞßRàÜGáâÛãäå

âçæRâÛãè ä

k = 1 k = 2 k = 3 k = 4

π (A,B) [a 1 , a 2 ] + [b 1 , b 3 , b 4 ]

| {z }

AB(1)

AB(2) = B AB(3) = A [a 1 , a 2 , a 3 ] + [b 3 , b 4 ]

| {z }

AB(4)

π (A,C) AC(1) = C [a 1 , a 2 ] + [c 2 , c 3 , c 4 ]

| {z }

AC (2)

[a 1 , a 2 , a 3 ] + [c 3 , c 4 ]

| {z }

AC (3)

AC(4) = A

π (B,C) BC(1) = C [b 1 , b 2 , b 3 ] + [c 3 , c 4 ]

| {z }

BC(2)

[b 1 , b 3 ] + [c 2 , c 3 , c 4 ]

| {z }

BC(3)

BC(4) = B

é¸êPëìíPîeìïeðëñ¦îeì

9

^òôó ïeîeìeõBé¸êPîkö

ó

ì£÷

òôó

ïeîmëì¸ïeøùPì£÷NðúPï ÷NîeíûñüDé¸êPîeøðîeýÏþPõÿ)ë÷¸ëì

F = [a ₁ , a ₂ ] + [b ₁ , b ₃ ] + [c ₂ , c ₄ ],

ñ¦îeï

ó

úPìîîkê

ó î

min{1, 2} ≥ max{1, 3, 4}

ëù÷NêPîmøðíPîeðëùPíPúPïeîeí ñoü

π _(A,B) , min{1, 2} ≥ max{2, 3, 4}

π _(A,C) , min{1, 3} ≥ max{2, 3, 4}

π _(B,C) .

îeùPïeî÷NêPîëùPíPîìî ÷êPëïRêvñ¦îeöøùì÷Nø

A

^ëì

{1 , 2},

÷NêPîëùPíPîvìî ÷êPëïRêvñ¦îeöøùì

÷Nø

B

^ëì

{1, 3, 4} ∩ {1, 3} = {1, 3},

^ó ^ì ^ò ^øööøì ^ò^ðøý ÷NêPîmëùPîú

ó

öë÷Nëîeì«ëýPöëîeíDñoü

÷NêPîmøðíPîeðìø

ò

π _(A,B)

^ó ^ùPí

π _(B,C) .

^ù ^ó ^{öøøúPìö}üv÷NêPîmìî ÷ø

ò

ëùPíPëïeîeì

{2, 4}

^{ñ¦îeöøùì}

÷Nø

C,

^ñîeï ^ó ^úPìî

{2 , 4} = Π ^B _C

\ {1 , 3}.

é¸êPîkïeøððîeì¦øùPíPëùì£ühì£÷Nîeý ø ò ëùPîú

ó

öë÷Nëîeì

ò

øð

A + B + C

^ëì

30x ₁ + 30x ₂ + 5x ₃ + 6x ₄ ≤ 300

5x 1 + 7x 2 + x ₃ + 10x 4 ≤ 95

5 x ₁ + 10 x ₂ + x ₃ + 2 x ₄ ≤ 70

5x ₁ + 10x ₂ + x ₃ + 5x ₄ ≤ 85

150x 1 + 210x 2 + 35x 3 + 42x 4 ≤ 1610

15 x ₁ + 21 x ₂ + 35 x ₃ + 42 x ₄ ≤ 665

15x ₁ + 21x ₂ + 35x ₃ + 105x ₄ ≤ 980

25x 1 + 30x 2 + 5x 3 + 6x 4 ≤ 255

10 x ₁ + 14 x ₂ + 7 x ₃ + 70 x ₄ ≤ 490

25x ₁ + 35x ₂ + 5x ₃ + 7x ₄ ≤ 260.

Computing the Minkowski sum of prisms

a ∈ R n

¯ a = ( ¯ a 1

1 , . . . , ¯ a 1

n )

Π = Π a ¯ = {x ∈ R n | h¯ a, xi ≤ 1 , x ≥ 0 }

A, B ⊂ R n

A + B = {z = a + b | a ∈ A, b ∈ B}

λ ∈ R

A ⊂ R n

λA = {x = λa | a ∈ A}.

A

B

A + B

λA

A

B

A + B

λA

?

?

hc, xi

(P 0 ) x = z 1 + z 2 + . . . + z k

ha i , z i i ≤ α i i ∈ {1, ..., k}, z i ≥ 0

c, a i , z i ∈ R n

α i ∈ R .

0 ∈ R n

e i

i

R n

i ∈ {1, ..., n}.

a = (a 1 , . . . , a n ) > 0 ∈ R n ,

Π a =

0, a 1 , . . . , a n

a i = a i e i , i ∈ {1, ..., n}.

Π a

a = (a 1 , . . . , a n ) > 0 ∈ R n .

Π a = {x ∈ R n | h¯ a, xi ≤ α

x ≥ 0 },

¯ a =

α a 1 , ..., a α

n

∈ R n

α > 0

Π a .

α = 1.

∆ a =

a 1 , . . . , a n

Π a .

a (1) , ..., a (K) ∈ R n

a (k) = a (k) 1 , . . . , a (k) n

> 0 ∈ R n ,

Π (k) = Π a (k) =

n

0, a (k),1 , . . . , a (k),n o

?

?

a (k),i = a (k) i e i , i ∈ {1, ..., n}, k ∈ {1, . . . , K }.

I = {1, . . . , n}

a i 1 , ..., a i r

Π a

a i 1 , . . . , a i r = [a i 1 , . . . , a i r ] = ∆ ( J a ) ,

J = {i 1 , . . . , i r }.

Π a (k) , k = 1, . . . , K,

∆ (k) J (k) = ∆ ( J a (k) ) (k)

n a (k),l o

l ∈J (k) ⊂ Π a (k)

J (k) ⊆ I

a = (a 1 , . . . , a n ) > 0 ∈ R n

Π a

S ⊂ ∆ a

J ⊆ I

S =

a i i ∈J = ∆ ( J a ) .

S ⊂ Π a

∆ a

Π a .

u, v ∈ Π a

[u, v]

S

a ∈ R ⁿ

¯ a = ( _¯ _a ¹

1 , . . . , _¯ _a ¹

Π = Π ^a ^¯ = {x ∈ R ⁿ | h¯ a, xi ≤ 1 , x ≥ 0 }

A, B ⊂ R ⁿ

A ⊂ R ⁿ

(P ₀ ) x = z ¹ + z ² + . . . + z ^k

ha ⁱ , z ⁱ i ≤ α i i ∈ {1, ..., k}, z ⁱ ≥ 0

c, a ⁱ , z ⁱ ∈ R ⁿ

α _i ∈ R .

0 ∈ R ⁿ

e ⁱ

R ⁿ

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ ,

Π ^a =

0, a ¹ , . . . , a ⁿ

a ⁱ = a _i e ⁱ , i ∈ {1, ..., n}.

Π ^a

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ .

Π ^a = {x ∈ R ⁿ | h¯ a, xi ≤ α

α a 1 , ..., _a ^α

∈ R ⁿ

Π ^a .

∆ ^a =

a ¹ , . . . , a ⁿ

Π ^a .

a ⁽¹⁾ , ..., a ^(K) ∈ R ⁿ

a ^(k) = a ^(k) ₁ , . . . , a ^(k) n

> 0 ∈ R ⁿ ,

Π ^(k) = Π ^a ^(k) =

0, a ^(k),1 , . . . , a ^(k),n o

a ^(k),i = a ^(k) _i e ⁱ , i ∈ {1, ..., n}, k ∈ {1, . . . , K }.

a ⁱ ¹ , ..., a ⁱ ^r

Π ^a

a ⁱ ¹ , . . . , a ⁱ ^r = [a ⁱ ¹ , . . . , a ⁱ ^r ] = ∆ ⁽ _J ^a ⁾ ,

Π ^a ^(k) , k = 1, . . . , K,

∆ ^(k) _J _(k) = ∆ ⁽ _J ^a _(k) ⁾ ^(k)

n a ^(k),l o

l ∈J ^(k) ⊂ Π ^a ^(k)

J ^(k) ⊆ I

a = (a 1 , . . . , a n ) > 0 ∈ R ⁿ

Π ^a

S ⊂ ∆ ^a

a ⁱ _i _∈J = ∆ ⁽ _J ^a ⁾ .

S ⊂ Π ^a

∆ ^a

Π ^a .

u, v ∈ Π ^a

Π ^a

a ⁱ _i∈J

Π ^a

a ⁱ

i ∈J = ∆ ⁽ _J ^a ⁾

a ⁽¹⁾ , . . . , a ^(K) ∈ R ⁿ

Π ^(k)

Π ^(k) = Π ^a ^(k)

n _F ∈ R ⁿ

F = Π ∩ {x ∈ Π | h n _F , xi = d ^∗ }, j = 1, ..., k,

d ^∗ = max

x∈ Π h n _F , xi > 0.

K ₀ ⊆ {1, . . . , K}

J ^(k) ⊆ I (k ∈ K ₀ )

∆ ^(k) _J _(k) + p

n Π ^(k) o

k / ∈K ₀ .

Π ⁽ⁱ ¹ ⁾ , ..., Π ⁽ⁱ ^r ⁾

F = S _i ₁ + ... + S _i _r + p

S _i _j = {x ∈ Π ⁽ⁱ ^j ⁾ | h n _F , xi = d ⁱ ^j }, j = 1, ..., r,

d ⁱ ^j = max

x∈ Π ^ij

h n _F , xi > 0

K ₀ = {i 1 , . . . , i _r }

S _k

Π ^(k) ,

S _k = ∆ ^(k) _J _(k)

J ^(k) ⊆ I

Π ^(k) = X

Π ^(k)

I _F =

K ₀ , h J ^(k) i