Aeaaa 4. EIIIA?AOEAIUA EA?U
, a /enei anaaiciiaeiuo eiaeeoeee ?aaii
= 2n ( 1.
Ec yoie oi?ioeu aeaeii, /oi /enei anaaiciiaeiuo eiaeeoeee cia/eoaeueii
?ano?o a caaeneiinoe io /enea anao ea?ieia a aeaiiie ea?a. Aeey
enneaaeiaaiey yoeo ea? iaiaoiaeeii o/eouaaoue ana aiciiaeiua eiaeeoeee,
e iiyoiio o?oaeiinoe enneaaeiaaiee aic?anoatho n ?inoii n. Ia?aciaaa
eiaeeoeeth, iiiaeanoai ea?ieia K aeaenoaoao eae iaeei ea?ie i?ioea
inoaeueiuo ea?ieia, e auea?uo yoie eiaeeoeee caaeneo io i?eiaiyaiuo
no?aoaaee eaaeaeui ec n ea?ieia.
Ooieoeey (, noaayuay a niioaaonoaea eaaeaeie eiaeeoeee K iaeaieueoee,
oaa?aiii iieo/aaiue aai auea?uo ((K), iacuaaaony oa?aeoa?enoe/aneie
ooieoeeae ea?u. Oae, iai?eia?, aeey aaneiaeeoeeiiiie ea?u n ea?ieia
((K) iiaeao iieo/eoueny, eiaaea ea?iee ec iiiaeanoaa K iioeiaeueii
aeaenoaotho eae iaeei ea?ie i?ioea inoaeueiuo N\K ea?ieia, ia?acothueo
ae?oaoth eiaeeoeeth (aoi?ie ea?ie).
Oa?aeoa?enoe/aneay ooieoeey ( iacuaaaony i?inoie, anee iia i?eieiaao
oieueei aeaa cia/aiey: 0 e 1. Anee oa?aeoa?enoe/aneay ooieoeey (
i?inoay, oi eiaeeoeee K, aeey eioi?uo ((K)=1, iacuaathony
auea?uaathueie, a eiaeeoeee K, aeey eioi?uo ((K) = 0, ( i?iea?uaathueie.
Anee a i?inoie oa?aeoa?enoe/aneie ooieoeee ( auea?uaathueie yaeythony oa
e oieueei oa eiaeeoeee, eioi?ua niaea?aeao oeene?iaaiioth iaionooth
eiaeeoeeth R, oi oa?aeoa?enoe/aneay ooieoeey (, iaicia/aaiay a yoii
neo/aa /a?ac (R, iacuaaaony i?inoaeoae.
Niaea?aeaoaeueii i?inoua oa?aeoa?enoe/aneea ooieoeee aicieeatho,
iai?eia?, a oneiaeyo aieiniaaiey, eiaaea eiaeeoeey yaeyaony
auea?uaathuae, anee iia niae?aao aieaa iieiaeiu aieinia (i?inoia
aieueoeinoai) eee ia iaiaa aeaoo o?aoae aieinia (eaaeeoeoee?iaaiiia
aieueoeinoai).
Aieaa neiaeiui yaeyaony i?eia? ioeaiee ?acoeueoaoia aieiniaaiey a Niaaoa
aaciianiinoe III, aaea auea?uaathueie eiaeeoeeyie yaeythony ana
eiaeeoeee, ninoiyuea ec anao iyoe iinoiyiiuo /eaiia Niaaoa iethn au?
oioy au iaeei iaiinoiyiiue /eai, e oieueei iie.
I?inoaeoay oa?aeoa?enoe/aneay ooieoeey iiyaeyaony, eiaaea a aieinothuai
eieeaeoeaa eiaaony iaeioi?ia (yae?i(, aieinothuaa n niaethaeaieai
i?aaeea (aaoi(, a aieina inoaeueiuo o/anoieeia ieacuaathony
ianouanoaaiiuie.
Iaicia/ei /a?ac (G oa?aeoa?enoe/aneoth ooieoeeth aaneiaeeoeeiiiie ea?u.
Yoa ooieoeey iaeaaeaao neaaeothueie naienoaaie :
ia?niiaeueiinoue
(G(() = 0,
o.a. eiaeeoeey, ia niaea?aeauay ie iaeiiai ea?iea, ie/aai ia auea?uaaao;
noia?aaeaeeoeaiinoue
(G(K(L) ( (G(K) + (G(L), anee K, L ( N, K(L ( (,
o.a. iauee auea?uo eiaeeoeee ia iaiueoa noiia?iiai auea?uoa anao
o/anoieeia eiaeeoeee;
aeiiieieoaeueiinoue
o.a. aeey aaneiaeeoeeiiiie ea?u n iinoiyiiie noiiie noiia auea?uoae
eiaeeoeee e inoaeueiuo ea?ieia aeieaeia ?aaiyoueny iauae noiia auea?uoae
anao ea?ieia.
?ani?aaeaeaiea auea?uoae (aeae?ae) ea?ieia aeieaeii oaeiaeaoai?youe
neaaeothuei anoanoaaiiui oneiaeyi: anee iaicia/eoue /a?ac xi auea?uo
i-ai ea?iea, oi, ai-ia?auo, aeieaeii oaeiaeaoai?youeny oneiaea
eiaeeaeaeoaeueiie ?aoeeiiaeueiinoe
o.a. ethaie ea?ie aeieaeai iieo/eoue auea?uo a eiaeeoeee ia iaiueoa, /ai
ii iieo/ee au, ia o/anoaoy a iae (a i?ioeaiii neo/aa ii ia aoaeao
o/anoaiaaoue a eiaeeoeee); ai-aoi?uo, aeieaeii oaeiaeaoai?youeny oneiaea
eieeaeoeaiie ?aoeeiiaeueiinoe
o.a. noiia auea?uoae ea?ieia aeieaeia niioaaonoaiaaoue aiciiaeiinoyi
(anee noiia auea?uoae anao ea?ieia iaiueoa, /ai ((N), oi ea?ieai iaca/ai
anooiaoue a eiaeeoeeth; anee aea iio?aaiaaoue, /oiau noiia auea?uoae
auea aieueoa, /ai ((N), oi yoi cia/eo, /oi ea?iee aeieaeiu aeaeeoue
iaaeaeo niaie noiio aieueooth, /ai o ieo anoue).
Oaeei ia?acii, aaeoi? x = (x1, …, xn), oaeiaeaoai?ythuee oneiaeyi
eiaeeaeaeoaeueiie e eieeaeoeaiie ?aoeeiiaeueiinoe, iacuaaaony aeaeaae?i
a oneiaeyo oa?aeoa?enoe/aneie ooieoeee (.
Nenoaia {N, (}, ninoiyuay ec iiiaeanoaa ea?ieia, oa?aeoa?enoe/aneie
ooieoeee iaae yoei iiiaeanoaii e iiiaeanoaii aeaeaaeae,
oaeiaeaoai?ythueo niioiioaieyi (2) e (3) a oneiaeyo oa?aeoa?enoe/aneie
ooieoeee, iacuaaaony eeanne/aneie eiiia?aoeaiie ea?ie.
Ec yoeo ii?aaeaeaiee iaiin?aaenoaaiii auoaeaao neaaeothuay
Oai?aia. *oiau aaeoi? x = (x1, …, xn) aue aeaeaae?i a eeanne/aneie
eiiia?aoeaiie ea?a {N, (},
iaiaoiaeeii e aeinoaoi/ii, /oiau
xi = (( i ) + (i, (i(N)
i?e/?i
(i ( 0 (i(N)
A aaneiaeeoeeiiiuo ea?ao enoiae oi?ie?oaony a ?acoeueoaoa aeaenoaee oao
naiuo ea?ieia, eioi?ua a yoie neooaoeee iieo/atho naie auea?uoe.
Enoiaeii a eiiia?aoeaiie ea?a yaeyaony aeae?ae, aicieeathuee ia eae
neaaenoaea aeaenoaey ea?ieia, a eae ?acoeueoao eo niaeaoaiee. Iiyoiio a
eiiia?aoeaiuo ea?ao n?aaieaathony ia neooaoeee, eae yoi eiaao ianoi a
aaneiaeeoeeiiiuo ea?ao, a aeaeaaee, e n?aaiaiea yoi iineo aieaa neiaeiue
oa?aeoa?.
Eiiia?aoeaiua ea?u n/eoathony nouanoaaiiuie, anee aeey ethauo eiaeeoeee
K e L auiieiyaony ia?aaainoai
((K) + ((L) yi aeey anao i(K (naienoai i?aaeii/oeoaeueiinoe)
Naienoai yooaeoeaiinoe icia/aao, /oi n?aaieaaaiue eiaeeoeeae aeae?ae x
aeieaeai auoue, ?aaeecoaiui yoie eiaeeoeeae: noiia auea?uoae eaaeaeiai
ec /eaiia eiaeeoeee ia aeieaeia i?aainoiaeeoue oaa?aiii iieo/aaiia ath
eiee/anoai. A i?ioeaiii neo/aa eiaeeoeey, ano?aoeaoenue n aeaeaae?i,
aeathuei ae noieueei, neieueei iia naiinoiyoaeueii ia a ninoiyiee
aeiaeoueny, aeieaeia niaeaneoueny ia iaai e ia caieiaoueny aai
n?aaiaieai n eaeeie eeai ae?oaeie aeaeaaeaie.
Oneiaea i?aaeii/oeoaeueiinoe io?aaeaao iaiaoiaeeiinoue (aaeeiiaeooey( a
i?aaeii/oaiee ni noi?iiu eiaeeoeee: anee oioy au iaeii ec ia?aaainoa xi
> yi aoaeao ia?ooaii, o.a. anee oioy au aeey iaeiiai ec /eaiia eiaeeoeee
K auea?uo a oneiaeyo aeaeaaea y aoaeao ia iaiueoei, /ai a oneiaeyo
aeaeaaea x, oi iiaeii aoaeao aiai?eoue i i?aaeii/oaiee aeaeaaea x
aeaeaaeo y ia anae eiaeeoeeae K, a oieueei oaie a? /eaiaie, aeey
eioi?uo niioaaonoaothuaa ia?aaainoai xi > yi niaethaeaaony.
Niioiioaiea aeiieie?iaaiey x iaae y ii eiaeeoeee K iaicia/aaony
/a?ac
.
Ii?aaeaeaiea. Aeae?ae x aeiieie?oao y, anee nouanoaoao oaeay
eiaeeoeey K, aeey eioi?ie aeae?ae x aeiieie?oao y. Yoi aeiieie?iaaiea
iaicia/aaony oae:
x > y.
Iaee/ea aeiieie?iaaiey x > y icia/aao, /oi a iiiaeanoaa ea?ieia N
iaeae?ony eiaeeoeey, aeey eioi?ie x i?aaeii/oeoaeueiaa y. Ioiioaiea
aeiieie?iaaiey ia iaeaaeaao iieiinoueth naienoaaie ?aoeaeneaiinoe,
neiiao?ee, o?aiceoeaiinoe, aiciiaeia oieueei /anoe/iay neiiao?ey e
o?aiceoeaiinoue. Niioiioaiea aeiieie?iaaiey aiciiaeii ia ii anyeie
eiaeeoeee. Oae, iaaiciiaeii aeiieie?iaaiea ii eiaeeoeee, ninoiyuae ec
iaeiiai ea?iea eee ec anao ea?ieia.
Ni?aaaaeeeaa neaaeothuay oai?aia.
.
I/aaeaeii, ana yaeaiey, iienuaaaiua a oa?ieiao aeiieie?iaaiey aeaeaaeae,
ioiinyony e eeannai no?aoaae/aneie yeaeaaeaioiinoe, iiyoiio aeinoaoi/ii
eco/aoue yoe eeannu (a ia naie ea?u) aeey nouanoaaiiuo ea? ii eo
(0,1)-?aaeooee?iaaiiie oi?ia, a aeey ianouanoaaiiuo ea? ( ii ioeaaui
ea?ai.
A ethaie ianouanoaaiiie ea?a eiaaony oieueei iaeei aeae?ae, iiyoiio
ieeaeeo aeiieie?iaaiee a iae iao.
?anniio?ei aeiieie?iaaiea aeaeaaeae a nouanoaaiiie ea?a ia neaaeothuai
i?eia?a.
I?eia?. Ionoue eiaaony (0,1)-?aaeooee?iaaiiay oi?ia nouanoaaiiie ea?u
o??o ea?ieia n iinoiyiiie noiiie (?aaiie 1). Iineieueeo aeiieie?iaaiea
iaaiciiaeii ie ii iaeiie ec iaeiiyeaiaioiuo eiaeeoeee 1,2,3, a oaeaea ii
eiaeeoeee, ninoiyuae ec anao o??o ea?ieia, oi aeiieie?iaaiea aiciiaeii
oieueei ii iaeiie ec aeaooyeaiaioiuo eiaeeoeee {1,2}, {1,3}, {2,3}.
Aeey iaaeyaeiinoe aeiieie?iaaiey aeaeaaeae aaaae?i iiiyoea
aa?ioeaio?e/aneeo eii?aeeiao. Inyie eii?aeeiao neoaeao o?e ine x1, x2,
x3, ninoaaeythuea iaaeaeo niaie iaeeiaeiaua oaeu 60i, inue x3 iaoiaeeony
ia ?annoiyiee aaeeieoeu io oi/ee ia?ana/aiey inae x1 e x2 (?en.1),
eii?aeeiaou oi/ee x = (x1, x2, x3) ( niioaaonoaaiii ?annoiyiey io yoie
oi/ee aei inae x1, x2, x3, acyoua n oaeeie ciaeaie, eae oeacaii ia
?en.1. (Iai?eia?, aeey oi/ee x ia ?en.1. x1 0, x3 > 0).
x1 x2
x1 = 1 x2 = 1
x2
x = (x1, x2,
x3)
x20 x3
x10
60i 60i x3>0 x3
x3=0
x3 y1, x2 > y2;
ii eiaeeoeee {1, 3}, anee x1 > y1, x3 > y3;
ii eiaeeoeee {2, 3}, anee x2 > y2, x3 > y3,
o.a. anee aeae?ae y iaoiaeeony a iaeiii ec caoo?eoiaaiiuo
ia?aeeaeia?aiiia (ca eneeth/aieai o??o a?aie/iuo i?yiuo, i?ioiaeyueo
/a?ac oi/eo x) ia ?en. 3, oi aeae?ae x aeiieie?oao aeae?ae y, a
anyeay oi/ea iaoiaeyuayny a ia caoo?eoiaaiiuo o?aoaieueieeao, yaeyaony
i?aaeii/oeoaeueiaa enoiaea x.
x3 = ( 1 x2 = ( 1
x = (x1, x2, x3)
x3 = 1 ( C3
x1 = 0
?en.3 x2 = 1 ( C2
?en. 4 x1 = 1 ( C1
Oaeei ia?acii, anee x e y ( aeaa enoiaea e ie iaeei ec ieo ia
i?aaeii/oeoaeueiaa ae?oaiai, oi niioaaonoaothuea oi/ee eaaeao ia i?yiie,
ia?aeeaeueiie iaeiie ec eii?aeeiaoiuo inae.
I?eia?. Ionoue eiaaony (0, 1)-?aaeooee?iaaiiay ea?a o??o ea?ieia n
iaioeaaie noiiie.
?anniio?ei nia/aea oneiaey aeiieie?iaaiey aeaeaaea x = (x1, x2, x3) iaae
aeaeaae?i y = (y1, y2, y3) ii eiaeeoeee {1, 2}. A yoii neo/aa eiaai :
Iineieueeo iiaeao auoue, /oi C3
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