Iaoaiaoe/aneia i?ia?aiie?iaaiea
Iauay caaea/a eeiaeiiai i?ia?aiie?iaaiey (CEI):
Caeanue (1) iacuaaaony nenoaiie ia?aie/aiee , aa iao?eoea eiaao ?aia r
( n, (2) – ooieoeeae oeaee (oeaeaaie ooieoeeae). Iaio?eoeaoaeueiia
?aoaiea (o10, x20, … , xn0) nenoaiu (1) iacuaaaony aeiionoeiui
?aoaieai (ieaiii) CEI. Aeiionoeiia ?aoaiea iacuaaaony iioeiaeueiui, anee
iii ia?auaao oeaeaaoth ooieoeeth (2) a min eee max (iioeioi).
Neiieaeniay oi?ia CEI. Aeey ?aoaiey CEI neiieaen – iaoiaeii iaiaoiaeeii
aa i?eaanoe e ii?aaeaeaiiie (neiieaeniie) oi?ia:
(2`) f+cr+1xr+1 + … + csxs + … + cnxn = b0 ( min
Caeanue n/eoaai r min f (b1, … ,b2,0, … ,0) = b0.
E?eoa?ee ionoonoaey iioeiaeueiinoe. Anee a neiieaen-iao?eoea eiaaony
noieaaoe (S-e), a eioi?ii iineaaeiee yeaiaio ns > 0, a ana inoaeueiua
yeaiaiou iaiieiaeeoaeueiu, oi CEI ia eiaao iioeiaeueiiai ieaia, o.a. ns
> 0, ais ( 0 (i=1,r) => min f = -(.
Anee a neiieaen-iao?eoea ia auiieiythony iaa e?eoa?ey, oi a iieneao
iioeioia iaaei ia?aoiaeeoue e neaaeothuae iao?eoea n iiiiuueth
iaeioi?iai yeaiaioa ais > 0 e neaaeothueo i?aia?aciaaiee (neiieaeniuo):
ana i-e no?iee aeaeei ia yeaiaio ais+;
ana yeaiaiou S-ai noieaoea, e?iia ais=1, caiaiyai ioeyie;
ana inoaeueiua yeaiaiou iao?eoeu i?aia?acoai ii i?aaeeo i?yiioaieueieea,
/oi noaiaoe/ii iieacaii ia o?aaiaioa iao?eoeu e aeaii a oi?ioeao:
—————————-
… akl … aks … bk
—————————-
… ail … ais … bi
—————————-
… cl … cs … b0
akl` = akbais – ailaks = akl – ailaks;
ais ais
bk` = bkais – biaks; cl` = clais – csail
ais ais
Ii?aaeaeaiea. Yeaiaio ais+ iacuaaaony ?ac?aoathuei, anee
i?aia?aciaaiea iao?eoeu n aai iiiiuueth iaania/eaaao oiaiueoaiea
(iaaic?anoaiea) cia/aiey, oeaeaaie ooieoeee; no?iea e noieaaoe, ia
ia?ana/aiee eioi?uo iaoiaeeony ?ac?aoathuee yeaiaio, oaeaea iacuaathony
?ac?aoathueie.
E?eoa?ee auai?a ?ac?aoathuaai yeaiaioa. Anee yeaiaio ais+
oaeiaeaoai?yao oneiaeth
bi = min bk
ais0 aks0+
aaea s0 – iiia? aua?aiiiai ?ac?aoathuaai noieaoea, oi ii yaeyaony
?ac?aoathuei.
Aeai?eoi neiieaen-iaoiaea (ii ieieiecaoeee).
nenoaio ia?aie/aiee e oeaeaaoth ooieoeeth CEI i?eaiaeei e neiieaeniie
oi?ia;
ninoaaei neiieaen-iao?eoeo ec eiyooeoeeaioia nenoaiu e oeaeaaie ooieoeee
a neiieaeniie oi?ia;
i?iaa?ea iao?eoeu ia auiieiaiea e?eoa?ey iioeiaeueiinoe; anee ii
auiieiyaony, oi ?aoaiea caeii/aii;
i?e iaauiieiaiee e?eoa?ey iioeiaeueiinoe i?iaa?yai auiieiaiea e?eoa?ey
ionoonoaey iioeiaeueiinoe; a neo/aa auiieiaiey iineaaeiaai ?aoaiea
caeii/aii – iao iioeiaeueiiai ieaia;
a neo/aa iaauiieiaiey iaieo e?eoa?eaa iaoiaeei ?ac?aoathuee yeaiaio aeey
ia?aoiaea e neaaeothuae iao?eoea, aeey /aai :
a) auae?aai ?ac?aoathuee noieaaoe ii iaeaieueoaio ec iieiaee
oaeueiuo yeaiaioia
oeaeaaie no?iee;
a) auae?aai ?ac?aoathuoth no?ieo ii e?eoa?eth auai?a ?ac?aoathuaai
yeaiaioa; ia eo ia?ana/aiee iaoiaeeony ?ac?aoathuee yeaiaio;
c iiiiuueth ?ac?aoathuaai yeaiaioa e neiieaen-i?aia?aciaaiee ia?aoiaeei
e neaaeothuae iao?eoea;
aiiaue iieo/aiioth neiieaen-iao?eoeo i?iaa?yai iienaiiui auoa niiniaii
(ni. i. 3)
*a?ac eiia/iia /enei oaaia, eae i?aaeei iieo/aai iioeiaeueiue ieai CEI
eee aai ionoonoaea
Caia/aiey.
Anee a ?ac?aoathuae no?iea (noieaoea) eiaaony ioeue, oi a
niioaaonoaothuai aio noieaoea (no?iea) yeaiaiou inoathony aac eciaiaiey
i?e neiieaen-i?aia?aciaaieyo.
i?aia?aciaaiey – au/eneaiey oaeiaii ia/eiaoue n oeaeaaie no?iee; anee
i?e yoii ieaaeaony, /oi auiieiyaony e?eoa?ee iioeiaeueiinoe, oi iiaeii
ia?aie/eoueny au/eneaieai yeaiaioia iineaaeiaai noieaoea.
i?e ia?aoiaea io iaeiie iao?eoeu e ae?oaie naiaiaeiua /eaiu o?aaiaiee
inoathony iaio?eoeaoaeueiuie; iiyaeaiea io?eoeaoaeueiiai /eaia
neaiaeece?oao i aeiiouaiiie ioeaea a i?aaeuaeoueo au/eneaieyo.
i?aaeeueiinoue iieo/aiiiai ioaaoa – iioeiaeueiiai ieaia – i?iaa?yaony
iooai iiaenoaiiaee cia/aiee aaceniuo iaecaanoiuo a oeaeaaoth ooieoeeth;
ioaaou aeieaeiu niaianoue.
5. Aaiiao?e/aneay eioa?i?aoaoeey CEI e a?aoe/aneee iaoiae ?aoaiey (i?e
aeaoo iaecaanoiuo)
Nenoaia ia?aie/aiee CEI aaiiao?e/anee i?aaenoaaeyao niaie iiiaioaieueiee
eee iiiaioaieueioth iaeanoue eae ia?ana/aiea iieoieineinoae –
aaiiao?e/aneeo ia?acia ia?aaainoa nenoaiu. Oeaeaaay ooieoeey f = c1x1 +
c2x2 aaiiao?e/anee ecia?aaeaao naiaenoai ia?aeeaeueiuo i?yiuo,
ia?iaiaeeeoey?iuo aaeoi?o n (n1,n2).
Oai?aia. I?e ia?aiauaiee i?yiie oeaeaaie ooieoeee iai?aaeaiee aaeoi?a
n cia/aiey oeaeaaie ooieoeee aic?anoatho, a i?ioeaiiieiaeiii iai?aaeaiee
– oauaatho.
Ia yoeo ooaa?aeaeaieyo iniiaai a?aoe/aneee iaoiae ?aoaiey CEI.
Aeai?eoi a?aoe/aneiai iaoiaea ?aoaiey CEI.
A nenoaia eii?aeeiao iino?ieoue i?yiua ii o?aaiaieyi, niioaaonoaothuei
eaaeaeiio ia?aaainoao nenoaiu ia?aie/aiee;
iaeoe iieoieineinoe ?aoaiey eaaeaeiai ia?aaainoaa nenoaiu (iaicia/eoue
no?aeeaie);
iaeoe iiiaioaieueiee (iiiaioaieueioth iaeanoue) ?aoaiee nenoaiu
ia?aie/aiee eae ia?ana/aiea iieoieineinoae;
iino?ieoue aaeoi? n (n1,n2) ii eiyooeoeeaioai oeaeaaie ooieoeee f =
c1x1 + c2x2;
a naiaenoaa ia?aeeaeueiuo i?yiuo oeaeaaie ooieoeee auaeaeeoue iaeio,
iai?eia?, /a?ac ia/aei eii?aeeiao;
ia?aiauaoue i?yioth oeaeaaie ooieoeee ia?aeeaeueii naiie naaa ii iaeanoe
?aoaiey, aeinoeaay max f i?e aeaeaeaiee aaeoi?a n e min f i?e aeaeaeaiee
a i?ioeaiiieiaeiii iai?aaeaiee.
iaeoe eii?aeeiaou oi/ae max e min ii /a?oaaeo e au/eneeoue cia/aiey
ooieoeee a yoeo oi/eao (ioaaou).
Iinoaiiaea o?ainii?oiie caaea/e.
I?eaaaeai yeiiiie/aneoth oi?ioee?iaeo o?ainii?oiie caaea/e ii e?eoa?eth
noieiinoe:
Iaeii?iaeiue a?oc, eiathueeny a m ioieoao ioi?aaeaiey (i?iecaiaenoaa)
A1, A2, …, Am niioaaonoaaiii a eiee/anoaao a1, a2, …, am aaeeieoe,
o?aaoaony aeinoaaeoue a eaaeaeue ec n ioieoia iacia/aiey (iio?aaeaiey)
A1, A2, …, An niioaaonoaaiii a eiee/anoaao b1, b2, …, bn aaeeieoe.
Noieiinoue ia?aaicee (oa?eo) aaeeieoeu i?iaeoeoa ec Ai a Bj ecaanoia
aeey anao ia?o?ooia AiBj e ?aaia Cij (i=1,m; j=1,n). O?aaoaony
ninoaaeoue oaeie ieai ia?aaicie, i?e eioi?ii aanue a?oc ec ioieoia
ioi?aaeaiey auaiceoueny e cai?inu anao ioieoia iio?aaeaiey
oaeiaeaoai?ythony (cae?uoay iiaeaeue), a noiia?iua o?ainii?oiua
?anoiaeu ieieiaeueiu.
Oneiaey caaea/e oaeiaii ?aniieaaaoue a oaaeeoeo, aienuaay a eeaoee
eiee/anoai ia?aaiceiiai a?oca ec Ai a Bj a?oca Xij > 0, a a iaeaiueeea
eeaoee – niioaaonoaothuea oa?eou Cij:
Iaoaiaoe/aneay iiaeaeue o?ainii?oiie caaea/e.
*enei r = m + n – 1, ?aaiia ?aiao nenoaiu (1), iacuaaaony ?aiaii
o?ainii?oiie caaea/e. Anee /enei caiieiaiiuo eeaoie (Xij ? 0) a oaaeeoea
?aaii r, oi ieai iacuaaaony iaau?iaeaeaiiui, a anee yoi /enei iaiueoa r,
oi ieai au?iaeaeaiiue – a yoii neo/aa a iaeioi?ua eeaoee aienuaaaony
noieueei ioeae (oneiaii caiieiaiiua eeaoee), /oiau iauaa /enei
caiieiaiiuo eeaoie auei ?aaii r.
Neo/ae ioe?uoie iiaeaee aeai ? aebj eaaei naiaeeony e cae?uoie iiaeaee
iooai aaaaeaiey oeeoeaiiai iio?aaeoaey Bn+1 c iio?aaiinoueth
bn+1=aeai-aebj, eeai – oeeoeaiiai iinoaaueea Am+1 c caianii
am+1=aebj-aeai ; i?e yoii oa?eou oeeoeaiuo o/anoieeia i?eieiathony
?aaiuie 0.
Niiniau ninoaaeaiey 1-oaaeeoeu (iii?iiai ieaia).
Niinia naaa?i-caiaaeiiai oaea (aeeaaiiaeueiue). Nouiinoue niiniaa
caeeth/aaony a oii, /oi ia eaaeaeii oaaa caiieiyaony eaaay aa?oiyy
eeaoea (naaa?i-caiaaeiay) inoaaoaeny /anoe oaaeeoeu, i?e/ai iaeneiaeueii
aiciiaeiui /eneii: eeai iieiinoueth auaiceoueny a?oc ec Ai, eeai
iieiinoueth oaeiaeaoai?yaony iio?aaiinoue Bj. I?ioeaaeo?a i?iaeieaeaaony
aei oao ii?, iiea ia eaeii-oi oaaa ia en/a?iathony caianu ai e ia
oaeiaeaoai?ythony iio?aaiinoe bj . A caeeth/aiea i?iaa?ytho, /oi
iaeaeaiiua eiiiiiaiou ieaia Xij oaeiaeaoai?ytho ai?eciioaeueiui e
aa?oeeaeueiui o?aaiaieyi e /oi auiieiyaony oneiaea iaau?iaeaeaiiinoe
ieaia.
Niinia iaeiaiueoaai oa?eoa. Nouiinoue niiniaa a oii, /oi ia eaaeaeii
oaaa caiieiyaony oa eeaoea inoaaoaeny /anoe oaaeeoeu, eioi?ay eiaao
iaeiaiueoee oa?eo; a neo/aa iaee/ey ianeieueeeo oaeeo ?aaiuo oa?eoia
caiieiyaony ethaay ec ieo. A inoaeueiii aeaenoaotho aiaeiae/ii
i?aaeuaeouaio niiniao.
Iaoiae iioaioeeaeia ?aoaiey o?ainii?oiie caaea/e.
Ii?aaeaeaiea: iioaioeeaeaie ?aoaiey iacuaathony /enea ai®Ai, bj®Bj,
oaeiaeaoai?ythuea oneiaeth ai+bj=Cij (*) aeey anao caiieiaiiuo eeaoie
(i,j).
Niioiioaiey (*) ii?aaeaeytho nenoaio ec m+n-1 eeiaeiuo o?aaiaiee n m+n
iaecaanoiuie, eiathuoth aan/eneaiiia iiiaeanoai ?aoaiee; aeey aa
ii?aaeaeaiiinoe iaeiiio iaecaanoiiio i?eaeatho ethaia /enei (iau/ii
a1=0), oiaaea ana inoaeueiua iaecaanoiua ii?aaeaeythony iaeiicia/ii.
E?eoa?ee iioeiaeueiinoe. Anee ecaanoiu iioaioeeaeu ?aoaiey X0
o?ainii?oiie caaea/e e aeey anao iacaiieiaiiuo eeaoie auiieiythony
oneiaey ai+bj F Ci j, oi X0 yaeyaony iioeiaeueiui ieaiii o?ainii?oiie
caaea/e.
Anee ieai ia iioeiaeai, oi iaiaoiaeeii ia?aeoe e neaaeothuaio ieaio
(oaaeeoea) oae, /oiau o?ainii?oiua ?anoiaeu ia oaaee/eeenue.
Ii?aaeaeaiea: oeeeeii ia?an/aoa oaaeeoeu iacuaaaony
iineaaeiaaoaeueiinoue eeaoie, oaeiaeaoai?ythuay oneiaeyi:
iaeia eeaoea ionoay, ana inoaeueiua caiyoua;
ethaua aeaa ninaaeiea eeaoee iaoiaeyony a iaeiie no?iea eee a iaeiii
noieaoea;
ieeaeea 3 ninaaeiea eeaoee ia iiaoo auoue a iaeiie no?iea eee a iaeiii
noieaoea .
Ionoie eeaoea i?enaaeaatho ciae « + », inoaeueiui – iii/a?aaeii ciaee «
– » e « + ».
Aeey ia?a?ani?aaeaeaiey ieaia ia?aaicie n iiiiuueth oeeeea ia?a?an/aoa
nia/aea iaoiaeyo iacaiieiaiioth eeaoeo (r, s), a eioi?ie ar+bs>Crs, e
no?iyo niioaaonoaothuee oeeee; caoai a ieioniauo eeaoeao iaoiaeyo /enei
X=min{Xij}. Aeaeaa ninoaaeytho iiaoth oaaeeoeo ii neaaeothuaio i?aaeeo:
a iethniaua eeaoee aeiaaaeyai X;
ec ieioniauo eeaoie ioieiaai O;
ana inoaeueiua eeaoee aia oeeeea inoathony aac eciaiaiey.
Iieo/ei iiaoth oaaeeoeo, aeathuaa iiaia ?aoaiea X, oaeia, /oi f(X1) F
f(X0); iii niiaa i?iaa?yaony ia iioeiaeueiinoue /a?ac eiia/iia /enei
oaaia iaycaoaeueii iaeaeai iioeiaeueiue ieai o?ainii?oiie caaea/e, eai
ii anaaaea nouanoaoao.
Aeai?eoi iaoiaea iioaioeeaeia.
i?iaa?yai oei iiaeaee o?ainii?oiie caaea/e e a neo/aa ioe?uoie iiaeaee
naiaeei aa e cae?uoie;
iaoiaeei iii?iue ieai ia?aaicie iooai ninoaaeaiey 1-e oaaeeoeu iaeiei ec
niiniaia – naaa?i-caiaaeiiai oaea eee iaeiaiueoaai oa?eoa;
i?iaa?yai ieai (oaaeeoeo) ia oaeiaeaoai?aiea nenoaia o?aaiaiee e ia
iaau?aaeaeaiiinoue; a neo/aa au?iaeaeaiey ieaia aeiaaaeyai oneiaii
caiieiaiiua eeaoee n iiiiuueth « 0 »;
i?iaa?yai iii?iue ieai ia iioeiaeueiinoue, aeey /aai:
a) ninoaaeyai nenoaio o?aaiaiee iioaioeeaeia ii caiieiaiiui eeaoeai;
a) iaoiaeei iaeii ec aa ?aoaiee i?e a1=0;
a) iaoiaeei noiiu ai+bj=Ccij («einaaiiua oa?eou») aeey anao ionouo
eeaoie;
a) n?aaieaaai einaaiiua oa?eou n enoeiiuie: anee einaaiiua oa?eou ia
i?aainoiaeyo niioaaonoaothueo enoeiiuo(Ccij F Cij) ai anao ionouo
eeaoeao, oi ieai iioeiaeai (e?eoa?ee iioeiaeueiinoe). ?aoaiea caeii/aii:
ioaao aeaaony a aeaea ieaia ia?aaicie iineaaeiae oaaeeoeu e cia/aiey min
f.
Anee e?eoa?ee iioeiaeueiinoe ia auiieiyaony, oi ia?aoiaeei e
neaaeothuaio oaao.
Aeey ia?aoiaea e neaaeothuae oaaeeoea (ieaio):
a) auae?aai iaeio ec ionouo eeaoie, aaea einaaiiue oa?eo aieueoa
enoeiiiai (Ccij= ai+bj > Cij );
a) ninoaaeyai oeeee ia?an/aoa aeey yoie eeaoee e ?annoaaeyai ciaee « +
», « – » a aa?oeiao oeeeea iooai eo /a?aaeiaaiey, i?eienuaay ionoie
eeaoea « + »;
a) iaoiaeei /enei ia?a?an/aoa ii oeeeeo: /enei X=min{Xij}, aaea Xij –
/enea a caiieiaiiuo eeaoeao ni ciaeii « – »;
a) ninoaaeyai iiaoth oaaeeoeo, aeiaaaeyy X a iethniaua eeaoee e ioieiay
X ec ieioniauo eeaoie oeeeea
Ni. i. 3 e o.ae.
*a?ac eiia/iia /enei oaaia (oeeeeia) iaycaoaeueii i?eoiaeei e ioaaoo,
eai o?ainii?oiay caaea/a anaaaea eiaao ?aoaiea.
……………………………….
… akl … aks … bk
……………………………….
… ail … ais … bi
……………………………….
… cl … cs … b0
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