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Решение задач – методы спуска

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Iaoiaeu nionea

Iauay noaia.

Ana iaoiaeu nionea ?aoaiey caaea/e aaconeiaiie ieieiecaoeee ?acee/athony
eeai auai?ii iai?aaeaiey nionea, eeai niiniaii aeaeaeaiey aaeieue
iai?aaeaiey nionea. Yoi iicaieyao iaienaoue iauoth noaio iaoiaeia
nionea.

?aoaaony caaea/a ieieiecaoeee ooieoeee ((x) ia an?i i?ino?ainoaa En.
Iaoiaeu nionea ninoiyo a neaaeothuae i?ioeaaeo?a iino?iaiey
iineaaeiaaoaeueiinoe {xk}. A ea/anoaa ia/aeueiiai i?eaeeaeaiey
auae?aaony ethaay oi/ea x0(En. Iineaaeiaaoaeueiua i?eaeeaeaiey x1, x2, …
no?iyony ii neaaeothuae noaia:

a oi/ea xk auae?atho iai?aaeaiea nionea – Sk;

iaoiaeyo (k+1)-a i?eaeeaeaiea ii oi?ioea xk+1=xk-pkSk.

Iai?aaeaiea Sk auae?atho oaeei ia?acii, /oiau iaania/eoue ia?aaainoai
((xk+1)<((xk) ii e?aeiae ia?a aeey iaeuo cia/aiee aaee/eiu pk. Ia aii?in, eaeiio ec niiniaia auai?a iai?aaeaiey nionea neaaeoao ioaeaoue i?aaeii/oaiea i?e ?aoaiee eiie?aoiie caaea/e, iaeiicia/iiai ioaaoa iao.*enei pk ii?aaeaeyao ?annoiyiea io oi/ee xk aei oi/ee ok+1. Yoi /enei iacuaaaony aeeeiie oaaa eee i?inoi oaaii. Iniiaiay caaea/a i?e auai?a aaee/eiu (k - yoi iaania/eoue auiieiaiea ia?aaainoaa ((xk+1)<((xk). Iaeiei ec yeaiaioa?iuo niiniaia auai?a oaaa yaeyaony niinia oaeaiaiey oaaa.Auae?atho (k=(k-1. Anee i?e yoii ((xk+1)<((xk), oi eeai ia?aoiaeyo e neaaeothuae (k+2)-e eoa?aoeee, eeai auae?atho (k=2(k-1. Anee cia/aiea ((o) iaiueoa aai i?aaeuaeouaai cia/aiey, oi i?ioeann oaeaiaiey iiaeii i?iaeieaeaoue aei oao ii?, iiea oauaaiea ia i?ae?aoeony. Anee ((xk+1)(((xk), oi auae?atho (k=0.5(k-1. Anee ((xk-0.5(k-1Sk)<((xk), oi iieaaatho xk+1=xk-0.5(k-1Sk e ia?aoiaeyo e neaaeothuae (k+2)-e eoa?aoeee. Anee aea ((xk-0.5(k-1Sk)(((xk), oi auae?atho (k=0.25(k-1 e o.ae.Iaoiae a?aaeeaioiiai nionea.Iaeiei ec naiuo ?ani?ino?ai?iiuo iaoiaeia ieieiecaoeee, naycaiiuo n au/eneaieai a?aaeeaioa, yaeyaony iaoiae nionea ii iai?aaeaieth aioea?aaeeaioa ieieiece?oaiie ooieoeee. A iieueco oaeiai auai?a iai?aaeaiey nionea iiaeii i?eaanoe neaaeothuea niia?aaeaiey. Iineieueeo aioea?aaeeaio, oi anoue (’(xk) a oi/ea xk oeacuaaao iai?aaeaiea iaenei?aeoaai oauaaiey ooieoeee, oi anoanoaaiiui i?aaenoaaeyaony nianoeoueny ec oi/ee xk ii yoiio iai?aaeaieth.Iaoiae nionea, a eioi?ii Sk=(’(xk), iacuaaaony iaoiaeii a?aaeeaioiiai nionea.Aaee/eia (k a iaoiaea a?aaeeaioiiai nionea o?aaeeoeeiiii au/eneyaony ioo?i i?eiaiaiey iaeiiai ec iaoiaeia iaeiiia?iie ieieiecaoeee ooieoeee ((()=((xk-((’(xk)), /oi ia eneeth/aao i?eiaiaiea e ae?oaeo niiniaia iouneaiey (k.Anee a ea/anoaa (k auae?atho oi/eo iaeiiia?iiai ieieioia ooieoeee ((()=((xk-(Sk) ?aeaenaoeeiiiue i?ioeann iacuaaaony iaoiaeii iaenei?aeoaai nionea: xk+1=xk-(k(’(xk), (k=arg min {((()=((xk-(Sk) | ((0}.Iaoiae iieii?aeeiaoiiai nionea.Iaeiei ec iaeaieaa i?inouo niiniaia ii?aaeaeaiey iai?aaeaiey nionea yaeyaony auai? a ea/anoaa Sk iaeiiai ec eii?aeeiaoiuo aaeoi?ia (e1, (e2, …, (en, aneaaenoaea /aai o xk ia eaaeaeie eoa?aoeee eciaiyaony eeoue iaeia ec eiiiiiaio.Nouanoaotho iiiai/eneaiiua aa?eaiou iieii?aeeiaoiiai nionea. Ii a ethaii ec yoeo iaoiaeia auae?atho a ea/anoaa -Sk oi ec aeaoo iai?aaeaiee, +ej, -ej, eioi?iio niioaaonoaoao ia?aaainoai[(’(xk), Sk] > 0.

=0, iieaaatho xk+1=xk e ia?aoiaeyo e neaaeothuae eoa?aoeee.

Iieoai ia?aue oeeee iaoiaea, ninoiyuee ec n eoa?aoeee. A i?iecaieueiie
oi/ea x0 auae?atho S0=(e, e ii?aaeaeyao aaee/eio (0 niiniaii oaeaiaiey
oae, /oiau auei ((x1)=((x0-(0S0)<((x0). Caoai auae?atho S1=(e2 e, iieaaay (=(0, oaeaiaieai au/eneytho (1 e oae aeaeaa. I?e yoii ia eaaeaeie eoa?aoeee no?aiyony ii?aaeaeaiea aaee/eiu oaaa iaoiaeii oaeaiaiey inouanoaeyoue n iaeiaiueoei /eneii au/eneaiee cia/aiee ooieoeee ((o). Oeeee caeai/eaaaony i?e k=n-1, iinea /aai ia/eiatho neaaeothuee oeeee, iieaaay Sn=(e1 e o.ae.I?aeoe/aneia caaeaieaIa i?aeoeea iai ioaeii auei iaeoe ieieioi ooieoeee z(x)=x2+y2-xy-3y c oi/iinoueth (, eniieuecoy iienaiiua auoa iaoiaeu.Iaoiaeaeaiea ieieioia iiae ooieoeee n iiiiuueth iaoiaea iieii?aeeiaoiiai nionea.Aeey iaoiaeaeaiey ieieioia iiae ooieoeee n iiiiuueth iaoiaea iieii?aeeiaoiiai nionea y eniieueciaae i?ia?aiio, i?aaenoaaeaiioth ieaea. Aoiaeiuie ia?aiao?aie yoie i?ia?aiiu yaeythony eii?aeeiaou ia/aeueiie oi/ee (y acye o=10, y=10), ia/aeueiue oaa ii o e ii y (y acye (o=0.5 e (y=0.5), a oae aea oi/iinoue ((=10-5; aieueooth oi/iinoue a?aoue ia eiaao niunea, iineieueeo ai a?aiy auiieiaiey i?ia?aiiu iaeaieeaaaony ioeaea e eneaaeaao aeaiiua oaeie oi/iinoe). Eoae, acya a ea/anoaa ia/aeueiuo oneiaee yoe cia/aiey y iieo/ee eii?aeeiaou oi/ee ieieioia:o= 1,00000977y= 1,99999931z=-3,00000142Aeey iieo/aiey ?acoeueoaoa i?ia?aiiie auei auiieiaii 24 eoa?aoeee.Iaoiaeaeaiea ieieioia n iiiiuueth iaoiaea a?aaeeaioiiai nionea.I?ia?aiia, eniieueciaaiiay iiie aeey auiieiaiey yoie caaea/e i?aaenoaaeaia ieaea.Iineieueeo aoiaeiua ia?aiao?u yoie i?ia?aiiu niaiaaeatho ni aoiaeiuie ia?aiao?aie caaea/e ?1, oi y acye eo oaeea aea, /oi e aeey ia?aie caaea/e, /oiau, n?aaiea iieo/aiiua ?acoeueoaou e eiee/anoai eoa?aoeee, iaiaoiaeeiuo aeey iienea ieieioia, y niia naeaeaoue eaeea-eeai auaiaeu i i?aeiouanoaao e iaaeinoaoeao iaaeo caaea/ ec i?aeoeee.Eoae, acya oa aea ia/aeueiua oneiaey y iieo/ee neaaeothuea ?acoeueoaou:x= 1,00000234y= 2,00000119z=-3,00000094Eiee/anoai eoa?aoeee, eioi?ia iio?aaiaaeinue aeey iaoiaeaeaiey oi/ee ieieioia ?aaii 20. Aeaeii, /oi eiee/anoai eoa?aoeee, iio?aaiaaaoaany ia?aie i?ia?aiia aieueoa, /ai eiee/anoai eoa?aoeee, iaiaoiaeeiuo aoi?ie i?ia?aiia. Yoi neaaeoao ec oiai, /oi aioea?aaeeaio oeacuaaao iai?aaeaiea iaenei?aeoaai oauaaiey ooieoeee.Ieaea oaeaea i?aaenoaaeai a?aoee noiaeeiinoe auoaiienaiiiai i?ioeanna aeey iiae ooieoeee e iieo ia/aeueiuo oneiaee.Iaiaoiaeeii oaeaea aeiaaaeoue ianeieueei aaaeiuo iiiaioia. Ai-ia?auo, ec oiai, /oi eiee/anoai eoa?aoeee, iio?aaiaaaoaany aeey iaoiaeaeaiey ieieioia a ia?aie caaea/a aieueoa, /ai ai aoi?ie ia neaaeoao oio oaeo, /oi aoi?ay i?ia?aiia ?aaioaao auno?aa, /ai ia?aay, iineieueeo aeey aoi?ie caaea/e iaiaoiaeeii au/eneyoue ia oieueei cia/aiea ooieoeee a eaeie-eeai oi/ea, ii e a? i?iecaiaeiie a yoie oi/ea, eioi?ay iiaeao auoue aieaa a?iiicaeea, /ai naia ooieoeey. Iaeiiaoe, aoi?ie iaoiae ieio au? e iioiio, /oi aeey i?iecaieueiie ooieoeee i?iecaiaeioth au/eneeoue iaaiciiaeii; i?eae?ony nia/aea aii?ieneie?iaaoue a?, a caoai eneaoue ieieioi (ca n/?o aii?ieneiaoeee cia/eoaeueii au?anoaao a?aiy e iia?aoiinoue ecia?aiee).

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