Imave periods ekuTvnis evklides amocana

savarjiSoebi

77. 
    .
    
78. 
    
    
79. 
    
    
80. 
    
    
81. 
    
    
82. 
    
    
83. 
    
    
84. 
    erTeulovan wreSi CavxazoT udidesi farTobis mqone marTkuTxedi.
    
85. 
    , Tu cnobilia, rom -simetriuli matricaa.
    
86. 
    ricxvi 8 gayaviT or nawilad ise, rom maTi namravlis namravli maT sxvaobaze iyos maqsimaluri (tartalia).
    
87. 
    CaxazeT wreSi samkuTxedi, romlisTvisac gverdebis kvadratebis jami aris maqsimaluri.
    
88. 
    sibrtyeze mocemulia sami wertili . moZebneT iseTi wertili, saidanac wertilebamde manZilebis kvadratebis jami aris minimaluri.
    

amozneqili eqstremaluri amocanebi

5.1gluvi eqstremaluri amocana Ria dasaSvebi simravliT

1. 
   amozneqili simravleebi da funqciebi
   

ganmarteba 1. vTqvaT X aris wrfivi sivrce da . vityviT, rom aris amozneqili simravle, Tu igi Tavis yovel or wertilTan erTad Seicavs maT SemaerTebel monakveTsac, anu:

da .

magaliTad: pirveli sami figura amozneqilia, meoTxe _ ara.

advili dasamtkicebelia, rom nebismieri raodenoba amozneqili simravleebis TanakveTa amozneqilia. sakmaod Zneli dasamtkicebelia, rom -Si ori TanaukveTi amozneqili simravlis gancaleba SeiZleba hipersibrtyiT (nax. 2).

ganmarteba 2. amozneqil simravleze gansazRvrul funqcias ewodeba amozneqili funqcia, Tu sruldeba e.w iensenis utoloba:

,

. (5.1)

Tu da -isTvis sruldeba ,

maSin -s ewodeba Cazneqili funqcia.

zogjer saWiro xdeba mivuTiToT, Tu romel simravlezea funqcia amozneqili an Cazneqili. magaliTad, araa arc amozneqili da arc Cazneqili _ze, magram -is garkveul qvesimravleebze _ aris amozneqili an Cazneqili.

SevniSnoT, rom -is amozneqiloba simravleze gulisxmobs (ganmartebis Tanaxmad), rom TviTon aris amozneqili.

amozneqil funqciebs aqvT didi praqtikuli da gamoyenebiTi mniSvneloba, magram amozneqilobis Semowmeba xSirad sakmaod rTulia. amitom mniSvnelovania amozneqiloba-Cazneqilobis garkvevisaTvis sxvadasxva kriteriumebis gamoyeneba. Semdegi winadadeba, romelic daumtkiceblad mogvyavs (Tumca misi damtkiceba rTuli araa), garkveul geometriul warmodgenas gviqmnis sakiTxze.

winadadeba 1. vTqvaT, amozneqilia da . maSin, -is amozneqilobisaTvis aucilebelia da sakmarisi misi grafikszeda simravlis

amozneqiloba, xolo -is CazneqilobisaTvis aucilebelia da sakmarisi misi grafiksqveda simravlis

amozneqiloba.

iensenis utoloba niSnavs, rom funqciis grafikis nebismieri ori wertilis SemaerTebeli qorda aucileblad grafikis zemoTaa mo-Tavsebuli, radgan nebismieria -dan (ix. nax.3).

gluvi funqciebis Semowmeba amozneqilobaze xdeba Semdegi Sedegebis safuZvelze.

Teorema 1. /gluvi funqciis amozneqilobis I rigis aucilebeli da sakmarisi piroba/. vTqvaT, aris Ria amozneqili simravle -Si da aris warmoebadi funqcia. maSin -is amozneqilobisTvis aucilebelia da sakmarisi Semdegi pirobis Sesruleba

. (5.2)

damtkiceba. vTqvaT, warmoebadi funqcia amozneqilia -ze. gamo-viyenoT amozneqilobis ganmarteba da aviRoT Zalian mcire dadebiTi parametri, romelic aRvniSnoT -Ti:

anu

radgan amitom

da roca es utoloba -is warmoebadobis ZaliT iZleva (5.2)-s.

vTqvaT axla, rom warmoebadi -isaTvis sruldeba (5.2). nebismierad aviRoT da . orjer gamoviyenoT (5.2), da wertilebisaTvis:

pirveli gavamravloT -ze, meore -ze da SevkriboT. miviRebT

rac amtkicebs amozneqilobas, radgan 

Semdeg TeoremaSi dagvWirdeba Semdegi

ganmarteba. vTqvaT, aris zomis kvadratuli simetriuli matrica. vityviT, rom aris

1. 
   dadebiTad naxevradgansazRvruli, Tu
   
2. 
   uaryofiTad naxevradgansazRvruli, Tu 
   

Teorema 2. /gluvi funqciis amozneqilobis II rigis aucilebeli da sakmarisi piroba/. vTqvaT, aris Ria amozneqili simravle -Si da aris orjer uwyvetad warmoebadi. maSin -is amozneqilobisTvis -ze aucilebelia da sakmarisi, rom heses matrica iyos dadebiTad naxevradgansazRvruli yoveli -isTvis.

damtkiceba. aucileblobis dasamtkiceblad, nebismierad aviRoT , aranulovani da vaCvenoT rom .

radgan Riaa da amozneqili, arsebobs iseTi rom amitom, Teorema1-is ZaliT

Tu gaviTvaliswinebT -is sigluves da gamoviyenebT teiloris formulas, gveqneba:

,

sadac

(5.3)

amitom

anu yoveli -sTvis sruldeba

,

rac (5.3)-is ZaliT gvaZlevs, rom .

axla vaCvenoT sakmarisoba. vTqvaT, yoveli da -isTvis sruldeba da vaCvenoT -is amozneqiloba. Tu teiloris formulis Tanaxmad arsebobs iseTi , rom sruldeba

(5.4)

(teiloris formula am saxiT, damatebiTi daSvebebis gareSe, samarTliania mxolod im SemTxvevaSi roca -is mniSvnelobebi namdvili ricxvebia). (5.4)-is marjvena mxare arauaryofiTia, amitom arauaryofiTia marcxena mxarec, rac Teorema 1-is ZaliT niSnavs -is amozneqilobas. 

magaliTi 1. SevamowmoT, amozneqilia Tu ara simravle



amoxsna. nebismierad aviRoT

da

rac -is gansazRvris Tanaxmad, niSnavs vaCvenoT rom

.

naxazi gviCvenebs, rom pasuxi dadebiTi unda iyos. amitom amozneqilobis Semowmeba unda gavagrZeloT. veqtorebze operaciebis gamoyenebiT,

am veqtorisTvis SevamowmoT -is ganmsazRvreli piroba:

(davamatoT da davakloT saWiro wevrebi)

,

rac niSnavs rom amozneqilia. 

magaliTi 2. SevamowmoT amozneqilia Tu ara simravle



amoxsna. SevecadoT warmovadginoT amozneqili simravleebis TanakveTis saxiT, rac misi amozneqilobis tolfasi iqneba. cxadia,

ganvixiloT meore, , romelic samkuTxedis Siga nawils warmoadgens. cnobilia rom samkuTxedi amozneqili figuraa, magram axla SevamowmoT es faqti. vTqvaT

da .

-s Sesabamisi wertili da boloebis mqone monakveTisa aris:

da radgan

amitom amozneqilia.

axla ganvixiloT simravle. geometriuli suraTi gviCvenebs, rom amozneqilia, amitom CavataroT formaluri Semowmeba, nebismierad aviRoT

da

e.i. da

veqtoris komponentebi dadebiTia, amitom saCvenebeli rCeba rom maTi namravli metia erTze.


2. amozneqili amocanebis specifika

rodesac amozneqili simravlea, xolo amozneqili funqciaa, maSin Semdeg minimizaciis amocanas

(5.5)

ewodeba amozneqili (minimizaciis) amocana.

Tavisi Sinaarsis gamo, Semdeg Tvisebas uwodeben globalur-lokalur Tvisebas.

winadadeba1. amozneqil amocanaSi lokaluri minimali amavdroulad aris globaluri minimali.

damtkiceba. vTqvaT (5.5) aris amozneqili amocana, xolo es niSnavs, rom raRac manZilisTvis sruldeba implikacia:

(5.6)

axla nebismierad aviRoT dasaSvebi wertili. roca ricxvi sakmaod mcirea, maSin

radgan amozneqilia, xolo

roca amitom (5.6)-is Tanaxmad axla gaviTvaliswinoT -is saxe da gamoviyenoT iensenis utoloba:

anu , rac -is nebismierobis ZaliT amtkicebs, rom

, ,

e.i. 

winadadeba 2. amozneqil amocanaSi minimumis wertilebis simravle amozneqilia.

damtkiceba. vTqvaT minimalebia (5.5)-Si. wina Sedegis ZaliT,

axla aviRoT :

e.i. minimalebis amozneqili kombinaciac minimalia. 

gansakuTrebiT mniSvnelovania SemTxveva, rodesac amozneqili amocana imavdroulad aris gluvic, radgan am dros minimumis aucilebeli pirobebi xdeba sakmarisic.

winadadeba 3. vTqvaT, aris Ria amozneqili simravle, aris warmoebadi da amozneqili funqcia. maSin yoveli wertili, sadac , warmoadgens minimals (5.5)-Si.

damtkiceba. nebismierad aviRoT . amozneqilobis I rigis aucilebeli da sakmarisi pirobis Tanaxmad, da - is gaTvaliswi-nebiT,

.

e.i. 

savarjiSoebi.

89. 
    daamtkiceT, rom amozneqili simravleebis nebismieri raodenobis TanakveTa amozneqilia.
    
90. 
    arsebobs Tu ara funqcia, romelic erTdroulad amozneqilia da Cazneqilic.
    
91. 
    vTqvaT, amozneqilia. ra SegviZlia vTqvaT _ funqciaze?
    
92. 
    aCveneT (geometriulad mainc), rom monakveTze amozneqili funqcia uwyvetia -ze.
    
93. 
    miiReT am paragrafis Sedegebis analogebi Cazneqili amocanisaTvis.
    
94. 
    vTqvaT amozneqilia. aCveneT rom aris amozneqili simravle.
    
95. 
    aCveneT, rom amozneqili simravleze gansazRvruli amozneqili funqciebis jami arauaryofiTi koeficientebiT aris amozneqili funqcia.
    
96. 
    SeamowmeT, amozneqilia Tu ara simravle :
    

a).

b).

g).

d).

e).

89. 
    aCveneT Semdegi funqciebis amozneqiloba:
    

a). , .

b). , .

g).

5.2 gluvi amozneqili funqciebis minimizacia ricxviTi meTodebiT

ganvixiloT minimizaciis amocana:

, (5.7)

sadac f aris amozneqili funqcia, romelsac aqvs pirveli an meore rigis uwyveti kerZo warmoebulebi. eqstremalurobis aucilebeli da sakmarisi pirobebis gamoyenebiT, rig SemTxvevebSi xerxdeba (5.7) amocanis amoxsna, magram, xSirad, saWiro xdeba minimalebis povna ricxviTi meTodebis gamoyenebiT. nebismieri ricxviTi meTodi gulisxmobs misi maxasiaTeblebis (miznis funqciis, dasaSvebi samravlis ganmsazRvravi funqciebis, maTi warmoebulebis) zusti an miaxloebiTi mniSvnelobebis gamoTvlas, da Semdeg, am informaciis safuZvelze amocanis amoxsnis ( minimalis an minimalebis wertilTa simravlis) miaxloebiTi mniSvnelobis povnas.

iseve, rogorc erT cvladze damokidebuli miznis funqciis mqone eqstremaluri amocanebis SemTxvevaSi, aqac, ganasxvaveben nulovani, pirveli, meore rigis (minimalis Zebna xdeba, Sesabamisad, mxolod funqciis mniSvnelobebis, pirveli da meore rigis warmoebulebis gamoyenebiT), agreTve, pasiur da mimdevrobiT ricxviT meTodebs.(ix. 3.2)

minimizaciis amocanis amosaxsnelad mimdevrobiTi ricxviTi meTodiT, Semdegi wesiT

aigeba wertilTa mimdevroba, romelic krebadia minimalisken, amasTan, yoveli konkretuli algoriTmi ganisazRvreba sawyisi miaxloebis wertilis, veqtoris, ricxvebis arCeviT, da agreTve, gaCerebis pirobiT.

sawyisi miaxloebis – wertilis arCevis raime zogadi wesi ar arsebobs, magram Tu amocanis specifikis an Sinaarsis gaTvaliswinebiT, cnobilia minimalis SesaZlo ganlageba, maSin sawyisi miaxloeba, bunebrivia, masTan axlos unda aviRoT.

veqtori gansazRvravs bijis mimarTulebas, xolo ricxvi _ bijis sigrZes.

konkretuli ricxviTi algoriTmis aRsawerad unda gvqondes gaCerebis piroba. praqtikaSi gamoiyeneba gaCerebis Semdegi pirobebi:

,

,

.

gamoTvlebis dawyebamde unda davasaxeloT da erTi, an ori gaCerebis piroba.

meTodis efeqturoba ganisazRvreba krebadobis siCqariT. vityviT, rom meTodi krebadia, Tu , sadac aris amocanis amonaxsni. meTodis efeqturoba xasiaTdeba krebadobis siCqariT.

vityviT, rom krebadia -ken wrfivad, Tu , rom

,

vityviT, rom krebadia -ken zewrfivad, Tu

,

vityviT, rom krebadia -ken kvadratulad, Tu , rom

,
am TavSi, ganvixilavT pirveli rigis minimizaciis mimdevrobiT meTodebs: gradientul meTodebs, niutonis meTods, gradientis proeqciis meTods, agreTve, SemTxveviTi Ziebis meTods.

1. gradientuli daSvebis meTodi

vTqvaT, f aris amozneqili, uwyveti kerZo warmoebulebis mqone funqcia, da vTqvaT aris minimali (5.7) amocanaSi, romlis moZebnac warmoadgens Cvens mizans.

nebismierad aviRoT sawyisi miaxloeba da avagoT mimdevroba Semdegnairad:

(5.8)

aq, veqtoris rolSi aviReT antigradienti, radganac antigradientis mimarTuleba mocemul wertilSi (roca ) emTxveva funqciis uswrafesi klebis mimarTulebas(ix.4.2, gv72). sidideebi (bijis sigrZe) SeiZleba avirCioT imdenad mcire, rom Sesruldes piroba

. (5.9)

im SemTxvevaSi, roca (5.9) piroba ar sruldeba, vanaxevrebT sidides da axlidan veZebT miaxloebas.

gaCerebis pirobad, Cveulebriv, gamoiyeneba piroba

, (5.10)

sadac winaswar mocemuli mcire ricxvia. Tu (5.10) sruldeba, viRebT: .

magaliTi 1. gradientuli daSvebis meTodiT sizustiT amoxseniT minimizaciis amocana:

.

amoxsna. aviRoT: , avagoT (5.8) mimdevroba da Sedegebi CavweroT cxrilSi:

k SeniSvna00011111-1-13,145-- =1–isaTvis (5.9) piroba ir-Rveva, amitom vanaxevrebT mas0001110,51-0,5-0,51,118--(5.9) piroba isev irRveva, amitom isev vanaxevrebT -s0001110,251-0,25-0,250,7940,106-0,3930,25(5.9) piroba sruldeba, (5.10)-ar sruldeba.2-0,2766326-0,15163260,7740.09830,04510,25(5.9) piroba sruldeba, (5.10)-ar sruldeba.3-0,3012259-0,16290960,7720,0262-0,023-(5.9),(5.10) pirobebi sruldeba sizuste miRweulia. amgvarad, . 

2. uswrafesi daSvebis meTodi

uswrafesi daSvebis meTodi imiT gansxvavdeba gradientuli daSvebis meTodisagan, rom amjerad SeirCeva pirobidan

, (5.11)

sadac . amgvarad, yovel bijze ixsneba minimizaciis erTganzomilebiani amocana. radganac f amozneqili fun-qciaa, romelsac aqvs pirveli rigis uwyveti kerZo warmoebulebi, amitom gantolebis amonaxsni warmoadgens (5.11) amocanis amonaxsns. am faqts aqvs lamazi geometriuli interpretacia: yoveli k –sTvis da veqtorebi urTierTmarTobulni arian (es gamomdinareobs pirobidan).

magaliTi. uswrafesi daSvebis meTodiT sizustiT amoxseniT Semdegi amocana:



amoxsna.

biji 1). aviRoT . maSin , ,

da funqciis minimumis mosaZebnad, dadebiT –ebs Soris, gamoviyenoT gadarCevis meTodi (es simartivisaTvis, radgan aseTi martivi saxis funqciis mniSvnelobebis gamoTvla problema araa) _idan dawyebuli bijiT 0,2:

0. . .0,180,200,220,240,26 1. . .0,79490,79030,78920,79160,7973e.i. =0,22, saidanac .

biji 2). . movaxdinoT -is minimizacia:

. . .0,280,300,320,340,36 . . .0,774010,773840,773800,773870,77408e.i. =0,32 da .

biji 3).

, -is minimizacia gvaZlevs:

. . .0,200,220,240,260,28 . . .0,772730,772410,772400,772410,77244e.i. =0,24, , amitom da . 

gavakeToT ramdenime SeniSvna.

₁) _k -s arCeva sakmaod Sromatevadi procesia, amitom sasurvelia erTxel SeirCes _s romelime mniSvneloba, romelic ar iqneba damokidebuli iteraciis nomerze da gamodgeba yovel iteraciaze.

₂) rodesac Zalian axlos aris -sTan, gradientuli meTodebis krebadobis siCqare neldeba. am dros, rekomendebulia ufro faqiz meTodze gadasvla, magaliTad iseTze, romelic iyenebs f –is kvadratul aproqsimacias.

3. niutonis meTodi

Tu amozneqili funqcia orjer uwyvetad warmoebadia, xolo , -s gamoTvla ar aris Zneli. maSin SesaZlebelia meore rigis mimdevrobiTi meTodebis gamoyeneba. vTqvaT, ukve gansazRvruli gvaqvs miaxloeba. -s midamoSi f –is nazrds aqvs saxe:

.

ganvixiloT nazrdis kvadratuli nawili:

(5.12)

da ganvsazRvroT miaxloeba pirobidan:

. (5.13)

(5.13) –is amosaxsnelad gamoiyeneba rogorc analizuri, aseve miaxloebiTi (specialurad kvadratuli funqciebisTvis gaTvaliswinebuli) meTodebi.

radganac , -s amozneqilobis gamo, dadebiTad naxevradgansazRvrulia, amitom amozneqili funqciaa (ix.5.1 Teorema 2). minimumis aucilebel da sakmaris pirobas (5.13)-Tvis aqvs saxe: ,

Tu amovxsniT am sistemas da mis amonaxsns CavTvliT miaxloebad, miviRebT

(5.14)

unda aRiniSnos, rom (5.13) amocanis amoxsna SeiZleba aRmoCndes sakmaod rTuli da gamoTvlebis sirTulis gaTvaliswinebiT, sawyisi amocanis sadari. amitom niutonis meTods iyeneben maSin, roca , -s gamoTvla da (5.13) gantolebis amoxsna ar aris dakavSirebuli siZneleebTan. niutonis meTodis Rirsebad iTvleba misi krebadobis maRali siCqare, magram am meTods aqvs mniSvnelovani naklic, misi krebadobisTvis sawyisi miaxloeba sakmarisad axlos unda iyos minimalTan, winaaRmdeg SemTxvevaSi, meTodi SeiZleba ar aRmoCndes krebadi.

SevniSnoT, rom (5.14)_Si rogorc mimarTuleba, aseve bijis sigrZe fiqsirebulia. niutonis meTodis sxvadasxva modifikacia mimarTulia iqiTken, rom am meTodis ZiriTadi Rirsebis (krebadobis maRali siCqare) SenarCunebiT, Semcirdes misi Sromatevadoba, rac dakavSirebulia , -s gamoTvlasTan da (5.13) gantolebis amoxsnasTan da Sesustdes moTxovna sawyisi miaxloebis amorCevaze.

niutonis meTodi bijis regulirebiT mdgomareobs SemdegSi:

roca , is emTxveva niutonis meTods. koeficientebis arCeva xdeba an mocemuli mimarTulebiT funqciis minimizaciis pirobidan, an bijis dayofiT (5.9) pirobis gaTvaliswinebiT.

magaliTi. magaliTi 1 –is pasuxi aviRoT sawyis miaxloebad da niutonis meTodiT ganvsazRvroT minimali Semdeg amocanaSi:



amoxsna. magaliTi 1 –is Sedegebis mixedviT, gvaqvs:

, ,

.

aqedan:

,

da (8) –is Tanaxmad:

.

radgan

amitom sasurveli sizuste miRweulia da . 

4. gradientis proeqciis meTodi

ganvixiloT minimizaciis amocana:

, (5.15)

sadac aris amozneqili Caketili simravle, xolo f aris amozneqili warmoebadi funqcia MM–ze.

Caketili MM–is SemTxvevaSi uSualod gradientuli meTodis gamoyeneba ar SeiZleba, Tu , radgan zogierTi SeiZleba gavides MM–idan. gradientuli proeqciis meTodis yoveli iteracia iTvaliswinebs Semdegi formuliT gansazRvruli gradientuli daSvebis miaxloebebis

,

dabrunebas dasaSveb MMsimravleSi, Tu . aseTi dabruneba xdeba -is MM–ze proeqtirebis saSualebiT, anu icvleba MMsimravlis im wertiliT, romelic misgan yvelaze axlosaa moTavsebuli.

ganmarteba. wertilis proeqcia simravleze ewodeba iseT wertils MM–idan, romelic akmayofilebs pirobas:

,

rogorc vxedavT, marjvena mxare gamoxatavs manZils x –idan MM–mde. 

cxadia, Tu , maSin , xolo roca , maSin MM–is Caketilobis gamo (vaierStrasis Teoremis Sedegis ZaliT) arsebobs da MM–is amozneqilobis ZaliT ki erTaderTia (savarjiSos saxiT, sasargebloa am faqtis geometriul SinaarsSi garkveva).

amgvarad, gradientis proeqciis meTodSi (5.15) amocanis minimalis mimdevrobiTi miaxloebebi aigeba Semdegi wesiT:

. (5.16)

-s SerCeva aqac sxvadasxvanairad SeiZleba; Sesabamisad miiReba gradientis proeqciis meTodis sxvadasxva varianti. maTgan yvelaze gavrcelebuli aris Semdegi ori.

1. 
   SeirCeva iseve, rogorc uswrafesi daSvebis meTodSi Ria MM–isTvis, anu SeirCeva Semdegi pirobidan:
   

,

sadac .

1. 
   SeirCeva ise, rom Sesruldes monotonurobis piroba . amisTvis Tavidan irCeven raime da (5.16)_Si iReben .Semdeg, amowmeben monotonurobis pirobas da Seusruleblobis SemTxvevaSi, yofen mas am pirobis Sesrulebamde.
   

unda aRiniSnos, rom roca , maSin -is gansazRvra xdeba minimizaciis Semdegi amocanis:

,

amoxsnis Sedegad. zogierTi kerZo saxis M simravlisaTvis (magaliTad, birTvi, naxevarsibrtye da zogierTi sxva) es amocana ixsneba analizurad lagranJis meTodis safuZvelze.

magaliTi 1. ipoveT wertilis proeqcia , Tu M simravles aqvs saxe:

(esaa Caketili birTvi -Si centriT 0 da radiusiT r).

amoxsna. aris Semdegi amocanis amonaxsni:

romelic SeiZleba SevcvaloT ekvivalenturi amocaniT:

(5.17)

ganvixiloT aratrivialuri SemTxveva . SevadginoT lagranJis funqcia:

da amovweroT pirobebi kritikuli wertilis gansazRvrisaTvis:

(5.18)

iZleva winaaRmdegobas II gantolebaSi: . e.i. . axla ganvixiloT ori SemTxveva -s mimarT.

a) Tu , anu aris (5.18) –is amonaxsni, magram maSin igi ar gansazRvravs kritikul wertils, radgan (radgan ).

b) Tu . maSin (5.18) iRebs saxes:

e.i. . miviReT, rom aris erTaderTi kritikuli wertili da, vaierStrasis Teoremis Sedegis ZaliT, igi aris agreTve globaluri minimali (5.17) –Si. 

magaliTi 2. gradientis proeqciis meTodiT amoxseniT Semdegi amocana:

.

gamoTvlebi SewyviteT, Tu Sesruldeba , an .

amoxsna. aviRoT , sadac

.

.

aviRoT (5.16) –Si . axla SegviZlia avagoT mimdevrobiTi miaxloebebi.