koba gelaSvili

daviT devaZe

beJan RvaberiZe

lela alxaziSvili

fridon dvaliSvili

Tbilisi 2009

saxelmZRvanelo warmoadgens im kursis gadamuSa-

vebul da Sevsebul variants, romelsac avtorebi wlebis

ganmavlobaSi kiTxuloben Tbilisis ivane javaxiSvilis

saxelobis da baTumis SoTa rusTavelis saxelobis

saxelmwifo universitetebSi. saxelmZRvaneloSi Sesulia

meTodebi, romlebsac garda Teoriulisa, gaaCniaT praq-

tikuli Rirebuleba da romelTa efeqtianoba aRiare-

bulia.

3

S e s a v a l i

adamianis moRvaweobis yvela sferoSi warmoiSoba

miznis miRwevis arsebuli saSualebebidan saukeTesos,

anu optimaluris arCevis problema. aseTi tipis amoca-

nebi pirvelad antikur epoqaSi daisva da isini maTema-

tikis ganviTarebis mTeli istoriis manZilze ipyrobdnen

yuradRebas da inarCunebdnen aqtualobas. maTematikis

enaze optimaluris arCeva daiyvaneba maqsimumis an mini-

mumis, anu eqstremumis povnaze. amitom, aseT amocanebs

maTematikaSi eqstremaluri amocanebi ewodeba. arsebobs

eqstremaluri amocanebi, romlebsac araviTari gamoye-

nebiTi Sinaarsi ar gaaCniaT da mxolod abstraqciis Se-

degs warmoadgenen. amaTgan gasxvavebiT, iseT eqstrema-

lur amocanebs, romlebic raime realuri amocanis maTe-

matikur models warmoadgenen, optimizaciis amocanebi

ewodeba. optimizaciis amocanis amoxsna mxolod trivia-

lur SemTxvevebSi aris advili. TiTqmis ar arsebobs iseTi

amocana, romelsac gamoyenebiTi mniSvneloba aqvs da

uceb, erTi formuliT ixsneba. maTematikuri progra-

mireba swavlobs optimizaciis amocanebis amoxsnis meTo-

debs, realuri amocanis gamokvlevis yvela etapis gaTva-

liswinebiT, am etapebidan ZiriTadi aris Semdegi:

1. amocanis sityvieri formulireba;

2. amocanis maTematikuri an kompiuteruli modelis

Sedgena;

3. amonaxsnis arsebobis sakiTxis garkveva;

4. amonaxsnis moZebnis algoriTmis (anu gegmis,

programis) gansazRvra;

5. uSualod amonaxsnis moZebna;

6. Sedegebis gaazreba sawyisi amocanis terminebSi.

amocanis sirTulis mixedviT, amaTgan erTi an ramde-

nime etapi SeiZleba Serwymulic iyos. termini maTemati-

kuri programireba SemoRebul iqna manamde, vidre kom-

4

piuterebi Seiqmneboda da igi ar gulisxmobs maincdamainc

programuli kodis Seqmnas. im dros programa ubralod

niSnavda algoriTms, gegmas.

maTematikuri daprogrameba iyofa ramdenime mimar-

Tulebad: wrfivi, arawrfivi, dinamikuri, mTelricxva,

geometriuli daprogramebebi da kidev ramdenime sxva.

amave dros, maTematikuri daprogrameba warmoadgens

kidev ufro vrceli disciplinis _ operaciaTa kvlevis

nawils, romelic agreTve iswavleba kompiuterul mec-

nierebaTa mimarTulebebze. amitom xSirad maTematikuri

daprogramebis zogierTi Tavi, garkveuli mosazrebebis

gamo, ikiTxeba operaciaTa kvlevis kursSi (magaliTad,

wrfivi daprogrameba).

maTematikur programirebas bevri saerTo aqvs kom-

piuteruli mecnierebis iseT fundamentur saganTan,

rogoricaa algoriTmebis ageba. bevr rameSi es ori sagani

avsebs erTmaneTs. kerZod, maTematikuri programirebis

ricxviTi algoriTmebis mniSvnelovani nawili damuSave-

bulia uwyveti (da ara diskretuli) maTematikis meTo-

debis safuZvelze.

dasasrul, aRvniSnoT maTematikuri programirebis

meTodebis ramdenime gamoyeneba:

operaciaTa kvleva: teqnikur-ekonomikuri siste-

mebis optimizacia (dagegmva, ekonometrika), sat-

ransporto da maragTa marTvis amocanebi da a.S.,

ricxviTi analizi: aproqsimacia, regresia, wrfivi

da arawrfivi sistemebis amoxsna da a.S.,

avtomatika: sistemaTa amocnoba, filtracia, war-

moebis marTva, robotebi da a.S.,

teqnika: struqturaTa optimizacia, informaciu-

li sistemebi, kompiuteruli qselebi, satranspor-

to, satelekomunikacio da kavSirgabmulobis qse-

lebis marTva.

5

s a r C e v i

nawili 1. ZiriTadi cnebebi Tavi 1. ZiriTadi aRniSvnebi da ganmartebebi ........ 9

aRniSvnebi ........................................................................ 11 ZiriTadi ganmartebebi ................................................ 13 SezRudvaTa tipebi ........................................................ 16 savarjiSoebi .................................................................. 18

Tavi 2. globaluri eqstremumis wertilebis

arsebobisD vaierStrasis sakmarisi

piroba ................................................................... 20 savarjiSoebi .................................................................. 27

nawili 2. erT cvladze damokidebuli

miznis funqcia

Tavi 3. erTi cvladis funqciis lokaluri

eqstremumebis Zebna ......................................... 30 1. mokle reziume ............................................................ 31 2. maTematikuri modelis Sedgena ............................. 31 3. kritikuli wertilebis gansazRvra ..................... 34 4. kritikuli wertilebis gamokvleva

vaierStrasis Teoremis Sedegis gamoyenebiT ................................................................. 36

5. kritikuli wertilebis gamokvleva ganmartebis safuZvelze ......................................... 37

6. kritikuli wertilebis gamokvleva eqstremalurobis II rigis sakmarisi pirobis safuZvelze ................................................................. 38

7. gluvi eqstremaluri amocana segmentis tipis dasaSvebi simravliT ..................................... 39

savarjiSoebi .................................................................. 40 Tavi 4. unimodaluri funqciebis minimizacia

intervalTa gamoricxvis meTodebiT ......... 41 1. ganmartebebi da intervalTa gamoricxvis

wesi ................................................................................ 41 2. intervalis sigrZis ganaxevrebis meTodi .......... 45 3. oqros kveTis meTodi ............................................... 48 savarjiSoebi .................................................................. 51

6

Tavi 5. unimodaluri funqciebis minimizacia

polinomialuri aproqsimaciiT da

wertilovani SefasebiT .................................. 53 1. Sefasebis meTodi, romelic iyenebs

kvadratul aproqsimacias ...................................... 53 2. pauelis meTodi .......................................................... 56 3. wertilovani aproqsimacia. Sua wertilis

meTodi .......................................................................... 57 savarjiSoebi .................................................................. 59

Tavi 6. lifSicuri funqciebis minimizacia

texilTa meTodiT ............................................. 60 savarjiSoebi .................................................................. 66

nawili 3. mraval cvladze damokidebuli

miznis funqcia

Tavi 7. gluvi eqstremaluri amocana Ria

dasaSvebi simravliT ........................................ 68 1. kritikuli wertilebis gansazRvra ..................... 68 2. kritikuli wertilebis gamokvleva

vaierStrasis Teoremis Sedegebis gamoyenebiT ................................................................. 69

3. kritikuli wertilebis gamokvleva ganmartebis safuZvelze ......................................... 71

4. kritikuli wertilebis gamokvleva eqstremalurobis II rigis sakmarisi

pirobis safuZvelze ................................................. 72 savarjiSoebi .................................................................. 75

Tavi 8. gluvi eqstremaluri amocana

tolobis tipis SezRudvebiT ........................ 77 1. mokle reziume ............................................................ 77 2. maTematikuri modelis Sedgena ............................. 78 3. kritikuli wertilebis gansazRvra ..................... 81 4. kritikuli wertilebis gamokvleva

vaierStrasis Teoremisa da misi Sedegebis gamoyenebiT ................................................................. 90

5. kritikuli wertilebis gamokvleva uSualod ganmartebis safuZvelze ...................... 91

7

6. kritikuli wertilebis gamokvleva eqstremalurobis meore rigis sakmarisi pirobis safuZvelze .............................................. 92

7. lagranJis mamravlebis maTematikuri da ekonomikuri interpretaciebi ........................... 97

8. funqciis uswrafesi cvlilebis mimarTulebis gansazRvra ................................... 99

savarjiSoebi ................................................................ 101 Tavi 9. gluvi eqstremaluri amocana

tolobisa da utolobis tipis

SezRudvebiT ..................................................... 103 1. mokle reziume .......................................................... 103 2. maTematikuri modelis Sedgena ........................... 105 3. kritikuli wertilebis gansazRvra ................... 109 4. kritikuli wertilebis gamokvleva ................... 114 5. lagranJis meTodis geometriuli

interpretacia ......................................................... 116 savarjiSoebi ................................................................ 120

nawili 4. amozneqili miznis funqcia Tavi 10. gluvi amozneqili eqstremaluri

amocanebi simravliT .................................... 122 1. amozneqili simravleebi da funqciebi .............. 122 2. amozneqili amocanebis specifika ....................... 130 3. qunisa da Taqeris Teoremis Camoyalibeba

da ganxilva ............................................................... 132 savarjiSoebi ................................................................ 135

Tavi 11. gluvi amozneqili funqciebis

minimizaciis ricxviTi meTodebi .............. 137 1. gradientuli daSvebis meTodi ............................ 139 2. uswrafesi daSvebis meTodi .................................. 140 3. niutonis meTodi ..................................................... 143 4. gradientis proeqciis meTodi ............................. 146 savarjiSoebi ................................................................ 151

gamoyenebuli literatura ........................................... 153 pasuxebi ................................................................................ 154

n a w i l i 1

ZiriTadi cnebebi

9

Tavi 1. ZiriTadi aRniSvnebi da

ganmartebebi

yoveli konkretuli amocanis gamokvleva Sinaarsob-

rivad gansxvavebuli ramdenime etapisagan Sedgeba. esenia:

amocanis maTematikuri modelis Sedgena, miRebuli eqs-

tremaluri amocanisTvis amoxsnis (optimizaciis) meTo-

dis SerCeva, amocanis amoxsna, Sedegebis analizi da in-

terpretireba. sxva sityvebiT, yovel konkretul amoca-

naSi saWiroa miznisa da saSualebebis garkveva. magaliTad,

Tu amocanaSi saqme exeba xarjebs, maSin bunebrivi mizania

xarjebis minimizacia, Tu laparakia mogebaze, maSin moge-

bis maqsimizacia, Tu vangariSobT saukeTeso, anu umokles

manZils, maSin isev minimizacia da a. S. maTematikis enaze,

amocanis mizans warmoadgens raime funqcionali (e. i.

funqcia mniSvnelobebiT namdvil ricxvTa simravleSi),

romelsac miznis funqcia ewodeba da romlis eqstre-

maluri mniSvnelobebic unda moiZebnos:

(x) min,f an (x) max,f

an orive erTad, anu

(x)f extr ,

bunebrivia, rom miznis funqciis mniSvnelobebi nam-

dvili ricxvebia, radgan sxva SemTxvevaSi ver avarCevT

eqstremalur (ukidures) mniSvnelobas. magram rogori

unda iyos misi gansazRvris are? magaliTisTvis warmovid-

ginoT, rom miznis funqcia aris :f X R da mis Sesaxeb

mxolod is viciT, rom X raRac simravlea. am SemTxvevaSi,

f funqciis eqstremaluri mniSvnelobebis mosaZebnad ar

gagvaCnia araviTari meTodi garda funqciis mniSvnelo-

bebis gadarCevisa, rac araviTar Sedegs ar mogvcems, Tu

f funqciis saxe usasrulo raodenoba wertilebs Sei-

cavs; am dros sasruli raodenoba moqmedebebiT amonaxsns

10

ver vipoviT. SemTxveviT, erTxelac, eqstremaluri mniSv-

neloba rom Segvxvdes, amas mainc ver mivxvdebiT, radgan

Sesadarebeli mniSvnelobebis raodenoba usasruloa.

umetes SemTxvevaSi, praqtikuli Sinaarsis mqone

amocanebis maTematikuri modelis Seqmna ar amoiwureba

miznis funqciis SerCeviT. am dros bunebrivad Cndeba

SezRudvebi, romlebic gviCvenebs, rom eqstremumi unda

veZioT ara argumentis yvela mniSvnelobisaTvis, aramed

gansazRvris aris romeliRac konkretul qvesimravleze,

romelsac dasaSvebi simravle ewodeba. dasaSvebi simrav-

lis elementebs, Cveulebriv, dasaSvebi wertilebi ewo-

debaT. sailustraciod, ganvixiloT piradi moxmarebis

Teoriis umartivesi amocana, rodesac bazarze aris

erTaderTi produqti da erTaderTi myidveli (e.w.

momxmarebeli), produqtis erTeulis fasia p , xolo mom-

xmareblis Semosavali, anu mis xelT arsebuli Tanxa aris

I . mocemulia agreTve zrdadi funqcia (am funqciis sxva

Tvisebebi amjerad saintereso araa) ( )U x , romelic gamo-

xatavs x raodenoba produqtis sargeblianobas moce-

muli myidvelisaTvis. amasTan, SesaZlebelia produqtis

erTeulis danawileba nebismieri proporciiT (magali-

Tad, erTeuli SeiZleba iyos 1kg Saqari, an milioni wyvili

fexsacmeli). piradi moxmarebis Teoriis umartives amo-

canas warmoadgens : (0, )U sargeblianobis maqsimi-

zacia, imis gaTvaliswinebiT, rom unda Sesruldes 0x

(radgan x raodenobaa da uaryofiTi ver iqneba) da

agreTve biujeturi SezRudva px I , rac niSnavs rom

daxarjulma Tanxam ar unda gadaaWarbos Semosavals.

sabolood, miiReba amocana:

( ) max,U x im pirobiT, rom 0x da px I .

amocanas, romelSic mravali cvladis funqciis

eqstremumebs veZebT da cvladebi garkveul pirobebs

11

(SezRudvebs) akmayofileben, maTematikuri programi-

rebis amocana ewodeba. maTematikuri programirebis

amocanebi miznis funqciisa da SezRudvebis struqturis

mixedviT iyofa ramdenime mimarTulebad: wrfivi, ara-

wrfivi, dinamikuri, mTelricxva, geometriuli daprog-

ramebebi da kidev ramdenime sxva. maTematikuri prog-

ramirebis zemoCamoTvlil meTodebs Soris yvelaze gan-

viTarebuli da TiTqmis dasrulebuli saxe aqvs miRebuli

wrfivi programirebis meTodebs. wrfivi programirebis

amocanebs, maTi Seswavlis meTodebis specifikurobis

gamo, SeviswavliT operaciaTa gamokvlevis kursSi.

warmodgenil kursSi, misi auditoriis interesebis,

meTodologiuri faqtorebis da koniunqturis gaTva-

liswinebiT, Setanilia maTematikuri programirebis amo-

canebis amoxsnis SedarebiT gavrcelebuli meTodebi,

romelTa efeqturoba dadasturebulia praqtikiT. isini

sxvadasxva wyarodanaa aRebuli, magram gadamuSavebulia

da gamoyenebulia aRniSvnebis erTiani sistema. gasaTva-

liswinebelia erTi garemoebac. radgan maTematikuri

programireba SedarebiT axalgazrda disciplinaa, zog-

jer, erTi da igive cnebis gamosaxatavad gamoiyeneba

sxvadasxva termini. Cven vimuSavebT SedarebiT gavrcele-

buli terminebiT da aucileblobis SemTxvevaSi mivu-

TiTebT sinonimebs.

aRniSvnebi

nR _ n-ganzomilebiani namdvili veqtor-striqo-

nebis sivrce; nR _ n-ganzomilebiani namdvili veqtor-svetebis

sivrce;

, , ,x y z _ kaligrafiuli mcire asoebi aRniSnaven ska-

larul (ricxviT) sidideebs;

12

x, y, z _ beWduri mcire asoebi aRniSnaven veqtorebs;

1 2x ( , , , )n

nx x x R _ veqtor-striqonis standartuli

Canaweri; ix _ x veqtoris i -uri komponenti (koordi-

nati);

x , y , ,i i

k j i ix y _ skalaruli sidide i -ur xarisxSi;

x ( , )x y da x ( , , )x y z _ organzomilebiani da

samganzomilebiani veqtoris tradiciuli

Canawerebi, ZiriTadad gamoiyeneba magali-

Tebis amoxsnis procesSi;

locmin(i) , an ubralod locmin (locmax, locextr) lokaluri

minimumis (maqsimumis, eqstremumis) werti-

lebis simravle ( i ) amocanaSi;

glmin(i) , an ubralod glmin (glmax, glextr) _ globaluri

minimumis (maqsimumis, eqstremumis) werti-

lebis simravle ( i ) amocanaSi;

xinf

Mf

_ :f M funqciis infimumi, anu am fun-

qciis mniSvnelobaTa simravlis zusti qveda

zRvari.

x _ x veqtoris evkliduri norma:

1/ 2

2

1

x xn

i

i

;

(x)rB _ Ria birTvi: (x) y .rB y x r

( )rB x _ Caketili birTvi: (x) y .rB y x r

R _ arauaryofiT namdvil ricxvTa simravle.

13

ZiriTadi ganmartebebi

ganmarteba. Semdeg Canawers:

(x) min, x ,f M (1.1)

sadac : ,f X R X Ria simravlea nR -Si da nM X R ,

ewodeba minimizaciis sasrulganzomilebiani amocana, an

ubralod minimizaciis amocana; f -s ewodeba miznis

funqcia anu kriteriumi, M -s ewodeba dasaSvebi

simravle. (1.1)-is amoxsna niSnavs, rom unda movZebnoT

rogorc minimumis wertilebi anu minimalebi, aseve

miznis funqciis minimaluri mniSvnelobebic.

(1.1)-is amoxsna gulisxmobs ori sakiTxis garkvevas:

arsebobs Tu ara minimumis wertilebi (1.1) amocanaSi, da

Tu arsebobs, rogor movZebnoT isini.

minimumis wertili SeiZleba iyos ori saxis.

ganmarteba. dasaSveb x M wertils ewodeba

globaluri minimali (1.1) amocanaSi, an ( )f _ funqciis

globaluri minimumis wertili M -ze (iwereba

x glmin(1.1) ), Tu sruldeba:

ˆ(x) (x), x .f f M

magaliTad:

marcxena naxazze funqcias aqvs lokaluri minimumis

wertilic.

2( )f x x

ˆ 0,x M R x M

globaluri minimumi

14

zogierT amocanaSi globaluri minimali arc arse-

bobs. magaliTad, Tu 3( ) , .f x x M R

SeniSvna. Sinaarsobrivad, samarTliania Semdegi

tolobebi: ”globaluri” = “absoluturi”, “lokaluri” =

“fardobiTi”, “minimumis wertili” = “minimali”.

ganmarteba. dasaSveb x M wertils ewodeba

lokaluri minimali (1.1) amocanaSi da iwereba

x locmin(1.1) , Tu romeliRac 0 -isaTvis sruldeba:

ˆ ˆ(x) (x), x (x)f f M B .

sxva sityvebiT, x locmin(1.1) maSin da mxolod maSin,

rodesac x M da arsebobs iseTi 0 , rom

ˆx M x x da ˆ(x) (x)f f .

cxadia, x glmin(1.1) x locmin(1.1) (am dros

), xolo Sebrunebuli debuleba samarTliani ar

aris, rac Cans Semdeg naxazze:

zogjer, eqstremaluri amocanis formulirebaSi mo-

nawileobas iReben garkveuli parametrebi; magaliTad,

gaCerebis piroba eqstremumis wertilis Ziebis procesSi,

iteraciebis maqsimaluri raodenoba an sizuste, risi

miRwevac sasurvelia mocemul amocanaSi. aseT amocanebs

SevxvdebiT maTematikuri daprogramebis ricxviTi meTo-

debis ganxilvisas.

f(x)

15

zogjer saWiro xdeba mkacri eqstremumis werti-

lebis ganxilva. mkacri minimumis SemTxvevaSi, ganmarte-

bebi igivea rac zemoT, magram niSani icvleba <-iT da

aRiniSneba, rom utolobaSi x unda gansxvavdebodes x –

isagan. magaliTad:

ganmarteba. dasaSveb x M wertils ewodeba

mkacri lokaluri minimali (1.1) amocanaSi, Tu romeliRac

0 –isaTvis sruldeba:

ˆ(x) (x),f f Tu ˆx x da ˆx (x)M B .

maqsimizaciis

(x) max, x ,f M (1.2)

amocanisaTvis, sadac : ,f X R Ra X Ria simravlea

n nR M X R Si da , f –s kvlav miznis funqcia

ewodeba, M –s kvlav dasaSvebi simravle ewodeba da

(1.2)-is amoxsna niSnavs, rom unda movZebnoT rogorc

maqsimumis wertilebi anu maqsimalebi, aseve miznis

funqciis maqsimaluri mniSvnelobebic.

ganmarteba. Semdeg Canawers:

(x) extr, x ,f M (1.3)

sadac : ,f X R X Ria simravlea nR -Si,

nM X R da

extr min,max , ewodeba sasrulganzomilebiani eqstre-

maluri amocana, an ubralod eqstremaluri amocana,

romelSic f aris miznis funqcia, M aris dasaSvebi

simravle. miznis funqciis minimumebs da maqsimumebs

ewodebaT misi eqstremumebi, xolo (1.1) da (1.2) amocanebis

minimalebi da maqsimalebi aris eqstremumis wertilebi

(1.3) amocanisTvis. SevniSnoT, rom Cven ar vxmarobT

termins eqstremali. (1.2) amocanis amoxsna, Cveulebriv,

16

niSnavs eqstremumis wertilebisa da eqstremumebis

moZebnas.

gavrcelebulia mosazreba, rom minimizaciis (1.1)

amocana zogadobis TvalsazrisiT ar Camouvardeba (1.3)

amocanas, radgan maqsimizaciis (1.2) amocana igivea, rac

(x) min , x ,f M (1.4)

saxis minimizaciis amocana. magram aq saWiroa jerovani

sifrTxile, radgan SesaZloa minimizaciis meTodi samar-

Tliani iyos miznis funqciebis iseTi klasisaTvis,

romelic araa Caketili niSnis Secvlis mimarT. amis

magaliTebs Cven SevxvdebiT unimodaluri da amozneqili

funqciebis minimizaciis amocanebSi. amdenad, gamoTqma

“zogadobis SeuzRudavad ganvixiloT minimizaciis amo-

cana” niSnavs Semdegs: rodesac (1.3) ekvivalenturia

minimizaciis ori ((1.1) da (1.4)) amocanisa erTobliobaSi,

maSin, simartivisaTvis da garkveulobisaTvis, mizanSe-

wonilia mxolod minimizaciis amocanebis ganxilva.

gamoyenebiTi Sinaarsis mqone amocanebis umeteso-

bisaTvis nM R . x M -s ewodeba piroba, an SezRudva,

amitom roca nM R , amboben, rom gvaqvs pirobiani anu

SezRudvebiani amocana. Tu nM R , maSin x M piroba

avtomaturad sruldeba da Sesabamis eqstremalur

amocanas ewodeba upirobo anu amocana SezRudvebis

gareSe.

SezRudvaTa tipebi

ganvixiloT eqstremaluri amocana:

(x) extr, xf M . (1.5)

rac ufro martivi saxe aqvs x M SezRudvas, miT ufro

advili amosaxsnelia (1.5). umartives SemTxvevaSi, roca

17

nM R , gvaqvs amocana SezRudvebis gareSe. SemdegSi

Cven SevxvdebiT Semdegi saxis SezRudvebs:

1. tolobis tipis SezRudvebi, rodesac

1 1x (x) , , (x)m mM f b f b ,

sadac yoveli :i nf R R funqcia, rogorc wesi,

uwyveti maincaa, xolo 1, , mb b mocemuli mudmivebia.

tolobis tipis SezRudvebis SemTxvevaSi, (1.5)

eqstremaluri amocanis Cawera xelsayrelia Semdegi

saxiT: 1 1 1

1 1(x , , x ) extr, (x , , x ) , , (x , , x ) .n n n

m mf f b f b

rogorc wesi, imisaTvis, rom Sesruldes M ,

viRebT m n .

2. utolobis tipis SezRudvebi, rodesac

1 1x (x) , , (x)

m mM g b g b ,

sadac yoveli :i ng R R uwyveti funqciaa.

utolobis tipis SezRudvebis SemTxvevaSi (1.5)

amocana iRebs Semdeg saxes, romelsac ewodeba maTema-

tikuri programirebis amocana: 1 1 1

1 1(x , , x ) extr, (x , , x ) , , (x , , x ) .n n n

m mf g b g b

SevniSnoT, rom garkveul SemTxvevebSi sakmarisia

aviRoT 10, , 0.

mb b

3. kombinirebuli SemTxveva, rodesac erTdroulad

gvaqvs tolobisa da utolobis tipis SezRudvebi.

Tumca, Teoriulad 1. da 2. aris 3.-is kerZo SemTxveva,

radgan

(x) 0 ( ) 0 & ( ) 0f f x f x .

18

savarjiSoebi

SeadgineT eqstremaluri amocana, romelSic:

#1. globaluri minimalebisa da maqsimalebis raode-

noba usasruloa.

#2. miznis funqcia SemosazRvrulia, globaluri

maqsimumi miiRweva, minimumi ara.

#3. miznis funqcia SemosazRvrulia, magram globa-

luri minimumi da maqsimumi ar miiRweva.

#4. miznis funqcia SemosazRvrulia, zogierT wer-

tilSi misi warmoebuli nulis tolia, magram

globaluri minimumi da maqsimumi ar miiRweva.

#5. miznis funqcia SemosazRvrulia, aqvs lokaluri

minimumebi da maqsimumebi, magram globaluri

minimumi da maqsimumi ar miiRweva.

#6. aris erTaderTi lokaluri eqstremumi, romelic

araa globaluri.

#7. aris lokaluri maqsimumebis usasrulo raode-

noba, magram ar aris arc erTi globaluri mini-

mumi.

miTiTeba: miznis funqcia SesaZloa iyos ori cvladis

funqcia.

Semdegi savarjiSoebisTvis SeadgineT Sesabamisi eqs-

tremaluri amocanebi tolobis an utolobis tipis Sez-

RudvebiT.

#8. moZebneT 2 1x x funqciis maqsimumi [-1,1000]

segmentze.

#9. moZebneT 2 2 5xy x y y funqciis minimumi

erTeulradiusian wreze centriT (-1,1)-Si.

#10. moZebneT umoklesi manZili sibrtyis mocemuli

(1,2) wertilidan 2 3 1x y wrfemde.

19

#11. CaxazeT wrewirSi samkuTxedi ise, rom gverdebis

kvadratebis jami iyos minimaluri.

#12. moZebneT sibrtyis is wertili, saidanac sam

mocemul wertilebamde manZilebis jami minima-

luria.

#13. CaxazeT wrewirSi maqsimaluri farTobis mqone

marTkuTxedi.

#14. gayaviT ricxvi 8 or nawilad ise, rom maTi nam-

ravlis namravli maT sxvaobaze iyos maqsimaluri

(tartalias amocana).

#15. ipoveT udidesi farTobis mqone marTkuTxa sam-

kuTxedi, romlis kaTetebis jami tolia moce-

muli ricxvis (fermas amocana).

#16. l sigrZis wiriT SemozRudeT maqsimaluri far-

Tobis mqone marTkuTxedi.

#17. (1.2) amocanisTvis ganmarteT globaluri da

lokaluri maqsimalebi.

20

Tavi 2. globaluri eqstremumis

wertilebis arsebobis

vaierStrasis sakmarisi piroba

pirveli kiTxva, romelic ismeba nebismieri eqstre-

maluri amocanis amoxsnis procesSi, exeba misi amonaxsnis

arsebobas, radgan sakmaod xSirad saukeTeso amonaxsni ar

arsebobs, xolo amonaxsnis ararsebobis SemTxvevaSi misi

Zebna drois da energiis fuW xarjvas niSnavs.

sakiTxis didi mniSvnelobis gamo, amJamad ramdenime

efeqturi sakmarisi meTodi arsebobs amonaxsnis arse-

bobis saCveneblad. lokaluri minimalebis arsebobis

damtkiceba, rogorc wesi, xdeba diferencialuri aRric-

xvis saSualebebis gamoyenebiT (magaliTad, mkacri mini-

mumis an maqsimumis arsebobis meore rigis sakmarisi

pirobebi, romlebsac SemdegSi SeviswavliT da gamovi-

yenebT). globaluri eqstremumis wertilebis arsebobis

dasadgenad dResdReobiT isev Seucvleli aris vaier-

Strasis klasikuri Teorema da misi Sedegebi.

amgvarad, vidre daviwyebT eqstremumis wertilebis

Ziebas, gavarkvioT maTi arsebobis sakiTxi.

miznis funqciis wyvetiloba da dasaSvebi simravlis

arakompaqturoba aris ori ZiriTadi faqtori, rac xels

uSlis globaluri minimalis arsebobasac da arsebobis

faqtis dadgenasac.

Teorema /vaierStrasis/. vTqvaT, M aris Caketili

da SemosazRvruli (anu kompaqturi) simravle nR -Si,

xolo :f M R uwyvetia. maSin

(x) extr, xf M , (2.1)

amocanaSi arsebobs globaluri minimali da maqsimali.

damtkiceba. aviRoT xinf (x)

Mk f

da SevniSnoT, rom

k -sTvis dasaSvebia mniSvnelobac. infimumis ganmar-

21

tebis ZaliT arsebobs iseTi mimdevroba 1

xi i

M

, rom

lim (x )i

if k

. radgan M kompaqturia, am mimdevrobidan

gamoiyofa qvemimdevroba 1

xj j

i M

, krebadi M –is

romeliRac 0x wertilisaken, rac ( )f -is uwyvetobis

ZaliT gvaZlevs:

0lim (x ) (x ).

jjif f

(2.2)

amave dros, qvemimdevroba 1

(x )j j

if

igive zRvriskenaa

krebadi, saiTkenac 1

(x )iif

, anu

lim (x )jj

if k

. (2.3)

zRvris erTaderTobis gamo, (2.2) da (2.3) gvaZlevs

0(x )k f , rac k –s ganmartebis Tanaxmad niSnavs:

0 0(x ) (x), x x glmin(2.1)f k f M .

zogjer vaierStrasis

Teoremas sityvierad ase

ayalibeben: uwyveti fun-

qcia kompaqtur simrav-

leze aRwevs Tavis glo-

balur minimums da maqsi-

mums.

ganvixiloT ramdeni-

me sailustracio maga-

liTi.

magaliTi 1. [ 2,2]M , 3( )f x x .

am magaliTSi M kompaqtia, ( )f x funqcia uwyveti,

amitom globaluri maqsimali da minimali miiRweva.

2 0 2

22

vaierStrasis Teorema gvaZlevs globaluri minimalis

da maqsimalis arsebobis sakmaris pirobebs, rac niSnavs,

rom Tu mocemuli pirobebi sruldeba, globaluri mini-

mali da maqsimali arsebobs. magram am pirobebis Sesru-

leba ar aris aucilebeli globaluri minimalis da

maqsimalis arsebobisTvis. sxva sityvebiT, aucilebeli

pirobebi asruleben filtris funqcias, romelic gamo-

ricxavs argumentis apriori araoptimalur mniSvnelo-

bebs da tovebs mxolod

im wertilebs, romlebic

warmoadgens amonaxsns,

an Tavisi garkveuli

TvisebebiT hgavs mas.

magaliTi 2.

( 1,2]M , 2( )f x x .

am magaliTSi, ( )f x funqcia uwyvetia, xolo M ar

aris kompaqti, magram globaluri minimali da maqsimali

miiRweva.

magaliTi 3.

[ 2,2]M ,

2

2

, 2,0 ,( )

6, 0,2 .

x xf x

x x

am magaliTSi M kom-

paqtia, magram ( )f x ar

aris uwyveti. miuxedavad

amisa, globaluri mini-

mali da maq-simali mainc

miiRweva, rac naCvenebia Sesabamis naxazze.

rogorc vnaxeT, meore da mesame magaliTebSi ar

sruldeba vaierStrasis Teoremis erT-erTi piroba,

3 2 1 0 1 2 3

5

3 2 1 0 1 2 3

5

4

3

2

1

1

2

3

4

5

6

7

23

magram globaluri minimali da maqsimali miiRweva. Semdeg

or magaliTSi (4 da 5) vaierStrasis Teoremis erT-erTi

pirobis darRveva iwvevs imas, rom globaluri minimali,

an maqsimali, an orive erTad, ar miiRweva.

magaliTi 4. [ 1,1]M ,

1/ , 0,( )

0, 0.

x xf x

x

am magaliTSi M kom-

paqtia, ( )f x funqcia ar

aris uwyveti. globaluri

maqsimali da minimali ar

miiRweva.

magaliTi 5.

( 1,1]M , ( )f x x .

M ar aris kompaq-

ti, ( )f x funqcia

uwyvetia. globalu-

ri maqsimali miiR-

weva, globaluri

minimali _ ara.

CamovayaliboT

vaierStrasis Teoremis erTi mniSvnelovani Sedegi,

romlebSic Sesustebulia SezRudva dasaSveb simravleze,

samagierod, damatebulia SezRudva miznis funqciaze. mas

CamovayalibebT orive tipis eqstremumisaTvis, amasTan,

ZiriTadi versia yalibdeba minimalisaTvis, maqsimalis

SemTxveva ki moyvanilia frCxilebSi.

Sedegi. vTqvaT, M aris Caketili simravle nR -Si,

xolo :f M R uwyvetia da

1 0 1

1

x

x

2 1 0 1 2

2

1

1

2

example 5 1

24

x , x ,lim (x) lim (x)

x xM Mf f

.

maSin (2.1) amocanaSi arsebobs globaluri minimali

(maqsimali).

damtkiceba. ganvixiloT minimumis SemTxveva; meore

analogiuria.

Tavidanve aRvniSnoT, rom xlim (x)f

niSnavs

Semdegs: yoveli 0 ricxvisaTvis arsebobs ricxvi

0r , iseTi rom

x , x (x)M r f . (2.4)

aviRoT: romelime 0x M da ricxvi 0

(x )f . axla

romelime 0r -isaTvis sruldeba (2.4). SegviZlia aviRoT

0r imdenad didi, rom 0x r . radgan (0)rM B

kompaqturi simravlea, amitom vaierStrasis Teoremis

ZaliT arsebobs x (0)rM B , iseTi rom

(x) (x), x (0).rf f M B (2.5)

kerZod, 0ˆ(x) (x )f f . garda amisa, (2.4) –is ZaliT

( ), ,Tuf x x M x r . (2.6)

(2.5) da (2.6) erTad gvaZlevs, rom

ˆ(x) (x), x .f f M

erTi cvladis SemTxvevaSi, rodesac 1n , piroba

lim (x) ( )x

f

niSnavs, rom lim (x) ( )x

f

da

lim (x) ( )x

f

.

kvlav ganvixiloT magaliTebi.

magaliTi 6. M R , 3( )f x x .

25

am magaliTSi M Caketi-

lia, ( )f x funqcia uwyvetia.

am SemTxvevaSi ar srul-

deba arc x ,

lim (x)xM

f

,

arc x ,

lim (x)xM

f

,

globaluri minimali da

maqsimali ar arsebobs.

magaliTi 7. M R , 4( )f x x .

M Caketilia, ( )f x

funqcia uwyvetia, srul-

deba x ,

lim (x)xM

f

piroba da amitom arse-

bobs globaluri mini-

mali.

magaliTi 8. 2M R , 2 2x ( , )f f x y x y .

am magaliTSi M aris mTeli sibrtye, e.i. Caketili

simravle, ( )f x funqcia uwyvetia. am SemTxvevaSic

sruldeba x , x

lim (x)M

f

piroba da amitom arsebobs

globaluri minimali, rac naCvenebia Semdeg naxazze.

2 0 2

2

2

10 0 10

0

50

100

x4

x

26

f

bunebrivad Cndeba kiTxva, ra xdeba im SemTxvevaSi,

rodesac arsebobs lim (x)x

f

, magram igi sasruli ricxvia.

es SedarebiT egzotikuri SemTxvevaa, amitom Zalian

mokled SegviZlia vTqvaT, rom rodesac sruldeba

vaierStrasis Teoremis Sedegis analogiuri pirobebi, anu

rodesac M aris Caketili simravle nR -Si, :f M

uwyvetia,

x ,lim (x)

xMf k

,

raime k R –sTvis, da damatebiT cnobilia agreTve, rom

0 0(x ) (x )f k f k romeliRac 0

x M -isaTvis, maSin

(2.1) amocanaSi arsebobs globa-

luri minimali (maqsimali).

magaliTi 9. M R , 2

( ) xf x e .

M Caketilia, ( )f x funqcia

uwyvetia, lim ( ) 0,x

f x

da

2 0 2

2

27

(0) 1 0 lim ( ).x

f f x

amitom arsebobs globaluri

minimali, magram ar arsebobs globaluri maqsimali.

magaliTi 10. M R , 2

( ) xf x xe .

M Caketilia, ( )f x funqcia uwyvetia, lim ( ) 0,x

f x

da (0) 0f . amasTan ar-

sebobs iseTi 0x M , rom

0(x ) 0f da arsebobs

iseTi 0x M , rom

0(x ) 0f , amitom ar-

sebobs globaluri mini-malic da globaluri maqsimalic.

savarjiSoebi

amocanebSi #18 – #28 vaierStrasis Teoremisa da misi Sedegebis safuZvelze daadgineT globaluri minima-lebisa da maqsimalebis arsebobis sakiTxi.

#18. extr, .x x R

#19. 22 3 5 extr, .x x x R

#20. 2 1 2 1 extr, .k kx x x R

#21. 2 2 1 1 extr, .k kx x x x R

#22. extr, .x x R

#23. extr, 4 1.xye x y

#24. 2 24 3 extr, 1.x y x y

#25. 2 2 2 8 8 8extr, 1.x y z x y z

#26. 2 25 4 extr, 1.x xy y x y

0

1

1

xex

2

x

28

#27. 2 2 3 2 min.x y xy x y

#28. 2 2 2 max{ , } min.x y x y

#29. amocanebSi #8 – #15 formalizebuli eqstremaluri amocanebisaTvis gamoikvlieT globaluri minimale-bisa da maqsimalebis arsebobis sakiTxi vaierStrasis

Teoremisa da misi Sedegebis safuZvelze.

#30. SeadgineT iseTi eqstremaluri amocana, romelSic ar sruldeba vaierStrasis Teoremis arc erTi pi-roba, magram globaluri minimali da maqsimali

arsebobs.

#31. aCveneT, rom wrfivi miznis funqciis mqone Semdeg minimizaciis amocanaSi

( ) min, [ , ] ,f x kx x a b R

roca 0k , arsebobs erTaderTi globaluri mini-

mali da igi aris a an b.

nimuSi. gavarCioT amocana: 2 2( , ) 2max{ , } min .f x y x y x y

radgan

1/ 22 2 2 2

2

( , ) 2

, 2 , , , 2 ,

f x y x y x y

x y x y x y x y

amitom marjvena mxare miiswrafis plius usasrulobi-

saken, roca ,x y . vaierStrasis Teoremis Sedegis

ZaliT arsebobs globaluri minimali.

n a w i l i 2

erT cvladze

damokidebuli miznis

funqcia

30

Tavi 3. erTi cvladis funqciis

lokaluri eqstremumebis Zebna

eqstremaluri amocanebi, romlebSic miznis funqcia

erT cvladzea damokidebuli, miekuTvneba eqstremaluri

amocanebis yvelaze martiv (advilad Sesaswavl) klass.

isini xSirad gvxvdeba, rogorc damoukidebeli amocanebi,

praqtikuli amocanebis amoxsnis dros, da rogorc dam-

xmare amocana, mravali cvladis funqciis eqstremumis

miaxloebiTi mniSvnelobis povnis procesSi. amdenad, maTi

Seswavla aqtualuria. lokaluri eqstremumis wertile-

bis moZebnisTvis gamoiyeneba diferencialuri aRricxvis

elementebi, rac bunebrivad zRudavs eqstremaluri

amocanis zogadobas: miznis funqcia unda iyos gluvi,

xolo dasaSvebi simravle Ria, naxevrad Ria an Caketili.

gavrcelebuli terminologiiT, eqstremalur amoca-

nas ewodeba gluvi, Tu miznis funqcia da dasaSvebi

simravlis ganmsazRvravi funqciebi uwyvetad warmoebadi

aris saWiro rigiT. amocanis sigluve amocanis ganzo-

milebisgan damoukidebuli cnebaa. am TavSi yvelgan,

sadac sawinaaRmdego araa aRniSnuli, vigulisxmebT, rom

miznis funqcia erTi cvladis funqcias warmoadgens,

xolo dasaSvebi simravle aris Ria.

erTi cvladis miznis funqciisTvis, gluvi eqstre-

maluri amocana Ria dasaSveb simravleze ismeba Semdeg-

nairad:

( ) extr,f x x M R , (3.1)

sadac M Ria simravlea, anu warmoidgineba Ria interva-

lebis sasruli an Tvladi gaerTianebis saxiT.

31

1. mokle reziume

gluvi eqstremaluri amocanis gamokvleva unda

moxdes Semdegi TanmimdevrobiT:

1. maTematikuri modelis Sedgena, anu amocanis

formulireba (3.1) saxiT;

2. (3.1) eqstremalur amocanaSi kritikuli werti-

lebis gansazRvris mizniT, 0f x gantolebis

amoxsna; 3. kritikuli wertilebis gamokvleva, maTgan eqs-

tremumis wertilebis amorCevis mizniT, vaier-

Strasis Teoremis Sedegebis, ganmartebis, an eqs-

tremalurobis sakmarisi pirobebis safuZvelze.

kritikuli wertilebis simravlis (rasac Cven Cveulebriv

aRvniSnavT K simboloTi) gansazRvra Zalian mniSvnelo-

vania, radgan xSirad es simravle aris sasruli, an am

simravleze miznis funqciis mniSvnelobebis raodenobaa

sasruli. nebismier SemTxvevaSi, Tu viciT kritikuli

wertilebis K simravle, maSin

( ) extr,f x x M R ,

amocana ekvivalenturia

( ) extr, ,f x x K

amocanis, rac (rogorc ukve aRvniSneT), umetes SemTxve-

vaSi gacilebiT advili amosaxsnelia.

2. maTematikuri modelis Sedgena

nimuSad ganvixiloT maragTa marTvis amocanis maTe-

matikuri modeli.

maragTa marTvis amocana ismeba im SemTxvevaSi, roca

aucilebelia Seiqmnas garkveuli produqciis maragi,

drois mocemul intervalze moTxovnis dakmayofilebis

mizniT. maragis uwyveti Sevseba dakavSirebulia drois da

resursebis araracionalurad gamoyenebasTan. didi mara-

32

gis Seqmna (rac maragis periodulad Sevsebas moiTxovs) ki

dakavSirebulia did kapitaldabandebasTan, rac zrdis

misi Senaxvis xarjebs. Cveni mizania, SevarCioT maragis

optimaluri moculoba.

ganvixiloT yvelaze martivi SemTxveva. vTqvaT, moT-

xovna mudmivia da udris erT./wel. erTi SekveTis (misi

sididis miuxedavad) Sesrulebis fasi aris K lari.

saqonlis erTeulis sawyisi fasia c lari. erTeuli

saqonlis Senaxvis fasia h lari/wel. vigulisxmoT, rom

moTxovna kmayofildeba dauyovnebliv da deficiti

akrZalulia (dakavSirebulia did jarimebTan), maSin

maragis donis droSi cvlilebis grafiks aqvs saxe:

vTqvaT, AB gverdis sigrZe warmoadgens A wertilSi

maragis moculobas. igi mcirdeba siCqariT erT./wel.

da iwureba C wertilSi. am dros Semodis saqonlis axali

Sevseba, maragis done aRdgeba da igi isev AB -s tolia.

ABC samkuTxedi, faqtobrivad, warmoadgens maragTa

marTvis erT cikls, romelic meordeba. Cveni mizania,

ganvsazRvroT SekveTis optimaluri moculoba da mara-

gis Sevsebis A da C wertilebs Soris drois optimaluri

intervali. Sesabamisi cvladebi aRvniSnoT Q -Ti da T -Ti.

B

A C dro

maragis moculoba

nax. 3.1

33

radgan T droSi Q maragi icleba siCqariT, amitom

/T Q .

unda aRiniSnos, rom roca Q mcirea, maSin T iRebs

mcire mniSvnelobebs . es niSnavs, rom SekveTebis sixSire

izrdeba, rac iwvevs SekveTebis Sesrulebaze xarjebis

zrdas da Senaxvis xarjebis Semcirebas. meore mxriv, roca

Q didia, T izrdeba, rac iwvevs SekveTebis sixSiris

Semcirebas e.i. mcirdeba SekveTebis Sesrulebaze gaweuli

xarjebi da izrdeba Senaxvis xarjebi. unda vipovoT Q -s

optimaluri mniSvneloba, romelsac Seesabameba minima-

luri wliuri xarji.

erTi ciklis ganmavlobaSi maragis Senaxvaze gaweuli

xarji aris T drois ganmavlobaSi saqonlis 2

Q erTe-

ulis Senaxvaze gaweuli xarjis toli da maTematikurad

samkuTxedis farTobis da h -is namravlis tolia.

erT ciklSi gaweuli xarji aris SekveTebis Sesrule-

baze gaweuli xarjisa da Senaxvaze gaweuli xarjis jami

2

2 2

Q hQK cQ hT K cQ

.

amrigad, wliuri danaxarji iqneba erT ciklze gawe-

uli danaxarjebisa da weliwadSi ciklebis raodenobis

namravlis toli (Q

wels Seesabameba 1 cikli, 1 wels ki

/Q ). amrigad, Semdeg eqstremalur amocanaSi

( ) extr,f Q Q M ,

miznis funqciaa ( )2

K hQf Q c

Q

, xolo (0, )M .

34

3. kritikuli wertilebis gansazRvra

Teorema 1 /eqstremalurobis I rigis aucilebeli

piroba/. vTqvaT, M R aris Ria simravle, :f M R

uwyvetad warmoebadi funqciaa x M wertilis romeli-

Rac midamoSi da x aris lokaluri eqstremumis wertili

( ) extr,f x x M (3.1)

amocanaSi. maSin, ˆ 0f x .

damtkiceba. ganvixiloT lokaluri minimumis Sem-

Txveva. vTqvaT, x aris lokaluri minimumis wertili da

( )f x funqcia x wertilis midamoSi uwyvetad warmoe-

badia. maSin SegviZlia gavSaloT ( )f x funqcia teiloris

mwkrivad am wertilis midamoSi:

2ˆ

ˆ ˆ( ) ( ) ( )df x

f x f x odx

, (3.2)

sadac 2( )o -iT aRniSnulia im wevrebis Gjami, sadac -is

xarisxi 1-ze metia. Tu x aris lokaluri minimumis

wertili M -Si, maSin gansazRvrebis Tanaxmad, arsebobs

x -is -midamo, iseTi, rom am midamos yoveli x -werti-

lisTvis sruldeba:

ˆ( ) ( )f x f x . (3.3)

(3.2)-dan da (3.3)-dan gamomdinareobs, rom

2ˆ

( ) 0df x

odx

.

sakmarisad mcire -Tvis marcxena mxaris niSans gan-

sazRvravs pirveli Sesakrebi, da radgan -ma SeiZleba

miiRos rogorc dadebiTi, ise uaryofiTi mniSvnelobebi,

utoloba Sesruldeba mxolod im SemTxvevaSi, roca

35

ˆ0

df x

dx .

analogiuria damtkiceba lokaluri maqsimumis Sem-

TxvevaSi.

zemoT moyvanili Teorema ekuTvnis p. fermas. igi

iZleva lokaluri minimalebis da maqsimalebis arsebobis

aucilebel pirobebs, anu, SesaZlebelia Teoremis

pirobebi Sesruldes,

magram funqcias ar

gaaCndes arc maq-

simali, arc minimali.

am SemTxvevis klasi-

kuri magaliTia miz-

nis funqcia 3(x)f x ,

romlisTvisac srul-

deba piroba 0 0f ,

magram 0x ar aris eqs-

tremumis wertili. igi

aris gadaRunvis wer-

tili.

SeniSvna. Cven am Tav-

Si vixilavT gluv fun-

qciebs, magram SesaZle-

belia funqcia ar iyos

warmoebadi da hqondes lokaluri minimali an maqsimali.

magaliTad, (x)f x . am funqcias 0x wertilSi warmoe-

buli ar aqvs, Tumca es wertili aris misi lokaluri

minimali.

ganmarteba. ganxiluli Teoremis pirobebSi,

( ) extr,f x x M ,

3 0 3x3

x

3 0 3

2

4

x

x

36

amocanisTvis kritikuli wertilebis simravle K ganisaz-

Rvreba ase:

ˆ ˆ 0K x M f x .

magaliTi. ipoveT 4 2( ) 8f x x x funqciis eqs-

tremumis wertilebi. amoxsna. CavweroT amocana standartuli saxiT:

4 2( ) 8 extr,f x x x x R . (3.4)

(3.4) amocanisaTvis ganvsa-

zRvroT kritikuli werti-

lebis simravle. amisTvis

amovxsnaT gantoleba

0df

dx , anu 34 2 0x x ,

saidanac vpoulobT:

0x , anu 0 .K Semdeg

punqtebSi avxsniT, Tu rogor xdeba kritikuli werti-lebis gamokvleva da davasrulebT am magaliTsac.

4. kritikuli wertilebis gamokvleva

vaierStrasis Teoremis Sedegis gamoyenebiT

vaierStrasis Teoremis Sedegis gamoyeneba SegviZlia

mxolod im SemTxvevaSi, roca dasaSvebi simravle aris

Caketili. Tu M R , igi erTdroulad Riac aris da

Caketilic.

vaierStrasis Teoremis Sedegebis gamoyenebiT Tu

davadgenT, rom

( ) extr,f x x R , (3.5)

amocanaSi glextr(3.5) , maSin SegviZlia SemovifargloT

gacilebiT martivi

( ) extr,f x x K , (3.6)

5 0 5

10

20

30

37

amocanis ganxilviT, romelsac aqvs eqstremumis igive

wertili, rac (3.5) –s, radgan

ˆ ˆ ˆglextr(3.5) glextr(3.6)x K x x da .

analogiurad vmoqmedebT globaluri maqsimalis Ziebis

SemTxvevaSic.

magaliTi. gavagrZeloT (3.4) magaliTis amoxsna: 4 2( ) 8 extr,f x x x x R .

rogorc vnaxeT, 0 .K radgan

( )limx

f x

, amitom glextr(3.4) .

radgan K erTelementiania, amitom 0x aris globaluri

minimali

( ) extr,f x x K ,

amocanaSi. e.i. 0 glmin(3.4) .

5. kritikuli wertilebis gamokvleva

ganmartebis safuZvelze

kvlav ganvixiloT (3.5). vTqvaT, sruldeba Teorema

1-is pirobebi da x K . yoveli x R warmovadginoT Sem-

degi saxiT: ˆ hx x da ganvixiloT sxvaoba:

ˆ ˆ ˆh .f f x f x f x f x

f -is niSnis h -ze damokidebulebis garkveva arsebiTia,

radgan:

1) Tu 0f yoveli SesaZlo h–isTvis, maSin

ˆ glmin(3.5)x ;

2) Tu 0f yoveli SesaZlo h–isTvis, maSin

ˆ glmax(3.5)x ;

3) Tu 0f yoveli sakmaod mcire h–isTvis, maSin

ˆ locmin(3.5)x ;

38

4) Tu 0f yoveli sakmaod mcire h–isTvis, maSin

ˆ locmax(3.5)x .

magaliTi. kvlav gavagrZeloT 4 2( ) 8 extr,f x x x x R

eqstremaluri amocanis ganxilva. aq 0K aris erTa-

derTi kritikuli wertili. Cvens SemTxvevaSi ˆ hx x

warmodgena aris 0x h , amitom ganvixiloT sxvaoba:

0f h f 4 2 8 8h h 4 2 0h h ,

rac x -is nebismierobis ZaliT niSnavs, rom ˆ 0x aris

globaluri minimali.

6. kritikuli wertilebis gamokvleva

eqstremalurobis II rigis sakmarisi pirobis

safuZvelze

Teorema 2 . vTqvaT, x R wertilSi :f M R

funqciis pirveli rigis warmoebuli nulis tolia, xolo

II rigis warmoebuli gansxvavebulia nulisagan. maSin,

a. Tu 2

2

ˆ0

d f x

dx , x aris lokaluri minimali.

b. Tu 2

2

ˆ0

d f x

dx , x aris lokaluri maqsimali.

damtkiceba. gavSaloT ( )f x funqcia teiloris

mwkrivad x wertilis midamoSi. radgan pirveli rigis

warmoebulebi nulis tolia, amitom miviRebT:

22

3

2

ˆˆ ˆ( ) ( ) ( )

2

d f xf x f x o

dx

(3.7)

(3.7)-is marjvena mxaris niSans, sakmarisad mcire -isTvis,

gansazRvravs misi pirveli Sesakrebi. TuU 2

2

ˆ0

d f x

dx ,

39

(3.7)-is marjvena mxaris pirveli Sesakrebi dadebiTia, e. i.

ˆ ˆ( ) ( )f x f x sxvaobac dadebiTia da x wertili aris

lokaluri minimali. TuU 2

2

ˆ0

d f x

dx , (3.7)-is marjvena

mxaris pirveli Sesakrebi uaryofiTia da ˆ ˆ( ) ( )f x f x

sxvaobac uaryofiTia da x wertili aris lokaluri

maqsimali.

magaliTi. ganxiluli 4 2( ) 8 extr,f x x x x R ,

magaliTisTvis davadgineT, rom kritikuli wertilia

0x . amovweroT sakmarisi pirobebi.

3x 4 2f x x da 0 0f ,

x 12 2f x da 0 2f ,

0 2f dadebiTia, e.i. 0x warmoadgens lokalur

minimals.

7. gluvi eqstremaluri amocana segmentis

tipis dasaSvebi simravliT

ganvixiloT amocana:

( ) extr, [ , ]f x x a b . (3.8)

am amocanis gamokvleva xdeba Semdegi sqemis mixedviT:

1) maTematikuri modelis Sedgena;

2) kritikuli wertilebis gansazRvra:

( , ) 0 ,K x a b f x a b ;

3) kritikuli wertilebis gamokvleva:

Tu ˆ ( , )x K a b , maSin gamokvleva unda Catardes

wina paragrafSi ganxiluli meTodebiT, xolo seg-

mentis boloebisTvis, warmoebulis ganmartebis

Tanaxmad, gvaqvs:

40

0 locmin(3.8)f a a ,

0 loc max(3.8)f a a ,

0 locmax(3.8)f b b ,

0 locmin(3.8)f b b .

savarjiSoebi

amoxseniT Semdegi eqstremaluri amocanebi:

#32. 3 6 extr.x x

#33. 3 218 27 extr.x x

#34. 2

1 extr.x x

#35. ( 1) extr.x xe

#36.

35 45 2 extr.

3

xx x

#37. 23 4 8 extr.x x

#38. 22 12 extr.x x

#39. 2

2 1 4 extr.x x

#40. 3 2 3 6 extr.x x x

#41. 3 26 31 extr.x x

#42. 2 3( 2) ( 5) extr.x x

#43. 3( 1) extr.x x

#44. 3 22 1 extr.x x x

#45. 2( 1) extr, x [0,5]x x

#46. 2(2 1) ( 4) extr, x [-1,1]x x

41

Tavi 4. unimodaluri funqciebis

minimizacia intervalTa

gamoricxvis meTodebiT

am TavSi ganvixilavT unimodaluri funqciebis mi-

nimizaciis e.w. pirdapir meTodebs, romlebic iyeneben

mxolod garkveul wertilebSi gamoTvlili funqciis

mniSvnelobebs. am meTodebisTvis funqciis uwyvetobac

araa aucilebeli.

1. ganmartebebi da intervalTa gamoricxvis

wesi

ganmarteba. ( )f x funqcias ewodeba unimodaluri

[ , ]a b segmentze, Tu igi uwyvetia [ , ]a b -ze da arsebobs

ricxvebi da , a b , iseTebi, rom:

1) Tu a , maSin , ]a a segmentze ( )f x mkacrad

klebadia;

2) Tu b , maSin [ , ]bb segmentze ( )f x mkacrad

zrdadia;

3) Tu ˆ [ , ]x , maSin, ˆ( ) min ( )a x b

f x f x

.

SevniSnoT, rom Semdegi sami segmentidan: ,a ,

[ , ] , [ , ]b , erTi an ori SeiZleba gadagvardes wer-

tilSi. nebismier SemTxvevaSi, unimodaluri funqciis

ganmartebidan gamomdinare, misi lokaluri minimalebis

simravle emTxveva globaluri minimalebis simravles da

is aris an erTi wertili ( SemTxvevaSi), an mTeli

[ , ] segmenti. qvemoT moyvanilia unimodaluri fun-

qciis ramdenime magaliTi:

42

da erTi araunimodaluri funqciis magaliTi:

arsebobs unimodaluri funqciebis sxva ganmarte-

bebic, romlebic gansxvavdeba aq moyvanilisgan (magali-

Tad [3]). kerZod, xSirad ixsneba uwyvetobis moTxovna.

[ , ]a b -ze unimodaluri funqciebis simravle aRvniS-

noT [ , ]Q a b -Ti. funqciis [ , ]Q a b -sadmi mikuTvnebis da-

sadgenad, xSirad sasargebloa Semdegi ori kriteriumis

gamoyeneba:

a b α

a b

a b

a b a b

43

a) Tu ( )f x funqcia warmoebadia [ , ]a b –ze da '( )f x ar

aris klebadi funqcia, maSin ( ) [ , ]f x Q a b .

b) Tu ( )f x funqcia orjer warmoebadia [ , ]a b –ze da

''( )f x , [ , ]x a b , maSin ( ) [ , ]f x Q a b .

unimodaluri funqciis minimizaciis ricxviTi meTo-

debi eyrdnobian unimodaluri funqciis Tvisebas, rom

minimalis marjvniv funqcia zrdadia, xolo minimalis

marcxniv _ klebadi. amitom, or gansxvavebul wertilSi

aseTi funqciis mniSvnelobebis SedarebiT SegviZlia

davadginoT, am wertilebiT da segmentis boloebiT Sed-

genil romel intervalSi ar aris moTavsebuli minimali,

gamovricxoT igi da amiT SevamciroT Ziebis intervali.

rodesac unimodaluri funqcia ar aris warmoebadi,

SegviZlia gamoviyenoT Ymxolod pirdapiri meTodebi,

romlebic saWiroeben miznis funqciis mniSvnelobebis

gamoTvlas specialurad SerCeul wertilebSi da ar

saWiroeben warmoebulis mniSvnelobebis gamoTvlas.

praqtikidan wamosul eqstremalur amocanebSi sak-

maod xSiria SemTxveva, rodesac funqciis mniSvnelobis

gamoTvla wertilSi niSnavs garkveuli fizikuri eqspe-

rimentis Catarebas (magaliTad, temperaturuli velis

gazomvas). roca mosaxerxebelia funqciis mniSvnelobebis

gamoTvla N wertilSi erTdroulad da maTi mniSvnelo-

bebis gamoTvla mimdevrobiT dakavSirebulia garkveul

siZneleebTan, gamoiyeneba pasiuri Ziebis meTodi. im

SemTxvevaSi, roca SesaZlebelia wina wertilebSi fun-

qciis mniSvnelobebis gamoyenebiT momdevno wertilebis

arCeva, maSin minimalis Zieba, cxadia, ufro efeqturi

iqneba. aseTi tipis meTodebs mimdevrobiTi meTodebi

ewodeba.

pirdapiri meTodebi bevria, amitom mniSvnelovania

gvesmodes, Tu ras niSnavs erTi pirdapiri meTodis upi-

44

ratesoba meoreze. sakiTxi ismeba ase: Tu gvaqvs saSualeba

mxolod N -jer gamovTvaloT [ , ]a b -ze unimodaluri f

funqciis mniSvneloba, maSin ukeTesia is algoriTmi, ro-

melic am pirobebSi mogvcems minimumis wertilis ukeTes

miaxloebas, anu minimalis Semcvel umcires intervals,

anu umcires maqsimalur cdomilebas. SeiZleba sakiTxi

asec davsvaT: ukeTesia is algoriTmi, romlisTvisac

mocemuli cdomilebiT minimalis miaxloebiTi mniSvne-

lobis sapovnelad, dagvWirdeba miznis funqciis mniSvne-

lobebis naklebi raodenobis gamoTvla.

umartivesi pirdapiri meTodi aris pasiuri Ziebis

meTodi, romlis mixedviTac [ , ]a b iyofa N tol nawilad

wertilebiT ( )

, 1, , 1i

b ax a i i N

N

. gamoiTvleba

miznis funqciis mniSvnelobebi maTSi da segmentis bolo-ebSi da sadac funqciis mniSvneloba iqneba minimaluri, is wertili gamocxaddeba minimalis saukeTeso miaxloebad.

am dros maqsimaluri cdomileba ar aRemateba b a

N

sidides, raSic gvarwmunebs Semdegi Teorema, romelsac ewodeba intervalTa gamoricxvis wesi.

Teorema /intervalTa gamoricxvis wesi/. vTqvaT, minimizaciis amocanaSi:

( ) minf x , a x b , (4.1)

[ , ]f Q a b da 1 2a x x b . maSin:

1. Tu 1 2( ) ( )f x f x , maSin 1glmin(4.1) [ , ]x b ,

Tu 1 2( ) ( )f x f x , maSin 2glmin(4.1) [ , ]a x .

2. Tu 1 2( ) ( )f x f x , maSin 1 2gl min(4.1) [ , ]x x .

3. Tu 1 2( ) ( )f x f x , maSin 2glmin(4.1) [ , ]a x ,

Tu 1 2( ) ( )f x f x , maSin 1glmin(4.1) [ , ]x b .

45

damtkiceba: ganvixiloT 1. vTqvaT, 1 2a x x b da

1 2( ) ( )f x f x , magram ˆ glmin(4.1)x ekuTvnis 1,a x –s.

maSin, unimodaluri funqciis ganmartebidan gamomdinare,

nebismieri minimalis marjvniv miznis funqcia zrdadia

(mkacrad zrdadobas ar vgulisxmobT). amitom unda

gvqondes 1 2( ) ( )f x f x , rac ewinaaRmdegeba 1 2( ) ( )f x f x

pirobas. e. i. nebismieri minimali moTavsebulia 1x -is

marjvniv. 1-is meore nawili analogiurad mtkicdeba.

ganvixiloT 2. 1 2( ) ( )f x f x niSnavs, rom an 1 2, [ , ]x x ,

an 1 [ , ]x a da 2 [ , ]x b . vTqvaT ˆ glmin(4.1)x iseTia,

rom 1x x , maSin, radgan minimalis marjvniv miznis fun-

qcia zrdadia, amitom 1 2, [ , ]x x (ix. unimodaluri fun-

qciis ganmarteba), amitom

1 2 1 2ˆ( ) ( ) ( ) , glmin(4.1)f x f x f x x x .

2x x SemTxveva analogiuria.

bolos, 3 warmoadgens 1-is da 2-is Sedegs.

intervalTa gamoricxvis wesi safuZvlad udevs mini-

mizaciis mimdevrobiT meTodebs, romlebSic funqciis

mniSvnelobebis gamoTvla xdeba ukve gamoTvlili mniS-

vnelobebis safuZvelze. aRvweroT ori mimdevrobiTi

meTodi, romlebic gamoirCevian efeqturobiT.

2. intervalis sigrZis ganaxevrebis meTodi

es meTodi xasiaTdeba imiT, rom yovel iteraciaze

(garda pirvelisa) gvWirdeba funqciis mniSvnelobis ga-

moTvla or wertilSi da es ori mniSvneloba saSualebas

gvaZlevs orjer SevamciroT Ziebis intervalis sigrZe,

romelSic moTavsebulia erTi globaluri minimali mainc,

romelsac x -iT aRvniSnavT. funqciis mniSvnelobebis

46

N -jer gamoTvlis Semdeg Cven gvrCeba 1

1 2

2( )

N

b a

sigr-

Zis monakveTi, romelSic moTavsebulia erTi mainc mini-

mali da gamoTvlilia agreTve funqciis mniSvneloba bo-

lo intervalis Sua mx wertilSi, romelsac viRebT x -is

saukeTeso miaxloebad. am meTodisaTvis maqsimaluri

cdomileba ar aRemateba

1

21 1( )

2 2

N

b a

sidides, rac

bevrad ukeTesi Sefasebaa, vidre gvqonda pasiuri Ziebis

meTodisTvis.

vTqvaT, :[ , ]f a b unimodaluria. CamovayaliboT

e.w. intervalis sigrZis ganaxevrebis algoriTmi, romlis

saSualebiTac miaxloebiT ganvsazRvravT ( ) minf x ,

,x a b , minimizaciis amocanis globalur minimals. mxed-

velobaSi unda viqonioT, rom arsebobs agreTve monak-

veTis Suaze gayofis algoriTmi, romelic aq ganxilul-

Tan SedarebiT naklebefeqturia.

meTodis fsevdokodi aseTia:

intervalis sigrZis ganaxevrebis meTodi (a,b,ε, f x )

1 2

m

a bx

; L b a ; gamovTvaloT mf x ;

2 while(1) 3 {

4

4l

Lx a ;

4m

Lx b ; gamovTvaloT lf x

da mf x ;

5 if l mf x f x //gamoiricxeba ( , ]mx b

6 {

47

7 mb x ; m lx x ; m lf x f x ;

8 } 9 // jer mxolod intervalis meoTxedis

gamoricxva SegviZlia 10 else 11 {

12 if r mf x f x //gamoiricxeba [ , )ma x

13 {

14 mb x ; m lx x ; m lf x f x ;

15 } 16 else //gamoiricxeba [ , )la x da ( , ]rx b

17 {

18 la x ; rb x ;

19 } 20 } 21 L b a ;

22 if 2L

23 return ˆ ˆ,m mx x f x f x ;

24 }

magaliTi: monakveTis ganaxevrebis meTodiT moZeb-

neT 2( ) (100 )f x x funqciis minimali 96 102x Sua-

ledSi. x gansazRvreT 0,5 sizustiT.

amoxsna: vimoqmedoT zemoT aRwerili algoriTmiT da

Sedegebi (iteraciebis mixedviT) warmovadginoT cxrilis

saxiT:

n a b L lx mx rx ( )lf x ( )mf x ( )rf x

1 96 102 6 97.5 99 100.5 6.25 1 0.25

2 99 102 3 99.75 100.5 101.25 0.063 0.25 1.563

3 99 100.5 1.5 99.375 99.75 100.125 0.391 0.063 0.016

4 99.75 100.5 0.75 99.938 100.125 0.016

48

radgan mesame iteraciis Semdeg miRebuli monakveTis

sigrZe naklebia 2 -ze, amitom ˆ 100.125mx x ,

ˆ( ) 0,016f x . –sizustiT minimalis miaxloebiTi mniS-

vnelobis gamosaTvlelad dagvWirda funqciis mniSvne-

lobis gamoTvla 7-jer. Sedegad miviReT 36 0.5 0.75

sigrZis intervali.

3. oqros kveTis meTodi

mimdevrobiTi meTodebidan optimaluria fibonaCis

meTodi. masTan SedarebiT gacilebiT martivia oqros

kveTis meTodi, romelic didi N –ebisTvis igive Sedegs

iZleva, rasac fibonaCis meTodi, anu aris asimptoturad

optimaluri. oqros kveTis meTodi, miznis funqciis

mniSvnelobebis N –jer gamoTvlis Semdeg gvitovebs 1( ) Nb a sigrZis monakveTs, romelic Seicavs minimals.

aq 0,618 aris 2 1 gantolebis dadebiTi amonax-

sni 5 1 2.0 0,61803... . da (1 ) ricxvebi, gan-

martebis mixedviT, axdenen [0,1] segmentis oqros kveTas.

monakveTis or aratol nawilad iseT dayofas, roca

mTeli monakveTis sigrZis Sefardeba misi didi nawilis

sigrZesTan tolia didi nawilis sigrZis Sefardebisa

mcire nawilis sigrZesTan, ewodeba am monakveTis oqros

kveTa. rac didia N , miT ufro vlindeba oqros kveTis

meTodis upiratesoba zemoT ganxilul meTodebTan

SedarebiT.

kvlav ganvixiloT minimizaciis amocana

( ) minf x (4.2)

[ , ]a b -ze unimodaluri f funqciiT da CamovayaliboT

oqros kveTis meTodis algoriTmi, rac saSualebas mogv-

49

cems mocemuli sizustiT davadginoT globaluri mini-

malis erT-erTi wertili (4.2) amocanaSi.

oqros kveTis meTodis fsevdokodi aseTia:

oqros kveTis meTodi (a,b,ε, f x )

1 5 1 2.0 , ( )lx b b a ; ( )rx a b a ;

gamovTvaloT ( )lf x da ( )rf x ;

2 while(1) 3 {

4 if ( )b a

5 {

6 return ( ˆlx x da ˆ( ) ( )lf x f x );

7 } 8 else 9 {

10 if l rf x f x //gamoiricxeba ( , ]rx b

Sualedi 11 {

12 rb x ; r lx x ; ( ) ( )r lf x f x ; ( )lx b b a ,

gamovTvaloT ( )lf x ;

13 }

14 else //axla gamoiricxeba [ , )la x

15 {

16 la x ; l rx x ; ( ) ( )l rf x f x ; ( )rx a b a ,

gamovTvaloT ( )rf x ;

17 } 18 } 29 }

aucilebelia yuradReba gavamaxviloT Semdeg momen-

tebze:

50

!) 0,5 , amitom l rx x (cxadia, ganvixilavT a b

SemTxvevas).

!!) Tu b a , maSin max | |la x b

x x

, kerZod ki

ˆ| |lx x . amitom SegviZlia, rom sizustiT

aviRoT ˆlx x .

!!!) Tu striqon 10-Si Sesrulda ( ) ( )l rf x f x , maSin

viTvliT mxolod ( )lf x , Tumca axal [ , ]a b -Si (anu

Zvel [ , ]ra x -Si) axali rx unda agveRo Cveulebrivi

wesiT: ra x a , Semdeg ki dagveTvala ( )rf x .

magram es mogvcemda:

2

(1 )( ) ( )

r

l

a x a

a a b a a a b a

a b a b b a x

(aq garkveuli analogiaa wina meTodis mx wertilTan

mimarTebaSi), rac niSnavs rom axali ( )rf x SegviZlia

pirdapir gavutoloT Zvel ( )lf x –s.

magaliTi. oqros kveTis meTodis gamoyenebiT,

0,5 sizustiT ganvsazRvroT globaluri minimalis

wertili 2( ) (100 ) minf x x , 96;102x ,

minimizaciis amocanaSi.

amoxsna. visargebloT aRwerili algoriTmiT da

Sedegebi warmovadginoT Semdeg cxrilSi, iteraciis

nomris miTiTebiT:

51

n a b ( )b a lx rx ( )lf x ( )rf x

1 96 102 3.708 98.292 99.708 2.917 0.085

2 98.292 102 2.292 99.708 100.584 0.085 0.341

3 98.292 100.584 1.416 99.168 99.708 0.692 0.085

4 99.168 100.584 0.875 99.708 100.043 0.085 0.0018

5 99.708 100.584 0.541 100.043 100.249 0.0018 0.062

6 99.708 100.249 0.334

amitom ˆ 100.043lx x , xolo ˆ( ) 0.0018f x . cxrilSi is-

rebiT miTiTebulia wertilebis transformaciis da maTi

mniSvnelobis SenarCunebis faqtebi. -sizustiT minima-

lis miaxloebiTi mniSvnelobis gamosaTvlelad dagvWir-

da funqciis mniSvnelobis gamoTvla 6-jer. 6-jer fun-

qciis mniSvnelobis gamoTvliT miviReT 56 0.618 0.541

sigrZis intervali.

savarjiSoebi

amocanebSi #47 – #49 intervalis sigrZis ganaxevre-

bis meTodiT, mocemuli sizustiT gansazRvreT f fun-

qciis globaluri minimumis x wertili.

#47. 4 3 2( ) 8 6 72 90f x x x x x , 1,5;2x , 0,05 .

#48. 6 2( ) 3 6 1f x x x x , 1;0x , 0,1 .

#49. 4 3 2( ) 3 10 21 12f x x x x x , 0;0,5x , 0,01 .

#50. ramdenjer unda gamoviTvaloT f funqciis mniSv-

neloba monakveTis sigrZis ganaxevrebis meTodiT,

rom globaluri minimumis wertili vipovoT winas-

war mocemuli sizustiT?

52

#51. ipoveT b –s maqsimaluri mniSvneloba, romlisTvi-

sac 2( ) 5 6f x x x funqcia iqneba unimodaluri

[ 5, ]b segmentze.

#52. ipoveT a–s maqsimaluri mniSvneloba, romlisTvisac 2( ) 8 9f x x x funqcia iqneba unimodaluri

[ 10, ]a segmentze.

#53. ipoveT c–s maqsimaluri mniSvneloba, romlisTvisac 2( ) 3 15 17f x x x funqcia iqneba unimodaluri

[0, ]c segmentze.

#54. vTqvaT, lx da rx aris [ , ]a b monakveTis oqros kveTis

wertilebi. aCveneT, rom lx aris [ , ]ra x monakveTis

oqros kveTis wertilebidan marjvena (udidesi),

xolo rx aris [ , ]lx b monakveTis oqros kveTis wer-

tilebidan marcxena (umciresi). ipoveT [ , ]ra x da

[ , ]lx b monakveTebis sigrZeebi.

#55. risi tolia maqsimaluri SesaZlo cdomileba oqros

kveTis meTodisTvis, Tu miznis funqciis mniSvnelo-

bebi gamoTvlil iqna N -jer?

#56. ramdenjer iqneba miznis funqciis mniSvnelobebis

gamoTvla saWiro, Tu viyenebiT oqros kveTis me-

Tods da gvinda globaluri minimumis wertilis

gansazRvra sizustiT?

Semdeg amocanebSi oqros kveTis meTodiT gansaz-

RvreT f funqciis globaluri minimumis wertili [ , ]a b

monakveTze sizustiT.

#57. 4 3( ) 2 4 1f x x x x , [ 1;0] , 0,1 .

#58. 4 2( ) ( 1) 2f x x x , [ 3; 2] , 0,05 .

#59. 5 3 2( ) 5 10 5f x x x x x , [ 3; 2] , 0,1

53

Tavi 5. unimodaluri funqciebis

minimizacia polinomialuri

aproqsimaciiT da

wertilovani SefasebiT

im ganmartebis mixedviT, romelsac Cveni kursi eyr-

dnoba, unimodaluri funqcia uwyvetia. vaierStrasis

erT-erTi Teoremis Tanaxmad, segmentze uwyveti fun-

qciis aproqsimacia SesaZlebelia polinomis saSualebiT

nebismieri sizustiT:

Teorema /vaierStrasis/. Tu f funqcia uwyvetia

,a b segmentze, maSin yoveli 0 -Tvis, arsebobs

algebruli polinomi p x iseTi, rom

( ) ( )f x p x , yoveli ,x a b -sTvis.

Tu aproqsimacia sakmaod zustia, miznis funqciis mini-

malad SegviZlia miviRoT polinomis globaluri mini-

mali. garda amisa, isev vaierStrasis Teoremis safuZ-

velze, miaxloebis sizuste SegviZlia gavaumjobesoT

ornairad: ufro maRali rigis polinomis gamoyenebiT an

saaproqsimacio Sualedis SemcirebiT. meore gza ufro

bunebrivia, radgan sawyisi Sualedi sakmaod swrafad

SegviZlia SevamciroT (magaliTad, oqros kveTis meTo-

diT), xolo Semdeg gamoviyenoT kvadratul (an kubur)

aproqsimaciaze dafuZnebuli meTodebi.

1. Sefasebis meTodi, romelic iyenebs

kvadratul aproqsimacias

ganvixiloT 1 3[ , ]x x segmentze unimodaluri f fun-

qciis minimalis moZebnis amocana mocemuli sizustiT:

( ) minf x , 1 3x x x .

54

vTqvaT, 1 2 3x x x da funqciis mniSvnelobebi am wer-

tilebSi aris: ( ), 1,2,3.i if f x i ganvsazRvroT 0 1 2, ,a a a

koeficientebi ise, rom kvadratuli polinomis

0 1 1 2 1 2p x a a x x a x x x x

grafikma gaiaros 1 1 2 2 3 3, , , , ,x f x f x f wertilebze.

pirvel rigSi, radgan 1 1 1 0f f x p x a , amitom

0 1a f . Semdeg, radgan

2 2 2 0 1 2 1 1 1 2 1f f x p x a a x x f a x x ,

amitom

2 11

2 1

f fa

x x

.

bolos, roca 3x x , gvaqvs:

2 13 3 3 1 3 1 2 3 1 3 2

2 1

f ff f x p x f x x a x x x x

x x

,

saidanac

3 1 2 12

3 2 3 1 2 1

1 f f f fa

x x x x x x

.

Tu 2 0a , maSin vixilavT axal amocanas

( ) minp x , 1 3x x x , (5.1)

da vipoviT ( )p x –is minimals mocemul segmentze. is

iqneba, an

1 2 2 2 1( ) 0p x a a x x a x x ,

gantolebis amonaxsni

11 2

2

1,

2 2

ax x x

a (5.2)

an segmentis erT-erTi bolo: 1x an 3x .

roca 1x , 2x , 3x wertilebi isea SerCeuli, rom

55

1 2( ) ( )f x f x , 3 2( ) ( )f x f x , (5.3)

maSin 2 0a da ( )p x saaproqsimacio mravalwevrs, rome-

lic gadis 1 1 2 2 3 3, , , , ,x f x f x f wertilebze aqvs globa-

luri minimali 1 3[ , ]x x -is SigniT da SegviZlia gamoviyenoT

pauelis meTodi.

Tu (5.3) piroba ar sruldeba, maSin viqceviT Sem-

degnairad: vamcirebT Ziebis intervals (mag, oqros kve-

Tis meTodiT) da vamowmebT (5.3)-s ( 2x -is rolSi SeiZleba

aviRoT oqros kveTis romelime wertili). rogorc ki

Sesruldeba (5.3) piroba, viyenebT kvadratul aproqsi-

maciaze damyarebul meTods. Tu piroba ar Sesrulda da

Ziebis intervali gaxda mocemul -ze naklebi, maSin

vasrulebT minimalis Ziebas oqros kveTis meTodiT.

magaliTi . kvadratuli aproqsimaciis gamoyenebiT

miaxloebiT gansazRvreT minimali

2 16( ) 2 min, 1 5,f x x x

x

amocanaSi 0.5 sizustiT.

amoxsna. aviRoT 1 3 21, 5, 3x x x ( 2x -ad Sua wer-

tils viRebT). gamoviTvaloT funqciis mniSvnelobebi:

1 2 318, 23,33, 53,2f f f

da agreTve maaproqsimirebeli polinomis koeficientebi:

1

23,33 18 8

3 1 3a

,

2

1 53,2 18 8 46

5 3 5 1 3 15a

.

( 0a , formalurad, ar monawileobs x -is warmodgenaSi).

bolos,

56

1 3 8/ 3

1,5652 2 46 /15

x

.

radgan 1<1,565<5, amitom viRebT ˆ 1,565x (Sedarebi-

saTvis, minimalis zusti mniSvneloba aris ˆ 1,5874x ).

2. pauelis meTodi

es meTodi mimdevrobiTia da yovel iteraciaze

iyenebs kvadratul aproqsimaciaze dafuZnebul meTods.

vTqvaT, 1 2 3, ,x x x wertilebi isea SerCeuli, rom 1 2 3x x x

da 1 2 3f f f (am utolobebis Sedegad miiReba 2 0a ).

meTodis algoriTmis fsevdokodi aseTia:

pauelis meTodi ( 1 2 3, ,x x x ,ε, f x )

1 while(1)

2 {

3 min 1 2 3min , ,f f f f , da minx iyos is wertili,

sadac miiRweva minf ;

4 { 1 2 3, ,x x x -is mixedviT gamoviTvaloT x , kvad-

ratul aproqsimaciaze dafuZnebuli meTodis gamoyenebiT; }

5 { optx -iT aRvniSnoT minx da x wertilebs Soris

is, romelSic miznis funqcia nakleb mniSvnelobas iRebs; }

6 if min min( )f f x x x &&

7 return ( ˆoptx x , ˆ( ) ( )optf x f x );

8 { optx wertilis orive mxares davalagoT masTan

yvelaze axlos mdgomi ori wertili da gadav-

nomroT es wertilebi zrdadobis mixedviT;}

9 }

57

gavakeToT ramdenime aucilebeli SeniSvna.

1) pirvel iteraciaze min 2f f .

2) zogjer iTxoven, rom SeCerebis piroba zedized

Sesruldes ramdenime momdevno iteraciisaTvis

(magaliTad, samjer), SemTxveviTobis Tavidan asa-

cileblad.

3) pauelis meTods minimalis siaxloves aqvs kvad-

ratuli krebadoba. es aris misi upiratesoba

oqros kveTis meTodTan, magram Tu sawyisi inter-

vali sakmaod didia, kvadratuli polinomiT miax-

loeba SeiZleba aRmoCndes sakmaod uxeSi, rac Seam-

cirebs pauelis meTodis krebadobas.

3. wertilovani aproqsimacia. Sua wertilis

meTodi

Tu unimodaluri miznis funqcia warmoebadic aris,

maSin SegviZlia gamoviyenoT wertilovan Sefasebaze da-

mokidebuli meTodebi. cxadia, Tu [ , ]f Q a b da a<x<b,

maSin:

a) ˆ0f x x x ,

b) ˆ0f x x x ,

g) ˆ0f x x x ,

sadac x aRniSnavs minimals min, ,f x a x b amo-

canaSi.

Sua wertilis meTodis algoriTmis fsevdokodi

aseTia:

Sua wertilis meTodi ( ,a b ,ε, f x )

1 while(1) 2 {

3 gamoviTvaloT / 2mx a b da mf x ;

58

4 if mf x

5 return ( ˆmx x ˆ( ) ( )mf x f x );

6 if 0mf x ma x ;

7 if 0mf x mb x ;

8 }

magaliTi . Sua wertilis meTodiT miaxloebiT gan-

sazRvreT minimali

2 16( ) 2 min, 1 5,f x x x

x

amocanaSi 0,05 sizustiT.

amoxsna. 21, 5, ( ) 4 16 / , ( ) 12 0,a b f x x x f a

( ) 20 16/5 0f b , rac niSnavs, rom minimali x aris [a,b]

segmentis Siga wertili da SegviZlia Sua wertilis

meTodis gamoyeneba.

I iteracia

striqoni 3). 3,mx da 12 16 / 9 10,22mf x ;

striqoni 4). mf x ;

striqoni 7). 0mf x , amitom 1, 3a b ;

II iteracia

striqoni 3). 2mx , 4 2 16 /(2 2) 4mf x ;

striqoni 4). mf x ;

striqoni 7). 0mf x , amitom 1, 2a b ;

III iteracia

striqoni 3). 3/ 2mx , 4 3/ 2 16 /(9 / 4) 6 7,11 1,11mf x ;

striqoni 4). mf x ;

striqoni 6). 0mf x , e.i. 3/ 2, 2a b ;

59

IV iteracia

striqoni 3). 1,75mx , 4 1,75 16 / 3,06 7 5,23 1,77mf x ;

striqoni 4). mf x ;

striqoni 7) 0mf x , e.i. 1,5, 1,75a b ;

V iteracia

striqoni 3). 1,63mx ,

4 1,63 16 / 2,64 6,52 6,06 0,46mf x .

striqoni 4). mf x , amitom ˆ 1,63x , xolo ˆ 40,45f x .

savarjiSoebi

#60. amocanebSi #47 – #49 da #57 – #59, moZebneT minimali

kvadratuli aproqsimaciis gamoyenebiT da SeadareT

adre miRebul Sedegebs.

#61. Sua wertilis meTodis gamoyenebiT gansazRvreT f

funqciis minimali [a,b] segmentze 0,5 sizustiT:

42( ) 8 12, 0 2

4

xf x x x x .

#62. amocanebSi, #47 – #49 da #57 – #59, moZebneT minimali

Sua wertilis meTodis gamoyenebiT da SeadareT

adre miRebul Sedegebs.

60

Tavi 6. lifSicuri funqciebis

minimizacia texilTa meTodiT

Tu unimodaluri funqciebisTvis rekomendirebul

meTodebs gamoviyenebT araunimodaluri funqciebisa-

Tvis, ukeTes SemTxvevaSi (uwyveti funqciebisaTvis) mi-

viRebT lokaluri minimalis miaxloebiT mniSvnelobas.

amave dros, unimodaluri funqciebis simravle aris Za-

lian viwro klasi uwyveti funqciebis simravlidan.

texilTa meTodi saSualebas iZleva miviRoT glo-

baluri eqstremumis wertilebis miaxloebiTi mniSvnelo-

bebi lifSicuri funqciebisaTvis. gavixsenoT

ganmarteba . vambobT, rom f funqcia akmayofilebs

lifSicis pirobas [a,b] monakveTze, Tu arsebobs iseTi

0L mudmivi, rom

( ) ( ) , , [ , ]f x f y L x y x y a b . (6.1)

umciress im L mudmivebs Soris, romlebic akmayofileben

(6.1) –s, uwodeben lifSicis mudmivs (igi damokidebulia f -

ze da [a,b] –ze), xolo f –s uwodeben lifSicuri funqcias

([a,b] –ze).

geometriulad (6.1) niSnavs, rom f funqciis grafikis

nebismieri ori wertilis SemaerTebeli qordis sakuTxo

koeficientis moduli ar aRemateba L–s. (6.1)-dan gamom-

dinareobs, rom lifSicuri funqcia uwyvetia [a,b] -ze,

amitom, vaierStrasis Teoremis ZaliT, igi aRwevs Tavis

globalur minimums da maqsimums [a,b] -ze.

araa aucilebeli, rom lifSicuri funqcia iyos war-

moebadi. magaliTad, , 1 1f x x x . magram Tu

funqcia warmoebadia monakveTze da warmoebuli Semosaz-

Rvrulia, maSin igi lifSicuria.

61

winadadeba . vTqvaT f funqcia warmoebadia [a,b] -ze

da f x , a x b , SemosazRvruli funqciaa. maSin f

funqcia [a,b]-ze akmayofilebs lifSicis pirobas

supa x b

L f x

mudmiviT.

damtkiceba. nebismieri , [ , ]x y a b wertilisaTvis

sasrul sxvaobaTa formula gvaZlevs:

f x f y f y y x y x ,

romeliRac [0,1] -isaTvis. aqedan da f -is Semosaz-

Rvrulobidan gamomdinareobs winadadebis daskvna.

vTqvaT, f aris lifSicuri funqcia [a,b] monakveTze

lifSicis L mudmiviT. cxadia, igive iqneba samarTliani –f

funqciisaTvis. amitom, lifSicuri funqciebisaTvis Seg-

viZlia zogadobis SeuzRudavad ganvixiloT mxolod

minimizaciis amocana:

min, [ , ]f x x a b , (6.2)

da vigulisxmoT L>0 (L=0 niSnavs f=const).

nebismierad aviRoT 0 [ , ]x a b da ganvsazRvroT

funqcia:

0 0( ) ,p x f x L x x a x b .

cxadia, es aris ori monakveTisagan Sedgenili texili,

0 0( )p x f x da misi monakveTebis sakuTxo koeficien-

tebia L . texilTa meTodi efuZneba im faqts, rom p–s

grafiki [a,b]–ze moTavsebulia f–is grafikis qvemoT.

marTlac, (6.1) –is ZaliT, roca a x b , gvaqvs:

0 0 0 0f x p x f x f x L x x f x f x L x x

1

0 0 0 0L f x f x x x x x

.

62

texilTa meTodi geometriulad niSnavs texilTa ise-

Ti mimdevrobis agebas, romelic qvemodan uaxlovdeba f-is

grafiks da nebismieri texilis nebismieri mdgenelis

sakuTxo koeficienti aris an +L, an –L. am texilebis

wveroebis erTi nawilis koordinatebis damaxsovreba

xdeba e.w. kritikuli wertilebis simravleSi, romelic

yovel iteraciaze erTi elementiT izrdeba. algoriTmis

aRweramde, moviyvanoT damxmare faqtebi. Tu ,a f a

wertilidan gavatarebT monakveTs –L sakuTxo koefi-

cientiT da ,b f b wertilidan gavatarebT monakveTs

+L sakuTxo koeficientiT (cxadia, orives grafiki f–is

grafikis qvemoTaa), maSin maTi gadakveTis wertili aris

* *

1 1,x p , sadac:

* *

1 1

1 1,

2 2x f a f b L a b p f a f b L a b

L ,

anu esaa pirveli texilis yvelaze qveda wertili (ix. na-

xazi 1).

*

1x

f

a b

1p

nax. 1

63

texilTa meTodis algoriTmi Semdegia.

biji 1). * *

1 1,x p wyvili gamoviyvanoT kritikuli werti-

lebis simravlidan da mis nacvlad SeviyvanoT

ori axali wyvili 1 1,x p , 1 1,x p (ix. nax. 2)

Semdegnairad:

* * * *

1 1 1 1 1 1 1 1 1

1, ,

2x x x x p f x p

,

sadac * *

1 1 1

1.

2f x p

L

biji 2). miRebuli ori 1 1,x p , 1 1,x p wyvilidan avir-

CioT is, romlis meore koordinati minimaluria

aRvniSnoT igi * *

2 2,x p -Ti da gamovricxoT kri-

tikuli wyvilebis simravlidan (cxadia, mocemul

bijze * *

2 2,x p -ad SegviZlia aviRoT nebismieri am

ori wyvilidan). * *

2 2,x p -is nacvlad, kritikuli

wyvilebis simravleSi davamatoT ori axali

nax. 2

1p

*

2 1x x *

1x a b 1x

f

64

wyvili 2 2,x p , 2 2,x p , romelTa komponentebi

iTvleba formulebiT:

* * * *

2 2 2 2 2 2 2 2 2

1, ,

2x x x x p f x p

,

sadac * *

2 2 2

1

2f x p

L

.

am bijis Sedegad, kritikuli wyvilebis simrav-

leSi aris sami wyvili (ix. nax. 3).

biji n). wina bijebis Sedegad, kritikuli wyvilebis sim-

ravleSi aris n cali ,x p saxis wyvili. maTgan

avirCioT is, romlis meore komponenta minima-

luria. aRvniSnoT igi * *,n nx p -Ti, gamovricxoT

igi kritikuli wyvilebis simravlidan da mis

nacvlad, kritikuli wyvilebis simravleSi, da-

vamatoT ori axali wyvili ,n nx p , ,n nx p , ro-

melTa komponentebi iTvleba formulebiT:

*

3x a b

f

2x 2x

2p

nax. 3

65

* * * *1, ,

2n n n n n n n n nx x x x p f x p

, (6.3)

sadac * *1

2n n nf x p

L

.

Cveulebriv, rodesac *

nf x sakmaod axlosaa

* mina x b

f f x

sididesTan, maSin viRebT *ˆnx x da

*

* nf f x . *f -Tan siaxlovis gansazRvrisaTvis gamoi-

yeneba Sefaseba:

*

*0 2 .n nf x f L (6.4)

e.i., roca sruldeba (6.4), maSin sruldeba Ziebis procesi.

SeniSvna . warmoebadi funqciisTvis ˆ 0f x , ami-

tom miznis funqciis mniSvnelobebi ufro swrafad midis

minimumisken, vidre *

nx dauaxlovdeba x -s. algoriTmis

gaCerebis pirobad (6.4)–Tan erTad SeiZleba *ˆnx x -is

simciris moTxovnac.

magaliTi . texilTa meTodiT [10;15] Sualedze

0.01 sizustiT moZebneT sin x

f xx

funqciis glo-

baluri minimumi *f da globaluri minimali x .

amoxsna. radgan

2 2 2

cos sincos sin 1( ) 0,11

x x xx x x xf x

x x x

roca [10;15]x , amitom SegviZlia aviRoT 0,11L (Tumca

es metobiTaa).

Tavidan vpoulobT * *

1 112,056, 0,281x p . Semdeg

aRwerili sqemis mixedviT (6.3) formulebiT vatarebT

gamoTvlebs. gamoTvlebis Sedegebi moyvanilia Semdeg

cxrilSi.

66

n

gamosaricxi wyvili (x, p)

2 nL dasamatebeli wyvilebi

*

nx *

np nx nx np

1 12,056 -0,281 0,240 10,963 13,149 -0,161

2 10,963 -0,161 0,070 10,646 11,280 -0,126

3 13,149 -0,161 0,203 12,227 14,071 -0,096

4 10,646 -0,126 0,038 10,474 10,818 -0,107

5 11,280 -0,126 0,041 11,094 11,466 -0,106

6 10,474 -0,107 0,024 10,364 10,584 -0,095

7 10,818 -0,107 0,160 10,745 10,891 -0,099

8 11,094 -0,106 0,016 11,020 11,168 -0,098

9 11,466 -0,106 0,028 11,338 11,594 -0,092

10 10,891 -0,099 0,008 < – – –

cxrilis mixedviT, *ˆ 10,891, (10,891) 0,091x f f . Sev-

niSnoT, rom miznis funqcia ar aris unimodaluri, amitom adre ganxiluli meTodebi ar gamogvadgeboda.

savarjiSoebi

#63. daamtkiceT, rom [a,b] segmentze lifSicuri funqcia uwyvetia amave segmentze.

#64. 3 212 5 6

3f x x x x funqcia lifSicuri aris

[0,1] da [0,10] Sualedebze. orive SemTxvevaSi moZeb-neT umciresi lifSicis mudmivebs Soris.

Semdeg amocanebSi, texilTa meTodiT ipoveT f fun-

qciis minimali da globaluri minimumi [a,b] segmentze mocemuli sizustiT:

#65. 2 4

0,9 1,1 , [0,8;1,2], 0,05f x x x x .

#66. 2

cos( ) , 7 11, 0,01

xf x x

x .

#67. 2 1 1f x x funqcia aris Tu ara lifSicuri

[–2,2]-ze? Tu aris, CaatereT pirveli 6 iteracia te-xilTa meTodiT.

n a w i l i 3

mraval cvladze

damokidebuli miznis

funqcia

68

Tavi 7. gluvi eqstremaluri amocana

Ria dasaSvebi simravliT

gamoyenebiT amocanebSi, TiTqmis yovelTvis, miznis

funqcia mravali cvladis funqcias warmoadgens. am tipis

amocanebSi yvelaze advilad gamosakvlevi aris Ria da-

saSvebi simravlis SemTxveva.

1. kritikuli wertilebis gansazRvra

Teorema 1 /eqstremalurobis I rigis aucilebeli

piroba/. vTqvaT, nM aris Ria simravle, :f M

uwyvetad warmoebadi funqciaa 1 nˆ ˆ ˆx , ,x x M werti-

lis romeliRac midamoSi da x aris lokaluri eqstre-mumis wertili

(x) extr, xf M

amocanaSi. maSin,

x

x 0 0, 1, ,x i

ff i n

.

damtkiceba. vTqvaT 1i . Tu 1 nˆ ˆ ˆx , ,x x aris f

funqciis lokaluri eqstremumis wertili, maSin 1x iqneba

1 1 2ˆ ˆ( , , , )ng x f x x x erTi cvladis funqciis lokaluri

eqstremumis wertili, amasTan, radgan 1 2( , , , )nf x x x

funqcia gansazRvrulia Ria simravleze, amitom 1g x

funqciac gansazRvrulia Ria simravleze da misTvis

SegviZlia gamoviyenoT fermas Teorema 1ˆ 0g x , anu

1 1

1 2 1 2

1 1

ˆ

ˆ ˆ ˆ ˆ ˆ, , , , , ,0

n n

x x

f x x x df x x x

x dx

analogiuria mtkiceba 2, ,i n -Tvis.

69

ganmarteba. ganxiluli Teoremis pirobebSi,

(x) extr, xf M ,

amocanisTvis kritikuli wertilebis simravle K gani-

sazRvreba Semdegnairad:

x x 0K M f .

magaliTi. ipoveT 2 2 2x y xy x y funqciis eqs-

tremumis wertilebi.

amoxsna. CavweroT amocana standartuli saxiT:

2 2

2( , ) 2 extr, ,f x y x y xy x y x y . (7.1)

(7.1) amocanisaTvis ganvsazRvroT kritikuli wertilebis

simravle. am mizniT SevadginoT da amovxsnaT sistema

/ 0, / 0f x f y . e.i.:

2 2 0,

2 1 0,

fx y

x

fy x

y

saidanac vpoulobT: 1, 0, 1,0 .x y K anu

Semdeg punqtebSi avxsniT, Tu rogor xdeba kritikuli

wertilebis gamokvleva da davasrulebT am magaliTsac.

2. kritikuli wertilebis gamokvleva

vaierStrasis Teoremis Sedegebis

gamoyenebiT

vaierStrasis Teoremis Sedegebi saSualebas gvaZlevs

gavarkvioT, Tu romeli kritikuli wertili aris globa-

luri eqstremumis wertili.

vaierStrasis Teoremis Sedegebis gamoyenebiT Tu

davadgenT, rom

(x) extr, xf M , (7.2)

70

amocanaSi glmin(7.2) , maSin SegviZlia Semovifar-

gloT gacilebiT martivi

(x) extr, xf K , (7.3)

amocanis ganxilviT, romelsac aqvs igive minimali, rac

(7.2) –s, radgan

x x glmin(7.2) x glmin(7.3)K da .

analogiurad vmoqmedebT globaluri maqsimumis Ziebis

SemTxvevaSic.

magaliTi. gavagrZeloT Semdegi magaliTis amoxsna:

2 2

2( , ) 2 extr, ,f x y x y xy x y x y .

rogorc vnaxeT, 1,0 .K radgan

( , )( , ) , glmin(7.1) .lim

x yf x y

amitom

radgan K erTelementiania, amitom (1,0) aris globaluri

minimali

( , ) extr, ,f x y x y K amocanaSi.

am meTodSi ZiriTadi sirTule mdgomareobs absolu-

turi eqstremumis wertilis arsebobis dadgenaSi. Cvens

magaliTSi, radgan

2 2

2 2 20 22 2

x yx y x y xy xy ,

amitom 2 2

2( , ) 2 , ( , )

2 2

x yf x y x y x y . (7.4)

axla cxadia, rom ( , )

( , ) ,limx y

f x y

radgan (7.4)–Si

kvadratuli wevrebis koeficientebi dadebiTia.

71

3. kritikuli wertilebis gamokvleva

ganmartebis safuZvelze

kvlav ganvixiloT (7.2). vTqvaT, sruldeba Teorema

1-is pirobebi da x K . yoveli x M warmovadginoT

Semdegi saxiT: x x h da ganvixiloT sxvaoba:

x h x x x .f f f f f

f -is niSnis h -ze damokidebulebis garkveva arsebiTia,

radgan:

5) Tu 0f yoveli SesaZlo h-isTvis, maSin

x glmin(7.2) ;

6) Tu 0f yoveli SesaZlo h-isTvis, maSin

x glmax(7.2) ;

7) Tu 0f yoveli sakmaod mcire h-isTvis, maSin

x locmin(7.2) ;

8) Tu 0f yoveli sakmaod mcire h-isTvis, maSin

x locmax(7.2) ;

magaliTi. kvlav gavagrZeloT

2 2

2( , ) 2 extr, ,f x y x y xy x y x y

eqstremaluri amocanis ganxilva. aq 1,0K aris

erTaderTi kritikuli wertili. Cvens SemTxvevaSi

x x h warmodgena aris

1 2 1 2, 1,0 h ,h 1 h ,hx y ,

amitom ganvixiloT sxvaoba:

, 1,0f x y f

2 2

1 2 1 2 1 21 h h 1 h h 2 1 h h 1 2

72

2 2 2 2

1 2 1 2 1 21h h h h h h 0

2

,

rac ,x y -is nebismierobis ZaliT niSnavs, rom x 1,0

aris globaluri minimali. SevniSnoT agreTve, rom bolo

formulidan martivad Cans ( , )

( , ) ,limx y

f x y

rac

arsebiTia vaierStrasis Teoremis tipis SedegebiT sar-

geblobisas.

4. kritikuli wertilebis gamokvleva

eqstremalurobis II rigis sakmarisi pirobis

safuZvelze

winaswar moviyvanoT ramdenime faqti, rac aucile-

belia am meTodis gamoyenebisas.

ganmarteba. vTqvaT, mocemulia kvadratuli

, 1

an

i j i jA

matrica. am matricas ewodeba:

a) dadebiTad gansazRvruli, Tu nebismieri aranu-

lovani 1y y , , yn veqtorisaTvis sruldeba:

T j

1 1

y y a y yn n

i

i j

i j

A

0 .

b) uaryofiTad gansazRvruli, Tu nebismieri aranu-

lovani 1y y , , yn veqtorisaTvis sruldeba:

T j

1 1

y y a y yn n

i

i j

i j

A

0 , anu Tu A aris dadebiTad

gansazRvruli.

silvestris cnobili Teorema (ix. [1]), romelsac aq daum-

tkiceblad moviyvanT, gvaZlevs advilad Sesamowmebel

kriteriums kvadratuli matricis gansazRvrulobis dad-

genisTvis. SevniSnoT, rom silvestris Teoremis sxva,

ufro informatiuli versiebic arsebobs.

73

Teorema /silvestris/. kvadratuli, simetriuli A

matricis dadebiTad gansazRvrulobisaTvis aucilebe-

lia da sakmarisi, rom misi yvela mTavari minori iyos

dadebiTi, anu yoveli 1,2, ,k n -isaTvis unda Sesrul-

des det 0k

A , sadac , 1

ak

k i j i jA

.

savarjiSo. CamoayalibeT kriteriumi A matricis

uaryofiTad gansazRvrulobisaTvis.

miTiTeba. A -s uaryofiTad gansazRvruloba igivea,

rac – A -s dadebiTad gansazRvruloba.

SemdegSi, xf -iT, Cveulebriv, aRvniSnavT :f M

funqciis meore warmoebulebisgan Sedgenil, anu heses

matricas x wertilSi:

2 2

1 2 1 n

2 2

n 1 n 2

(x ) x xx

x x (x )

f f

ff f

.

Teorema 2 /eqstremalurobis II rigis sakmarisi

piroba/. vTqvaT nM aris Ria simravle, :f M

funqcias aqvs meore rigis uwyveti kerZo warmoebulebi.

maSin, imisaTvis rom 1 nˆ ˆ ˆx x , , x M iyos mkacri loka-

luri maqsimali (minimali) amocanaSi

(x) extr, xf M ,

sakmarisia Semdegi pirobebis Sesruleba:

ˆ ˆx 0 xdaf f uaryofiTadaa gansazRvruli

( ˆ ˆx 0 xdaf f dadebiTadaa gansazRvruli).

74

damtkiceba. damtkicebis idea martivia. radgan samar-

Tliania teiloris formula:

ˆ(x) (x)f f 2T1ˆ ˆ ˆ ˆ ˆx x x x x x x x

2f f o ,

amitom roca x, x sakmaod axlos arian erTmaneTTan,

marjvena mxareSi, pirveli wevri nulia, xolo mesamis

ugulebelyofa SegviZlia (ufro maRali rigis usas-

rulod mcirea), amitom marcxena mxaris niSani emTxveva

marjvena mxareSi meore wevris niSans, rac amtkicebs

Teoremas.

axla davasruloT Teorema 2-is damtkiceba lokalu-

ri minimalisaTvis. radgan xf dadebiTadaa gansaz-

Rvruli, amitom xA f matricis yvela mTavari

minori dadebiTia: det 0k

A , 1,2, ,k n . determi-

nanti aris matricis elementebis uwyveti funqcia, xolo,

Tavis mxriv, matricis elementebi 2 x

x xi j

f

aris x -is

uwyveti funqciebi indeqsebis yoveli ,i j wyvilisaTvis.

amgvarad, mTavari minorebi aris x M -is uwyveti

funqciebi da es funqciebi dadebiTia x -Si. amitom x -is

raRac midamoSi isini SeinarCuneben niSans. Tu 0 ise

aris SerCeuli, rom x -is ˆx x midamoSi xf -is

mTavari minorebi inarCuneben niSans, maSin

ˆx x (x)f dadebiTadaa gansazRvruli.

0 imdenad mcire SegviZlia aviRoT, rom agreTve

Sesruldes

ˆx x x M .

teiloris formulis Tanaxmad, roca ˆx x , gvaqvs

75

ˆ(x) (x)f f

T1

ˆ ˆ ˆ ˆ ˆx x x x x x x x2

f f , (7.5)

sadac [0,1] damokidebulia x-ze. SevniSnoT, rom

teiloris formulis es saxe samarTliania mxolod

ricxviTi mniSvnelobebis mqone funqciebisaTvis. pirobis

Tanaxmad, x 0f , xolo radgan

ˆ ˆ ˆ ˆx x x x x x ,

amitom ˆ ˆx x xf dadebiTadaa gansazRvruli, anu

T

ˆ ˆ ˆ ˆx x x x x x x 0f .

amgvarad (7.5) gvaZlevs:

ˆ ˆ ˆx x (x) (x) 0 (x) (x)f f f f .

savarjiSoebi

amoxseniT Semdegi eqstremaluri amocanebi:

#68. 2 25 4 16 12 extr.x xy y x y

#69. 2 23 4 8 12 extr.x xy y x y

#70. 2 22 extr.x xy y

#71. 22 21 10 extr.x y x

#72. 3 2 2 3 6 extr.x y z yz x y

#73. 2 3 22 extr.x xy y z

#74. 2 2 2 2 extr.x y z xy x z

#75. 3 22 10 extr.x xy y

#76. 2 22 1 extr.y x x

76

#77. 50 20

extr.xyx y

#78. 2 2 4 6 extr.x y x y

#79. 2 2 2 27 extr.x y z xy x

#80. 3 3 3 extr.x y xy

#81. 2 3 2 28 6 3 extr.

x ye x xy y

77

Tavi 8. gluvi eqstremaluri amocana

tolobis tipis SezRudvebiT

1. mokle reziume

vTqvaT, yoveli :i nf R R , 0,1, ,i m aris uwyvetad

warmoebadi funqcia. Semdeg eqstremalur amocanas

0 1(x) extr, (x) 0, , (x) 0mf f f , (8.1)

ewodeba gluvi amocana tolobis tipis SezRudevebiT.

aq, 0f aris miznis funqcia, xolo

1x (x) (x) 0n mM R f f

aris dasaSvebi simravle. cxadia, M aris Caketili.

ganmartebis Tanaxmad, x aris lokaluri minimali (8.1) –Si

(e.i. x locmin(8.1) ), Tu romelime 0 ricxvisaTvis

1 0 0ˆ ˆx x & (x) (x) 0 (x) (x).mf f f f

(8.1) amocanisTvis SemoviRoT lagranJis funqcia:

0 0 0 1 1(x, , ) x x xm mf f f L ,

sadac 1, , m , 1x x , ,xn , j -ebs ewodebaT

lagranJis mamravlebi, 0 -s ewodeba lagranJis mTavari

mamravli. uSualod ganmartebebidan gamomdinareobs, rom

(8.1) amocanis dasaSveb simravleze miznis funqcia da

lagranJis funqcia erTmaneTis tolia mudmivi Tanamam-

ravlis sizustiT, anu

0 0 0(x, , ) xf L , roca 1 x 0, , x 0.mf f

am paragrafSi Cven davasabuTebT, rom tolobis tipis

SezRudvebiani gluvi amocanis gamokvleva xdeba Semdegi

TanmimdevrobiT:

1. maTematikuri modelis Sedgena, anu amocanis formu-

lireba (8.1) saxiT;

78

2. lagranJis funqciis Sedgena (8.1)) amocanis monace-

mebis mixedviT:

0 0 0 1 1(x, , ) x x xm mf f f L ;

3. (8.1) eqstremalur amocanaSi kritikuli wertilebis

(anu iseTi wertilebis, romlebic SesaZloa warmoad-

gendnen (8.1)-is amonaxsns) gansazRvris mizniT ganto-

lebaTa sistemis Sedgena

0

0

(x, , )0, 1, , ,

x

(x, , )0, 1, , .

i

j

i n

j m

L

L (8.2)

4. (8.1) amocanisaTvis kritikuli wertilebis simrav-

lis ganisazRvra:

0 1

0

, ,

( , , )

m

n

RK x

x

arsebobs aranulovani iseTi

rom aris (8.2)-is amonaxsni

;

gadamwyveti mniSvneloba aqvs, rom

locmin(8.2), locmax(8.2) .K

5. kritikuli wertilebis gamokvleva maTgan eqstre-

mumis wertilebis amorCevis mizniT.

SevniSnoT, rom amocanis gamokvleva moxdeba igive

sqemiT, Tu yvela funqcia: miznis funqcionali da Sez-

Rudvebi gansazRvrulebi arian raime Ria simravleze.

2. maTematikuri modelis Sedgena

sailustraciod, ganvixiloT piradi moxmarebis neo-

klasikuri amocana. vTqvaT, kerZo pirma an firmam (zoga-

dad momxmarebeli) piradi moxmarebis mizniT unda Sei-

Zinos n sxvadasxva dasaxelebis produqtis 1 2x , x , , xn

raodenobebi ise, rom sargeblianobis 1 2U x , x , , xn

79

funqciam, romelic aRwers momxmareblis gemovnebas,

miiRos maqsimaluri mniSvneloba. simartivisTvis vgulis-

xmobT, rom TiToeuli saxis produqtis erTeuli aris

usasrulod danawilebadi (praqtikulad aseTia 1 kg. Saqa-

ri an milioni wyvili fexsacmeli) da rom valis aReba

SeuZlebelia. es ori daSveba garantias gvaZlevs, rom

yoveli 0,ix . cnobilia produqtebis fasebi

1 2, , , np p p da Tanxa I , romelic gaaCnia momxmarebels.

formalurad, moxmarebis neoklasikuri amocana aris

amocana utolobis tipis SezRudvebiT:

1 2U x , x , , x maxn , ( 1 2

1 2x x xn

np p p I da

yoveli x 0i ).

Tu davukvirdebiT amocanis Sinaarss, davrwmundebiT rom

igi sinamdvileSi ufro martivia. arsebiTia is garemoeba,

rom funqcia 1 2U x , x , , xn , romelic aRwers momxmareb-

lis gemovnebas, ver iqneba nebismieri. arsebobs bunebrivi

SezRudvebi, romelsac akmayofilebs momxmareblis ge-

movneba. am SezRudvebs moxmarebis TeoriaSi aqsiomebs

uwodeben. aRvweroT ori maTgani.

1. Tu x =y , 1,... 1, 1,...,i i i j j n da x >yj j, maSin

1 1 1 1 1 1U x ,..., x , x ,x ,..., x U ,..., x , ,x , , xj j j n j j j nx y

anu, Tu erTi produqtis raodenoba imatebs, xolo

danarCeni produqtebis raodenoba fiqsirebulia,

maSin krebulis sargeblianoba izrdeba. maTemati-

kurad, es niSnavs, rom 1 2U x , x , , xn funqcia

zrdadia TiToeuli argumentis mimarT:

1 2 1 2

jU x , x , , x U x , x , , x 0n n

jx

, 1,...,j n .

80

ekonomikaSi amas uwodeben zRvruli sargebliano-

bebis dadebiTobis kanons.

2. Tu x =y , 1,... 1, 1,...,i i i j j n da x >yj j, maSin

1 1 1 1 1 1

j jU x ,..., x , x ,x ,..., x <U ,..., x , ,x , , xj j j n j j j nx y

anu, Tu erTi produqtis raodenoba imatebs, xolo

danarCeni produqtebis raodenoba fiqsirebulia,

maSin im erTi produqtis zRvruli sargeblianoba

mcirdeba:

2

1 2

2U x , x , , x 0

( )

n

jx

, 1,...,j n .

ekonomikaSi amas uwodeben kanons zRvruli sargeb-

lianobis klebadobis Sesaxeb.

am Tvisebebidan gamomdinareobs, rom zRvruli sar-

geblianoba yovelTvis dadebiTia, Tumca klebadia.

es niSnavs, rom 1 2U x , x , , xn funqcia zrdadia,

magram misi zrdis tempi mcirdeba argumentebis

gazrdasTan erTad.

1 2

1 2x x xn

np p p I biujetur SezRudvaSi SeiZ-

leba mxolod toloba gvqondes, radgan Tu Tanxa darCa,

misi daxarjviT SeiZleba raime produqtis mcire dozis

SeZena mainc da amgvarad sargeblianobis gazrda. e.i.

moxmarebis neoklasikur amocanaSi realurad sruldeba

tolobis tipis SezRudva 1 2

1 2x x xn

np p p I . maSa-

sadame, moxmarebis amocanas aqvs saxe:

1 2U x , x , , x maxn , 1 2

1 2x x xn

np p p I , x 0i .

81

3. kritikuli wertilebis gansazRvra

imisaTvis, rom tolobis tipis SezRudvebian eqstre-

malur amocanaSi

0 1(x) extr, (x) 0, , (x) 0mf f f (8.3)

ganvsazRvroT kritikuli wertilebi, unda Camovayali-

boT da davamtkicoT eqstremalurobis pirveli rigis

aucilebeli piroba. amisaTvis, winaswar davamtkicoT

erTi damxmare faqti, romelsac damoukidebeli mniSvne-

lobac aqvs.

lema 1 . vTqvaT, , , :m nW M g W M da

:f M funqciebi uwyvetebia da x aris lokaluri eqstre-

mumis wertili

(x) extr, xf M (8.4)

amocanaSi. maSin yoveli iseTi w W , romlisTvisac sruldeba

ˆ ˆ(w) xg piroba, warmoadgens lokaluri eqstremumis wertils

(w) extr, wh W (8.5)

amocanaSi, sadac h f g .

damtkiceba. vTqvaT, x locmin(8.4) . arsebobs 1 0 iseTi,

rom sruldeba:

1ˆ ˆx x & x (x) (x).M f f (8.6)

radgan g uwyvetia w -Si, am 1 -isaTvis arsebobs 0 iseTi, rom

1ˆ ˆw w & w (w) (w) .W g g

es ki ˆ ˆ(w) xg tolobisa da (4.11) –is Tanaxmad niSnavs, rom

ˆ ˆw w & w (x) (w)W f f g ,

anu w locmin(8.5) . SemTxveva w locmax(8.5) analogiuria.

Teorema 1 /eqstremalurobis pirveli rigis auci-

lebeli piroba/. vTqvaT, x aris dasaSvebi wertili (8.3)

amocanaSi da yoveli if , 0,1, ,i m , uwyvetad war-

82

moebadia x -is raRac midamoSi. Tu x aris lokaluri

eqstremumis wertili (8.3) amocanaSi, maSin arseboben

erTdroulad nulis aratoli lagranJis mamravlebi

1ˆ ˆ ˆ, , m da 0 , iseTebi rom

0ˆ ˆˆ(x, , )

0, 1, , ,xi

i n

L

(8.7)

anu

0 0 1 1ˆ ˆ ˆˆ ˆ ˆx x x 0m mf f f . (8.71)

damtkiceba. garkveulobisTvis aviRoT x locmin(8.3) .

Teorema davamtkicoT induqciiT SezRudvebis raodenobis, anu

m –is mimarT. rodesac 0m , maSin Teorema samarTliania wina paragrafis Teoremis /eqstremalurobis pirveli rigis aucile-beli pirobis/ ZaliT. axla, davuSvaT Teoremis samarTlianoba

1m raodenoba SezRudvebisTvis da davamtkicoT m –isTvis.

zogadobis SeuzRudavad SegviZlia vigulisxmoT, rom 1 xf

aranulovania (winaaRmdeg SemTxvevaSi aviRebdiT 1ˆ 1 , danarCen

mamravlebs ki nulis tols da Teoremis daskvna trivialurad

Sesruldeboda). vigulisxmoT agreTve, rom 1

1

x0.

x

f

maSin,

1 n

1(x , , x ) 0f gantoleba, romelic sruldeba 1 nˆ ˆ ˆx (x , , x )

wertilSi mainc, amoixsneba 1x cvladis mimarT. es niSnavs iseTi

1x =2 n(x , , x )g funqciis arsebobas, romelic gansazRvrulia

2 nˆ ˆ(x , , x ) wertilis raime midamoSi, amave midamoSi yvelagan

sruldeba

2 n 2 n

1 (x , , x ), x , , x 0f g (8.8)

da 1 2 nˆ ˆ ˆx (x , , x )g .

Canawerebis simartivis mizniT, gamoviyenoT Semdegi aRniS-vnebi:

83

x

i

j

f

–Ti aRvniSnoT kerZo warmoebuli

1 nˆ ˆ ˆx (x , , x ) wer-

tilSi, 1, ,i m , 1, ,j n .

x j

g

–Ti aRvniSnoT kerZo warmoebuli

2 nˆ ˆ(x , , x ) wer-

tilSi, 2, ,j n .

maSin, rTuli funqciis gawarmoebis wesis Tanaxmad, (8.8)–is gawar-

moeba i–uri cvladis mimarT gvaZlevs:

1 1

10

x x xi i

f fg

anu

1

1 1

1x x xi i

f fg

. (8.9)

lema 1–is ZaliT, (8.3) SegviZlia gadavweroT Semdegi saxiT: 2 n 2 n 2 n

0 2(x , , x ) extr, (x , , x ) 0, , (x , , x ) 0mF F F , (8.10)

sadac 2 n 2 n 2 n(x , , x ) (x , , x ), x , , xi iF f g .

(8.10) amocanaSi SezRudvebis raodenoba saSualebas gvaZlevs gamoviyenoT induqciuri daSveba. amitom, arseboben erTdroulad

nulis aratoli lagranJis mamravlebi 0 2ˆ ˆ ˆ, , , m da, iseTebi

rom yoveli 2, ,i n -isTvis sruldeba:

2 n 2 n2 n

0 20 2

ˆ ˆ ˆ ˆˆ ˆ(x , , x ) (x , , x )(x , , x )ˆ ˆ ˆ 0x x x

mmi i i

F FF

. (8.11)

(8.11) igivea, rac:

iiii x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f 2

1

1

11

1

22

0

1

1

11

1

00

ˆˆ

i

m

i

mm

x

f

x

f

x

f

x

f1

1

11

1

= 0 1 20 1 2

ˆ ˆ ˆ ˆx x x x

mmi i i i

f ff f

= 0, (8.12)

sadac 1 1 1

0 1 2 1 11 0 21 1 1 1 1 1ˆ ˆ ˆ ˆ

x x x x x x

mm

f ff f f f

.

cxadia, (8.12) amtkicebs Teoremas.

84

es Teorema Zalian mniSvnelobania rogorc Teoriuli,

aseve gamoyenebiTi TvalsazrisiT, amitom misi gaazrebis-

Tvis sasargeblo iqneba, Tu yuradRebas gavamaxvilebT

Semdeg faqtorebze.

SeniSvna 1. (8.71) niSnavs, rom n –ganzomilebiani veq-

torebis

0 1ˆ ˆ ˆ(x), (x), , (x)mf f f (8.13)

sistema wrfivad damokidebulia. amgvarad, es Teorema

(eqstremumis I rigis aucilebeli piroba) amtkicebs, rom

tolobis tipis SezRudvebis SemTxvevaSi, lokaluri eqs-

tremumis aucilebel pirobas warmoadgens (8.13) sistemis

wrfivad damokidebuleba. am kuTxiT, es Teorema warmoad-

gens fermas Teoremis mravalganzomilebiani analogis

ganzogadebas, radgan roca 0m , maSin 0ˆ(x)f -is wrfi-

vad damokidebuleba niSnavs, rom 0ˆ(x) 0f .

SeniSvna 2. roca m n , maSin n –ganzomilebiani veq-

torebis (8.13) sistema wrfivad damokidebulia da Teo-

remis daskvna trivialurad sruldeba. amitom, xSirad

amocanis dasmaSi gulisxmoben xolme, rom m n .

SeniSvna 3. didi xnis ganmavlobaSi Tvlidnen, rom

rodesac x locmin(8.3) an x locmax(8.3) , maSin 0ˆ(x)f

gamoisaxeba 1ˆ ˆ(x), , (x)mf f sistemis veqtorebis wrfivi

kombinaciis saxiT raime damatebiTi pirobebis gareSe.

amitom iyenebdnen mxolod mamravlebs 1ˆ ˆ, , m . Semdeg

moiZebna magaliTebi (ix. Semdegi SeniSvna), romlebic gviCveneben, rom eqstremumis aucilebel pirobas war-moadgens (8.13) sistemis wrfivad damokidebuleba, magram

85

SesaZloa am dros 1ˆ ˆ(x), , (x)mf f sistemac wrfivad damo-

kidebuli iyos. amitom lagranJis funqciis argumentebs

daemata 0 cvladic, oRond boloSi: 0(x, , ) L , rac

arRvevs argumentebis mimdevrobas.

SeniSvna 4. lagranJis Semdeg Zalian didxans Tvlid-

nen, rom yovelTvis sruldeba 0ˆ 1 . es umetes SemTxve-

vebSi marTlac asea, magram Tu SezRudvebi aris “gadagva-

rebuli”, maSin aRmoCndeba xolme 0ˆ 0 . magaliTad, gan-

vixiloT: 2 2min, 0x x y .

am amocanaSi aris erTaderTi dasaSvebi wertili (M

gadagvarda wertilSi) da, maSasadame, igi aris amonaxsnic:

ˆ ˆ, 0,0x y . meore mxriv, Tu lagranJis funqciaSi

2 2

0 1x x y aviRebT 0 1 da SevadgenT sistemas

kritikuli wertilis gansazRvrisaTvis, vnaxavT rom igi

araTavsebadia:

1

1

2 2

1 2 0,

2 0,

0.

x

y

x y

radgan

000

x x(x, , ), 1, , ,

x x x

m

mi i i

f fi n

L

da

0(x, , )x , 1, , ,j

j

f j m

L

amitom Teorema 1, lagranJis funqciis terminebSi,

amtkicebs Semdegs: Tu x warmoadgens lokaluri eqs-

tremumis wertils (8.3) amocanaSi, maSin arsebobs erT-

86

droulad nulis aratoli ricxvebi 0 1ˆ ˆ ˆ, , , m , iseTebi

rom 0ˆ ˆˆ(x, , ) sameuli (sadac 1

ˆ ˆ ˆ, , m ) warmoadgens

Semdegi sistemis amonaxsns:

0

0

(x, , )0, 1, , ,

x

(x, , )0, 1, , .

i

j

i n

j m

L

L (8.14)

es niSnavs, rom (8.3) sistemis eqstremumis wertilebi unda

veZeboT (8.14)–is iseT amonaxsnebs Soris, romelTa

lagranJis mamravlebi erTdroulad nulis toli ar

xdeba. nebismierad aRebuli lagranJis mamravlebisaTvis

lagranJis funqcia dasaSveb simravleze (da kerZod

eqstremumis wertilebSi) miznis funqciis tolia mudmivi

Tanamamravlis sizustiT, xolo Tu lagranJis mamravlebi

SerCeulia (8.14) sistemis amonaxsnebis saSualebiT, maSin

aseTi lagranJis funqciis kerZo warmoebulebi nulis

toli xdeba eqstremumis wertilebSi. (SevniSnoT, rom

(8.14) sistemaSi gvaqvs n m gantoleba da 1n m

ucnobi) Tu dasaSvebi simravle Ria araa, TviTon miznis

funqciis warmoebuli SesaZloa ar gaxdes nulis toli

eqstremumis wertilebSi. rogorc vxedavT, lagranJis

meTodi (eqstremumis wertilebis gansazRvra (8.14)–is

amonaxsnebis saSualebiT) warmoadgens SezRudvaTa mox-

snis saSualebas eqstremalur amocanaSi tolobis tipis

SezRudvebiT. amitom, Teorema 1–is pirobebSi, (8.3) amo-

canisaTvis kritikuli wertilebis simravle ganisazRvre-

ba Semdegnairad:

0 1

0

, ,

( , , )

m

nK xx

arsebobs aranulovani iseTi

rom aris (4.18)-is amonaxsni

.

87

agebis Tanaxmad, loc min(8.3), loc max(8.3) .K

magaliTi. amovxsnaT amocana: 2 2extr, 1.xy x y

amocanas mivceT standartuli saxe:

0 ,f x y xy , 2 2

1 , 1f x y x y da

2 2

1 0 0 1( , , , ) 1x y xy x y L .

kritikuli wertilebis gansazRvris mizniT Sevad-

ginoT sistema:

0 1

0 1

2 2

1

2 0,

2 0,

1 0.

y xx

x yy

x y

L

L

L

(8.15)

aq sami gantolebaa oTxi ucnobiT. radgan gantolebebi

0 1, cvladebis mimarT erTgvarovnebia, amitom cvla-

debis ricxvis Semcireba SegviZlia ori SesaZlo Sem-

Txvevis cal-calke ganxilvis xarjze.

a). davuSvaT 0 0 . maSin 1 0 0x y rac ewinaaR-

mdegeba (8.15) sistemis mesame gantolebas. e.i. es Sem-

Txveva kritikul wertils ar iZleva.

b). ganvixiloT 0 0 SemTxveva. SegviZlia 0 -ze gavyoT

(8.15)-is pirveli ori gantoleba, ris Semdegac 1

0

SegviZlia CavTvaloT erT cvladad da is aRvniSnoT

1 -iT. (8.15) sistema miiRebs saxes:

88

1

1

2 2

2 0,

2 0,

1.

y x

x y

x y

(8.16)

Tu SevkrebT I da II gantolebebs, miviRebT

12 1 0x y , saidanac 1 1/ 2 , an x y .

jer ganvixiloT 1 1/ 2 . maSin (8.16)- is I da II

gantoleba gvaZlevs x y , xolo III gantolebidan

miviRebT: 2

.2

x y e.i. (8.15)–is amonaxsnebia

2 2 1, 1

2 2 2, , da radgan 1/ 2,1 0 , amitom

2 2

2 2, K . Tu x y , wina SemTxvevis msgavsad

miviRebT 2 2

2 2, K , radgan 2 2 1

, 12 2 2

, , aris

(8.15)–is amonaxsnebi.

geometriulad, 0 ,f x y xy -is grafiki warmo-

adgens zedapirs samganzomilebian sivrceSi, xolo

2 2

1 , 1 0f x y x y SezRudva wrewirs amave siv-

rceSi.

89

gavixsenoT, rom f x funqciis donis wirebi ewodeba

Semdegi saxis simravleebs: ,L x f x R .

sibrtyeze gamovsaxoT dasaSvebi simravle 1 ,f x y

2 2 1 0x y , romelic warmoadgens erTeulradiusian

wrewirs centriT koordinatTa saTaveSi da miznis fun-

qciis 0 ,f x y xy donis wirebi, romlebic warmoadgenen

hiperbolebs.

Tu 0f x warmoebadi funqciaa, maSin 0f x gradients

aqvs funqciis uswrafesi zrdis mimarTuleba, xolo anti-gradients uswrafesi klebis mimarTuleba. (ix. punqti funqciis uswrafesi cvlilebis mimarTulebis gansaz-

Rvris Sesaxeb). Cvens amocanaSi 0 ,f x y x (ix. wina na-

xazi). geometriulad, am amocanis amoxsna niSnavs dasaSveb

simravleze (anu 2 2 1x y wrewirze) im wertilebis pov-

nas, romlebic Seesabamebian minimaluri (maqsimaluri)

mniSvnelobis mqone L donis wirebs. (donis wiris yovel

wertilSi miznis funqciis mniSvneloba erTi da igivea). naxazidan Cans, rom, magaliTad pirvel meoTxedSi, miT

metia miznis funqciis mniSvneloba, rac ufro Sors aris

donis wiri koordinatTa saTavidan da maT Soris yvelaze

Sors myofi da Tan iseTi, romelsac saerTo wertili aqvs

90

dasaSveb simravlesTan, aris is wiri, romelic exeba wre-

wirs (ix. naxazi). Sexebis wertilis koordinatebi SeiZleba

vipovoT Semdegnairad: Sexebis wertilSi hiperbolis da

wrewiris mxebebis sakuTxo koeficientebi emTxvevian

erTmaneTs. Tu ganvixilavT y -s rogorc x -is aracxad

funqcias da gavawarmoebT 2 2 1x y da xy gantole-

bebs miviRebT

2 2 0x yy , 0y xy .

Tu maT davumatebT amocanis SezRudvas, saZiebeli

wertilisTvis miviRebT Semdeg sistemas:

2 2

2 2 0,

0,

1.

x yy

y xy

x y

2 2

,

1.

x y

y x

x y

miviReT igive kritikuli wertilebi:

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2, , , , , , , .

4. kritikuli wertilebis gamokvleva

vaierStrasis Teoremisa da misi Sedegebis

gamoyenebiT

es midgoma Zalze zogadia, amitom ganvixiloT kon-

kretul magaliTze.

magaliTi. 2 2extr, 1.xy x y /amoxsnis gagrZe-

leba/. rogorc zemoT miviReT, 2 2 2 2, ,

2 2 2 2,K ,

anu gvaqvs oTxi kritikuli wertili. 2 2 1x y simravle

gansazRvravs wrewirs, rac kompaqturi simravlea, xolo

0f xy uwyveti funqciaa. maSasadame, vaierStrasis Teo-

remis ZaliT, Cvens amocanaSi arsebobs globaluri maqsi-

malebi da minimalebi.

91

radgan glextr locextr K , amitom K –Si aris ro-

gorc globaluri maqsimali, aseve globaluri minimali.

Semdegi Casmebi:

0 02 / 2, 2 / 2 1/ 2, 2 / 2, 2 / 2 1/ 2,f f

gviCveneben, rom miznis funqcia dasaSveb simravleze

Rebulobs mxolod or gansxvavebul mniSvnelobas.

amitom,

2 / 2, 2 / 2 gl max, 2 / 2, 2 / 2 gl min .

5. kritikuli wertilebis gamokvleva uSualod

ganmartebis safuZvelze

gamovikvlioT, magaliTad, 2 / 2, 2 / 2 . warmovadgi-

noT 2R -is nebismieri ,x y wertili Semdegi saxiT:

1 2, 2 / 2, 2 / 2 h ,hx y anu

1 22 / 2 h , 2 / 2 hx y .

Tu ,x y wertili akmayofilebs 2 2 1x y SezRudvas,

maSin 1 2h ,h agreTve izRudebian, rasac male vnaxavT. jer

SevafasoT:

1 2

0 0 0, 2 / 2, 2 / 2 2 / 2 h 2 / 2 h 2 / 2 2 / 2f f x y f

= 1 2 1 2h +h 2 / 2 h h . (8.17)

(8.17) mxolod im SemTxvevaSi mogvcems saWiro informa-

cias, Tu gaviTvaliswinebT 2 2 1x y SezRudvas:

2 2 2 2

1 2 1 2 1 22 / 2 h 2 / 2 h 1 h h 2 / 2 h h 0

2 2

1 2 1 2h h 2 / 2 h h

,

ris safuZvelzec (8.17) gvaZlevs:

92

2 2 2

1 2 1 2 1 2

0

1 1h h 2h h h h 0

2 2f

,

e.i. 2 / 2, 2 / 2 gl max .

6. kritikuli wertilebis gamokvleva

eqstremalurobis meore rigis sakmarisi

pirobis safuZvelze

kvlav ganvixiloT amocana tolobis tipis SezRud-

vebiT

0 1(x) extr, (x) 0, , (x) 0mf f f ,

sadac 0 1, , , mf f f funqciebi orjer uwyvetad warmoe-

badia. aRvniSnoT M –iT dasaSvebi simravle

1x (x) (x) 0 .n mM f f

vidre Teoremas (eqstremalurobis meore rigis sakmaris

pirobas) CamovayalibebT da davamtkicebT, moviyvanoT

ramdenime martivi faqti.

yoveli sasrulganzomilebiani 1, , m veqto-

risaTvis ganvsazRvroT funqcia

0

1

(x) (x) (x), x ,m

i i n

i

l f f R

romelsac, M simravlis gansazRvris Tanaxmad, aqvs Sem-

degi mniSvnelovani Tviseba

0(x) (x), x .l f M (8.18)

Tu x aris eqstremumis wertili, an Tundac kriti-

kuli wertili, xolo ( 1)m -cali lagranJis mamravli

11, , , m isea SerCeuli, rom veqtori 1

1x , ,x , , , ,1n

m

akmayofilebs kritikuli wertilis (8.11) pirobas, maSin

93

0

1

(x) (x) (x), x ,m

i i n

i

l f f

funqcia, garda (8.18)–isa, akmayofilebs agreTve x 0l

tolobas.

Teorema 2 /eqstremalurobis meore rigis sakmarisi

piroba/. vTqvaT, :i nf R R funqciebs, 0,1, , ,i m aqvT

uwyveti meore rigis kerZo warmoebulebi, x M da

sruldeba Semdegi ori piroba:

a) 1ˆ ˆ ˆ, , m , iseTi, rom x 0l , sadac

0

1

ˆ(x) (x) (x), x ;m

i i n

i

l f f

b) ˆ ˆh x h 0 h x h 0T Tl l sruldeba yoveli iseTi

aranulovani h n veqtorisaTvis, romelic akmayo-

filebs SezRudvebs:

1

xx h 0, 1, , h 0, 1, , .

x

njT k

j kk

ff j m j m

maSin, x aris mkacri lokaluri maqsimali (minimali)

Semdeg eqstremalur amocanaSi:

0 1(x) extr, (x) 0, , (x) 0mf f f .

damtkiceba . davamtkicoT mkacri lokaluri maqsi-

malis arseboba. lokaluri minimalis arseboba analogiu-

rad mtkicdeba.

davuSvaT sawinaaRmdego. vTqvaT, Teoremis pirobebi

sruldeba, magram x araa mkacri lokaluri maqsimali.

maSin, arsebobs veqtorTa mimdevroba 1

xi iM

iseTi,

rom

94

0 0ˆ(x ) (x),if f i (8.19)

da ˆlimx xii

. SemoviRoT aRniSvnebi:

1

ˆ ˆx x , h x x , .i i i i

i

i

rogorc vxedavT, ˆx x+ h , .i i i i cxadia, yoveli hi

veqtori Zevs erTeulovan sferoze, romelic kompaqturi

simravlea da amitom arsebobs 1

hi i

mimdevrobis qvemim-

devroba, krebadi romeliRac h -isaken. radgan nR -is

norma uwyvetadaa damokidebuli veqtorebze, h 1 . zo-

gadobis SeuzRudavad SegviZlia vigulisxmoT, rom Tvi-

Ton 1

hi i

aris krebadi h -isaken.

teiloris formulis Tanaxmad, , 1, ,i j m

ricxvebisTvis gvaqvs:

ˆ ˆ ˆx h x x h hT

j i i j j i i i if f f o . (8.20)

pirobis Tanaxmad, x 0, x 0,j j if f amitom (8.20)

igivea, rac

ˆ0 x h T i

j i

i

of

. (8.21)

Tu (8.21)–Si gadavalT zRvarze roca i , miviRebT:

x h 0, 1, , ,T

jf j m

anu am h -isaTvis SegviZlia gamoviyenoT Teoremis b)

piroba, romlis ZaliTac ˆh x h 0Tl . aRvniSnoT

ˆh x h 0Tl da / 2 . radgan x 0l , tei-

loris formula gvaZlevs:

95

21ˆ ˆx x h x h h

2

T

i i i i i i il l l o .

radgan hi i i , amitom

2

2 2

2ˆ ˆx x h x h 2

iT

i i i

i i

ol l l

. (8.22)

radgan limh hii

, amitom arsebobs nomeri 0i , iseTi rom

ˆh x h / 2.T

i il Cven SegviZlia aviRoT 0i imdenad

didi, rom roca 0i i , Sesruldes

2

22

2

i

i

o

(radgan

2

2lim 0

i

ii

o

).

sabolood, (8.22)–dan vRebulobT, rom yoveli 0i i -

isaTvis

2

2ˆx x 0i

i

l l

0 0ˆ ˆx x 0 x xi il l f f

rac ewinaaRmdegeba (8.19)–s.

magaliTi. davasruloT 2 2extr, 1xy x y amo-

canis ganxilva, romelSic 2 22 2, ,

2 22 2,K .

2 2,

2 2 wertilisaTvis gvqonda 0 1

ˆ ˆ1, 1/ 2 , ami-

tom 1 2 2

2, 1l x y xy x y . matrica 2 2

,2 2

l 1 1

1 1

araa gansazRvruli arc dadebiTad, arc uaryofiTad.

2 2

1 , 1f x y x y , 2 2,1 2 2

2, 2f .

96

ganvsazRvroT 1 2h h ,h veqtorebi 1 x h 0Tf

pirobidan:

1

1 2

2

h2, 2 0 h h ,

h

anu h , , 0. dasasrul, radgan

21 1

, 4 0, 0,1 1

amitom 2 2,

2 2locmax .

kritikuli wertilebis meore wyvilisaTvis:

2 2,

2 2 , gvqonda 0 1

ˆ ˆ1, 1/ 2, amitom

1 2 2

2, 1l x y xy x y , 2 2

,2 2

l 1 1

1 1

da verafers vambobT gansazRvrulobaze.

2 2

1 , 1f x y x y , 2 2,1 2 2

2, 2f .

Semdeg,

1

1 2

2

h2, 2 0 h h ,

h

anu CvenTvis saintereso veqtorebs aqvT saxe:

h , , 0.a dasasrul,

21 1

, 4 0, 0,1 1

a

amitom 2 2,

2 2locmin . rogorc vxedavT, II rigis piro-

bebis gamoyeneba mxolod lokalur informacias gvaZlevs

kritikuli wertilebis Sesaxeb. miuxedavad amisa, es yve-

laze mZlavri meTodia da xSirad gamoiyeneba.

97

7. lagranJis mamravlebis maTematikuri da

ekonomikuri interpretaciebi

ganvixiloT minimizaciis amocana

0 1 1(x) min, (x) , , (x) ,m mf f b f b

anu standartuli saxiT:

0 1 1(x) min, (x) 0, , (x) 0m mf b f b f . (8.23)

cxadia, (8.23)–is yoveli amonaxsni damokidebulia

1b , , mb b parametr– veqtoris mniSvnelobaze. magali-

Tad, Tu 0 1 ,f x f x x x , maSin 0 1( ) min, ( )f x f x b

amocanas aqvs trivialuri amonaxsni ˆ .x b

davuSvaT, , 1 ,if i m , funqciebi aris imdenad kar-

gi, rom b veqtoris yoveli SesaZlo mniSvnelobisaTvis

b B , (8.23) amocanas aqvs erTaderTi amonaxsni x b , da

rom calsaxad ganisazRvrebian lagranJis mamravlebi

11, b , , bm , romelTaTvisac Teorema 1–is ZaliT

0

1

x b b x b 0m

i i

i

f f

. (8.24)

radgan TviTon lagranJis funqcia am mamravlebiT aris

0

1

x b xm

i i i

i

f b f

,

amitom yoveli x–isaTvis, romelic akmayofilebs (8.23)-is

SezRudvebs

(x) 0, 1, ,i ib f i m , (8.25)

sruldeba

0 0

1

x x b xm

i i i

i

f f b f

.

kerZod, roca x=x b , gvaqvs

98

0 0

1

x b x b b x bm

i i i

i

f f b f

. (8.26)

gavawarmooT (8.26)–is orive mxare jb -Ti:

k0 0

1

x b x b x b

x

n

kkj j

f f

b b

1 j

bx b

b

mi

i i

i

b f

k

1 1

x b x bb

x

m nii

i ki kj j

fb

b b

(marjvena mxareSi meore wevri nulis tolia (8.25)–is gamo)

k

0

1 1

x b x b x bb b

x x

n mi

j ik kk i j

f f

b

= ((8.24) –is ZaliT) bj .

amgvarad,

0 x bb , 1, , .j

j

fj m

b

(8.27)

rogorc vxedavT, lagranJis optimaluri mamravlebi

gviCveneben, Tu ramdenadaa mgrZnobiare miznis funqciis

mniSvneloba SezRudvis parametrebis cvlilebis mimarT.

magaliTad, Tu lagranJis romelime mamravli nulis toli

aRmoCnda, Sesabamisi parametris mcire cvlileba gavle-

nas ar iqoniebs miznis funqciis eqstremalur mniSvnelo-

baze.

lagranJis mamravlebis interpretacia gansakuTre-

biT Sinaarsiania ekonomikur amocanebSi. maragis ganawi-

lebis ekonomikur amocanebSi miznis funqcias aqvs Rire-

bulebis ganzomileba (Rirebuleba aris fasis namravli

99

produqciis moculobaze), xolo SezRudvebis saSua-

lebiT dgindeba produqciis gansazRvruli moculoba.

(8.27)–is mixedviT, lagranJis mamravls aqvs fasis ganzo-

mileba. am mizezis gamo, lagranJis mamravlebs uwodeben

Crdilovan fass mocemul produqciaze.

8. funqciis uswrafesi cvlilebis

mimarTulebis gansazRvra

vTqvaT, nO R

simravlea; x O da

:f O aris war-

moebadi funqcia x -

Si. Cveni amocanaa x

wertilSi f fun-

qciis uswrafesi cvlilebis mimarTulebebis gansazRvra.

Cveulebriv, mimarTuleba ewodeba nR -is erTeulovani

sferos elementebs. mag. Tu h nR aris erT-erTi

mimarTuleba (e.i 1Thh ) maSin sakmaod mcire 0

ricxvebisaTvis x h O da h mimarTulebiT f –is

cvlilebis siCqare x -Si aris:

0

1ˆ ˆlim (x h) (x)f f

,

radgan f warmoebadia x -Si, amitom es sidide aris

x Tf h . amgvarad, Cven gvainteresebs is mimarTulebebi,

romlisTvisac x Tf h iRebs eqstremalur mniSvnelobas.

ganmarteba: x wertilSi f-is uswrafesi cvlilebis

mimarTulebebi, warmoadgenen Semdegi eqstremaluri

amocanis

O

x

x h

x h

100

x Tf h extr , 1Thh , (8.28)

eqstremumis wertilebs.

amocana rom aratrivialuri iyos, unda vigulisxmoT

xf ≠0 (aranulovani veqtoria). axla amovxsnaT (8.28).

SevadginoT lagranJis funqcia:

1 0 0 1ˆ( , , ) ( ) 1T Th f x h hh L .

amovweroT sistema kritikuli wertilebis gansazRvri-

saTvis:

1

0 11 1

0 1

0 1

ˆ(x)2 h 0,

h x

ˆ(x)2 h 0,

h x

ˆ(x)2 h 0,

h x

hh 1,

i

i i

n

n n

T

f

f

f

L

L

L

anu veqtorulad

0 1ˆ(x) 2 h 0,

hh 1.T

f

(8.29)

SevamciroT cvladebis ricxvi SemTxvevaTa ricxvis zrdis

xarjze.

1. vTqvaT 0 0 maSin 1 0 0h rac araa Tavsebadi

hhT=1-Tan.

2. ganvixiloT 0 0 SemTxveva. SegviZlia 0 -ze gavyoT

(8.29)–is pirveli gantoleba, ris Semdegac 1 / 0 ,

SegviZlia CavTvaloT erT cvladad da is aRvniSnoT

101

1 -iT. (8.29) sistema iRebs saxes (λ=0 gamoiwvevda

x 0f -s):

1ˆ(x) 2 h 0,

hh 1,T

f

saidanac,

2

2

1x 1

4f

anu 1

1x

2f ,

e.i (8.29)-is amonaxsnebia

x 1

ˆ, x ,1x 2

ff

f

, rom-

lebic gvaZleven or kritikul wertils:

K= ˆ ˆx / xf f .

gamovikvlioT es wertilebi. vaierStrasis Teoremis

gamoyeneba SegviZlia, radgan erTeulovani sfero kom-

paqturia, xolo miznis funqcia _ uwyveti. amitom kri-

tikul wertilebs Soris erTi aris globaluri minimali,

meore globaluri maqsimali. miznis funqciaSi Casma gvaZ-

levs, rom gradientis mimarTuleba, anu ˆ ˆx / xf f

aris globaluri maqsimali (8.28)- Si, anu f-is uswrafesi

zrdis mimarTuleba, xolo antigradientis mimarTuleba

ˆ ˆx / xf f aris f-is uswrafesi klebis mimarTuleba

x -Si.

savarjiSoebi

#82. 2 3min, 0y x x y .

#83. 2 24 extr, 1x y x y

#84. 2 2 extr, 4 3 1x y x y

#85. extr, 1xye x y

102

#86. 2 25 4 extr, 1x xy y x y

#87. 3 2min, 0x x y

#88. 2 2 4 4extr, 1x y x y

#89. erTeulovan wreSi CavxazoT udidesi farTobis mqone

marTkuTxedi.

#90. x x extr, x 1TA , Tu cnobilia, rom

, 1

x ,n

n i j i jA a

-simetriuli matricaa.

#91. ricxvi 8 gayaviT or nawilad ise, rom maTi namravlis

namravli maT sxvaobaze iyos maqsimaluri (tartalia).

#92. CaxazeT erTeulovan wreSi samkuTxedi, romlisTvisac

gverdebis kvadratebis jami aris maqsimaluri.

#93. sibrtyeze mocemulia sami wertili 1 2 3x ,x ,x . moZebneT

iseTi x wertili, saidanac 1 2 3x ,x ,x wertilebamde man-

Zilebis kvadratebis jami aris minimaluri.

103

Tavi 9. gluvi eqstremaluri amocana

tolobisa da utolobis tipis

SezRudvebiT

1. mokle reziume

gluvi amocana tolobisa da utolobis tipis SezRud-

vebiT ewodeba Semdeg eqstremalur amocanas:

0 1 1(x) extr, (x) 0, , (x) 0, (x)=0, , (x)=0. k mg g g f f (9.1)

sadac yoveli : , : , ( 0, , 1, , )j n i ng f j k i m

aris sakmarisad gluvi funqcia. radgan eqstremumis gan-

martebaSi monawileoben utolobebi, amitom miznis fun-

qcia da utolobebis ganmsazRvreli funqciebi erTi da

igive simboloTia aRniSnuli. (9.1)-is amomwurav analizs

Cven ar moviyvanT, radgan Tavisi sirTulisa da specifi-

kis gamo es ufro speckursis Temaa. praqtikuli amocane-

bis amoxnisas xelsayrelia (9.1)-is nacvlad ganvixiloT

ori minimizaciis amocana (erTgan 0 ( ) ming x , meoregan

ki 0 ( ) ming x ), radgan minimizaciisa da maqsimizaciis

amocanebisaTvis algoriTmebi gansxvavdeba mcire deta-

lebiT, rac advilad iwvevs Secdomas. amitom zogadobis

SeuzRudavad, SemdegSi ganvixilavT minimizaciis gluv

amocanas:

0 1 1(x) min , (x) 0, , (x) 0, (x)=0, , (x)=0. k mg g g f f (9.2)

am amocanaSi miznis funqciaa 0g ( )x , xolo dasaSvebi sim-

ravle aris:

1 1M= x (x) 0, , (x) 0, (x)=0, , (x)=0.n k mg g f f

am paragrafSi Cven davasabuTebT, rom tolobis da

utolobis tipis SezRudvebis mqone gluvi amocanis

gamokvleva unda moxdes Semdegi TanmimdevrobiT:

104

1. maTematikuri modelis Sedgena, anu amocanis formu-

lireba (9.2) saxiT;

2. (9.2) amocanisaTvis lagranJis funqciis Sedgena:

00 1

( , , , ) ( ) ( )k m

j j i ij i

x g x f x

L , (9.3)

sadac i da j R lagranJis mamravlebia, xolo

1( , , )k , 1( , , )m .

3. (9.2) amocanisaTvis kritikuli wertilebis simravlis

gansazRvra:

0, 1,..., ,x

(x) 0, 1,... ,

(x) 0, 1,..., ,

i

j j

S

i n

g j k

f s m

L

(9.4)

gantolebaTa sistemis 0(x, , , ) amonaxsnebs Soris

SevarCioT iseTebi, rom Sesruldes:

a) 0( , , ) 0,

b) 0( , ) 0,

g) 1, ,(x) 0,jg kj .

aseTi 0(x, , , ) amonaxsnebis x -ebis simravle aR-

vniSnoT K -Ti da mas vuwodoT (9.2) amocanis kriti-

kuli wertilebis simravle. lagranJis principis Ta-

naxmad locmin(9.2) Sedis K -Si.

4. kritikuli wertilebis gamokvleva im mizniT, rom

maTgan amovarCioT minimumis wertilebi.

105

2. maTematikuri modelis Sedgena

sailustraciod, ganvixiloT fasiani qaRaldebis

portfelis Sedgenis amocana.

erT-erTi umniSvnelovanesi amocana, romlis gadaW-

rac uwevT sainvesticio firmebs (bankebs, fondebs, sadaz-

Rvevo kompaniebs) aris fasiani qaRaldebis optimaluri

portfelis Sedgenis amocana. portfeli niSnavs sxvada-

sxva saxis fasian qaRaldebSi (obligaciebi, aqciebi, saban-

ko sadepozito sertifikatebi da sxva) Cadebuli Tanxebis

moculobebis erTobliobas. am amocanis gadasaWrelad,

misi maTematikuri sirTulisa da ekonomikuri mniSvnelo-

bis gamo, damuSavebulia sxvadasxvanairi modelebi.

vigulisxmoT, rom damdeg sainvesticio periodSi

SeiZleba naRdi C kapitalis Cadeba n saxis fasian qaRal-

debSi. vTqvaT, x j aris j -uri saxis fasian qaRaldSi Cade-

buli kapitalis moculoba (dolarebSi), 1, ,j n . maSin,

cxadia, rom 1 2x x x , x 0, 1, ,n jC j n .

vTqvaT, arsebobs statistikuri monacemebi bolo T

wlis ganmavlobaSi TiToeuli saxis kapitaldabandebis

Sesaxeb, romlebic asaxaven am periodSi fasebis meryeobas

da dividendebis gacemas. am monacemebis saSualebiT,

SesaZlebelia Sefasdes kapitaldabandebebidan miRebuli

Semosavali fasiani qaRaldis yoveli saxisTvis. vTqvaT,

jr t aris t welSi j saxis fasian qaRaldSi dabandebuli

erTi dolarisgan miRebuli mTliani Semosavali. maSin:

1j j j

j

j

p t p t d tr t

p t

,

106

sadac jp t aris j saxis fasiani qaRaldis fasi wlis

(zogadad, sainvesticio periodis) dasawyisSi, jd t – t

welSi miRebuli jamuri dividendebi.

SevniSnoT, rom jr t SesaZloa Zlier icvlebodes

wlidan wlamde da SeiZleba hqondes nebismieri niSani.

imisaTvis, rom Sefasdes j saxis fasian qaRaldSi daban-

debis mizanSewoniloba, unda Sefasdes j saxis fasian

qaRaldSi dabandebuli erTi dolarisgan miRebuli Semo-

savlis saSualo anu mosalodneli mniSvneloba:

1

1 T

j j

t

r tT

.

mosalodneli Semosavlis mTliani sidide iqneba:

1

x xn

j T

j

j

E

,

sadac 1, , n , 1x x , , xn .

(SevniSnoT, rom ar aris kavSirSi amave asoTi aRniSnul

lagranJis mamravlTan).

modeli 1. am amocanisTvis SedarebiT martivad gamo-

sakvlev (Sesabamisad, naklebad adekvatur) maTematikur

models aqvs saxe:

1

x maxn

j

j

j

z

,

im pirobiT, rom

1

x , x 0n

j j

j

C

.

sxva sityvebiT, gvinda movaxdinoT mTliani mosalodneli

mogebis maqsimizacia imis gaTvaliswinebiT, rom SezRu-

dulia investiciebis mTliani moculoba da valis aReba

ar daiSveba. amave models SeuZlia damatebiT gaiTvalis-

107

winos sxva SezRudvebi. ganvixiloT SedarebiT gavrcele-

buli ori maTgani.

sainvesticio firmebis umravlesoba zRudavs kapi-

taldabandebaTa moculobas Cveulebrivi aqciebisTvis,

radgan maTgan Semosavals axasiaTebs mniSvnelovani rxe-

vebi (anu Zlieri cvlilebebi). aseTi SezRudva SeiZleba

ACaiweros formuliT:

1

1

j

j J

x b

,

sadac 1J Sedgeba Cveulebrivi aqciebis sxvadasxva saxee-

bis indeqsebisgan, xolo 1b aRniSnavs Cveulebriv aqciebSi

maqsimalur dasaSveb kapitaldabandebas.

mravali sainvesticio firmis mmarTveloba aucileb-

lad miiCnevs kapitalis nawilis SenarCunebas naRd fulad

an mis ekvivalentur raime formad, meanabreTa moTxovni-

lebebis dakmayofilebis mizniT. es SezRudva SeiZleba

ACaiweros formuliT:

2

2

j

j J

x b

,

sadac 2J simravlis indeqsebi Seesabameba fasian qaRal-

debs, romlebic naRdi fulis ekvivalenturia (magaliTad,

Semnaxvel angariSebs, mimdinare angariSebs); 2b aris auci-

lebeli naRdi saSualebebis minimaluri raodenoba.

am martivi modelis ZiriTadi nakli imaSi mdgomare-

obs, rom investiciebTan dakavSirebuli riskis gaTvalis-

wineba ar xdeba. fasiani qaRaldebis portfeli, romelsac

SevadgenT am modelis gamoyenebiT, SeiZleba gvpirde-

bodes maRal saSualo mosalodnel Semosavals, magram

investiciebTan dakavSirebuli riskic aseve maRali

iqneba. maRali riskis gamo, realuri Semosavali SesaZloa

gacilebiT dabali aRmoCndes, vidre Teoriulad mosa-

lodneli.

108

modeli 2. am modelSi xdeba fasiani qaRaldebis

TiToeul saxesTan dakavSirebuli riskis gaTvaliswineba.

zogierTi fasiani qaRaldis, e.w. “spekulaciur aqciebs”

aqvT tendencia Zlieri rxevebisadmi, rac zrdis riskis

faqtors, Tumca saSualo mosalodneli mogebac maRalia,

radgan aseTi qaRaldebis kursma SesaZloa Zlier aiwios.

meore mxriv. “usafrTxo investiciebi”, rogoricaa mim-

dinare angariSebi an sabanko depozituri sertifikatebi,

iZleva nakleb Semosavals.

sainvesticio riskis zomad gamoiyeneba Semosavlis

gadaxra misi saSualo mniSvnelobidan bolo T wlis

ganmavlobaSi. aRvniSnoT jj -Ti j saxis fasiani qaRaldis

dispersia (sainvesticio riski) romelic gamoiTvleba

formuliT:

2

1

1 T

jj j j

t

r tT

.

gasaTvaliswinebelia, rom fasiani qaRaldebis romelime

jgufis kursi SesaZloa damokidebuli iyos ekonomikis

garkveuli sferos mdgomareobaze; am sferos Cavardna

gamoiwvevs am jgufis fasiani qaRaldebis kursis dacemas.

iseTi fasiani qaRaldebis magaliTebs, romelTa kursebi

erTdroul rxevas eqvemdebareba, warmoadgenen saavto-

mobilo da sanavTobo firmebis aqciebi, komunaluri mom-

saxurebis sawarmoTa aqciebi da sxva. msgavsi riskis Sesam-

cireblad, investiciebi unda ganawildes fasiani qaRal-

debis sxvadasxva jgufebze. aseT ganawilebaSi gamoiyeneba

Semosavlis doneTa Tanafardobis Sefasebebi fasiani

qaRaldebis yoveli wyvilisTvis. es Tanafardoba gamoisa-

xeba kovariaciis ij koeficientebiT, romlebic gamoiT-

vleba bolo wlebis statistikuri monacemebis safuZ-

velze:

109

1

1 T

ij i i j j

t

r t r tT

.

SevniSnoT, rom roca i j , es sidide gadadis j saxis

fasiani qaRaldis dispersiaSi. amgvarad, sainvesticio

riskis sazomad SegviZlia gamoviyenoT sidide:

1 1

x x x xn n

i j T

ij

i j

V Q

,

sadac ijQ aris kovariaciis n n zomis matrica,

Sedgenili n saxis fasiani qaRaldisaTvis.

portfelis gansazRvris procesSi fasiani qaRalde-

bis mflobeli SesaZloa dainteresebuli iyos garkveuli

moculobis saSualo mosalodneli mogebis miRebiT mini-

maluri riskis pirobebSi. Sesabamis optimizaciis amoca-

nas aqvs saxe:

x x minTV Q ,

im pirobiT, rom

1

x , x 0, xn

j j T

j

C R

,

sadac R aris mogebis minimaluri saSualo mosalodneli

mniSvneloba portfelis arCevis dros. modelSi, SesaZ-

loa, agreTve iyos firmis politikasTan dakavSirebuli

SezRudvebi, rogorc wina modelSi iyo ganxiluli.

3. kritikuli wertilebis gansazRvra

Semdeg Teoremas, romelic warmoadgens minimumis pir-

veli rigis aucilebel pirobas minimizaciis

0 1 1(x) min , (x) 0, , (x) 0, (x)=0, , (x)=0. k mg g g f f (9.5)

amocanaSi, tradiciulad uwodeben lagranJis mamravlTa

wess gluvi amocanebisTvis tolobis da utolobis tipis

SezRudvebiT.

110

Teorema 1 /minimumis pirveli rigis aucilebeli

piroba/. vTqvaT, jg da if funqciebi ( 0, , , 1, , )j k i m

uwyvetad warmoebadebia x n wertilis raime midamoSi,

xolo x aris lokaluri minimumis wertili (9.2) amoca-

naSi. maSin arsebobs erTdroulad nulis aratoli lag-

ranJis mamravlebi 1 1 0ˆ ˆ ˆ ˆ ˆ, , , , , , ,m k romelTaTvisac

sruldeba Semdegi pirobebi:

a) x -is mimarT lagranJis funqciis stacionalurobis

piroba:

01 1

ˆˆ ˆ ˆ(x, , , ) ˆ ˆ ˆ ˆ ˆ ˆ0, 1, , , , , , ,x

m kii n

L;

b) niSnebis SeTanxmebis piroba: yoveli 0, ,j k -sTvis

0j ;

g) ˆ ˆ(x)=0, 1, , .j jg j k

damtkiceba. Tu aRvniSnavT: ˆ xi ib g , maSin x aris

lokaluri minimali Semdeg amocanaSi:

0 1 1 k 1ˆ ˆ(x) min , (x) , , (x)=b (x)=0, , (x)=0. k mg g b g f f

amitom SegviZlia gamoviyenoT wina paragrafis analo-

giuri Teorema da davaskvnaT, rom arsebobs erTdroulad

nulis aratoli lagranJis mamravlebi 1 1 0ˆ ˆ ˆ ˆ ˆ, , , , , , ,m k

romelTaTvisac sruldeba x -is mimarT lagranJis fun-

qciis stacionalurobis piroba:

0ˆˆ ˆ ˆ(x, , , )

0, 1, ,xi

i n

L

.

amgvarad, dasamtkicebeli rCeba b) da g). simartivisTvis,

maT davamtkicebT kerZo SemTxvevisTvis, romelic, prin-

cipSi, srulad amowuravs Teoremis praqtikul gamoyene-

bebs.

111

davuSvaT, sruldeba igive pirobebi, rac sakmarisia

lagranJis mamravlebis ekonomikuri Sinaarsis gasarkve-

vad da vigulisxmoT, rom 1b , , mb b parametr – veqto-

ris yoveli mniSvnelobisTvis, romelic mcired ganxvav-

deba 1ˆ ˆ ˆb , , mb b -sgan, Semdeg amocanas

0 1 1(x) min , (x) 0, , (x) 0, (x)=0, , (x)=0,k mg g g f f

aqvs erTaderTi amonaxsni x b . yoveli 1, ,j k –sTvis

SesaZlebelia ori SemTxveva: ˆ 0jb (e.i. x 0jg ) an

ˆ 0jb (e.i. x 0jg ). pirvel SemTxvevaSi erTdroulad

sruldeba Semdegi ori faqti: erTi mxriv, 0 x

ˆj

j

g

b

,

xolo meore mxriv erTi cvladis funqcia 0 x jg b , gan-

xiluli ˆ ˆ,j jb b

naxevradRia SualedSi sakmaod mcire

-sTvis, maqsimums aRwevs ˆjb -Si; amitom, 0j . meore

SemTxvevaSi, 0 x jg b , ganxiluli ˆ ˆ,j jb b Ria Sua-

ledSi sakmaod mcire -sTvis, maqsimums aRwevs ˆjb -Si da

amitom, 0j .

magaliTi. erTeulovan wrewirSi CavxazoT maqsima-

luri farTobis mqone marT-

kuTxedi.

amoxsna. x -iT aRvniSnoT

horizontaluri gverdis

naxevari, y -iT vertikalu-

ris naxevari, maSin farTobi

aris 4xy , SezRudvebi aris

y

x

nax. 1

112

2 2x +y 1 da x 0, y 0 . amgvarad, gvaqvs:

0

1

2

2 2

1

( , ) min,

( , ) 0,

( , ) 0,

( , ) 1 0.

g x y xy

g x y x

g x y y

f x y x y

(9.6)

esaa gluvi eqstremaluri amocana tolobis da uto-

lobis tipis SezRudvebiT. SevadginoT misTvis lagranJis

funqcia:

2 2

1 2 0 0 1 2( , , , , , ) ( 1)x y xy x y x y L .

amovweroT gantolebebi, romlebic gansazRvraven kriti-

kul wertilebs:

0 1

0 2

1

2

2 2

/ 2 0,

/ 2 0,

0,

0,

1.

x y x

y x y

x

y

x y

L

L

(9.7)

es aris arawrfivi sistema xuTi gantolebiTa da eqvsi

cvladiT.

movZebnoT kritikuli wertilebi. am mizniT Cven ve-

ZebT (9.7)-is iseT 1 2 0( , , , , , )x y amonaxsnebs, romlebic

akmayofileben zemoT aRweril a), b), g) SezRudvebs.

visargebloT lagranJis funqciis erTgvarovnebiT mam-

ravlebis mimarT da SevamciroT cvladebis raodenoba

((9.7)-Si SemTxvevaTa ricxvis gazrdis xarjze):

1) jer ganvixiloT 0 0 . (9.7) miiRebs saxes:

2 2

1 1 2 22 0 1, 2 0x x x y y y ,

113

e.i. 2 2

1 22 0, 2 0,x x y y saidanac 2 22 ( ) 0x y

da radgan 2 2x +y 1, amitom 0 . 1 2 0, e.i. Tu

0 0 , rogori 1 2( , , , , ,0)x y amonaxsnic ar unda

hqondes (9.7)-s, yovelTvis 1 2( , , ,0) 0 , rac winaaR-

mdegobaSia a) SezRudvasTan. e.i 0 0 SemTxveva arc

erT kritikul wertils ar iZleva.

2) aviRoT 0 1 . axla SevecadoT (9.7)-Si movxsnaT

( , ) 0j jg x y saxis arawrfivobebi. amisaTvis cal-

calke unda ganvixiloT SemTxvevebi 0j da 0j

(b) pirobis ZaliT meti SemTxveva ar aris). 1 0 Tu

maSin

2

2

2 2

2 ,

2 ,

0,

1

y x

x y

y

x y

axla, Tu 2 0 , maSin 0y , 2 0x da winaaRmde-

gobaa g)-sTan. e.i. 1 0 -is dros

2 0 qveSemTxveva dag-

vrCa gansaxilveli.

aviRoT 2 0 . maSin

2 22 , 2 , 1x y y x x y

2 2 2 2 2 2 2 2 2 2 24 , 4 ,1 4 ,x y y x x y x y

e.i. 2 =1/4, da

2 2x =y , anu x y (g)–is ZaliT).

114

sabolood, 2 2x +y 1 gvaZlevs 1/ 2x y , e.i.

(9.7)-is amonaxsnia 1/ 2,1/ 2,1/ 2,0,0,1 , romlisaTvisac

sruldeba:

a) (1/ 2,0,1) 1

b) (0,0,1) 0

g) 1/ 2 0, 1/ 2 0 .

amitom 1/ 2,1/ 2 K .

amiT 2 0 qveSemTxvevis ganxilva dasrulda.

1 –is SemTxvevebi simetriulia ganxilulisa da iZle-

va igive kritikul wertils. sabolood, Cvens amocanaSi

aris erTaderTi kritikuli wertili: 1/ 2,1/ 2K .

4. kritikuli wertilebis gamokvleva

rodesac eqstremaluri amocana Seicavs utolobis

tipis SezRudvebs, misi kritikuli wertilebis gamokvle-

va rTulia. ZiriTad instruments warmoadgens vaierStra-

sis Teorema da misi Sedegebi, xolo, ukidures SemTxve-

vaSi, ganmartebis Semowmeba unda vcadoT. Tu funqciebs

aqvT damatebiTi Tvisebebi (amozneqiloba), maSin Cndeba

damatebiTi SesaZleblobebic.

magaliTi /gagrZeleba/. 2 2min, 0, 0, 1xy x y x y da ukve viciT,

rom kritikuli wertili erTaderTia 1/ 2,1/ 2K .

rogorc vxedavT, miznis funqcia 0g xy uwyvetia,

xolo dasaSvebi simravle

115

2 2

2, 1, 0, 0M x y x y x y

aris erTeulovani wrewiris

Caketili erTi meoTxedi, anu

kompaqturi simravle. amitom

amocanaSi arsebobs globalu-

ri minimali da igi moTavse-

bulia kritikul wertilebs

Soris. magram kritikuli wer-

tili erTaderTia, amitom

1/ 2,1/ 2 glmin(9.6) .

pasuxi: maqsimaluri farTobi aqvs kvadrats, romlis

gverdis sigrZea 2 .

es magaliTi sakmaod martivia, amitom uSualod

ganmartebis safuZvelzec SegviZlia gamovikvlioT

1/ 2,1/ 2 kritikuli wertili. amisTvis ganvixiloT

1 2

0 0 0(1/ 2 h ,1/ 2 h ) (1/ 2,1/ 2)g g g

1 2 1 2 1 2(1/ 2 h )(1/ 2 h ) 1/ 2 1/ 2 h h h h ; (9.8)

magram wertili 1 2(1/ 2 h ,1/ 2 h ) unda akmayofileb-

des SezRudvebs:

1 2 2 2 1 2 2 2 1 2(1/ 2 h ) (1/ 2 h ) 1 (h ) (h ) 2 (h h ) 0 ,

saidanac 1 2 1 2 2 21h h ((h ) (h ) )

2 . rac gvaZlevs:

2

1 21 2 2 2 1 2

0

1 h h(h ) (h ) h h 0

2 2 2g

.

amitom, 1/ 2,1/ 2 glmin(6.7) .

nax. 2

116

5. lagranJis meTodis geometriuli

interpretacia

rodesac 1n , maSin eqstremaluri amocanisaTvis

TvalsaCino geometriuli interpretaciis micema advi-

lia, radgan dasaSvebi simravle da miznis funqcia gamoi-

saxebian sibrtyeze,

rogorc nax. 3 –ze.

rodesac 2n ,

geometriuli inter-

pretaciisaTvis viye-

nebT xoy sibrtyes,

sadac dasaSvebi sim-

ravle Cveulebriv

gamoisaxeba, miznis

funqcias ki gamovsa-

xavT donis wirebisa da uswrafesi zrdis mimarTulebis

saSualebiT.

ganmartebis mixedviT, 2:g funqciis donis

wiri (igives niSnavs ganurCevlobis wiri) ewodeba sib-

rtyis iseTi wertilebis erTobliobas, sadac am funqciis

mniSvnelobebi erTi da igivea, anu simravles

2, ( , ) constx y g x y ,

romelsac xSirad aigiveben mis ganmsazRvrel ganto-

lebasTan:

( , ) constg x y .

amasTan, gansxvavebuli konstantebi warmoSoben gansxva-

vebuli donis wirebs. donis wirebis kargad cnobil

magaliTebs warmoadgenen erTi da igive simaRlis doneebi

topografiul rukaze da erTi da igive barometruli

wnevis wirebi amindis rukaze. es magaliTebi, kerZod,

gviCveneben rom erTi da igive c konstanta SeiZleba gan-

sazRvravdes mraval donis wirs.

x

dasaSvebi simravle

miznis funqcia

nax. 3

117

c1

c2

c3

ˆ 1x

3

2

ˆ ˆx (x)g

y

nax. 4

rac Seexeba funqciis uswrafesi zrdis mimarTulebas,

Cven ukve viciT, rom igi emTxveva gradientis

1

(x) (x)(x) , ,

x xn

g gg

mimarTulebas yoveli x–isaTvis.

geometriulad, pirobiTi maqsimumis amocanis amoxsna

niSnavs iseTi wertilis an wertilTa jgufis moZebnas

dasaSveb simravleSi, romelzec gadis yvelaze maRali

donis wiri, ganlagebuli yvelaze Sors uswrafesi zrdis

mimarTulebiT. nax. 4–ze naCvenebia

2( , ) extr, ( , ) 3 0g x y xy f x y y x (9.9)

maqsimizaciis amocanis amonaxsni, romelSic donis wire-

bis konstantebi 1 2 3, ,c c c izrdebian g-s uswrafesi zrdis

mimarTulebiT, xolo (0,3) da 3,0 wertlebze gama-

vali parabola warmoadgens dasaSveb simravles.

rogorc vxedavT, dasaSvebi simravle da 2c donis

wiri, romelic dasaSveb simravleSi miznis funqciis

uswrafesi zrdis mimarTulebiT yvelaze Sors aris

moTavsebuli, erTmaneTs exebian x (1,2) maqsimalSi

(termini “wirebis Sexeba” Cven jer ar gvaqvs ganmar-

tebuli, magram

mas male mivani-

WebT azrs).

vaCvenoT,

rom dasaSvebi

simravlisa da

donis wiris Se-

xeba maqsimumis

wertilSi (to-

lobis tipis Sez-

118

Rudvebis SemTxvevaSi minimalisTvisac igive mdgomareo-

baa) aris zogadi faqti

( , ) max, ( , ) 0g x y f x y (9.10)

saxis amocanebSi da es faqti warmoadgens lagranJis

meTodis geometriul Sinaarss.

imisaTvis, rom dasaSvebi simravle da donis wiri

marTlac wirebs warmoadgendnen, unda moviTxovoT

ˆ ˆ(x) 0, (x) 0g f , raTa Sesruldes aracxadi funqciis

Sesaxeb Teoremis pirobebi.

davuSvaT sawinaaRmdego, rom arsebobs (9.10) saxis

amocana (ix. nax. 5), iseTi, rom x maqsimalSi ikveTeba

dasaSvebi simravle 2, ( , ) 0M x y f x y da x -ze

gamavali donis wiri 2( , )g x y c . maSin, f da g funqcie-

bis sigluvis gamo,

iarsebebs 1c donis wi-

ri ˆ(x)g antigradi-

entis mimarTulebiT

x -dan, romelic sak-

maod axlosaa 2c do-

nis wirTan da kveTs

M-s, da arsebobs 3c

donis wiri ˆ(x)g gra-

dientis mimarTulebiT x -dan, romelic sakmaod axlosaa

2c donis wirTan da aseve kveTs M-s. cxadia, 1 2 3c c c da

amitom 3 2ˆ(x) (x)g c c g , sadac x aris 3c donis wirisa

da M–is gadakveTis wertili. amgvarad, miviReT winaaR-

mdegoba daSvebasTan, e.i. donis wiri da dasaSvebi

simravle maqsimalSi ar kveTen erTmaneTs. rodesac or

x

ˆ ˆx (x)g

M

c1

c2

c3

x

nax. 5

119

gluv wirs aqvs saerTo wertili, magram isini ar kveTen

erTmaneTs, es niSnavs, rom isini exebian erTmaneTs.

(9.9) amocanaSi, x maqsimalis gansazRvrisaTvis lag-

ranJis meTodi iZleva pirobas (x) (x) 0g f , rac

niSnavs, rom ˆ ˆ(x), (x)g f veqtorebi moTavsebulia erT

wrfeze, anu, dasaSvebi simravle da 2c donis wiri ki ar

kveTs erTmaneTs x -Si, aramed exeba.

rodesac SezRudvebi tolobis tipisaa, -s niSani

SezRuduli ar aris, ˆ(x)g da ˆ(x)f SeiZleba mimarTuli

iyos rogorc erTsa da imave, aseve sxvadasxva mxares.

sxva mdgomareobaa utolobis tipis SezRudvis Sem-

TxvevaSi, ris geometriul axsnasac axla moviyvanT (ix.

nax. 6).

simartivisaTvis vigulisxmoT, rom 0( , )g x y aris

wrfivi funqcia, 1( , )g x y ki kvadratuli, iseTi rom

1( , ) 0g x y warmoad-

gens elifss, romlis

SigniT (gareT) 1g Rebu-

lobs uaryofiT (dade-

biT) mniSvnelobebs, ra-

sac miuTiTebs Sesaba-

misad ganlagebuli + da

– niSnebi. am pirobebSi,

ganvixiloT amocana

0 1( , ) min, ( , ) 0.g x y g x y (9.11)

vTqvaT, x locmin(9.11) . maSin, lagranJis meTodis

Tanaxmad, 0 0 , 1 0 da amitom ˆ(x)g da ˆ(x)f gra-

dientebi erT wrfeze sxvadasxva mxaresaa mimarTuli

0ˆ ˆx (x)g

g(x,y)=const

1ˆ ˆx (x)g

nax. 6

120

(sxvadasxva niSani aqvT). amaTgan, x –Si modebuli 1ˆ(x)g

veqtori aucileblad elifsis gareT aris mimarTuli,

radgan 1ˆ(x)g uswrafesi zrdis mimarTulebaa, elifsi

nulovani donea, dadebiTi doneebi ki mxolod elifsis

gareTaa. x –Si modebuli 0ˆ(x)g veqtori aseve gareT rom

yofiliyo mimarTuli, maSin 0g miiRebda ufro mcire

mniSvnelobebs elifsis SigniT.

savarjiSoebi

gansazRvreT kritikuli wertilebi da amoxseniT Sem-

degi amocanebi:

#94. 2 2min, 1.xy x y

#95. 2 2extr, 50.xy x y

#96. 2 2 min , 2 - 5, 3x y x y x y .

#97. 4 4 6 6extr, 1.x y x y

#98. 4 4 2 2extr, 1.x y x y

#99. 2 2 6 6extr, 1.x y x y

#100. 2 2 2 4 4 4extr, 1.x y z x y z

#101. 4 4 4 2 2 2extr, 1.x y z x y z

#102. 6 6 6 2 2 2extr, 1.x y z x y z

#103. 2 2 2extr, 1.xyz x y z

n a w i l i 4

amozneqili miznis

funqcia

122

Tavi 10. gluvi amozneqili

eqstremaluri amocanebi

simravliT

1. amozneqili simravleebi da funqciebi

ganmarteba. vTqvaT X aris wrfivi sivrce da

M X . vityviT, rom M aris amozneqili simravle, Tu

igi Tavis yovel or wertilTan erTad Seicavs maT

SemaerTebel monakveTsac, anu:

x, y M da 0,1 1 x y M .

mag.: pirveli sami figura amozneqilia, meoTxe _ ara.

advili dasamtkicebelia, rom nebismieri raodenoba

amozneqili simravleebis TanakveTa amozneqilia. sakmaod

Zneli dasamtkicebelia,

rom n -Si ori TanaukveTi

amozneqili simravlis gan-

caleba SeiZleba hipersib-

rtyiT (nax. 2).

ganmarteba . amozne-

qil M simravleze gansaz-

Rvrul :f M funqcias ewodeba amozneqili fun-

qcia, Tu sruldeba e.w iensenis utoloba:

1 x y 1 x yf f f ,

x, y ,M 0,1 . (10.1)

Tu x,y M da 0,1 -isTvis sruldeba

1 x y 1 x yf f f ,

nax. 1

nax. 2

123

maSin f -s ewodeba Cazneqili funqcia.

zogjer saWiro xdeba mivuTiToT, Tu romel sim-

ravlezea funqcia amozneqili an Cazneqili. magaliTad,

sin( )x araa arc amozneqili da arc Cazneqili R -ze, magram

R -is garkveul qvesimravleebze aris amozneqili an Caz-

neqili.

SevniSnoT, rom f -is amozneqiloba M simravleze

gulisxmobs (ganmartebis Tanaxmad), rom M TviTon aris

amozneqili.

amozneqil funqciebs aqvT didi praqtikuli da ga-

moyenebiTi mniSvneloba, magram amozneqilobis Semowmeba

xSirad sakmaod rTulia. amitom mniSvnelovania amozne-

qiloba-Cazneqilobis garkvevisaTvis sxvadasxva krite-

riumebis gamoyeneba. Semdegi winadadeba, romelic daum-

tkiceblad mogvyavs (Tumca misi damtkiceba rTuli araa),

garkveul geometriul warmodgenas gviqmnis sakiTxze.

winadadeba 10.1. vTqvaT, M amozneqilia da

:f M . maSin, f -is amozneqilobisaTvis aucilebelia

da sakmarisi misi grafikszeda simravlis

,x x ,M xf

amozneqiloba, xolo f -is CazneqilobisaTvis aucile-

belia da sakmarisi misi grafiksqveda simravlis

,x x ,M xf

amozneqiloba.

124

f

1 x y

y x

1f x y

1 f x f y

nax. 3

iensenis utoloba niSnavs, rom funqciis grafikis

nebismieri ori wertilis SemaerTebeli monakveTi auci-

leblad grafikis zemoT aris moTavsebuli, radgan

nebismieria 0,1 -dan (ix. nax.3).

gluvi funqciebis Semowmeba amozneqilobaze xdeba

Semdegi Teoremebis safuZvelze.

Teorema 1. /gluvi funqciis amozneqilobis I rigis

aucilebeli da sakmarisi piroba/. vTqvaT, M aris Ria

amozneqili simravle n -Si da :f M aris warmoe-

badi funqcia. maSin f -is amozneqilobisTvis aucile-

belia da sakmarisi Semdegi pirobis Sesruleba

y x x y x ,T

f f f x,y M . (10.2)

damtkiceba. vTqvaT, warmoebadi f funqcia amozneqi-

lia M -ze. gamoviyenoT amozneqilobis ganmarteba da

aviRoT Zalian mcire dadebiTi parametri, romelic

aRvniSnoT dt -Ti:

1 x y 1 x y ,f dt dt dt f dt f

125

anu x y x x y x ,f dt f dt f f

radgan 0,dt amitom

x y x xy x

f dt ff f

dt

da roca 0dt es utoloba f -is warmoebadobis ZaliT

iZleva (10.2)-s.

vTqvaT axla, rom warmoebadi :f M -isaTvis sru-

ldeba (10.2). nebismierad aviRoT x,y M da 0,1 . or-

jer gamoviyenoT (10.2), ,x x y x da y, x y x

wertilebisaTvis:

x x y x x y x x y ,T

f f f

y x y x 1 x y x x y ,T

f f f

pirveli gavamravloT 1 -ze, meore -ze da Sevkri-

boT. miviRebT

1 x y x x ,f f f y

x y

f x y x

f y f x

nax. 4

f

126

rac amtkicebs amozneqilobas, radgan

x y x 1 x y.

ganmarteba. vTqvaT, A aris n n zomis kvadratuli

simetriuli matrica. vityviT, rom A aris

1) dadebiTad naxevradgansazRvruli, Tu h h 0,TA

h ,n

2) uaryofiTad naxevradgansazRvruli, Tu h h 0,TA

h n .

Teorema 2. /gluvi funqciis amozneqilobis II rigis

aucilebeli da sakmarisi piroba/. vTqvaT, M aris Ria

amozneqili simravle n -Si da :f M aris orjer uw-

yvetad warmoebadi. maSin f -is amozneqilobisTvis M -ze

aucilebelia da sakmarisi, rom heses matrica xf iyos

dadebiTad naxevradgansazRvruli yoveli x M -isTvis.

damtkiceba. aucileblobis dasamtkiceblad, nebis-

mierad aviRoT x M , aranulovani h n da vaCvenoT,

rom h h 0Tf x .

radgan M Riaa da

amozneqili, arsebobs

0, iseTi rom

0 x h .M

amitom, Teorema 1-is Za-

liT

0 x h x x h .Tf f f

Tu gaviTvaliswinebT f -is sigluves da gamoviyenebT

teiloris formulas, gveqneba:

x h

x+h

M

x h

x

nax. 5

127

2

2x h x x h h x h h2

T Tf f f f o

,

sadac

2

2 20

hlim 0.

h

o

(10.3)

amitom

2

20 h x h h2

Tf o

anu yoveli 0, -sTvis sruldeba

2

2

2 2

2 h0 h x h h

h

To

f

,

rac (10.3)-is ZaliT gvaZlevs, rom h x h 0Tf .

axla vaCvenoT sakmarisoba. vTqvaT, yoveli x M da

h n -isTvis sruldeba h x h 0Tf da vaCvenoT

f -is amozneqiloba. Tu x, y ,M teiloris formulis

Tanaxmad arsebobs iseTi 0,1 , rom sruldeba

1

y x y x y x x y x y x2

Tf f f x f (10.4)

(teiloris formula am saxiT, damatebiTi daSvebebis

gareSe, samarTliania mxolod im SemTxvevaSi roca f -is

mniSvnelobebi namdvili ricxvebia). (10.4)-is marjvena

mxare arauaryofiTia, amitom arauaryofiTia marcxena

mxarec, rac Teorema 1-is ZaliT niSnavs f -is amozneqi-

lobas.

magaliTi 1. SevamowmoT, amozneqilia Tu ara sim-

ravle

2, | 2 1 .H x y x y

128

amoxsna. nebismierad aviRoT

0,1 da 1 1 2 2, , , ,x y x y H

rac H -is gansazRvris Ta-

naxmad, niSnavs 22 1, 1,2.i ix y i

vaCvenoT rom

1 1 2 21 , ,x y x y H .

naxazi gviCvenebs, rom pasuxi

dadebiTi unda iyos. amitom amozneqilobis Semowmeba

unda gavagrZeloT. veqtorebze operaciebis gamoyenebiT,

1 1 2 2 1 2 1 21 , , 1 , 1 .x y x y x x y y

am veqtorisTvis SevamowmoT H -is ganmsazRvreli piroba:

2

1 2 1 2

2 2 2 2

1 2 1 2 1 2

1 2 1

1 2 1 2 4 1

x x y y

x x y y y y

(davamatoT da davakloT saWiro wevrebi)

2 2

1 1 2 21 2 1 2x y x y

2 2 2 2 2 2

1 1 2 2 1 22 1 1 2 4 1y y y y y y

2 2

1 1 2 21 2 1 2 1 1y y y y

2

1 21 2 1 1y y ,

rac niSnavs, rom H amozneqilia.

magaliTi 2. SevamowmoT amozneqilia Tu ara sim-

ravle

, | 1, 4, 0, 0 .H x y xy x y x y

amoxsna. SevecadoT H warmovadginoT amozneqili

simravleebis TanakveTis saxiT, rac misi amozneqilobis

tolfasi iqneba. cxadia,

y

1

nax. 6

x

129

, | 1, 0, 0 , | 4, 0, 0 .H x y xy x y x y x y x y

ganvixiloT meore, 2 , | 4, 0, 0H x y x y x y , ro-

melic samkuTxedis Siga na-

wils warmoadgens. cnobi-

lia, rom samkuTxedi amoz-

neqili figuraa, magram ax-

la SevamowmoT es faqti.

vTqvaT,

0,1 da

1 1 2 2 2, , ,x y x y H .

-s Sesabamisi wertili

1 1,x y da 2 2,x y boloebis mqone monakveTisa aris:

1 1 2 2 1 2 1 21 , , 1 , 1x y x y x x y y

da radgan 1 21 0,x x 1 21 0,y y

1 2 1 2 1 1 2 21 1 1 4 1 4,x x y y x y x y

amitom 2H amozneqilia.

axla ganvixiloT 1 , | 1, 0, 0H x y xy x y sim-

ravle. geometriuli suraTi

gviCvenebs, rom 1H amozneqilia,

amitom CavataroT formaluri

Semowmeba, nebismierad aviRoT

0,1 da 1 1 2 2 1, , , ;x y x y H

e.i. 0, 0i ix y da 0,i ix y

1,2.i

1 1 2 2 1 2 1 21 , , 1 , 1x y x y x x y y

veqtoris komponentebi dadebiTia, amitom saCvenebeli

rCeba rom maTi namravli metia erTze.

nax. 7 y

4

x o 4

y

x

nax. 8

130

1 2 1 2

2 2

1 1 1 2 2 1 2 2

1 1

1 1

x x y y

x y x y x y x y

2 22 2

1 2 2 11 2 1 1 2 1.x y x y

2. amozneqili amocanebis specifika

minimizaciis amocanas

x min,f x ,M (10.5)

sadac nM R amozneqili simravlea, xolo :f M R

amozneqili funqciaa ewodeba (minimizaciis) amozneqili

amocana.

Tavisi Sinaarsis gamo, Semdeg Tvisebas uwodeben

globalur-lokalur Tvisebas.

winadadeba 1. gluv amozneqil amocanaSi lokaluri

minimali amave dros aris globaluri minimali.

damtkiceba. vTqvaT (10.5) aris amozneqili amocana,

xolo x locmin(10.5) es niSnavs, rom raRac 0 ric-

xvisTvis sruldeba implikacia:

ˆ ˆx & x x x x .M f f (10.6)

axla nebismierad aviRoT

dasaSvebi 0x M wertili.

sakmaod mcire 0,1 -

Tvis, Tu aRvniSnavT

0ˆx 1 x x M ,

maSin

ˆx x

radgan M amozneqilia, xo-

lo ˆx x roca 0 ,

x

x

0x

M

nax. 9

131

amitom (10.6)-is Tanaxmad x x .f f axla gaviTvalis-

winoT x -is saxe da gamoviyenoT iensenis utoloba:

0 0ˆ ˆ ˆx 1 x x 1 x x ,f f f f

anu 0x xf f , rac 0x M -is nebismierobis ZaliT

amtkicebs, rom x xf f , x M , e.i.

loc min 10.5 gl min 10.5 loc min 10.5 gl min 10.5 .

winadadeba 2. gluv amozneqil amocanaSi minimumis

wertilebis simravle amozneqilia.

damtkiceba. vTqvaT 1 2x , x minimalebia (10.5)-Si. wina

Sedegis ZaliT, 0 1 0 2x x .f f axla aviRoT 0,1 :

0 1 2 0 1 0 2 0 1 0 21 x x 1 x x x xf f f f f

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, M aris Ria amozneqili sim-

ravle, :f M aris warmoebadi da amozneqili fun-

qcia. maSin yoveli x M wertili, sadac x 0f , war-

moadgens minimals (10.5)-Si.

damtkiceba. nebismierad aviRoT x M . amozneqilo-

bis I rigis aucilebeli da sakmarisi pirobis Tanaxmad, da

x 0f - is gaTvaliswinebiT,

ˆ ˆ ˆx x x x x 0T

f f f .

e.i. ˆx xf f .

132

3. qunisa da Taqeris Teoremis Camoyalibeba da

ganxilva

ganvixiloT zogadi saxis amozneqili amocana utolo-

bis tipis SezRudvebiT:

0 x min,g x 0ig , 1,...,i k , 0x X , (10.7)

sadac 0X aris romeliRac X wrfivi sivrcis amozneqili

qvesimravle da yoveli :ig X , 0,1,...,i m aris amoz-

neqili funqcia.

radgan 0g -is amozneqilobidan ar gamomdinareobs

– 0g -is amozneqiloba, amitom arsebiTia rom (10.7) aris

minimizaciis amocana.

(10.7) amocanaSi dasaSvebi simravlis gansazRvraSi

monawileoben 1,..., kg g funqciebi da 0X amozneqili sim-

ravle. dasaSvebi simravle aq aris:

0x | xM X X da 1 x 0,... x 0kg g .

ig -s amozneqilobidan martivad gamomdinareobs

x | x 0iX g simravlis amozneqiloba. amitom M

SegviZlia warmovadginoT amozneqili simravleebis Ta-

nakveTis saxiT:

0 1x x 0 ... x | x 0kM X X g X g .

zogadi saxis amozneqili amocanisaTvis Cveulebrivad

ganisazRvreba lagranJis funqcia:

0

0

x, , xk

i i

i

g

L , sadac 1,..., k ;

aRsaniSnavia, rom igi ver iTvaliswinebs 0x X Sez-

Rudvas.

Semdegi Teorema damtkicebulia 1951 wels. igi mra-

valmxrivaa saintereso, aRsaniSnavia, rom Teoremis piro-

133

bebSi mxolod amozneqilobis cneba monawileobs, araa

laparaki normirebul sivrceebsa da uwyvet da warmoebad

asaxvebze.

saqme imaSia, rom arsebobs Sinagani kavSirebi uwyve-

tobasa da amozneqilobas, warmoebadobasa da amozneqi-

lobas Soris. magaliTad, Tu : ,g a b amozneqilia,

igi uwyvetia ,a b -ze, xolo amozneqil : ng asax-

vas yovel wertilSi gaaCnia mimarTulebiTi warmoe-

bulebi.

Teorema /quni–Taqeri/. 1). vTqvaT, X aris wrfivi

sivrce, 0X amozneqili qvesimravlea X -Si da :ig X ,

0,1,...,i k amozneqili funqciebia X -ze.

Tu x aris minimali (10.7) amocanaSi, maSin moiZebneba

iseTi erTdroulad nulis aratoli lagranJis mamrav-

lebi 1 0ˆ ˆ ˆ, , ,k rom:

a) 0

0 0x

ˆ ˆ ˆ ˆ ˆmin x, , x, ,X

L L (minimumis principi);

b) ˆ 0,i 0,1,...,i k (arauaryofiTobis piroba);

g) ˆ x 0i ig , 1,...,i k .

2) Tu 0ˆ 0, maSin a)–g) pirobebi sakmarisia, raTa da-

saSvebi x wertili iyos minimali (10.7)-Si;

3) imisaTvis rom 0ˆ 0, sakmarisia moiZebnos iseTi

0x X , romlisTvisac Sesruldeba (sleiteris) piro-

bebi: x 0ig 1,...,i k .

am Teoremis damtkiceba Cvens gegmaSi ar Sedis, TviT

quni–Taqeris Teoremac mimoxilvis mizniT mogvyavs, mag-

ram saWirod migvaCnia ramdenime sasargeblo SeniSvnis

gakeTeba.

134

(i) a) piroba niSnavs, rom lagranJis mamravlebis

SerCevis xarjze zogadi saxis amozneqili amocana, ro-

melic Seicavs utolobis tipis SezRudvebs, miiyvaneba

amozneqil amocanaze utolobis tipis SezRudvebis

gareSe:

0ˆ ˆmin x, , min L , 0x X . (10.8)

sxva sityvebiT, lagranJis funqciis gamoyenebiT SegviZ-

lia “movxsnaT” utolobis tipis SezRudvebi, rac arse-

biTia, radgan (10.8)-is amoxsna gacilebiT advilia, vidre

(10.7)-isa; magaliTad, Tu 0 nX X da ig warmoebade-

bic arian, maSin a) miiRebs Cveul saxes

0

xˆ 0

x

mi

i ji

g

,

0

ˆ ˆ1,..., x 0m

i i

i

j n g

,

radgan 0ˆ ˆx, , L amozneqilia n -ze x -is mimarT (Teo-

remis pirobebSi) da amitom misTvis upirobo minimumis

aucilebeli piroba aris sakmarisic.

(ii) roca 0ˆ 0, SegviZlia yvela mamravli erTdro-

ulad gavyoT 0ˆ 0, 0 -ze da mivaRwioT 0

ˆ 1 -s. 0ˆ 0

migviTiTebs dasaSvebi simravlis gadagvarebaze. marT-

lac, Tu sleiteris piroba Sesrulda, maSin dasaSvebi

simravle araa gadagvarebuli da 0ˆ 0 .

(iii) Teoremis damtkiceba efuZneba amozneqili ana-

lizis erT-erT ZiriTad Sedegs _ gancalkevebis Teo-

remas, romlis formulirebas erT kerZo SemTxvevaSi axla

moviyvanT.

Teorema /simravlis da wertilis gancalkevebis

Sesaxeb sasrulganzomilebian SemTxvevaSi/. vTqvaT, M

amozneqilia n -Si da 0 M . maSin moiZebneba iseTi

1,..., n ricxvebi, rom

135

1

x 0n

i

i

i

, 1x x ,...,xn M

(sxva sityvebiT, 0-ze gamavali hipersibrtye

1

x 0n

i

i

i

mTlian

sivrces hyofs or nawilad, romelTagan erTSi sruladaa moTav-

sebuli M ).

savarjiSoebi

#104. daamtkiceT, rom amozneqili simravleebis nebis-

mieri raodenobis TanakveTa amozneqilia.

#105. arsebobs Tu ara funqcia, romelic erTdroulad

amozneqilia da Cazneqilic.

#106. vTqvaT, f amozneqilia. ra SegviZlia vTqvaT – f

funqciaze?

#107. aCveneT (geometriulad mainc), rom ,a b mona-

kveTze amozneqili funqcia uwyvetia ,a b -ze.

#108. miiReT am paragrafis Sedegebis analogebi Cazne-

qili amocanisaTvis.

#109. vTqvaT :g X R amozneqilia. aCveneT rom

| 0x X g x aris amozneqili simravle.

#110. aCveneT, rom amozneqili simravleze gansazRvru-

li amozneqili funqciebis jami arauaryofiTi

koeficientebiT aris amozneqili funqcia.

#111. SeamowmeT, amozneqilia Tu ara simravle 2H R :

a). 2 2, | 5 .H x y x y

b). , | 1, 0, 0 .H x y xy x y

g). , | .xH x y e y

136

d). 2 2, | 2, 4 .H x y x y x y

e). , | 2 2,2 2, 0 .H x y x y x y y

#112. aCveneT Semdegi funqciebis amozneqiloba:

a). 2 2 2, , 3 2 2 2 2 2 6 4 2f x y z x y z xy xz yz x y z ,

3, ,x y z R .

b). xf x e , x R ,

g). 2 2, 2 sin( )f x y x y x y , x R .

137

Tavi 11. gluvi amozneqili funqciebis

minimizaciis ricxviTi

meTodebi

ganvixiloT minimizaciis amocana:

x min, x nf R , (11.1)

sadac f aris amozneqili funqcia, romelsac aqvs pir-

veli an meore rigis uwyveti kerZo warmoebulebi. eqs-

tremalurobis aucilebeli da sakmarisi pirobebis gamo-

yenebiT, rig SemTxvevebSi xerxdeba (11.1) amocanis amoxsna,

magram, xSirad, saWiro xdeba minimalebis povna ricxviTi

meTodebis gamoyenebiT. nebismieri ricxviTi meTodi gu-

lisxmobs misi maxasiaTeblebis (miznis funqciis, dasaSve-

bi samravlis ganmsazRvravi funqciebis, maTi warmoebu-

lebis) zusti an miaxloebiTi mniSvnelobebis gamoTvlas,

da Semdeg, am informaciis safuZvelze amocanis amonax-

snis ( x 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.

mimdevrobiTi ricxviTi meTodiT minimizaciis amo-

canis amoxsna niSnavs Semdegi formuliT

k+1x , 0, , 0,1,k k k k kx h R k

wertilTa iseTi mimdevrobis agebas, romelic krebadia x

minimalisken. amasTan, sxvadasxva algoriTmi ermaneTisgan

gansxvavdeba sawyisi miaxloebis 0x wertilis, kh veqto-

ris, k ricxvebis arCeviT, da agreTve gaCerebis pirobiT.

138

sawyisi miaxloebis – 0x wertilis arCevis raime zoga-

di wesi ar arsebobs. magram Tu amocanis specifikis an

Sinaarsis gaTvaliswinebiT cnobilia minimalis SesaZlo

ganlageba, maSin sawyisi miaxloeba, bunebrivia, masTan

axlos unda aviRoT.

kh veqtori gansazRvravs ( k+1 )-e bijis mimarTule-

bas, xolo k ricxvi _ bijis sigrZes.

konkretuli ricxviTi algoriTmis aRsawerad unda

gvqondes gaCerebis piroba. praqtikaSi gamoiyeneba gaCe-

rebis Semdegi pirobebi:

1 1k kx x ,

1 2( ) ( )k kf x f x ,

1 3( )kf x .

gamoTvlebis dawyebamde unda davasaxeloT da gaCe-

rebis erTi an ori piroba.

vityviT, rom meTodi krebadia, Tu ˆ,kx x k ,

sadac x aris x min, x nf R amocanis amonaxsni.

meTodis efeqturoba xasiaTdeba krebadobis siC-

qariT.

vityviT, rom kx krebadia x -sken wrfivad, Tu

0(0,1), 0q k , rom

1ˆ ˆ

k kx x q x x , 0k k .

vityviT, rom kx krebadia x -sken zewrfivad, Tu

1 1ˆ ˆ ,k k kx x q x x 0 ,kq k .

vityviT, rom kx krebadia x -sken kvadratulad, Tu

00, 0C k , rom

139

2

1ˆ ˆ

k kx x C x x , 0k k .

am TavSi, ganvixilavT minimizaciis mimdevrobiT me-

Todebs: gradientul meTodebs, niutonis meTods, gra-

dientis proeqciis meTods, agreTve, SemTxveviTi Ziebis

meTods.

1. gradientuli daSvebis meTodi

vTqvaT, f aris amozneqili, uwyveti kerZo warmoe-

bulebis mqone funqcia, da vTqvaT x nR aris minimali

(11.1) amocanaSi, romlis moZebnac warmoadgens Cvens

mizans.

nebismierad aviRoT sawyisi miaxloeba 0x M da

avagoT mimdevroba 1

xk kM

Semdegnairad:

1x x x , 0,1, ,k k k kf k (11.2)

aq, kh veqtoris rolSi aviReT antigradienti, radganac

antigradientis mimarTuleba xkf mocemul wertilSi

(roca x 0kf ) emTxveva funqciis uswrafesi klebis

mimarTulebas (ix. Tavi 8). k sidideebi (bijis sigrZe)

SeiZleba avirCioT imdenad mcire, rom Sesruldes piroba

1x < x , 0,1,k kf f k . (11.3)

im SemTxvevaSi, roca (11.3) piroba ar sruldeba, vana-

xevrebT k sidides da axlidan veZebT 1x k miaxloebas.

gaCerebis pirobad, Cveulebriv, gamoiyeneba piroba

x, 1, ,

x

k

i

fi n

, (11.4)

140

sadac winaswar mocemuli mcire ricxvia. Tu (11.4)

sruldeba, viRebT: ˆ ˆx x , x xk kf f .

magaliTi 1 . gradientuli daSvebis meTodiT 0,05

sizustiT amoxseniT minimizaciis amocana:

2 2

2, 2 min, ,x y

f x y x y e x y

.

amoxsna. aviRoT: 0 0 0 0x , 0,0 , 1x y , avagoT

(11.2) mimdevroba da Sedegebi CavweroT cxrilSi:

k kx ky xkf xkf

x

xkf

y

k SeniSvna

0 0 0 1 1 1 1

1 –1 –1 3,145 – – 0

1 -saTvis

(11.3) piroba ir-Rveva, amitom vana-xevrebT mas

0 0 0 1 1 1 0,5

1 –0,5 –0,5 1,118 – –

(11.3) piroba isev irRveva, amitom isev vanaxevrebT

0 -s

0 0 0 1 1 1 0,25

1 –0,25 –0,25 0,794 0,106 –0,393 0,25 (11.3) piroba sru-ldeba, (11.4) – ar sruldeba.

2 –0,2766326 –0,1516326 0,774 0.0983 0,0451 0,25 (11.3) piroba sru-ldeba, (11.4) – ar sruldeba.

3 –0,3012259 –0,1629096 0,772 0,0262 –0,023 –

(11.3), (11.4) – pi-robebi sruldeba sizuste miRweu-lia.

amgvarad, ˆ ˆx 0,301, 0,163 , x 0,772f .

2. uswrafesi daSvebis meTodi

uswrafesi daSvebis meTodi imiT gansxvavdeba gra-

dientuli daSvebis meTodisagan, rom k amjerad SeirCeva

pirobidan

141

0

mink k k

, (11.5)

sadac x xk k kf f . amgvarad, yovel bijze

ixsneba minimizaciis erTganzomilebiani amocana. radga-

nac f amozneqili funqciaa, romelsac aqvs pirveli rigis

uwyveti kerZo warmoebulebi, amitom 0k k ganto-

lebis amonaxsni warmoadgens (11.5) amocanis amonaxsns. am

faqts aqvs lamazi geometriuli gaazreba: yoveli k–

sTvis xkf da 1xkf veqtorebi urTierTmarTo-

bulni arian (es gamomdinareobs

1x x ( ( )) x ( )k k k k k k kf f f x f f x

pirobidan).

magaliTi . uswrafesi daSvebis meTodiT 0,05

sizustiT amoxseniT Semdegi amocana:

2 2

2, 2 min, ,x y

f x y x y e x y

2xk

1x k

x xk kf

x k

f(x)=const

142

amoxsna.

biji 1). aviRoT 0 0,0x . maSin

0x 1;1f , 2 2

0 0 ,0 3f e ,

da 0 funqciis minimumis mosaZebnad, dadebiT –

ebs Soris, gamoviyenoT gadarCevis meTodi (es si-

martivisaTvis, radgan aseTi martivi saxis fun-

qciis mniSvnelobebis gamoTvla problema araa)

-dan dawyebuli bijiT 0,2:

0 . . . 0,18 0,20 0,22 0,24 0,26

0 1 . . . 0,7949 0,7903 0,7892 0,7916 0,7973

e.i. =0,22, saidanac

1x 0;0 0,22 1;1 0,22; 0,22 .

biji 2). 1x 0,204; 0,236 ,f

2 2

1 0,22 0,204 2 0,22 0,236

0,44 0,032e . movaxdinoT 1 -is minimizacia:

. . . 0,28 0,30 0,32 0,34 0,36

1 . . . 0,77401 0,77384 0,77380 0,77387 0,77408

e.i. =0,32 da

2x 0,22; 0,22 0,32 0,204; 0,236 0,2853; 0,1445 .

biji 3). 2

2x 8,007;7,268 10 ,f

2

2

2 0,2853 9,007 10

2

22 0,1445 7,268 10 20,420 15,27510e

,

2 -is minimizacia gvaZlevs:

143

. . . 0,20 0,22 0,24 0,26 0,28

2 . . . 0,77273 0,77241 0,77240 0,77241 0,77244

e.i. =0,24,

2

3 3x 0,3045; 0,1619 , x 1,821; 2,051 10f ,

amitom 3x x da 3x xf f .

gavakeToT ramdenime SeniSvna.

1) k -s arCeva sakmaod Sromatevadi procesia, amitom

sasurvelia erTxel SeirCes -s romelime mniSvne-

loba, romelic ar iqneba damokidebuli iteraciis

nomerze da gamodgeba yovel iteraciaze.

2) rodesac x k Zalian axlos aris x -sTan, gradien-

tuli meTodebis krebadobis siCqare neldeba. am

dros, rekomendebulia ufro faqiz meTodze ga-

dasvla, magaliTad iseTze, romelic iyenebs f–is

kvadratul aproqsimacias.

3. niutonis meTodi

Tu amozneqili : nf funqcia orjer uwyvetad

warmoebadia, xolo xf , xf -s gamoTvla ar aris

Zneli. maSin SesaZlebelia meore rigis mimdevrobiTi

meTodebis gamoyeneba. vTqvaT, ukve gansazRvruli gvaqvs

x k miaxloeba. x k -s midamoSi f -is nazrds aqvs saxe:

21x x x x x x x x x x x x

2

T T

k k k k k k kf f f f o .

ganvixiloT nazrdis kvadratuli nawili:

1

x x x x x x x x x2

T T

k k k k k kf f (11.6)

da ganvsazRvroT 1x k miaxloeba pirobidan:

144

n

1x

x min xk k k

. (11.7)

(11.7)–is amosaxsnelad gamoiyeneba rogorc anali-

zuri, aseve miaxloebiTi (specialurad kvadratuli fun-

qciebisTvis gaTvaliswinebuli) meTodebi.

radganac x ( )k kf x , xf -s amozneqilobis ga-

mo, kxf dadebiTad naxevradgansazRvrulia, amitom

xk amozneqili funqciaa. minimumis aucilebel da

sakmaris pirobas (11.7)-Tvis aqvs saxe:

x x x x 0k k kf f .

Tu amovxsniT am sistemas da mis amonaxsns CavTvliT 1x k

miaxloebad, miviRebT

1

1x x x x .k k k kf f

(11.8)

unda aRiniSnos, rom (11.7) amocanis amoxsna SeiZleba

aRmoCndes sakmaod rTuli da gamoTvlebis sirTulis

gaTvaliswinebiT, sawyisi amocanis sadari. amitom niuto-

nis meTods iyeneben maSin, roca xf , xf -is gamoTvla

da (11.7) 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 (11.8)-Si rogorc mimarTuleba, aseve

bijis sigrZe fiqsirebulia. niutonis meTodis sxvadasxva

modifikacia mimarTulia iqiTken, rom am meTodis Ziri-

Tadi Rirsebis (krebadobis maRali siCqare) SenarCunebiT,

Semcirdes misi Sromatevadoba, rac dakavSirebulia

145

xf , xf -s gamoTvlasTan da (11.7) gantolebis amox-

snasTan da Sesustdes moTxovna sawyisi miaxloebis

amorCevaze.

niutonis meTodi bijis regulirebiT mdgomareobs

SemdegSi:

1

1x x x x .k k k k kf f

roca k 1 , is emTxveva niutonis meTods. k koefi-

cientebis arCeva xdeba an mocemuli mimarTulebiT fun-

qciis minimizaciis pirobidan, an bijis dayofiT (11.2)

pirobis gaTvaliswinebiT.

magaliTi . magaliTi 1–is pasuxi aviRoT sawyis miax-

loebad da niutonis meTodiT ganvsazRvroT minimali

Semdeg amocanaSi:

2 2

2, 2 min, ,x y

f x y x y e x y

amoxsna. magaliTi 1–is Sedegebis mixedviT, gvaqvs:

0x 0,3012259, 0,1629096 ,

2

0x 2,622655; 2,296005 10f ,

0

0,39319151 0,62967835x

0,62867835 0,22329787f

.

aqedan:

2

1

0 2

0,39319151 5,3404226 10x

5,3404226 10 0,22329787f

,

da (11.2) –is Tanaxmad:

2

1 1 1x , 0,3012259; 0,1629096 10 2,622655; 2,296005x y

2

2

0,39319151 5,3404226 100,3127641; 0,1563821

5,3404226 10 0,22329787

.

146

radgan

6 6

1x 7,9 10 ;7,9 10 ;f

amitom sasurveli sizuste miRweulia da

x 0,3127641; 0,1563821 .

4. gradientis proeqciis meTodi

ganvixiloT minimizaciis amocana:

x min, xf M , (11.9)

sadac nM aris amozneqili Caketili simravle, xolo

f aris amozneqili warmoebadi funqcia MM –ze.

Caketili MM –is SemTxvevaSi uSualod gradientuli

meTodis gamoyeneba ar SeiZleba, Tu nM , radgan zo-

gierTi x k SeiZleba gavides MM –idan. gradientuli pro-

eqciis meTodis yoveli iteracia iTvaliswinebs Semdegi

formuliT gansazRvruli gradientuli daSvebis miax-

loebebis

1x x xk k k kf ,

dabrunebas dasaSveb MM simravleSi, Tu 1xk M . aseTi

dabruneba xdeba 1x k -is MM –ze proeqtirebis saSuale-

biT, anu 1x k icvleba MM simravlis im wertiliT, romelic

misgan yvelaze axlosaa moTavsebuli.

ganmarteba . x n wertilis proeqcia nM sim-

ravleze ewodeba iseT xMP wertils MM –idan, romelic

akmayofilebs pirobas:

y

x x min y xMM

P

.

rogorc vxedavT, marjvena mxare gamoxatavs manZils x-dan

MM –mde.

147

cxadia, Tu x M , maSin x xMP , xolo roca x M ,

maSin MM –is Caketilobis gamo (vaierStrasis Teoremis Se-

degis ZaliT) xMP arsebobs da MM –is amozneqilobis

ZaliT ki xMP erTaderTia (savarjiSos saxiT, sasargeb-

loa am faqtis geometriul SinaarsSi garkveva).

amgvarad, gradientis proeqciis meTodSi (11.9) amo-

canis x minimalis mimdevrobiTi miaxloebebi aigeba Sem-

degi wesiT:

1x x f x , 0,1,2,k M k k kP k . (11.10)

k -s SerCeva aqac sxvadasxvanairad SeiZleba; Sesa-

bamisad miiReba gradientis proeqciis meTodis sxvadasxva

varianti. maTgan yvelaze gavrcelebuli aris Semdegi

ori.

1) k SeirCeva iseve, rogorc uswrafesi daSvebis me-

TodSi Ria MM –isTvis, anu SeirCeva Semdegi piro-

bidan:

0

mink k k

,

sadac x xk k kf f .

2) k SeirCeva ise, rom Sesruldes monotonurobis pi-

roba 1k kf f . amisTvis Tavidan irCeven raime 0

da (11.10)-Si iReben k . Semdeg, amowmeben monoto-

nurobis pirobas da Seusruleblobis SemTxvevaSi,

yofen mas am pirobis Sesrulebamde.

unda aRiniSnos, rom roca x M , maSin xMP -is

gansazRvra xdeba minimizaciis Semdegi amocanis

y x y min, yg M ,

148

amoxsnis Sedegad. zogierTi kerZo saxis M simravlisaTvis

(magaliTad, birTvi, naxevarsibrtye da zogierTi sxva) es

amocana ixsneba analizurad lagranJis meTodis safuZ-

velze.

magaliTi 1 . ipoveT z n wertilis proeqcia

zMP , Tu M simravles aqvs saxe:

22 2x xx xT

nM r r

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

amoxsna. zMP aris Semdegi amocanis amonaxsni:

2

0

1

22

1

1

x x z min,

x x 0.

nj j

j

nj

j

g

g r

romelic SeiZleba SevcvaloT ekvivalenturi amocaniT:

2

0

1

22

1

1

x x z min,

x x 0.

nj j

j

nj

j

g

g r

(11.11)

ganvixiloT aratrivialuri SemTxveva z , 0M r . Se-

vadginoT lagranJis funqcia:

T 2

0 0x, , x z x z xxT

L r

da amovweroT pirobebi kritikuli wertilis gansaz-

RvrisaTvis:

0

T 2

2 x z +2 x=0x

xx =0

L

r

(11.12)

149

0 0 0 x 0 iZleva winaaRmdegobas II gantole-

baSi: 2 0r . e.i. 0 1 . axla ganvixiloT ori SemTxveva

-s mimarT.

a) Tu 0 x z , anu z,0,1 aris (11.12) –is amonaxsni,

magram maSin igi ar gansazRvravs kritikul wertils,

radgan 2 2zz zT r (radgan z M ).

b) Tu 0 . maSin (11.12) iRebs saxes:

T 2

x z + x=0

xx r

e.i.

2

2

z1 1x z zz 1 x z

1 z1

T rr

r

.

miviReT, rom x zz

r aris erTaderTi kritikuli wer-

tili da, vaierStrasis Teoremis Sedegis ZaliT, igi aris

agreTve globaluri minimali (11.11) –Si.

magaliTi 2 . gradientis proeqciis meTodiT amox-

seniT Semdegi amocana:

2 2 2, min, 1g x y x y x y .

gamoTvlebi SewyviteT, Tu Sesruldeba x 0,01kg , an

1x x 0,01k k .

amoxsna. aviRoT 0x 0;0,5 M , sadac

2 2

2, 1M x y x y .

aviRoT (11.10) –Si 0,75 0,1k . axla SegviZlia avagoT

mimdevrobiTi miaxloebebi.

150

biji 1). 0x 1;1g , amitom (11.10) gvaZlevs:

1 0 0x x x 0;0,5 0,75 1;1 0,75; 0,25 .M M MP g P P

radgan 0,75; 0,25 M , amitom 1x 0,75; 0,25 .

saWiro sizuste jer ar aris miRweuli, radgan

1x 1; 0,5g da 1 0x x 1,06 0,01 .

biji 2). 1x 1; 0,5g ,

2x 0,75; 0,25 0,75 1; 0,5 1,5;0,125M MP P ,

magram 1,5;0,125 M , radgan

2 2 21,5;0,125 1,5 0,125 2,266 1 .

radgan M aris Caketili birTvi centriT 0-Si da

radiusiT 1r , SegviZlia gamoviyenoT wina maga-liTis Sedegi da aviRoT:

2

1,5;0,125x 1,5;0,125 0,9965;0,08304

2,266MP .

saWiro sizuste jer kidev ar aris miRweuli,

radgan 2 1x x 0,298 0,01 . Semdegi bijebis Se-

degebi moyvanilia cxrilSi.

k xk 1x xk k xkg x xk kg

2 (0,9965;0,08304) 0,298 (-1;0,1661) (1,7465;-0,0415)

3 (0,9997;0,0238) 0,107 (-1;-0,0476) (1,7497;0,0119)

4 (0,99998;0,00679) 0,031 (-1;0,0136) (1,74998;-0,00339)

4 (0,9998;0,00194) 0,0087 sizuste miRweulia

rogorc vxedavT,

5 5ˆ ˆx x 0,9998;0,00194 , x x 1g g .

SevniSnoT, rom am amocanis zusti amonaxsni aris

ˆ ˆx 1;0 , x 1g .

151

savarjiSoebi

#113 – #115 amocanebSi ganaxorcieleT gradientuli

daSvebis erTi biji da SeadareT erTmaneTs 0xf da

1xf .

#113. 2 2

2, 2 min, x ,x yf x y x y e x y ,

0x 1;1 ,

a) 0 0,1a , b) 0 0,265a , g) 0 0,5a ;

#114. 2 2

2, 2 min, ,f x y x y xy x y x y ,

0x 0;0 ,

a) 0 0,1a , b) 0 0,5a , g) 0 1a ;

#115. 2 2

3, , min, , ,xf x y z e x y z x y z ,

0x 1;1;1 ,

a) 0 0,1a , b) 0 0,21268a , g) 0 1a .

#116 – #117 amocanebSi, ipoveT 0x wertilidan uswra-

fesi daSvebis meTodis bijis sigrZe 0a .

#116. 2 2

2, 2 min, ,f x y x y xy x y x y ,

0x 0;0 ,

#117. 4 2 2

3, , min, , ,f x y z x y z xy yz x y z ,

0x 0;1;0 .

#118 – #120 amocanebSi, moaxdineT kvadratuli miznis

funqciebis minimizacia da daasruleT gamoTvlebi maSin,

rodesac Sesruldeba piroba

x0,01, 1,2,

x

k

j

fj n

.

152

#118. 2 2

2, 4 4 6 17 min, ,f x y x xy y x x y ;

#119. 2 2

2, 10 3 10 min, ,f x y x xy y y x y ;

#120. 2 2

2, 2 6 min, ,f x y x xy y x y x y ;

#121 – #124 amocanebSi, moZebneT z n wertilis

zMP proeqcia Semdegi saxis M simravleze:

#121. n–ganzomilebiani sivrcis arauaryofiT oqtantze:

x x 0, 1, ,i

nM i n

#122. n–ganzomilebian paralelepipedze:

x x , 1, ,i

n i iM a b i n ;

#123. Caketil r radiusian birTvze centriT 0x -Si:

2

2

0

1

x x xn

i i

n

i

M r

.

#124. naxevarsivrceze:

1

1

x a x , a a , ,a 0n

i i n

n

i

M b

amocanebSi #125 – #128, gamoiyeneT wina amocanebis

pasuxebi da gradientis proeqciis meTodiT amoxseniT

minimizaciis amocanebi:

#125. 2 24 3 2 16 min, 0, 0x y xy x y x y .

#126. 2 29 12 3616 min, 1 4,1 2x y x y x y .

#127. 2 2

2 min, 4 2 1x y x y .

#128. 2 2

2, 4 17 5 min, ,f x y x xy y y x y .

153

gamoyenebuli literatura

1. В.М. Алексеев, Э.М. Галеев, В.М.Тихомиров. Сборник задач по

оптимизации. Москва, Наука, 1984.

2. Э. Маленво. Лекции по микроэкономическому анализу.

Москва, Наука, 1985.

3. G.V. Reklaitis, A. Ravindran and K.M. Regsdell. Engineering

optimization, Wiley, New York , 1983.

4. А.Б. Горстко, Ю.А. Домбровский, С.В. Жак. Методы

оптимизации. Методические указания. Москва, МГУ, 1981.

5. a. gagniZe, k. gelaSvili. optimizaciis meTodebi da

TamaSTa Teoria. Tbilisi, 2002.

6. Сборник задач по математике. Под редакцией А.В. Ефимова.

Москва, Наука, 1990.

7. Ф.П. Васильев. Численные методы решения экстремальных

задач. Москва, Наука, 1980.

8. Д.В. Кирьянов. MathCad 13 в подлиннике. СПб.: БХВ, 2005.

9. M. Minoux. Mathematical Programming: theory and algorithms.

Wiley, New York, 1986.

154

pasuxebi

#1. ( ) sin( )f x x

#2. 2

1( )

1f x

x

#3. ( ) ( )f x arctg x

#4. 3( ) ( )f x arctg x

#5. ( ) sin( ) ( )f x x arctg x

#6. 22 2( ) 2 xf x x y e

#7. 2( ) sin( )f x y x

#8.

2 1 max,

1 1000 .

x x

x

#9.

2 2

2 2

5 min,

( 1) ( 1) 1.

xy x y y

x y

#10.

2 2( 1) ( 2) min,

2 3 1.

x y

x y

#11.

2 2 2 2 2 2

1 2 1 2 2 3 2 3 1 3 1 3

2 2

( ) ( ) ( ) ( ) ( ) ( ) min,

1, 1,2,3.i i

x x y y x x y y x x y y

x y i

#12. ( ) ( ) ( ) ( ) ( ) ( ) min,

( , ) , , , , mocemuli wertilebia.i i

x x y y x x y y x x y y

x y i

2 2 2 2 2 21 1 2 2 3 3

1 1 2 3

#13. 2 2

max,

1,

0, 0.

xy

x y

x y

#14. ( ) max,

8 .

xy x y

x y

155

#15.

max,

,

0, 0.

xy

x y const

x y

#16.

max,

,2

0, 0.

xy

lx y

x y

#18. glmin ,glmax .

#19. glmin .

#20. glmin ,glmax .

#21. glmin .

#22. glmin .

#23. glmax .

#24. glmin ,glmax .

#25. glmin ,glmax .

#26. glmin .

#27. glmin .

#28. glmin .

#29. yvelgan arsebobs amonaxsni.

#30.

2

2

( 10,0),

( ) ( 10, ).110, .

1

xxe x

f x Mx

x

#32. K .

#33. 0 locmin, 12 locmax.

#34. 1

1 locmin, locmax.3

#35. 1

locmin glmin.2

156

#36. 2 29 2 29

locmin, locmin.25 25

#37. 2

locmin glmin.3

#38. 1

locmax glmax.4

#39. 5 1

locmin, locmax.2 2

#40. 1 10 1 10

locmin, locmax.3 3

#41. 4 locmin, 0 locmax.

#42. 16

locmin,2 locmax.5

#43. 1

locmin glmin.4

#44. 1

1 locmin, locmax.3

#45. 1

0;1 locmin, ;5 locmax.3

#46. 11;1 locmin, locmax.2

#47. ˆ ˆ1.6030, ( ) 2.1376.x f x

#48. ˆ ˆ0.7549, ( ) 3.6347.x f x

#49. ˆ ˆ0.3684, ( ) 2.4154.x f x

#50. 12 1

2ln (ln ) .2b a

#51. 2.5.b

#52. 4.a

#53. 2.5.c

#57. ˆ ˆ0.6823, ( ) 0.5814.x f x

#58. ˆ ˆ2.3247, ( ) 7.7290.x f x

157

#59. ˆ ˆ2.2340, ( ) 61.1806.x f x

#62. 3ˆ( ) 1.393 10 .f x

#63. 2ˆ( ) 1.152 10 .f x

#68. (20,46) locmin.

#69. ( 16, 26) K.

#70. (0,0) locmin.

#71. (1,1) locmin.

#72. (1, 4,2) locmin,( 1, 4,2) K.

#73. 2 2

( , ,0) locmin,(0,0,0) K.3 3

#74. 2 1

( , ,1) locmin.3 3

#75. 25 125

( , ) locmin,(0,0) K.3 3

#76. (1,1) locmin.

#77. (5,2) locmin.

#78. (2,3) locmin.

#79. 2 1

( , ,0) locmin.3 3

#80. (1,1) locmin,(0,0) K.

#81. 1 1

(0,0) locmin,( , ) K.4 2

#82. 8 2

(0,0) locmin,( , ) locmax.27 3

#83. 4 1 4 1

( , ) glmin,( , ) glmax.17 17 17 17

#84. 4 3

( , ) glmin.25 25

#85. 1 1

( , ) locmax.2 2

158

#86. 1 3

( , ) glmin.2 2

#87. ar aqvs amonaxsni.

#88. 1 1 1 1

4 4 4 4(0, 1), ( 1,0) glmin,( 2 , 2 ), ( 2 , 2 ) glmax.

#89.

1 1 1 1 1 1 1 1( , ), ( , ) gl min, ( , ), ( , ) gl max.

2 2 2 2 2 2 2 2

udidesi farTobi aqvs kvadrats.

#90. umciresi (udidesi) sakuTrivi ricxvis Sesabamisi sakuT-rivi veqtori funqcionals aniWebs minimalur (maqsima-lur) mniSvnelobas.

#91. 4 4

(4 ,4 ) glmax.3 3

#92. wesieri samkuTxedi.

#93. 1 2 31 2 3

x x xx , x , xTu mocemuli wertilebia , maSin amonaxsnia .

3

#94. 2 2 2 2

, , , min .2 2 2 2

gl

#95. 5, 5 , 5,5 min.gl

#96. 3 3

, min.2 2

gl

#97. 0,0 min, 0, 1 , 1,0 max.gl gl

#98. 0,0 min, 0, 1 , 1,0 max.gl gl

#99. 1 1

6 60,0 min, 2 , 2 max.gl gl

#100. 1 1 1

4 4 40,0,0 min, 3 , 3 , 3 max.gl gl

159

#101. 0,0,0 min, 1,0,0 , 0, 1,0 0,0, 1 , max.gl gl

#102. 0,0,0 min, 1,0,0 , 0, 1,0 0,0, 1 , max.gl gl

#103. 1 1 1 1 1 1 1 1 1 1 1 1

, , , , , , , , , , , min,3 3 3 3 3 3 3 3 3 3 3 3

1 1 1 1 1 1 1 1 1 1 1 1, , , , , , , , , , , max.

3 3 3 3 3 3 3 3 3 3 3 3

gl

gl

#105. wrfivi funqcia.

#111. amozneqilia a), b), d), e).

#113.

1 1 0

1 1 0

1 1 0

(0.0611; 0.1389) ( ) ( )

( 1.4881; 2.0187) ( ) ( )

( 3.6945; 4.6983) ( ) ( )

a)

b)

g)

x f x f x

x f x f x

x f x f x

#114.

1 1 0

1 1 0

1 1 0

( 0.1; 0.1) ( ) ( )

( 0.5;0.5) ( ) ( )

( 1; 1) ( ) ( )

a)

b)

g)

x f x f x

x f x f x

x f x f x

#115.

1 1 0

1 1 0

1 1 0

( 0.1437;0.4;0.4) ( ) ( )

( 1.4323; 0.2761; 0.2761) ( ) ( )

( 10.4366; 5; 5) ( ) ( )

a)

b)

g)

x f x f x

x f x f x

x f x f x

#116. 0 0.25a

#117. 0 0.236a

#118. ˆ ˆ(2.55; 0.85), ( ) 21.675.x f x

#119. ˆ ˆ( 0.9677; 6.4516), ( ) 32.2581.x f x

#120. ˆ ˆ(0.5;0), ( ) 0.25.x f x

#121. 1( ) ( , , ), max 0, , 1, , .sadacn j j

MP z z z z z j n

160

#122. 1

, ,

( ) ( , , ), , , 1, , .

, ,

Tu

sadac Tu

Tu

j

j j

n j j

M j j

j

j j j

a z a

P z z z z b z b j n

z a z b

#123. 0

0

0

, ,

( ) ( ), .

Tu

TuM

z z M

P z r z xx z M

z x

#124. ( ) max 0, .T

M

aP z z b az

a