Imave periods ekuTvnis evklides amocana

biji 1). , amitom (5.16) gvaZlevs:

radgan , amitom . saWiro sizuste jer ar aris miRweuli, radgan

da .

biji 2). , , magram , radgan . radgan M aris Caketili birTvi centriT 0 –Si da radiusiT , SegviZlia gamoviyenoT wina magaliTis Sedegi da aviRoT:

.

saWiro sizuste jer kidev ar aris miRweuli, radgan . Semdegi bijebis Sedegebi moyvanilia cxrilSi.

kx_k 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)0,0087sizuste miRweuliarogorc vxedavT, . SevniSnoT, rom am amocanis zusti amonaxsni aris . 
savarjiSoebi

amocanebSi #98-100 ganaxorcieleT gradientuli daSvebis erTi biji da SeadareT erTmaneTs da .

#98 , ,

a) , b) , g) ;

#99 , ,

a) , b) , g) ;

#100 , ,

a) , b) , g) .

amocanebSi #101-102, ipoveT wertilidan uswrafesi daSvebis meTodis bijis sigrZe .

#101 , ,

#102 , .

amocanebSi #103 - 105, moaxdineT kvadratuli miznis funqciebis minimizacia da daasruleT gamoTvlebi maSin, rodesac Sesruldeba piroba .

#103 ;

#104 ;

#105 ;

amocanebSi #106 -109, moZebneT wertilis proeqcia Semdegi saxis M simravleze:

#106 n _ ganzomilebiani sivrcis arauaryofiT oqtantze:

#107 n _ ganzomilebiani paralelepipedze:

;

#108 Caketil r radiusian birTvze centriT -Si:

.

#109 naxevarsivrceze:

amocanebSi #110 – 113, gamoiyeneT wina amocanebis pasuxebi da gradientis proeqciis meTodiT amoxseniT minimizaciis amocanebi:

#110 .

#111 .

#112 .

#113 .

amocanebSi #114 -117, moZebneT wertilis proeqcia Semdegi saxis M simravleze:

#114 n _ ganzomilebiani sivrcis arauaryofiT oqtantze:

#115 n _ ganzomilebiani paralelepipedze:

;

#116 Caketil r radiusian birTvze centriT -Si:

.

#117 naxevarsivrceze:

amocanebSi #118 – 120, gamoiyeneT wina amocanebis pasuxebi da gradientis proeqciis meTodiT amoxseniT minimizaciis amocanebi:

#118 .

#119 .

#120 .

gluvi eqstremaluri amocanebi tolobis da utolobis tipis SezRudvebiT
1. mokle reziume

gluvi amocana tolobisa da utolobis tipis SezRudvebiT ewodeba Semdeg eqstremalur amocanas:

(6.1)

sadac yoveli g_j da f_i (j=0,...,k, i=1,…,m) aris sakmarisad gluvi funqcia . radgan eqstremumis ganmartebaSi monawileoben utolobebi, amitom miznis funqcia da utolobebis ganmsazRvreli funqciebi erTi da igive simboloTia aRniSnuli. (6.1)-is amomwurav analizs Cven ar moviyvanT, radgan Tavisi sirTulisa da specifikis gamo es ufro speckursis Temaa. praqtikuli amocanebis amoxnisas xelsayrelia (6.1)-is nacvlad ganvixiloT ori minimizaciis amocana (erTgan g₀(x)→min, meoregan ki _g₀(x)→min), radgan minimizaciisa da maqsimizaciis amocanebisaTvis algoriTmebi gansxvavdeba mcire detalebiT, rac advilad iwvevs Secdomas, amitom zogadobis SeuzRudavad, SemdegSi ganvixilavT minimizaciis gluv amocanas:

g₀(x)→min, g₁(x)≤0, …, g_k(x)≤0, f₁(x)=0,…, f_m(x)=0. (6.2)

am amocanaSi miznis funqciaa ,xolo dasaSvebi simravle aris:

am paragrafSi Cven davasabuTebT, rom tolobis da utolobis tipis SezRudvebis mqone gluvi amocanis gamokvleva unda moxdes Semdegi TanmimdevrobiT:

1. 
   maTematikuri modelis Sedgena, anu amocanis formulireba (6.2) saxiT;
   
2. 
   (6.2) amocanisaTvis lagranJis funqciis Sedgena:
   

(6.3)

sadac λ_i da μ_i є R lagranJis mamravlebia, xolo μ=( μ₁,…, μ_k), λ=( λ₁,…, λ_m).

1. 
   (6.2) amocanisaTvis kritikuli wertilebis simravlis gansazRvra:
   

gantolebaTa sistemis:

(6.5)

amonaxsnebs Soris SevarCioT iseTebi, rom Sesruldes:

a)

b)

g) .

aseTi amonaxsnebis _ebis simravle aRvniSnoT K-Ti da vuwodoT kritikuli wertilebis simravle (6.2) amocanaSi. lagranJis principis Tanaxmad locmin(6.2) Sedis K-Si.

1. 
   kritikuli wertilebis gaamokvleva im mizniT, rom maTgan amovarCioT minimumis wertilebi.
   

2. maTematikuri modelis Sedgena

sailustraciod, ganvixiloT fasiani qaRaldebis portfelis Sedgenis amocana.

erT–erTi umniSvnelovanesi amocana, romlis gadaWrac uwevT sainvesticio firmebs (bankebs, fondebs, sadazRvevo kompaniebs) aris fasiani qaRaldebis optimaluri portfelis Sedgenis amocana. portfeli niSnavs sxvadasxva saxis fasian qaRaldebSi (obligaciebi, aqciebi, sabanko sadepozito sertifikatebi da sxva) Cadebuli Tanxebis moculobebis erTobliobas. am amocanis gadasaWrelad, misi maTematikuri sirTulisa da ekonomikuri mniSvnelobis gamo, damuSavebulia sxvadasxvanairi modelebi.

vigulisxmoT, rom damdeg sainvesticio periodSi SeiZleba naRdi C kapitalis Cadeba n saxis fasian qaRaldebSi. vTqvaT, aris j– uri saxis fasian qaRaldSi Cadebuli kapitalis moculoba (dolarebSi), . maSin, cxadia rom

.

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, aris t welSi j saxis fasian qaRaldSi dabandebuli erTi dolarisgan miRebuli mTliani Semosavali. maSin:

,

sadac aris j saxis fasiani qaRaldis fasi wlis (zogadad, sainvestcio periodis) dasawyisSi, - t welSi miRebuli jamuri dividendebi.

SevniSnoT, rom SesaZloa Zlier icvlebodes wlidan wlamde da SeiZleba hqondes nebismieri niSani. imisaTvis, rom Sefasdes j saxis fasian qaRaldSi dabandebis mizanSewoniloba, unda Sefasdes j saxis fasian qaRaldSi dabandebuli erTi dolarisgan miRebuli Semosavlis saSualo anu mosalodneli mniSvneloba:

.

mosalodneli Semosavlis mTliani sidide iqneba:

,

sadac , .

modeli 1. am amocanisTvis SedarebiT martivad gamosakvlev (Sesabamisad, naklebad adekvatur) maTematikur models aqvs saxe:

,

im pirobiT, rom

.

sxva sityvebiT, gvinda movaxdinoT mTliani mosalodneli mogebis maqsimizacia imis gaTvaliswinebiT, rom SezRudulia investiciebis mTliani moculoba da valis aReba ar daiSveba. amave models SeuZlia damatebiT gaiTvaliswinos sxva SezRudveb. ganvixiloT ori SedarebiT gavrcelebuli maTgani.

sainvesticio firmebis umravlesoba zRudavs kapitaldabandebaTa moculobas Cveulebrivi aqciebisTvis, radgan maTgan Semosavals axasiaTebs mniSvnelovani rxevebi (anu Zlieri cvlilebebi). aseTi SezRudva SeiZleba ACaiweros formuliT:

,

sadac Sedgeba Cveulebrivi aqciebis sxvadasxva saxeebis indeqsebisgan, xolo aRniSnavs Cveulebriv aqciebSi maqsimalur dasaSveb kapitaldabandebas.

mravali sainvesticio firmis mmarTveloba aucileblad miiCnevs kapitalis nawilis SenarCunebas naRd fulad an mis ekvivalentur raime formad, meanabreTa moTxovnilebebis dakmayofilebis mizniT. es SezRudva SeiZleba ACaiweros formuliT:

,

sadaac simravlis indeqsebi Seesabameba fasian qaRaldebs, romlebic naRdi fulis ekvivalenturia (magaliTad, Semnaxvel angariSebs, mimdinare angariSebs); aris aucilebeli naRdi saSualebebis minimaluri raodenoba.

am martivi modelis ZiriTadi nakli imaSi mdgomareobs, rom investiciebTan dakavSirebuli riskis gaTvaliswineba ar xdeba. fasiani qaRaldebis pirtfeli, romelsac SevadgenT am modelis gamoyenebiT, SeiZleba gvpirdebodes maRal saSualo mosalodneli Semosavals, magram investiciebTan dakavSirebuli riskic aseve maRali iqneba. maRali riskis gamo, realuri Semosavali SesaZloa gacilebiT dabali aRmoCndes, vidre Teoriulad mosalodneli.

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 mimdinare angariSebi an sabanko depozituri sertifikatebi, _ iZleva nakleb Semosavals.

sainvesticio riskis zomad gamoiyeneba Semosavlis gadaxra misi saSualo mniSvnelobidan bolo T wlis ganmavlobaSi. aRvniSnoT -Ti j saxis fasiani qaRaldis dispersia (sainvesticio riski) romelic gamoiTvleba formuliT:

.

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 saavtomobilo da sanavTobo firmebis aqciebi, komunaluri momsaxurebis sawarmoTa aqciebi da sxva. msgavsi riskis Sesamcireblad, investiciebi unda ganawildes fasiani qaRaldebis sxvadasxva jgufebze. aseT ganawilebaSi gamoiyeneba Semosavlis doneTa Tanafardobis Sefasebebi fasiani qaRaldebis yoveli wyvilisTvis. es Tanafardoba gamoisaxeba kovariaciis koeficientebiT, romlebic gamoiTvleba bolo wlebis statistikuri monacemebis safuZvelze:

.

SevniSnoT, rom roca , es sidide gadadis j saxis fasiani qaRaldis dispersiaSi. amgvarad, sainvesticio riskis sazomad SegviZlia gamoviyenoT sidide:

,

sadac aris kovariaciis matrica, zomis, Sedgenili n saxis fasiani qaRaldisaTvis.

portfelis gansazRvris procesSi fasiani qaRaldebis mflobeli SesaZloa dainteresebuli iyos garkveuli moculobis saSualo mosalodneli mogebis miRebiT minimaluri riskis pirobebSi. Sesabamis optimizaciis amocanas aqvs saxe:

,

im pirobiT, rom

,

sadac R aris mogebis minimaluri saSualo mosalodneli mniSvneloba portfelis arCevis dros. modelSi, SesaZloa, agreTve iyos firmis politikasTan dakavSirebuli SezRudvebi, rogorc wina modelSi iyo ganxiluli.

3. kritikuli wertilebis gansazRvra

Semdeg Teoremas, romelic warmoadgens minimumis pirveli rigis aucilebel pirobas minimizaciis

g₀(x)→min, g₁(x)≤0, …, g_k(x)≤0, f₁(x)=0,…, f_m(x)=0. (6.6)

amocanaSi, tradiciulad uwodeben lagranJis mamravlTa wess gluvi amocanebisTvis tolobis da utolobis tipis SezRudvebiT.

Teorema 1 /minimumis pirveli rigis aucilebeli piroba/. vTqvaT, g_j da f_i funqciebi (j=0,…,k, i=1,…,m) uwyvetad warmoebadebia wertilis raime midamoSi, xolo aris lokaluri minimumis wertili (6.2) amocanaSi. maSin arseboben erTdroulad nulis aratoli lagranJis mamravlebi romelTaTvisac sruldeba Semdegi pirobebi:

a) x-is mimarT lagranJis funqciis stacionalurobis piroba:

;

b) niSnebis SeTanxmebis piroba: yoveli _sTvis ;

g)

damtkiceba. Tu aRvniSnavT: , maSin aris lokaluri minimali Semdeg amocanaSi:

g₀(x)→min, g₁(x)= , …, g_k(x)= , f₁(x)=0,…, f_m(x)=0.

amitom SegviZlia gamoviyenoT wina paragrafis analogiuri Teorema da davaskvnaT, rom arseboben erTdroulad nulis aratoli lagranJis mamravlebi romelTaTvisac sruldeba x-is mimarT lagranJis funqciis stacionalurobis piroba:

.

amgvarad, dasamtkicebeli rCeba b) da g). simartivisTvis, maT davamtkicebT kerZo SemTxvevisTvis, romelic, principSi, srulad amowuravs Teoremis praqtikul gamoyenebebs.

davuSvaT, sruldeba igive pirobebi, rac sakmarisia lagranJis mamravlebis ekonomikuri Sinaarsis gasarkvevad da vigulisxmoT, rom parametr – veqtoris yoveli mniSvnelobisTvis, romelic mcired ganxvavdeba -sgan, sadac, Semdeg amocanas

g₀(x)→min, g₁(x)= , …, g_k(x)= , f₁(x)=0,…, f_m(x)=0

aqvs erTaderTi amonaxsni . yoveli –sTvis SesaZlebelia ori SemTxveva: (e.i. ) an (e.i. ). pirvel SemTxvevaSi erTdroulad sruldeba Semdegi ori faqti: erTis mxriv, , xolo meores mxriv erTi cvladis funqcia , ganxiluli naxevradRia SualedSi sakmaod mcire -sTvis, maqsimums aRwevs -Si; amitom, . meore SemTxvevaSi, , ganxiluli Ria SualedSi sakmaod mcire -sTvis, maqsimums aRwevs -Si da amitom, . ℃

magaliTi. erTeulovan wrewirSi CavxazoT maqsimaluri farTobis mqone marTkuTxedi.

amoxsna. x-iT aRvniSnoT horizo-ntaluri gverdis naxevari, y-iT vertikalis naxevari, maSin farTobi aris 4xy, SezRudvebi aris x²+y²=1 da x≥0, y≥0. amgvarad , gvaqvs:

(6.7)

esaa gluvi eqstremaluri amocana tolobis da utolobis tipis SezRudvebiT. SevadginoT misTvis lagranJis funqcia:

.

amovweroT gantolebebi, romlebic gansazRvraven kritikul wertilebs:

(6.8)

es aris arawrfivi sistema xuTi gantolebiTa da eqvsi cvladiT.

movZebnoT kritikuli wertilebi. am mizniT Cven veZebT (6.8)-is iseT amonaxsnebs, romlebic akmayofileben zemoT aRweril a), b), g) SezRudvebs. visargebloT lagranJis funqciis erTgvarovnebiT mamravlebis mimarT da SevamciroT cvladebis raodenoba ((6.8)-Si SemTxvevaTa ricxvis gazrdis xarjze):

1. 
   jer ganvixiloT μ₀=0. (6.8) miiRebs saxes:
   

μ₁₌ 2λx μ₁x=0, x²+y²=1, μ₂ =2λy, μ₂y=0,

e.i saidanac da radgan x²+y²=1, amitom λ=0  μ₁=μ₂=0, e.i Tu μ₀=0, rogori amonaxsnic ar unda hqondes (6.8)-s, yovelTvis , rac winaaRmdegobaSia a) SezRudvasTan. e.i μ₀=0 SemTxveva arc erT kritikul wertils ar iZleva.

1. 
   aviRoT μ₀=1. axla SevecadoT (6.8)-Si movxsnaT saxis arawrfivobebi. amisaTvis cal-calke unda ganvixiloT SemTxvevebi da . (b) pirobis ZaliT meti ar aris). Tu μ₁=0 maSin
   

axla, Tu

μ₂>0, maSin y=0, <0 da winaaRmdegobaa g)-sTan. e.i. μ₁=0-is dros μ₂=0 qveSemTxveva dagvrCa gansaxilavi.

μ₂=0. maSin

e.i. λ²=1/4 da x² y², anu x y (g) -ZaliT).

sabolood, x²+y² 1 gvaZlevs x y , e.i (6.8)-is amonaxsnia , romlisaTvisac sruldeba:

a) (1/2,0,0,1)≠1

b) (0,0,1)≥0

g) .

amitom .

amiT μ₂=0 qveSemTxvevis ganxilva dasrulda.

μ₁ –is SemTxvevebi simetriulia ganxilulisa da iZleva igive kritikul wertils. sabolood, Cvens amocanaSi aris erTaderTi kritikuli wertili: .

4. kritikuli wertilebis gamokvleva

rodesac eqstremaluri amocana Seicavs utolobis tipis SezRudvebs, misi kritikuli wertilebis gamokvleva rTulia. ZiriTad instruments warmoadgens vaierStrasis Teorema da misi Sedegebi, xolo, ukidures SemTxvevaSi, ganmartebis Semowmeba unda vcadoT. Tu funqciebs aqvT damatebiTi Tvisebebi (amozneqiloba), maSin Cndeba damatebiTi SesaZleblobebic.

magaliTi /gagrZeleba/. da ukve viciT, rom kritikuli wertili erTaderTia . rogorc vxedavT, miznis funqcia uwyvetia, xolo dasaSvebi simravle

aris erTeulovani wrewiris Caketili erTi meoTxedi, anu kompaqturi simravle. amitom amocanaSi arsebobs globaluri minimali da igi moTavsebulia kritikul wertilebs Soris. magram kritikuli wertili erTaderTia, amitom .

pasuxi: maqsimaluri farTobi aqvs kvadrats, romlis gverdis sigrZea .

es magaliTi sakmaod martivia, amitom uSualod ganmartebis safuZvelzec SegviZlia gamovikvlioT kritikuli wertili . amisTvis ganvixiloT

(6.9)

magram wertili unda akmayofilebdes SezRudvebs:

,

saidanac . rac gvaZlevs:

.

amitom, .

5. lagranJis meTodis geometriuli interpretacia

rodesac , maSin eqstremaluri amocanisaTvis TvalsaCino geometriuli interpretaciis micema advilia, radgan dasaSvebi simravle da miznis funqcia gamoisaxebian sibrtyeze, rogorc nax. 6 –ze.

rodesac , geometriuli in-terpretaciisaTvis viyenebT sibr-tyes, sadac dasaSvebi simravle Cveu-lebriv gamoisaxeba, miznis funqcias ki gamovsaxavT donis wirebisa da uswra-fesi zrdis mimarTulebis saSualebiT.

ganmartebis mixedviT, funqciis donis wiri (igives niSnavs ganurCevlobis wiri) ewodeba sibrtyis iseTi wertilebis erTobliobas, sadac am funqciis mniSvnelobebi erTi da igivea, anu simravles

,

romelsac xSirad aigiveben mis ganmsazRvrel gantolebasTan:

.

amasTan, gansxvavebuli konstantebi warmoSoben gansxvavebuli 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 konstanta SeiZleba gansazRvravdes mraval donis wirs.

rac Seexeba funqciis uswrafesi zrdis mimarTulebas, Cven ukve viciT, rom igi emTxveva gradientis

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. 7 _ze naCvenebia

(6.10)

maqsimizaciis amocanis amonaxsni, romelSic donis wirebis konstantebi izrdebian g-s uswrafesi zrdis mimarTulebiT, xolo da wertlebze gamavali parabola warmoadgens dasaSveb simravles.

rogorc vxedavT, dasaSvebi simravle da donis wiri, romelic dasaSveb simravleSi miznis funqciis uswrafesi zrdis mimarTulebiT yvelaze Sors aris moTavsebuli, erTmaneTs exebian maqsimalSi (termini “wirebis Sexeba” Cven jer ar gvaqvs ganmartebuli, magram mas male mivaniWebT azrs).

vaCvenoT, rom dasaSvebi simravlisa da donis wiris Sexeba maqsimumis wertilSi (tolobis tipis SezRudvebis SemTxvevaSi minimalisTvisac igive mdgomareobaa) aris zogadi faqti

(6.11)

saxis amocanebSi da es faqti warmoadgens lagranJis meTodis geometriul Sinaarss.

imisaTvis, rom dasaSvebi simravle da donis wiri marTlac wirebs warmoadgendnen, unda moviTxovoT , raTa Sesruldes aracxadi funqciis Sesaxeb Teoremis pirobebi.

davuSvaT sawinaaRmdego, rom arsebobs (6.11) saxis amocana (ix. nax. 8), iseTi, rom maqsimalSi ikveTeba dasaSvebi simravle da -ze gamavali donis wiri . maSin, f da g funqciebis sigluvis gamo, iarsebebs donis wiri antigradientis mimarTulebiT -idan, romelic sakmaod axlosaa donis wirTan da kveTs M–s, da arsebobs donis wiri gradientis mimarTulebiT -idan, romelic sakmaod axlosaa donis wirTan da aseve kveTs M-s. cxadia, da amitom , sadac aris donis wirisa da M–is gadakveTis wertili. amgvarad, miviReT winaaRmdegoba daSvebasTan, e.i. donis wiri da dasaSvebi simravle maqsimalSi ar kveTen erTmaneTs, Tumca es jer kidev ar niSnavs, rom isini exebian erTmaneTs.

(10) amocanaSi, maqsimalis gansazRvrisaTvis lagranJis meTodi iZleva pirobas , rac niSnavs, rom veqtorebi moTavsebulia erT wrfeze, anu, dasaSvebi simravle da donis wiri ki ar kveTs erTmaneTs _Si, aramed exeba. magaliTad, (9) amocanisTvis naxazze Tu gamovsaxavT veqtors, davinaxavT rom igi Zevs da wertilebze gamaval wrfeze.

axla davamtkicoT, rom dasaSveb simravlesa da maqsimalSi gamaval donis wirs aqvT erTi da igive mxebi (es niSnavs maT “Sexebas”), romelic orTogonaluria rogorc , aseve veqtorisa.

jer ganvixiloT M–is mxebi maqsimalSi. aviRoT gluvi wiri , ganmartebuli wertilis raRac midamoSi da iseTi, rom:

1. 
   ;
   
2. 
   
   

b) niSnavs, rom h aris M–is (anu donis wiris) mxebi _Si. aseTi funqcia arsebobs aracxadi funqciis Sesaxeb Teoremis Tanaxmad. vaCvenoT, rom

(6.12)

anu _Si gradientis mimarTuleba orTogonaluria mxebi veqtorisa. marTlac, radgan yoveli , amitom e.i.

e.i. sruldeba (6.12).

analogiurad mtkicdeba, rom Tu aris _ze gamavali donis wiris mxebi veqtori (cxadia, am dros ), maSin:

.

lagranJis meTodis Tanaxmad, da moTavsebulia erTi da igive wrfeze, amitom h da w agreTve erTi da igive wrfeze arian moTavsebulebi, rac niSnavs, rom dasaSvebi simravle da donis wiri erTmaneTs exebian (aqvT erTi da igive mxebi) _Si.

rodesac SezRudvebi tolobis tipisaa, -s niSani SezRuduli ar aris, da SeiZleba mimarTuli iyos rogorc erTi da igive, aseve sxvadasxva mxares.

sxva mdgomareobaa utolobis tipis SezRudvis SemTxvevaSi, ris geometriul axsnasac axla moviyvanT (ix. nax. 10).

simartivisaTvis vigulisxmoT, rom aris wrfivi funqcia, ki kvadratuli, iseTi rom warmoadgens elifss, romlis SigniT (gareT) Rebulobs uaryofiT (dadebiT) mniSvnelobebs, rasac miuTiTebs Sesabamisad ganlagebuli + da – niSnebi. am pirobebSi, ganvixiloT amocana

(6.13)

vTqvaT, . maSin, lagranJis meTodis Tanaxmad, , da amitom da gradientebi erT wrfeze sxvadasxva mxaresaa mimarTuli (sxvadasxva niSani aqvT). amaTgan, _Si modebuli veqtori aucileblad elifsis gareT aris mimarTuli, radgan uswrafesi zrdis mimarTulebaa, elifsi nulovani donea, dadebiTi doneebi ki mxolod elifsis gareTaa. _Si modebuli veqtori aseve gareT rom yofiliyo mimarTuli, maSin miiRebda ufro mcire mniSvnelobebs elifsis SigniT.

6. qunisa da Taqeris Teoremis Camoyalibeba da ganxilva

ganvixiloT zogadi saxis amozneqili amocana utolobis tipis SezRudvebiT:

, , , (6.14)

sadac aris romeliRac wrfivi sivrcis amozneqili qvesimravle da yoveli , aris amozneqili funqcia.

radgan -is amozneqilobidan ar gamomdinareobs _ -is amozneqiloba, amitom arsebiTia rom (6.14) aris minimizaciis amocana.

(16.14) amocanaSi dasaSvebi simravlis gansazRvraSi monawileoben funqciebi da amozneqili simravle. dasaSvebi simravle aq aris:

da .

-s amozneqilobidan martivad gamomdinareobs simravlis amozneqiloba. amitom SegviZlia warmovadginoT amozneqili simravleebis TanakveTis saxiT:

.

zogadi saxis amozneqili amocanisaTvis Cveulebrivad ganisazRvreba lagranJis funqcia:

, sadac ;

aRsaniSnavia, rom igi ver iTvaliswinebs SezRudvas.

Semdegi Teorema damtkicebulia 1951 wels. igi mravalmxrivaa saintereso, aRsaniSnavia, rom Teoremis pirobebSi mxolod amozneqilobis cneba monawileobs, araa laparaki normirebul sivrceebsa da uwyvet da warmoebad asaxvebze.

saqme imaSia, rom arsebobs Sinagani kavSirebi uwyvetobasa da amozneqilobas, warmoebadobasa da amozneqilobas Soris.. magaliTad, Tu amozneqilia, igi uwyvetia -ze, xolo amozneqil

asaxvas yovel wertilSi gaaCnia mimarTulebiTi warmoebulebi.

Teorema /quni_Taqeri/. 1). vTqvaT, aris wrfivi sivrce, amozneqili qvesimravlea -Si da , amozneqili funqciebia -ze

Tu aris minimali (1) amocanaSi, maSin moiZebneba iseTi erTdroulad nulis aratoli lagranJis mamravlebi rom:

a) (minimumis principi);

b) (arauaryofiTobis piroba);

g) , .

2) Tu maSin a)_g) pirobebi sakmarisia, raTa dasaSvebi wertili iyos minimali (1)-Si;

3) imisaTvis rom sakmarisia moiZebnos iseTi , romlisTvisac Sesruldeba (sleiteris) pirobebi: .

am Teoremis damtkiceba Cvens gegmaSi ar Sedis, TviT quni_Taqeris Teoremac mimoxilvis mizniT mogvyavs. magram ramdenime sasargeblo SeniSvnis gakeTeba SegviZlia.

(i) a) piroba niSnavs, rom lagranJis mamravlebis SerCevis xarjze zogadi saxis amozneqili amocana, romelic Seicavs utolobis tipis SezRudvebs, miiyvaneba amozneqil amocanaze utolobis tipis SezRudvebis gareSe:

, . (6.15)

sxva sityvebiT, lagranJis funqciis gamoyenebiT SegviZlia “movxsnaT” utolobis tipis SezRudvebi, rac arsebiTia, radgan (6.15)-is amoxsna gacilebiT advilia, vidre (6.14)-isa; magaliTad, Tu da warmoebadebic arian, maSin a) miiRebs Cveul saxes

, ,

radgan amozneqilia -ze -is mimarT (Teoremis pirobebSi) da amitom misTvis upirobo minimumis aucilebeli piroba aris sakmarisic.

(ii) roca SegviZlia yvela mamravli erTdroulad gavyoT -ze da mivaRwioT -s. migviTiTebs dasaSvebi simravlis gadagvarebaze. marTlac, Tu sleiteris piroba Sesrulda, maSin dasaSvebi simravle araa gadagvarebuli da .

(iii) Teoremis damtkiceba efuZneba amozneqili analizis erT-erT ZiriTad Sedegs _ ganacalkevebis Teoremas, romlis formulirebas erT kerZo SemTxvevaSi axla moviyvanT.

Teorema. /gancalkevebis Sesaxeb sasrulganzomilebian SemTxvevaSi/. vTqvaT amozneqilia -Si da . maSin moiZebnebian iseTi ricxvebi, rom

,

(sxva sityvebiT, 0-ze gamavali hipersibrtye mTlian sivrces hyofs or nawilad, romelTagan erTSi sruladaa moTavsebuli ).

savarjiSoebi

vTqvaT uwyvetad warmoebadi funqciaa, gansazRvreT kritikuli wertilebi Semdeg amocanebSi:

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