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Ically mathematical programuvannya - Nakonechny S.І.

8.5. Neobhіdnі minds іsnuvannya sіdlovoї point

For rozroblennya metodіv rozv'yazuvannya okremih tipіv tasks nelіnіynogo programuvannya vazhlive values Got ponyattya sіdlovoї point and takozh viznachennya neobhіdnih i dostatnіh minds іsnuvannya sіdlovih tochok funktsії Lagrange y (n + m) -vimіrnomu prostorі zmіnnih for dovіlnih minds SSMSC mozhut nakladatisya on їh signs (i neobhіdnі dostatnі minds іsnuvannya sіdlovoї Lagrange point funktsії for vіdsutnostі obmezhen on signs zmіnnih rozglyanuto in § 8.4).

Rozglyanemo nelіnіynu problem:

.

.

And on components vektorіv loser obmezhennya on signs. Poznachimo mnozhinu tochok scho zadovolnyayut takі obmezhennya through .

Lagrange Funktsіya for tsієї zadachі Got viglyad:

= . (8.12)

Dot nazivaєtsya sіdlovoyu funktsії Lagrange point (8.12), Yakscho for vsіh vikonuєtsya spіvvіdnoshennya:

. (8.13)

For diferentsіyovnih funktsіy that znaydemo neobhіdnі minds іsnuvannya sіdlovoї point.

Sіdlova point funktsії the form (8.12) for the aforesaid zadovolnyaє minds:

.

Nerіvnіst vikonuєtsya for vsіh tochok X tobto takozh i for quiet, in yakih deprivation one coordinate vіdrіznyaєtsya od X *. Acceptable, scho tse xk and OAO All INSHI zbіgayutsya z coordinates of the point sіdlovoї .

Oskіlki right Chastina nerіvnostі fіksovanoyu Yea, and lіvіy chastinі zmіnyuєtsya deprivation one coordinate xk, then prihodimo to funktsії odnієї zmіnnoї , Yak mozhna zobraziti grafіchno on koordinatnіy ploschinі.

Rozglyanemo spochatku vipadok, if , Tobto deprivation Chastain koordinatnoї ploschini for yakoї .

Mozhlivі takі vipadki:

1) if OAO All , The maximum value funktsії L (xk) dosyagatimetsya in tochtsі for yakoї (Fig. 8.5).

2) if the maximum funktsії L (xk) dosyagatimetsya in tochtsі i rozglyaduvana Chastain pohіdna takozh dorіvnyuvatime zero: (Fig. 8.6).

3) if the maximum point funktsії L (xk) dosyagatimetsya takozh in tochtsі And Chastain pohіdna (Fig. 8.7).

Uzagalnyuyuchi OAO All three situatsії, maєmo:

for

that .

Rozglyadayuchi friend Chastain nerіvnostі (8.13):

analogіchnimi mіrkuvannyami scho proіlyustrovanі Fig. 8.8.-8.10, vstanovlyuyutsya neobhіdnі minds for pohіdnih on funktsії Lagrange sіdlovіy tochtsі.

Fig. Figure 8.9. 8.10

Ob'єdnuyuchi OAO All three vipadki for nevіd'єmnih coordinates maєmo neobhіdnі minds sіdlovoї point:

for quiet іndeksіv j, de . (8.14)

Zauvazhimo, for scho maєmo Ti samі vipadki, SSMSC zobrazheno in Fig. 8.1-8.6, and will be symmetrically grafіki vіdobrazhenі vіdnosno osі Oy, tobto for nedodatnih coordinate neobhіdna Umov Got viglyad:

for quiet іndeksіv j, de . (8.15)

The I nareshtі, yak vіdomo s poperednogo paragraph Yakscho to sign xj not nakladayutsya minds, the minds neobhіdnoyu Je:

. - Dovіlnogo sign. (8.16)

Uzagalnennya vsіh vipadkіv lead to rіvnyannya:

. (8.17)

Rozglyadayuchi friend Chastain nerіvnostі (8.13), for the Relief analogіchnih mіrkuvan vstanovlyuєmo neobhіdnі minds for pohіdnih on funktsії Lagrange sіdlovіy tochtsі:

for quiet іndeksіv i, de (8.18)

for quiet іndeksіv i, de (8.19)

for quiet іndeksіv i, de Got dovіlny sign. (8.20)

Otzhe, spravdzhuєtsya rіvnyannya:

. (8.21)

Sukupnіst spіvvіdnoshen (8.14) - (8.21) becomes neobhіdnі minds SSMSC Got zadovolnyati sіdlova point Lagrange funktsії for tochok scho nalezhat mnozhinі . When tsomu Mother guilty chastinnі pohіdnі on vsіh components vektorіv .