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Mathematics programmers - Nakonechny S.I.

8.4. Classic method of optimization. The Lagrange multiplication method

Yak already zazadulos, for rozvjazuvannya tasks nelinynogo programvannya not існує universal method, tobto before them neobhіdno zastosovuvati broadly in the number of methods and computational algorithms. Wongs in the main base on a densified number of numbers і to lie in the specific setting of tasks and the formation of an economic and mathematical model.

Methods of outlining the tasks of non-linear programming are straightforward and indirect. For the direct approach of the methods of knowledge of the optimal plans for building a straight line of the Nishvidshoi zbil'shennia (zmenshennya) the meaning of the tsil'ovoy function. Tipovim predstavnikom tsієї grupi methodів є градієнтні. The technique of applying indirect methods in front of the task of the tasks to the point, the optimum of the syllable is known by the simplest methods. The middle of indirect nikrasche rozroblennymi є method of rozv'yazvanya tasks quadratic and separable programvannya.

Naiprostіshimi for rozv'yazuvannya і tasks nelinіynogo programvannya, in some system obmezhen sdakadaetsya lishe z rivnyan.

8.4.1. Wit and frenzy ekstremumi functions

In the theory of functional functions, the task of solving the extremes is not to hide the unnatural addictions of minds, and to lie down to the tasks of the insane extravagance of functions. Local that global ekstremumi toti viznachayutsya from neobihdnyh ta dostatnіh minds існування екстремуму функції.

Nagadaєmo, scho neobhіdna umova іsnuvannya local ekstremumu funktsії dvuh zmіnnih sformuyutsya so: in order to point the point Bula point of local ekstremumu, neobhidno, shcheob function Bullet, non-reusable, and differential in the location of the part and in the first part of the field for zmіnimi That In this case, they have zeroed:

.

Dot Nazivayutsya critical.

Dostatna umova іsnuvannya local ekstremumu funktsії dvuh zmіnnih sformulyatsya so: to order the critical point Bula point of local ekstremumu, dostatno, shcheob funktsiia The boule is marked in the critical points That is small in part in the non-partial partial order.

Todi, yakshtoo

,

Then in point Function Mae extremum, prichomu, yakshto

,

Тоді - point of local maximum function , But

,

Тоді - the point of local minimum function .

At different times, yakshchoo

,

Then in point Function Ekstremumu not має.

Yakshto

,

Then the nutriment about esnuvannya ekstremumu zalishaetsya vidkrimim.

Yakshchoo the task of polygas in the localized chi of the global ekstremumu deykoy funktsii for the purpose, but on zmінні такої functції накладаються додаткові обмеження, маємо task пошуку умовного екстремуму функції. The term "cleansing" means, but the tasks of zeminnyi mait your zadovolnyati activities.

Rozglyanemo taku task for vypadku two zmіnnih:

Know (8.4)

For the name of . (8.5)

The simplest way to solve problems in this kind of field is to get the sword and the weariness (8.5) to know the visions of one's mind through the ins. Primir, viznachayut across . Otrimany vizaz vidom Pідставляваюю у функцію (8.4), що після цього стаь functцією однієї змінії , І даі знаходить її неумовний екстремум.

Acting point Є point of exclusion function , Then the point Точ the point of the intelligent extra function (8.4) for the state (8.5).

However, you do not have to go into the vidushakti analitic viraz oneniyeh zmіnії through іншу в умові (8.5). Often, it is not important to prick importantly. Takozh Inodi folds usazhalniti danii sposib for the function n zmіnnih, yaki impersonated m obmezhen. It is described to be simple to perform a simple task of solving problems in the form of a witty extravagance of functions of many zmіnnih to tasks and to the frenzied extremity of function of one's memory can not be the basis of the universal method of isolating tasks for wiping out extremes. The classical method for assigning problems to type (8.4), (8.5) was to string Lagrange out .