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Mathematics programmers - Nakonechny S.I.
3.3. The basic theorems of the duality of economical zmist
Zv'azok Mizh optimal rozv'yazkami straightforward dyadystoy tasks vstavlyayutyu lemi teorii dvuhistosti. The problems (3.1) - (3.3) and (3.4) - (3.6) can be solved visually using the economic inte- gration invoked in § 3.1.
Lema 3.1 ( basic nerves of the theory of binary ). Yakshto That - allow rozv'yazki vidpodіdno direct ya dvoystoy tasks, then vikonuetsya nervivnist
Abo . (3.7)
Beforehand . Multiply to the skin of the system (3.2) on the basis of the two tasks:
Маємо:
Pіsumuvavshi pravosі і ліві частини неівностей, отримаємо:
. (3.8)
Analogically, the system of interchange (3.5) of dual tasks:
Підсумувавши після множення тут також ліві та праваі частини, отримаємо нерівність:
(3.9)
Ліві частини нерівностей (3.8) та (3.9) zbigayutsya, the annunciation:
.
Nervesity (3.7) is brought.
Lema 3.2 ( availability of the optimum ). Yakshto That - permissibles rozv'yazki vidpodіdno direct ya dyhistoї problems, for yakih vikonuєtsya рівність
(3.10)
Then X * , Y * - the optimal rozvyazyki vidpodіdnih tasks.
Beforehand . Nehai - an admissible plan of direct problems (3.1) - (3.3). Тоді on підставі нерівності (3.7) маємо: . For the purpose of tasks , Annealing
(3.11)
Oskilki for admissions - dovіlny permissible plan for direct tasks, then the nerves (3.11) vikonuetsya for be-ya zmolivih rozv'yakiv. Otzhe, maemo, scho with Tsіlова functція (3.1) is typing the nibble value, tobto is the optimal rozv'yakom pochatkoy tasks.
In the analogous manner, - The optimal plan for the two tasks.
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