?????? ???? ???? ?? ??????? ????? ?? ?????? ??? ????? ??????? ? ?????? ???? ?? ????? ???????. ????? ??? ???? ????? ?????? ?? ?? ????? ??????? ?? ????? ?????? ??????. ????? ??? ???? ???? ??????? ?? ?? ????? ?????? ???? ?? ???????? ????? ????? ?? ???? ??????? ????? ???? ???????????. ?? ???? ?????? ??????? ?????? ???? ?? ????????? ?????? ??????? ???? ?????? ????????????? ??? ?? ?? ????? ????? ??? ???? ?? ??? ????????. ??? ???? ???? ??????? ?? ?? ???????? ????? ???????? ????? ??????? ?????? ????? ?? ??? ?? ????? ?? ????? ????? ??? ?????????? ??????? ?????? ? ?????? ?????.

????? ??? ???? [ ?????? ]

??? ???? ?????? ??? ???? ???? ?? ?????? ??? ???? ??? ? ??? ?? ????? ?? ??? ???? ?? ??? ≪????? ????? ???????≫ ???? ??. ????? ????????? ?????? ?? ????? ???? (???) ? ??? ??? (????) ????? ?? ??? ????? ???? ????? ???? ???? ??? ????. ^[?] ????? ??? ???? ???? ??????? ?? ??? ??? ???? ????? ??????? ????? ??????? ?????? ???? ?? ????? ????? ????? ?? ???????? ?? ?????? ??? ?? ????? ??????? ?? ????? ?????? ??????. ??? ????? ??????? ????? ??????? ???? ??? ???? ???? ????? ?? ??? ????? ???. ??? ???? ???? ??????? ?? ????? ??? ?????? ????? ???? ????? ???. ?? ???? ?? ????? ???? ??????. ?????? ?? ???? ????? ?? ??? ?????? ??? ???? ??? ???? ?? ?? ??? ???? ???? ?? ??? ??? ?? ??? ??? ?? ?? ????? ???: ??? ????? ??????? ???? ??? ? ??????? ?????? ??? ? ???? ?????? ???? ??????? ???? ??????. (??? ??? ??? ????? ?? ?????? ????? ??? ?? ????? ??? ???? ?? ??????? ??????? ????? ??? ??????) ??????? ??? ???? ?? ???? ??????? ????? ?? ?????? ????????? ???? ???????. ^[?]

???? ??? ???? [ ?????? ]

??? ???? ?? ???? ????? ????? ??? ???. ???? ? ????? (????) ? ???? (????) ??? ????? ????? ????? ??? ?? ???? ???? ?? ??? ??? ????? ???? ????? ??? ???? ?? ?????? ??? (????) ????? ??. . ^[?] ??? ???? ???? ??????? ?? ??? ???? ????? ????? ???? ????? ? ???? ????? ??????? ???? ? ????? ?????? ????? ????? ????? ?? ????? ?????? ???? ?? ???? ??????? ????? ???? ??????????? ??? ?? ?????? ????? ???????? ????? ???. ??? ???????? ???? ?? ???? ??? ?????? ????? ?? ?? ??? ? ???? ?????? ???? ?? ?????? ???? ????. ??? ???? ?????? ? ????? ????? ?? ?? ?????? ????? ?????? ?? ???? ?????????? ????? ??????? ?? ?? ??? ???? ????????? ?????? ???? ???? ??????? ?????? ???? ????? ???????? ????? ????? ?? ?????? ???? ?? ??? ?? ????? ??? ?? ???????? ????? ??? ????? ?????? ???? ?? ???? ????? ??? ?? ??????? ??????? ????? ?? ?? ??? ????. ^[?]

????? ????? [ ?????? ]

???? ??? ???? [ ?????? ]

???? ???? ???? ???? ??? ???:

??? ??????? ???????????? ????? ?????? ? ???(x*,y*,p)?? ????? ????? ?????? ????? ????? ????? (*x*,y) ?????? ???? ???. ?????? ??? ?? ????? ?????? ??????? ?????? ???? ???. ?????: ??? ???? ${\displaystyle (x^{*},y^{*},p)}$ ?? ????? ????? ??????? ???? ?????? ???? ????? ?? ?? ${\displaystyle (w_{1},...,w_{I})}$ ???? ?? ?????? ?? ${\displaystyle \Sigma _{i}w_{i}=p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}^{*}}$ ?????????? ??????? ???? ???? ?? ????? ????? ???? ???????:

${\displaystyle ifx_{i}\succ x_{i}^{*}}$????? ${\displaystyle p\cdot x_{i}>w_{i}}$

???? ?? ?????? ?? x ?? ??????????? i ?? ?? ?? ${\displaystyle x_{i}^{*}}$ ????? ?????????? ???? ???. ????? ??????? ????? ???? ?????? ??:

${\displaystyle ifx_{i}\succeq x_{i}^{*}}$ ????? ${\displaystyle p\cdot x_{i}\geq w_{i}}$

???? ?????? ?? ????? ?? ???? ${\displaystyle x_{i}^{*}}$ ???? ?????? ??? ??? ?????? ??? ???? ????.

??? ??? ???? ????? ${\displaystyle (x,y)}$ ?? ???? ?????? ???? ?? ????? ${\displaystyle (x^{*},y^{*})}$ ???? ????. ???? ${\displaystyle x_{i}\succeq x_{i}^{*}}$ ???? ???? i?? ? ${\displaystyle x_{i}\succ x_{i}^{*}}$ ???? ?? i??. ?? ????? ${\displaystyle p\cdot x_{i}\geq w_{i}}$ ???? ???? i?? ? ${\displaystyle p\cdot x_{i}<w_{i}}$ ???? ???? i??? ????????

${\displaystyle \Sigma _{i}p\cdot x_{i}>\Sigma _{i}w_{i}=p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}^{*}}$ ?? ????? ?? ???? ?? ${\displaystyle y_{j}^{*}}$ ???????????? ??? ????? j ?? ?? ????? ???? p ??? ?????:

${\displaystyle p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}^{*}\geq p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}}$

?? ?????

${\displaystyle \Sigma _{i}p\cdot x_{i}>p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}}$

??? ?? ??? ???? (x,y) ???? ??????? ????. ?? ????? ${\displaystyle \Sigma _{i}x_{i}={\bar {w}}+\Sigma _{j}y_{j}}$ ???? ${\displaystyle \Sigma _{i}p\cdot x_{i}=p\cdot {\bar {w}}+\Sigma _{j}p\cdot y_{j}}$ ?? ?? ????? ?? ????? ???????. ?? ????? ????? ?????? ${\displaystyle (x^{*},y^{*})}$ ?????? ??????. ^[?]

????? ??? ???? [ ?????? ]

?? ??????? ?? ?????? ${\displaystyle (\{X_{i},\succeq _{i}\},Y_{j},{\bar {w}})}$ ????? ${\displaystyle (x^{*},y^{*})}$ ? ????? ???? ${\displaystyle p=(p_{1},...,p_{L})\neq 0}$ ?? ??? ????? ????? ?????? ?? ??????? ??? ????? ???? ${\displaystyle (w_{1},...,w_{I})}$ ?? ???? ${\displaystyle \Sigma _{i}w_{i}=p\cdot {\bar {w}}+p\cdot \Sigma p\cdot y_{j}^{*}}$ ???? ????? ???? ??????? ??:

(i)???? ???? j??? ${\displaystyle y_{j}^{*}}$ ??? ?? ?? ${\displaystyle Y_{j}}$ ?????? ?????? ????

${\displaystyle p\cdot y_{j}\leq p\cdot y_{j}^{*}}$ ???? ???? ${\displaystyle y_{j}\in Y_{j}}$

(ii) ???? ?? i ??? ${\displaystyle x_{i}\succ x_{i}^{*}}$ ????? ${\displaystyle p\cdot x_{i}\geq w_{i}}$

(iii) ${\displaystyle \Sigma _{i}x_{i}^{*}={\bar {w}}+\Sigma _{j}y_{j}^{*}}$

???? ???? ???? ???? ??? ???:

??????? ?? ?????? ${\displaystyle (\{X_{i},\succeq _{i}\},Y_{j},{\bar {w}})}$?? ?? ??? ?????? ? ??? ???? ${\displaystyle Y_{j}}$ ???? ??? ? ?????? ??????? ${\displaystyle \succeq _{i}}$ ???? ? ?????? ????? ????? ?????? ???? ????? ???? ??????? ?????? ???? ${\displaystyle (x^{*},y^{*})}$ ????? ???? ${\displaystyle p=(p_{1},...,p_{L})\neq 0}$ ???? ?????????????? ${\displaystyle (x^{*},y^{*},p)}$ ?? ??? ????? ????? ?????????.

???? ?? i ???? ${\displaystyle V_{i}}$ ?? ?? ????? ???? ???? ${\displaystyle x_{i}}$????? ?? ?? ${\displaystyle x_{i}^{*}}$ ????? ???? ????? ?? ????: ${\displaystyle V_{i}=\{x_{i}\in X_{i};x_{i}\succ x_{i}^{*}\}\subset \mathbb {R} ^{L}}$ ??? ????? ???????:

${\displaystyle V=\Sigma V_{i}=\{\Sigma x_{i}\in \mathbb {R} ^{L};x_{1}\in V_{1},...,x_{I}\in V_{I}\}}$

${\displaystyle Y=\Sigma _{j}Y_{j}=\{\Sigma y_{j}\in \mathbb {R} ^{L};y_{1}\in Y_{1},...,y_{J}\in Y_{J}\}}$

?? ????? V ????? ???? ????????? ?? ??? I ??? ????? ?????? ?? ?? ???? ???? ?????????? ?? ?? ${\displaystyle x_{i}^{*}}$ ????? ????. ???? Y ???? ?? ??????? ????? ????? ???.

??? ?. ???? ????? ${\displaystyle V_{i}^{*}}$ ????????.??? ???? ${\displaystyle x_{i}\succ x_{i}^{*}}$ ? ${\displaystyle x_{i}'\succ x_{i}^{*}}$ ?? ???? ?? ${\displaystyle 0\leq \alpha \leq 1}$ ?? ?????? ???? ???? ${\displaystyle \alpha x_{i}+(1-\alpha )x_{i}'\succeq x_{i}^{*}}$ . ?? ?????? ?? ???? ??????? ???? ??? ???? ????? ?? ???? ??????? ??? ??? ${\displaystyle x_{i}'\succeq x_{i}^{*}}$ ?? ????? ?? ???? ???? ???? ??????? ?????: ${\displaystyle \alpha x_{i}+(1-\alpha )x_{i}'\succeq x_{i}}$ ?? ?? ??? ??????? ????? ${\displaystyle \alpha x_{i}+(1-\alpha )x_{i}'\succ x_{i}^{*}}$.

????. ?????????? V ? ${\displaystyle Y+\{{\bar {w}}\}}$ ????????. ??? ?? ?? ???? ???? ???? ???.

????. ${\displaystyle V\cap Y+\{{\bar {w}}\}=\varnothing }$ ??? ???????? ?? ?????? ???? ?? ${\displaystyle (x^{*},y^{*})}$ ???. ??? ?????? ?? ?? V ? ?? ?? ${\displaystyle Y+\{{\bar {w}}\}}$ ???? ???? ??????? ?? ?? ???????? ? ????? ????? ??????? ????? ?????? ?? ??? ???? ?? ?? ???? ???? ??????? ????? ???????? ???? ?? ?? ?? ?? ${\displaystyle x_{i}^{*}}$ ????? ?? ????.

????. ????? ???? ${\displaystyle p=(p_{1},..,p_{L})\neq 0}$ ? r ???? ???? ?????????? ${\displaystyle p\cdot z\geq r}$ ?? ???? ?? ${\displaystyle z\in V}$ ? ${\displaystyle p\cdot z\leq r}$ ?? ???? ?? ${\displaystyle z\in Y+\{{\bar {w}}\}}$ ??? ???? ?? ????? ???? ????? ??? ????? ???????? ???? ????? ???.

????.??????? ??i ${\displaystyle x_{i}\succeq x_{i}^{*}}$ ????? ${\displaystyle p\cdot (\Sigma _{i}x_{i})\geq r}$.??? ???? ???? ??i ${\displaystyle x_{i}\succeq x_{i}^{*}}$ ?? ??? ????? ??????? ????? ???? ?? ??????????? i ??? ???? ${\displaystyle {\hat {x_{i}}}}$ ?? ??????? ???? ????? ?? ${\displaystyle x_{i}}$ ???? ???? ?????????? ${\displaystyle {\hat {x_{i}}}\succeq x_{i}}$ ????????? ${\displaystyle {\hat {x_{i}}}\in V_{i}}$.?? ????? ${\displaystyle \Sigma {\hat {x_{i}}}\in V}$ ? ${\displaystyle p\cdot (\Sigma {\hat {x_{i}}})\geq r}$ ?? ????? ?? ${\displaystyle {\hat {x_{i}}}\rightarrow x_{i}}$ ????? ?????? ${\displaystyle p\cdot (\Sigma _{i}x_{i})\geq r}$.

??? ?. ${\displaystyle p\cdot (\Sigma x_{i}^{*})=p{\dot {(}}{\bar {w}}+\Sigma _{j}y_{j}^{*})=r}$ ?? ??? ? ????? ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})\geq r}$ ?? ????? ?? ???? ?? ${\displaystyle \Sigma _{i}x_{i}^{*}={\bar {w}}\Sigma _{j}y_{j}^{*}\in Y+\{{\bar {w}}\}}$ ?? ????? ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})\leq r}$ ???????? ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})=r}$ ? ??? ${\displaystyle \Sigma _{i}x_{i}^{*}={\bar {w}}\Sigma _{j}y_{j}^{*}}$ ????? ${\displaystyle p{\dot {(}}{\bar {w}}+\Sigma _{j}y_{j}^{*})=r}$

????. ???? ?? j ????? ${\displaystyle p\cdot y_{j}\leq p\cdot y_{j}^{*}}$ ?? ???? ?? ${\displaystyle y_{j}\in Y_{j}}$. ???? ?? ????? j ????? ${\displaystyle y_{j}+\Sigma _{j\neq h}y_{h}^{*}\in Y}$ ?? ????? ${\displaystyle p\cdot ({\bar {w}}+y_{j}+\Sigma _{h\neq j}y_{h}^{*})\leq r=p\cdot ({\bar {w}}+y_{j}+\Sigma _{h\neq j}y_{h}^{*})}$ ?? ????? ${\displaystyle p\cdot y_{j}\leq p\cdot y_{j}^{*}}$

????. ?? ???? ?? i ??? ${\displaystyle x_{i}\succ x_{i}^{*}}$ ????? ${\displaystyle p\cdot x_{i}\geq p\cdot x_{i}^{*}}$. ??? ???? ????? ${\displaystyle x_{i}\succ x_{i}^{*}}$ ?? ??? ? ? ? ????? ?? ?????:

${\displaystyle p\cdot (x_{i}+\Sigma _{k\neq i}x_{k}^{*})\geq r=p\cdot (x_{i}^{*}+\Sigma _{k\neq i}x_{k}^{*})(}$ ?? ????? ${\displaystyle p\cdot x_{i}\geq p\cdot x_{i}^{*}}$

??? ?. ??? ???? ${\displaystyle w_{i}=p\cdot x_{i}^{*}}$ ?? ???? ?? ${\displaystyle i=1,...,I}$ ????? ${\displaystyle (x^{*},y^{*},p)}$ ?? ?? ????? ?? ??? ????? ????? ?????? ???????? ??????. ^[?]

????? [ ?????? ]