❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ P❤↕♠ ❚❤à ❉✉♥❣ ❚❾P ▲➬■ ❱⑨ Ù◆● ❉Ö◆● ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵✱ t❤→♥❣ ✺✱ ♥➠♠ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ P❤↕♠ ❚❤à ❉✉♥❣ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❙÷ ♣❤↕♠ t♦→♥ ❤å❝ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❤❙✳ ◆❣✉②➵♥ ◗✉è❝ ❚✉➜♥ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵✱ t❤→♥❣ ✺✱ ♥➠♠ ✷✵✶✽ ▼ö❝ ❧ö❝ ✶ ❈⑩❈ ❑❍⑩■ ◆■➏▼ ❈❒ ❇❷◆ ❈Õ❆ ❚❾P ▲➬■ ✼ ✶✳✶ ❚➟♣ ❛❢❢✐♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ P❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ✈➔ ❜❛♦ ✤â♥❣ t÷ì♥❣ ✤è✐ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✺ ❙✐➯✉ ♣❤➥♥❣ ✈➔ ❝→❝ ✤à♥❤ ❧➼ t→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✻ ▼➦t ❝ü❝ ❜✐➯♥ ✈➔ ✤✐➸♠ ❝ü❝ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ❙Ü ❇■➎❯ ❉■➍◆ ❈Õ❆ ❚❾P ▲➬■ ✷✶ ỹ t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸ ❚❾P ▲➬■ ✣❆ ❉■➏◆ ✷✻ ✸✳✶ t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ▼➦t ❝õ❛ ♠ët ✤❛ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸ ✣➾♥❤ ✈➔ ❝↕♥❤ ❝õ❛ ❦❤è✐ ✤❛ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹ ❈ü❝ ❝õ❛ ✤❛ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ❙ü ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❝→❝ ❦❤è✐ ✤❛ ❞✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹ ❇✃ ✣➋ ❋❆❘❑❆❙ ❱⑨ Ù◆● ❉Ö◆● ✐ ✸✻ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✹✳✶ ❇ê ✤➲ ❋❛r❦❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ❜ê ✤➲ ❋❛r❦❛s ✈➔♦ ❣✐↔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✸ Ù♥❣ ❞ö♥❣ ❝õ❛ ❜ê ✤➲ ❋❛r❦❛s ✈➔♦ ❜➔✐ t♦→♥ ❝èt ❧ã✐ ❝õ❛ trá ❝❤ì✐ t÷ì♥❣ t→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ✹✵ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲❮■ ❈❷▼ ❒◆ t õ ữủ õ ợ t ỗ ự trữợ t tổ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ●✐↔✐ t➼❝❤✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ q✉❛✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛② ❣✐→♦ ❚❤❙✳ ◆❣✉②➵♥ ố ữớ trỹ t ữợ õ õ ỵ qỵ tổ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ✤➲ t➔✐ ♥➔② ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟② tỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝↔♠ t❤ỉ♥❣ ỳ ỵ õ õ t ổ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ tỉ✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵✱ t❤→♥❣ ✺✱ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ P❤↕♠ ❚❤à ❉✉♥❣ ✶ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ❝õ❛ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ❝→❝ t❤➛② ❝ỉ tr t t sỹ ữợ t t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦ ❚❤❙✳ ◆❣✉②➵♥ ◗✉è❝ ❚✉➜♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❦❤â❛ ❧✉➟♥ tæ✐ ❝â t❤❛♠ ❦❤↔♦ ♥❤ú♥❣ t➔✐ ❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥ ✤➣ ✤÷đ❝ ❤➺ t❤è♥❣ tr♦♥❣ ♠ư❝ t t õ ỗ ự ❞ư♥❣✧ ❦❤ỉ♥❣ ❝â trò♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ❦❤â❛ ❧✉➟♥ ❦❤→❝✳ ❍➔ ◆ë✐✱ ♥❣➔② ✶✵✱ t❤→♥❣ ✺✱ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ P❤↕♠ ❚❤à ❉✉♥❣ ✷ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▼Ð ✣❺❯ ❚ø r➜t ❧➙✉✱ ❦❤→✐ ♥✐➺♠ ỗ ữủ t tr ổ ữ ✤♦↕♥ t❤➥♥❣✱ ✤÷í♥❣ t❤➥♥❣✱ t❛♠ ❣✐→❝✱ ❤➻♥❤ trá♥✳✳✳ ◆❣➔② ♥❛②✱ t ỗ ố tữủ ự t♦→♥ tè✐ ÷✉ ❤♦→✱ ❝➙♥ ❜➡♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❉ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t✱ ❝➜✉ tró❝ ❝õ❛ t➙♣ ỗ ú t qt t tr t♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❝✉ë❝ sè♥❣ ❦❤→ ❤✐➺✉ q✉↔ ✈➔ ✤ë❝ ✤→♦✳ ❱➲ ♠➦t ❤➻♥❤ ❤å❝✱ t❛ t❤➜② t➟♣ ỗ t ự tt ữớ t ố ❤❛✐ ✤✐➸♠ ❜➜t ❦➻ tr♦♥❣ ♥â✳ ❱➲ ♠➦t ❣✐↔✐ t➼❝❤✱ t t t ỗ õ t ữủ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✭❤ú✉ ❤↕♥ ❤♦➦❝ ✈ỉ ữ t t trú t ỗ ợ trú t ữỡ tr t✉②➳♥ t➼♥❤✳ ◆➠♠ ✶✽✾✹✱ ❋❛r❦❛s ✤➣ ❝æ♥❣ ❜è ❝æ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ✧◆❣✉②➯♥ ❧➼ ❋♦✉r✐❡r ✈➔ ù♥❣ ❞ö♥❣✧ ✭❆ ❋♦✉r✐❡r✲❢➨❧❡ s t r ữ ữ ữủ ợ t♦→♥ ❤å❝ q✉❛♥ t➙♠✳ ✣➳♥ ♥➠♠ ✶✾✵✷✱ ➷♥❣ ❝æ♥❣ ❜è t✐➳♣ ❜➔✐ ❜→♦ ☎ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✏ ❯❜❡r ❞✐❡ ❚❤❡♦r✐❡ ❞❡r ❊✐♥❢❛❝❤❡♥ ❯♥❣❧❡✐❝❤✉♥❣❡♥✑ ❜➡♥❣ t✐➳♥❣ ✣ù❝✳ ▲ó❝ ♥➔②✱ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ỉ♥❣ ♠ỵ✐ ữủ t ợ t rở r t q✉↔ ❝õ❛ ❋❛r❦❛s ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ❝➛✉ ♥è✐ ❣✐ú❛ t ỗ t ữỡ tr t❤➜② t➛♠ q✉❛♥ trå♥❣ ❦➳t q✉↔ ❝õ❛ ❋❛r❦❛s✱ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➲ ♥â✳ ❚ø ✤â✱ ❝❤ó♥❣ t❛ ✤➣ ❝â ♥❤✐➲✉ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❦➳t q✉↔ ✤â ♥❤÷ ❝õ❛✿ ❈✳ ●✳ ❇r♦②❞❡♥ ♥➠♠ ✶✾✽✽❀ ❆✳ ❉❛① ♥➠♠ ✶✾✼✼❀ ❱✳ ❈❤❛♥❞r✉✱ ❈✳▲❛ss❡③✱ ❏✳ ▲✳ ▲❛ss❡③ rt ữợ sỹ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❝õ❛ t❤➛② ❝ỉ✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❚❤❙✳ ◆❣✉②➵♥ ◗✉è❝ ❚✉➜♥✱ ❝ò♥❣ ✈ỵ✐ sü ✤❛♠ ♠➯ ❝õ❛ ❜↔♥ t❤➙♥✱ tæ✐ ✤➣ ✸ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ♠↕♥❤ ❞↕♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t ỗ ự ỹ tr ỳ ❦➳t q✉↔ ✤➣ ❝â ✈➔ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ õ q tợ t ỗ õ ❧✉➟♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜è♥ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ❈→❝ ỡ t ỗ ữỡ ự t ỗ tr ởt số õ q t ỗ s ♣❤➥♥❣ ✈➔ ❝→❝ ✤à♥❤ ❧➼ t→❝❤✳ ❈❤÷ì♥❣ ✷✳ ❙ü ❜✐➸✉ t ỗ r ữỡ ú tổ ❝ù✉ ✈➲ ❝➜✉ tró❝ ❤➻♥❤ ❤å❝ ✈➔ sü ❜✐➸✉ ❞✐➵♥ t ỗ t ởt số ỡ ổ ỹ t ỗ ữỡ ỗ ữỡ ự t ỗ ữ ởt số ỡ t t t ỗ tứ õ ró t ỗ ữỡ ❇ê ✤➲ ❋❛r❦❛s ✈➔ ù♥❣ ❞ư♥❣✳ ❈❤ó♥❣ tỉ✐ ♥➯✉ ❧↕✐ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❜ê ✤➲ ❋❛r❦❛s ✧❈❤♦ A ∈ Rm×n ✈➔ b ∈ Rm ✳ ●✐↔ sû F = {x ∈ Rn+ : Ax = b} ✈➔ G = {y ∈ Rm : yA ≥ 0, yb ≤ 0}✳ ❑❤✐ ✤â✱ F = ∅ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ G = ∅✧✳ ❈✉è✐ ❝ò♥❣✱ t→❝ ❣✐↔ ♥➯✉ ù♥❣ ❞ư♥❣ ❜ê ✤➲ ❋❛r❦❛s ✤➸ ①➨t ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❦❤ỉ♥❣ ➙♠ ❤❛② ❦❤ỉ♥❣ ✈➔ ù♥❣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❝èt ❧ã✐ trá ❝❤ì✐ t÷ì♥❣ t→❝✳ ▼➦❝ ❞ò ❦❤â❛ ❧✉➟♥ ❤♦➔♥ t❤➔♥❤ ✈ỵ✐ sü ❝è ❣➢♥❣ ❝õ❛ ❜↔♥ t❤➙♥✱ s♦♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ✤➙② ❝ơ♥❣ ❧➔ ✈➜♥ ✤➲ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❜↔♥ t❤➙♥ tỉ✐✱ ♥➯♥ tr♦♥❣ q✉→ tr➻♥❤ ✐♥ ➜♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt ổ ữủ ỵ õ õ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❈✉è✐ ❝ò♥❣✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❚❤❙✳ ◆❣✉②➵♥ ◗✉è❝ ✹ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt t t ữợ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥✳ ✺ P õ tốt ỵ ❤✐➺✉ t♦→♥ ❤å❝ R ❚➟♣ t➜t ❝↔ ❝→ sè t❤ü❝✳ Rn ❚➟♣ t➜t ❝↔ ❝→❝ ✈❡❝t♦r ❝â n ❝❤✐➲✉✳ ·, ã ổ ữợ ỳ tỷ cl C ❇❛♦ ✤â♥❣ ❝õ❛ C ✳ int C P❤➛♥ tr♦♥❣ ❝õ❛ C conv E ỗ E cone E ◆â♥ s✐♥❤ ❜ð✐ t➟♣ E ✳ ri C P❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ t➟♣ C ✳ aff D ❇❛♦ ❛❢❢✐♥❡ ❝õ❛ t➟♣ D✳ ✻ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✸✳✹ ❈ü❝ ❝õ❛ ✤❛ ❞✐➺♥ ❈❤♦ x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❝æ♥❣ t❤ù❝ ✭✸✳✶✮ t❤❡♦ ✤â Ax0 ≤ b✳ ❚❛ ❝â A x − x0 ≤ b − Ax0 ✱ ✈ỵ✐ b − Ax0 ≥ 0✳ ❇ð✐ ✈➟②✱ ❞♦ sü t❤❛② ✤ê✐ ❣è❝ tå❛ ✤ë s❛♥❣ ✈à tr➼ ✈➔ sü ♣❤➙♥ ❝❤✐❛ ❤➺ ✭✸✳✶✮ ❝â t❤➸ ❝❤✉②➸♥ t❤➔♥❤ ❞↕♥❣ , x ≤ 1, i = 1, 2, p, , x ≤ 0, i = p + 1, , m ✭✸✳✻✮ ▼➺♥❤ ✤➲ ✸✳✹✳ ❈❤♦ P ❧➔ ✤❛ ❞✐➺♥ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✸✳✻✮✱ ❝ü❝ ❝õ❛ P ❧➔ ♠ët ✤❛ ❞✐➺♥ Q = conv 0; a1 , , ap + cone ap+1 , , am ✭✸✳✼✮ ❱➔ ♥❣÷đ❝ ❧↕✐ ❝ü❝ ❝õ❛ Q ❧➔ ✤❛ ❞✐➺♥ P ✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ y ∈ Q✱ tù❝ ❧➔ p m i y= i=1 t❤➻ y, x = λi ≤ 1, λi ≥ 0, i = 1, , m λi a , m i=1 λi i=1 , x ≤ 1✱ ✈ỵ✐ ♠å✐ x ∈ P ✱ ❞♦ ✤â y ∈ P o ✭❝ü❝ ❝õ❛ P ✮✳ ❱➟② Q ⊂ P o ✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ tr t ú ỵ r ợ y Qo ❧✉ỉ♥ ❝â y, x ≤ ✈ỵ✐ ♠å✐ x ∈ Q✳ ✣➦❝ ❜✐➺t✱ ♥➳✉ i ≤ p✱ x = t❤➻ , y ≤ 1, i = 1, , p✳ ◆➳✉ i ≥ p + x = ợ số ữỡ ợ tũ ỵ θ > t❤➻ , y ≤ 0, i = p + 1, , m✳ ◆❤÷ ✈➟②✱ Qo ⊂ P ❞♦ ✤â P o ⊂ (Qo )o = Q✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ Q = P o ✈➔ t❛ ❝â ✤➥♥❣ t❤ù❝ s❛✉ Qo = P oo = P ✳ ❍➺ q✉↔ ✸✳✸✳ ◆➳✉ ❤❛✐ ✤❛ ❞✐➺♥ ✤❛ ❝❤✐➲✉ Q✱ P ❝❤ù❛ ❣è❝ ❧➔ ❝ü❝ ❝õ❛ ✤❛ ❞✐➺♥ ❦✐❛ t❤➻ ❝â ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ù♥❣ ✶✲✶ ❣✐ú❛ ❝→❝ ♠➦t ❝õ❛ P ❦❤æ♥❣ ✸✷ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤ù❛ ❣è❝ ✈➔ ❝→❝ t➟♣ ❝❤ù❛ ❝→❝ ✤➾♥❤ ❦❤→❝ ❝õ❛ Q✳ ◆❤÷ ✈➟②✱ ♥â ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ù♥❣ ✶✲✶ ❣✐ú❛ P ự ố ợ ữỡ ỹ ❜✐➯♥ ❝õ❛ Q✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❣✐↔ sû P ✱ Q ①→❝ ✤à♥❤ t❤❡♦ ♠➺♥❤ ✤➲ ✸✳✹ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ak , x ≤ 1, k ∈ {1, , p} ❧➔ ❞÷ tr♦♥❣ ✭✸✳✻✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈❡❝t♦r ak ❝â t❤➸ ❜ä tø ✭✸✳✼✮ ♠➔ ❦❤æ♥❣ ❧➔♠ t❤❛② ✤ê✐ Q✳ ❇ð✐ ✈➟②✱ ✤➥♥❣ t❤ù❝ ak , x = ①→❝ ✤à♥❤ ♠ët ❞✐➺♥ ❝õ❛ P ❦❤æ♥❣ ❝❤ù❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ak ❧➔ ♠ët ✤➾♥❤ ❝õ❛ Q✳ ữỡ tỹ ố ợ q ỹ ố ỵ ợ ố D ổ tỗ t ❤❛✐ t➟♣ ❤ú✉ ❤↕♥ V = {v i , i ∈ I} ✈➔ U = {uj , j ∈ J} s❛♦ ❝❤♦ ✭✸✳✽✮ D = conv V + cone U õ D ỗ õ x = ợ i = i àj ≥ 0✱ i ∈ I ✱ j ∈ J ✳ i∈I λi v i + j∈J µj uj ✱ i∈I ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ L ❧➔ lineality ❝õ❛ D t❤➻ D =L+C ✭✸✳✾✮ ❱ỵ✐ C := D ∩ L⊥ ❧➔ ❦❤è✐ ✤❛ ❞✐➺♥ ❦❤ỉ♥❣ ❝❤ù❛ ✤÷í♥❣ t❤➥♥❣✳ ❚❤❡♦ ✤à♥❤ ❧➼ ✷✳✶ C = conv V + cone U1 ✸✸ ✭✸✳✶✵✮ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚r♦♥❣ ✤â✱ V ❧➔ t➟♣ ❤ñ♣ ❝→❝ ✤➾♥❤ ✈➔ U1 ❧➔ t➟♣ ❤đ♣ ❝→❝ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ C ✳ ❇➙② ❣✐í ❝❤♦ U0 ❧➔ ❝ì sð ❝õ❛ L✱ ✈➻ ✈➟② L = cone{U0 ∪ (−U0 )}✳ ❚ø ✭✸✳✾✮ ✈➔ ✭✸✳✶✵✮ ❝❤ó♥❣ t❛ ❦➳t ❧✉➟♥ D = conv V + cone U1 + cone{U0 ∪ (−U0 )}, tø ✭✸✳✽✮ ✤÷đ❝ t❤✐➳t ❧➟♣ ❜ð✐ U = U0 ∪ (−U0 ) ∪ U1 ỗ t ỳ ❝→❝ ♥û❛ ✤÷í♥❣ t❤➥♥❣ ①✉➜t ♣❤→t tø ❣è❝ ❧➔ ♠ët õ ỗ õ t ỳ t tr U Rn tỗ t ♠❛ tr➟♥ A ❝➜♣ m × n s❛♦ ❝❤♦ cone U = {x|Ax ≤ 0} ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ U ✭✸✳✶✶✮ = {uj , j ∈ J}, M = cone{uj , j ∈ J}✱ tø ♠➺♥❤ ✤➲ ✸✳✹ t❛ ❝â M o = {x| v j , x ≤ 0, j J} tỗ t a1 , , am s❛♦ ❝❤♦ M o = {x = m i i=1 |λi a |λi ≥ 0, i = 1, , m}✳ ❈✉è✐ ❝ò♥❣ t❛ ❝â (M o )o = {x| , x ≤ 0, i = 1, , m} sỹ ố ú ỵ r M oo = M ỵ t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ ❜➜t ❦➻ V ✈➔ U tr♦♥❣ Rn tỗ t ởt tr A m ì n ✈➔ ♠ët ✈❡❝t♦r b ∈ Rm s❛♦ ❝❤♦ conv V + cone U = {x|Ax ≤ b} ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ V = {v i , i ∈ I}, U = {uj , j ∈ J}✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ V = ∅✱ ❞♦ tr÷í♥❣ ❤đ♣ V = ∅ ✤➣ ✤÷đ❝ ①û ❧➼✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥✱ ✸✹ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ①❡♠ ①➨t t➟♣ λi (v i , 1) + K = {(x, t) = i∈I µj (uj , 0)| λi ≥ 0, µj ≥ 0} j∈J ❚➟♣ ❤đ♣ ♥➔② ❝❤ù❛ tr♦♥❣ ♥û❛ ❦❤ỉ♥❣ ❣✐❛♥ {(x, t) ∈ Rn × R| t ≥ 0} ❝õ❛ Rn+1 ✈➔ ♥➳✉ ❝❤ó♥❣ t❛ ❝â D = conv V + cone U t❤➻ x ∈ D t÷ì♥❣ ✤÷ì♥❣ (x, 1) ∈ K ✳ ❚ø ❜ê ✤➲ õ ỗ tự tỗ t↕✐ (A, b) s❛♦ ❝❤♦ K = {(x, t)|Ax − tb ≤ 0} ❉♦ ✤â D = {x|Ax ≤ b} ữủ ự q ỗ ✤â♥❣ D ❝❤➾ ❝â ❤ú✉ ❤↕♥ ❝→❝ ♠➦t ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤è✐ ✤❛ ❞✐➺♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû r➡♥❣ D ❦❤ỉ♥❣ ❝❤ù❛ ✤÷í♥❣ t❤➥♥❣✳ ❑❤✐ ✤â D ❝â ♠ët t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ V ❝õ❛ ❝→❝ ✤✐➸♠ ❝ü❝ ❜✐➺♥ ✈➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ U ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥✱ tø ✤à♥❤ ❧➼ ✷✳✶ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ tr♦♥❣ ❞↕♥❣ conv V + cone U ✱ ♥➯♥ ❞♦ ✤â ♥â ❧➔ ♠ët ❦❤è✐ ✤❛ ❞✐➺♥ t❤❡♦ ✤à♥❤ ❧➼ tr➯♥✳ ✸✺ ❈❤÷ì♥❣ ✹ ❇✃ ✣➋ ❋❆❘❑❆❙ ❱⑨ Ù◆● ❉Ư◆● ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♣❤→t ❜✐➸✉ ❜ê ✤➲ ❋❛r❦❛s ✈➔ ù♥❣ ❞ư♥❣ tr♦♥❣ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❜➔✐ t♦→♥ trá ❝❤ì✐ t÷ì♥❣ t→❝ ✹✳✶ ❇ê ✤➲ ❋❛r❦❛s ✣à♥❤ ♥❣❤➽❛ ✹✳✶✳ ●✐↔ sû A ∈ Rm×n ✱ ❝❤♦ (a1 , a2 , , an ) ❧➔ ❝→❝ ❝ët ❝õ❛ A✳ ❚➟♣ t➜t ❝↔ ❝→❝ ❝→❝ tê ❤đ♣ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ ➙♠ ❝→❝ ❝ët ❝õ❛ A ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝õ❛ A✱ ❦➼ ❤✐➺✉ cone A✳ cone(A) = {b ∈ Rm : Ax = b x ∈ Rn+ }✳ ❱➼ ❞ö ✹✳✶✳✶✳ ❳➨t ♠❛ tr➟♥ A= 1 ✸✻ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍➻♥❤ ✹✳✶✿ ❍➻♥❤ t❤➸ ❤✐➺♥ ♠ët ♥â♥ tr♦♥❣ R2✳ ①❂✭✶✱ ✷✱ ✷✮✱ t❤➻ ❝❤ó♥❣ t❛ t❤✉ ✤÷đ❝ b ∈ cone(A)✳ b= −7 ❇ê ✤➲ ✹✳✶✳ ❈❤♦ A ∈ Rm×n t❤➻ cone(A) ❧➔ ♠ët t➟♣ ỗ ự b1, b2 cone(A) b = λb1 + (1 − λ)b2 ✈ỵ✐ λ ∈ (0, 1)✳ ❈❤♦ b1 = Ax1 ✈➔ b2 = Ax2 ✈ỵ✐ x1 , x2 ∈ Rn+ ✱ t❤➻ b = λAx1 + (1 − λ)Ax2 = A[λx1 ] + (1 − λ)x2 ], ❞♦ ✤â b ∈ cone A✳ ❱➟② cone A t ỗ A Rmìn ✈➔ b ∈ Rm✳ ●✐↔ sû F = {x ∈ Rn+ : Ax = b} ✈➔ G = {y ∈ Rm : yA ≥ 0, yb ≤ 0}✳ ❑❤✐ ✤â✱ F G = ∅✳ =∅ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû F = ∅✱ ❝❤♦ x ∈ F ✱ ❝❤å♥ y ∈ Rm s❛♦ ❝❤♦ yA ≥ t❤➻ yb = y(Ax) = (yA)x ≥ 0✳ ❱➟② G = ∅ ✸✼ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ●✐↔ sû G = ∅ ✈➔ ♣❤↔♥ ❝❤ù♥❣ F = ∅✱ ♥❣❤➽❛ ❧➔ {x ∈ Rn+ : Ax = b, b / cone(A)} cone(A) ỗ ✈➔ ✤â♥❣✱ ♥➯♥ ❝❤ó♥❣ t❛ ❝â t❤➸ t→❝❤ b r❛ r❛ ❦❤ä✐ cone(A) ❜ð✐ s✐➯✉ ♣❤➥♥❣ (y, α) s❛♦ ❝❤♦ yb < α ✈➔ yz > α ✈ỵ✐ ♠å✐ z ∈ cone(A)✱ ❞♦ ✤â α < ♥➯♥ yb < 0✳ ❈❤♦ aj ❧➔ ♠ët ❝ët ❝õ❛ ✈➨❝ tì A✱ ❝❤ó♥❣ t❛ s➩ ❝❤➾ r❛ r➡♥❣ yaj ≥ 0✳ ❳➨t λ > ♥➯♥ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ λaj ∈ cone(A)✱ ❞♦ ✤â y(λaj ) > α✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ yaj < ✈ỵ✐ λ > ❜➜t ❦➻ ✤õ ❧ỵ♥ t❛ ❝â t❤➸ ❝❤å♥ λ s❛♦ ❝❤♦ y(λaj ) < α✱ tr→✐ ✈ỵ✐ ❣✐↔ sû✳ ❉♦ ✤â yaj ≥ 0✱ ♥➯♥ yA ≥ ✈➔ yb < α < 0✳ ❱➟② G = ∅✳ ✹✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ❜ê ✤➲ ❋❛r❦❛s ✈➔♦ ❣✐↔✐ t♦→♥ ❍➺ s❛✉ ❝â ♥❣❤✐➺♠ ❦❤æ♥❣ ➙♠ ❤❛② ❦❤æ♥❣❄ x1 −5 × x2 = x3 ⑩♣ ❞ư♥❣ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ G t❛ ✤÷đ❝✿ y + y2 < y + y2 ≥ 3y1 ≥ −5y1 + 2y2 ≥ ✸✽ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ t ữỡ tr ố õ t t ữợ ❞↕♥❣ ♠❛ tr➟♥ ♥❤÷ s❛✉✿ 1 y ≥ 0 × y2 −5 ❱➻ y1 ≥ 0✱ tø ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ t❛ ❝â y2 < tr ợ t ữỡ tr ố ❉♦ ✤â ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ G ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✳ ❱➟② →♣ ❞ư♥❣ ❜ê ✤➲ ❋❛r❦❛s t❤➻ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ ❝â ♥❣❤✐➺♠✳ ❱➼ ❞ö ✹✳✷✳✶✳ ❍➺ s❛✉ ❝â ♥❣❤✐➺♠ ❦❤æ♥❣ ➙♠ ❦❤æ♥❣❄ 1 0 1 1 x 2 1 × x = 2 1 x3 1 t ữỡ tr G ữủ t ữợ tr➟♥ ❧➔ y 1 y2 ≥ 1 × y3 1 y4 1 2y1 + 2y2 + 2y3 + y4 < 0✳ ▼ët ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ G ❝â t❤➸ ❧➔ y1 = y2 = y3 = −1 ✈➔ y4 = 1✳ ❱➟② ❤➺ ❜❛♥ ✤➛✉ ❦❤æ♥❣ ❝â ♥❣❤✐➯♠ ❦❤æ♥❣ ➙♠✳ ❚rü❝ q✉❛♥✱ ♥➳✉ ♥❤➙♥ −1 ✈➔♦ ✸ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ợ ữỡ tr ố s ✤â ❝ë♥❣ t➜t ❝↔ ❝❤ó♥❣ ❧↕✐✳ ❚❛ ❝â = −2 ✭✈æ ❧➼✮✳ ✸✾ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✹✳✸ Ù♥❣ ❞ö♥❣ ❝õ❛ ❜ê ✤➲ ❋❛r❦❛s ✈➔♦ ❜➔✐ t♦→♥ ❝èt ❧ã✐ ❝õ❛ trá ❝❤ì✐ t÷ì♥❣ t→❝ ❚rá ❝❤ì✐ t÷ì♥❣ t→❝ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ♠ët t➟♣ ❝õ❛ n ♥❣÷í✐ ❝❤ì✐ N ✈➔ ❣✐→ trà ❤➔♠ v : 2N → R t❤➸ ❤✐➺♥ ❣✐→ trà sü t÷ì♥❣ t ữớ ỡ ợ ộ tr tữỡ t S ⊂ N t❤➻ ❣✐→ trà v(S) ❧➔ t❤➸ ❤✐➺♥ sü t÷ì♥❣ t→❝✳ ❇ë (N, v) ✤à♥❤ ♥❣❤➽❛ ♠ët trá ❝❤ì✐ t÷ì♥❣ t→❝✳ ❈❤ó♥❣ t❛ ✤➣ ✈ø❛ ✤à♥❤ ♥❣❤➽❛ trá ❝❤ì✐ t÷ì♥❣ t→❝ sû ❞ư♥❣ ❣✐→ trà ❤➔♠✳ ❙❛✉ ✤➙② ❧➔ ♠ët ✈➼ ❞ö ✈➲ ❣✐→ trà ❤➔♠✳ ❇→♥ ♠ët s↔♥ ♣❤➞♠✱ ①❡♠ ①➨t sü ❝❤✐❛ s➫ ♠ët ♠➦t ❤➔♥❣ ❣✐ú❛ ❤❛✐ ♥❣÷í✐ ❜→♥ ✈➔ ♠✉❛✳ ❚➟♣ ❝õ❛ ♥❣÷í✐ ❝❤ì✐ ❝â t❤➸ ❦➼ ❤✐➺✉ ❜ð✐ N = {1, 2, s} ợ s ữớ tr t ✤➸ ♥❣÷í✐ ❝❤ì✐ ❝â t❤➸ ♠✉❛ ❧➔ ✺ ✈➔ ✶✵✳ ữớ ổ õ tr ợ t rỏ ❝❤ì✐ t÷ì♥❣ t→❝ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ v(∅) = v({s}) = 0❀ v({1}) = v({2}) = v({1, 2}) = 0; v({1, s}) = 5; v({2, s}) = v({1, 2, s}) = 10✳ ✣à♥❤ ♥❣❤➽❛ ♠ët trá ❝❤ì✐ t÷ì♥❣ t→❝ ❦❤æ♥❣ ♥â✐ ✈➲ ❝→❝❤ ❝❤✐❛ ❣✐→ trà ❝õ❛ trá ❝❤ì✐ ❣✐ú❛ ❝→❝ ♥❣÷í✐ ❝❤ì✐✳ ▼ët ✈❡❝t♦r x ∈ Rn ữủ ố ỗ iN xi = v(N ) ✈➔ xi ≥ v({i})✳ ❚❛ ❝â t❤➸ ❝♦✐ ố ỗ tr v(N ) ❝✉♥❣ ❝➜♣ ❝❤♦ ♠é✐ ♥❣÷í✐ trà ➼t ♥❤➜t ✈➔ ❣✐→ trà ✤â ♥❤✐➲✉ ♥❤÷ ❣✐→ trà ♥❣÷í✐ ❝❤ì✐ s➩ ✤÷đ❝ ♥❤➟♥ ❦❤✐ ❝❤➾ ♠ët ♠➻♥❤✳ ❑❤→✐ q✉→t ❤â❛ ✤✐➲✉ ♥➔② ợ ộ sỹ tữỡ t ỳ ú t❛ ❝â ❦❤→✐ ♥✐➺♠ ❝èt ❧ã✐ ❝õ❛ trá ❝❤ì✐✱ ❦➼ ❤✐➺✉ ❧➔ core✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✷✳ ❈èt ❧ã✐ ❝õ❛ trá ❝❤ì✐ t÷ì♥❣ t→❝ (N, v) ❧➔ t➟♣ C(N, v) = {x ∈ Rn ❈èt ❧ã✐ ❝õ❛ trá ❝❤ì✐ ❧✐➯♥ ❦➳t ✈ỵ✐ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ê♥ ✤à♥❤✳ ◆â q✉② ✤à♥❤ ✹✵ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ r➡♥❣ ♠å✐ sü t÷ì♥❣ t→❝ ❝õ❛ ❝→❝ ✤↕✐ ❧➼ ♣❤↔✐ ➼t ♥❤➜t ❤å s➩ ♥❤➟♥ ✤÷đ❝ ♥➳✉ ❤å t❤➔♥❤ ❧➟♣ ❧✐➯♥ ♠✐♥❤ ❝õ❛ r✐➯♥❣ ❤å✳ ❳➨t ✈➼ ❞ö ð tr➯♥✱ t➟♣ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ core ❧➔ x1 + x2 + xs = 10 x1 + x2 ≥ x1 + xs ≥ x2 − xs ≥ 10 xi ≥ ∀i ∈ 1, 2, s ❈❤♦ sü t❤❛② t❤➳ ♣❤ö x2 ≤ ✈➔ x1 ≤ ♠➔ x1 ≥ ♥➯♥ x1 = s✉② r❛ xs ≥ 5✳ ❱➟② x1 = 0, ≤ x2 ≤ ✈➔ ≤ xs ≤ 10 ✈ỵ✐ x2 + xs = 10 ❝➜✉ t❤➔♥❤ core ❝õ❛ trá ❝❤ì✐ ♥➔②✳ ◆❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ t➜t ❝↔ ✤➲✉ ❝â core✳ ❱➼ ❞ư✱ ①➨t ♠ët trá ❝❤ì✐ ✈ỵ✐ ❤❛✐ ✤↕✐ ❧➼ {1, 2} ✈ỵ✐ v({1}) = = v({2}) ♥❤÷♥❣ v({1, 2}) = 3✳ ❚rá ❝❤ì✐ ♥➔② ❝â core ré♥❣ ✈➻ ❦❤ỉ♥❣ ❝â (x1 , x2 ) t❤ä❛ ♠➣♥ x1 ≤ 1, x2 ≤ ✈➔ x1 + x2 = 3✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ trá ❝❤ì✐ ❝â core t❛ ❝â t❤➸ sû ❞ö♥❣ ❜ê ✤➲ ❋❛r❦❛s✳ ❈❤♦ B(N ) ❧➔ t➟♣ ❝â t❤➸ ❣✐↔✐ t❤❡♦ ❤➺ yS = 1, ∀i ∈ N, ys ≥ 0, ∀S S N :i∈S N ys ❝â t❤➸ ❝♦✐ ♥❤÷ trå♥❣ ❧÷đ♥❣ ✤÷đ❝ ❝❤♦ ❜ð✐ sü ❧✐➯♥ ❦➳t ❝õ❛ S ✳ ◆â ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ B(N ) = ∅✳ ❱➼ ❞ö✱ ❝❤♦ ys = ✈ỵ✐ ♠å✐ S ❝â |S| = ✈➔ t❤✐➳t ❧➟♣ yS = s➩ ❝❤♦ ❣✐→ trà y ∈ B(N )✳ ✹✶ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ỵ r C(N, v) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ v(N ) ≥ S N v(S)ys , ∀y ∈ B(N )✳ ✭◆➳✉ ♠ët trá ❝❤ì✐ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② t❤➻ ✤÷đ❝ ❣å✐ trá ❝❤ì✐ ❝➙♥ ❜➡♥❣✱ ❝èt ❧ã✐ ❝õ❛ ♠ët trá ❝❤ì✐ ❧➔ ❦❤ỉ♥❣ ré♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥➳✉ trá ❝❤ì✐ ✤â ❝➙♥ ❜➡♥❣✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➺ ❧✐➯♥ ❦➳t t÷ì♥❣ ù♥❣ ❝õ❛ core ❝õ❛ ♠ët trá ❝❤ì✐✿ i∈N xi = v(N ) i∈N xi ≥ v(S) xi tü ❞♦ ✭❈❖❘❊✮ ∀S N ∀i ∈ N ❍➺ ❜➜t ♣❤÷ì♥❣ ♣❤÷ì♥❣ tr➻♥❤ G ❝õ❛ ✭❈❖❘❊✮ ❧➔✿ v(N )yN − yN − S N v(S)yS < S N :i∈s yS =0 yS ≥ (BAL) ∀i ∈ N ∀S N yN tü ❞♦✳ ❇➙② ❣✐í ❣✐↔ sû C(N, v) = ∅ t❤➻ ❈❖❘❊ ❝â ♠ët ♥❣❤✐➺♠ ✈➔ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (BAL) ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✳ ❇➙② ❣✐í sû ❜➜t ❦➻ y ∈ B(N ) ❝❤ó♥❣ t❛ ❝❤♦ yN = ✈➔ ❤❛✐ ❧✐➯♥ ❦➳t ❝✉è✐ ❝õ❛ (BAL) ❧➔ t❤ä❛ ♠➣♥✳ ❱➻ (BAL) ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥➯♥ ❝❤ó♥❣ t❛ ❝â v(N ) − v(S)ys ≥ 0✳ ✣✐➲✉ S N ♥➔② ❝❤ù♥❣ tä trá ❝❤ì✐ ❝➙♥ ❜➡♥❣✳ ❇➙② ❣✐í ❣✐↔ sû trá ❝❤ì✐ ❝➙♥ ❜➡♥❣ t❤➻ ✈ỵ✐ ♠å✐ y ∈ B(N )✱ ❝❤ó♥❣ t❛ ❝â v(N ) ≥ v(S)ys ✳ ❈❤ó♥❣ t❛ s➩ ❝❤➾ r❛ r➡♥❣ (BAL) ❦❤æ♥❣ ❝â S N ♥❣❤✐➺♠✱ ✭❈❖❘❊✮ ❝â ♥❣❤✐➺♠ ✈➔ C(N, v) = 0✳ P❤↔♥ ❝❤ù♥❣ (BAL) ❝â ♠ët ♥❣❤✐➺♠ y ✳ ❘ã r➔♥❣ yN = ✭♠é✐ yS = t tr ợ t ữỡ tr tr (BAL) ♥❣❤➽❛ yS = ✹✷ yS yN , ✈ỵ✐ ♠å✐ S ⊆ N ✳ ❱➻ y P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (BAL) ♥➯♥ ❝â v(S)yS v(N ) < S N yS = ∀i ∈ N yS = ∀S S N :i∈S N y B(N ) tr ợ t ữỡ tr➻♥❤ ✤➛✉ ♥➯♥ trá ❝❤ì✐ ❧➔ ❝➙♥ ❜➡♥❣✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ trá ❝❤ì✐ tr♦♥❣ ✈➼ ❞ư ✭trá ❝❤ì✐ ởt t ú ỵ r trá ❝❤ì✐ ♥➔② ❝â ❞✉② ♥❤➜t ♠ët sü ❤đ♣ t→❝✱ r sỹ ủ t ợ õ tr ữỡ ❧➔ {1, s} ✈➔ {2, s}✳ ❱➟② ❝❤ó♥❣ t❛ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ✈ỵ✐ ♠é✐ y ∈ B(n) ❝â v({1, 2, s}) = 10 ≥ v({1, s})y{1,s} + v({2, s})y{2,s} = 5y{1,s} + 10y{2,s} ◆❤÷♥❣ y ∈ B(N ) t❤➻ ❝â y{1,s} + y{2,s} ≤ 1✱ ♥➯♥ ❝❤ó♥❣ t❛ ❝â trá ❝❤ì✐ ♥➔② ❧➔ ❝➙♥ ❜➡♥❣✳ ❚❤➟t ✈➟②✱ ❞➵ ❞➔♥❣ ♣❤→t ❜✐➸✉ ❦➳t q✉↔ ❝õ❛ sü t÷ì♥❣ t→❝ ❝❤✉♥❣ ❧➔ ✧trá ❝❤ì✐ t❤à tr÷í♥❣✧✳ ▼ët trá ❝❤ì✐ t❤à tr÷í♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽ ❜ð✐ ♠ët ♥❣÷í✐ ❜→♥ s ✈➔ ♠ët t➟♣ ♥❣÷í✐ ♠✉❛ B ✳ ❱➟② t➟♣ ♥❣÷í✐ ❝❤ì✐ ❧➔ N = B {s}✳ ❈❤ù❝ ♥➠♥❣ ❝❤➼♥❤ ❝õ❛ trá ❝❤ì✐ t trữớ t tr ợ ộ S ⊆ N t❛ ✤à♥❤ ♥❣❤➽❛ v(S) ❧➔ ❣✐→ trà ❧➔ ❣✐→ trà t❤à tr÷í♥❣ ❝õ❛ ♥❣÷í✐ ❝❤ì✐ ❝õ❛ t➟♣ S ✳ ✣➦t v(S) = 0✱ ♥➳✉ s ∈ / S✳ ▼ët trá ❝❤ì✐ t❤à tr÷í♥❣ ✤ì♥ ✤✐➺✉ ♥➳✉ v(S) ≤ v(T ) ✈ỵ✐ ♠å✐ S ⊆ T ⊆ N ✳ ✹✸ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ỵ trỏ ỡ t trữớ ỡ ✤✐➺✉ ✤➲✉ ❝➙♥ ❜➡♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② ❜➜t ❦➻ y ∈ B(N ) t❛ ❝â v(S)yS = S N v(S)yS S N :s⊂S ≤ v(N ) yS , S N :s⊂S yS = 1✳ ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② r➡♥❣ ✈➻ y ∈ B(N )✱ ❝❤ó♥❣ t❛ ❝â S N :s⊂S v(S)yS ✳ ❱➟② trá ❝❤ì✐ t❤à tr÷í♥❣ ✤ì♥ ✤✐➺✉ ❧➔ ❝➙♥ ❜➡♥❣✳ S N :s⊂S ❚❤➟t ✈➟② ♠é✐ ♣❤➛♥ tû t➛♠ t❤÷í♥❣ tr♦♥❣ ❝èt ❧ã✐ ❝õ❛ trá ❝❤ì✐ t❤à tr÷í♥❣ ✤ì♥ ✤✐➺✉ ❧➔ xs = v(N ) ✈➔ xi = ♥➳✉ i = s✳ ✹✹ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥✱ tæ✐ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ t➟♣ ỗ ự rs ỹ tr ❜→♦ ❬✷❪✳ ❈ư t❤➸✿ ❚ê♥❣ ❤đ♣✱ ❧✐➺t ❦➯ ❝→❝ ❦✐➳♥ tự q t ỗ t ỗ ▲➔♠ rã ❝→❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ✤à♥❤ ❧➼ ✶✳✷✱ ♠➺♥❤ ✤➲ ✷✳✸✳ ❚ü ✤÷❛ r❛ ✈➔ ❣✐↔✐ ❝→❝ ✈➼ ❞ư ✹✳✶✳✶✱ ✈➼ ❞ư ✹✳✶✳✷✳ ❱➻ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝â ❤↕♥ ♥➯♥ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❝á♥ ♥❤✐➲✉ t❤✐➳✉ sât õ tr ọ qỵ t ổ õ ỵ õ ữủ t ỡ ổ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✺ P❍❸▼ ❚❍➚ ❉❯◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❍♦➔♥❣ ❚ö② ✭✶✾✾✼✮✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ▲♦♥ ❉♦♥✳ ❬✷❪ ❉❡❜❛s✐s ▼✐s❤r❛♥ ✭✷✵✶✶✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♥✈❡① s❡ts ✇✐t❤ ❛♣♣❧✐✲ ❝❛t✐♦♥ t♦ ❡❝♦♥♦♠✐❝s✱ ■♥❞✐❛♥ ❙t❛t✐st✐❝❛❧ ■♥st✐t✉t❡✱ ■❞✐❛♥✳ ❬✸❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✸✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠ ✭●✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐✮✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✹❪ P●❙✳ ❚❙ ✣é ❱➠♥ ▲÷✉ ✲ P●❙✳ ❚❙ P t ỗ ✈➔ ❦➽ t❤✉➟t✳ ❬✺❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✸✮✱ ▲➼ t❤✉②➳t tè✐ ÷✉ ✲ ❇➔✐ ❣✐↔♥❣ ❧ỵ♣ ❝❛♦ ❤å❝✱ ❱✐➺♥ t♦→♥ ❤å❝ ❍➔ ◆ë✐✳ ❬✻❪ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❍✉②➲♥ ✭✷✵✶✼✮✱ ▼ët sè ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ ❋❛r❦❛s✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐ ✷✳ ❬✼❪ ◆❣✉②➵♥ ❚❤à r ỗ ởt số ự t ỗ õ tốt ữ P❤↕♠ ❍➔ ◆ë✐ ✷✳ ✹✻