Định lí tách với điều kiện về phần trong tựa tương đối và áp dụng cho điều kiện tối ưu và đối ngẫu

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❚❍➚ ❍❷■ ❆◆❍ ✣➚◆❍ ▲➑ ❚⑩❈❍ ❱❰■ ✣■➋❯ ❑■➏◆ ❱➋ P❍❺◆ ❚❘❖◆● ❚Ü❆ ❚×❒◆● ✣➮■ ❱⑨ ⑩P ❉Ư◆● ❈❍❖ ✣■➋❯ ❑■➏◆ ❚➮■ ×❯ ❱⑨ ✣➮■ ◆●❼❯ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✹ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❚❍➚ ❍❷■ ❆◆❍ ✣➚◆❍ ▲➑ ❚⑩❈❍ ❱❰■ ✣■➋❯ ❑■➏◆ ❱➋ P❍❺◆ ❚❘❖◆● ❚Ü❆ ❚×❒◆● ✣➮■ ❱⑨ ⑩P ❉Ư◆● ❈❍❖ ✣■➋❯ ❑■➏◆ ❚➮■ ×❯ ❱⑨ ✣➮■ ◆●❼❯ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ Ù◆● ❉Ö◆● ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữợ P ì ◆❣✉②➯♥ ✲ ✷✵✶✹ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ✶ ✣à♥❤ t ợ tr tỹ tữỡ ố ỵ tt ố ✶✳✶ P❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ✣à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✶✳✸ ⑩♣ ❞ư♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤ì♥ ♠ư❝ t✐➯✉ ✸ ✽ ✶✵ ✷ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ♥â♥ ✶✺ ✷✳✶ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ⑩♣ ❞ư♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ⑩♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ✳ ✳ ✷✺ ❑➳t ❧✉➟♥ ✷✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✵ ✐ ▼ð ✤➛✉ ❚r♦♥❣ ❣✐↔✐ t➼❝❤ ỗ ỹ ữ t ❣✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥✱ tè✐ ÷✉ ❤â❛ ✳✳✳✱ ❝→❝ ✤à♥❤ t t ỗ õ ởt tr rt q trồ r t ỗ õ t→❝❤ ❝❤➼♥❤✱ ✭①❡♠ ❬✶❪✮✳ ❚r♦♥❣ ✤à♥❤ ❧➼ t→❝❤ t❤ù ♥❤➜t t sỷ ởt tr t ỗ ♣❤↔✐ ❝â ♣❤➛♥ tr♦♥❣ ❦❤→❝ ré♥❣✳ ❈➙✉ ❤ä✐ ✤÷đ❝ ✤➦t r tr t ỗ ✤➲✉ ❜➡♥❣ ré♥❣ t❤➻ ❧✐➺✉ ❝â t❤➸ t→❝❤ ✤÷đ❝ ❤❛✐ t ỗ ổ tữỡ tr ổ ổ ❝❤✐➲✉ ❤❛② ❦❤ỉ♥❣❄ ❈➙✉ tr↔ ❧í✐ ❧➔ ❝â✳ ▼ỵ✐ ✤➙②✱ ❈❛♠♠❛r♦t♦ ✈➔ ❉✐ ❇❡❧❧❛ ❬✺❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤à♥❤ ❧➼ t→❝❤ ♠ỵ✐ ❞ü❛ tr➯♥ ❦❤→✐ ♥✐➺♠ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✤➸ t❤❛② t❤➳ ❝❤♦ ♣❤➛♥ tr♦♥❣✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ❝→❝ ❦➳t q✉↔ ♠ỵ✐ ✈➲ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉✳ ✣➙② ❧➔ ✤➲ t➔✐ ✤÷ñ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤➼♥❤ ✈➻ t❤➳ tæ✐ ❝❤å♥ ✤➲ t➔✐✿ ✧ ✣à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ❞ư♥❣ ❝❤♦ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉✧✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✤à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tỹ tữỡ ố tr ỵ tt ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉✳ ▲✉➟♥ ✈➠♥ ❜❛♦ ỗ ữỡ t ♠ư❝ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶✳ ✣à♥❤ ❧➼ t ợ tr tỹ tữỡ ố ỵ tt ố r ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ❝õ❛ ❇♦r✇❡✐♥ ✲ ▲❡✇✐s ❬✷❪ ✈➔ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❈❛♠♠❛r♦t♦ ✲ ❉✐ ❇❡❧❧❛ ❬✺❪ ✈➲ ✤à♥❤ ❧➼ t→❝❤✱ tr♦♥❣ ✤â ♣❤➛♥ tr♦♥❣ ✤÷đ❝ t❤❛② t❤➳ ❜➡♥❣ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tố ữ õ r tr trữớ ủ ỗ ổ ❤↕♥ ❝❤✐➲✉ ✈ỵ✐ ♠ët ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② t❤❛② t❤➳ ❝❤♦ ✤✐➲✉ ✶ ❦✐➺♥ ❙❧❛t❡r t❤ỉ♥❣ t❤÷í♥❣✳ ❈❤÷ì♥❣ ✷✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ♥â♥ ❚r➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝õ❛ ❈❛♣➠t➠ ❬✻❪ ✈➲ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ♥â♥ ✈➔ ❛❢❢✐♥❡ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✤à♥❤ ❧➼ t→❝❤ ❞ü❛ tr➯♥ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐✳ ❈❤÷ì♥❣ ✷ ❝ơ♥❣ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ✤÷đ❝ →♣ ❞ư♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ✈ỵ✐ r➔♥❣ ❜✉ë❝ ♥â♥ ✈➔ ❛❢❢✐♥❡ ✈➔ ❝→❝ ❦➳t q✉↔ ✈➲ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ữợ sỹ ữợ t t P ộ ữ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✈➲ sü t➟♥ t➙♠ ✈➔ ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❤➛② tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❡♠ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ♣❤á♥❣ ✣➔♦ t↕♦ ❑❤♦❛ ❤å❝ ✈➔ ◗✉❛♥ ❤➺ q✉è❝ t➳✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✈➔ ỗ ú ù tr q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ t ổ ữủ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✸ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✹ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ữỡ t ợ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ỵ tt ố ữỡ tr ❝→❝ ❦➳t q✉↔ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ❝õ❛ ❇♦r✇❡✐♥ ✲ ▲❡✇✐s ❬✷❪ ✈➔ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❈❛♠♠❛r♦t♦ ✲ ❉✐ ❇❡❧❧❛ ❬✺❪ ✈➲ ✤à♥❤ ❧➼ t→❝❤✱ tr♦♥❣ ✤â ♣❤➛♥ tr♦♥❣ ✤÷đ❝ t❤❛② t❤➳ ❜➡♥❣ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tố ữ õ r tr trữớ ủ ỗ ổ ❤↕♥ ❝❤✐➲✉ ✈ỵ✐ ♠ët ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② t❤❛② t❤➳ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r t❤ỉ♥❣ t❤÷í♥❣✳ ✶✳✶ P❤➛♥ tr♦♥❣ tü❛ tữỡ ố r t tố ữ ỗ ổ ❤↕♥ ❝❤✐➲✉✱ ❝â t❤➸ ①↔② r❛ tr÷í♥❣ ❤đ♣ ❝→❝ ✤à♥❤ ❧➼ t→❝❤ t❤ỉ♥❣ t❤÷í♥❣ ❦❤ỉ♥❣ t❤➸ sû ❞ư♥❣ ✤÷đ❝✱ ❝❤➥♥❣ ❤↕♥✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❤➻♥❤ ♥â♥ ❞÷ì♥❣ tr♦♥❣ Lp ✱ C = {u ∈ Lp (T, µ) : u(t) ≥ 0, h.k.n}, rộ ỵ ợ ởt t ỗ r s ỹ ♥✐➺♠ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐✳ ✣â ❧➔ sü ♠ð rë♥❣ ❝õ❛ ❦❤→✐ ♥✐➺♠ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❈❤ó♥❣ t❛ s➩ ❜➢t ✤➛✉ ợ t ỗ tr Rn ởt ổ ❣✐❛♥ ✸ ✈❡❝tì X ✈➔ t➟♣ C ⊂ X ✱ t❛ ❦➼ ❤✐➺✉ ♥â♥ s✐♥❤ ❜ð✐ C ❧➔✿ coneC = {λx | x ∈ C, λ ∈ R, λ ≥ 0} ◆❤➢❝ ❧↕✐ ❬✶❪✿ P❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ✭r❡❧❛t✐✈❡ ✐♥t❡r✐♦r✮ ❝õ❛ t➟♣ A ⊂ Rn ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ A tr♦♥❣ ❛❢❢ A✱ ❦➼ ❤✐➺✉ ❧➔ riA✱ tr♦♥❣ ✤â ❛❢❢ A ❧➔ ❜❛♦ ❛❢❢✐♥❡ ❝õ❛ t➟♣ A✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉✿ ▼➺♥❤ ✤➲ ✶✳✶✳✶✳ ●✐↔ sû C ❧➔ ♠ët t ỗ õ x riC ♥➳✉ cone(C − x¯) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ⊂ Rn ●✐↔ sû x ¯ ∈ riC ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ❧➙♥ ❝➟♥ N ❝õ❛ x¯ t❛ ❝â✿ N ∩ ❛❢❢C ⊂ C ❉♦ ✤â✱ cone(C − x ¯) ❂ ❛❢❢C − x¯ ✈➔ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✳ ▼➦t ❦❤→❝✱ ❣✐↔ sû x ¯∈ / riC ✳ ❑❤✐ ✤â ❝â t❤➸ t→❝❤ ❤♦➔♥ t♦➔♥ x¯ ✈ỵ✐ C ✿ ∃y ∈ Rn s❛♦ ❝❤♦ y T x¯ ≤ y T x, ∀x ∈ C, ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤➦t ①↔② r❛ ✈ỵ✐ x ♥➔♦ ✤â t❤✉ë❝ C ✭❬✶✵❪✱ ✣à♥❤ ❧➼ ✶✶✳✸✮✳ ◆❤÷ ✈➟②✱ y T z ≥ 0, ∀z ∈ cone(C − x¯), ✈➔ x − x¯ ∈ cone(C − x¯) ◆❤÷♥❣ x ¯−x ∈ / cone(C − x¯)✱ ❝❤♦ ♥➯♥ cone(C − x¯) ❦❤æ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✳ ❑➼ ❤✐➺✉ clC ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ C ✳ ❇ê ✤➲ ✶✳✶✳✶✳ ●✐↔ sû C ⊂ Rn ởt t ỗ õ C ởt ổ ❣✐❛♥ ❝♦♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ clC ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✳ ✹ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ C ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t❤➻ clC = C ✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ C = clC ✈➔ clC ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t❤➻ C ♥➡♠ tr♦♥❣ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣ ❝õ❛ clC ✭❬✶✵❪✱ ❍➺ q✉↔ ✶✶✳✺✳✷✮✱ ♠➔ ✤✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸✳ ❉♦ ✤â✱ tr♦♥❣ ♠➺♥❤ ✤➲ ✶✳✶✳✶✱ t❛ ❝â t❤➸ t❤❛② t❤➳ ❜➡♥❣ cone(C − x ¯) ✤â♥❣✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐✳ ❚ø ✤➙②✱ X s➩ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ❍❛✉s❞♦r❢❢ X ổ tổổ ố ỗ t➜t ❝↔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ X ✳ P❤➛♥ tû ❦❤ỉ♥❣ ❝õ❛ X ∗ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ θX ∗ ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❬✷❪ ●✐↔ sỷ C ởt t ỗ X P❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✭q✉❛s✐ r❡❧❛t✐✈❡ ✐♥t❡r✐♦r✮ ❝õ❛ C ❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû x ∈ C ♠➔ cl(cone(C − x)) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ qriC ✳ ◆â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ C t↕✐ x ¯ ∈ C ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ NC (¯ x) := {φ ∈ X ∗ : φ(x − x¯) ≤ 0, ∀x ∈ C} ❇➙② ❣✐í✱ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❤ú✉ ➼❝❤ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐✱ ①❡♠ ❬✷❪ ✈➔ ❬✸❪✳ ✣à♥❤ ❧➼ ✶✳✶✳✶✳ ❬✷❪ ●✐↔ sû C ởt t ỗ X x ∈ C ✳ ❑❤✐ ✤â✱ x¯ ∈ qriC ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ NC (¯x) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ∗✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû K ⊂ X ✈➔ K ❧➔ ♠ët ♥â♥✱ ❝ü❝ ❝õ❛ K ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ K o = {φ ∈ X ∗ | φ(x) ≤ 1, ∀x ∈ K} = {φ ∈ X ∗ | φ(x) ≤ 0, ∀x ∈ K} ữỡ tỹ ợ õ L X t ❝â✿ o L = {x ∈ X | φ(x) ≤ 0, ∀φ ∈ L} ◆❣❛② ❧➟♣ tù❝ t❛ t❤➜② r➡♥❣✿ ♥➳✉ K ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t❤➻ t❛ ❝â K o ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥✳ ◆➳✉ L ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t❤➻ t❛ ✺ ❝â o L ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥✳ ❇➙② ❣✐í✱ t❛ ❝â✿ ❱ỵ✐ φ ∈ X ∗ ✱ φ(x − x ¯) ≤ 0✱ ✈ỵ✐ ♠å✐ x ∈ C ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ φ(u) ≤ 0✱ ✈ỵ✐ ♠å✐ u ∈ cl(cone(C − x ¯))✱ ❞♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ φ✳ ❱➟②✱ NC (¯ x) = cl(cone(C − x¯))o ▼➦t ❦❤→❝✱ t❤❡♦ ✤à♥❤ ❧➼ ❧÷ï♥❣ ❝ü❝ ✭①❡♠ ❬✾❪✮✱ t❛ ❝â✿ o NC (¯ x) = o (cl(cone(C − x¯))o ) = cl(cone({0} ∪ cl(cone(C − x¯)))) = cl(cone(C − x¯)) ❱➟②✱ ✤à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ❧➼ ✶✳✶✳✷✳ ❬✺❪ ●✐↔ sỷ C D t ỗ X s❛♦ ❝❤♦ qriC = ∅✱ qriD = ∅ ✈➔ x0 ∈ C ✱ x¯ ∈ qriC ✱ α ∈ R ✈➔ λ ∈ [0, 1)✳ ❑❤✐ ✤â✱ ✭❛✮ qriC + qriD ⊆ qri(C + D)✱ ✭❜✮ αqriC = qri(αC)✱ ✭❝✮ qri(C × D) = qriC × qriD✱ ✭❞✮ cl(qriC) = cl(C)✱ ✭❡✮ (1 − λ)x0 + λ¯x ∈ qriC ✱ ✭❢✮ qriC = qri(qriC)✱ ✭❣✮ qri(C − x) = qriC − x ✭ ∀x ∈ X ✮✱ ✭❤✮ cl[cone(qriC)] = cl(coneC)✱ ♥➳✉ qriC = ∅✳ ✣➸ ❧➔♠ rã ❤ì♥ ✈➲ ❦❤→✐ ♥✐➺♠ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ❝→❝ ✤à♥❤ ♥❣❤➽❛ s❛✉ ✭①❡♠ ❬✻❪✮✳ ✣à♥❤ ♥❣❤➽❛ sỷ C t ỗ X ✳ ✭✐✮ ❍↕❝❤ ✭❝♦r❡✮ ❝õ❛ C ❧➔ coreC := {x ∈ C | cone(C − x) = X} ✻ ✭✐✐✮ ❍↕❝❤ ❝❤➢❝ ❝❤➢♥ ✭✐♥tr✐♥s✐❝ ❝♦r❡✮ ❝õ❛ C ❧➔ irc(C) := {x ∈ C | cone(C − x) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ ❳ } ✭✐✐✐✮ P❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ♠↕♥❤ ✭str♦♥❣ ✲ q✉❛s✐ r❡❧❛t✐✈❡ ✐♥t❡r✐♦r✮ ❝õ❛ C ❧➔ sqriC := {x ∈ C | cone(C − x) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ✤â♥❣ ❝õ❛ ❳ } ✭✐✈✮ ❚ü❛ ♣❤➛♥ tr♦♥❣ ✭q✉❛s✐ ✲ ✐♥t❡r✐♦r✮ ❝õ❛ C ❧➔ qiC := {x ∈ C | cl[cone(C − x)] = X} ❈→❝ ❜❛♦ ❤➔♠ s❛✉ ✤ó♥❣ ✭①❡♠ ❬✻❪✮✿ intC ⊆ coreC ⊆ sqriC ⊆ icrC ⊆ qriC ⊆ C, ✈➔ intC ⊆ coreC ⊆ qiC ⊆ qriC ⊆ C ◆➳✉ X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ t❤➻ qriC = sqriC = icrC = riC, ✈➔ coreC = qiC = intC, tr♦♥❣ ✤â✱ riC ❧➔ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ C ✱ ♥❣❤➽❛ ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ C tr♦♥❣ ❜❛♦ ❛❢❢✐♥❡ ❝õ❛ ♥â✳ ✣à♥❤ ❧➼ ✶✳✶✳✸✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ỗ ữỡ õ tự tỹ ữủ õ ỗ C ợ cl(C C) = X ✈➔ X ∗ ✤÷đ❝ ①→❝ ✤à♥❤ t❤ù tü ❜ë ♣❤➟♥ ❜ð✐ C ∗✳ ❑❤✐ ✤â✱ x¯ ∈ qriC ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ φ(¯x) > ✈ỵ✐ ♠å✐ φ ∈ X ∗ \ {0}✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x ¯ qriC ữ ợ C φ = t❛ ❝â✿ φ(¯ x) = 0✳ ❑❤✐ ✤â✱ −φ(x − x¯) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C S ỗ f C ❤➔♠ t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐✱ ❝❤ó♥❣ t❛ s✉② r❛ ty1 + (1 − t)y2 ∈ f (a, tx1 + (1 − t)x2 ) + intC ✭✷✳✶✮ ▼➦t ❦❤→❝✱ ❞♦ K S ỗ g ởt K ✈➔ h ❧➔ ❛❢❢✐♥❡ ♥➯♥ t❛ ❝â✿ tz1 + (1 − t)z2 ∈ g(tx1 + (1 − t)x2 ) + K, ✭✷✳✷✮ tw1 + (1 − t)w2 = h(tx1 + (1 − t)x2 ) ✭✷✳✸✮ ✈➔ ❉♦ ✤â✱ tø ✭✷✳✶✮✱✭✷✳✷✮ t s r M ỗ tt qri((g, h)(S) + K × {0}) = ∅✱ ❝❤♦ ♥➯♥ tỗ t (z0 , w0 ) qri((g, h)(S) + K × {0}) ⊆ (g, h)(S) + K × {0} õ ợ (z0 , w0 ) tỗ t x ∈ S ✈➔ y0 ∈ Y t❤ä❛ ♠➣♥ y0 ∈ f (a, x) + intC, z0 ∈ g(x) + K ✈➔ w0 = h(x), tù❝ ❧➔ (y0 , z0 , w0 ) ∈ M ✣➸ ❝❤➾ r❛ qriM ❦❤æ♥❣ ré♥❣✱ t❛ ❧➜② (y ∗ , z ∗ , w ∗ ) ∈ NM (y0 , z0 , w0 )✱ tù❝ ❧➔ y ∗ (l − y0 ) + z ∗ (h − z0 ) + w∗ (p − w0 ) ≤ 0, ✈ỵ✐ ♠å✐ (l, h, p) ∈ M ✭✷✳✹✮ ❚❛ ❝â (y0 + c, z0 , w0 ) ∈ M ✱ ✈ỵ✐ ♠å✐ c ∈ intC ✳ ⑩♣ ❞ö♥❣ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✹✮ t❛ ❝â✿ y ∗ (c) ≤ 0, ✈ỵ✐ ♠å✐ c ∈ intC ✭✷✳✺✮ ❱➻ y0 ∈ f (a, x) + intC ✈➔ f (a, x) + intC ởt t tỗ t ởt ❧➙♥ ❝➟♥ ❝➙♥ U ❝õ❛ ✤✐➸♠ ✵ tr♦♥❣ Y s❛♦ ❝❤♦ y0 − c ∈ f (a, x) + intC, ✈ỵ✐ ♠å✐ c ∈ intC ∩ U ❉♦ ✤â✱ (y0 − c, z0 , w0 ) ∈ M ✱ ✈ỵ✐ ♠å✐ c ∈ intC ∩ U ✳ ❚ø ✭✷✳✹✮✱ t❛ ❝â✿ y ∗ (c) ≥ 0, ✈ỵ✐ ♠å✐ c ∈ intC ∩ U ✶✽ ✭✷✳✻✮ ❈è ✤à♥❤ c0 ∈ intC U õ tỗ t V ❝õ❛ ✵ tr♦♥❣ Y s❛♦ ❝❤♦ c0 + V ⊆ intC ∩ U ❍ì♥ ♥ú❛✱ ❣✐↔ sû e ∈ Y õ tỗ t t > s ❝❤♦ c0 − te ∈ intC ∩ U, ✈➔ c0 + te ∈ intC ∩ U ❚ø ✭✷✳✺✮ ✈➔ ✭✷✳✻✮✱ t❛ ❝â✿ y ∗ (c0 − te) ≤ ✈➔ y ∗ (c0 + te) ≤ 0, ✈➔ y ∗ (c0 − te) ≥ ✈➔ y ∗ (c0 + te) ≥ ❈→❝ q✉❛♥ ❤➺ ♥➔② ❝❤➾ r❛ r➡♥❣ y ∗ (e) = 0✱ ✈ỵ✐ ♠å✐ e ∈ Y ✱ ❝â ♥❣❤➽❛ ❧➔ y ∗ = 0✳ ❱ỵ✐ y ∗ = 0✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✹✮ trð t❤➔♥❤ z ∗ (h − z0 ) + w∗ (p − w0 ) ≤ 0, ✈ỵ✐ ♠å✐ (h, p) ∈ (g, h)(S) + K ì {0} tữỡ ữỡ ợ (z ∗ , w ∗ ) ∈ N(g,h)(S)+K×{0} (z0 , w0 )✳ ❱➻ N(g,h)(S)+K×{0} (z0 , w0 ) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ Z ∗ ×W ∗ ✱ t❛ ❝â✿ −(z ∗ , w∗ ) ∈ N(g,h)(S)+K×{0} (z0 , w0 ), ✈➔ (−y ∗ , −z ∗ , −w∗ ) ∈ NM (y0 , z0 , w0 ) ❱➟②✱ NM (y0 , z0 , w0 ) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ Z ∗ × W ∗ ✈➔ (y0 , z0 , w0 ) ∈ qriM (0, 0, 0) M tỗ t↕✐ ♠ët ✤✐➸♠ x ∈ S s❛♦ ❝❤♦ f (a, x) ∈ −intC, g(x) ∈ −K ✈➔ h(x) = ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ a ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮✳ ❱➻ ✈➟②✱ (0, 0, 0) ∈ / M✳ ✶✾ ❉♦ qriM ⊆ M ✱ t❛ ❝â✿ (0, 0, 0) / qriM q tỗ t (y ∗ , z ∗ , w∗ ) ∈ Y ∗ × Z ∗ × W ∗ \ {(0, 0, 0)}, s❛♦ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ✤ó♥❣ y ∗ (y) + z ∗ (z) + w∗ (w) ≥ 0, ✈ỵ✐ ♠å✐ (y, z, w) ∈ M ✭✷✳✽✮ ❱ỵ✐ ♠é✐ c ∈ intC ✈➔ ✈ỵ✐ ♠é✐ t > 0✱ (y + tc, z, w) ❝ô♥❣ t❤✉ë❝ M ✳ ❚ø ✭✷✳✽✮✱ t❛ ❝â✿ y ∗ (y) + z ∗ (z) + w∗ (w) + y ∗ (c) ≥ 0, t ✈ỵ✐ ♠å✐ c ∈ intC ✈➔ t > 0✳ ❈❤♦ t → ∞ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ ❞♦ cl(intC) = C ✱ t❛ ♥❤➟♥ ✤÷đ❝ y ∗ ∈ C ∗ ✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠é✐ k ∈ K ✈➔ ♠é✐ t > 0✱ (y, z + tk, w) ∈ M ✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✽✮ t❛ ❝â✿ y ∗ (y) + z ∗ (z) + w∗ (w) + z ∗ (k) ≥ 0, t ✈ỵ✐ ♠å✐ k ∈ K ✈➔ t > 0✳ ❈❤♦ t → ∞ t❛ ữủ z (k) ợ k K ✳ ❱➟②✱ z ∗ ∈ K ∗ ✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣ r➡♥❣ y ∗ = 0✳ ❑❤✐ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✽✮ ❜✐➳♥ ✤ê✐ t❤➔♥❤ z ∗ (z) + w∗ (w) ≥ 0, ✈ỵ✐ ♠å✐ (z, w) ∈ (g, h)(S) + K × {0} ✭✷✳✾✮ ❚ø ✤✐➲✉ ❦✐➺♥ ✤➲✉ ✭❘❈✮✱ ✈ỵ✐ (z , w ) tỗ t x0 S s❛♦ ❝❤♦ z ∗ (g(x0 )) + w∗ (h(x0 ))) < ✭✷✳✶✵✮ ▲➜② z0 := g(x0 ) ✈➔ w0 := h(x0 )✳ ❑❤✐ ✤â✱ (z0 , w0 ) ∈ (g, h)(S)+K ì{0} t ợ ợ ♠å✐ c ∈ intC ✱ k ∈ K ✈➔ t > 0✱ t❛ ❝â✿ (f (a, a) + tc, g(a) + tk, h(a)) ∈ M ✷✵ ❚ø ✭✷✳✽✮✱ t❛ ❝â✿ t[y ∗ (c) + z ∗ (k)] + z ∗ (g(a)) ≥ ❈❤♦ t → ∞✱ t❛ ♥❤➟♥ ✤÷đ❝ z ∗ (g(a)) ≥ ▼➦t ❦❤→❝✱ ✈➻ g(a) ∈ −K ✈➔ z ∗ ∈ K ∗ ✱ t❛ ❝â✿ z ∗ (g(a)) ≤ ❱➟② z ∗ (g(a)) = 0✳ ❚ø ✭✷✳✽✮ ✈➔ sü ❦✐➺♥✿ ❱ỵ✐ ❜➜t ❦➻ x ∈ S t❤➻ (f (a, x) + tc, g(x), h(x)) ∈ M ✈ỵ✐ ♠å✐ t > 0, ✈➔ c ∈ intC ✱ t❛ s✉② r❛ r➡♥❣ y ∗ (f (a, x)) + z ∗ (g(x)) + w∗ (h(x)) ≥ = y ∗ (f (a, a)) + z ∗ (g(a)) + w (h(a)), ợ x S ữ t r r tỗ t (y , z ∗ , w ∗ ) ∈ C \ {0} × K ∗ × W ∗ s❛♦ ❝❤♦ y ∗ (f (a, a)) + z ∗ (g(a)) + w∗ (h(a)) = y ∗ {(f (a, x)) + z ∗ (g(x)) + w∗ (h(x))} x∈S ✭⇐= ✮ ❈❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✿ ●✐↔ sû r➡♥❣ a ∈ A ❦❤æ♥❣ ♣❤↔✐ tỗ t b A s❛♦ ❝❤♦ f (a, b) ∈ −intC ❚❤❡♦ ❣✐↔ tt tỗ t (y , z , w ∗ ) ∈ C ∗ \ {0} × K ∗ × W ∗ t❤ä❛ ♠➣♥ z ∗ (g(a)) = 0, ✈➔ = y ∗ (f (a, a)) + z ∗ (g(a)) + w∗ (h(a)) = y ∗ {(f (a, x)) + z ∗ (g(x)) + w∗ (h(x))} x∈S ∗ ≤ y (f (a, b)) + z ∗ (g(b)) + w∗ (h(b)) < ✭ ✈æ ❧➼✦✮ ✷✶ ❉♦ ✤â✱ a ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮ ✈➔ ✤à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤♦ h ≡ 0✳ ❚ø ✤à♥❤ ❧➼ ✷✳✶✳✶ ð tr➯♥ t❛ s✉② r❛ ✤à♥❤ ❧➼ ✸✳✶ ❝õ❛ ●♦♥❣ ❬✽❪✳ ❍➺ q✉↔ ✷✳✶✳✶✳ ●✐↔ sû ❣✐↔ tt A tọ tỗ t x0 S s❛♦ ❝❤♦ g(x0) ∈ −intK ✳ ❑❤✐ ✤â✱ x ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tỗ t (y, z) C \ {0} ì K ∗ s❛♦ ❝❤♦ z ∗ (g(a)) = 0, ✈➔ y ∗ (f (a, a)) + z ∗ (g(a)) = min{y ∗ (f (a, x)) + z ∗ (g(x))} x∈S ❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❤➺ q✉↔ ♥➔②✱ t❛ ♣❤↔✐ ❝❤➾ r❛ qri(g(S) + K) = ∅ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ ✤ó♥❣ ✈ỵ✐ h ≡ 0✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ z ∗ ∈ K ∗ \ {0}✱ t❛ ❝â✿ z ∗ (g(x0 )) < 0, ✈➔ ♥❤÷ ✈➟② ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ ✤➣ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ tø ♠➺♥❤ ✤➲ ✷✳✷ ❝õ❛ ❬✶✶❪✱ tê♥❣ ❝õ❛ ♠ët t➟♣ ✈➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ ❦❤→❝ ❧➔ t➟♣ ❝♦♥ ❝õ❛ ♣❤➛♥ tr♦♥❣ ❝õ❛ tê♥❣ ❤❛✐ t➟♣✱ t❛ ❝â✿ g(S) + intK ⊆ int(g(S) + K) ❚ø q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ ❣✐ú❛ ♣❤➛♥ tr♦♥❣ ✈➔ tü❛ ♣❤➛♥ tr♦♥❣✱ t❛ s✉② r❛ r➡♥❣ qri(g(S) + K) = ∅✳ ◆❤➟♥ ①➨t ✷✳✶✳✶✳ ✣à♥❤ ❧➼ ✷✳✶✳✶ ❧➔ ♠ët tê♥❣ q✉→t ❝õ❛ ✤à♥❤ ❧➼ ✸✳✶ ❝õ❛ ●♦♥❣ ❬✽❪✳ ✣à♥❤ ❧➼ ✷✳✶✳✶ ð ✤➙② t❛ ①➨t ❜➔✐ t♦→♥ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ♥â♥ ✈➔ ❛❢❢✐♥❡✱ ❝á♥ ✤à♥❤ ❧➼ ✸✳✶ t r õ tr tỗ t↕✐ x0 ∈ S t❤ä❛ ♠➣♥ g(x0 ) ∈ −intK ✱ ✤÷đ❝ t❤❛② ❜➡♥❣ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ ②➳✉ ❤ì♥✳ ❚➼♥❤ rộ õ ỗ K t qri(g(S) + K) ❦❤ỉ♥❣ ré♥❣✱ ❝á♥ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ✷✷ ✷✳✷ ⑩♣ ❞ư♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ❇➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✭❊❑❋✮ ❜❛♦ ỗ t tố ữ tỡ ụ ữ ❜➔✐ t♦→♥ ❦❤→❝✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ❦➼ ❤✐➺✉ minw S ❧➔ t➟♣ ❝→❝ ❝ü❝ t✐➸✉ ②➳✉ ❝õ❛ S t❤❡♦ ♥â♥ C ✱ tù❝ ❧➔ z0 ∈ minw S ♥❣❤➽❛ ❧➔ z0 ∈ S, ✈➔ S ∩ (z0 − intC) = ∅ ●✐↔ sû F : S → Z ✳ ❍➔♠ ✈❡❝tì f : S × S → Z ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ f (a, b) := F (b) − F (a)✳ ❑❤✐ ✤â ✭❊❑❋✮ trð t❤➔♥❤ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉✿ ❚➻♠ a ¯ ∈ A s❛♦ ❝❤♦ F (b) − F (¯ a) ∈ / −intC, ✈ỵ✐ ♠å✐ b ∈ A ✭❲❱▼P✮ ✣✐➸♠ a ¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❲❱▼P✮✱ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ F (¯ a) ∈ minw F (A) ❱ỵ✐ ❜➔✐ t♦→♥ ♥➔②✱ ❣✐↔ t❤✐➳t ✭A✮ trð t❤➔♥❤✿    ❋ ❧➔ ♠ët ❈ ✲ ❤➔♠, ❣ ❧➔ ♠ët ✲ ❑ ✲ ❤➔♠,   ❤ ❧➔ ❛❢❢✐♥❡ ❚ø ✤à♥❤ ❧➼ ✷✳✶✳✶✱ t❛ ❝â ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✭❲❱▼P✮✳ ✣à♥❤ ❧➼ ✷✳✷✳✶✳ ●✐↔ sû qri(g, h)(S)+K ×{0} = ∅ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ t❤ä❛ ♠➣♥✳ ✣✐➸♠ a ∈ A ❧➔ ♥❣❤✐➺♠ ❝õ❛ P tỗ t (y , z ∗ , w∗ ) ∈ C ∗ \ {0} × K ∗ × W ∗ s❛♦ ❝❤♦ z ∗ (g(a)) = 0, ✈➔ y ∗ (F (a)) = y ∗ (F (a)) + z ∗ (g(a)) + w∗ (h(a)) = min{y ∗ (F (x)) + z ∗ (g(x)) + w∗ (h(x))} x∈S ✷✸ ✣à♥❤ ❧➼ t✐➳♣ t❤❡♦ ❧➔ ♠ët trữ t P ữợ ổ ♥❣ú ♥❤➙♥ tû ▲❛❣r❛♥❣❡✳ ❚❛ ❦➼ ❤✐➺✉ L+ (Z, Y ) ❧➔ t➟♣ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ①→❝ ✤à♥❤ tr♦♥❣ Z ✈ỵ✐ ❣✐→ trà tr♦♥❣ Y t❤❡♦ t➼♥❤ ❝❤➜t✿ ❈❤ó♥❣ →♥❤ ①↕ K ✈➔♦ C ❀ L(W, Y ) ❧➔ t➟♣ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ①→❝ ✤à♥❤ tr♦♥❣ Z ✈ỵ✐ ❣✐→ trà tr♦♥❣ Y ✳ ✣à♥❤ ❧➼ ✷✳✷✳✷✳ ●✐↔ sû qri((g, h)(S) + K × {0}) = ∅ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ t❤ä❛ ♠➣♥✳ ✣✐➸♠ a ∈ S ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭❲❱▼P✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ a ∈ A ✈➔ tỗ t tỷ r (T, M ) L+(Z, Y ) × L(W, Y ) t❤ä❛ ♠➣♥ F (a) ∈ minw F (x) + T (g(x)) + M (h(x)) , x∈S ✈➔ T (g(a)) = ❈❤ù♥❣ ♠✐♥❤✳ ✭=⇒✮ ●✐↔ sû a ∈ S ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮✳ ❱➟②✱ a ∈ A✳ ❇ð✐ ✈➻ ❝→❝ ❣✐↔ t❤✐➳t ✤à♥❤ ❧➼ tọ tỗ t (y , z ∗ , w ∗ ) ∈ C ∗ \{0}×K ∗ ×W ∗ s❛♦ ❝❤♦ z ∗ (g(a)) = 0, ✈➔ y ∗ (F (a)) = y ∗ (F (a)) + z ∗ (g(a)) + w∗ (h(a)) = min{y ∗ (F (x)) + z ∗ (g(x)) + w∗ (h(x))} ✭✷✳✶✶✮ x∈S ●✐↔ sû c0 ∈ intC t❤ä❛ ♠➣♥ y ∗ (c0 ) = 1✳ ❑❤✐ ✤â✱ t❛ ✤à♥❤ ♥❣❤➽❛ T : Z → Y ✈➔ M : W → Y ❜ð✐ T (z) := z ∗ (z)c0 ✈➔ M (w) := w∗ (w)c0 ❉➵ t❤➜② T (g(a)) = z ∗ (g(a))c0 = 0✳ ❍ì♥ ♥ú❛✱ tø ✭✷✳✶✶✮✱ t❛ ❝â✿ y ∗ (F (a)) = min{y ∗ (F (x) + z ∗ (g(x))c0 + w∗ (h(x))c0 )} x∈S ✣✐➲✉ ✤â t÷ì♥❣ ữỡ ợ y (F (a)) = min{y (F (x) + T (g(x)) + M (h(x)))}, ✈ỵ✐ ♠å✐ x ∈ S x∈S ✷✹ ❱➻ t❤➳✱ ❦➳t ❤đ♣ ✈ỵ✐ y ∗ ∈ C ∗ \ {0}✱ t❛ s✉② r❛ F (x) + T (g(x)) + M (h(x)) − F (a) ∈ / −intC, ✈ỵ✐ ♠å✐ x ∈ S ❱➟②✱ F (a) ∈ minw F (x) + T (g(x)) + M (h(x)) x∈S ✭⇐=✮ ●✐↔ sû a ∈ A ❧➔ ữủ ợ x S t ❦➻✱ t❛ ❝â✿ F (x) + T (g(x)) + M (h(x)) − F (a) ∈ / −intC ✭✷✳✶✷✮ ✣➦❝ ❜✐➺t✱ ✭✷✳✶✷✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ A✳ ❱ỵ✐ x ∈ A✱ t❛ ❝â✿ g(x) ∈ −K ✈➔ h(x) = ❱➟②✱ T (g(x)) ∈ −C ✈➔ M (h(x)) = 0, ✈ỵ✐ ♠å✐ x ∈ A ❚ø ✭✷✳✶✷✮ ✈➔ ♥❤ú♥❣ ♥❤➟♥ ①➨t tr➯♥✱ t❛ s✉② r❛ F (x) − F (a) ∈ / −intC ✈ỵ✐ ♠å✐ x ∈ A ❱➻ ✈➟②✱ a ∈ A ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭❲❱▼P✮✳ ✷✳✸ ⑩♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ❱ỵ✐ ❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡✱ t❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ②➳✉ ϕ ✈ỵ✐ ϕ : L+ (Z, Y ) × L(W, Y ) ⇒ Y ❝õ❛ ✭❲❱▼P✮ ❜ð✐ F (x) + T (g(x)) + M (h(x)) , ϕ(T, M ) := minw x∈S ✈ỵ✐ ♠å✐ (T, M ) ∈ L+ (Z, Y ) × L(W, Y )✳ ❇➙② ❣✐í✱ ✈ỵ✐ ❜➔✐ t♦→♥ ✭❲❱▼P✮ t❛ ①➨t ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✿ max ϕ(T, M ) s❛♦ ❝❤♦ (T, M ) ∈ L+ (Z, Y ) × L(W, Y ) ✷✺ ✭❉P✮ ▼ët ♥❣❤✐➺♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ (T0 , M0 ) L+ (Z, Y ) ì L(W, Y ) ữủ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ✭❉P✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ϕ(T0 , M0 ) ∩ maxw ϕ(T, M ) = ∅ (T,M )∈L+ (Z,Y )×L(W,Y ) ✣à♥❤ ❧➼ ✷✳✸✳✶✳ ●✐↔ sû qri(g, h)(S) + K × {0} = ∅ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ t❤ä❛ ♠➣♥✳ ◆➳✉ a ∈ A ởt P t tỗ t ②➳✉ (T0 , M0 ) ∈ L+ (Z, Y ) × L(W, Y ) ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✭❉P✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ❧➼ ✷✳✷✳✷ tọ tỗ t (T0 , M0 ) L+ (Z, Y ) × L(W, Y ), s❛♦ ❝❤♦ F (x) + T0 (g(x)) + M0 (h(x)) = ϕ(T0 , M0 ) ✭✷✳✶✸✮ F (a) ∈ minw x∈S ❱➻ a ∈ A ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭❲❱▼P✮✱ t❛ ❝â✿ F (a) = F (a) + T0 (g(a)) + M0 (h(a)) ∈ (F (x) + T0 (g(x)) + M0 (h(x))) x∈S ❱ỵ✐ ♠é✐ y ∈ ϕ(T0 , M0 )✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ϕ(T0 , M0 )✱ t❛ ❝â✿ F (a) − y ∈ / −intC ❱➟②✱ t➼♥❤ ✤è✐ ♥❣➝✉ ②➳✉ ✤ó♥❣✱ tù❝ ❧➔ F (a) ∈ maxw ϕ(T0 , M0 ) ✭✷✳✶✹✮ ◗✉❛♥ ❤➺ ✭✷✳✶✹✮ ✤ó♥❣ ✈ỵ✐ ♠é✐ ♥❣❤✐➺♠ ❝❤➜♣ ♥❤➟♥ ữủ (T, M ) L+ (Z, Y ) ì L(W, Y )✳ ❱➟② F (a) ❧➔ ✤✐➸♠ ❝ü❝ ✤↕✐ ②➳✉ ❝õ❛ t➟♣ ϕ(T, M ) t❤❡♦ ♥â♥ C ✱ tr♦♥❣ ✤â (T, M ) ∈ L+ (Z, Y ) × L(W, Y )✱ tù❝ ❧➔ (ϕ(T, M ) − F (a)) ∩ intC = ∅, ✷✻ ✈ỵ✐ ♠å✐ (T, M ) ∈ L+ (Z, Y ) × L(W, Y )✳ ❱➻ t❤➳✱ t❛ s✉② r❛ F (a) ∈ maxw ϕ(T, M ) ✭✷✳✶✺✮ (T,M )∈L+ (Z,Y )×L(W,Y ) ◗✉❛♥ ❤➺ ✭✷✳✶✸✮ ✈➔ ✭✷✳✶✺✮ ❝❤♦ t❛ ϕ(T0 , M0 ) ∩ maxw ϕ(T, M ) = ∅ (T,M )∈L+ (Z,Y )×L(W,Y ) ❱➟② (T0 , M0 ) ❧➔ ♥❣❤✐➺♠ t ố P rữợ t t❤ó❝ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ t❛ ①➨t tr÷í♥❣ ❤đ♣ h ≡ 0✱ Y := R ✈➔ C := R+ ✳ ❑❤✐ ✤â✱ ❝→❝ ❜➔✐ t♦→♥ ✭❲❱▼P✮ ✈➔ ✭❉P✮ trð t❤➔♥❤✿ inf F (x), ✭P✮ sup inf F (x) + λ(g(x)) , ✭❉✮ x∈A ✈➔ λ∈K ∗ x∈S tr♦♥❣ ❦❤✐ ✤â ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✭❘❈✮ ❝â ❞↕♥❣ s❛✉✿ ❱ỵ✐ ♠å✐ z K \ {0} tỗ t x S s❛♦ ❝❤♦ z ∗ (g(x)) < 0✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ ✤à♥❤ ❧➼ ✷✳✸✳✶ trð t❤➔♥❤ ✤à♥❤ ❧➼ s❛✉✳ ✣à♥❤ ❧➼ ✷✳✸✳✷✳ ●✐↔ sû qri(g(S) + K) = ∅ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ t❤ä❛ ♠➣♥✳ ◆➳✉ a ∈ A ởt P t tỗ t λ ∈ K ∗ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✭❉✮ ✈➔ ❣✐→ trà ❝õ❛ ✭P✮ ❜➡♥❣ ❣✐→ trà ❝õ❛ ✭❉✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ♣❤↔✐ ❝❤➾ r❛ r➡♥❣ ❣✐→ trà ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ❜➡♥❣ ❣✐→ trà ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ứ s r tỗ t λ ∈ K ∗ ❧➔ ♠ët ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✭❉✮✳ ❍ì♥ ♥ú❛✱ ❞♦ ϕ(λ) = inf F (x) + λ(g(x)) , x∈S ✈➔ q✉❛♥ ❤➺ ✭✷✳✶✸✮ t❛ t❤➜② ❣✐→ trà ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ❜➡♥❣ ❣✐→ trà ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✳ ✷✼ ❑➳t q✉↔ s❛✉ ✤➙② ❝â tr♦♥❣ ❬✹❪✱ ❝❤♦ t❛ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ t➼♥❤ ❦❤æ♥❣ ré♥❣ ❝õ❛ t➟♣ qri(g(S) + K) ❇ê ✤➲ ✷✳✸✳✶✳ ●✐↔ sû cl(K − K) = Z ✈➔ tỗ t x0 S s g(x0) qri(K) ❑❤✐ ✤â ∈ qi(g(S) + K)✳ ❍➺ q✉↔ t✐➳♣ t❤❡♦ ❧➔ ♠ët ❝↔✐ t✐➳♥ ❝õ❛ ❤➺ q✉↔ ✹✳✶ ❬✹❪✱ tr♦♥❣ ✤â t→❝ ❣✐↔ ①➨t ♠ët ✤✐➲✉ ❦✐➺♥ ❦❤æ♥❣ ❝➛♥ t❤✐➳t ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❤➺ q✉↔ ✤÷đ❝ ✤➣ ❦➸ tr➯♥✳ ❍➺ q✉↔ ✷✳✸✳✶✳ ●✐↔ sû cl(K − K) = Z ✈➔ x ∈ S t❤ä❛ ♠➣♥ g(x) ∈ −qriK ✳ ◆➳✉ a ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭P✮ t❤➻ tỗ t K ởt ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✭❉✮ ✈➔ ❣✐→ trà ❝õ❛ ✭P✮ ❜➡♥❣ ❣✐→ trà ❝õ❛ ✭❉✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ♣❤↔✐ ❝❤➾ r❛ ❣✐↔ t❤✐➳t ❝❤➼♥❤ q✉② ✭❘❈✮ t❤ä❛ ♠➣♥✳ ●✐↔ sỷ z K \ {0} tũ ỵ cl(K K) = Z tỗ t ởt ♣❤➛♥ tû x ∈ S s❛♦ ❝❤♦ g(x) ∈ −qriK ✱ tø ✤à♥❤ ❧➼ ✶✳✶✳✸ t❛ ❝â✿ z ∗ (g(x)) < ❱➟②✱ ✤✐➲✉ ❦✐➺♥ ✭❘❈✮ t❤ä❛ ♠➣♥ ✈➔ ❦➳t q✉↔ ✤÷đ❝ s✉② r❛ tø ❤➺ q✉↔ ✷✳✸✳✶ ✈➔ ✤à♥❤ ❧➼ ✷✳✸✳✷✳ ✷✽ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ✤à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉✳ ❈→❝ ♥ë✐ ❞✉♥❣ ✤÷ñ❝ tr➻♥❤ ❜➔② tr ỗ P tr tỹ tữỡ ✤è✐ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t❀ ✲ ✣à♥❤ ❧➼ t→❝❤ ợ tr tỹ tữỡ ố →♣ ❞ư♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤ì♥ ♠ư❝ t✐➯✉❀ ✲ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ♥â♥ ✈➔ ❛❢❢✐♥❡❀ ✲ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ tè✐ ÷✉ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ♥â♥ ✈➔ ❛❢❢✐♥❡❀ ✲ ❈→❝ ❦➳t q✉↔ ✤è✐ ♥❣➝✉ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉✳ ✣à♥❤ ❧➼ t ợ tr tỹ tữỡ ố ✈➔ ❝→❝ →♣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ tè✐ ÷✉ ❤â❛ ❧➔ ✈➜♥ ✤➲ t❤í✐ sü ✤➣✱ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ tr ữợ qố t q t ự ✷✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ộ ữ P t ỗ ◆❳❇ ❑❤♦❛ ❤å❝ ✈➔ ❑ÿ t❤✉➟t✱ ❍➔ ◆ë✐✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ P❛rt✐❛❧❧② ❢✐♥✐t❡ ❝♦♥✈❡① ♣r♦✲ ❣r❛♠♠✐♥❣✱ P❛rt ■✿ ◗✉❛s✐ r❡❧❛t✐✈❡ ✐♥t❡r✐♦rs ❛♥❞ ❞✉❛❧✐t② t❤❡♦r②✱ ❬✷❪ ❇♦r✇❡✐♥✱ ❏✳ ▼✳✱ ▲❡✇✐s✱ ❆✳ ❙✳ ✭✶✾✾✷✮✿ ▼❛t❤✳ Pr♦❣r❛♠✐♥❣✳ ✺✼✱✶✺✲✹✽✳ ❬✸❪ ❇♦r✇❡✐♥✱ ❏✳ ▼✳✱ ●♦❡❜❡❧✱ ❘✳✭✷✵✵✸✮✿ ◆♦t✐♦♥s ♦❢ r❡❧❛t✐✈❡ ✐♥t❡r✐♦r ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❏✳ ▼❛t❤✳ ❙❝✐✳ ✶✶✺✱ ✷✺✹✷✲✷✺✺✸✳ ❘❡✈✐s✐♥❣ s♦♠❡ ❞✉❛❧✐t② t❤❡♦r❡♠s ✈✐❛ t❤❡ q✉❛s✐r❡❧❛t✐✈❡ ✐♥t❡r✐♦r ✐♥ ❝♦♥✈❡① ♦♣t✐♠✐③❛✲ t✐♦♥✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✸✾✱ ✻✼✲✽✹✳ ❬✺❪ ❈❛♠♠❛r♦t♦✱ ❋✳✱ ❉✐ ❇❡❧❧❛✱ ❇✳ ✭✷✵✵✺✮✿ ❙❡♣❛r❛t✐♦♥ t❤❡♦r❡♠ ❜❛s❡❞ ♦♥ t❤❡ q✉❛s✐r❡❧❛t✐✈❡ ✐♥t❡r✐♦r ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❞✉❛❧✐t② t❤❡♦r②✱ ❏✳ ❬✹❪ ❇♦☛t✱ ❘✳■✳✱ ❈s❡t♥❡❦✱ ❊✳❘✳✱ ▼♦❧❞♦✈❛♥✱ ❆✳ ✭✷✵✵✽✮✿ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✷✷✺✱ ✷✷✸✲✷✷✾✳ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❡①t❡♥❞❡❞ ❑② ❋❛♥ ✐♥✲ ❡q✉❛❧✐t② ✇✐t❤ ❝♦♥❡ ❛♥❞ ❛❢❢✐♥❡ ❝♦♥str❛✐♥ts ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✱ ❏✳ ❬✻❪ ❈❛♣➠t➠✱ ❆✳✭✷✵✶✷✮✿ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✺✷✱ ✻✻✶✲✻✼✹✳ ❬✼❪ ❏❛❤♥✱ ❏✳ ✭✶✾✾✻✮✿ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ◆♦♥❧✐♥❡❛r ❖♣t✐✲ ♠✐③❛t✐♦♥✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ❬✽❪ ●♦♥❣✱ ❳✳❍✳ ✭✷✵✵✽✮✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ✈❡❝t♦r ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✹✷✱ ✶✹✺✺✲✶✹✻✻✳ ✸✵ ❬✾❪ ❍♦❧♠❡s✱ ❘✳❇✳ ✭✶✾✼✺✮✿ ●❡♦♠❡tr✐❝ ❋✉♥t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣✲ ♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✳ ❬✶✵❪ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳ ❚✳ ✭✶✾✼✵✮✿ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✳ ❬✶✶❪ ❚❛♥❛❦❛✱ ❚✳✱ ❑✉r♦✐✇❛✱ ❉✳ ✭✶✾✾✸✮✿ ❚❤❡ ❝♦♥✈❡①✐t② ♦❢ A ❛♥❞ B ❛ss✉r❡s t❤❛t intA + B = int(A + B)✳ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳ ✻✱ ✽✸✲✽✻✳ ✸✶ ❳⑩❈ ◆❍❾◆ ❈❍➓◆❍ ❙Û❆ ▲❯❾◆ ❱❿◆ ❳→❝ ♥❤➟♥ ✤➣ ❝❤➾♥❤ sû❛ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ◆❣✉②➵♥ ❚❤à ❍↔✐ ❆♥❤✳ ❚➯♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✣à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐ ✈➔ →♣ ❞ư♥❣ ❝❤♦ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷ ❇↔♦ ✈➺ ♥❣➔② ✷✷✳✵✻✳✷✵✶✹ ✣➣ ❝❤➾♥❤ sỷ t ữ t ỗ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ữợ P ộ ữ ✸✷