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◆
ữợ ồ
P ❱❿◆ ▲×❯

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✹

▼ư❝ ❧ư❝
▼ð ✤➛✉

✶

✶ ✣à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ tr tỹ tữỡ ố
ử ỵ tt ✤è✐ ♥❣➝✉
✸
✶✳✶

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♣ ❞ö♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✸

⑩♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ②➳✉ ✳ ✳

✷✺

❑➳t ❧✉➟♥

✷✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✵

✐

r t ỗ ỹ ♥❤÷ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❣✐↔✐
t➼❝❤ ♣❤✐ t✉②➳♥✱ tè✐ ÷✉ ❤â❛ t t ỗ õ ởt tr
rt q trồ r t ỗ õ ✤à♥❤ ❧➼ t→❝❤ ❝❤➼♥❤✱ ✭①❡♠ ❬✶❪✮✳
❚r♦♥❣ ✤à♥❤ ❧➼ t→❝❤ t❤ù t t sỷ ử ởt tr t
ỗ ♣❤↔✐ ❝â ♣❤➛♥ tr♦♥❣ ❦❤→❝ ré♥❣✳ ❈➙✉ ❤ä✐ ✤÷đ❝ ✤➦t r
tr ừ t ỗ rộ t õ t t ữủ
t ỗ ❦❤ỉ♥❣ t÷ì♥❣ ❣✐❛♦ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ ❤❛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✉↔ ợ
tố ữ ố ✤➲ t➔✐ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠
♥❣❤✐➯♥ ❝ù✉✳ ❈❤➼♥❤ ✈➻ t❤➳ tæ✐ ❝❤å♥ ✤➲ t➔✐✿ ✧ ✣à♥❤ ❧➼ t→❝❤ ợ
tr tỹ tữỡ ố ❞ư♥❣ ❝❤♦ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✈➔ ✤è✐
♥❣➝✉✧✳
▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✤à♥❤ ❧➼ t→❝❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ♣❤➛♥ tr♦♥❣ tỹ
tữỡ ố ử tr ỵ tt tố ữ ố
ỗ ♠ð ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣✱ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝
❝→❝ t➔✐ t
ữỡ t ợ tr tỹ tữỡ ố
ử ỵ t❤✉②➳t ✤è✐ ♥❣➝✉
❚r➻♥❤ ❜➔② ❝→❝ ❦➳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✐ ✈➔
→♣ ❞ư♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t tố ữ õ r ở 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♥❣ ✣↕✐ ❤å❝ ồ
ồ ữợ sỹ ữợ t t➻♥❤ ❝õ❛ P●❙✳ ❚❙ ✣é ❱➠♥
▲÷✉✳ ❊♠ 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
ở ú ù tr q tr➻♥❤ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳
❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤
❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt rt ữủ sỹ õ ỵ ừ ❝→❝ t❤➛②
❝ỉ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✸ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✹
❚→❝ ❣✐↔

◆❣✉②➵♥ ❚❤à ❍↔✐ ❆♥❤

✷


ữỡ

t ợ
tr tỹ tữỡ ố ử
ỵ tt ố
ữỡ tr➻♥❤ ❜➔② ❝→❝ ❦➳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♦→♥ tè✐ ữ õ r ở
tr trữớ ủ ỗ ổ ✈ỵ✐ ♠ët ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② t❤❛②
t❤➳ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r t❤ỉ♥❣ t❤÷í♥❣✳

✶✳✶ P❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐
❚r♦♥❣ ❝→❝ ❜➔✐ t tố ữ ỗ ổ õ t 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 ✤➛✉ ✈ỵ✐ ❝→❝ 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 ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ 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 tt ❝↔ ❝→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ X ✳
P❤➛♥ tû ❦❤ỉ♥❣ ❝õ❛ X ∗ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ θX ∗ ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❬✷❪

●✐↔ sû C ❧➔ ♠ët t ỗ ừ X P tr tỹ tữỡ ✤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 ❧➔ ♠ët t➟♣ ỗ ừ 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 ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ t❤➻ t❛ ❝â

K o ❝ơ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✳ ◆➳✉ 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♥❤ ❧➼ ✶✳✶✳✷✳ ❬✺❪

●✐↔ sû C ✈➔ D ❧➔ ✷ t ỗ ừ 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❡♠



sỷ C t ỗ ừ 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 ở ữủ
õ ỗ C ợ cl(C C) = X ✈➔ X ∗ ✤÷đ❝ ①→❝ ✤à♥❤ t❤ù tü
❜ë ♣❤➟♥ ❜ð✐ C ∗✳ ❑❤✐ ✤â✱ x¯ ∈ qriC ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ φ(¯x) > 0 ✈ỵ✐ ♠å✐
φ ∈ X ∗ \ {0}
ự
sỷ x
qriC ữ ợ φ ∈ C ∗ ✱ φ = 0 t❛ ❝â✿ φ(¯
x) = 0✳
❑❤✐ ✤â✱

−φ(x − x¯) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C.
✼

❱➟②✱ −φ ∈ NC (¯
x)✳
❚ø ✤à♥❤ ❧➼ ✶✳✶✳✶✱ t❛ ❝â φ ∈ NC (¯
x)✱ ♥➯♥

φ(x − x¯) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C,
❤❛② ❧➔

φ(x) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C.
◆❤÷♥❣ ✈➻ φ ∈ C ∗ ✱ ♥➯♥ φ(x) = 0✱ ✈ỵ✐ ♠å✐ x ∈ C ✳
❱➻ ✈➟②✱ φ(x) = 0✱ ✈ỵ✐ ♠å✐ x ∈ cl(C − C) = X ✳ ❉♦ ✤â✱ φ = 0✳ ✣✐➲✉
♥➔② ♠➙✉ t❤✉➝♥ ợ = 0
ữủ x

/ qriC ứ ✤à♥❤ ❧➼ ✶✳✶✳✶✱ ✈ỵ✐ φ ♥➔♦ ✤â t❤✉ë❝ X ∗ ✱
t❛ ❝â✿

−φ(x − x¯) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C,
✈ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤➦t ❝❤♦ x
¯ ♥➔♦ ✤â t❤✉ë❝ C ✳ ❚ø ✤â s✉② r❛ φ = 0 ✈➔

φ ≥ 0✳ ◆❤÷ ✈➟②✱ ♥➳✉ φ(¯
x) > 0 t❛ ❝â✿
1
−φ( x x) > 0.
2
ổ ỵ φ(¯
x) = 0✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✣à♥❤ ❧➼ ✶✳✶✳✸ ❧➔ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ✈ỵ✐ ♠ët ✤✐➸♠ t❤✉ë❝ tr
tỹ tữỡ ố ừ ởt õ ỗ õ s ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝

❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ữỡ t t

t ợ ♣❤➛♥ tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② ✤à♥❤ ❧➼ t→❝❤ ❧✐➯♥ q✉❛♥ ✤➳♥
♣❤➛♥ tr♦♥❣ tü❛ tữỡ ố ừ ởt t ỗ



sỷ S T ỳ t ỗ ổ rộ ừ X ✈ỵ✐ qriS = ∅✱
qriT = ∅ t❤ä❛ ♠➣♥ cl[cone(qriS − qriT )] ❦❤æ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥
t✉②➳♥ t➼♥❤ ừ X õ tỗ t X ∗ \ {θX } s❛♦ ❝❤♦
Φ(s) ≤ Φ(t), ✈ỵ✐ ♠å✐ s ∈ S, t ∈ T.
∗

✽


ự
t t ỗ C C = qriS qriT ✳ ❚❤❡♦ ❝→❝ ♣❤➛♥ ✭❛✮ ✈➔ ✭❢✮ ❝õ❛ ✤à♥❤
❧➼ ✶✳✶✳✷✱ t❛ ❝â✿

qriS − qriT = qri(qriS) − qri(qriT )
⊆ qri(qriS − qriT )
⊆ qriS − qriT.
❉♦ ✤â✱

qri(qriS − qriT ) = .
rữợ t X C t X ∈ C \ qriC ✈➻ cl(cone(qriS − qriT ))
❦❤æ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ✳ ❱➟②✱ NC (θX ) ❦❤æ♥❣
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ X ∗ ✱ ♥❣❤➽❛ ❧➔ NC (θX ) = {θX ∗ }✳ ❑❤✐

✤â✱ ∃Φ ∈ NC (θX ) s❛♦ ❝❤♦ Φ = θX ∗ ✳ ❱➻ ✈➟②✱

Φ(x) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ C.
◆➳✉ θX ∈ X \ C ✱ t❛ ❝❤å♥

A = conv[qriC ∪ {θX }].
❚❛ ❝â✿

cl(coneA) = cl(cone(qriC)).
❙✉② r❛✱ θX ∈ A \ qriA✳
❚ø tr÷í♥❣ ❤đ♣ tr➯♥✱ t❛ ❝â ∃Φ ∈ X ∗ \ {θX ∗ } s❛♦ ❝❤♦

Φ(x) ≤ 0, ✈ỵ✐ ♠å✐ x ∈ A.
❉♦ ✤â✱

Φ(x) ≤ 0, ợ ồ x C.
ợ trữớ ủ tø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ s✉② r❛

sup Φ ≤ inf Φ.
qriT

qriS

❇➙② ❣✐í✱ tø ✤à♥❤ ❧➼ ✶✳✶✳✷ ✭❛✮ t❛ ❝â✿

qriS ⊆ S ⊆ clS ⊆ cl(qriS),
✾

✭✶✳✶✮

qriT ⊆ T ⊆ clT ⊆ cl(qriT ).
❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝✱ t❛ ❝â✿

sup Φ = sup Φ ✈➔ inf Φ = inf Φ.
qriS

qriT

S

T

❱➻ ✈➟②✱ tø ✭✶✳✶✮ t❛ ❝â✿

sup Φ ≤ inf Φ.
T

S

❉♦ ✤â✱ Φ ❧➔ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ t→❝❤ S ✈➔ T ✳

❍➺ q✉↔ ✶✳✷✳✶✳

●✐↔ sỷ M ởt t ỗ ổ rộ ừ X ✈➔ x0 ∈ X t❤ä❛ ♠➣♥
qriM = ∅ ✈➔ x0
/ qriM.
õ tỗ t x X \ {θX } s❛♦ ❝❤♦
x∗ (x) ≥ x∗ (x0 ), ✈ỵ✐ ♠å✐ x ∈ M.
∗

◆❤➟♥ ①➨t ✶✳✷✳✶✳

❙ü ❦✐➺♥ s❛✉ ♥â✐ ổ ú tỗ t X \ {θX ∗ } t→❝❤

S ✈➔ T t❤➻ cl[cone(qriS − qriT )] ❦❤æ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥
t➼♥❤ ❝õ❛ X ✳
✣➸ t❤➜② ✤✐➲✉ ♥➔②✱ ❝❤ó♥❣ t❛ ①➨t ♠ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥✳ ❈❤♦

X = R2 , S = {(x, y) ∈ R : 2x + 3y ≥ 0}, T = {(0, 0)}.
ó r S T t ỗ ✈➔ qriT = {(0, 0)}✳ ❍ì♥ ♥ú❛✱ ❤➔♠ t✉②➳♥
t➼♥❤ ❧✐➯♥ tö❝

Φ(x, y) = 2x + 3y, ∀(x, y) ∈ R2 ,
t→❝❤ S ✈➔ T ✱ ♥❤÷♥❣ cl[cone(qriS − qriT )] = S ❦❤æ♥❣ ❧➔ ❦❤æ♥❣ ❣✐❛♥
❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ R2 ✳

✶✳✸ ⑩♣ ❞ö♥❣ ❝❤♦ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤ì♥ ♠ư❝ t✐➯✉
❇➙② ❣✐í✱ ❝❤ó♥❣ t❛ →♣ ❞ư♥❣ ✤à♥❤ ❧➼ ✶✳✷✳✶ ❝❤♦ ♠ët ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â
r➔♥❣ ❜✉ë❝✳
✶✵


●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ t✉②➳♥ t➼♥❤ t❤ü❝ ✈➔ S ❧➔ t➟♣ ❝♦♥ ❦❤æ♥❣
ré♥❣ ❝õ❛ X ❀ (Y, . ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤ü❝ ❝â t❤ù tỹ ở
ữủ ởt õ ỗ C ❀ f : S → R ✈➔ g : S → Y ❧➔ ❤❛✐
❤➔♠ t❤ä❛ ♠➣♥ (f, g) : S → R × Y ❧➔ ❝♦♥✈❡①❧✐❦❡ t❤❡♦ ♥â♥ R+ × C ❝õ❛

R × Y ✭♥❣❤➽❛ ❧➔✱ t➟♣ (f (S) + [0, +)) ì (g(S) + C) ỗ t r➔♥❣
❜✉ë❝ T = {x ∈ S : g(x) ∈ −C} ❧➔ ❦❤ỉ♥❣ ré♥❣✳

❳➨t ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â r➔♥❣ ❜✉ë❝ s❛✉✿

min f (x),

✭P ✮

max∗ inf [f (x) + u(g(x))],

✭P ∗ ✮

x∈T

u∈C x∈S

tr♦♥❣ ✤â C ∗ ❧➔ ♥â♥ ✤è✐ ♥❣➝✉ ❝õ❛ C ✳
❚❤❡♦ ❬✼❪ ♥➳✉ C ❝â ♣❤➛♥ tr♦♥❣ ❦❤æ♥❣ ré♥❣ ✭intC = ∅✮✱ ❜➔✐ t♦→♥ ✭P ✮
❧➔ ❣✐↔✐ ✤÷đ❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r s✉② rë♥❣ t❤ä❛ ♠➣♥❀ tù❝ ❧➔ tỗ t
ởt tỡ x
S ợ g(
x) −intC t❤➻ ❜➔✐ t♦→♥ ✭P ∗ ✮ ❧➔ ❣✐↔✐ ✤÷đ❝ ✈➔
❣✐→ trà ❤➔♠ ♠ö❝ t✐➯✉ ❝õ❛ ✷ ❜➔✐ t♦→♥ ❧➔ ❜➡♥❣ ♥❤❛✉✳
❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r ❝â t❤➸ ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ❱➻ t❤➳✱ ❝❤ó♥❣
t❛ sû ❞ư♥❣ ✤à♥❤ ❧➼ ✶✳✷✳✶ ✤➸ ①➙② ❞ü♥❣ ✤à♥❤ ❧➼ ✤è✐ ♥❣➝✉ ❝❤♦ tr÷í♥❣ ❤đ♣

intC ❝â t❤➸ ❜➡♥❣ ∅ ✈➔ t❤❛② ✈➔♦ ✤â t❛ sû ❞ö♥❣ qriC ✳

✣à♥❤ ❧➼ ✶✳✸✳✶✳

●✐↔ sû
• X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ t✉②➳♥ t➼♥❤ t❤ü❝ ✈➔ S ❧➔ t➟♣ ❝♦♥ ❦❤æ♥❣ ré♥❣

❝õ❛ X ❀
• ✭Y, . ✮ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ tỹ ợ tự tỹ ở ữủ
ởt õ ỗ C
ã f : S R g : S → Y ❧➔ ✷ ❤➔♠ s❛♦ ❝❤♦ ❤➔♠ (f, g) : S → R × Y
❧➔ ❝♦♥✈❡①❧✐❦❡ t õ R+ ì C ừ R ì Y
ã qri[g(S) + C] = ∅❀
• cl{cone[qri(g(S) + C)]} ❦❤ỉ♥❣ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤
❝õ❛ Y ❀
✶✶


t➟♣ T = {x ∈ S : g(x) ∈ −C} ❦❤æ♥❣ ré♥❣✳
▼➦t ❦❤→❝✱ ❣✐↔ sû r➡♥❣ qriC = ∅ ✈➔ cl(C − C) = Y ✳
◆➳✉ ❜➔✐ t♦→♥ ✭P ✮ ữủ tỗ t x S ợ g(¯x) ∈ −qriC t❤➻
❜➔✐ t♦→♥ ✭P ∗✮ ❝ơ♥❣ ❣✐↔✐ ✤÷đ❝ ✈➔ ❣✐→ trà ❤➔♠ ♠ö❝ t✐➯✉ ❝õ❛ ✷ ❜➔✐ t♦→♥
❧➔ ❜➡♥❣ ♥❤❛✉✳
❈❤ù♥❣ ♠✐♥❤✳
•

❚❤❡♦ ❣✐↔ t❤✐➳t✱ t➟♣

(f (S) + [0, +∞)) ì (g(S) + C) := A ì B,
ỗ ỡ ♥ú❛✱ t❤❡♦ ♣❤➛♥ ✭❝✮ ❝õ❛ ✤à♥❤ ❧➼ ✶✳✶✳✷✱

qri(A × B) = qriA × qriB = intA × qri[g(S) + C].
❉♦ intA = ∅✱ t❛ ❝â qri(A × B) = ∅✳
❇ð✐ ✈➻ ✭P ✮ ❣✐↔✐ ✤÷đ❝✱ ∃x0 ∈ T s❛♦ ❝❤♦

f (x0 ) ≤ f (x), ∀x ∈ T.
❚❛ ❝â cl{cone[qri(A × B) − (f (x0 ), θY )]} ❦❤æ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥

❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ R × Y ✳ ⑩♣ ❞ư♥❣ ✤à♥❤ ❧➼ ✶✳✷✳✶✱ ∃µ ✈➔ ∃u ∈ Y ợ

(à, u) = (0, Y )
àf (x0 ) ≤ µ(f (x1 ) + α) + u(g(x2 ) + y),

✭✶✳✷✮

✈ỵ✐ ♠å✐ x1 , x2 ∈ S ✱ α ≥ 0✱ y ∈ C ✳
❱ỵ✐ x1 = x2 = x0 ✈➔ α = 0✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮✱ t❛ ❝â✿

u(y) ≥ −u(g(x0 )), ∀y ∈ C.

✭✶✳✸✮

●✐↔ sû r➡♥❣✱ tỗ t y C u(y) < 0 ✤â✱ ✈ỵ✐ λy ∈ C ✭tr♦♥❣ ✤â

λ > 0 ✤õ ❧ỵ♥✮✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ ❦❤ỉ♥❣ ✤ó♥❣ ♥ú❛✳ ❚ø ✤â✱ t❛ ❝â✿
u(y) ≥ 0, ∀y ∈ C.
❙✉② r❛✱ u ∈ C ∗ ✳
❱ỵ✐ y = θY ✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✸✮✱ t❛ ❝â✿

u(g(x0 )) ≥ 0.
✶✷


❱➻ g(x0 ) ∈ −C ✈➔ u ∈ C ∗ ✱ t❛ ❝â✿

u(g(x0 )) ≤ 0.
❱➟②✱ u(g(x0 )) = 0✳
❈❤å♥ x1 = x2 = x0 ✈➔ y = θY ✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮✱ ❝❤ó♥❣ t❛ ❝â

µα ≥ 0✱ ợ ồ 0 õ à 0
sû µ = 0✳ ❑❤✐ ✤â✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮ ✈ỵ✐ y = θY ✱ t❛ ❝â✿
u(g(x)) ≥ 0, ∀x ∈ S.

✭✶✳✹✮

❚❤❡♦ ❣✐↔ t❤✐➳t✱ ∃¯
x ∈ S s❛♦ ❝❤♦ −g(¯
x) ∈ qriC ✳ ❑➳t ❤đ♣ ✈ỵ✐ ❜➜t ✤➥♥❣
t❤ù❝ ✭✶✳✹✮✱ t❛ ❝â✿

u(g(¯
x)) = 0.
❉♦ ✤â✱

u(y + g(¯
x)) ≥ 0, ∀y ∈ C.
❑❤✐ ✤â✱

−u ∈ NC (−g(¯
x)).
❚ø ✤â✱ t❛ ❝â u ∈ NC (−g(¯
x))✳ ❱➻ ✈➟②✱

u(y) ≤ 0, ∀y ∈ C.
❙✉② r❛✱ u(C) = {0} ✈➔ ❞♦ ✤â✱

u(cl(C − C)) = {0},
❝â ♥❣❤➽❛ u = θY ∗ ✳ ❱➟②✱ ♥➳✉ µ = 0 ú t ởt t

ợ (à, u) = (0, θY )✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ❝â µ > 0 ✈➔ tø ❜➜t ✤➥♥❣ t❤ù❝
✭✶✳✷✮✱ ✈ỵ✐ α = 0 ✈➔ y = θY ✱ t❛ ❝â✿

f (x0 ) ≤ f (x) + (1/µ)u(g(x)), ∀x ∈ S.
◆➳✉ t❛

u :=

1
u C ,
à

ợ u
(g(x0 )) = 0 t❛ ♥❤➟♥ ✤÷đ❝

inf [f (x) + u¯(g(x))] ≥ f (x0 ) + u¯(g(x0 )).

x∈S

✶✸


◆❤÷♥❣ t❛ ❝â✿

inf [f (x) + u(g(x))] ≤ f (x0 ) + u(g(x0 )) = f (x0 ), ∀u ∈ C ∗ .

x∈S

❑❤✐ ✤â✱

sup inf [f (x) + u(g(x))] ≤ f (x0 ) = f (x0 ) + u¯(g(x0 )).

u∈C ∗ x∈S

❙✉② r❛✱

f (x0 ) = max∗ inf [f (x) + u(g(x))].
u∈C x∈S

❱➟②✱ u
¯ ∈ C ∗ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (P ∗ ) ✈➔ ❣✐→ trà
❤➔♠ ♠ö❝ t✐➯✉ ❝õ❛ ✷ ❜➔✐ t♦→♥ ❧➔ ❜➡♥❣ ♥❤❛✉✳

✶✹


❈❤÷ì♥❣ ✷

❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❜➜t ✤➥♥❣
t❤ù❝ ❑② rở ợ r
ở õ
r ữỡ ú tæ✐ tr➻♥❤ ❜➔② ♠ët ❞↕♥❣ ♠ð rë♥❣ ❝õ❛ ❜➜t
✤➥♥❣ t❤ù❝ t ỗ t tè✐ ÷✉ ✈❡❝tì✱
❝→❝ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈❡❝tì✱ ❝→❝ ❜➔✐ t♦→♥ ❜ị ✈❡❝tì ✈➔
❝→❝ ❜➔✐ t♦→♥ ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥â♥ ♥❤÷ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✳ ◆❤✐➲✉
❦➳t q✉↔ ✤÷đ❝ t❤✐➳t ❧➟♣ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ sỹ tỗ t
t t t ở ♥❤↕② ✈➔ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉✳
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉
❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣ ✭❊❑❋✮ ✈ỵ✐ r➔♥❣ ❜✉ë❝
♥â♥ ✈➔ ❛❢❢✐♥❡ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✤à♥❤ ❧➼ t t ỗ ỹ tr

tr tỹ tữỡ ố ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ✶✳ ❈→❝ ❦➳t q✉↔ ✤÷đ❝
→♣ ử t tố ữ tỡ ợ r ở õ ỵ
tt ố ừ t tè✐ ÷✉ ✈❡❝tì ②➳✉✳ ❈→❝ ❦➳t q✉↔ ✤÷đ❝ tr➻♥❤
❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧➔ ❝õ❛ ❈❛♣➠t➠ ❬✻❪✳

✷✳✶ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉
●✐↔ sû X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝❀ Y, Z W
ổ ỗ ữỡ t ❍ì♥ ♥ú❛✱ ❣✐↔ sû S ⊆ X ❧➔ ♠ët t➟♣ ỗ
ổ rộ f : S ì S Y ✱ g : S → Z ✈➔ h : S → W ❧➔ ❝→❝ ❤➔♠✳
✶✺


❑❤ỉ♥❣ ❣✐❛♥ Y ✤÷đ❝ tr❛♥❣ ❜à t❤ù tü ❜ë ♣❤➟♥ õ C ồ ỗ
õ intC = Z ữủ tr tự tỹ õ ỗ K Y ∗ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥
✤è✐ ♥❣➝✉ ❝õ❛ Y ✳
❈❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥ ♠ð rë♥❣✿
❚➻♠ a
¯ ∈ A s❛♦ ❝❤♦ f (¯
a, b) ∈
/ −intC, ✈ỵ✐ ♠å✐ b ∈ A,

✭❊❑❋✮

tr♦♥❣ ✤â A := {x ∈ S | g(x) ∈ −K ✈➔ h(x) = 0} ❧➔ t➟♣ ❝→❝ ♥❣❤✐➺♠
❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭❊❑❋✮✳
▼ư❝ t✐➯✉ ❝õ❛ ♣❤➛♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ✈➔ ✤õ
✤➸ ♠ët ✤✐➸♠ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❊❑❋✮✳ ❈❤ó♥❣ t❛ s➩ ❞ü❛
tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➛♥ tr♦♥❣ tỹ tữỡ ố ừ õ ỗ
t ợ tr tỹ tữỡ ố ổ rộ ❝õ❛
❈❛♠♠❛r♦t♦ ✈➔ ❉✐ ❇❡❧❧❛ ❬✺❪ ✤➣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ✶ ✤➸ ❝❤ù♥❣

♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤✳
●✐↔ sû M t ỗ ừ Y qriM tr♦♥❣ tü❛ t÷ì♥❣ ✤è✐
❝õ❛ M ✳ ◆â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ t ỗ M ừ Y t y0 Y ✤÷đ❝ ①→❝
✤à♥❤ ♥❤÷ s❛✉

NM (y0 ) := {y ∗ ∈ Y ∗ | y ∗ (y − y0 ) ≤ 0, ∀y ∈ M }.
◆❤➢❝ ❧↕✐✿ ψ : A → Y ✤÷đ❝ ❣å✐ ❧➔ C ✲ ❤➔♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉✱ ✈ỵ✐ ♠å✐

a1 ✱a2 ∈ A ✈➔ ♠å✐ t ∈ [0, 1]✱
ψ(ta1 + (1 − t)a2 ) ≤C tψ(a1 ) + (1 − t)ψ(a2 ).
❑❤✐ C ❝❤ù❛ ✭❤♦➦❝ ❜➡♥❣✱ ❤♦➦❝ ✤÷đ❝ ❝❤ù❛ tr♦♥❣✮ ♦rt❤❛♥t ❦❤ỉ♥❣ ➙♠✱ t❤➻
✧C ✲ ❤➔♠✧ ✤÷đ❝ ồ C ỗ ỗ C ỗ t

C ự ữủ ự tr ♦rt❤❛♥t ❦❤ỉ♥❣ ❞÷ì♥❣✱ t❤➻
✧C ✲ ❤➔♠✧ ✤÷đ❝ ❣å✐ ❧➔ ✧C ✲ ❧ã♠✧ ✭❤♦➦❝ ✧❧ã♠✧✱ ❤♦➦❝ ✧C ✲ ❧ã♠ ❝❤➦t✧✮✳
❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ✤÷❛ ✈➔♦ ❣✐↔ t❤✐➳t s❛✉✿



f (x, x) = 0, ✈ỵ✐ ♠å✐ x ∈ S;



❢ ❧➔ ♠ët ❈ ✲ ❤➔♠ t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐;
●✐↔ t❤✐➳t ✭A✮ :

❣ ❧➔ ♠ët ✲ ❑ ✲ ❤➔♠;





❤ ❧➔ ❛❢❢✐♥❡,
✶✻


✈➔ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉②✿
❱ỵ✐ ♠å✐ (z ∗ , w ∗ ) ∈ K ∗ × W ∗ \ {(0, 0)}, tỗ t x S s

z (g(x)) + w∗ (h(x)) < 0.

✭❘❈✮

✣à♥❤ ❧➼ ✷✳✶✳✶✳

●✐↔ sû qri((g, h)(S) + K × {0}) = ∅✳ ✣✐➸♠ a ∈ A ởt
ừ tỗ t ✭y∗✱ z ∗✱ w∗✮ ∈ C ∗ \ {0} × K ∗ × W ∗
s❛♦ ❝❤♦
z ∗ (g(a)) = 0,

✈➔
0 = y ∗ (f (a, a)) + z ∗ (g(a)) + w∗ (h(a))
= min{y ∗ (f (a, x)) + z ∗ (g(x)) + w∗ (h(x))}.
x∈S

❈❤ù♥❣ ♠✐♥❤✳
✭=⇒✮ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ❝→❝❤ →♣ ❞ư♥❣ ❤➺ q✉↔ ✶✳✷✳✶
✈ỵ✐ t➟♣ M ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
●✐↔ sû a ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮ ✈➔

M := {(y, z, w) ∈ Y × Z × W : ∃x ∈ S s❛♦ ❝❤♦

y ∈ f (a, x) + intC, z ∈ g(x) + K ✈➔ w = h(x)}.
❱ỵ✐ ♠é✐ x ∈ S ✱ ❝❤å♥ y := f (a, x) + c✱ z := g(x) + k ✈➔ w := h(x)✱
tr♦♥❣ ✤â c ∈ intC ✈➔ k ∈ K ✱ t❛ ❝â✿

M = ∅,
✈➔

(z, w) ∈ (g, h)(S) + K × {0}.
✣➸ r t ỗ ừ t M sỷ (y1 , z1 , w1 )✱ (y2 , z2 , w2 ) ∈ M ✈➔

t ∈ [0, 1]✳ ❱➟②✱ t❛ ❝â x1 , x2 ∈ S s❛♦ ❝❤♦
y1 ∈ f (a, x1 ) + intC, z1 ∈ g(x1 ) + K ✈➔ w1 = h(x1 ),
✈➔

y2 ∈ f (a, x2 ) + intC, z2 ∈ g(x2 ) + K ✈➔ w2 = h(x2 ).
✶✼


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 ♠ët ✲ 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 ì {0}) = tỗ t

(z0 , w0 ) ∈ qri((g, h)(S) + K × {0})
⊆ (g, h)(S) + K ì {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 ♠ët
❧➙♥ ❝➟♥ ❝➙♥ 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 ✳ ❑❤✐ ✤â✱ tỗ t V ừ tr

Y s ❝❤♦
c0 + V ⊆ intC ∩ U.
❍ì♥ ♥ú❛✱ ❣✐↔ sû e Y õ tỗ t t > 0 s❛♦ ❝❤♦

c0 − te ∈ intC ∩ U,
✈➔

c0 + te ∈ intC ∩ U.
❚ø ✭✷✳✺✮ ✈➔ ✭✷✳✻✮✱ t❛ ❝â✿

y ∗ (c0 − te) ≤ 0 ✈➔ y ∗ (c0 + te) ≤ 0,
✈➔

y ∗ (c0 − te) ≥ 0 ✈➔ y ∗ (c0 + te) ≥ 0.
❈→❝ 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 ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ 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) = 0.
✣✐➲✉ ♥➔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
ữủ y C
t ợ ♠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) 0 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 ✤✐➲✉ ❦✐➺♥ ✤➲✉ ợ (z , w ) tỗ t x0 ∈ S s❛♦ ❝❤♦

z ∗ (g(x0 )) + w∗ (h(x0 ))) < 0.

✭✷✳✶✵✮

▲➜② z0 := g(x0 ) ✈➔ w0 := h(x0 )✳
❑❤✐ ✤â✱ (z0 , w0 ) ∈ (g, h)(S)+K ì{0} t ợ
ợ ồ 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)) ≥ 0.
❈❤♦ t → ∞✱ t❛ ♥❤➟♥ ✤÷đ❝

z ∗ (g(a)) ≥ 0.
▼➦t ❦❤→❝✱ ✈➻ g(a) ∈ −K ✈➔ z ∗ ∈ K ∗ ✱ t❛ ❝â✿

z ∗ (g(a)) ≤ 0.

❱➟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)) ≥ 0
= y ∗ (f (a, a)) + z (g(a)) + w (h(a)),
ợ ồ 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))
= min y ∗ {(f (a, x)) + z ∗ (g(x)) + w∗ (h(x))}.
x∈S

✭⇐= ✮ ❈❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✿ ●✐↔ sû r➡♥❣ a ∈ A ❦❤æ♥❣ ♣❤↔✐
❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮✳ ❱➻ 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,
✈➔

0 = y ∗ (f (a, a)) + z ∗ (g(a)) + w∗ (h(a))
= min y ∗ {(f (a, x)) + z ∗ (g(x)) + w∗ (h(x))}
x∈S
∗

≤ y (f (a, b)) + z ∗ (g(b)) + w∗ (h(b)) < 0 ✭ ✈æ ❧➼✦✮.
✷✶


❉♦ ✤â✱ a ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭❊❑❋✮ ✈➔ ✤à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣
♠✐♥❤✳
❈❤♦ h ≡ 0✳ ❚ø ✤à♥❤ ❧➼ ✷✳✶✳✶ ð tr➯♥ t❛ s✉② r❛ ✤à♥❤ ❧➼ ✸✳✶ ❝õ❛ ●♦♥❣
❬✽❪✳

❍➺ q✉↔ ✷✳✶✳✶✳

●✐↔ sû ❣✐↔ t❤✐➳t ✭A✮ t❤ä❛ ♠➣♥ tỗ t x0 S s g(x0)
intK ✳ ❑❤✐ ✤â✱ x ∈ A ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ tỗ
t (y, z) C ∗ \ {0} × 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♥❤
❧➼ ✸✳✶ ❬✽❪ ❝❤➾ ①➨t r➔♥❣ ❜✉ë❝ õ tr tỗ t x0 S tọ
g(x0 ) ∈ −intK ✱ ✤÷đ❝ t❤❛② ❜➡♥❣ ✤✐➲✉ ❦✐➺♥ ỡ
rộ ừ õ ỗ K ❜↔♦ t➟♣ qri(g(S) + K) ❦❤ỉ♥❣ ré♥❣✱ ❝á♥
✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳
✷✷