Phương pháp chiếu giải bài toán cân bằng hai cấp

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❉÷ì♥❣ ❱➠♥ ❚❤✐

P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆
❈❹◆ ❇➀◆● ❍❆■ ❈❻P

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥✱ ♥➠♠ ✷✵✶✻


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❉÷ì♥❣ ❱➠♥ ❚❤✐

P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆
❈❹◆ ❇➀◆● ❍❆■ ❈❻P

❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ ❚➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

●❙✳❚❙❑❍ ◆●❯❨➍◆ ❳❯❹◆ ❚❻◆
❚❤→✐ ◆❣✉②➯♥✱ ♥➠♠ ✷✵✶✻

▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝✱ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣
❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

❉÷ì♥❣ ❱➠♥ ❚❤✐

✐


▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ tr♦♥❣ ❦❤â❛ ✷✷ ✤➔♦ t↕♦ ❚❤↕❝ s➽ ❝õ❛ tr÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ ●❙✳❚❙ ◆❣✉②➵♥
❳✉➙♥ ❚➜♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ t❤➛②
❤÷î♥❣ ❞➝♥✱ ♥❣÷í✐ ✤➣ t↕♦ ❝❤♦ tæ✐ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱
t✐♥❤ t❤➛♥ ❧➔♠ ✈✐➺❝ ♥❣❤✐➯♠ tó❝ ✈➔ ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥✱ ❝æ♥❣ sù❝ ❤÷î♥❣
❞➝♥ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❚æ✐ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ s➙✉ s➢❝ tî✐ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❝õ❛ tr÷í♥❣
✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱
❦❤➼❝❤ ❧➺✱ ✤ë♥❣ ✈✐➯♥ tæ✐ ✈÷ñt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥ tr♦♥❣ ❤å❝ t➟♣✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❧➣♥❤ ✤↕♦ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐✱ ❣✐ó♣
✤ï tæ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ tæ✐ ❤å❝ t➟♣✳
❈✉è✐ ❝ò♥❣✱ tæ✐ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ♥❣÷í✐ t❤➙♥ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱
õ♥❣ ❤ë tæ✐ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳

❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

❉÷ì♥❣ ❱➠♥ ❚❤✐
✐✐


▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ö❝ ❧ö❝

✐✐✐

▼ët sè ❦þ ❤✐➺✉ ✈✐➳t t➢t

✈

▼ð ✤➛✉

✶

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

✶✳✷

▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❧ç✐ ✳ ✳ ✳ ✳ ✳

✹

✶✳✶✳✶

❑❤→✐ ♥✐➺♠ ✈➲ t➟♣ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✶✳✷

✣↕♦ ❤➔♠ ✈➔ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➔ ❝→❝ tr÷í♥❣ ❤ñ♣ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✶

❇➔✐ t♦→♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✷✳✷

❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✷✳✸

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✷✳✹

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤ tr♦♥❣ trá ❝❤ì✐ ❦❤æ♥❣ ❤ñ♣ t→❝

✶✺

✐✐✐


✶✳✷✳✺

❙ü tç♥ t↕✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✸

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✶✳✹

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✶✳✹✳✶

❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✶✳✹✳✷

❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ♥❣❤✐➺♠
❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣
✷✳✶

✷✷

✷✹

❚❤✉➟t t♦→♥ ❝❤✐➳✉ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❣✐↔
✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✷

❚❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❣✐↔ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳

✸✶

✷✳✸

⑩♣ ❞ö♥❣ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✷✳✸✳✶

❚➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ ❝❤✉➞♥ ❊✉❝❧✐❞❡ tr➯♥ t➟♣ ♥❣❤✐➺♠
❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❣✐↔ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✸✳✷

✹✷

●✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr➯♥ t➟♣ ♥❣❤✐➺♠
❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸

❑➳t ❧✉➟♥

✻✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✼✵

✐✈


▼ët sè ❦þ ❤✐➺✉ ✈✐➳t t➢t
R

t➟♣ sè t❤ü❝✳

N

t➟♣ sè tü ♥❤✐➯♥✳

H

❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳

Rn

❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n ❝❤✐➲✉✳

x, y = xT y

x =

x, x

t➼❝❤ ✈æ ❤÷î♥❣ ❝õ❛ ❤❛✐ ✈➨❝tì x ✈➔ y ✳
❝❤✉➞♥ ❝õ❛ ✈➨❝tì x✳

domf

♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ❤➔♠ f ✳

imF

♠✐➲♥ ↔♥❤ ❝õ❛ →♥❤ ①↕ F ✳

epif

tr➯♥ ✤ç t❤à ❝õ❛ ❤➔♠ f ✳

ϕ (x) =

ϕ(x) ✤↕♦ ❤➔♠ ❝õ❛ ϕ t↕✐ x✳

ϕ (x; d)

✤↕♦ ❤➔♠ t❤❡♦ ❤÷î♥❣ d ❝õ❛ ϕ t↕✐ x✳

∂ϕ(x)

❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ϕ t↕✐ x✳

x f (x, y)

✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ f (., y) t↕✐ x✳

y f (x, y)

✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ f (x, .) t↕✐ y ✳

∂f (x, x)

❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ f (x, .) t↕✐ x✳

intC

♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ C ✳

riC

♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ t➟♣ C ✳

xk → x

❞➣② xk ❤ë✐ tö tî✐ x✳

PC (x)

❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ t➟♣ C ✳

✈

NC (x)

♥â♥ ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ C t↕✐ x✳

B[a, r]

q✉↔ ❝➛✉ ✤â♥❣ t➙♠ a ❜→♥ ❦➼♥❤ r✳

C

❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ C ✳

lim = lim inf

❣✐î✐ ❤↕♥ ❞÷î✐✳

lim = lim sup

❣✐î✐ ❤↕♥ tr➯♥✳

EP (C, f )

❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳

V IP (C, f )

❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✤ì♥ trà✮✳

Sf

t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ EP (C, f )✳

SF

t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (C, F )✳

BEP (C, f, g)

❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❤❛✐ ❝➜♣✳

M N EP (C, f )

❜➔✐ t♦→♥ t➻♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ ❝❤✉➞♥ tr➯♥ t➟♣ Sf ✳

V IEP (C, f, F ) ❜➔✐ t♦→♥ V IP (Sf , F )✳
BV IP (C, F, G) ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳

✈✐


▼ð ✤➛✉
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐
❇➔✐ t♦→♥ tè✐ ÷✉✿

min f (x),
x∈D

✭✶✮

✈î✐ D ⊂ Rn ❧➔ ❜➔✐ t♦→♥ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ù♥❣ ❞ö♥❣ t♦→♥
❤å❝ ✈➔♦ ❝✉ë❝ sè♥❣✳ ❑❤✐ f ❝â ✤↕♦ ❤➔♠ ✭✶✮ ❧✐➯♥ q✉❛♥ tî✐✿

f (x), x − x ≥ 0, ∀x ∈ D.

✭✷✮

◆➠♠ ✶✾✻✵ ❙t❛♠♣❛❝❝❤✐❛ ✤÷❛ r❛ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ ❈❤♦ F : D → Rn
❚➻♠ x ∈ D s❛♦ ❝❤♦ F (x), x − x ≥ 0, ∀x ∈ D.
❇➔✐ t♦→♥ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❈❤♦ D ❧➔ t➟♣
❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ✱ f : D × D → R ❧➔ s♦♥❣ ❤➔♠ ❝➙♥ ❜➡♥❣✳
❳➨t ❜➔✐ t♦→♥✿
❚➻♠ x ∈ D s❛♦ ❝❤♦ f (x, x) ≥ 0, ∀x ∈ D.
❇➔✐ t♦→♥ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳ ❈❤➼♥❤ ①→❝✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣
✤÷ñ❝ ✤÷❛ r❛ ❧➛♥ ✤➛✉ ❜ð✐ ❍✳ ◆✐❦❛✐❞♦ ✈➔ ❑✳ ■s♦❞❛ ♥➠♠ ✶✾✺✺ ❦❤✐ tê♥❣ q✉→t
❤â❛ ❜➔✐ t♦→♥ ❝➙♥ ◆❛s❤ tr♦♥❣ trá ❝❤ì✐ ❦❤æ♥❣ ❤ñ♣ t→❝ ✈➔ ✤÷ñ❝ ❑② ❋❛♥ ❣✐î✐
✶


t tữớ ữủ ồ t tự t
ợ t q tở ữ t tố ữ t
t tự t t ở t s
tr ỵ tt trỏ ỡ ổ ủ t t q t ữủ
t ữủ ử trỹ t t t ừ õ
ữợ ự t rt tr õ
ự ỹ ữỡ ữ t ồ qt
t r tr tỹ t
P trồ t ừ tr ởt ữỡ
t ỡ ử ợ t

trú ỗ ữỡ
ữỡ tự ỡ ừ t ỗ ữủ sỷ ử
tr ữỡ s t ợ t t t
tữỡ ữỡ t
ữỡ r tt t t t tự
ỡ t ỡ ử t


ử ự
ử ừ ỹ ữỡ t
ỡ ử ởt ợ t




✸✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ✈➔ ①➙② ❞ü♥❣ t❤✉➟t t♦→♥ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥✿ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣
❣✐↔ ✤ì♥ ✤✐➺✉✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❣✐↔ ✤ì♥ ✤✐➺✉ ✈➔ →♣ ❞ö♥❣ ❣✐↔✐
❜➔✐ t♦→♥ ❤❛✐ ❝➜♣✳

✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
❚❤❛♠ ❦❤↔♦ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➣ ❝æ♥❣ ❜è tr➯♥ ❝→❝ t↕♣ ❝❤➼ ✈➔ s→❝❤ ❣✐→♦
❦❤♦❛ ❝❤✉②➯♥ ❦❤↔♦✳ ❚ê♥❣ ❤ñ♣✱ ♣❤➙♥ t➼❝❤✱ ✤→♥❤ ❣✐→ ✈➔ sû ❞ö♥❣ ❝→❝ ❦➳t q✉↔
❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ tr♦♥❣ ❧✉➟♥ ✈➠♥✳

✺✳ ❉ü ❦✐➳♥ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉
❚r➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➔ ①➙② ❞ü♥❣ t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐
t♦→♥ ❝➙♥ ❜➡♥❣ ❣✐↔ ✤ì♥ ✤✐➺✉✳

✸

ữỡ

ởt số tự
ữỡ s tự t tr ởt số
t q tt t t ỗ tự ợ t
t trữớ ủ r ừ õ ũ ởt số
sỹ tỗ t ừ t P t t tr
t tữỡ ữỡ P ố ũ tr t
ởt số trữớ ủ r ừ t tự tr
ữỡ ữủ tứ t t

ởt số t q ỡ ừ t ỗ
t ỗ ỗ
ởt số ỡ ừ t ỗ

sỷ X ởt ổ tỡ tr R t C X
ữủ ồ
ỗ x, y C 0 1 x + (1 )y C



õ õ t > 0, x C x C
õ ỗ õ ứ õ õ t ứ ởt t ỗ

x, y C, , à > 0 x + ày C.
t ỗ õ ố ợ ởt số t ữ ở
ợ ởt số tỹ ự C D t ỗ tr X t

C D, C + D ụ t ỗ ợ ồ , R

sỷ C ởt t ỗ rộ tr ổ
rt tỹ H x0 C õ t

NC (x0 ) = { H : , x x0 0, x C},
ữủ ồ õ t ừ C t x0 t NC (x0 ) ữủ ồ
õ t tr ừ C t x0
0 NC (x0 ) tứ tr t t NC (x0 ) ởt õ
ỗ õ

sỷ C = ổ t tt ỗ ởt t ừ
ổ rt H y H ởt tỡ t ý ồ

dC (y) = inf x y
xC

tứ y C tỗ t PC (y) s

dC (y) = y PC (y) ,
t t õ PC (y) ừ y tr C



ứ tr t t PC (y) ừ y tr C ừ
t tố ữ

min
xC

1
xy
2

2

.

õ t ừ y tr C õ t ữ t
ỹ t ừ x y

2

tr C

ỵ C ởt t ỗ õ
rộ ừ ổ rt H õ
x H PC (x) ừ x tr C ổ tỗ t t
= PC (x) x , y 0, y C
x PC (x) õ t t
PC (x) PC (y) x y , x, y H (t ổ )
PC (x) PC (y)
x PC (x)

2

2

PC (x) PC (y), x y , x, y H (t ỗ ự)

x PC (x), x y , y C

sỷ X ổ tỡ tổổ ỗ ữỡ tỹ
C X ởt t ỗ f : C R {+} õ
f ữủ ồ ỗ tr C

f (x + (1 )y) f (x) + (1 )f (y), x, y C, [0; 1] ;
f ữủ ồ ỗ t tr C

f (x + (1 )y) < f (x) + (1 )f (y), x, y C, x = y, (0; 1) ;




✭❝✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ç✐ ♠↕♥❤ tr➯♥ C ✈î✐ ❤➺ sè δ > 0 ♥➳✉

f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) − λ(1 − λ)δ y − x 2 ,
∀x, y ∈ C, ∀λ ∈ [0; 1] ;
✭❞✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ ✭❧ã♠ ❝❤➦t✱ ❧ã♠ ♠↕♥❤✮ tr➯♥ C ♥➳✉ −f ❧➔
❧ç✐ ✭❧ç✐ ❝❤➦t✱ ❧ç✐ ♠↕♥❤✮ tr➯♥ C ❀
✭❡✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ tü❛ ❧ç✐ tr➯♥ C ♥➳✉ ∀λ ∈ R t➟♣ ♠ù❝

{x ∈ C : f (x) ≤ λ}
❧➔ t➟♣ ❧ç✐❀
✭❢✮ ❈→❝ t➟♣

domf = {x ∈ C : f (x) < +∞} ,
epif = {(x, t) ∈ C × R : f (x) ≤ t} ,
❧➛♥ ❧÷ñt ❧➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ✈➔ tr➯♥ ✤ç t❤à ❝õ❛ f ❀
✭❣✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ domf = ∅ ✈➔ f (x) > −∞ ✈î✐

♠å✐ x ∈ C ✳

✣à♥❤ ❧þ ✶✳✶✳✻✳ ✭①❡♠ ❬✶❪✱ ✣à♥❤ ❧þ ✷✳✸✮✳ ●✐↔ sû f : C −→ R ∪ {+∞} ❧➔ ❤➔♠
❧ç✐ ✈➔ α ∈ [−∞, +∞]✳ ❑❤✐ ✤â ❝→❝ t➟♣ ♠ù❝

L0α (f ) = {x ∈ X : f (x) < α} ;
Lα (f ) = {x ∈ X : f (x) ≤ α}
❧➔ ❝→❝ t➟♣ ❧ç✐✳
✼


sỷ f : H R õ
f ữủ ồ ỷ tử ữợ t x0 H

lim f (x) f (x0 ).
xx0

f ữủ ồ ỷ tử ữợ tr C õ ỷ tử
ữợ t ồ x C f ữủ ồ ỷ tử tr tr C f
ỷ tử ữợ tr C f ữủ ồ tử tr C õ ứ ỷ
tử ữợ ỷ tử tr tr C

ỵ ỵ sỷ f ỗ tữớ
tr H x0 H õ s tữỡ ữỡ
f tử t x0
f tr tr ởt ừ x0
int(epif ) = ;
int(domf ) = f tử tr int(domf ) ỗ tớ

int(epif ) = {(x, t) H ì R : x int(domf ), f (x) < t}.

ữợ ừ ỗ
sỷ f : H R, x H d H\{0} õ f
ữủ ồ
rt t x tỗ t tỡ x H s

lim

yx

f (y) f (x) x , y x
= 0,
yx




❦❤✐ ✤â x∗ ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ f t↕✐ x ✈➔ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔

f (x) ❤♦➦❝

f (x)❀
✭❜✮ ❈â ✤↕♦ ❤➔♠ t❤❡♦ ❤÷î♥❣ d t↕✐ x ♥➳✉ tç♥ t↕✐ ❣✐î✐ ❤↕♥

f (x; d) = lim+
t→0

f (x + td) − f (x)
.
t

❈â t❤➸ t❤➜② r➡♥❣ ♥➳✉ ❤➔♠ f ❦❤↔ ✈✐ t↕✐ x t❤➻ ♥â ❝â ✤↕♦ ❤➔♠ t❤❡♦ ♠å✐
❤÷î♥❣ t↕✐ x ✈➔ t❛ ❝â f (x; d) =

f (x), d , ∀d ∈ H.

✣à♥❤ ❧þ ✶✳✶✳✶✵✳ ✭①❡♠ ❬✷❪✱ ▼➺♥❤ ✤➲ ✶✶✳✻✮✳ ❈❤♦ f : Rn → R ∪ {+∞} ❦❤↔ ✈✐✱
C ⊂ Rn ❧➔ t➟♣ ❧ç✐ ✤â♥❣✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭❛✮ f ❧➔ ❤➔♠ δ ❧ç✐ ♠↕♥❤ tr➯♥ C ❀
✭❜✮ δ y − x

2

≤

✭❝✮ f (y) − f (x) ≥

f (y) − f (x), y − x , ∀x, y ∈ C;
f (x), y − x + δ y − x 2 , ∀x, y ∈ C.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✶✳ ●✐↔ sû f : H → R ∪ {+∞} ❧➔ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣
tr➯♥ H✱ ω ∈ H ✤÷ñ❝ ❣å✐ ❧➔ ❞÷î✐ ✤↕♦ ❤➔♠ ❝õ❛ f t↕✐ x ♥➳✉

f (y) ≥ ω, y − x + f (x), ∀y ∈ H.
❚➟♣ t➜t ❝↔ ❝→❝ ❞÷î✐ ✤↕♦ ❤➔♠ ❝õ❛ f t↕✐ x ✤÷ñ❝ ❣å✐ ❧➔ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛

f t↕✐ x✱ ❦➼ ❤✐➺✉ ❧➔ ∂f (x)✳ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥ t↕✐ x ♥➳✉
∂f (x) = ∅.

✣à♥❤ ❧þ ✶✳✶✳✶✷✳ ✭①❡♠ ❬✷❪✱ ▼➺♥❤ ✤➲ ✶✶✳✸✮✳ ❈❤♦ f : Rn → R ∪ {+∞} ❧➔ ❤➔♠

❧ç✐ ❝❤➼♥❤ t❤÷í♥❣✳ ❑❤✐ ✤â✿
✭❛✮ ◆➳✉ x ∈
/ domf t❤➻ ∂f (x) = ∅✳
✾


x int(domf ) t f (x) = t

ỵ ỵ f ỗ tữớ tr
Rn õ s tữỡ ữỡ

f (x);
f (x, d) , d .

ỵ sỷ f ỗ tr Rn õ
tr ỳ tr t ỗ C {fk } ởt ỗ ỳ tr C
s

lim fk (x) = f (x), x C.

k

x C {xk } C s limk xk = x t ợ t y Rn t
{y k } ở tử y t õ

lim sup fk (xk ; y k ) f (x; y).
k

ỡ ỳ ợ t số

> 0 tỗ t số k0 s

fk (xk ) f (x) + B[0; 1], k k0 ,
ợ B[0; 1] ỡ õ tr Rn

ỵ sỷ C Rn ởt ỗ õ
rộ f : Rn R {+} ỗ õ ồ ỹ t
ữỡ ừ f tr C ỹ t t ử r t
ỹ t argminxC f (x) ừ f tr C ởt t ỗ ỡ ỳ f ỗ



t t số õ ổ q ởt ỹ t tr C f ỗ
t số ổ õ t ởt ỹ t t ử tr C

ỵ sỷ C Rn ởt ỗ
rộ f : Rn R {+} ỗ ữợ tr C
õ x0 ỹ t ừ f tr C

0 f (x0 ) + NC (x0 ).

q

ợ tt ữ tr ỵ 1.1.15 t x0

intC ỹ t ừ f tr C 0 f (x0 ) t
f t tr t

f (x0 ) = 0

t trữớ ủ r
sỷ C t ỗ õ rộ tr ổ rt H f :

C ì C R tọ f (x, x) = 0 ợ ồ x C ởt f ữ ữủ
ồ s t ữ s
x C s f (x , y) 0, y C.
t EP (C, f ) t ừ õ Sf .
t õ ỡ ữ õ ữủ
ợ t q trồ tở ỹ ữ t tố
ữ t ỹ t t tự t
t ở t s õ t ởt tố
t ỗ ở t t ỗ tứ



♥❤❛✉✱ ❤ñ♣ ♥❤➜t ❝❤ó♥❣ tr♦♥❣ ♠ët t❤➸ t❤è♥❣ ♥❤➜t ❝❤✉♥❣ r➜t t❤✉➟♥ t✐➺♥ ❝❤♦
✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✳
❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ö ✈➲ ♥❤ú♥❣ ❜➔✐ t♦→♥ q✉❡♥ t❤✉ë❝ ❝â t❤➸ ✤÷ñ❝ ♠æ t↔
❞÷î✐ ❞↕♥❣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳

✶✳✷✳✶ ❇➔✐ t♦→♥ tè✐ ÷✉
❈❤♦ C ❧➔ t➟♣ ❧ç✐ ✤â♥❣ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈➔

g : C → R ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ C ✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ tè✐ ÷✉ ✤÷ñ❝ ♣❤→t
❜✐➸✉ ♥❤÷ s❛✉✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ g(x∗ ) ≤ g(y), ∀y ∈ C.
✣➦t f (x, y) := g(y) − g(x) t❤➻ ❜➔✐ t♦→♥ tè✐ ÷✉ tr➯♥ ✤÷ñ❝ ✤÷❛ ✈➲ ❜➔✐ t♦→♥
❝➙♥ ❜➡♥❣ EP (C, f )✳

✶✳✷✳✷ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥

❈❤♦ C ⊂ H ❧➔ t➟♣ ✤â♥❣ ❦❤→❝ ré♥❣ ✈➔ F : C → H ❧➔ ♠ët →♥❤ ①↕ ✤ì♥ trà✳
❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✤ì♥ trà✮ V IP (C, F ) ❧➔ ❜➔✐ t♦→♥✿



❚➻♠ x∗ ∈ C s❛♦ ❝❤♦


 F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C.

❚❛ ✤➦t f (x, y) = F (x), y − x ✱ t❤➻ ✤÷❛ ✤÷ñ❝ ❜➔✐ t♦→♥ V IP (C, F ) ✈➲ ❜➔✐
t♦→♥ EP (C, f )✳
▼ët tr÷í♥❣ ❤ñ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ V IP (C, F ) ❧➔ ❦❤✐ C ❧➔ ♠ët
♥â♥ ❧ç✐ ✤â♥❣ ❦❤→❝ ré♥❣ tr♦♥❣ Rn ✳ ❑þ ❤✐➺✉ C + = {x ∈ Rn : x, y ≥ 0, ∀y ∈
✶✷


C} ❧➔ ♥â♥ ❝ü❝ ❝õ❛ C ✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (C, F )
trð t❤➔♥❤ ❜➔✐ t♦→♥ ❜ò CP (C, F ) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ❧➔ ❜➔✐ t♦→♥✿



❚➻♠ x∗ ∈ C s❛♦ ❝❤♦


F (x∗ ) ∈ C + , F (x∗ ), x∗ = 0.

❚ê♥❣ q✉→t✱ ①➨t ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤❛ trà M V IP (C, F ) s❛✉✿




❚➻♠ x∗ ∈ C, u∗ ∈ F (x∗ ) s❛♦ ❝❤♦


 u∗ , y − x∗ ≥ 0, ∀y ∈ C,

✈î✐ C ⊆ H ❧➔ ♠ët t➟♣ ❧ç✐ ✤â♥❣ ✈➔ F : C → 2H ❧➔ →♥❤ ①↕ ✤❛ trà✳ ❑❤✐ ✤â ✈î✐
♠é✐ x ∈ C ✱ F (x) ❧➔ ♠ët t➟♣ ❧ç✐ ❝♦♠♣❛❝t ✈➔ ❦❤→❝ ré♥❣ t❛ ✤➦t

f (x, y) = sup u, y − x .
u∈F (x)

❑❤✐ ✤â✱ x∗ ∈ C ✈➔ u∗ ∈ F (x∗ ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ M V IP (C, F ) ❦❤✐
✈➔ ❝❤➾ ❦❤✐ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ EP (C, f ) ✭①❡♠ ❬✶✵❪✱ tr❛♥❣
✶✶✻✵✮✳
▼ët tr÷í♥❣ ❤ñ♣ r✐➯♥❣ q✉❡♥ t❤✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ M V IP (C, F ) ❧➔ ❜➔✐ t♦→♥
q✉② ❤♦↕❝❤ ❧ç✐ CO(C, h)✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ h(x∗ ) ≤ h(x), ∀y ∈ C,
✈î✐ h ❧➔ ♠ët ❤➔♠ ❧ç✐ ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥ tr➯♥ C ✳ ◆❤÷ ✤➣ ❜✐➳t✱ ✤✐➸♠ x∗ ∈ C ❧➔
♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ COP (C, h) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥

✶✸


❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤❛ trà M V IP (C, ∂h) s❛✉✿



❚➻♠ x∗ ∈ C, u∗ ∈ ∂h(x∗ ) s❛♦ ❝❤♦



 u∗ , y − x∗ ≥ 0, ∀y ∈ C ✳

❳➨t ✈➲ ❦❤➼❛ ❝↕♥❤ ❦✐♥❤ t➳ t❤➻ ❜➔✐ t♦→♥ M V IP (C, F ) ❝❤➼♥❤ ❧➔ ❜➔✐ t♦→♥ t➻♠
♣❤÷ì♥❣ →♥ s↔♥ ①✉➜t x∗ tr♦♥❣ t➟♣ ❝→❝ ♣❤÷ì♥❣ →♥ s↔♥ ①✉➜t C ✭❤❛② t➟♣ ❝❤✐➳♥
❧÷ñ❝✮ ✈➔ ✈➨❝tì ❣✐→ u∗ tr♦♥❣ t➟♣ ❝→❝ ❣✐→ t❤➔♥❤ F (x∗ ) ù♥❣ ✈î✐ ♣❤÷ì♥❣ →♥ s↔♥
①✉➜t x∗ s❛♦ ❝❤♦ ❝❤✐ ♣❤➼ s↔♥ ①✉➜t ❧➔ t❤➜♣ ♥❤➜t✳

✶✳✷✳✸ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
●✐↔ sû C ∈ H ❧➔ ♠ët t➟♣ ❧ç✐ ✤â♥❣ ❦❤→❝ ré♥❣ ✈➔ F : C → C ❧➔ →♥❤ ①↕ ✤ì♥
trà✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ F P (C, F ) ❧➔ ❜➔✐ t♦→♥✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = F (x∗ ).
◆➳✉ ✤➦t f (x, y) = x − F (x), y − x , ∀x, y ∈ C, t❤➻ ❜➔✐ t♦→♥ F P (C, F ) trð
t❤➔♥❤ ❜➔✐ t♦→♥ EP (C, f )✳ ❚ê♥❣ q✉→t ❤ì♥✱ t❛ ①➨t ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤❛
trà M F P (C, F )✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = F (x∗ ),
ð ✤â F : C → 2C ❧➔ →♥❤ ①↕ ✤❛ trà ❝â ❣✐→ trà ❧ç✐ ❝♦♠♣❛❝t ❦❤→❝ ré♥❣✳ ✣➦t

f (x, y) = max x − u, y − x , ∀x, y ∈ C,
u∈F (x)

❦❤✐ ✤â x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ M F P (C, F ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x∗ ❧➔ ♥❣❤✐➺♠
❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ EP (C, F ) ✭①❡♠ ❬✷❪✱ tr❛♥❣ ✷✷✶✲✷✷✷✮✳
✶✹


✶✳✷✳✹ ❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤ tr♦♥❣ trá ❝❤ì✐ ❦❤æ♥❣ ❤ñ♣ t→❝
❳➨t ♠ët trá ❝❤ì✐ ❦❤æ♥❣ ❤ñ♣ t→❝ ❣ç♠ ❝â p ✤➜✉ t❤õ✱ ✤➜✉ t❤õ t❤ù i ❝â t➟♣
❝❤✐➳♥ ❧÷ñ❝ ❧➔ Ci ⊆ Rni ✈➔ ❝â ❤➔♠ ❝❤✐ ♣❤➼ ❧➔ fi : C → R ✈î✐ C = C1 × ... × Cp
t÷ì♥❣ ù♥❣✱ tù❝ ❧➔ ♥➳✉ ✤è✐ t❤õ t❤ù ♥❤➜t✱ t❤ù ❤❛✐✱ ✳✳✳ t❤ù p✱ ❧➛♥ ❧÷ñt ❝❤å♥ ❝❤✐➳♥

❧÷ñ❝ ❝❤ì✐ ❧➔ x1 ∈ C1 , x2 ∈ C2 , ..., xp ∈ Cp ✱ t❤➻ ❝❤✐ ♣❤➼ ❝õ❛ ♠é✐ ✤è✐ t❤õ t÷ì♥❣
ù♥❣ s➩ ❧➔

f1 (x1 , x2 , ..., xp ), f2 (x1 , x2 , ..., xp ), ...fp (x1 , x2 , ..., xp ).
▼ö❝ t✐➯✉ ❝õ❛ ♠é✐ ✤è✐ t❤õ ❧➔ t➻♠ ❦✐➳♠ ♠ët ❝❤✐➳♥ ❧÷ñ❝ ❝❤ì✐ tr♦♥❣ t➟♣ ❝❤✐➳♥ ❧÷ñ❝
❝❤ì✐ t÷ì♥❣ ù♥❣ ✤➸ ❧➔♠ ❝ü❝ t✐➸✉ ❝❤✐ ♣❤➼ ❝õ❛ ♠➻♥❤✳ ❑þ ❤✐➺✉ x = (x1 , x2 , ..., xp )✱
♠ët ✤✐➸♠ x∗ ∈ C ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ◆❛s❤ ♥➳✉ ❦❤æ♥❣ tç♥ t↕✐ ♠ët
✤è✐ t❤õ ♥➔♦ ❝â t❤➸ ❣✐↔♠ ✤÷ñ❝ ❝❤✐ ♣❤➼ ❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ ❝❤✐➳♥ ❧÷ñ❝ ❝❤ì✐ ❝õ❛
♠➻♥❤ tr♦♥❣ ❦❤✐ ❝→❝ ✤è✐ t❤õ ❦❤→❝ ✈➝♥ ❣✐ú ♥❣✉②➯♥ ❝❤✐➳♥ ❧÷ñ❝ ❝õ❛ ❤å✳ ❱➲ ♠➦t
t♦→♥ ❤å❝✱ ✤✐➸♠ x∗ ∈ C ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ◆❛s❤ ♥➳✉✿

fi (x∗1 , ..., x∗i−1 , x∗ , x∗i+1 , ..., x∗p ) ≤ fi (x∗1 , ..., x∗i−1 , yi , x∗i+1 , ..., x∗p ),
✈î✐ ♠å✐ yi ∈ Ci ✈➔ ✈î✐ ♠å✐ i = 1, 2, ..., p.
❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ◆❛s❤ x∗ ✤÷ñ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤✳
❇➡♥❣ ❝→❝❤ ✤➦t
p

[fi (x1 , ..., xi , ..., xp ) − fi (x1 , ..., yi , ..., xp )] ,

f (x, y) =
i=1

t❛ ✤÷❛ ✤÷ñ❝ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤ ✈➲ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ EP (C, f )✳

✶✺


ỹ tỗ t ừ t
r t tr ởt số sỹ tỗ t
t t ừ t ừ t EP (C, f )

ỵ ỵ C ởt t ỗ t
rộ s f : C ì C R {+} õ t t
s
f (., y) ỷ tử tr ợ ồ y C;
f (x, .) ỗ ỷ tử ữợ ữợ tr C ợ ồ x C
õ t EP (C, f ) õ

q

C t ỗ õ ổ t s

f : C ì C R {+} sỷ ự (C1 ) s ữủ
tọ

C B = , x C\B, y C B : f (x, y) < 0.
õ t EP (C, f ) õ
ỵ tr ởt trữớ ủ r ừ ỵ s

ỵ s r sỷ C ởt t ỗ õ
rộ tr ổ rt H f : C ì C R {+} s
tr C f tọ tt
f (., y) ỷ tử tr ợ ồ y C;
f (x, .) tỹ ỗ tr C ợ ộ x C



t❤➻ ❜➔✐ t♦→♥ EP (C, f ) ❝â ♥❣❤✐➺♠✱ ♥➳✉ ♥❤÷ C ❧➔ t➟♣ ❝♦♠♣❛❝t ❤♦➦❝ ✤✐➲✉ ❦✐➺♥
❜ù❝ (C1 ) ✤÷ñ❝ t❤ä❛ ♠➣♥✳
✣➸ ①➨t t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ t❛ ♥❤➢❝ ❧↕✐ ❝→❝ ✤à♥❤

♥❣❤➽❛ ✈➲ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ s♦♥❣ ❤➔♠✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ●✐↔ sû C

⊂ H✳ ❙♦♥❣ ❤➔♠ ❝➙♥ ❜➡♥❣ f : C × C →

R ∪ {+∞} ✤÷ñ❝ ❣å✐ ❧➔

✭❛✮ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈î✐ ❤➺ sè γ > 0 ♥➳✉

f (x, y) + f (y, x) ≤ −γ x − y 2 , ∀x, y ∈ C;
✭❜✮ ✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ♥➳✉

f (x, y) + f (y, x) < 0, ∀x, y ∈ C, x = y;
✭❝✮ ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉

f (x, y) + f (y, x) ≤ 0, ∀x, y ∈ C;
✭❞✮ ❣✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉

f (x, y) ≥ 0 ⇒ f (y, x) ≤ 0, ∀x, y ∈ C;
✭❡✮ ❣✐↔ ✤ì♥ ✤✐➺✉ t❤❡♦ x∗ tr➯♥ C ♥➳✉

∀y ∈ C, f (x∗ , y) ≥ 0 ⇒ f (y, x∗ ) ≤ 0;
✭❢✮ ❣✐↔ ✤ì♥ ✤✐➺✉ t❤❡♦ t➟♣ S tr➯♥ C ♥➳✉ f ❣✐↔ ✤ì♥ ✤✐➺✉ t❤❡♦ x∗ tr➯♥ C ✈î✐
♠å✐ x∗ ∈ S.
❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ t❛ s✉② r❛✿ (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e), ∀x∗ ∈ C.
✶✼