Phương pháp chiếu giải bài toán cân bằng hai cấp

ì ì P

ữỡ

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

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥✱ ♥➠♠ ✷✵✶✻




ì ì P

ữỡ

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

❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐
số


ữớ ữợ ❦❤♦❛ ❤å❝✿

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

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝✱ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ tr
ữủ ró ỗ ố

t❤→♥❣ ✹ ♥➠♠ ✷✵✶✻
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

❉÷ì♥❣ ❱➠♥ ❚❤✐

✐


▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ tr♦♥❣ ❦❤â❛ ✷✷ ✤➔♦ t↕♦ ❚❤↕❝ s➽ ❝õ❛ tr÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ✲ ồ ữợ sỹ ữợ ừ ◆❣✉②➵♥
❳✉➙♥ ❚➜♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ t tợ t
ữợ ữớ t ❝❤♦ tỉ✐ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱
t✐♥❤ t❤➛♥ ❧➔♠ ✈✐➺❝ ♥❣❤✐➯♠ tó❝ ✈➔ ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥✱ ổ sự ữợ
tổ t
ổ ụ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❝õ❛ tr÷í♥❣
✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱
❦❤➼❝❤ ❧➺✱ ✤ë♥❣ ✈✐➯♥ tỉ✐ ✈÷đt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥ tr♦♥❣ ❤å❝ t➟♣✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❧➣♥❤ ✤↕♦ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐✱ ❣✐ó♣
✤ï tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ tỉ✐ ❤å❝ t➟♣✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ♥❣÷í✐ t❤➙♥ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱
õ♥❣ ❤ë tæ✐ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳

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

❉÷ì♥❣ ❱➠♥ ❚❤✐
✐✐


ử ử
ớ



ớ ỡ



ử ử



ởt số ỵ ✈✐➳t t➢t

✈

▼ð ✤➛✉

✶

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

✶✳✷

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





t ỗ ỗ





ữợ ừ ỗ ✳ ✳

✽

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

✶✶

✶✳✷✳✶

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

✶✷

✶✳✷✳✷

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

✶✷

✶✳✷✳✸

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

✶✹

✶✳✷✳✹

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

✶✺

✐✐✐




ỹ tỗ t ừ 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♣ ❞ư♥❣ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✷✳✸✳✶

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

✷✳✸✳✷

✹✷

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

✺✸

❑➳t ❧✉➟♥

✻✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✼✵

✐✈


ởt số ỵ t tt
R

t số tỹ

N

t số tỹ ♥❤✐➯♥✳

H

❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳

Rn

❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n ❝❤✐➲✉✳

x, y = xT y

x =

x, x

t ổ ữợ ừ tỡ x ✈➔ y ✳
❝❤✉➞♥ ❝õ❛ ✈➨❝tì x✳

domf

♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ❤➔♠ f

imF

ừ F

epif

tr ỗ t ❝õ❛ ❤➔♠ f ✳

ϕ (x) =

ϕ(x) ✤↕♦ ❤➔♠ ❝õ❛ ϕ t x

(x; d)

t ữợ d ừ t x

(x)

ữợ ừ t x

x f (x, y)

✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ f (., y) t↕✐ x✳

y f (x, y)

✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ f (x, .) t↕✐ y

f (x, x)

ữợ ừ f (x, .) t x✳

intC

♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ C ✳

riC

♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ t➟♣ C ✳

xk → x

❞➣② xk ❤ë✐ tư tỵ✐ 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

ợ ữợ

lim = lim sup

ợ ❤↕♥ 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❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳

✈✐



ỵ ồ t
t tố ữ

min f (x),
x∈D

✭✶✮

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

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 ✤÷đ❝ ❣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 ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳ ❈❤➼♥❤ ①→❝✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣
✤÷đ❝ ✤÷❛ r❛ ❧➛♥ ✤➛✉ ❜ð✐ ❍✳ ◆✐❦❛✐❞♦ ✈➔ ❑✳ ■s♦❞❛ ♥➠♠ ✶✾✺✺ ❦❤✐ tê♥❣ q✉→t
❤â❛ ❜➔✐ t♦→♥ ❝➙♥ ◆❛s❤ tr♦♥❣ trá ❝❤ì✐ ❦❤ỉ♥❣ ❤đ♣ t→❝ ✈➔ ✤÷đ❝ ❑② ❋❛♥ ❣✐ỵ✐
✶


t❤✐➺✉ ♥➠♠ ✶✾✼✷ ✭t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❑② ❋❛♥✮✳ ❇➔✐ t♦→♥ ❝➙♥
❜➡♥❣ ❜❛♦ ❤➔♠ ♥❤✐➲✉ ❧ỵ♣ ❜➔✐ t♦→♥ q✉❡♥ t❤✉ë❝ ♥❤÷ ❜➔✐ t♦→♥ tè✐ ÷✉✱ ❜➔✐ t♦→♥
❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t s
tr ỵ tt trỏ ỡ ổ ủ t→❝ ✳✳✳ ❱➻ ✈➟②✱ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ✈➲
❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✤÷đ❝ →♣ ❞ư♥❣ trü❝ t✐➳♣ ❝❤♦ ❝→❝ t t ừ õ
ữợ ự t♦→♥ ❝➙♥ ❜➡♥❣ r➜t ✤❛ ❞↕♥❣✱ tr♦♥❣ ✤â ✈✐➺❝ ♥❣❤✐➯♥
❝ù✉ ①➙② ❞ü♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ✤➣ ✤÷❛ t♦→♥ ❤å❝ ✈➔♦ ❣✐↔✐ q✉②➳t ♥❤✐➲✉
✈➜♥ ✤➲ ✤➦t r❛ tr♦♥❣ t❤ü❝ t➳✳
P❤➛♥ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉
❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❣✐↔ ✤ì♥ ✤✐➺✉ ✈➔ →♣ ❞ư♥❣ ✈➔♦ ❧ỵ♣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣

❤❛✐ ❝➜♣✳ trú ỗ ữỡ
ữỡ tự ỡ ừ t ỗ ữủ sỷ ử
tr ữỡ s t ợ t t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ t÷ì♥❣ ✤÷ì♥❣ ✈➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❤❛✐ ❝➜♣✳
❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥
♣❤➙♥ ❣✐↔ ✤ì♥ ✤✐➺✉✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❣✐↔ ✤ì♥ ✤✐➺✉ ✈➔ →♣ ❞ö♥❣ ❣✐↔✐ ❜➔✐ t♦→♥
❝➙♥ ❜➡♥❣ ❤❛✐ ❝➜♣✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣
❣✐↔ ✤ì♥ ✤✐➺✉ ✈➔ →♣ ❞ư♥❣ ✈➔♦ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❤❛✐ ❝➜♣✳

✷


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

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

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

✸

❈❤÷ì♥❣ ✶

▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✣➸ ❝❤✉➞♥ ❜à ❝❤♦ ❝❤÷ì♥❣ s❛✉✱ ♣❤➛♥ t❤ù ♥❤➜t tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠
✈➔ ❦➳t q✉↔ ❝➛♥ t❤✐➳t ✈➲ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❣✐↔✐ t ỗ tự ợ t
t ❜➡♥❣ ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ♥â ❝ị♥❣ ♠ët số
sỹ tỗ t ừ t ❝➙♥ ❜➡♥❣✳ P❤➛♥ t✐➳♣ t❤❡♦ tr➻♥❤ ❜➔② ✈➲
❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ t÷ì♥❣ ✤÷ì♥❣✳ P❤➛♥ ❝✉è✐ ❝ị♥❣ tr➻♥❤ ❜➔② ✈➲ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ ❤❛✐ ❝➜♣ ✈➔ ♠ët sè tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♥➔②✳ ❑✐➳♥ t❤ù❝ tr♦♥❣
❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❧➜② tø ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✼❪✱ ❬✽❪✱ ❬✶✶❪✳

✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ừ t ỗ
t ỗ ỗ
ởt số ỡ ừ t ỗ

sỷ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì tr➯♥ R✱ t➟♣ C X
ữủ ồ
ỗ x, y C ✈➔ 0 ≤ λ ≤ 1 ⇒ λx + (1 − λ)y ∈ C ❀
✹


✭❜✮ ◆â♥ ❝â ✤➾♥❤ t↕✐ ✵ ♥➳✉ ∀λ > 0, x C x C
õ ỗ ♥➳✉ ♥â ✈ø❛ ❧➔ ♥â♥ ❝â ✤➾♥❤ t↕✐ ✵ ✈ø❛ ởt t ỗ

x, y C, , µ > 0 ⇒ λx + µy ∈ C.
❈→❝ t➟♣ ỗ õ ố ợ ởt số t ữ ♣❤➨♣ ❣✐❛♦✱ ♣❤➨♣ ❝ë♥❣✱
♣❤➨♣ ♥❤➙♥ ✈ỵ✐ ♠ët sè t❤ü❝✳ ❚ù❝ C D t ỗ tr X t❤➻

C ∩ D, λC + βD ❝ô♥❣ ❧➔ ❝→❝ t ỗ ợ ồ , R

sỷ C ởt t ỗ rộ tr ❦❤æ♥❣ ❣✐❛♥
❍✐❧❜❡rt t❤ü❝ H ✈➔ x0 ∈ C ✳ ❑❤✐ ✤â t➟♣

NC (x0 ) = {ω ∈ H : ω, x − x0 ≤ 0, ∀x ∈ C},
✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ C t↕✐ x0 ✈➔ t➟♣ −NC (x0 ) ✤÷đ❝ ❣å✐ ❧➔
♥â♥ ♣❤→♣ t✉②➳♥ ✭tr♦♥❣✮ ❝õ❛ C t↕✐ x0 ✳
❍✐➸♥ ♥❤✐➯♥ 0 ∈ NC (x0 ) ✈➔ tø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ t❤➜② NC (x0 ) ởt õ
ỗ õ

sỷ C = ổ t tt ỗ ởt t ừ
ổ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔ y ∈ H ❧➔ ♠ët ✈➨❝ tì ❜➜t ❦ý✱ ❣å✐

dC (y) = inf x − y
x∈C

❧➔ tứ y C tỗ t PC (y) s❛♦ ❝❤♦

dC (y) = y − PC (y) ,
t❤➻ t❛ ♥â✐ PC (y) ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ y tr➯♥ C ✳
✺


❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ t❤➜② ❤➻♥❤ ❝❤✐➳✉ PC (y) ❝õ❛ y tr➯♥ C ❧➔ ♥❣❤✐➺♠ ❝õ❛
❜➔✐ t♦→♥ tè✐ ÷✉

min
x∈C

1
x−y
2

2

.

◆â✐ ❝→❝❤ ❦❤→❝✱ ✈✐➺❝ t➻♠ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ y tr➯♥ C ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ t➻♠
❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ x y

2

tr C

ỵ C ởt t ỗ õ
rộ ừ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H✳ ❑❤✐ ✤â✿
✭❛✮ ∀x ∈ H ❤➻♥❤ PC (x) ừ x tr C ổ tỗ t ✈➔ ❞✉② ♥❤➜t❀
✭❜✮ ω = PC (x) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x − ω, y − ω ≤ 0, ∀y ∈ C ❀
✭❝✮ ⑩♥❤ ①↕✿ x → PC (x) ❝â t➼♥❤ ❝❤➜t
✶✳ PC (x) − PC (y) ≤ x − y , ∀x, y ∈ H (t➼♥❤ ❦❤æ♥❣ ❣✐➣♥)❀
✷✳ 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 ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ●✐↔ sû X ổ tỡ tổổ ỗ ữỡ tỹ
C X ởt t ỗ f : C −→ R ∪ {+∞}✱ ❦❤✐ ✤â
✭❛✮ ❍➔♠ f ✤÷đ❝ ồ ỗ tr C

f (x + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), ∀x, y ∈ C, ∀λ ∈ [0; 1] ;
✭❜✮ ❍➔♠ f ữủ ồ ỗ t tr C

f (λx + (1 − λ)y) < λf (x) + (1 − λ)f (y), ∀x, y ∈ C, x = y, ∀λ ∈ (0; 1) ;

✻


f ữủ ồ ỗ tr C ✈ỵ✐ ❤➺ 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 ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ ✭❧ã♠ ❝❤➦t✱ ❧ã♠ ♠↕♥❤✮ tr➯♥ C f
ỗ ỗ t ỗ tr C
f ữủ ồ tỹ ỗ tr C ♥➳✉ ∀λ ∈ R t➟♣ ♠ù❝

{x ∈ C : f (x) }
t ỗ
t

domf = {x ∈ C : f (x) < +∞} ,
epif = {(x, t) ∈ C × R : f (x) ≤ t} ,
ữủt ỳ tr ỗ t ❝õ❛ f ❀
✭❣✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ domf = ∅ ✈➔ f (x) > −∞ ✈ỵ✐

♠å✐ x C

ỵ ỵ ✷✳✸✮✳ ●✐↔ sû f : C −→ R ∪ {+∞}
ỗ [, +] õ t➟♣ ♠ù❝

L0α (f ) = {x ∈ X : f (x) < α} ;
Lα (f ) = {x ∈ X : f (x) }
t ỗ



✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ●✐↔ sû f : H → R✳ ❑❤✐ ✤â
✭❛✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tư❝ ữợ t x0 H

lim f (x) f (x0 ).
x→x0

✭❜✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tư❝ ữợ tr C õ ỷ tử
ữợ t ♠å✐ x ∈ C ✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ tr➯♥ C ♥➳✉ −f ❧➔
♥û❛ ❧✐➯♥ 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 ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭❛✮ f ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠ x0 ❀
✭❜✮ f ❜à ❝❤➦♥ tr➯♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♠ð ❝õ❛ x0 ❀
✭❝✮ int(epif ) = ∅;
✭❞✮ int(domf ) = ∅ ✈➔ f ❧✐➯♥ tö❝ tr➯♥ int(domf ) ỗ tớ

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

✶✳✶✳✷ ữợ ừ ỗ
✶✳✶✳✾✳ ●✐↔ sû f : H → R, x ∈ H ✈➔ d ∈ H\{0}✳ ❑❤✐ ✤â ❤➔♠ f
✤÷đ❝ ❣å✐
rt t x tỗ t tỡ x∗ ∈ H s❛♦ ❝❤♦

lim

y→x

f (y) − f (x) − x∗ , y − x
= 0,
y−x

✽


❦❤✐ ✤â x∗ ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ f t x ữủ ỵ

f (x)

f (x)
õ t ữợ d t x tỗ t↕✐ ❣✐ỵ✐ ❤↕♥

f (x; d) = lim+
t→0

f (x + td) − f (x)
.
t

❈â t❤➸ t❤➜② r➡♥❣ ♥➳✉ ❤➔♠ f ❦❤↔ t x t õ õ t ồ
ữợ t↕✐ x ✈➔ t❛ ❝â f (x; d) =

f (x), d , d H.

ỵ ✤➲ ✶✶✳✻✮✳ ❈❤♦ f : Rn → R ∪ {+∞}
C Rn t ỗ õ õ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭❛✮ f ❧➔ ❤➔♠ ỗ tr C
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 {+} ỗ tữớ
tr H H ữủ ồ ữợ ừ f t↕✐ x ♥➳✉

f (y) ≥ ω, y − x + f (x), y H.
tt ữợ ừ f t x ữủ ồ ữợ ✈✐ ♣❤➙♥ ❝õ❛

f t↕✐ x✱ ❦➼ ❤✐➺✉ ❧➔ ∂f (x)✳ f ữủ ồ ữợ t x
f (x) = .

ỵ ✤➲ ✶✶✳✸✮✳ ❈❤♦ f : Rn → R ∪ {+∞}

ỗ tữớ õ
x
/ domf t❤➻ ∂f (x) = ∅✳
✾


✭❜✮ ◆➳✉ x ∈ int(domf ) t❤➻ ∂f (x) = t

ỵ ỵ f ỗ tữớ tr
Rn ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣

✭❛✮ ω ∈ ∂f (x);
f (x, d) , d .

ỵ ✭①❡♠ ❬✷❪✱ ▼➺♥❤ ✤➲ ✶✶✳✽ ✮✳ ●✐↔ sû 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❤➻ ✈ỵ✐ ❜➜t ❦➻ y ∈ Rn ✈➔ ❜➜t
❦➻ ❞➣② {y k } ❤ë✐ tö ✈➲ y t❛ ❝â✿

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

❍ì♥ ♥ú❛✱ ✈ỵ✐ t số

> 0 tỗ t số k0 s ❝❤♦

∂fk (xk ) ⊂ ∂f (x) + B[0; 1], ∀k ≥ k0 ,
✈ỵ✐ B[0; 1] ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à õ tr Rn

ỵ sỷ C Rn ởt ỗ ✤â♥❣
❦❤→❝ ré♥❣ ✈➔ f : Rn → R ∪ {+∞} ỗ õ ồ ỹ t
ữỡ ❝õ❛ f tr➯♥ C ✤➲✉ ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ t♦➔♥ ❝ö❝✱ ♥❣♦➔✐ r❛ t➟♣ ❝→❝ ✤✐➸♠
❝ü❝ t✐➸✉ argminx∈C f (x) ừ f tr C ởt t ỗ ỡ ỳ f ỗ



❝❤➦t t❤➻ ❤➔♠ sè ❝â ❦❤æ♥❣ q✉→ ♠ët ✤✐➸♠ ❝ü❝ t tr C f ỗ
t sè ❧✉æ♥ ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉ t♦➔♥ ử tr C

ỵ ✶✶✳✶✷ ✮✳ ●✐↔ sû C ❦❤✐ ⊆ Rn✱ ❧➔ ♠ët ỗ
rộ f : Rn R {+} ỗ ữợ tr C ❑❤✐
✤â x0 ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ f tr➯♥ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

0 ∈ ∂f (x0 ) + NC (x0 ).

q

ợ tt ữ tr ỵ 1.1.15 t x0

intC ỹ t ❝õ❛ f tr➯♥ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ 0 ∈ ∂f (x0 )✳ ✣➦❝ ❜✐➺t✱ ♥➳✉
❤➔♠ f ❦❤↔ ✈✐ t❤➻ ✤✐➲✉ ❦✐➺♥ ♥➔② trð t❤➔♥❤

f (x0 ) = 0✳

✶✳✷ ❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣
●✐↔ sû C t ỗ õ rộ tr ổ rt H ✈➔ f :

C × C → R t❤ä❛ ♠➣♥ f (x, x) = 0 ✈ỵ✐ ♠å✐ x ∈ C ❀ ♠ët ❤➔♠ f ♥❤÷ ✈➟② ✤÷đ❝
❣å✐ ❧➔ s♦♥❣ ❤➔♠ ❝➙♥ ❜➡♥❣✳ ❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ♥❤÷ s❛✉✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ f (x∗ , y) ≥ 0, ∀y ∈ C.
❚❛ ❦➼ ❤✐➺✉ ❜➔✐ t♦→♥ ♥➔② ❧➔ EP (C, f ) ✈➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♥â ❧➔ Sf .
❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ❝â ❞↕♥❣ ❦❤→ ✤ì♥ ❣✐↔♥ ♥❤÷♥❣ ♥â ữủ
ợ t q trồ tở ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷✿ ❜➔✐ t♦→♥ tè✐
÷✉✱ ❜➔✐ t♦→♥ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥
✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤❀ ♥â t ởt tố
t ỗ ở t t ỗ tứ



♥❤❛✉✱ ❤đ♣ ♥❤➜t ❝❤ó♥❣ tr♦♥❣ ♠ët t❤➸ t❤è♥❣ ♥❤➜t ❝❤✉♥❣ r➜t t❤✉➟♥ t✐➺♥ ❝❤♦
✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✳
❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ö ✈➲ ♥❤ú♥❣ ❜➔✐ t♦→♥ q✉❡♥ t❤✉ë❝ ❝â t❤➸ ữủ ổ t
ữợ t

t tố ữ
C t ỗ õ rộ tr ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈➔

g : C → R ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ C ✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ tè✐ ÷✉ ✤÷đ❝ ♣❤→t
❜✐➸✉ ♥❤÷ s❛✉✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ g(x∗ ) ≤ g(y), ∀y ∈ C.
✣➦t f (x, y) := g(y) − g(x) t❤➻ ❜➔✐ t♦→♥ tè✐ ÷✉ tr➯♥ ✤÷đ❝ ✤÷❛ ✈➲ ❜➔✐ 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❤➻ ✤÷❛ ✤÷đ❝ ❜➔✐ t♦→♥ V IP (C, F ) ✈➲ ❜➔✐
t♦→♥ EP (C, f )✳
▼ët tr÷í♥❣ ❤ñ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ V IP (C, F ) C ởt
õ ỗ õ rộ tr Rn ỵ C + = {x ∈ Rn : x, y ≥ 0, ∀y ∈
✶✷


C} ❧➔ ♥â♥ ❝ü❝ ❝õ❛ C ✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (C, F )
trð t❤➔♥❤ ❜➔✐ t♦→♥ ❜ị CP (C, F ) ✤÷đ❝ ✤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,

ợ C H ởt t ỗ õ F : C → 2H ❧➔ →♥❤ ①↕ ✤❛ trà✳ ❑❤✐ ✤â ✈ỵ✐
♠é✐ x ∈ C ✱ F (x) ❧➔ ởt t ỗ t rộ 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♥❣ ❤đ♣ r✐➯♥❣ q✉❡♥ t❤✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ M V IP (C, F ) t
q ỗ CO(C, h)
x ∈ C s❛♦ ❝❤♦ h(x∗ ) ≤ h(x), ∀y ∈ C,
ợ h ởt ỗ ữợ 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➟♣ ❝❤✐➳♥
❧÷đ❝✮ ✈➔ ✈➨❝tì ❣✐→ u∗ tr♦♥❣ t➟♣ ❝→❝ ❣✐→ t❤➔♥❤ F (x∗ ) ù♥❣ ợ ữỡ s
t x s s ①✉➜t ❧➔ t❤➜♣ ♥❤➜t✳

✶✳✷✳✸ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
●✐↔ sû C H ởt t ỗ õ rộ ✈➔ 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 tr õ tr ỗ 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✐ ❦❤ỉ♥❣ ủ t
t ởt trỏ ỡ ổ ủ t ỗ õ p ✤➜✉ t❤õ✱ ✤➜✉ t❤õ t❤ù i ❝â t➟♣
❝❤✐➳♥ ❧÷đ❝ ❧➔ Ci ⊆ Rni ✈➔ ❝â ❤➔♠ ❝❤✐ ♣❤➼ ❧➔ fi : C R ợ C = C1 ì ... ì Cp
tữỡ ự tự ố từ tự ♥❤➜t✱ t❤ù ❤❛✐✱ ✳✳✳ t❤ù p✱ ❧➛♥ ❧÷đt ❝❤å♥ ❝❤✐➳♥

❧÷đ❝ ❝❤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 ❝❤✐➳♥ ❧÷đ❝ ❝❤ì✐ tr♦♥❣ t➟♣ ❝❤✐➳♥ ❧÷đ❝
❝❤ì✐ t÷ì♥❣ ù♥❣ ✤➸ ❧➔♠ ❝ü❝ t✐➸✉ ừ ỵ x = (x1 , x2 , ..., xp )✱
♠ët ✤✐➸♠ x∗ ∈ C ✤÷đ❝ ồ s ổ tỗ t ♠ët
✤è✐ t❤õ ♥➔♦ ❝â t❤➸ ❣✐↔♠ ✤÷đ❝ ❝❤✐ ♣❤➼ ❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ ❝❤✐➳♥ ❧÷đ❝ ❝❤ì✐ ❝õ❛
♠➻♥❤ tr♦♥❣ ❦❤✐ ❝→❝ ✤è✐ t❤õ ❦❤→❝ ✈➝♥ ❣✐ú ♥❣✉②➯♥ ❝❤✐➳♥ ❧÷đ❝ ❝õ❛ ❤å✳ ❱➲ ♠➦t
t♦→♥ ❤å❝✱ ✤✐➸♠ x∗ ∈ C ✤÷đ❝ ❣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 ),
✈ỵ✐ ♠å✐ yi ∈ Ci ✈➔ ✈ỵ✐ ♠å✐ i = 1, 2, ..., p.
❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ◆❛s❤ x∗ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤✳
❇➡♥❣ ❝→❝❤ ✤➦t
p

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

f (x, y) =
i=1

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

✶✺


ỹ tỗ t ừ t
r ♣❤➛♥ ♥➔②✱ t❛ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ✈➲ sỹ tỗ t
t t ừ t ừ t EP (C, f )

ỵ ỵ C ởt t ỗ t
rộ s f : C × C → R ∪ {+∞} ❝â ❝→❝ t➼♥❤ ❝❤➜t
s❛✉✿
✭✐✮ f (., y) ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈ỵ✐ ♠å✐ y C;
f (x, .) ỗ ỷ tử ữợ ữợ tr C ợ ồ x ∈ C ✳
❑❤✐ ✤â ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ EP (C, f ) ❝â ♥❣❤✐➺♠✳

❍➺ q✉↔ ✶✳✷✳✷✳

❈❤♦ C ❧➔ t➟♣ ỗ õ ổ t s

f : C × C → R ∪ {+∞}✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ❜ù❝ (C1 ) s❛✉ ✤➙② ✤÷đ❝
t❤ä❛ ♠➣♥✿

C ∩ B = ∅, ∀x ∈ C\B, ∃y ∈ C ∩ B : f (x, y) < 0.
❑❤✐ ✤â ❜➔✐ 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 × C → R ∪ {+∞} ❧➔ s♦♥❣
❤➔♠ ❝➙♥ ❜➡♥❣ ①→❝ ✤à♥❤ tr➯♥ C ✳ ◆➳✉ f t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t✿
✭✐✮ f (., y) ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈ỵ✐ ♠å✐ y ∈ C;
✭✐✐✮ f (x, .) tỹ ỗ tr C ợ ộ x C ✱
✶✻


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

♥❣❤➽❛ ✈➲ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ s♦♥❣ ❤➔♠✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ●✐↔ sû C

⊂ H✳ ❙♦♥❣ ❤➔♠ ❝➙♥ ❜➡♥❣ f : C × C →

R ∪ {+∞} ✤÷đ❝ ❣å✐ ❧➔

✭❛✮ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈ỵ✐ ❤➺ 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 ✈ỵ✐
♠å✐ x∗ ∈ S.
❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ t❛ s✉② r❛✿ (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e), ∀x∗ ∈ C.
✶✼