Một phương pháp chiếu giải bài toán bất đẳng thức biến phân hai cấp

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❍❖⑨◆● ❚❍➚ ❚❍❷❖

▼❐❚ P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆
❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❍❆■ ❈❻P

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✷✵


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❍❖⑨◆● ❚❍➚ ❚❍❷❖

▼❐❚ P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆
❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❍❆■ ❈❻P
❈❤✉②➯♥ ♥❣➔♥❤✿
▼➣ sè✿

❚❖⑩◆ Ù◆● ❉Ö◆●
✽✹✻✵✶✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆

ở ữợ ồ
P ế

◆●❯❨➊◆✱ ◆❿▼ ✷✵✷✵




ử ử
ớ ỡ



ỵ



s







ữỡ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

✻

✶✳✶

✶✳✷

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶ ❙ü ❤ë✐ tö ②➳✉✱ ❤ë✐ tö ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ❚♦→♥ tû ❝❤✐➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✸ ◆â♥ ♣❤→♣ t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✹ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥
✶✳✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ▼ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ✤÷đ❝ ♠ỉ t ữợ t tự
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✸ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✹ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳

✳ ✻
✳ ✻
✳ ✼
✳ ✽
✳ ✽
✳ ✶✶
✳ ✶✶
✳ ✶✷
✳ ✶✹
✳ ✶✻

❈❤÷ì♥❣ ✷✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
✷✷

✷✳✶

❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✶✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✶✳✷ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸


✐✈

✷✳✶✳✸
✷✳✷

❚❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣
♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣
✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳
✷✳✷✳✷ ❙ü ❤ë✐ tư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❜✐➳♥
✳ ✳ ✳
✳ ✳ ✳
✳ ✳ ✳
✳ ✳ ✳

✳
✳
✳
✳

✷✹
✷✻
✷✻
✷✼

❑➳t ❧✉➟♥

✸✷

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✸


✶

▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐
◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝
❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ tỉ✐ ✤÷đ❝ t❤❛♠ ❣✐❛ ❤å❝ t➟♣✱ ♥❣❤✐➯♥
❝ù✉✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ✤➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥
✲ ❚✐♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✈➔ qỵ t ổ trỹ t ợ ồ
❑✶✷❆ ✭❦❤â❛ ✷✵✶✽ ✕ ✷✵✷✵✮ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ỳ tự qỵ
ụ ữ t tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳
✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët tổ ổ ữủ sỹ ữợ

❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨✳ ❚ỉ✐ ①✐♥
tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝ỉ ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉
❝ỉ ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ t t tợ ỗ ✤➣
❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔
t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✷✵
❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

❍♦➔♥❣ ❚❤à ❚❤↔♦




ỵ
H
C
Ã, Ã
(F, C)
NC (x0 )
S(F,C)
P(F, C)
P(F, C)

PC
(F, G, C)
S(G,C)


ổ rt tỹ

ởt t ỗ õ rộ ừ H
t ổ ữợ
t t tự ✈ỵ✐ →♥❤ ①↕ ❣✐→ F ✈➔
t➟♣ r➔♥❣ ❜✉ë❝ C
♥â♥ ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ C t↕✐ x0
t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
❱■(F, C)
❜➔✐ t♦→♥ tè✐ ÷✉
❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝❤✐➳✉ H ❧➯♥ C
❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣
t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(G, C)
t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❇❱■(F, G, C)


✸

❉❛♥❤ s→❝❤ ❜↔♥❣
✶✳✶
✶✳✷
✶✳✸

❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (5, 5, 5)T ∈ R3 ✱ ❝❤å♥ µ = 1/(k + 2) ✳ ✳ ✳ ✷✶
❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 2) ✳ ✷✶
❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 4) ✳ ✷✶


✹

▼ð ✤➛✉

❈❤♦ H ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈ỵ✐ t ổ ữợ Ã, Ã Ã
C ởt t ỗ õ rộ ừ H ✈➔ →♥❤ ①↕ F : C → H t❤÷í♥❣
✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❣✐→ ✭tr♦♥❣ ♠ët ✈➔✐ tr÷í♥❣ ❤đ♣✱ F ✤✐ tø H tỵ✐ H✮✳ ❇➔✐ t♦→♥
❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✤ì♥ trà✮ tr♦♥❣ H✱ ✈✐➳t t➢t ❱■(F, C)✱ ✤÷đ❝ ♣❤→t ❜✐➸✉
♥❤÷ s❛✉✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ F (x∗ ), x − x∗ ≥ 0 ✈ỵ✐ ♠å✐ x ∈ C.
❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(F, C) ✤÷đ❝ ❣✐ỵ✐ t❤✐➺✉ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦
♥➠♠ ✶✾✻✻ ❜ð✐ ●✳❏✳ ❍❛rt♠❛♥ ✈➔ ●✳ ❙t❛♠♣❛❝❝❤✐❛✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈✐➺❝ ❣✐↔✐ ❜➔✐
t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❬✼❪✳
❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â q✉❛♥ ❤➺ ♠➟t t❤✐➳t ✈ỵ✐ ♥❤✐➲✉ ❜➔✐ t♦→♥ t❤ü❝
t✐➵♥ ♥❤÷ ♠ỉ ❤➻♥❤ ❝➙♥ ❜➡♥❣ ♠↕♥❣ ❣✐❛♦ t❤ỉ♥❣✱ ❜➔✐ t tỹ t ỷ
ỵ
✶✾✼✶✱ ▼✳ ❙✐❜♦♥② ❬✶✸❪ ✤➣ ①➨t ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣
tr÷í♥❣ ❤đ♣ t➟♣ r➔♥❣ ❜✉ë❝ C ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû ✤ì♥ ✤✐➺✉✳
❈ơ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ trữớ ủ
t t ợ t➟♣ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕
❦❤æ♥❣ ❣✐➣♥ ✭tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❦❤✐ C ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû
✤ì♥ ✤✐➺✉✮✳
◆❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧➔ ♠ët ✤➲ t➔✐ ✤÷đ❝
♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ t➼♥❤ ù♥❣ ❞ö♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♥➔②
tr♦♥❣ ♠ët sè ♥❣➔♥❤ ❦❤♦❛ ❤å❝✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤÷đ❝ ♥❣❤✐➯♥
❝ù✉ ♠ð rë♥❣ t❤➔♥❤ ❝→❝ ❞↕♥❣ tê♥❣ q✉→t ❤ì♥ ♥❤÷ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥
♣❤➙♥ ✤❛ trà ✈ỵ✐ →♥❤ ①↕ F ❧➔ →♥❤ ①↕ ✤❛ trà✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ t➻♠


✺

✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱
❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳ ✳ ✳
▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ♠ët ữỡ ởt ợ t t

tự ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪✳ ◆ë✐
❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ✧❇➜t ✤➥♥❣ t❤ù❝
❜✐➳♥ ♣❤➙♥ tr♦♥❣ ổ rt ữỡ ợ t t tỷ ❝❤✐➳✉✱
→♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ t♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ò♥❣ ♠ët sè t➼♥❤
❝❤➜t❀ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt❀ ❣✐ỵ✐
t❤✐➺✉ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣
✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❈❤÷ì♥❣ ✷ ✧▼ët ♣❤÷ì♥❣
♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ rt
ữỡ ợ t t t tự ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt ❝ị♥❣ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥❀ ♠ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t
✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư
❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳


✻

❈❤÷ì♥❣ ✶

❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ ❍✐❧❜❡rt
❈❤÷ì♥❣ ♥➔② ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣
❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✱ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔
♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛
❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð tê♥❣ ❤đ♣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✷✱ ✺✱ ✽✱ ✶✵✱ ✶✶✱ ✶✷✱ ✶✺✱ ✶✼❪✳

✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ C ❧➔ ♠ët t➟♣ ỗ õ
rộ ừ H ỵ t ổ ữợ Ã, Ã tữỡ ự ữủ ✤à♥❤
❜ð✐ x =

✶✳✶✳✶

x, x ✈ỵ✐ ♠å✐ x ∈ H✳

❙ü ❤ë✐ tö ②➳✉✱ ❤ë✐ tö ♠↕♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ✭①❡♠ ❬✶❪✮✳ ▼ët ❞➣② {xk } ⊂ H ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ở
tử tợ x H ỵ xk → x∗ ✭t÷ì♥❣ ù♥❣ xk

x∗ ✮✱ ♥➳✉ xk − x∗ 0

tữỡ ự u, xk x 0 ợ ♠å✐ u ∈ H✮ ❦❤✐ k → ∞✳
▼ët ❞➣② {xk } ⊂ H ❤ë✐ tư ♠↕♥❤ ✤➳♥ x∗ t❤➻ ❝ơ♥❣ ❤ë✐ tư ②➳✉ ✤➳♥ x∗ ✱ ♥❤÷♥❣
✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❚✉② ♥❤✐➯♥✱ t➼♥❤ ❝❤➜t ❑❛❞❡❝✕❑❧❡❡ ❝❤➾ r❛ r➡♥❣

xk → x∗
❈❤♦ C = ∅, C ⊂ H✳

✈➔ xk

x∗ =⇒ xk → x∗ .


✼

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭①❡♠ ❬✷❪✮✳ ⑩♥❤ ①↕ S : C → H ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ✤â♥❣ t↕✐
✵✱ ♥➳✉ {xk } ❧➔ ♠ët ❞➣② tr♦♥❣ C s❛♦ ❝❤♦ xk

x¯ ✈➔ (I − S)(xk ) → 0✱ t❤➻

(I − S)(¯
x) = 0 ✳
❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝❤✉➞♥✱ t❛ ❝â t➼♥❤ ❝❤➜t s❛✉

❇ê ✤➲ ✶✳✶✳✸ ✭①❡♠ ❬✶❪✮✳ ❱ỵ✐ ♠é✐ x, y ∈ H✱ t❛ ❝â
= x

2

− y

(ii) tx + (1 − t) y

2

=t x

(i) x − y

✶✳✶✳✷

2

2

− 2 x − y, y .
2

+ (1 − t) y

2

− t (1 − t) x − y

2

, ∀t ∈ [0, 1] .

❚♦→♥ tû ❝❤✐➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

❍➻♥❤ ❝❤✐➳✉ ❝õ❛ ♠ët ✤✐➸♠ x ∈ H tr C ỵ PC (x) ởt t❤✉ë❝

C ✈➔ ❣➛♥ ✤✐➸♠ x ♥❤➜t✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
PC (x) = ❛r❣♠✐♥ { x − y : y ∈ C}.

✭✶✳✶✮

P❤➨♣ ❝❤✐➳✉ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✶✮ ❝â ❝→❝ t➼♥❤ ❝❤➜t s

ỵ C ởt t ỗ
õ rộ ừ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✳ ❑❤✐ ✤â
✭❛✮ x − PC (x), y − PC (x) ≤ 0, ∀y ∈ C, x ∈ H;
✭❜✮ ❤➻♥❤ ❝❤✐➳✉ PC (x) ❝õ❛ x tr➯♥ C ổ tỗ t t
PC (x) PC (y)

2

≤ PC (x) − PC (y), x − y , x, y H t ỗ ự

PC (x) − PC (y) ≤ x − y , ∀x, y ∈ H ✭t➼♥❤ ❦❤æ♥❣ ❣✐➣♥✮✳

▼➺♥❤ ✤➲ ✶✳✶✳✺ ✭❬✶✶❪✱ ▼➺♥❤ ✤➲ C ởt t ỗ õ
rộ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✳ ❑❤✐ ✤â
✭❛✮ x − PC (x)

2

≤ x−y

2

− y − PC (x) 2 , ∀x ∈ H, y ∈ C ❀

✭❜✮ PC (x) − PC (y)

2

≤ x−y

✭❝✮ x − PC (x − y)

2

≤ y , ∀x, y ∈ H;

✭❞✮ z − PC (x − y)

2

≤ x−z

2

2

− PC (x) − x + y − PC (y) 2 , ∀x, y ∈ H;

− 2 x − z, y + 5 y 2 , ∀x, z ∈ C, y ∈ H.






õ t

sỷ C t ỗ ré♥❣ 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 ✳
◆➳✉ f : H → R {+} ỗ tữớ tr H w H ữủ ồ
ữợ ừ f t↕✐ x ♥➳✉

f (y) ≥ w, y − x + f (x), y H.
tt ữợ ừ f t x ữủ ồ ữợ ừ

f t x ỵ f (x) f ữủ ồ ữợ t↕✐ x ♥➳✉
∂f (x) = ∅✱ ✈➔ f ❧➔ ❦❤↔ ữợ tr t ỗ õ C H ♥➳✉ ∂f (x) = ∅✱

✈ỵ✐ ♠å✐ x ∈ C ✳
❚ø õ t ữợ ừ ❤➔♠ f tr➻♥❤ ❜➔② ð tr➯♥✱
t❛ ❝â ✤÷đ❝ ❦➳t q✉↔ s

ỵ sỷ C t ỗ õ rộ
tr H f : H R {+} ỗ ữợ tr C õ

x0 rf (x) ✈ỵ✐ ♠å✐ x ∈ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
0 ∈ ∂f (x0 ) + NC (x0 ).

✶✳✶✳✹

⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ t♦→♥ tû ✤ì♥ ✤✐➺✉

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼ ✭①❡♠ ❬✷❪✮✳ ▼ët →♥❤ ①↕ S : C → H✱ ✤÷đ❝ ❣å✐ ❧➔
✭❛✮ ❦❤æ♥❣ ❣✐➣♥✱ ♥➳✉

S(x) − S(y) ≤ x − y

∀x, y ∈ C;

✭❜✮ tü❛ ❦❤æ♥❣ ❣✐➣♥✱ ♥➳✉ ❋✐①(S) = ∅ ✈➔

S(x) − x∗ ≤ x − x∗

∀(x, x∗ ) ∈ C × ❋✐①(S);

tỹ (S) = tỗ t β ∈ (0, 1) t❤ä❛ ♠➣♥

S(x) − x∗ ≤ β x − x∗

∀(x, x∗ ) ∈ C × ❋✐①(S);

✭❞✮ ♥û❛ (S) = tỗ t [0, 1) t❤ä❛ ♠➣♥

S(x) − x∗

2

≤ x − x∗

2

+ β x − S(x)

∀(x, x∗ ) ∈ C × ❋✐①(S);

2

✭❡✮ ✤â♥❣ ②➳✉ tr➯♥ C ♥➳✉ {xk } ⊂ C, xk

x ✈➔ S(xk )

✭❢✮ ❧✐➯♥ tö❝ ②➳✉ ♥➳✉ xn

S(x)✳

x✱ t❤➻ S(xn )

w t❤➻ w = S(x);

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ✭①❡♠ ❬✷❪✮✳ ❍➔♠ f : H → R ∪ {±∞} ✤÷đ❝ ❣å✐ ❧➔
✭❛✮ ♥û❛ tử ữợ t x C ợ ồ ❞➣② {xk } ⊂ C ❤ë✐ tö ♠↕♥❤ ✤➳♥ x
t❤➻ lim inf k→∞ f (xk ) ≥ f (x).
✭❜✮ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ x ∈ C ♥➳✉ ✈ỵ✐ ♠å✐ ❞➣② {xk } ⊂ C ❤ë✐ tö ♠↕♥❤ ✤➳♥ x
t❤➻ lim supk→∞ f (xk ) ≤ f (x).
❍➔♠ f ❧➔ ỷ tử ữợ ỷ tử tr tr C f ỷ tử
ữợ ỷ tử tr t ♠å✐ x ∈ C ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✾ ✭①❡♠ ❬✷❪✮✳ ▼ët →♥❤ ①↕ F : C → H ✤÷đ❝ ❣å✐ ❧➔
✭❛✮ γ ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✱ ♥➳✉

F (x) − F (y), x − y ≥ γ x − y

2

∀x, y ∈ C;

✭❜✮ ✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉

F (x) − F (y), x − y ≥ 0 ∀x, y ∈ C;
✭❝✮ ❣✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉

F (y), x − y ≥ 0 ⇒ F (x), x − y ≥ 0 ∀x, y ∈ C;
✭❞✮ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝ tr➯♥ C ✱ ♥➳✉

F (x) − F (y), x − y ≥ β F (x) − F (y)

2

∀x, y ∈ C;


✶✵

✭❡✮ ♣❛r❛✲✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉ F ✤ì♥ ✤✐➺✉ tr➯♥ C ✈➔

F (x) − F (y), x − y = 0 ⇒ F (x) = F (y) ∀x, y ∈ C;
✭❢✮ ♣❛r❛✲✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ S ⊂ C ✱ ♥➳✉ F ❣✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ✈➔

{x ∈ S, y ∈ C, F (y), x − y = 0} ⇒ y ∈ S;
✭❣✮ L✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✱ ♥➳✉

F (x) − F (y) ≤ L x − y

∀x, y ∈ C.

❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✾✱ ♥➳✉ F ❧➔ →♥❤ ①↕ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝ t❤➻ F ❧➔

L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➡♥❣ sè L =

1
β

✈➔ ✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ✈➔ t❛ ❝â q✉❛♥ ❤➺

(a) ⇒ (b) ⇒ (c)✳ ◆❤÷♥❣ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣

q✉→t✳ ❈❤➥♥❣ ❤↕♥ F : C → R ①→❝ ✤à♥❤ ❜ð✐ F (x) = x2 ❧➔ ❣✐↔ ✤ì♥ ✤✐➺✉✱ ♥❤÷♥❣
❦❤ỉ♥❣ ✤ì♥ ✤✐➺✉ tr➯♥ C = R❀ ❧➔ ✤ì♥ ✤✐➺✉✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥

C = [0, 1]✳
❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❜ê ✤➲ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ❝õ❛
t❤✉➟t t♦→♥ tr♦♥❣ ❝❤÷ì♥❣ s❛✉✳

❇ê ✤➲ ✶✳✶✳✶✵ ✭❬✶✼❪✱ ❇ê ✤➲ ✸✳✶✮✳ ❈❤♦ A : H → H ❧➔ t♦→♥ tû β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤
✈➔ L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ λ ∈ (0, 1] ✈➔ µ ∈ (0, L2β2 )✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ x ∈ H✱ →♥❤
①↕ T (x) = x − λµA(x) t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝

T (x) − T (y) ≤ (1 − λτ ) x − y
✈ỵ✐ τ = 1 −

∀x, y ∈ H,

1 − µ(2β − µL2 ) ∈ (0, 1]✳

❇ê ✤➲ ✶✳✶✳✶✶ ✭❬✶✵❪✱ ❇ê ✤➲ ✷✳✶✮✳ ❈❤♦ {λn } ✈➔ {βn } ❧➔ ❞➣② ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥
∞

∞

λn = ∞,
n=0

∞

λ2n
n=0

< ∞, ✈➔

λn n < .
n=0

õ

(i) ỗ t {nk } ⊂ {βn } t❤ä❛ ♠➣♥ limk→∞ βnk = 0.
(ii) ◆➳✉ {λn } ✈➔ {βn } t❤ä❛ ♠➣♥ βn+1 − βn < θλn , ∀θ > 0✱ t❤➻ {βn } t❤ä❛ ♠➣♥
limn→∞ βn = 0.


✶✶

✶✳✷ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♠ët sè ❜➔✐ t♦→♥
❧✐➯♥ q✉❛♥
✶✳✷✳✶

❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥

❈❤♦ C t ỗ rộ tr ởt ổ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈➔
→♥❤ ①↕ F : C → H✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ①→❝ ✤à♥❤ ❜ð✐ ♠✐➲♥ r➔♥❣
❜✉ë❝ C ✈➔ →♥❤ ①↕ ❣✐→ F ✱ ỵ (F, C) t
x C

F (x∗ ), x − x∗ ≥ 0 ∀x ∈ C.

s❛♦

ừ t (F, C) ữủ ỵ S(F,C) ỹ tỗ t
ừ t (F, C) ✤÷đ❝ s✉② r❛ tø t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ F ✈➔ ✤✐➲✉ ❦✐➺♥ t➟♣ C ❧➔
❝♦♠♣❛❝t✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ t➟♣ C ổ t t ỵ t ở rr
ổ ỏ õ t ử ữủ õ sỹ tỗ t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❱■(F, C)
❝â t❤➸ ✤÷đ❝ t❤✐➳t ❧➟♣ ❞ü❛ ✈➔♦ t➼♥❤ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ❝õ❛ →♥❤
①↕ ❣✐→ F ✳

▼➺♥❤ ✤➲ ✶✳✷✳✶ ✭①❡♠ ❬✹❪✮✳ ◆➳✉ F : C → H ❧➔ →♥❤ ①↕ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥
C ✈➔ L✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C t❤➻ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(F, C)
❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳

❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ 0 < µ <

2β
L2

✈➔ ①➨t →♥❤ ①↕ T : C → C ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

T (x) = PC (x − µF (x)) ∀x ∈ C.
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x, y ∈ C ✱ t❛ ❝â✿

T (x) − T (y)

2

= PC (x − µF (x)) − PC (y − µF (y))
≤ x − µF (x) − y + µF (y)

= x−y

2

2

2

− 2µ F (x) − F (y), x − y + µ2 F (x) − F (y) 2 .

❉♦ F ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ✈➔ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✱ ♥➯♥

T (x) − T (y)

2

≤ x−y

2

− 2µβ x − y

2

+ µ2 L2 x − y

=(1 − 2µβ + µ2 L2 ) x − y 2 .

2

✶✷

❉♦ ✤â

T (x) − T (y) ≤ (1 − µ(2β + µL2 ) x − y .
=ρ x − y ,
tr♦♥❣ ✤â✱ ρ =

(1 − µ(2β + µL2 ) ∈ [0, 1). ❱➟② T : C → C ❧➔ →♥❤

ỵ tỗ t ❞✉② ♥❤➜t x∗ ∈ C s❛♦ ❝❤♦ T (x∗ ) = x∗ ✳ ❉♦
✤â✱ x∗ ∈ S(F,C) ✳

✶✳✷✳✷

▼ët ❜➔✐ t♦→♥ tỹ t ữủ ổ t ữợ t tự
♣❤➙♥

▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ tr♦♥❣ ❧➽♥❤ ✈ü❝ ❦✐♥❤ t➳ ✭❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ ❦✐♥❤ t➳✮ ✤÷đ❝ ổ õ ữợ t t tự ❜✐➳♥ ♣❤➙♥✳
▼ët sè ♠ỉ ❤➻♥❤ ❦✐♥❤ t➳ ✤÷đ❝ ①➙② ❞ü♥❣ st
ỗ ❝➛✉ ❝õ❛ ♠ët ❧♦↕✐ ❤➔♥❣ ❤â❛ ♥➔♦ ✤â✱ ❝→❝ ♠æ ❤➻♥❤ ♥➔② t❤ü❝ ❝❤➜t
❝❤➼♥❤ ❧➔ ♠æ ❤➻♥❤ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳ ❑❤→✐ ♥✐➺♠ ❝➙♥ ❜➡♥❣ ❦✐♥❤ t➳ ❝â t❤➸
❜✐➸✉ t ữợ ố q ờ s ỳ ỗ ữủt q
ừ ộ ❤â❛✳ ❉♦ ✤â✱ ❤➛✉ ❤➳t ❝→❝ ♠æ ❤➻♥❤ ❝➙♥ ❜➡♥❣ t
õ t t ữợ t♦→♥ ❜ò ❤♦➦❝ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ✣➸ ♠✐♥❤
❤å❛ ❝❤♦ ❦❤➥♥❣ ✤à♥❤ ♥➔②✱ t❛ ♠æ t↔ ♠ët tr♦♥❣ ♥❤ú♥❣ ổ t tờ qt
t rs ữ r
rữợ t✐➯♥ t❛ ①➨t ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ t❤à tr÷í♥❣ ✤÷đ❝ ổ t ữợ

t ữỡ tr ởt ỡ s s t ố n t ỵ ❧➛♥
❧÷đt ❜ð✐ i, i = 1, 2, . . . , n m ỵ t tử ộ ỵ ữủ ỵ

j, j = 1, 2, . . . , m ỵ p tỡ n t ừ ộ t
ỗ t p = (p1 , p2 , . . . , pn ) sỷ ữủ ố ợ t
tự i ừ tt ỵ di ◆â✐ ❝❤✉♥❣ di ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ ❝õ❛ t➜t ❝↔ ❝→❝
♠➦t ❤➔♥❣✱ tù❝ ❧➔ di = di (p)✳ ❑❤✐ ✤â t❛ ❝â
m

di (p) =

dij (p),
j=1

tr♦♥❣ ✤â dij (p) ❧➔ ❝❤➾ ♥❤✉ ❝➛✉ ✤è✐ ✈ỵ✐ ♠➦t ❤➔♥❣ t❤ù i ❝õ❛ ✤↕✐ ỵ tự j


✶✸

❚÷ì♥❣ tü t❛ ❝â ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ♠➦t ❤➔♥❣ t❤ù i tt ỵ ỵ
si ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ ❝õ❛ t➜t ❝↔ ❝→❝ ♠➦t ❤➔♥❣✱ tù❝ ❧➔
m

si (p) =

sij (p),
j=1

tr♦♥❣ ✤â sij (p) ❧➔ ❧÷đ♥❣ ừ t tự i ỵ tự j ợ tỡ
p

õ t tờ ủ ữủ ✤è✐ ✈ỵ✐ t➜t ❝↔ ❝→❝ ♠➦t ❤➔♥❣ t❤➔♥❤ ♠ët ✈➨❝✲tì
❝ët n✲❝❤✐➲✉ d ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥ {d1 , d2 , . . . , dn } ✈➔ ❧÷đ♥❣ ❝✉♥❣ ✤è✐ ✈ỵ✐ n ♠➦t
❤➔♥❣ t❤➔♥❤ ♠ët ✈➨❝✲tì ❝ët n✲❝❤✐➲✉ s ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥ {s1 , s2 , . . . , sn }✳
✣✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ ❝õ❛ t❤à tr÷í♥❣ ②➯✉ ❝➛✉ ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ♠é✐ ♠➦t ❤➔♥❣
♣❤↔✐ ❜➡♥❣ ❧÷đ♥❣ ❝➛✉ ừ t õ ợ tỡ p tữỡ ữỡ ợ
ữỡ tr s

s(p ) = d(p ).
ữỡ tr tr õ t ữủ ữợ tờ q✉→t ♥➳✉ t❛ ①→❝
✤à♥❤ ✈➨❝✲tì x ≡ p ✈➔ F (x) s(p) d(p) ú ỵ r ợ ❜➔✐ t♦→♥ ❣✐↔✐ ❤➺
♣❤÷ì♥❣ tr➻♥❤ ❝❤÷❛ ✤õ tê♥❣ q✉→t ✤➸ ❜↔♦ ✤↔♠ ❝❤♦ ❜➔✐ t♦→♥ ✤❛♥❣ ①➨t✱ ❝❤➥♥❣ ❤↕♥
tr÷í♥❣ ❤đ♣ p∗ ≤ 0✳
❚✐➳♣ t❤❡♦✱ t❛ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥ ❝❤♦ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ t❤à
tr÷í♥❣ ð tr➯♥✳ t trữớ ủ ữủ trữợ ❤➔♠ ❝✉♥❣✳ ❑❤✐
✤â✱ t❤❛② ✈➻ ✤✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ t❤à tr÷í♥❣ ✤÷đ❝ ①➨t tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤➺ ♣❤÷ì♥❣
tr➻♥❤✱ t❛ ①➨t ✤✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ s❛✉✿ ✈ỵ✐ ♠é✐ ♠➦t ❤➔♥❣ t❤ù i✱ i = 1, 2, . . . , n✳

s(p∗ ) − d(p∗ ) = 0 ♥➳✉ p∗ > 0,
i
s(p∗ ) − d(p∗ ) ≥ 0 ♥➳✉ p∗ = 0.
i

✣✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ♥➳✉ ❣✐→ ❝õ❛ ♠é✐ ♠➦t ❤➔♥❣ ❧➔ ❞÷ì♥❣ tr♦♥❣
tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ t❤➻ ❧÷đ♥❣ ❝✉♥❣ ♣❤↔✐ ❜➡♥❣ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ❧♦↕✐ ♠➦t ❤➔♥❣ ✤â✳
▼➦t ❦❤→❝✱ ♥➳✉ ❣✐→ ❝õ❛ ♠➦t ❤➔♥❣ ð tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❧➔ ❜➡♥❣ 0 t❤➻ ❦❤✐ ✤â ❧÷đ♥❣
❝✉♥❣ ✈÷đt q ữủ ố ợ t õ tự ❧➔ s(p∗ ) − d(p∗ ) > 0✱
❤❛② t❤à tr÷í♥❣ ♥❣ø♥❣ ❤♦↕t ✤ë♥❣✳ ❍ì♥ ♥ú❛✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ❜➜t ♣❤÷ì♥❣
tr➻♥❤ ♥➔② ❜↔♦ ✤↔♠ r➡♥❣ ❣✐→ ❝õ❛ ❝→❝ ♠➦t ❤➔♥❣ ❦❤æ♥❣ ❧➜② ❣✐→ trà ➙♠✳ ❑❤✐ ✤â✱ ♠æ

✶✹

❤➻♥❤ t♦→♥ ❞↕♥❣ ❜➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥ ❝❤♦ ❜➔✐ t♦→♥ ①➨t ð tr➯♥ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥
♥❤÷ s❛✉✿ ①→❝ ✤à♥❤ p∗ ∈ Rn+ t❤ä❛ ♠➣♥

s(p∗ ) − d(p∗ ) ≥ 0 ✈➔

s(p∗ ) − d(p∗ ), p∗ = 0.

❍ì♥ ♥ú❛✱ t❛ ❜✐➳t r➡♥❣ ❜➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥ ❧➔ ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛
❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❉♦ ✤â✱ t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ❜➔✐ t♦→♥ ❜ò ♣❤✐ t ữợ
t tự ữ s t ✤✐➸♠ p∗ ∈ Rn+ t❤ä❛ ♠➣♥

s(p∗ ) − d(p∗ ), p − p∗ ≥ 0 ∀p ∈ Rn+ .

✶✳✷✳✸

▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥

❙❛✉ ✤➙② ❧➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(F, C)✳

❇➔✐ t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ♥➳✉ H = Rn ✈➔ C ❧➔ t♦➔♥ ❜ë ❦❤æ♥❣ Rn t
t (F, C) tữỡ ữỡ ợ t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû F (x∗ ) = 0.

▼➺♥❤ ✤➲ ✶✳✷✳✷ ✭①❡♠ ❬✽❪✮✳ ◆➳✉ H = Rn ✱ C = Rn ✈➔ →♥❤ ①↕ F : Rn → Rn t❤➻
x∗ ∈ Rn ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(F, C) ❦❤✐ ✈➔ ❝❤➾
❦❤✐ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû F (x∗ ) = 0.

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ F (x∗ ) = 0 t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮ ①↔② r❛ ❞➜✉ ❜➡♥❣✳ ❉♦

✤â✱ t❛ ❝â x∗ ∈ S(F,C) .
◆❣÷đ❝ ❧↕✐✱ ♥➳✉ x∗ ∈ S(F,C) t❤➻ F (x∗ ), x − x∗ ≥ 0 ✈ỵ✐ ♠å✐ x ∈ Rn . ❈❤å♥

x = x∗ − F (x∗ ), t❛ ✤÷đ❝ F (x∗ ), x − x∗ ≥ 0 ❤❛② − F (x∗ )

2

≥ 0. ❉♦ ✤â

F (x∗ ) = 0.

❇➔✐ t♦→♥ tè✐ ÷✉
❈❤♦ C ❧➔ t➟♣ ỗ õ rộ ừ H F : C R ỗ
ỷ tử ữợ t tố ữ ỵ P(F, C) t♦→♥
❚➻♠ x∗ ∈ C

s❛♦ ❝❤♦

F (x∗ ) ≤ F (y) ✈ỵ✐ ♠å✐ y ∈ C.


✶✺

❱➼ ❞ö ✶✳✷✳✸✳ ❈❤♦ f ❧➔ ❤➔♠ sè ❦❤↔ ✈✐ tr➯♥ [a, b] ⊂ R✳ ❚➻♠ x∗ ∈ [a, b] s❛♦ ❝❤♦
f (x∗ ) = min f (x).
x∈[a,b]

✭❛✮ ◆➳✉ x∗ ∈ (a, b) t❤➻ f (x∗ ) = 0✳
✭❜✮ ◆➳✉ x∗ = a t❤➻ f (x∗ ) ≥ 0✳
✭❝✮ ◆➳✉ x∗ = b t❤➻ f (x∗ ) ≤ 0✳

❚r♦♥❣ ❝↔ ❜❛ tr÷í♥❣ ❤đ♣ t❛ ✤➲✉ ❝â f (x∗ )(x − x∗ ) ≥ 0✳ ✣➙② ❧➔ ♠ët ❜➜t ✤➥♥❣
t❤ù❝ ❜✐➳♥ ♣❤➙♥✳

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
❈❤♦ C ⊂ H ❧➔ ♠ët ởt t ỗ õ rộ ỡ trà F : C → C ✳
❑❤✐ ✤â ❜➔✐ t♦→♥ t ở ỵ P(F, C) t♦→♥
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = F (x∗ ).

✭✶✳✸✮

▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣
✤÷đ❝ ♥➯✉ tr♦♥❣ ♠➺♥❤ ữợ

sỷ C ởt t rộ ỗ õ ừ
ổ ❍✐❧❜❡rt t❤ü❝ H✳ ❑❤✐ ✤â x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t (F, C)
ợ ộ à > 0✱ x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ PC (I − µF ) : C → C,
tù❝ ❧➔

x∗ = PC (x∗ − µF (x∗ )).

✭✶✳✹✮

❈❤ù♥❣ ♠✐♥❤✳ ✣➦t T (x) = PC (x − λF (x))✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✤✐➸♠ ❜➜t ✤ë♥❣
✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✹✱ t❛ ❝â

x∗ ∈ ❋✐①(T ) ⇔ x∗ = T (x∗ )
⇔ x∗ = PC (x∗ − λF (x∗ ))
⇔ x∗ − λF (x∗ ) − x∗ , z − x∗ ≤ 0,
⇔ λF (x∗ ), z − x∗ ≥ 0,
⇔ F (x∗ ), z − x∗ ≥ 0,

⇔ x∗ ∈ S(F,C) .

∀z ∈ C
∀z ∈ C

∀z ∈ C


✶✻

◆❤➟♥ ①➨t ✶✳✷✳✺✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝ơ♥❣ ❝â t❤➸ ✤÷❛ ✈➲ ❜➔✐
t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣

x∗ = F (x∗ ),

✭✶✳✺✮

F (x) = x − g(x) + PC [g(x) − ρ(A(x) − T (x))],

✭✶✳✻✮

ð ✤➙②

✈ỵ✐ ρ ❧➔ ♠ët t❤❛♠ sè ❞÷ì♥❣✳
❚ø ♥❤➟♥ ①➨t ♥➔②✱ t❛ ①➙② ❞ü♥❣ t❤✉➟t t♦→♥ s❛✉ ✤➙② ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐
t♦→♥ ❱■(F, C)✳

✶✳✷✳✹

▼ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥

❚❤✉➟t t♦→♥ ✶✳✷✳✻ ✭①❡♠ ❬✶✷❪✮✳ ❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ x0 ∈ H✳ ❉➣②
❧➦♣ {xk+1 } ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉

xk+1 = xk − g(xk ) + PC [g(xk ) − ρ(A(xk ) − T (xk ))],

k = 0, 1, 2, . . . ✭✶✳✼✮

tr♦♥❣ ✤â ρ > 0 ❧➔ ❤➡♥❣ sè✳
▼ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t
✶✳ ◆➳✉ g(x) = x ∈ C t❤➻ ❞➣② ❧➦♣ ✭✶✳✼✮ ❝â ❞↕♥❣

x0 ∈ H,

xk+1 = PC [ρ(A(xk ) − T (xk ))],

k = 0, 1, 2, . . .

✭✶✳✽✮

✷✳ ◆➳✉ T (x) = 0 t❤➻ ❞➣② ❧➦♣ ✭✶✳✼✮ ❝â ❞↕♥❣

x0 ∈ H,

xk+1 = xk − g(xk ) + PC [g(xk ) − ρA(xk )],

k = 0, 1, 2, . . .
✭✶✳✾✮

✸✳ ◆➳✉ T (x) = 0 ✈➔ g = I ✱ →♥❤ ①↕ ✤ì♥ ✈à tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✱

t❤➻ ❞➣② ❧➦♣ ✭✶✳✼✮ ❝â ❞↕♥❣

x0 ∈ H,

xk+1 = PC [xk − ρA(xk )],

k = 0, 1, 2, . . .

✭✶✳✶✵✮




ỵ A, g : H −→ H ❧➔ →♥❤ ①↕ ✤ì♥
✤✐➺✉ ♠↕♥❤ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ t÷ì♥❣ ù♥❣✳ ◆➳✉ →♥❤ ①↕ T : C → C ❧➔ ❧✐➯♥ tö❝
▲✐♣❝❤✐t③ t❤➻

xk+1 → x∗ tr➯♥ H,
✈ỵ✐

ρ−

α + γ(k − 1))
<
β2 − γ2

α + γ(k − 1)2 − (β 2 − γ 2 )k(2 − k))
,
β2 − γ2

α > γ(1 − k) +

(β 2 − γ 2 )k(2 − k) ✈➔ γ(1 − k) < α,

k<1

tr♦♥❣ ✤â {xk+1 } ❧➔ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✼✮ ✈➔ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t
✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥

A(x∗ ), g(x) − g(x∗ ) ≥ T (x∗ ), g(x) − g(x∗ )

∀g(x) ∈ C.

✭✶✳✶✶✮

✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ t ờ s

ờ C ởt t ỗ tr ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ H t❤➻ x∗ ∈ H ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✶✶✮ ❦❤✐
✈➔ ❝❤➾ ❦❤✐ x∗ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤

g(x∗ ) = PC [g(x∗ ) − ρ(A(x∗ ) − T (x∗ ))],

✭✶✳✶✷✮

tr♦♥❣ ✤â ρ > 0 ❧➔ ❤➡♥❣ sè✱ PC ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝❤✐➳✉ H ❧➯♥ C ✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✽✱ ♥❣❤✐➺♠ x∗ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥
♣❤➙♥ ✭✶✳✶✶✮ ❝â t❤➸ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➸♠ ❜➜t ✤ë♥❣ ✭✶✳✶✷✮✳ ❉♦ ✤â✱
tø ✭✶✳✶✷✮ ✈➔ ✭✶✳✻✮✱ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❦❤æ♥❣ ❣✐➣♥ ❝õ❛ →♥❤ ①↕ PC t❛ ♥❤➟♥ ✤÷đ❝

xk+1 − x∗

= xk − x∗ − (g(xk ) − g(x∗ ))
+ PC [g(xk ) − ρ(A(xk ) − T (x∗ ))]
− PC [g(x∗ ) − ρ(A(x∗ ) − T (x∗ ))]


✶✽

❤❛②

xk+1 − x∗

≤ xk − x∗ − (g(xk ) − g(x∗ ))
+ PC [g(xk ) − ρ(A(xk ) − T (x∗ ))]
− PC [g(x∗ ) − ρ(A(x∗ ) − T (x∗ ))]
xk − x∗ − (g(xk ) − g(x∗ ))

≤2
+

xk − x∗ − ρ(A(xk ) − A(x∗ )) + ρ(T (xk ) − T (x∗ ))

.

✭✶✳✶✸✮
❱➻ A ❧➔ →♥❤ ①↕ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ g ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ♥➯♥
2

xk − x∗ − (g(xk ) − g(x∗ ))

≤ (1 − 2δ + σ 2 ) xk − x∗

2

✭✶✳✶✹✮

✈➔

xk − x∗ − ρ(A(xk ) − A(x∗ ))

2

≤ (1 − 2ρα + ρ2 β 2 ) xk − x∗

2

.

✭✶✳✶✺✮

❚ø ✭✶✳✶✸✮✱ ✭✶✳✶✹✮✱ ✭✶✳✶✺✮ ✈➔ →♣ ❞ö♥❣ t➼♥❤ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ❝õ❛ T ✱ t❛ ✤÷đ❝

xk+1 − x∗ ≤ {(2 1 − 2δ + σ 2 ) + ργ +

1 − 2αρ + ρ2 β 2 } xk − x∗

= {k + ργ + t(p)} xk − x∗
= θ xk − x∗ ,

tr♦♥❣ ✤â

k = 2 1 − 2δ + σ 2 ,
t(ρ) =

1 − 2αρ + ρ2 β 2 ,

✈➔

θ = k + ργ + t(ρ).
●✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ t(ρ) ❧➔ ρ¯ = α/β 2 ✈ỵ✐ t(ρ) =

1 − α2 /β 2 ✳ ❚❛ s➩ ❝❤➾ r❛

θ < 1✳ ❚❤➟t ✈➟②✱ ❝❤♦ ρ = ρ¯, k + ργ + t(¯
ρ) < 1 ❦➨♦ t❤❡♦
k < 1 ✈➔ α > γ(1 − k) +

(β 2 − γ 2 )k(2 − k).

❉♦ ✤â✱ θ = k + ργ + t(ρ) < 1 ✈ỵ✐ ♠å✐ ρ ✈ỵ✐

ρ−

α + γ(k − 1))
<
β2 − γ2

α + γ(k − 1)2 − (β 2 − γ 2 )k(2 − k))
,

β2 − γ2

k<1


✶✾

(β 2 − γ 2 )k(2 − k) ✈➔ γ(1 − k) < α.

α > γ(1 − k) +

❇ð✐ ✈➻ θ < 1 ♥➯♥ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ✈➔ ❞♦
✤â✱ ❞➣② ❧➦♣ xn+1 t❤✉ ✤÷đ❝ ❤ë✐ tư ✤➳♥ x∗ ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ừ t t
tự

ú ỵ ✶✳✷✳✾✳ ❚❛ ❝â ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t s❛✉ ✤➙②✿
✶✳ ◆➳✉ g(x) = x ∈ C t❤➻ ❜➔✐ t♦→♥ ✭✶✳✶✶✮ tữỡ ữỡ ợ t t
tỷ x C s❛♦ ❝❤♦

A(x∗ ), x − x∗ ≥ T (x∗ ), x − x∗

∀x ∈ C.

✭✶✳✶✻✮

✷✳ ◆➳✉ T (x) = 0 t t tữỡ ữỡ ợ t t ♣❤➛♥ tû

x∗ ∈ C s❛♦ ❝❤♦ g(x∗ ) ∈ C ✈➔ t❤ä❛ ♠➣♥
A(x∗ ), g(x) − g(x∗ ) ≥ 0 ∀g(x) ∈ C.

✭✶✳✶✼✮

✸✳ ◆➳✉ T (x) = 0 ✈➔ g = I ✱ →♥❤ ①↕ ✤ì♥ ✈à tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ rt tỹ H
t t tữỡ ữỡ ợ ❜➔✐ t♦→♥ t➻♠ ♣❤➛♥ tû x∗ ∈ C s❛♦ ❝❤♦

A(x∗ ), x − x∗ ≥ 0 ∀x ∈ C.

✭✶✳✶✽✮

✣➙② ❝❤➼♥❤ ❧➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✷✮ ✈ỵ✐ →♥❤ ①↕ ❣✐→ F ✤÷đ❝
t❤❛② ❜ð✐ A✳

◆❤➟♥ ①➨t ✶✳✷✳✶✵ ✭①❡♠ g = I ỗ ♥❤➜t✳ ❚r♦♥❣ tr÷í♥❣
❤đ♣ ♥➔② k = 0 ✈➔ ✭✶✳✻✮ trð t❤➔♥❤

F (x) = PC [x − ρ(Ax − T (x))],
✈➔ θ = ργ + t(p) < 1 ✈ỵ✐ 0 < ρ < 2(α − γ)/(β 2 − γ 2 ), ργ < 1 ✈➔ γ < α.
❉♦ ✤â✱ →♥❤ ①↕ F (x) ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✻✮✳
✷✳ ◆➳✉ g = I ✱ →♥❤ ①↕ ỗ t T (x) = 0 r trữớ ủ ♥➔②✱ k = 0✱

γ = 0 ✈➔ ✭✶✳✻✮ trð t❤➔♥❤
F (x) = PC [x − ρAx],


✷✵

✈ỵ✐ θ = t(ρ) < 1 ✈ỵ✐ 0 < ρ < 2α/β 2 ✳ ❉♦ ✤â →♥❤ ①↕ F (x) ❝â ♠ët ✤✐➸♠ ❜➜t
✤ë♥❣✱ ✤â ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✽✮✳
✸✳ ◆➳✉ T (x) ≡ 0✱ t❤➻ γ = 0 ✈➔ ✭✶✳✻✮ trð t❤➔♥❤

F (x) = x − g(x) + PC [g(x) − ρA(x)],
✈➔

θ = k + t(ρ) < 1 ✈ỵ✐ k < 1, α > β
✈➔

α
ρ− 2 <
β

k(k − 2),

α2 − β 2 (2k − k 2 )
.
β2

❉♦ ✤â →♥❤ ①↕ F (x) ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✱ ✤â ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✼✮✳
❙❛✉ ✤➙② ❧➔ ♠ët ✈➼ ❞ö ♠✐♥❤ ❤å❛✳

❱➼ ❞ö ✶✳✷✳✶✶✳ ❳➨t ❜➔✐ t♦→♥ ❝ü❝ trà ❝â r➔♥❣ ❜✉ë❝
ϕ(x∗ ) = min ϕ(x),
x∈C

✭✶✳✶✾✮

✈ỵ✐ ❤➔♠ ϕ : R3 → R ①→❝ ✤à♥❤ ❜ð✐

ϕ(x) = (x1 − 1)2 + (x2 − 2)2 + (x3 − 3)2
✈➔

C = {x = (x1 , x2 , x3 ) ∈ R3 : x1 + 2x2 − x3 ≤ 2}
❧➔ t➟♣ ❝♦♥ ❦❤→❝ rộ ỗ õ tr ổ R3 ✤â✱ t❛ ❝â ❣r❛❞✐❡♥t

ϕ : R3 → R3 ❝õ❛ ❤➔♠ ϕ ❧➔
ϕ(x) = 2x − a
✈ỵ✐ x = (x1 , x2 , x3 )T ∈ R3 ✈➔ a = (2, 4, 6)T ∈ R3 ✳ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❜➔✐ t♦→♥
✭✶✳✶✾✮ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✿

ϕ(x∗ ), x − x∗ ≥ 0 ∀x ∈ C.
◆❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✾✮ ❧➔ ✤✐➸♠ x∗ = (1, 2, 3)T ∈ C ⊂ R3 ✳

✭✶✳✷✵✮


✷✶

❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✶✵✮ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥
✭✶✳✷✵✮ ❝ô♥❣ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✾✮ ✈ỵ✐ A(x) =

ϕ(x) ❝â t➼♥❤ ❝❤➜t

2✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ 2✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✳
❑➳t q✉↔ t➼♥❤ t♦→♥ tr➯♥ ▼❆❚▲❆❇ ✤÷đ❝ ❝❤♦ tr♦♥❣ ❝→❝ ❇↔♥❣ ✶✳✶✕✶✳✸✳

◆❤➟♥ ①➨t ✶✳✷✳✶✷✳ ❚ø ❇↔♥❣ ✶✳✶✕✶✳✸ ♥❤➟♥ t❤➜② ✈ỵ✐ ①➜♣ ①➾ ❜❛♥
ợ sỹ ỹ ồ t số à ũ ❤đ♣✱ t❛ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❦❤→ tèt
❝❤♦ ú ừ t s ữợ
❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐

x0 = (5, 5, 5)T ∈ R3 ✱

xk

❙❛✐ sè ✭

k

✭sè ❧➛♥ ❧➦♣✮
✶✵
✺✵

(1.0727, 2.0545, 3.0364)T
T

(1.0031, 2.0024, 3.016)

xk − x∗

✮

❚✐♠❡

✵✳✵✾✼✾

✵✳✵✼✸s

✵✳✵✵✹✷

✵✳✵✸✶s

(1.0008, 2.0006, 3.0004)T

✵✳✵✵✶✶

✵✳✵✷✼s

✺✵✵

(1.0000, 2.0000, 3.0000)T

✹✳✷✾✾✺❡✲✵✺

✵✳✾✽✻s

T

✶✳✵✼✻✵❡✲✵✺

✶✳✵✵✻s

(1.0000, 2.0000, 3.0000)

❇↔♥❣ ✶✳✷✿ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐

✭sè ❧➛♥ ❧➦♣✮

x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 2)

xk

❙❛✐ sè ✭

xk − x∗

✮

❚✐♠❡

✶✵

(0.6182, 0.8727, 2.4000)T

✶✳✸✸✷✾

✵✳✵✺✾s

✺✵

T

✵✳✵✺✼✺

✵✳✵✸✼s

✶✵✵

(0.9958, 1.9877, 2.9935)T

✵✳✵✶✹✺

✵✳✵✸✹s

✺✵✵

(0.9998, 1.9995, 2.9999)T

✺✳✸✷✽✹❡✲✵✹

✶✳✵✹✺s

T

✶✳✸✸✸✹❡✲✵✹

✶✳✶✸✸s

✶✵✵✵

(0.9835, 1.9514, 2.9741)

(1.0000, 1.9999, 3.0000)

❇↔♥❣ t t ợ

k

à = 1/(k + 2)

k

ồ

số ❧➦♣✮

x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 4)

xk

❙❛✐ sè ✭

xk − x∗

✮

❚✐♠❡

✶✵

(−0.6154, −2.7692, 2.0000)T

✺✳✶✸✸✼

✵✳✵✶✸s

✺✵

(0.9086, 1.7300, 2.9434)T

✵✳✷✾✵✻

✵✳✵✸✼s

✶✵✵

(0.9760, 1.9292, 2.9852)

T

✵✳✵✼✻✷

✵✳✵✽✶s

✺✵✵

(0.9990, 1.9971, 2.9994)T

✵✳✵✵✸✷

✶✳✵✼✼s

✼✳✾✻✽✼❡✲✵✹

✶✳✶✸✽s

✶✵✵✵

(0.9997, 1.9993, 2.9998)

T