ĐẠI HỌC ĐÀ NẴNG
TRƯỜNG ĐẠI HỌC SƯ PHẠM
KHOA TỐN

LÊ HỒNG NHUẬN

CƠ SỞ LÝ THUYẾT
CHO BÀI TỐN TỐI ƯU CĨ ĐIỀU KIỆN
CHUYÊN NGÀNH: SƯ PHẠM TOÁN HỌC

Giảng viên hướng dẫn: T.S Phạm Quý Mười


✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
❑❍❖❆ ❚❖⑩◆
✖✖✖ ✯ ✖✖✖
▲➊ ❍❖⑨◆● ◆❍❯❾◆

❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚
❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❈➶ ✣■➋❯ ❑■➏◆
❈❍❯❨➊◆ ◆●⑨◆❍✿ ❙× P❍❸▼ ❚❖⑩◆ ❍➴❈

❑❍➶❆ ▲❯❾◆ ❚➮❚ P
ữợ P ỵ ữớ

✶✷ ♥➠♠ ✷✵✶✾


▼Ö❈ ▲Ö❈
▲❮■ ◆➶■ ✣❺❯

✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺
✶✳✻
✶✳✼
✶✳✽

▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
●✐→ trà r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✶ ●✐→ trà r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ❉↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❚æ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ sè ❧✐➯♥ tö❝ tr➯♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỵ r tr tr
t ỗ ỗ
ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✷ ❍➔♠ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈→❝ ✤à♥❤ ỵ t s s tỹ


































































































































































é ▲➑ ❚❍❯❨➌❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❈➶ ✣■➋❯ ❑■➏◆

✷✳✶
✷✳✷
✷✳✸
✷✳✹

❇➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ ✳
❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët
❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ✳
❇➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳
✳
✳
✳

✳
✳
✳
✳

❑➌❚ ▲❯❾◆
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✳
✳
✳
✳

✳
✳

✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✷
✹

✹
✺
✽
✽
✶✵
✶✶
✶✷

✶✺
✶✺
✶✺
✶✼
✷✵

✷✸
✷✸
✷✺
✸✹
✸✾

✹✶
✹✶

✶



ỵ tt tố ữ ởt ỹ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝
✈➜♥ ✤➲ ✈➲ ❝ü❝ trà ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ✤í✐ sè♥❣ t❤ü❝ t➳ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❝→❝ ♥❣➔♥❤
❦❤♦❛ ❤å❝✳ ❚✉② ♥❤✐➯♥ ♣❤↔✐ ✤➳♥ ♥❤ú♥❣ ♥➠♠ ✸✵✱ ✹✵ ❝õ❛ t❤➳ ❦➾ tố ữ ợ ữủ
t ợ tữ ởt ỵ tt ở ợ ữợ ❝ù✉ ❦❤→❝ ♥❤❛✉
✈➔ ♥❣➔② ❝➔♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ ❤ì♥ tr♦♥❣ ❝✉ë❝ sè♥❣ ♥❤í sü ♣❤→t tr✐➸♥ ❝õ❛ ❝ỉ♥❣
♥❣❤➺ t❤ỉ♥❣ t✐♥✱ ✤➦❝ ❜✐➺t ❧➔ ♠→② t➼♥❤✳
❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ❝ì s ỵ tt t tố ữ trỡ õ ✤✐➲✉ ❦✐➺♥✳ ❈ö
t❤➸✱ ❦❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✱ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➲✉
❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ tè✐ ữ õ õ ỗ
ữỡ q✉→t ♥❤ú♥❣ ✈➜♥ ✤➲ ❝❤✉♥❣ ♥❤➜t ❦❤✐ t✐➳♣ ❝➟♥ ✈ỵ✐ ỵ tt t tố
ữ õ


ữỡ ❑✐➳♥ t❤ù❝ ❝ì sð

✿
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ❝→❝ ❦➼ ❤✐➺✉ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣
ỗ ởt số tr r t ỗ ỗ số tử tr Rn
ỵ r ỵ tr tr ụ ữ ỵ t t ỡ s
t ❝❤♦ ✈✐➺❝ t✐➳♣ ❝➟♥ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ð ❝→❝ ❝❤÷ì♥❣ s❛✉✳

❈❤÷ì♥❣ ✷✿ ❈ì sð ❧➼ t❤✉②➳t ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥

❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ tè✐ ù✉ ♣❤✐ t✉②➳♥ ❝â ✤✐➲✉ ❦✐➺♥✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔
❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ù✉ ❝â ✤✐➲✉ ❦✐➺♥✳
❙❛✉ ♠ët t❤í✐ ❣✐❛♥ t➼❝❤ ❝ü❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ự ữợ sỹ t t ừ t
ữợ ❞➝♥✱ ❜↔♥ t❤➙♥ ❡♠ ✤➣ t➼❝❤ ❧ơ② ✤÷đ❝ ❝❤♦ ♠➻♥❤ r➜t ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤
♥❣❤✐➺♠ ❝ơ♥❣ ♥❤÷ ♥✐➲♠ ✤❛♠ ố ợ ởt ỵ tt ợ tr ồ ✈➔ ✤➳♥ ♥❛②
❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤✳
❊♠ ①✐♥ ỷ ớ ỡ t tợ trữớ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱
❚r÷í♥❣ ✣❍❙P✱ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣ ✤➣ t↕♦ ❝ì ❤ë✐ ❝❤♦ ❡♠ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tợ t P ỵ ữớ t
t ữợ ❞➝♥ ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❧➔♠ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣
❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ❝❤ó♥❣ ❡♠ tr♦♥❣ s✉èt
❜è♥ ♥➠♠ ❤å❝ ✈ø❛ q✉❛✳ ❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣
✤ï✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳

✷


✣➔ ◆➤♥❣✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥


▲➯ ❍♦➔♥❣ ◆❤✉➟♥

✸


❈❍×❒◆● ✶

❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð
P❤➛♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♠❛ tr➟♥✱ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✱ ❣✐→ trà
r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❚ỉ♣ỉ✱ ❤➔♠ sè ❧✐➯♥ tư❝✱ ❝→❝ ✤à♥❤ ✈➲ ❣✐→
trà tr t ỗ ỗ ỵ t ❝ü❝ ✤✐➸♠ ✳✳✳ ♥❤÷ ♠ët ❝ỉ♥❣ ❝ư ✤➸ t✐➳♣
❝➟♥ ❝→❝ ❦✐➳♥ t❤ù❝ ð ❝❤÷ì♥❣ s❛✉✳

✶✳✶ ▼❛ tr➟♥

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ♠❛ tr➟♥ ❝ï m × n ❧➔ ♠ët ❜↔♥❣ sè ỳ t ỗ m
n ởt ồ ♠❛ tr➟♥ m × n✱ ❦➼ ❤✐➺✉ ❧➔ Am×n✱ ❝â ❞↕♥❣✿


Am×n

a11 a12 . . . a1n
 a21 a22 . . . a2n

=  ✳✳
 ✳
am1 am2 . . . amn





.


▼❛ tr ù m ì n ữủ ồ m ì n✲♠❛ tr➟♥✳
◆➳✉ m ✈➔ n ✤➣ rã t❤➻ ♠❛ tr➟♥ Amìn ữủ ồ A (aij )mìn


ởt

m ì n tr ữủ ồ
i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , n}.

tr➟♥ ❦❤æ♥❣✱ ❦➼ ❤✐➺✉ ❧➔ 0✱ ♥➳✉

aij = 0

✭✐✐✮ ▼ët ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ✭n × n✲♠❛ tr➟♥✮ ♠➔ ❝→❝ ♣❤➛♥ tû aij = 0 ✈ỵ✐ i = j ✈➔
aii = 1, ∀i ∈ {1, 2, . . . , n} ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n✱ ❦➼ ❤✐➺✉ ❧➔ In ❤♦➦❝ I✳
❑❤✐ ✤â✿


1 0
 0 1


In =  0 0
 ✳✳
 ✳

0 0

0 ... 0
0 ... 0 

1 ... 0 



0 ... 1

❈→❝ ♣❤➨♣ t♦→♥ ✈➲ ♠❛ tr➟♥ t❤æ♥❣ t❤÷í♥❣ ✤÷đ❝ ❤✐➸✉ ✈➔ t❤ü❝ ❤✐➺♥ ♥❤÷ ❜➻♥❤ t❤÷í♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳

✹


✭❛✮ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ m × n✲♠❛ tr➟♥ A✱ ❦➼ ❤✐➺✉ ❧➔ AT ❧➔ ♠ët n × m✲♠❛ tr
ợ aTij = aji
tr ổ A ữủ ồ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥➳✉ AT = A✳
✭❝✮ ▼❛ tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ♥➳✉ ❝â ♠❛ tr➟♥ A−1✳ ❑❤✐ ✤â ♠❛ tr➟♥ A
✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✱ t❤ä❛ ♠➣♥ A−1A = I = AA−1✳
❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ A ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â ❦❤ỉ♥❣
✤÷đ❝ tr➻♥❤ ❜➔②✱ ♥❣÷í✐ ✤å❝ ❝â t❤➸ t❤❛♠ ❦❤↔♦ t❤➯♠ tr♦♥❣ ❝→❝ ❣✐→♦ tr➻♥❤ ✤↕✐ sè t✉②➳♥
t➼♥❤✳

✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ R

n


❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n✲❝❤✐➲✉ ❧➔ Rn ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët ✤✐➸♠ x tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ R ❧➔ ♠ët ❜ë n sè t❤ü❝ ✤÷đ❝
n

s➢♣ ①➳♣ t❤❡♦ t❤ù tü ữủ t ữợ ởt


x1
x2


x = ✳✳  ,
 ✳ 
xn


✈ỵ✐ ♠é✐ sè xi ∈ R✱ i ∈ {1, 2, . . . , n} ✤÷đ❝ ❣å✐ ❧➔ tå❛ ✤ë t❤ù i ❝õ❛ ✤✐➸♠ x✳
✣➲ t❤✉➟♥ t t q ữợ




x = (x1 , x2 , . . . , xn )T = 


x1
x2
✳✳

✳




.


xn
❑➼ ❤✐➺✉ 0 = (0, 0, . . . , 0)T ∈ Rn ❧➔ ❣è❝ tå❛ ✤ë ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Rn ✳
▼é✐ ✤✐➸♠ x t❤✉ë❝ Rn ①→❝ ✤à♥❤ ♠ët ✈➨❝tì tr♦♥❣ Rn ✈ỵ✐ ✤✐➸♠ ❣è❝ ❧➔ 0 ✈➔ ✤✐➸♠ ♥❣å♥
❧➔ x✳ ❱➨❝tì ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ ✈➟② ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ x✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ✣♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ✭✈➨❝tì✮ x ✈➔ y tr♦♥❣ R ✱ ❦➼ ❤✐➺✉ ❧➔ [x, y]✱
n

❧➔ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ✭✈➨❝tì✮ ❝â ❞↕♥❣

αx + (1 − α) y,

∀ 0 ≤ α ≤ 1.

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ ❤❛✐ ✈➨❝tì x = (x , x , . . . , x ) ✈➔ y = (y , y , . . . , y ) tr
T

Rn

1

2


T

n

õ
ổ ữợ ❝õ❛ ❤❛✐ ✈➨❝tì✱ ❦➼ ❤✐➺✉ ❧➔ xT y✱ yT x ❤♦➦❝
n

xT y = yT x = x, y =

xi y i .
i=1

✺

1

x, y

2

n

✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐


✭✐✐✮ ❍❛✐ ✈➨❝tì x ✈➔ y trü❝ ❣✐❛♦ ♥➳✉ xT y = yT x = x, y = 0✳
✭✐✐✐✮ ✣ë ❞➔✐ ❤❛② ❝❤✉➞♥ ❝õ❛ ✈➨❝tì x✱ ❦➼ ❤✐➺✉ ❧➔ x ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
n

T

x = x x

1/2

x2i .

=
i=1

❍➻♥❤ ✶✳✶✿ ✣♦↕♥ t❤➥♥❣

◆❤➟♥ ①➨t ✶✳✶✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì C ✱ t ổ ữợ ừ tỡ x, y õ
n

t t s
ổ ữợ ừ tỡ ❧➔

x, y

✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐

n

xi y i .

x, y =
i=1


✭✐✐✮
✭✐✐✐✮
✭✐✈✮

x ≥0

✈➔

x, y = y, x

x =0

❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳

✳

✈ỵ✐ λ ∈ C✳
x, λy = λ x, y ✈ỵ✐ λ ❧➔ ❧✐➯♥ ❤đ♣ ❝õ❛ λ✳
❚➼♥❤ ❝❤➜t ✶✳✶✳ ▼ët sè t➼♥❤ ❝❤➜t ✤➣ ❜✐➳t ✈➲ ❝❤✉➞♥ ❝õ❛ ♠ët ✈➨❝tì✿
✭❛✮ x ≥ 0 ✈➔ x = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳
✭❜✮ αx =| α | x ✱ ✈ỵ✐ ♠å✐ α ∈ R✳
✭❝✮ x + y ≤ x + y ✱ ✈ỵ✐ ♠å✐ ✈➨❝tì x, y tr♦♥❣ Rn✳
✭❞✮ ❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③
λx, y = λ x, y

|xT y| ≤ x

✈ỵ✐ ♠å✐ ✈➨❝tì x, y tr♦♥❣ Rn✳
✻


.

y ,


✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❈→❝ ✈➨❝tì a , a , . . . , a ✤÷đ❝ ❣å✐ ❧➔✿
1

2

k

✭✐✮ P❤ư t❤✉ë❝ t✉②➳♥
t➼♥❤ tỗ t số tỹ 1, 2, . . . , k ổ ỗ tớ
k
0 s i=1 λi ai = 0✳
✭✐✐✮ ✣ë❝ ❧➟♣ t✉②➳♥ t➼♥❤k ♥➳✉ ❦❤æ♥❣ tỗ t số tỹ 1, 2, . . . , k ổ ỗ tớ
0 s i=1 iai = 0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳

✭✐✮ ▼ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ✈➨❝tì a1, a2, . . . , ak ❧➔ ♠ët ✈➨❝tì ❝â ❞↕♥❣ ki=1 λiai
✈ỵ✐ λ1, λ2, . . . , λk ∈ R✳
❑❤✐ ✤â t➟♣ ❤đ♣ ❝→❝ ✈➨❝tì ❞↕♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❤đ♣ ❝→❝ ✈➨❝tì s✐♥❤ ❜ð✐ a1, a2, . . . , ak ✳
✭✐✐✮ ❚➟♣ ❤đ♣ ❝→❝ ✈➨❝tì ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ s✐♥❤ r❛ Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝ì sð ❝õ❛ Rn✳
▼é✐ ❝ì sð ❝õ❛ Rn ❝❤ù❛ ✤ó♥❣ n ✈➨❝tì✳
✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❍↕♥❣ ❝õ❛ m × n✲♠❛ tr➟♥ A ❜➡♥❣ sè ❝ët ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥ ♥❤➜t
❝õ❛ ♠❛ tr➟♥ A ✈➔ ❝ơ♥❣ ❜➡♥❣ sè ❤➔♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥ ♥❤➜t ❝õ❛ ♠❛ tr➟♥ A✳
❍ì♥ ♥ú❛✱ ♥➳✉ ❤↕♥❣ ❝õ❛ m × n✲♠❛ tr➟♥ A ❜➡♥❣ min{m, n} t❤➻ ♠❛ tr➟♥ A ✤÷đ❝ ❣å✐
❧➔ ❝â ❝➜♣ ✤➛② ✤õ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❝õ❛ E n ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣ ✤è✐ ợ
ở ổ ữợ tự
a + µb ∈ M, ∀a, b ∈ M, ∀λ, µ ∈ R✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ❙è ❝❤✐➲✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❧➔ sè ✈➨❝tì ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥
♥❤➜t tr♦♥❣ M ✳
✣à♥❤ ♥❣❤➽❛⊥✶✳✶✷✳ ❱ỵ✐ M ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ E n✱ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ M ✱
❦➼ ❤✐➺✉ M t ủ ỗ tỡ trỹ ❣✐❛♦ ✈ỵ✐ ❝→❝ ✈➨❝tì t❤✉ë❝ M ✳
❉ü❛ ✈➔♦ ❦➳t q✉↔ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♣❤➛♥ ❜ị trü❝ ❣✐❛♦ ❝õ❛ M
❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ E n ✱ ❤ì♥ ♥ú❛ M ✈➔ M ⊥ ❝↔♠ s✐♥❤ ❦❤ỉ♥❣ ❣✐❛♥ E n ✳ ◆â✐
❝→❝❤ ❦❤→❝ ✈ỵ✐ ♠é✐ ✈➨❝tì x ∈ E n ✤➲✉ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ❞✉② t ữợ x = a + b ợ
a M, b ∈ M ⊥ ✳ ▲ó❝ ✤â✱ a, b ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝❤✐➳✉ trü❝ ❣✐❛♦ ❝õ❛ ✈➨❝tì x ❧➛♥
❧÷đt ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ M ✈➔ M ⊥ ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳ ▼ët q✉❛♥ ❤➺ t÷ì♥❣ ù♥❣ A ❣→♥ ♠é✐ ✤✐➸♠ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ X ✈ỵ✐

♠ët ✤✐➸♠ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Y ✤÷đ❝ ❣å✐ ❧➔ ♠ët →♥❤ ①↕ tø X ✤➳♥ Y ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔
A:X →Y✳
⑩♥❤ ①↕ A ❝â t❤➸ ❧➔ t✉②➳♥ t➼♥❤ ❤♦➦❝ ♣❤✐ t✉②➳♥ t➼♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✹✳ ❈❤✉➞♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ A✱ ❦➼ ❤✐➺✉
A = max

x ≤1

Ax

.

x


.

❚➼♥❤ ❝❤➜t ✶✳✷✳ ❱ỵ✐ ♠å✐ ✈➨❝tì x✱ t❛ ❝â✿
Ax ≤ A
✼

A

✱ ✤÷đ❝ ✤✐♥❤ ♥❣❤➽❛ ❧➔✿


✶✳✸ ●✐→ trà r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣
✶✳✸✳✶

●✐→ trà r✐➯♥❣

✣à♥❤ ♥❣❤➽❛ ✶✳✶✺✳ ❈❤♦ ♠ët n × n✲♠❛ tr➟♥ ✈✉ỉ♥❣ A✳ tỗ t ởt ổ ữợ R
ởt ✈➨❝tì x = 0 t❤ä❛ ♠➣♥ Ax = λx t❤➻ ữủ ồ ởt tr r ự ợ
tỡ r✐➯♥❣ x ❝õ❛ A✳
◆❤➟♥ ①➨t ✶✳✷✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ λ ∈ R ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A ❧➔ ♠❛
tr➟♥ A − λI s✉② ❜✐➳♥✱ tù❝ ❧➔ ❞❡t (A − λI) = 0.
❚ø ♥❤➟♥ ①➨t tr➯♥ t❛ ❝â t❤➸ ♥❤➟♥ t❤➜② ♠ët ❣✐→ trà r✐➯♥❣ λ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✤❛
t❤ù❝
det (λI − A) = λn + an−1 λ + . . . + a1 λ + a0 = 0.
✣❛ t❤ù❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤
trữ ừ tr ổ A

ỵ sỷ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❞❡t (λI − A) = 0 ❝â n ♥❣❤✐➺♠ t❤ü❝ ♣❤➙♥
❜✐➺t λ1, λ2, . . . , n õ tỗ t n tỡ ở t✉②➳♥ t➼♥❤ x1, x2, . . . , xn ❧➔ ❝→❝ ✈➨❝tì
r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A✱ tù❝ ❧➔ ❝→❝ ✈➨❝tì ♥➔② t❤ä❛ ♠➣♥✿

Axi = λi xi ,

∀i ∈ {1, 2, . . . , n}.

❈❤ù♥❣ ♠✐♥❤✳
❚❛ ❝â✿ ❞❡t (λxi − A) = 0 tỗ t tỡ xi = 0 s❛♦ ❝❤♦ Axi = λi xi ✈ỵ✐
i ∈ {1, 2, . . . , n}✳
❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❤➺ n ✈➨❝tì x1 , x2 , . . . , xn ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳
❚❤➟t ✈➟②✿
●å✐ c1 , c2 , . . . , cn ❧➔ ❝→❝ ✈æ ữợ s ni=1 ci xi = 0
t tr B = (λ2 I − A)(λ3 I − A) . . . (λn I − A)✳
❑❤✐ ✤â✿

Bxn = (λ2 I − A)(λ3 I − A) . . . (λn I − A)xn
= (λ2 I − A)(λ3 I − A) . . . (λn xn − Axn )
Bxn = 0.
✈➻ λn xn − Axn = 0✳
❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿

Bxi = 0, ∀i = 2, 3, . . . , n✳

✽


▼➦t ❦❤→❝ t❛ ❝ô♥❣ ❝â✿

(λ2 I − A)(λ3 I − A) . . . (λn−1 I − A)(λn I − A)x1
(λ2 I − A)(λ3 I − A) . . . (λn−1 I − A)(λn x1 − Ax1 )
(λ2 I − A)(λ3 I − A) . . . (λn−1 I − A)(λn x1 − λ1 x1 )
(λ2 I − A)(λ3 I − A) . . . (λn−1 I − A)(λn − λ1 )x1

(λ2 I − A)(λ3 I − A) . . . (λn−1 x1 − Ax1 )(λn − λ1 )
(λ2 I − A)(λ3 I − A) . . . (λn−1 − λ1 )x1 (λn − λ1 )
···
= (λ2 I − A)(λ3 − A)x1 . . . (λn−1 − λ1 )(λn − λ1 )
= (λ2 − λ1 )(λ3 − λ1 ) . . . (λn−1 − λ1 )(λn − λ1 )x1 .

Bx1 =
=
=
=
=
=

Bx1
❉♦ ✈➟②✿
n

B

n

=

ci x i
i=1

ci Bxi
i=1

= ci Bx1

= c1 (λ2 − λ1 )(λ3 − λ1 ) . . . (λn−1 − λ1 )(λn − λ1 )x1 = 0.
▼➔ ❝→❝ λi ♣❤➙♥ ❜✐➺t ✈➔ ❝→❝ ✈➨❝tì xi = 0 ♥➯♥ tø ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝ c1 = 0✳
❚÷ì♥❣ tü ♥❤÷ ❝→❝❤ ①➨t ♠❛ tr➟♥ B ♥❤÷ tr➯♥ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ữủ

ci = 0 ợ i = 1, 2, . . . , n✳
❉♦ ✤â n ✈➨❝tì x1 , x2 , . . . , xn ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳
❱➟② ỵ ữủ ự

ỵ t ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ❧➔ ❝→❝ sè t❤ü❝✳
❈❤ù♥❣ ♠✐♥❤✳
●å✐ x = 0 ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ A✳
❑❤✐ ✤â✿ A, x = λ, x
⇒ Ax, x = λx, x = λ x
▼➦t ❦❤→❝✱ t❛ ❝â✿ Ax, x = x, AT x = x, Ax = x, λx = x, x =
õ ợ ú ỵ x > 0✱ t❛ ❝â✿

λ

x =λ

x ⇒ (λ − λ)

x

x =0⇒λ=λ

❱➟② R ỵ ữủ ự

ỵ ▼å✐ n × n✲♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤ü❝ ❝â n ✈➨❝tì r✐➯♥❣ ♣❤➙♥ ❜✐➺t t❤➻ ✤ỉ✐ ♠ët
trü❝ ❣✐❛♦✳

❈❤ù♥❣ ♠✐♥❤✳

✾


●✐↔ sû Axi = λi xi ✱ Axj = λj xj ✈ỵ✐ λi = λj ✈➔ A = AT ✳
❚❛ ❝â✿
xTi Axj = xTi AT xj = λi xTi xj .
▲↕✐ ❝â✿

xTi Axj = (Axj )T xi = xTj AT xi = λj xTj xi = λj xTi xj .

❉♦ ✤â✿

λi xTi xj = λj xTi xj ✳
▼➔ λi = λj ♥➯♥ tø ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r❛ xTi xj = 0✳
❱➟② xi ✈➔ xj trü❝ ❣✐❛♦✳

◆❤➟♥ ①➨t ✶✳✸✳ ◆❣÷í✐ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣✱ ♥➳✉ A ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ t❤➻ A ❝â

❣✐→ trà r✐➯♥❣ ✭❦➸ ❝↔ ❜ë✐✮ ✈➔ ❝â n ✈➨❝tì r✐➯♥❣ t÷ì♥❣ ù♥❣ x1, x2, . . . , xn t❤➻ n ✈➨❝tì r✐➯♥❣
♥➔② t↕♦ t❤➔♥❤ ♠ët ❝ð sð trü❝ ❣✐❛♦ ❝õ❛ Rn✳
❍ì♥ ♥ú❛✱ ♥➳✉ ❝→❝ ✈➨❝tì r✐➯♥❣ ♥➔② ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ✭ tù❝ ♠é✐ ✈➨❝tì ❝â ❝❤✉➞♥
✤ì♥ ✈à✮ t❤➻ ♠❛ tr➟♥

n

Q = [x1 , x2 , . . . , xn ]

t❤ä❛ ♠➣♥

QT Q = I✱

❤❛②
QT = Q−1 ✳

▼❛ tr➟♥ Q ♥❤÷ ✈➟② ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ trü❝ ❣✐❛♦✳
◆❣♦➔✐ r❛✱ t❛ ❝ô♥❣ ❝â
Q−1 AQ = QT AQ = QT [Ax1 , Ax2 , . . . , Axn ] = QT [λ1 x1 , λ2 x2 , . . . , λn xn ]

❤❛②





λ1
λ2



Q−1 AQ = 




.


✳✳✳
λn


▲ó❝ ✤â✱ ♠❛ tr➟♥ A ✤÷đ❝ ❝❤➨♦ ❤â❛✳
✶✳✸✳✷

❉↕♥❣ t♦➔♥ ♣❤÷ì♥❣

▼ët ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ f : Rn → R ❧➔ ♠ët ❤➔♠ sè ❝â ❞↕♥❣

f (x) = xT Qx
ợ Q n ì n tr tỹ ✤è✐ ①ù♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✾✳ ✭✐✮ ▼❛ tr➟♥ ✤è✐ ①ù♥❣ Q ✤÷đ❝ ❣å✐ ❧➔ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥➳✉ ❞↕♥❣
t♦➔♥ ♣❤÷ì♥❣ ❝õ❛ ♥â xT Qx > 0 ✈ỵ✐ ♠å✐ ✈➨❝tì x = 0✳
✶✵


✭✐✐✮ ❚÷ì♥❣ tü✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ ✤à♥❤ ♥❣❤➽❛ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ Q ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✱
①→❝ ✤à♥❤ ➙♠✱ ♥û❛ ①→❝ ✤à♥❤ ➙♠ ♥➳✉ xT Qx ≥, <, ≤ 0 ợ ồ tỡ x = 0
ú ỵ r ú t❛ ❦❤æ♥❣ ✤à♥❤ ♥❣❤➽❛ t➯♥ ❝❤♦ ♠❛ tr➟♥ Q ♥➳✉ t ữỡ
ừ õ ữỡ ợ ởt số tỡ x ✈➔ ➙♠ ✈ỵ✐ ❝→❝ ✈➨❝tì ❝á♥ ❧↕✐✳

◆❤➟♥ ①➨t ✶✳✹✳ ▼❛ tr➟♥ Q ❧➔ ♠❛ tr➟♥ Q ❧➔ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✭❤❛② ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✮

❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t➜t ❝↔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ Q ❧➔ ❞÷ì♥❣ ✭❤❛② ❦❤ỉ♥❣ ➙♠✮✳
❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ ✈➨❝tì x ✤➦t y = Q−1x ợ Q ữủ ữ tr t
✤â✿
n
xT Ax = xT QT AQy =

λi yi2 ,

i=1

✈ỵ✐ yi tũ ỵ x tũ ỵ ứ õ t õ ♥❤➟♥ ①➨t tr➯♥✳
◆❤➟♥ ①➨t ✶✳✺✳ ▼ët
♠❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ A ❝â ♠ët ❝➠♥ ❜➟❝ ✷ ✤è✐ ①ù♥❣✱ ♥û❛
1/2
①→❝ ✤à♥❤ ❞÷ì♥❣ A t❤ä❛ ♠➣♥ A1/2A1/2 = A = A✳
◆❣♦➔✐ r ợ tr Q ữ tr ①➨t ✶✳✸ t❛ ①→❝ ✤à♥❤ ✤÷đ❝✿

A

1/2

1/2



λ1

1/2

λ2



=Q 

T

✳✳✳


1/2



 Q.


λn

❚ø ✤â✱ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ xT Qx ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ t❤➔♥❤

Q1/2 x

2

✶✳✹ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❚æ♣æ

✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✵✳ ▼ët ❞➣② ❝→❝ ✈➨❝tì x ✱x ✱. . .✱x ✱. . .✱ ❦➼ ❤✐➺✉ ❧➔ {x }

❤♦➦❝ ✤ì♥
❣✐↔♥ ❧➔ {xk }✱ ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tö ✤➳♥ x ♥➳✉ xk − x → 0 ❦❤✐ k → ∞ ✭tù❝ ❧➔ ✈ỵ✐ ♠é✐
ε > 0 tỗ t N s xk x < ε ✈ỵ✐ ♠å✐ k ≥ N ✮✳
❑❤✐ ✤â t❛ ✈✐➳t xk → x ❤♦➦❝ limk→∞ xk ❂ x✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✶✳ ▼ët ✤✐➸♠ x ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ tư ừ tỡ {xk } tỗ
t ởt ❝õ❛ ❞➣② {xk } ❤ë✐ tö ✤➳♥ x✳
0


1

k

∞
k k=0

❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ t❤➜②✱ x ❧➔ ♠ët ✤✐➸♠ tö ❝õ❛ {xk } tỗ t ởt t K
ừ t ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ {xk }k∈K ❤ỉ✐ tư ✤➳♥ x✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✷✳

✭✐✮ ▼ët ❤➻♥❤ ❝➛✉ ♠ð t➙♠ x✱ ❜→♥ ❦➼♥❤ ε > 0 ❧➔ ♠ët t➟♣ ❤ñ♣ ❝â ❞↕♥❣
S(x, ε) = {y ∈ Rn |

y − x < ε}.

✭✐✐✮ ▼ët ❤➻♥❤ ❝➛✉ ✤â♥❣ t➙♠ x✱ ❜→♥ ❦➼♥❤ ε > 0 ❧➔ ♠ët t➟♣ ❤ñ♣ ❝â ❞↕♥❣
S(x, ε) = {y ∈ Rn |

y − x ≤ ε}.

▼ët ❤➻♥❤ ❝➛✉ ♠ð t➙♠ x ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x ❜→♥ ❦➼♥❤ ε✳
✶✶


✣à♥❤ ♥❣❤➽❛ ✶✳✷✸✳ ▼ët t➟♣ ❝♦♥ P ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ♠ð ♥➳✉ ♠å✐ ✤✐➸♠ t❤✉ë❝
n

✤➲✉ ❧➔♠ t➙♠ ❝õ❛ ♠ët ❤➻♥❤ ❝➛✉ ♠ð ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ P ✳

❚ù❝ ❧➔✱ P ♠ð ♥➳✉ ✈ỵ✐ ♠é✐ ✤✐➸♠ x P tỗ t > 0 s S(x, ε) ⊂ P.
✣à♥❤ ♥❣❤➽❛ ✶✳✷✹✳ ▼ët t➟♣
❝♦♥ P ❝õ❛ E n ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ✤â♥❣ ♥➳✉ ♣❤➛♥ ❜ị ❝õ❛
♥â ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Rn✳
◆❤➟♥ ①➨t ✶✳✻✳ ▼ët tữỡ ữỡ ởt t P õ ợ ♠é✐ ❞➣② {xk } ⊂ P
❤ë✐ tö ✤➳♥ x t❤➻ x ∈ P ✳
P

▼ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ t t õ

ỵ

ừ ỳ ❤↕♥ ❝→❝ t➟♣ ♠ð ❧➔ ♠ët t➟♣ ♠ð✳ ❍ñ♣ ❝õ❛ ♠ët ❤å ❜➜t ❦➻ ♥❤ú♥❣ t➟♣
♠ð ❝ô♥❣ ❧➔ ♠ët t➟♣ ♠ð✳
✭✐✐✮ ❍ñ♣ ❝õ❛ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ✤â♥❣ ❝ô♥❣ ❧➔ ♠ët t➟♣ ✤â♥❣✳ ●✐❛♦ ❝õ❛ ♠ët ❤å
❜➜t ❦➻ ♥❤ú♥❣ t➟♣ ✤â♥❣ ♥➔♦ ❝ô♥❣ ❧➔ ♠ët t➟♣ ✤â♥❣✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✻✳ ✣✐➸♠ x ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ t➟♣ ủ P tỗ t ởt
ừ x trå♥ tr♦♥❣ P ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✼✳ P❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ P ✱ ❦➼ ❤✐➺✉ P ❧➔ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ✤✐➸♠ tr♦♥❣
o

❝õ❛ P ✳

❉➵ t❤➜②✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ P ❧➔ t➟♣ ♠ð ❧ỵ♥ ♥❤➜t ❝❤ù❛ tr♦♥❣ P ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✽✳ ●✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ✤â♥❣ ❝❤ù❛ P ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ P ✱
❦➼ ❤✐➸✉ ❧➔ P

ỵ õ P ừ P ủ ❝õ❛ P ✈➔ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ P ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✵✳ ✭✣à♥❤ ♥❣❤➽❛ ❍❡♥✐❡✲❇♦r❡❧✮ ▼ët t➟♣ ❝♦♥ P ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔
n

♠ët t➟♣ ❝♦♠♣❛❝t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ✈ø❛ ✤â♥❣ ✈➔ ❜à ❝❤➦♥ ✭❣✐ỵ✐ ♥ë✐✮✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ P
❝♦♠♣❛❝t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ P ✤â♥❣ ✈➔ ợ ồ x P tỗ t M > 0 s x M
ỵ ❲❡✐❡rstr❛ss✮ ◆➳✉ P ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t ✈➔ ❞➣② {xk } ⊂ P
t❤➻ ❞➣② {xk } ❝â ♠ët ✤✐➸♠ tö tở P tự tỗ t ởt ừ {xk } ❤ë✐ tö ✤➳♥
♠ët ✤✐➸♠ t❤✉ë❝ P ✮✳

✶✳✺ ❍➔♠ sè ❧✐➯♥ tö❝ tr➯♥ R

n

✣à♥❤ ♥❣❤➽❛ ✶✳✸✷✳ ▼ët ❤➔♠ sè f ①→❝ ✤à♥❤ tr➯♥ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥
n

tö❝ t↕✐ x ♥➳✉ xk → x t❤➻ f (xk ) → f (x)✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ f ❧✐➯♥ tö❝ t↕✐ x ♥➳✉✿
∀ε > 0✱ ∃δ > 0 : ∀y t❤ä❛ ♠➣♥ y − x < δ t❤➻ f (y) f (x) <
ỵ ỵ rstrss ởt f tử tr ởt t ❝♦♠♣❛❝t
P
∗
t❤➻ ❝â ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉ tr♦♥❣ P ✱ ♥❣❤➽❛ tỗ t x P f (x) f (x ) ✈ỵ✐ ♠å✐
x ∈ P✳
✶✷


✣à♥❤ ♥❣❤➽❛ ✶✳✸✹✳ ▼ët t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ t❤ü❝ f , f , . . . , f tr♦♥❣ R ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔
1

2


n

m

♠ët ❤➔♠ ✈➨❝tì f = (f1, f2, . . . , fm)✳
❚ø ✤â t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ sè f (x) = (f1(x), f2(x) . . . , fm(x)) tr♦♥❣ Rn ✈ỵ✐ ♠é✐ ✈➨❝tì
x ∈ Rn ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✸✺✳ ▼ët ❤➔♠ ❣✐→ trà ✈➨❝tì f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ ♥➳✉ ♠é✐ ❤➔♠ t❤➔♥❤
♣❤➛♥ ❝õ❛ ♥â ❧➔ ❧✐➯♥ tö❝✳
◆❣♦➔✐ r❛✿
✭✐✮ ◆➳✉ ♠é✐ ❤➔♠ t❤➔♥❤ ♣❤➛♥ ❝õ❛ f = (f1, f2, . . . , fm) ❧✐➯♥ tö❝ tr♦♥❣ ♠ët t➟♣ ♠ð ❝õ❛
E n t❤➻ t❛ ✈✐➳t f ∈ C ✳
✭✐✐✮ ◆➳✉ ♠é✐ ❤➔♠ t❤➔♥❤ ♣❤➛♥ ❝â ✤↕♦ ❤➔♠ ❝➜♣ ♠ët ❧✐➯♥ tö❝ tr➯♥ t➟♣ ♥➔② t❤➻ t❛ ✈✐➳t
f ∈ C 1✳
✭✐✐✐✮ ▼ët ❝→❝❤ tê♥❣ q✉→t✱ ♥➳✉ ♠é✐ ❤➔♠ t❤➔♥❤ ♣❤➛♥ ❝â ✤↕♦ ❤➔♠ t✐➯♥❣ ❝➜♣ p ❧✐➯♥ tö❝ t❤➻
t❛ ✈✐➳t f ∈ C p✳
✣à♥❤ ♥❣❤➽❛ ✶✳✸✻✳ ◆➳✉ f ❧➔ ♠ët ❤➔♠ t❤ü❝ tr♦♥❣ Rn✱ ✭f (x) = f (x1, x2, . . . , xn)✮✱ t❛
✤à♥❤ ♥❣❤➽❛ ❣r❛❞✐❡♥t ❝õ❛ f ❧➔ ✈➨❝tì
∇f (x) =

∂f (x)
∂f (x) ∂f (x)
,
,...,
∂x1
∂x2
∂xn

.


❚r♦♥❣ t➼♥❤ t♦→♥ ♠❛ tr➟♥✱ ❣r❛❞✐❡♥t ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ✈➨❝tì ✤÷đ❝ ✈✐➳t t❤❡♦ ❞↕♥❣
❤➔♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✼✳ ◆➳✉ f ∈ C t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ ♠❛ tr➟♥ ❍❡ss✐❛♥ ❝õ❛ f t↕✐ x ữ
2

ởt n ì n tr ữủ ❧➔ F(x) ❤♦➦❝ ∇2f (x)✿
∂ 2 f (x)
.
∂xi ∂xj

F(x) =

◆❤➟♥ ①➨t ✶✳✼✳ ❇ð✐ ✈➻

∂ 2f
∂ 2f
=
,
∂xi ∂xj
∂xj ∂xi

♥➯♥ ♠❛ tr➟♥ ❍❡ss✐❛♥ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳
❱ỵ✐ ♠é✐ ❤➔♠ ❣✐→ trà ✈➨❝tì f = (f1 , f2 , . . . , fm ) ♥❤ú♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❧➔ t÷ì♥❣ tü✳

◆❤➟♥ ①➨t ✶✳✽✳

✭✐✮ ◆➳✉ f ∈ C 1 t❤➻ ✤↕♦ ❤➔♠ ởt ữủ m ì n tr
f (x) =


∂fi (x)
.
∂xj

✭✐✐✮ ◆➳✉ f ∈ C 2 t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ m ♠❛ tr➟♥ ❍❡ss✐❛♥ F1(x), F2(x), . . . , Fm(x) t÷ì♥❣
ù♥❣ ❧➔ m t❤➔♥❤ ♣❤➛♥ ❝õ❛ ❤➔♠ sè ♥➔②✳
✶✸


✭✐✐✐✮ ❱ỵ✐ λT = (λ1, λ2, . . . , λm) ∈ Rm t❤➻ ❤➔♠ sè λT f ❝â ❣r❛❞✐❡♥t ❧➔ λT ∇f (x) ✈➔ ❝â
♠❛ tr➟♥ ❍❡ss✐❛♥ λT F(x) ✈➔ ❜➡♥❣
m
T

λ F(x) =

λi Fi (x).
i=1

✣à♥❤ ♥❣❤➽❛ ✶✳✸✽✳ ❍➔♠ sè f ①→❝ ✤à♥❤ tr➯♥ ♠ët t➟♣ ♠ð P t❤✉ë❝ R ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔
n

✈✐ t↕✐ ✤✐➸♠ x ∈ P ♥➳✉ tỗ r ừ f t ồ ✈➔ ✈ỵ✐ ♠å✐ d ∈ Rn✱
d ✤õ ♥❤ä ✈➔ x + d ∈ P t❛ ❝â✿
f (x + d) = f (x) + ∇f (x)d + o( d ),

tr♦♥❣ ✤â o(

❧➔ ✈ỉ ❝ị♥❣ ❜➨ ❜➟❝ ❝❛♦ ❤ì♥ d ❦❤✐

◆❤➟♥ ①➨t tự tr tữỡ ữỡ ợ
d )

d 0

f (x + d) − f (x) − ∇f (x)d
= 0.
→0
d

lim
d

✣à♥❤ ♥❣❤➽❛ ✶✳✸✾✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ tr➯♥ t➟♣ ♠ð P t❤✉ë❝ E ♥➳✉ f ❦❤↔ ✈✐ t↕✐
n

♠å✐ ✤✐➸♠ t❤✉ë❝ P ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✵✳ ❈❤♦ ❤➔♠ sè f ①→❝ ✤à♥❤ tr➯♥ R ✈➔ ♠ët ✈➨❝tì d ∈ R \{0}
n

ợ

lim+

t0

n

f (x + td) f (x)

,
t

tỗ t ✭❤ú✉ ❤↕♥ ❤♦➦❝ ✈ỉ ❤↕♥✮ ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ t ữợ ừ f t
x Rn ữủ f (x, d)
ỵ f ①→❝ ✤à♥❤ tr➯♥ Rn ✈➔ ✤✐➸♠ x ∈ Rn✳ ◆➳✉ f ❦❤↔ ✈✐ t↕✐ x t❤➻
f (x, d) = ∇f (x)d,

✈ỵ✐ ♠å✐ d ∈ Rn\{0}✳
❈❤ù♥❣ ♠✐♥❤✳ f ❦❤↔ ✈✐ t↕✐ x ♥➯♥ ✈ỵ✐ ♠å✐ d ∈ Rn\{0}✱ t❛ ❝â✿
lim+

t→0

f (x + td) − f (x) − t.∇f (x)d
= 0.
t d

❉♦ ✤â✿

f (x, d) − ∇f (x)d
= 0.
d

❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✶✹


ỵ r tr tr✉♥❣ ❣✐❛♥

▼ët ❦➳t q✉↔ ❝➛♥ t❤✐➳t ✈➔ ✤÷đ❝ sû ❞ư♥❣ tr tố ữ ỵ r
tr tr

ỵ ❣✐→ trà tr✉♥❣ ❣✐❛♥✮

✭✐✮ ◆➳✉ f ∈ C 1 tr♦♥❣ ự t [x1, x2] t tỗ t trà θ, 0 ≤ θ ≤ 1
s❛♦ ❝❤♦
f (x2 ) = f (x1 ) + ∇f (θx1 + (1 − θ)x2 )(x2 − x1 ).

✭✐✐✮ ❍ì♥ ♥ú❛✱ ♥➳✉ f ∈ C 2 t tỗ t tr , 0 θ ≤ 1 s❛♦ ❝❤♦
1
f (x2 ) = f (x1 ) + ∇f (x1 )(x2 − x1 ) + (x2 − x1 )T F(θx1 + (1 − θ)x2 )(x2 − x1 ).
2

Ð ✤➙② F ❧➔ ♠❛ tr➟♥ ❍❡ss✐❛♥ ❝õ❛ ❤➔♠ f

t ỗ ỗ
r ú t ởt số tt t ỗ
ỗ ♥❤÷♥❣ s➩ ❦❤ỉ♥❣ ✤÷❛ r❛ ❝❤ù♥❣ ♠✐♥❤✱ ❝❤ó♥❣ t❛ ❝â t t ự tr



ỗ

R ữủ ồ t ỗ
n

(1 λ)x1 + λx2 ∈ Ω✱

∀x1 , x2 ∈ Ω, 0 1


ỗ t ổ ỗ

t R b R , b = 0✳ ❑❤✐ ✤â ❝→❝ t➟♣ s❛✉ t ỗ tr
n

Rn

H = {x ∈ Rn| x, b

= β}✳
✶✺


✭✐✐✮ ◆û❛ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣
{x ∈ Rn | x, b ≤ β} ✈➔
{x ∈ Rn | x, b ≥ β}✳
✭✐✐✐✮ ◆û❛ ❦❤æ♥❣ ❣✐❛♥ ♠ð
{x ∈ Rn | x, b < β} ✈➔
{x ∈ Rn | x, b > β}.

✭✐✈✮ ◆➳✉ C D t ỗ t t s ụ t ỗ
C + D = {x + y| x ∈ C, y ∈ D}✱
αC = {αx| x ∈ C}, α ∈ R✳
✭✐✈✮ ●✐❛♦ ❝õ❛ ♠ët ❤å tũ ỵ t ỗ ởt t ỗ

t ừ t ỗ

ởt t ừ R ỗ ừ ❦➼ ❤✐➺✉ ❧➔ co(Ω)✱ ❧➔
n


❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ỗ ự ỗ õ ừ ữủ ♥❣❤➽❛ ❧➔ ❜❛♦ ✤â♥❣
❝õ❛ co(Ω)✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✺✳ ▼ët t➟♣ Ω ❧➔ ♠ët ♥â♥ ♥➳✉ x ∈ Ω t❤➻ αx ∈ Ω ✈ỵ✐ ♠å✐ α > 0✳ ❍ì♥
♥ú❛✱ ♠ët ♥â♥ ❧➔ ởt t ỗ ữủ ồ ởt õ ỗ





ỗ

số f tr t ỗ ữủ ồ ỗ ợ ộ
x1 , x 2 ∈ Ω

✈➔ ✈ỵ✐ ♠å✐ α✱ 0 ≤ α ≤ 1 t❛ ❝â✿

f (αx1 + (1 − α)x2 ) ≤ αf (x1 ) + (1 − α)f (x2 ).

❍ì♥ ♥ú❛✱ ♥➳✉ ✈ỵ✐ ♠å✐ α✱ 0 < α < 1✱ ✈➔ x1 = x2 t❛ ❝â
f (αx1 + (1 − α)x2 ) < αf (x1 ) + (1 − )f (x2 ),

t f ữủ ồ ỗ t
g tr t ỗ ữủ ❣å✐ ❧➔ ❧ã♠ ♥➳✉ ❤➔♠ sè f = −g
❧➔ ❤➔♠ ỗ số g ữủ ồ ỗ t g ỗ t

ỗ

t f , f ỗ tr t ỗ t f
tr


1

2

1

+ f2

ụ ỗ

t f ỗ tr t ỗ t f ụ ỗ tr ợ ồ
0

t f ỗ tr t ỗ t t F

c

ỗ ợ ồ c

= {x |f (x) c}

ỵ số f C õ
1

f ỗ tr t ỗ
f (y) f (x) + ∇f (x)(y − x),

✈ỵ✐ ♠å✐ x, y ∈ Ω✳
✶✼


❧➔ t➟♣


ó

f ỗ t tr t ỗ
f (y) > f (x) + ∇f (x)(y − x),

✈ỵ✐ ♠å✐ x, y ∈ Ω ✈➔ x = y✳
❈❤ù♥❣ ♠✐♥❤✳
✭✐✮ ✧⇒✧ ●✐↔ sû f ỗ õ ợ ồ 0 ≤ 1 t❛ ❝â✿

f (αy + (1 − α)x) ≤ αf (y) + (1 − α)f (x).
❇✐➳♥ ✤ê✐ ✈➔ ❝❤✉②➸♥ ✈➳ t❛ ✤÷đ❝

f (x + α(y − x))
≤ f (y) − f (x).
α
❈❤♦ α → 0 t❛ ✤÷đ❝

∇f (x)(y − x) ≤ f (y) − f (x).
✧⇐✧ ●✐↔ sû

f (y) ≥ f (x) + ∇f (x)(y − x),
✈ỵ✐ ♠å✐ x, y ∈ Ω✳
❈è ✤à♥❤ x1 ✱ x2 ∈ Ω ✈➔ ✤➦t x = αx1 + (1 − α)x2 ✳
❑❤✐ ✤â ✈ỵ✐ ♠å✐ α ♠➔ 0 ≤ α ≤ 1✱ t❛ ❝â✿

f (x1 ) ≥ f (x) + ∇f (x)(x1 − x)

f (x2 ) ≥ f (x) + ∇f (x)(x2 − x)

✭✶✳✶✮
✭✶✳✷✮

◆❤➙♥ ✷ ✈➳ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮ ✈ỵ✐ α✱ ✭✶✳✷✮ ✈ỵ✐ (1 − α) ✈➔ ❝ë♥❣ t❤❡♦ ✈➳ t❛ ✤÷đ❝✿

αf (x1 ) + (1 − α)f (x2 ) ≥ f (x) + ∇f (x)[αx1 + (1 − α)x2 − x].
❱➻ x = αx1 + (1 − α)x2 ✱ t❛ t❤✉ ✤÷đ❝✿

αf (x1 ) + (1 − α)f (x2 ) ≥ f (αx1 − (1 − α)x2 ).
✶✽


✭✐✐✮ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭✐✐✮ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ tự ổ r tr
f ỗ ❝❤➦t✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ x = y, α ∈ (0; 1)✱ ❣✐↔ sû ❞➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ f ỗ
t
t z = 21 x + 12 yứ õ t ❝â✿

1
1
1
f (z) < f (x) + f (y) = f (x) + ∇f (x)(y − x).
2
2
2

✭✶✳✸✮

✣➦t t = βx + (1 − β)z ✈ỵ✐ β ∈ (0; 1)✱ t❤❡♦ t➼♥❤ t ừ ỗ t t õ


1
f (t) < βf (x) + (1 − β)f (z) < f (x) + (1 − β)∇f (x)(y − x).
2

✭✶✳✹✮

▲↕✐ ❝â t − x = (1 − β)(z − x) = 12 (1 − β)(z − x) = 12 (1 − β)(y − x)✳ ❉♦ ✤â✱ tø ✭✶✳✹✮ s✉②
r❛✿
f (t) < f (x) + ∇f (x)(t − x).
❚r→✐ ✈ỵ✐ ✭✐✮✱ ❞♦ ✤â t õ ự

ỵ f C õ
2

f ỗ tr Ω ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠❛ tr➟♥ ❍❡ss✐❛♥ F(x) ♥û❛ ữỡ ợ
ồ x
f ỗ t tr tr ss F(x) ữỡ ợ
ồ x
ự
f ∈ C 2 ♥➯♥ ✈ỵ✐ ♠å✐ x, y ∈ Ω✱ t❛ ❝â

1
f (y) = f (x) + ∇f (x)(y − x) + (y − x)T F(x)(y − x),
2
✶✾


✈ỵ✐ x ∈ [x, y]✳
✰ ◆➳✉ F ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ t❤➻ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ tr♦♥❣ ✤➠♥❣ t❤ù❝ tr➯♥ ❦❤ỉ♥❣

õ t ỵ f ỗ
ữủ f ỗ tr ss F ổ ỷ ữỡ t
tỗ t ởt ✈➨❝tì d s❛♦ ❝❤♦
dT F(x)d < 0.
❈❤å♥ y = x + εd ✈ỵ✐ ε > 0✳ ◆➳✉ ε ✤õ ♥❤ä t❤➻ y, x ✤õ ❣➛♥ x s❛♦ ❝❤♦

dT F(x)d < 0,
❞♦ ♠❛ tr➟♥ ❍❡ss✐❛♥ ❧✐➯♥ tư❝✳
❑❤✐ ✤â ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ tr♦♥❣ ✤➥♥❣ t❤ù❝ ➙♠ ♥➯♥ f (y) < f (x) tr ợ ỵ
F ữỡ t tữỡ tỹ ữ tr ỵ t õ f ỗ t

ỵ t s ✈➔ s✐➯✉ ♣❤➥♥❣ tü❛

✣à♥❤ ♥❣❤➽❛ ✶✳✺✵✳ ▼ët t➟♣ V tr♦♥❣ R ữủ ồ ởt t t ợ ♠å✐
n

t❤➻ λx1 + (1 − λ)x2 ∈ V ✈ỵ✐ ♠å✐ λ ∈ R✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✶✳ ▼ët s✐➯✉ ♣❤➥♥❣ tr♦♥❣ Rn ❧➔ ♠ët (n − 1)✲❝❤✐➲✉ ✤❛ t✉②➳♥ t➼♥❤✳
❚➼♥❤ ❝❤➜t ✶✳✻✳ ❈❤♦ a ❧➔ ♠ët ✈➨❝tì ❝ët n✲❝❤✐➲✉ ❦❤→❝ ✈➨❝tì 0✱ ✈➔ ♠ët sè t❤ü❝ c✳ ❑❤✐
✤â t➟♣ ❤ñ♣✿
H = {x ∈ Rn |aT x = c}✱
❧➔ ♠ët s✐➯✉ ♣❤➥♥❣ tr♦♥❣ Rn✳
❈❤ù♥❣ ♠✐♥❤✳ ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ H ❧➔ t➟♣ ✤❛ t✉②➳♥ t➼♥❤✳

x1 , x 2 ∈ V

●♦à x1 ❧➔ ♠ët ✈➨❝tì ❜➜t ❦➻ t❤✉ë❝ H ❦❤✐ ✤â t➟♣ M = H − x1 ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥
❝♦♥ ✤❛ t✉②➳♥ t➼♥❤ ❝õ❛ Rn ✳
❑❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❝❤ù❛ t➜t ❝↔ ❝→❝ ✈➨❝tì x t❤ä❛ ♠➣♥ aT x = 0 ❤❛② ❝→❝ ✈➨❝tì trü❝ ❣✐❛♦
✈ỵ✐ ✈➨❝tì a✳ ❉♦ ✈➟② ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ (n − 1)✲❝❤✐➲✉✳


❚➼♥❤ ❝❤➜t ✶✳✼✳ ❈❤♦ H ❧➔ ♠ët s✐➯✉ ♣❤➥♥❣ tr♦♥❣ R ✳ ❑❤✐ ✤â✱ tỗ t ởt tỡ n
n

ởt số tỹ c s ❝❤♦

H = {x ∈ Rn |aT x = c}✳

❈❤ù♥❣ ♠✐♥❤✳

●å✐ x1 ∈ H ✈➔ ✤➦t M = H − x1 ✳ ❱➻ H ❧➔ ♠ët s✐➯✉ ♣❤➥♥❣ ♥➯♥ M ❧➔ ♠ët
❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ (n − 1)✲❝❤✐➲✉✳
●å✐ a ❧➔ ♠ët ✈➨❝tì ❜➜t ❦➻ trü❝ ❣✐❛♦ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì M ✳
❑❤✐ ✤â a ∈ M ⊥ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✶ ❝❤✐➲✉ ✈➔ M = {x ∈ Rn |aT x = 0}✳
✣➦t c = aT x1 t❛ t❤➜② ♥➳✉ x2 ∈ H t❤➻ x2 − x1 ∈ M ✈➔ ❞♦ ✤â aT x2 − aT x1 = 0✳
❚ø ✤â s✉② r❛ aT x2 = c✳
❉♦ ✈➟② H ⊂ {x ∈ Rn |aT x = c}✳
▲↕✐ ❝â H ❧➔ ❦❤æ♥❣ ❣✐❛♥ (n − 1)✲❝❤✐➲✉ ✈➔ t❤❡♦ t➼♥❤ ❝❤➜t ✶✳✻ {x ∈ Rn |aT x = c} ❝ơ♥❣ ❧➔
♠ët ❦❤ỉ♥❣ ❣✐❛♥ (n − 1)✲❝❤✐➲✉ ♥➯♥ H = {x ∈ Rn |aT x = c}✳
✷✵


❍➻♥❤ ✶✳✻✿ ❙✐➯✉ ♣❤➥♥❣

❚ø ❚➼♥❤ ❝❤➜t ✶✳✻ ✈➔ ❚➼♥❤ ❝❤➜t ✶✳✼✱ t❛ t❤➜② ♠ët s✐➯✉ ♣❤➥♥❣ ❧➔ ♠ët t➟♣ ❤ñ♣ ❝→❝
♥❣❤✐➺♠ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✷✳ ❈❤♦ a ❧➔ ♠ët ✈➨❝tì ❦❤→❝ ✈➨❝tì 0 tr♦♥❣ R ✈➔ c ởt số tỹ
n


õ tữỡ ự ợ s H = {x ∈ Rn|aT x = c} ❧➔✿
✭✐✮ ◆û❛ ❦❤ỉ♥❣ ❣✐❛♥ ❞÷ì♥❣ ✤â♥❣ H+ = {x ∈ Rn|aT x ≥ c}.
✭✐✐✮ ◆û❛ ❦❤æ♥❣ ❣✐❛♥ ➙♠ ✤â♥❣ H− = {x ∈ Rn|aT x ≤ c}.
✭✐✐✐✮ ◆û❛ ❦❤ỉ♥❣ ❣✐❛♥ ❞÷ì♥❣ ♠ð H += {x ∈ Rn|aT x > c}.
o

✭✐✈✮ ◆û❛ ❦❤æ♥❣ ❣✐❛♥ ➙♠ ♠ð H −= {x ∈ Rn|aT x < c}.
ỵ C ởt t ỗ y ❧➔ ♠ët ✤✐➸♠ ♥❣♦➔✐ ❜❛♦ ✤â♥❣ ❝õ❛ C ✳ õ
tỗ t ởt tỡ a s
o

aT y < inf aT x.
x∈C

❈❤ù♥❣ ♠✐♥❤✳

✣➦t

δ = inf |x − y|.
x∈C

❑❤✐ ✤â tỗ t x0 C s |x0 y| = δ ✈➻ ❤➔♠ sè f (x) − |x − y| ①→❝ ✤à♥❤
tr➯♥ ♠ët t➟♣ ✤â♥❣ ✈➔ ❜à ❝❤➦♥ t tỗ t tr ọ t ỡ ỳ ❣✐→ trà x ✤â ❧➔
❣✐❛♦ ❝õ❛ ❜❛♦ ✤â♥❣ ❝õ❛ C ✈ỵ✐ ❤➻♥❤ ❝➛✉ t➙♠ y ❜→♥ ❦➼♥❤ 2δ ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✈➨❝tì
a = x0 − y t❤ä❛ ♠➣♥ ừ ỵ
t x C ợ ♠å✐ α ♠➔ 0 ≤ α ≤ 1 t❛ ❝â✿

x0 + α(x − x0 ) ∈ C ✳
❉♦ ✤â


|x0 + α(x − x0 − y)2 ≥ |x0 − y|2 ✳
❑❤❛✐ tr✐➸♥ t❤✉ ✤÷đ❝✿
✷✶


2α(x0 − y)T (x − x0 ) + α2 |x − x0 |2 ≥ 0✳
❈❤♦ α → 0 t❛ ✤÷đ❝

(x0 − y)T (x − x0 ) ≥ 0.

❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r❛✿

(x0 − y)T x ≥ (x0 − y)T x0 = (x0 − y)T y + (x0 − y)T (x0 − y)
= (x0 − y)T y + δ 2
❚ø ✤â t❛ ✤➣ ❝❤ù♥❣ ✤÷đ❝ ✈➨❝tì a = x0 − y tọ ỵ

ỵ C ởt t ỗ y ừ C õ tỗ t ởt

s ự y C tr♦♥❣ ♠ët ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣ ❝õ❛ ♥â✳
❈❤ù♥❣ ♠✐♥❤✳ ●å✐ {yk } ❧➔ ♠ët ❞➣② ✈➨❝tì ❦❤ỉ♥❣ ♥➡♠ tr♦♥❣ ❜❛♦ ✤â♥❣ ❝õ❛ C ✈➔ ❤ë✐ tö

✤➳♥ y✳ ❑❤✐ ✤â tỗ t ởt tỡ {ak } tữỡ ự s {yk } t ỵ
|ak | = 1 t❤ä❛ ♠➣♥
aTk yk < inf aTk x.
x∈C

❱➻ ❞➣② {ak } tỗ t {ai } i ∈ I ❤ë✐ tư ✤➳♥ a✳
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ ✈➨❝tì x ∈ C t❛ ❝â✿

aT y = lim aTi x ≤ lim aTi x = ax.

i∈I

i∈I

✣à♥❤ ♥❣❤➽❛ ✶✳✺✺✳ ▼ët s ự ởt t ỗ C tr ởt ỷ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣

❝õ❛ ♥â ✈➔ ❝❤ù❛ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ C ✤÷đ❝ ❣å✐ ❧➔ s✐➯✉ ♣❤➥♥❣ tü❛ ❝õ❛ C ✳

✷✷


❈❍×❒◆● ✷

❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯
❈➶ ✣■➋❯ ❑■➏◆
❈❤÷ì♥❣ ♥➔② t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ tè✐ ữ õ ỗ
ỡ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ✤✐➲✉
❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳ ❚➜t ❝↔ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤➲✉ ✤÷đ❝
❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t✳ ❈→❝ ❦➳t q✉↔ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ð ✤➙② ✤÷đ❝ t❤❛♠ ❦❤↔♦
❝❤õ ②➳✉ tø t➔✐ ❧✐➺✉ ❬✾❪✳

✷✳✶ ❇➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥
❇➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ ❧➔ ❜➔✐ t♦→♥ ❝â ❞↕♥❣✿

min f (x)

✭✷✳✶✮

s.t. ci (x) = 0, ; i = 1, . . . , me ,
ci (x) ≥ 0, i = me + 1, . . . , m.


✭✷✳✷✮
✭✷✳✸✮

x∈Rn

❑❤✐ ✤â✱ f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♠ư❝ t✐➯✉✱ ci (x), (i = 1, . . . , m) ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠
✤✐➲✉ ❦✐➺♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ❣✐↔ sû r➡♥❣ ❤➔♠ ♠ư❝ t✐➯✉ f (x) ✈➔ ❝→❝ ❤➔♠
✤✐➲✉ ❦✐➺♥ ci (x), (i = 1, . . . , m) ❧➔ ❝→❝ ❤➔♠ trì♥✱ ♥❤➟♥ ❣✐→ trà t❤ü❝ tr➯♥ Rn ✈➔ ➼t ♥❤➜t
♠ët ❤➔♠ ❧➔ ♣❤✐ t✉②➳♥✱ me ✈➔ m ❧➔ ♥❤ú♥❣ sè ♥❣✉②➯♥ ❦❤æ♥❣ ợ 0 me m. ú
t tữớ t
E = {1, . . . , me } ✈➔ I = {me + 1, . . . , m}

❧➛♥ ❧÷đt ❧➔ t➟♣ ❝❤➾ sè ❝õ❛ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝ ✈➔ t➟♣ ❝❤➾ sè ❝õ❛ ❝→❝ r➔♥❣ ❜✉ë❝ ❜➜t
✤➥♥❣ t❤ù❝✳ ◆➳✉ m = 0✱ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮ trð t❤➔♥❤ ♠ët ❜➔✐ t♦→♥ tè✐ ÷✉ ❦❤ỉ♥❣ ✤✐➲✉
❦✐➺♥❀ ♥➳✉ me = m = 0, ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❜➔✐ t♦→♥ tố ữ ợ r
ở tự tt ❤➔♠ ci(x), (i = 1, . . . , m) ✤➲✉ ❧➔ t✉②➳♥ t➼♥❤✱ t❤➻ ❇➔✐ t♦→♥
✭✷✳✶✮✲✭✷✳✸✮ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ t✉②➳♥ t➼♥❤✳ ❙❛✉ ✤➙② t❛ ✤÷❛ r❛ ♠ët
sè ❦❤→✐ ♥✐➺♠✿

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ✣✐➸♠ x ∈ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➳✉ x t❤ä❛ ♠➣♥
n

❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✮✲✭✷✳✸✮✳ ❚➟♣ ❤đ♣ ỗ tt ữủ ữủ ồ ❧➔ t➟♣
❤đ♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳
✷✸