Hàm lồi suy rộng và ứng dụng

ổ tọ ỏ t ỡ s s tợ P
trữớ ồ sữ ở ữợ t t
tổ t
ổ t ỡ ổ ừ trữớ ồ ữ
ở tr tử tự tổ tr sốt q tr ồ t
ứ q
ổ ỡ ỡ q ỗ s
ú ù ở t ồ t ủ tổ t


ở t


ứ ụ


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▲❮■ ❈❆▼ ✣❖❆◆
▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷

❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ P●❙✳❚❙ ◆❣✉②➵♥ ◆➠♥❣ ❚➙♠✳
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐ ✳
tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tæ✐ ✤➣ ❦➳ t❤ø❛
t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳
❈→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳

❍➔ ◆ë✐✱ ♥❣➔② ✵✺ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✸
❚→❝ ❣✐↔

✣✐♥❤ ❚❤ø❛ ❱ô


✷

▼ö❝ ❧ö❝
▲í✐ ❝↔♠ ì♥

✶

▲í✐ ❝❛♠ ✤♦❛♥

✶

❇↔♥❣ ❦➼ ❤✐➺✉

✹

▼ð ✤➛✉

✺

✶

✼

▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺
✷

❑❤æ♥❣ ❣✐❛♥ ❒❝❧✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➼♥❤ ❧✐➯♥ tö❝ ✈➔ t➼♥❤ ❦❤↔ ✈✐ ❝õ❛ ❤➔♠ sè ✳
❚➟♣ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❇➔✐ t♦→♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❍➔♠ ❧ç✐ s✉② rë♥❣

✳
✳
✳
✳
✳

✳
✳

✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳

✳
✳
✳

✳
✳
✳
✳
✳

✳ ✼
✳ ✽
✳ ✶✶
✳ ✶✷
✳ ✶✽
✷✶

✷✳✶ ❍➔♠ tü❛ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✷ ❍➔♠ ❣✐↔ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺


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✸

✷✳✸ ▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ♥❤ú♥❣ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸

Ù♥❣ ❞ö♥❣ ✈➔♦ ❧þ t❤✉②➳t tè✐ ÷✉

✹✶

✸✳✶ Ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈î✐ r➔♥❣ ❜✉ë❝ ❤➻♥❤ ❤å❝ ✳ ✳ ✹✶
✸✳✷ Ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ ✹✺
❑➳t ❧✉➟♥

✹✻

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✽


❇↔♥❣ ❦➼ ❤✐➺✉
R
Rn
R = R ∪ {−∞, +∞}
f :X→R
int A
A
dom(f )
epi(f )
ϕ (x)
∇f (x)

ϕ (x)
∇2 f (x)
||.||
|x|
af f (A)
coA
(x, y) = {λx + (1 − λ)y | λ ∈ (0, 1)}
(x, y] = {λx + (1 − λ)y | λ ∈ (0, 1]}
[x, y] = {λx + (1 − λ)y | λ ∈ [0, 1]}
L(f, α) = {x ∈ X | f (x) α}

✤÷í♥❣ t❤➥♥❣ t❤ü❝
❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ ♥ ✲ ❝❤✐➲✉
t➟♣ sè t❤ü❝ s✉② rë♥❣
→♥❤ ①↕ ✤✐ tø ❳ ✈➔♦ R
♣❤➛♥ tr♦♥❣ ❝õ❛ A
❜❛♦ ✤â♥❣ ❝õ❛ A
♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ f
tr➯♥ ✤ç t❤à ❝õ❛ f
✤↕♦ ❤➔♠ ❝õ❛ ϕ t↕✐ x
❣r❛❞✐❡♥t ❝õ❛ f t↕✐ x
✤↕♦ ❤➔♠ ❜➟❝ ❤❛✐ ❝õ❛ varphi t↕✐ x
♠❛ tr➟♥ ❍❡ss✐❛♥ ❝õ❛ f t↕✐ x
❝❤✉➞♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn
trà t✉②➺t ✤è✐ ❝õ❛ sè x
❜❛♦ ❧ç✐ ❛❢❢✐♥❡ ❝õ❛ A
❜❛♦ ❧ç✐ ❝õ❛ A
✤♦↕♥ t❤➥♥❣ ♠ð ♥è✐ x ✈➔ y
✤♦↕♥ t❤➥♥❣ ♠ð ♥è✐ x ✈➔ y
✤♦↕♥ t❤➥♥❣ ✤â♥❣ ♥è✐ x ✈➔ y

t➟♣ ♠ù❝ ❞÷î✐


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✺

▼Ð ✣❺❯
✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐

❈→❝ ❤➔♠ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣
❧þ t❤✉②➳t tè✐ ÷✉ ❤♦→ ✭①❡♠ ❬✽❪✱ ❬✶✵❪ ✈➔ ♥❤ú♥❣ t➔✐ ❧✐➺✉ tr➼❝❤ ❞➝♥ tr♦♥❣ ✤â✮✳
❍➔♠ ❧ç✐ s✉② rë♥❣ ✤➣ ✤÷ñ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔
t❤✉ ✤÷ñ❝ ♥❤✐➲✉ ❦➳t q✉↔ s➙✉ s➢❝✳ ❈→❝ ❤➔♠ tü❛ ❧ç✐✱ ❤➔♠ ❣✐↔ ❧ç✐ ✤➣ ✤÷ñ❝
▼❛♥❣❛s❛r✐❛♥ tr➻♥❤ ❜➔② tr♦♥❣ ❬✶✵❪ ✳ ❉✳ ❆✉ss❡❧ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t
✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❤➔♠ tü❛ ❧ç✐ ✈➔ ❣✐↔ ❧ç✐ ❦❤æ♥❣ trì♥ q✉❛ t➼♥❤ tü❛ ✤ì♥
✤✐➺✉ ✈➔ ❣✐↔ ✤ì♥ ✤✐➺✉ ❝õ❛ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ✤â ✈➔ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛
❝→❝ ❦❤→✐ ♥✐➺♠ ♥➔② tr♦♥❣ ❬✺❪✱ ❬✻❪✳ ❆✳ ❉❛♥✐✐❧✐❞✐s ✈➔ ◆✳ ❍❛❞❥✐s❛✈✈❛s ♥❣❤✐➯♥
❝ù✉ ❝→❝ ❤➔♠ tü❛ ❧ç✐ ❝❤➦t ✈➔ tü❛ ❧ç✐ ❜→♥ ❝❤➦t ❦❤æ♥❣ trì♥ ❬✷❪✳✳✳ ❙❛✉ ❦❤✐
✤÷ñ❝ ❤å❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ❚♦→♥ ❣✐↔✐ t➼❝❤✱ ✈î✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉
s➙✉ ❤ì♥ ✈➲ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✤➣ ❤å❝✱ ♠è✐ q✉❛♥ ❤➺ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ❝❤ó♥❣✱
tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✿ ✏❍➔♠ ❧ç✐ s✉② rë♥❣ ✈➔ ù♥❣ ❞ö♥❣✑ ✳
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉

❍✐➸✉ ❜✐➳t tê♥❣ q✉❛♥ ✈➲ ❤➔♠ ❧ç✐ s✉② rë♥❣✱ ♥➢♠ ✤÷ñ❝ ♥❤ú♥❣ t➼♥❤ ❝❤➜t
❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✈➔ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ✈➔♦ tè✐ ÷✉ ❤â❛✳
✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉

❈→❝ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❧ç✐ s✉② rë♥❣✱ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠
❧ç✐ s✉② rë♥❣✱ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝õ❛ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝


✻

❜✐➳♥ ♣❤➙♥ ✈➔ tè✐ ÷✉ ❤â❛✳
✹✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

❍➔♠ ❧ç✐ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞✳
✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

❚❤✉ t❤➟♣ t➔✐ ❧✐➺✉ ✈➲ ❤➔♠ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ s✉② rë♥❣✳ ✣å❝✱ ♣❤➙♥ t➼❝❤ ✈➔
tê♥❣ ❤ñ♣ ✤➸ ✤÷ñ❝ ♠ët ♥❣❤✐➯♥ ❝ù✉ tê♥❣ q✉❛♥ ✈➲ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✈➔ ù♥❣
❞ö♥❣✳
✻✳ ❉ü ❦✐➳♥ ✤â♥❣ ❣â♣ ♠î✐ ❝õ❛ ✤➲ t➔✐

✰ ◆❣❤✐➯♥ ❝ù✉ tê♥❣ q✉❛♥ ✈➲ ❤➔♠ ❧ç✐ s✉② rë♥❣ ✈➔ ù♥❣ ❞ö♥❣✳


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✼

❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
♥❤➜t ❝õ❛ t➟♣ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ Rn ❝ò♥❣ ✈î✐ ♥❤ú♥❣ t➼♥❤
❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♥â✳ ◆❤ú♥❣ ❦✐➳♥ t❤ù❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔②
✤÷ñ❝ ❝❤å♥ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✽❪✳

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❒❝❧✐t
❚➟♣ ❤ñ♣
Rn := {x = (x1 , ..., xn )T : x1 , ..., xn ∈ R},

tr♦♥❣ ✤â





x1
x 
 2
x = (x1 , ..., xn )T :=  ✳ 
✳
xn

✈î✐ ❤❛✐ ♣❤➨♣ t♦→♥
(x1 , ..., xn )T + (y1 , ..., yn )T := (x1 + y1 , ..., xn + yn )T

λ(x1 , ..., xn )T := (λx1 , ..., λxn )T ,

λ∈R


✽

❧➟♣ t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈➨❝ tì ❒❝❧✐t n−❝❤✐➲✉✳
◆➳✉ x = (x1, ..., xn)T ∈ Rn t❤➻ xi ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ ❤♦➦❝ tå❛ ✤ë t❤ù
i ❝õ❛ x✳ ❱➨❝ tì ❦❤æ♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♥➔② ❣å✐ ❧➔ ❣è❝ ❝õ❛ Rn ✈➔ ✤÷ñ❝ ❦➼
❤✐➺✉ ✤ì♥ ❣✐↔♥ ❧➔ 0✱ ✈➟② 0 = (0, ..., 0)T ✳
❚r♦♥❣ Rn t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ✈æ ❤÷î♥❣ ❝❤➼♥❤ t➢❝ ., . ♥❤÷ s❛✉✿ ✈î✐
x = (x1 , ..., xn )T , y = (y1 , ..., yn )T ∈ Rn ✱
n

x, y =

❑❤✐ ✤â ✈î✐ ♠å✐ x = (x1, ..., xn)T ∈

xi y i .

i=1
Rn

t❛ ✤à♥❤ ♥❣❤➽❛
n

x :=

(xi )2

x, x =
i=1

✈➔ ❣å✐ ❧➔ ❝❤✉➞♥ ❊✉❝❧✐❞ ❝õ❛ ✈➨❝ tì x✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ x0 ∈ Rn ✱ ε > 0✱ t❛ ❣å✐ t➟♣
B(x0 , ε) := {x ∈ Rn :

x − x0 < ε}

❧➔ ❤➻♥❤ ❝➛✉ ♠ð tr♦♥❣ Rn ❝â t➙♠ t↕✐ x0✱ ❜→♥ ❦➼♥❤ ε✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❚➟♣ U ⊂ Rn ❣å✐ ❧➔ ♠ð ♥➳✉ ✈î✐ ♠å✐ x0 ∈ U ✱ tç♥ t↕✐
ε > 0 s❛♦ ❝❤♦ B(x0 , ε) ⊂ U.
❚➟♣ F ⊂ Rn ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ U := Rn \ F ❧➔ ♠ð✳
❚➟♣ V ⊂ Rn ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x ∈ Rn ♥➳✉ tç♥ t↕✐ ε > 0 s❛♦ ❝❤♦
B(x, ε) ⊂ V.

✶✳✷ ❚➼♥❤ ❧✐➯♥ tö❝ ✈➔ t➼♥❤ ❦❤↔ ✈✐ ❝õ❛ ❤➔♠ sè
✐✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ t↕✐ x ∈ Rn
✭✈î✐ f (x) < ∞✮✱ ♥➳✉ ✈î✐ ♠å✐ ε > 0✱ tç♥ t↕✐ ❧➙♥ ❝➟♥ U ❝õ❛ x s❛♦ ❝❤♦
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳


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✾

f (x) − ε ≤ f (y)

(∀y ∈ U ) .

✐✐✮ ◆➳✉ f (x) = +∞✱ t❤➻ ❢ ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ t↕✐ x✱ ♥➳✉ ✈î✐
♠å✐ N > 0✱ tç♥ t↕✐ ❧➙♥ ❝➟♥ U ❝õ❛ x s❛♦ ❝❤♦✿ f (y) ≥ N (∀y ∈ U ) .
✐✐✐✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ ♥➳✉ f ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ t↕✐
♠å✐ x ∈ Rn✳
❈❤♦ f : X → R✳ ❚❛ ♥â✐ f ❧✐➯♥ tö❝ t↕✐ x0 ∈ X ♥➳✉
✈î✐ ♠å✐ ε > 0✱ tç♥ t↕✐ δ > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ x ∈ X ∩ B(x0, δ) t❛ ❝â
f (x0 ) ∈ B(x0 , ε)✳
❚❛ ♥â✐ f ❧✐➯♥ tö❝ tr➯♥ X ♥➳✉ f ❧✐➯♥ tö❝ t↕✐ ♠å✐ x0 ∈ X ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳

❈❤♦ U ⊂ Rn ❧➔ t➟♣ ♠ð✱ ❤➔♠ sè f : U → R✱ ✈➔
x0 = (x01 , ..., x0n )T ∈ U ✳ ❑❤✐ ✤â tç♥ t↕✐ δ > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ h ∈ R ♠➔
|h| < δ t❛ ❝â x(h) = (x0 , ..., x0i−1 , x0i + h, x0i+1 , ..., x0n ) ∈ U ✳
◆➳✉ tç♥ t↕✐ ❣✐î✐ ❤↕♥
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

f (x0 , ..., x0i−1 , x0i + h, x0i+1 , ..., x0n ) − f (x0 )
lim
h
h→0

t❤➻ t❛ ❣å✐ ♥â ❧➔ ✤↕♦ ❤➔♠ r✐➯♥❣ ✭❝➜♣ 1✮ t❤❡♦ ❜✐➳♥ xi ❝õ❛ f t↕✐ x0✱ ❦➼ ❤✐➺✉
∂f 0

(x )
∂xi

◆➳✉ f ❝â ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣
(

❤♦➦❝

fxi (x0 ).

✱ i = 1, ..., n✱ t❤➻ ✈➨❝ tì

∂f
0
∂xi (x )

∂f 0
∂f 0 T
(x ), ...,
(x ))
∂x1
∂xn

❣å✐ ❧➔ ❣r❛❞✐❡♥t ❝õ❛ f t↕✐ x0 ✈➔ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔
∇f (x0 ) = (

∂f 0
∂f 0 T
(x ), ...,
(x )) .

∂x1
∂xn

∂f
∂f
◆➳✉ ✤↕♦ ❤➔♠ r✐➯♥❣ ∂x
(x) tç♥ t↕✐ t↕✐ ♠å✐ x ∈ U t❤➻ t❛ ❝â ❤➔♠ ∂x
:U →
∂f
R ①→❝ ✤à♥❤ ❜ð✐ q✉② t➢❝ x → ∂x
(x)✳ ◆➳✉ tç♥ t↕✐ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤❡♦ ❜✐➳♥
i

i

i


✶✵
∂f
t❤ù j ❝õ❛ ❤➔♠ ∂x
t↕✐ x0 t❤➻ t❛ ❣å✐ ♥â ❧➔ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ 2 t❤❡♦ ❜✐➳♥
xi ✈➔ xj ❝õ❛ f t↕✐ x0 ✱ ❦➼ ❤✐➺✉ ❧➔
i

∂ 2f
(x0 )
∂xi ∂xj

▼❛ tr➟♥ ∂x∂ ∂xf (x0) ✱ i,

x0 ✱ ❦➼ ❤✐➺✉ ❧➔ ∇2 f (x0 )✳
2

i

j

❤♦➦❝

j = 1, ..., n

fxi xj (x0 ).

❣å✐ ❧➔ ♠❛ tr➟♥ ❍❡ss✐❛♥ ❝õ❛ f t↕✐

❈❤♦ U ⊂ Rn ❧➔ t➟♣ ♠ð✱ ❤➔♠ sè f : U → R✱ x0 = (x01, ..., x0n)T ∈ U ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❚❛ ♥â✐ f ❦❤↔ ✈✐ t↕✐ x0 ♥➳✉ ✈î✐ ♠å✐ i = 1, ..., n tç♥ t↕✐
✤↕♦ ❤➔♠ r✐➯♥❣ fx (x0) ✈➔ ✈î✐ ♠å✐ h ∈ Rn \ {0} ♠➔ x0 + h ∈ U t❛ ❝â
i

f (x0 + h) = f (x0 ) + ∇f (x0 ), h + r( h ),

tr♦♥❣ ✤â r( hh ) → 0 ❦❤✐ h → 0✳
◆➳✉ f ❦❤↔ ✈✐ t↕✐ ♠å✐ x ∈ U t❤➻ t❛ ♥â✐ f ❦❤↔ ✈✐ tr➯♥ U ✳
◆➳✉ f ❦❤↔ ✈✐ tr➯♥ U ✈➔ ❝→❝ ❤➔♠ fx (.) : U → R✱ i = 1, ..., n✱ ✤➲✉
❧✐➯♥ tö❝ tr➯♥ U ✱ t❤➻ t❛ ♥â✐ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ U ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❚❛ ♥â✐ f ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ t↕✐ x0 ♥➳✉ ✈î✐ ♠å✐ i, j = 1, ..., n
tç♥ t↕✐ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐ fx x (x0) ✈➔ ✈î✐ ♠å✐ h ∈ Rn \ {0} ♠➔
x0 + h ∈ U t❛ ❝â
i

i j

h, ∇2 f (x0 )h
f (x + h) = f (x ) + ∇f (x ), h +
+ r( h 2 ),
2
0

0

0

tr♦♥❣ ✤â r( hh ) → 0 ❦❤✐ h → 0✳
◆➳✉ f ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ t↕✐ ♠å✐ x ∈ U t❤➻ t❛ ♥â✐ f ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr➯♥
2

2

U✳

◆➳✉ f ❦❤↔ ✈✐ ❤❛✐ ❧➛♥ tr➯♥ U ✱ ❝→❝ ❤➔♠ fx (.) : U → R ✈➔ fx x (.) :
U → R✱ i, j = 1, ..., n✱ ✤➲✉ ❧✐➯♥ tö❝ tr➯♥ U ✱ t❤➻ t❛ ♥â✐ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝
❤❛✐ ❧➛♥ tr➯♥ U ✳
i

i j


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✶✶

❈❤♦ f : Rn → R ✈➔ x0 ∈ Rn✳ ❑❤✐ ✤â
✭✐✮ ◆➳✉ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛
h ∈ Rn ♠➔ h ✤õ ♥❤ä t❛ ❝â

✣à♥❤ ❧➼ ✶✳✶✳

x0

t❤➻ ✈î✐

f (x0 + h) = f (x0 ) + ∇f (x0 ), h + r( h ),

tr♦♥❣ ✤â r( hh ) → 0 ❦❤✐ h → 0✳
✭✐✐✮ ◆➳✉ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❤❛✐ ❧➛♥ tr➯♥ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛
✈î✐ h ∈ Rn ♠➔ h ✤õ ♥❤ä t❛ ❝â

t❤➻

h, ∇2 f (ν)h
,
f (x + h) = f (x ) + ∇f (x ), h +

2
ν = tx0 + (1 − t)h✱ t ∈ (0, 1)✳
0

tr♦♥❣ ✤â

x0

0

0

✶✳✸ ❚➟♣ ❧ç✐
✣à♥❤ ♥❣❤➽❛ ✶✳✽✳

❚➟♣ X ⊂ Rn ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐✱ ♥➳✉

∀x, y ∈ X, ∀λ ∈ R : 0 ≤ λ ≤ 1 ⇒ λx + (1 − λ) y ∈ X.

❈❤♦ Xα ⊂ Rn (α ∈ I) ❧➔ ❝→❝ t➟♣ ❧ç✐✱ ✈î✐ I ❧➔ t➟♣ ❝❤➾ sè
❜➜t ❦➻✳ ❑❤✐ ✤â X = Xα ❝ô♥❣ ❧ç✐✳
▼➺♥❤ ✤➲ ✶✳✶✳

α∈I

❈❤♦ ❝→❝ t➟♣ Xi ⊂ Rn ❧ç✐✱ λi ∈ R (i = 1, 2, ..., m)✳ ❑❤✐
✤â λ1X1 + ... + λmXm ❝ô♥❣ ❧➔ t➟♣ ❧ç✐✳
▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ ❝→❝ t➟♣ Xi ⊂ Rn ❧ç✐✱ (i = 1, 2, . . . , m)✳ ❑❤✐ ✤â t➼❝❤
✣➲ ❝→❝ X1 × ... × Xm ❧➔ t➟♣ ❧ç✐ tr♦♥❣ Rn × ... × Rn .
✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❱➨❝ tì x ∈ Rn ✤÷ñ❝ ❣å✐ ❧➔ tê ❤ñ♣ ❧ç✐ ❝õ❛ ❝→❝ ✈➨❝tì

m
n
x1 , ..., xm t❤✉ë❝ R ✱ ♥➳✉ ∃λi ≥ 0 (i = 1, 2, ..., m) ,
λi = 1 s❛♦ ❝❤♦

▼➺♥❤ ✤➲ ✶✳✷✳

i

1

m

i=1

m

x=

λi xi .
i=1


✶✷

❈❤♦ t➟♣ X ⊂ Rn ❧ç✐❀ x1, ..., xm ∈ X ✳ ❑❤✐ ✤â X ❝❤ù❛ t➜t
❝↔ ❝→❝ tê ❤ñ♣ ❧ç✐ ❝õ❛ x1, ..., xm✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ❈❤♦ X ⊂ Rn ✳ ●✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ❧ç✐ ❝❤ù❛ X
✤÷ñ❝ ❣å✐ ❧➔ ❜❛♦ ❧ç✐ ✭❝♦♥✈❡① ❤✉❧❧✮ ❝õ❛ t➟♣ X ✱ ❦➼ ❤✐➺✉ ❧➔ coX ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ●✐↔ sû X ⊂ Rn ✳ ●✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ❧ç✐ ✤â♥❣

❝❤ù❛ X ✤÷ñ❝ ❣å✐ ❧➔ ❜❛♦ ❧ç✐ ✤â♥❣ ❝õ❛ t➟♣ X ✈➔ ❦➼ ❤✐➺✉ ❧➔ coX ✳
▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ X ⊂ Rn ❧ç✐✳ ❑❤✐ ✤â✱
✐✮ P❤➛♥ tr♦♥❣ intX ✈➔ ❜❛♦ ✤â♥❣ X ❝õ❛ X ❧➔ ❝→❝ t➟♣ ❧ç✐❀
✐✐✮ ◆➳✉ x1 ∈ ✐♥tX, x2 ∈ X ✱ t❤➻
{λx1 + (1 − λ)x2 : 0 < x1 ≤ 1} ⊂ ✐♥tX.
✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳ ⑩♥❤ ①↕ f : Rn → Rm ✤÷ñ❝ ❣å✐ ❧➔ ❛❢❢✐♥❡ ♥➳✉
✣à♥❤ ❧➼ ✶✳✷✳

∀x, y ∈ E1 , λ ∈ R; f ((1 − λ) x + λy) = (1 − λ) f x + λf y.

P❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ A ⊂ E ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛
A tr♦♥❣ ❛❢❢A❀ ❦➼ ❤✐➺✉ ❧➔ r✐ A✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳

❈→❝ ✤✐➸♠ t❤✉ë❝ r✐ A ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ t➟♣ A✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✹✳ ❚➟♣ A \ r✐A ✤÷ñ❝ ❣å✐ ❧➔ ❜✐➯♥ t÷ì♥❣ ✤è✐ ❝õ❛ A✳
❚➟♣ ❆ ✤÷ñ❝ ❣å✐ ❧➔ ♠ð t÷ì♥❣ ✤è✐✱ ♥➳✉ r✐ A = A✳

✶✳✹ ❍➔♠ ❧ç✐
❈❤♦ ❤➔♠
R ∪ {−∞, +∞}✱ ❝→❝ t➟♣
✣à♥❤ ♥❣❤➽❛ ✶✳✶✺✳

f : X → R✱

tr♦♥❣ ✤â

epi(f ) = {(x, α) ∈ X × R| f (x) ≤ α} ,
dom(f ) = {x ∈ X| f (x) < +∞}

X ⊂ Rn ✱ R =


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✶✸

✤÷ñ❝ ❣å✐ ❧➛♥ ❧÷ñt ❧➔ tr➯♥ ✤ç t❤à ✈➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ f ✳
❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ❧ç✐✱ f : X → R✳
❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ tr➯♥ X ♥➳✉ tr➯♥ ✤ç t❤à epi(f ) ❝õ❛ ♥â ❧➔ ♠ët
t➟♣ ❧ç✐ tr♦♥❣ Rn × R✳
◆➳✉ dom f = ∅ ✈➔ −∞ < f (x) ✈î✐ ♠å✐ x ∈ X t❛ ♥â✐ ❤➔♠ f ❧➔ ❝❤➼♥❤
t❤÷í♥❣✳
❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❧ã♠ tr➯♥ X ♥➳✉ −f ❧➔ ❤➔♠ ❧ç✐ tr➯♥ X ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✻✳

❱➼ ❞ö

✣➦t

✭❍➔♠ ❝❤➾✮✳ ❈❤♦ C = ∅ ❧➔ ♠ët t➟♣ ❧ç✐ tr♦♥❣ Rn✳
δC (x) :=

0 khi x ∈ C,

+∞ khi x ∈
/ C.

❚❛ ♥â✐ δC ❧➔ ❤➔♠ ❝❤➾ ❝õ❛ C ✳
✰ ∀x, y ∈ C, ∀λ ∈ (0, 1)✱ t❛ ❝â✿ δC (x) = 0, δC (y) = 0.
❉♦ C ❧ç✐ ♥➯♥ λx + (1 − λ)y ∈ C.
❙✉② r❛ δC [λx + (1 − λ)y] = 0 = λδC (x) + (1 − λ)δC (y) .
✰ ∀x ∈ C, ∀y ∈/ C, ∀λ ∈ (0, 1)✱ t❛ ❝â✿
δC (x) = 0, δC (y) = +∞, δC [λx + (1 − λ)y] ≤ +∞.

❙✉② r❛ δC [λx + (1 − λ)y] ≤ λδC (x) + (1 − λ)δC (y) .
✰ ∀x, y ∈/ C, ∀λ ∈ (0, 1)✱ t❛ ❝â✿
δC (x) = +∞, δC (y) = +∞, δC [λx + (1 − λ)y] ≤ +∞.

❙✉② r❛ δC [λx + (1 − λ)y] ≤ λδC (x) + (1 − λ)δC (y) .
❱➼ ❞ö ✭❍➔♠ tü❛✮✳ ❈❤♦ C = ∅ ❧➔ ♠ët t➟♣ ❧ç✐ tr♦♥❣ Rn ✳
✣➦t SC (y) := supx∈C y, x ✈î✐ y ∈ Rn✳ ❚❛ ♥â✐ SC ❧➔ ❤➔♠ tü❛ ❝õ❛


✶✹

C✳

❱î✐ ♠å✐ x, y ∈ C ✈➔ ✈î✐ ♠å✐ , λ ∈ (0, 1)✱ t❛ ❝â
SC [λx + (1 − λ) y] = sup λx + (1 − λ) y, z
z∈C

= sup { λx, z + (1 − λ) y, z }
z∈C

≤ sup λx, z + sup (1 − λ) y, z
z∈C

z∈C

= λsup x, z + (1 − λ) sup y, z
z∈C

z∈C

= λSC (x) + (1 − λ) SC (y) .

❱➟② SC ❧➔ ❤➔♠ ❧ç✐ tr➯♥ C ✳
●✐↔ sû f1, ..., fm ❧➔ ❝→❝ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣ tr➯♥ X ✳ ❑❤✐
✤â✱ tê♥❣ f1 + ... + fm ❧➔ ♠ët ❤➔♠ ❧ç✐✳

✣à♥❤ ❧➼ ✶✳✸✳

✈✐✳

❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ✤➦❝ tr÷♥❣ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧ç✐ ♠ët ❜✐➳♥ ❦❤↔

❈❤♦ ϕ : (a, b) → R✳
✐✮ ◆➳✉ ϕ ❦❤↔ ✈✐ tr➯♥ (a, b) t❤➻ ϕ ❧ç✐ tr➯♥ (a, b) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ϕ ❦❤æ♥❣
❣✐↔♠ tr➯♥ (a, b)✳
✐✐✮ ◆➳✉ ϕ ❝â ✤↕♦ ❤➔♠ ❜➟❝ ❤❛✐ tr➯♥ (a, b) t❤➻ ϕ ❧ç✐ tr➯♥ (a, b) ❦❤✐ ✈➔
❝❤➾ ❦❤✐ ϕ (t) 0 ✈î✐ ♠å✐ t ∈ (a, b)✳
✐✐✐✮ ◆➳✉ ϕ ❧ç✐ tr➯♥ [a, b] t❤➻ ϕ ❧✐➯♥ tö❝ tr➯♥ (a, b)✳

✣à♥❤ ❧➼ ✶✳✹✳

❈❤♦ X ❧➔ t➟♣ ❧ç✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn ✈➔ f : X → R✳ ❑❤✐
✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ f (λx + (1 − λ) y) ≤ λf (x) + (1 − λ) f (y) ∀λ ∈ [0, 1] , ∀x, y ∈ X.
❜✮ f (λx + (1 − λ) y) λf (x)+(1 − λ) f (y) ∀λ > 1, ∀x, y ∈ X s❛♦ ❝❤♦
✣à♥❤ ❧➼ ✶✳✺✳

λx + (1 − λ) y ∈ X.


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✶✺

❝✮ f (λx + (1 − λ) y)

λf (x)+(1 − λ) f (y) ∀λ < 0, ∀x, y ∈ X

s❛♦ ❝❤♦

λx + (1 − λ) y ∈ X.

❞✮✭❇➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥✮ ❱î✐ ❜➜t ❦➻ x1, . . . , xm ∈ X ✱ i = 1, . . . , m ✈➔ ✈î✐
m

❜➜t ❦➻ λi ∈ [0, 1], i = 1, . . . , m, λi = 1 ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤ó♥❣✿
i=1

f (λ1 x1 + ... + λm xm ) ≤ λ1 f (x1 ) + ... + λm f (xm ) .

❡✮ ❱î✐ ♠å✐ x ∈ X ✱ ✈î✐ ♠å✐ y ∈ Rn✱ ❤➔♠ ϕx,y (t) = f (x + ty) ❧➔ ❤➔♠ ❧ç✐
tr➯♥ ✤♦↕♥ Tx,y = {t ∈ R | x + ty ∈ X}.
❢✮ ❱î✐ ♠å✐ x, y ∈ X ✱ ❤➔♠ ψx,y (λ) = f (λx + (1 − λ)y) ❧ç✐ tr➯♥ ✤♦↕♥ [0, 1].
❣✮ ❚r➯♥ ✤ç t❤à ❝õ❛ f ❧➔ t➟♣ ❧ç✐ tr♦♥❣ Rn+1✳
●✐↔ sû X ⊂ Rn ❧➔ ♠ët t➟♣ ❧ç✐ ♠ð✱ f : X → R✳ ❑❤✐ ✤â✱ f
❧ç✐ tr➯♥ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
❤✮ ✈î✐ ♠å✐ x0 ∈ X ✱ tç♥ t↕✐ x∗ ∈ Rn s❛♦ ❝❤♦
✣à♥❤ ❧➼ ✶✳✻✳

f (x) − f (x0 )

x∗ (x − x0 ),

x ∈ X.

❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ❧ç✐ ✈➔ f : X → R✳ ❑❤✐ ✤â✱ ♥➳✉ f
❧ç✐ tr➯♥ X t❤➻✱ ✈î✐ ♠å✐ α ∈ R t➟♣ ♠ù❝ ❞÷î✐ ❝õ❛ f
✣à♥❤ ❧➼ ✶✳✼✳

L(f, α) = {x ∈ X | f (x) ≤ α}

❧➔ t➟♣ ❧ç✐✳

❳➨t ❤➔♠ sè f : R → R ①→❝ ✤à♥❤ ❜ð✐ f (x) = x3✳ ❚❛ ❝â f ❦❤æ♥❣
❧ç✐ tr➯♥ R✱ tr♦♥❣ ❦❤✐ L(f, α) = {x ∈ R | x3 ≤ α} = {x ∈ R | x ≤ α1/3}

❧➔ t➟♣ ❧ç✐ ✈î✐ ♠å✐ α ∈ R✳

❱➼ ❞ö✳


✶✻

❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ♠ð ✈➔ f : X → R ❦❤↔ ✈✐ tr➯♥ X ✳
❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ f ❧ç✐ tr➯♥ X
❜✮ ❱î✐ ♠å✐ x ∈ X ✈➔ ✈î✐ ♠å✐ y ∈ Rn ❤➔♠
✣à♥❤ ❧➼ ✶✳✽✳

ϕx,y (t) = y, ∇f (x + ty) ,

❜✐➳♥ t✱ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ ✤♦↕♥ Tx,y = {t ∈ R | x + ty ∈ X}.
❝✮ ❱î✐ ♠å✐ x, y ∈ X ✱ ❤➔♠
ψx,y (λ) = (x − y), ∇f (λx + (1 − λ)y) ,

❜✐➳♥ λ✱ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ ✤♦↕♥ [0, 1] ✳
❞✮ ❱î✐ ♠å✐ x, y ∈ X ✱ f (x) − f (y) (x − y), ∇f (y) ✳
❡✮ ❱î✐ ♠å✐ x, y ∈ X ✱ f (x) − f (y) (x − y), ∇f (x) ✳
❢✮ ❱î✐ ♠å✐ x, y ∈ X ✱ (x − y), ∇f (x) − ∇f (y) 0✳
❈❤♦ f : X → R ❧➔ ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❤❛✐ ❧➛♥ tr➯♥ t➟♣
❧ç✐ ♠ð X ⊂ Rn✳ ❑❤✐ ✤â✱ f ❧ç✐ tr➯♥ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠❛ tr➟♥ ❍❡ss✐❛♥
∇2 f (x) ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈î✐ ♠å✐ x ∈ X ✳
✣à♥❤ ❧➼ ✶✳✾✳

❈❤♦ X ❧➔ t➟♣ ❧ç✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn✱ f : X → R✳
❚❛ ♥â✐ f ❧ç✐ ❝❤➦t tr➯♥ X ♥➳✉

✣à♥❤ ♥❣❤➽❛ ✶✳✶✼✳

f (λx + (1 − λ) y) < λf (x)+(1 − λ) f (y) ∀λ ∈ [0, 1] , ∀x, y ∈ X, x = y.

❈❤♦ X ❧➔ t➟♣ ❧ç✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Rn ✈➔ f : X → R✳ ❑❤✐
✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ f ❧ç✐ ❝❤➦t tr➯♥ X
❜✮ ❱î✐ ♠å✐ x ∈ X ✱ ✈î✐ ♠å✐ y ∈ Rn✱ ❤➔♠ ϕx,y (t) = f (x + ty) ❧➔ ❤➔♠ ❧ç✐
❝❤➦t tr➯♥ ✤♦↕♥ Tx,y = {t ∈ R | x + ty ∈ X}.
✣à♥❤ ❧➼ ✶✳✶✵✳


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✶✼

❝✮ ❱î✐ ♠å✐ x, y ∈ X ✱ ❤➔♠ ψx,y (λ) = f (λx + (1 − λ)y) ❧ç✐ tr➯♥ ✤♦↕♥ [0, 1]
✳
❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ♠ð ✈➔ f : X → R ❦❤↔ ✈✐ tr➯♥
X ✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ f ❧ç✐ ❝❤➦t tr➯♥ X
❜✮ ❱î✐ ♠å✐ x, y ∈ X ✱ x = y✱ f (x) − f (y) > (x − y), ∇f (y) ✳
❝✮ ❱î✐ ♠å✐ x, y ∈ X ✱ x = y✱ f (x) − f (y) < (x − y), ∇f (x) ✳
❞✮ ❱î✐ ♠å✐ x, y ∈ X ✱ (x − y), ∇f (x) − ∇f (y) > 0✳

✣à♥❤ ❧➼ ✶✳✶✶✳

❈❤♦ f : X → R ❧➔ ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❤❛✐ ❧➛♥ tr➯♥ t➟♣
❧ç✐ ♠ð X ⊂ Rn✳ ❑❤✐ ✤â✱ ♥➳✉ ♠❛ tr➟♥ ❍❡ss✐❛♥ ∇2f (x) ①→❝ ✤à♥❤ ❞÷ì♥❣
✈î✐ ♠å✐ x ∈ X ✱ ♥❣❤➽❛ ❧➔ ✈î✐ ♠å✐ x ∈ X ✱ y, ∇2f (x)y > 0 ✈î✐ ♠å✐
y ∈ Rn , y = 0✱ t❤➻ f ❧ç✐ ❝❤➦t tr➯♥ X ✳
✣à♥❤ ❧➼ ✶✳✶✷✳

✣✐➲✉ ❦✐➺♥ ♥➯✉ tr➯♥ ❝❤➾ ✤õ ❝❤ù ❦❤æ♥❣ ❝➛♥ ✤➸ f ❧ç✐ ❝❤➦t✳ ❱➼ ❞ö ♥❤÷✱
f (x) = x4 ❧ç✐ ❝❤➦t tr➯♥ R✱ ♥❤÷♥❣ ∇2 f (x) = 12x2 ❦❤æ♥❣ ①→❝ ✤à♥❤ ❞÷ì♥❣
tr➯♥ R✱ ✈➻ ∇2f (0) = 0✳
❍➔♠ f : X → R ①→❝ ✤à♥❤ tr➯♥ t➟♣ ❧ç✐ X ⊂ Rn ✤÷ñ❝
❣å✐ ❧➔ ❤➔♠ ❛♣❤✐♥ tr➯♥ X ♥➳✉ ♥â ✈ø❛ ❧ç✐ ✈ø❛ ❧ã♠ tr➯♥ X ✱ ♥❣❤➽❛ ❧➔
✣à♥❤ ♥❣❤➽❛ ✶✳✶✽✳

f (λx + (1 − λ) y) = λf (x) + (1 − λ) f (y) ∀λ ∈ [0, 1] , ∀x, y ∈ X.

❧➔ ❤➔♠ ❛♣❤✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐
c ∈ Rn ✈➔ sè α ∈ R s❛♦ ❝❤♦ f (x) = c, x + α✳
✣à♥❤ ❧➼ ✶✳✶✸✳

❍➔♠

f : Rn → R

●✐↔ sû ❢ ❧➔ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣ tr➯♥
❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✐✮ f ❜à ❝❤➦♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x ❀
✣à♥❤ ❧➼ ✶✳✶✹✳

Rn ✳

❑❤✐ ✤â ❝→❝




f tử t x
int(epif ) =
int(domf ) = f tử tr int(domf )
ỗ tớ t {(x, à) X ì R : x int(domf ), f (x) < à} .

t tố ữ
t tố ữ t ữủ ổ t ữợ
ợ x X,

min f (x)

P

tr õ f : Rn R ởt trữợ X Rn ởt t
trữợ Rn ổ n
ỏ t t (P ) ữ s
min{f (x) : x X}.

t (P ) ữủ ồ ởt t tố ữ
f ồ ử t X t r ở ữủ
ừ (P ) tỷ ừ X ữủ ồ tỷ ữủ
ừ (P )
X = Rn t t õ (P ) ởt t ổ õ r ở

ữủ (P ) t õ r ở




x X
f (x )

f (x)

ợ ồ

xX

ữủ ồ tố ữ tố ữ t ử
ỹ t t ử ừ t (P )


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õ x X ởt tố ữ ữỡ ỹ
t ữỡ ừ (P ) tỗ t ởt U ừ x s

f (x) f (x)

ợ ồ

x X U.

õ x X ởt tố ữ ữỡ t
ỹ t ữỡ t ừ (P ) tỗ t ởt U ừ x s

f (x) < f (x)
ợ ồ x X U, x = x.
r tố ữ t ử ừ t P
tố ữ ữỡ ừ P ữ r ữủ ỳ
ử ự tọ r ữủ ổ ú ụ ú
X ởt t ỗ f ởt ỗ tr X t ồ
tố ữ ữỡ ừ P tố ữ t ử ừ P
s ự õ
X Rn t ỗ f : X R ởt ỗ
õ x ởt tố ữ ừ t P



min{f (x) : x X},

t õ ụ ởt tố ữ t ử ừ t õ
ự sỷ x tố ữ ữỡ ừ P õ
x X tỗ t V tr Rn ừ x s
f (x )

f (x) x V X.

x X tũ ỵ X ỗ ợ ồ t (0, 1) ừ ọ t õ
xt := tx + (1 t)x V X.


✷✵

❉♦ f ❧ç✐ tr➯♥ X ♥➯♥
f (x∗ )

s✉② r❛ f (x∗)
❝ö❝ ❝õ❛ ✭P✮✳

f (xt )

f (x)✳ ❱➻ x ∈ X

tf (x) + (1 − t)f (x∗ ),

tò② þ✱ t❛ s✉② r❛ x∗ ❧➔ ♥❣❤✐➺♠ tè✐ ÷✉ t♦➔♥

❈❤♦ X ⊂ Rn ❧➔ t➟♣ ❧ç✐✱ f : X → R ❧➔ ♠ët ❤➔♠ ❧ç✐ tr➯♥
X ✳ ❑❤✐ ✤â✱ t➟♣ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✿

✣à♥❤ ❧➼ ✶✳✶✻✳

min{f (x) : x ∈ X},

❧➔ ♠ët t➟♣ ❧ç✐✳
❈❤♦ X ⊂ Rn ❧➔ t➟♣ ❧ç✐✱ f : X → R ❧➔ ♠ët ❤➔♠ ❧ç✐ ❝❤➦t

tr➯♥ X ✳ ❑❤✐ ✤â✱ ♥➳✉ x∗ ❧➔ ♠ët ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✿
✣à♥❤ ❧➼ ✶✳✶✼✳

min{f (x) : x ∈ X},

t❤➻ ♥â ❝ô♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ tè✐ ÷✉ t♦➔♥ ❝ö❝ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✤â✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû r➡♥❣ ❜➔✐ t♦→♥ ❝â ❤❛✐ ♥❣❤✐➺♠ tè✐ ÷✉ x1 ✈➔ x2 ✈➔
f (x1 ) = f (x2 ) = α✳ ❚❤❡♦ ✤à♥❤ ❧þ tr➯♥✱ ✈î✐ ♠å✐ λ ∈ (0, 1) t❛ ❝â f (λx1 +
(1 − λ)x2 ) = α✱ ♠➙✉ t❤✉➝♥ ✈î✐ ✤✐➲✉ f ❧ç✐ ❝❤➦t✳

❑➳t ❧✉➟♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ ✤➣ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ♠ët sè t➼♥❤
❝❤➜t ❝ì ❜↔♥ ❝õ❛ t➟♣ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐ ❝ò♥❣ ✈î✐ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛✳


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ữỡ
ỗ s rở
ữỡ tr ởt số ở ừ ỗ s rở
tỹ ỗ ừ ỗ ỳ ở tr tr
ữỡ ừ ồ tứ t

tỹ ỗ


tỹ ỗ

X Rn t ỗ f : X R ồ

f (x + (1 )y)

max{f (x), f (y)},

x, y X,

[0, 1].

ởt tữỡ ữỡ ừ tỹ ỗ f : X R
tr õ X Rn ởt t ỗ
t

x, y X,

f (x)

f (y) = f (x + (1 )y)

ồ ỗ f
sỷ f ỗ õ
t

f (x + (1 )y)

:X R

f (y),

[0, 1]

tỹ ỗt

f (x) + (1 )f (y)
max{f (x), f (y)},

x, y X,

[0, 1].


✷✷

❱➼ ❞ö s❛✉ ❝❤ù♥❣ tä r➡♥❣✱ ✤✐➲✉ ♥❣÷ñ❝ ❧↕✐ tr♦♥❣ ♥❤➟♥ ①➨t tr➯♥ ❦❤æ♥❣
✤ó♥❣✳
❱➼ ❞ö ✷✳✶ ✳ ▲➜② X = {(x, y) ∈ R2 | x, y
0}✱ f : X → R❀ f (x, y) = −xy ✳
❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ❧ç✐ ✈➔
✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
❛✮ f ❧➔ ❤➔♠ tü❛ ❧ç✐ tr➯♥ X ✱ ♥❣❤➽❛ ❧➔
✣à♥❤ ❧➼ ✷✳✶✳

f (λx + (1 − λ)y)

max{f (x), f (y)},

f : X → R✳

∀x, y ∈ X,

❑❤✐ ✤â✱ ❝→❝

∀λ ∈ [0, 1].

❜✮ ❱î✐ ♠å✐ x ∈ X ✈➔ ✈î✐ ♠å✐ y ∈ Rn ❤➔♠ sè gx,y (t) = f (x + ty) ❧➔ tü❛
❧ç✐ tr➯♥ ✤♦↕♥ Tx,y = {t ∈ R | x + ty ∈ X}✳
❝✮ ❱î✐ ♠å✐ x, y ∈ X ❤➔♠ hx,y (λ) = f (λx + (1 − λ)y) ❧➔ tü❛ ❧ç✐ tr➯♥ ✤♦↕♥
[0, 1]✳
❞✮❱î✐ ♠å✐ α ∈ R t➟♣ ♠ù❝ ❞÷î✐
L(f, α) = {x ∈ X | f (x)

α}

❧➔ ❧ç✐ ✭❝â t❤➸ ré♥❣✮✳
❡✮ ❱î✐ ♠å✐ α ∈ R✱ t➟♣ ♠ù❝ ❞÷î✐ ❝❤➦t
SL(f, α) = {x ∈ X | f (x) < α}

❧➔ t➟♣ ❧ç✐ ✳
❈❤ù♥❣ ♠✐♥❤✳
a) =⇒ b) :

❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷ñ❝ r➡♥❣✱ Tx,y ❧➔ t➟♣ ❧ç✐✳ ▲➜② t1, t2 ∈

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✷✸

Tx,y

✈➔ λ ∈ [0, 1]✳ ❚❛ ❝â λt1 + (1 − λ)t2 ∈ Tx,y ✳ ❱➻ f tü❛ ❧ç✐ tr➯♥ X ✱ t❛ ❝â
gx,y (λt1 + (1 − λ)t2 ) = f (x + λt1 + (1 − λ)t2 y)
= f (λ(x + t1 y) + (1 − λ)(x + t2 y))
≤ max{f (x + t1 y), f (x + t2 y)}
= max{g(t1 ), g(t2 )}
b) =⇒ c) :

▲➜② x, y ∈ X ✈➔ λ ∈ [0, 1]✳ ❚❛ ❝â
hx,y (λ) = f (λx + (1 − λ)y)
= f (y + λ(x − y))
= gy,x−y (λ).

❱➻ gy,x−y (λ) ❧ç✐✱ t❛ ❝â hx,y (λ) ❧ç✐✳
c) =⇒ d) : ▲➜② tò② þ x, y ∈ X ✈➔ λ ∈ [0, 1]✳ ❚ø t➼♥❤ ❝❤➜t tü❛ ❧ç✐ ❝õ❛
hx,y t❛ ❝â
hx,y (λ)

max{hx,y (1), hx,y (0)},

♥❣❤➽❛ ❧➔
f (λx + (1 − λ)y)

max{f (x), f (y)}

α

✈➔ ❞♦ ✤â✱ λx + (1 − λ)y ∈ L(f, α)✱ ♥❣❤➽❛ ❧➔ L(f, α) ❧ç✐✳
d) =⇒ a) : ▲➜② tò② þ x, y ∈ X ✱ λ ∈ [0, 1] ✈➔ ✤➦t α = max{f (x), f (y)}✳
❱➻ x, y ∈ L(f, α) ✈➔ L(f, α) ❧ç✐✱ t❛ ❝â λx + (1 − λ)y ∈ L(f, α)✱ ♥❣❤➽❛ ❧➔
f (λx + (1 − λ)y) α ✈➔ ♥❤÷ ✈➟② f tü❛ ❧ç✐ tr➯♥ X ✳
d) =⇒ e) : ▲➜② tò② þ x, y ∈ SL(f, α)✱ λ ∈ [0, 1] ✈➔ ✤➦t
α0 = max{f (x), f (y)} < α.

❑❤✐ ✤â✱ x, y ∈ L(f, α0) ✈➔ λx + (1 − λ)y ∈ L(f, α0)✳ ❉♦ ✤â✱ f (λx + (1 −
λ)y) α0 < α✱ ♥❣❤➽❛ ❧➔ λx + (1 − λ)y ∈ SL(f, α)✳




e) = d) :

tũ ỵ x, y L(f, ) [0, 1] õ
x, y SL(f, + )

ợ ồ

>0

x + (1 )y SL(f, + ) ợ ồ > 0 õ f (x + (1 )y) <
+ ợ ồ > 0 f (x + (1 )y)
X Rn ởt t ỗ f : X R ởt
tr X õ f tỹ ỗ tr X


x, y X,

f (x)

f (y) (x y), f (y)

ự sỷ f tỹ ỗ f (x)

f (y)

f (y + (x y)) = f (x + (1 )y)

0.

õ
f (y) [0, 1]

ữ
lim
0

f (y + (x y)) f (y)
= (x y), f (y)

0.

ữủ sỷ tr ỵ tọ f (x) f (y)
+ (1 )y)

sỷ ự (0, 1) f (x
> f (y)
ợ
hx,y = f (x + (1 )y)

t õ
> hx,y (0)
hx,y ()

hx,y (1).

õ tứ t ừ h s r h tử t
| h() = h(0)}
{ [0, ]
s h()
=
õ õ tỷ ợ t õ tỗ t [0, ]
]
ỵ tr tr
h(0) h() > h(0) ợ ồ (,