Các phép toán dưới vi phân của hàm lồi

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❈⑩❈ ◗❯❨ ❚➁❈ ❚➑◆❍ ❚❖⑩◆ ❉×❰■ ❱■ P❍❹◆ ❈Õ❆ ❍⑨▼ ▲➬■

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽


❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❈⑩❈ ◗❯❨ ❚➁❈ ❚➑◆❍ ❚❖⑩◆ ❉×❰■ ❱■ P❍❹◆ ❈Õ❆ ❍⑨▼ ▲➬■
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍➶❆ ▲❯❾◆ ✿

❚❙✳ ◆●❯❨➍◆ ❱❿◆ ❚❯❨➊◆

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽



ữủ ỷ ớ ỡ tợ t ổ trữớ ồ ữ
ở t ổ ú ù tr q
tr ồ t t trữớ t t t õ
tốt
t tọ ỏ t ỡ s s tợ t

t t ú ù tr sốt q tr ồ t ự
t õ
r q tr ự ổ tr ọ ỳ t sõt
ữủ sỹ õ õ ỵ ừ t
ổ t t ồ t ữủ t ỡ
t ỡ

ở t


r


▲❮■ ❈❆▼ ✣❖❆◆
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ ◆❣✉②➵♥ ❱➠♥

❚✉②➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❦❤æ♥❣ trò♥❣ ✈î✐ ❜➜t ❦➻ ✤➲
t➔✐ ♥➔♦ ❦❤→❝✳
❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤
tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈î✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳

❍➔ ◆ë✐✱ ♥❣➔② ✵✼ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

✐✐


▼ö❝ ❧ö❝
▲í✐ ♠ð ✤➛✉

✶

✶

✷

❍➔♠ ❧ç✐
✶✳✶

❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷

❍➔♠ ❧ç✐ trì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✸

✣↕♦ ❤➔♠ t❤❡♦ ❤÷î♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✷ ❚➼♥❤ t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥

✶✻

✷✳✶

❉÷î✐✲❣r❛❞✐❡♥t ✈➔ ❞÷î✐ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✷

❈→❝ q✉② t➢❝ t➼♥❤ t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✵

✷✳✸

❉÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♠❛①

✸✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥

✹✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✶

✐


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

▲í✐ ♠ð ✤➛✉
●✐↔✐ t➼❝❤ ❧ç✐ ❧➔ ♠ët ❜ë ♠æ♥ q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥ ❤✐➺♥
✤↕✐✳ ●✐↔✐ t➼❝❤ ❧ç✐ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ❦❤➼❛ ❝↕♥❤ ❣✐↔✐ t➼❝❤ ❝õ❛ t➟♣ ❧ç✐ ✈➔
❤➔♠ ❧ç✐✳ ❉÷î✐ ✈✐ ♣❤➙♥✱ ♠ët ♠ð rë♥❣ ❝❤♦ ✤↕♦ ❤➔♠ ❦❤✐ ❤➔♠ ❦❤æ♥❣ ❦❤↔
✈✐✱ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❧ç✐✳ ❱✐➺❝ ❦❤↔♦ s→t ❝→❝ q✉② t➢❝
t➼♥❤ t♦→♥ ❝õ❛ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ❤➔♠ ❧ç✐ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣
❧þ t❤✉②➳t tè✐ ÷✉ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳
❱î✐ ♠♦♥❣ ♠✉è♥ ✤÷ñ❝ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❤➔♠ ❧ç✐ ✈➔ ♣❤➨♣ t➼♥❤ ❞÷î✐
✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐✱ tæ✐ ✤➣ ❝❤å♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐✿ ✏❈→❝ q✉② t➢❝ t➼♥❤
t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐✑✳
▼ö❝ ✤➼❝❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣✱ ❝→❝ ❦✐➳♥
t❤ù❝ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ♥❤➜t ✈➲ ❤➔♠ ❧ç✐ ✈➔ ❝→❝ q✉② t➢❝ t➼♥❤ t♦→♥ ❞÷î✐
✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐✳
❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✤÷ñ❝ tr➻♥❤ ❜➔② ❞ü❛ tr➯♥ ❝✉è♥

❝❤✉②➯♥ ❦❤↔♦ ❬✸✱ ❈❤❛♣t❡r ✷❪✳
❑❤â❛ ❧✉➟♥ ❣ç♠ ❤❛✐ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛
❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ t➟♣ ❧ç✐ ✈➔ ❤➔♠ ❧ç✐
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❝→❝ q✉② t➢❝ t➼♥❤ t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥✳ ▼ö❝ ✷✳✶
♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❞÷î✐✲❣r❛❞✐❡♥t ✈➔ ❞÷î✐ ✈✐ ♣❤➙♥✳ ▼ö❝
✷✳✷ tr➻♥❤ ❜➔② ♠ët sè q✉② t➢❝ t➼♥❤ t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥✳ ▼ö❝ ✷✳✸ tr➻♥❤ ❜➔②
✈➲ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ♠❛①✳

✶


❈❤÷ì♥❣ ✶
❍➔♠ ❧ç✐

✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
❑➼ ❤✐➺✉ R := R ∪ {±∞} ✈➔ ❣å✐ ❧➔ t➟♣ sè t❤ü❝ ♠ð rë♥❣✳
❈❤♦ f : Rn → R ❧➔ ♠ët ❤➔♠ sè✳

▼✐➲♥ ❤ú✉ ❤✐➺✉ ✈➔ t➟♣ tr➯♥ ✤ç t❤à

❝õ❛ f t÷ì♥❣ ù♥❣ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❜ð✐✿

domf = {x ∈ Rn : f (x) < +∞} ,
epif = {(x, v) ∈ Rn × R : v ≥ f (x)} .

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët t➟♣ X ∈ Rn ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ ♥➳✉ ✈î✐ ♠å✐ x1 , x2 ∈ X
✈➔ α ∈ [0, 1]✱ t❛ ❝â (1 − α)x1 + αx2 ∈ X ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❇❛♦ ❧ç✐ ❝õ❛ ♠ët t➟♣ X ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ conv X ❧➔ ❣✐❛♦

❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ❧ç✐ ❝❤ù❛ X ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ X ❧➔ ♠ët t➟♣ ❧ç✐ ✤â♥❣ tr♦♥❣ Rn ✈➔ x ∈ Rn ✳ ▼ët
✤✐➸♠ t❤✉ë❝ X ❣➛♥ x ♥❤➜t ✤÷ñ❝ ❣å✐ ❧➔

❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ X ✈➔ ❦➼ ❤✐➺✉

❧➔ ΠX (x)✳
❚❤❡♦ ❬✸✱ ❚❤❡♦r❡♠ ✷✳✶✵❪✱ t❛ ❝â ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ ♠ët ✤✐➸♠ ❧➯♥ ♠ët t➟♣


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❧ç✐ ✤â♥❣ ❧✉æ♥ tç♥ t↕✐ ✈➔ ❞✉② ♥❤➜t✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët t➟♣ K ⊂ Rn ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

♥â♥ ♥➳✉ αx ∈ K

✈î✐ ♠å✐ α > 0 ✈➔ x ∈ K ✳

❇ê ✤➲ ✶✳✶✳

●✐↔ sû X ❧➔ ♠ët t➟♣ ❧ç✐✳ ❑❤✐ ✤â t➟♣
cone(X) = {γx : x ∈ X, γ ≥ 0}

❧➔ ♠ët ♥â♥ ❧ç✐✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ K ❧➔ ♠ët ♥â♥✳ ❚➟♣ ❤ñ♣

K ◦ := {y ∈ Rn : y, x ≤ 0, ∀x ∈ K}
✤÷ñ❝ ❣å✐ ❧➔

♥â♥ ❝ü❝ ❝õ❛ K ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ X ❧➔ ♠ët t➟♣ ❧ç✐ ✤â♥❣ ✈➔ x ∈ X ✳ ❚➟♣ ❤ñ♣
NX (x) = {v ∈ Rn : ΠX (x + v) = x}
✤÷ñ❝ ❣å✐ ❧➔

♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ X t↕✐ x✳

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ r➡♥❣

NX (x) = [cone(X − x)]◦ .

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ▼ët ❤➔♠ sè f ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ ♥➳✉ epif ❧➔ ♠ët t➟♣ ❧ç✐✳

✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❱➼ ❞ö ✶✳✶✳ ▼ët ✈➼ ❞ö ✈➲ ❤➔♠ ❧ç✐✿




x ln(x) − x ♥➳✉ x > 0,




f (x) = 0
♥➳✉ x = 0,





+∞
♥➳✉ x < 0.

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ▼ët ❤➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❧ã♠ ♥➳✉ −f ❧ç✐✳
✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ▼ët ❤➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ f (x) > −∞
✈î✐ ♠å✐ x ✈➔ f (x) < +∞ ✈î✐ ➼t ♥❤➜t ♠ët x✳

❇ê ✤➲ ✶✳✷✳ ▼ët ❤➔♠ f

t❛ ❝â

❧➔ ❧ç✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐ x1✱ x2 ✈➔ 0 ≤ α ≤ 1

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

✭✶✳✶✮

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ x1 ∈/ domf ❤♦➦❝ x2 ∈/ domf ✱ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ t➛♠
t❤÷í♥❣✳ ◆➳✉ x1 ∈ domf ✈➔ x2 ∈ domf ✳ ❑❤✐ ✤â ❝→❝ ✤✐➸♠





1



2



x
x
 ∈ epif.
 ✈➔ 

f (x2 )
f (x1 )
◆➳✉ f ❧ç✐ t❤➻



1

2



αx + (1 − α)x


 ∈ epif.
1
2
αf (x ) + (1 − α)f (x )
❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t➟♣ tr➯♥ ✤ç t❤à✱ t❛ ❝â ✭✶✳✶✮✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû t❛ ❝â ✭✶✳✶✮✱ (xi , v i ) ∈ epif ✱ i = 1, 2✱ ✈➔ α ∈ [0, 1]✳

✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❑❤✐ ✤â✱ t❤❡♦ ✭✶✳✶✮✱ t❛ ❝â

f (αx1 + (1 − α)x2 ) ≤ αf (x1 ) + (1 − α)f (x2 )
≤ αv 1 + (1 − α)v 2 .
❉♦ ✤â✱ (αx1 + (1 − α)x2 , αv 1 + (1 − α)v 2 ) ∈ epif ✳ ✣✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔

epif ❧➔ ♠ët t➟♣ ❧ç✐✳
❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮ ❝â t❤➸ ✤÷ñ❝ sû ❞ö♥❣ ♥❤÷ ♠ët ✤à♥❤ ♥❣❤➽❛ ❦❤→❝
✈➲ ❝→❝ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣✳

❱➼ ❞ö ✶✳✷✳ ❍➔♠
f (x) = x
ð ✤â

·

♦

♦,

❧➔ ♠ët ❝❤✉➞♥ tr♦♥❣ Rn ✱ ❧➔ ♠ët ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣✳ ❚❤➟t

✈➟②✱ ✈î✐ ♠å✐ x, y ∈ Rn ✈➔ α ∈ [0, 1]✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✱ t❛ ❝â

αx + (1 − α)y

♦

≤ αx

♦

+ (1 − α)y

♦

=α x

♦

+ (1 − α) y

♦.

❱➼ ❞ö ✶✳✸✳ ●✐↔ sû Z ❧➔ ♠ët t➟♣ ❧ç✐ ✤â♥❣ tr♦♥❣ Rn ✳ ❑❤♦↔♥❣ ❝→❝❤ tî✐ Z ✱

f (x) = min x − z
z∈Z

ð ✤â

·

♦

♦,

❧➔ ♠ët ❝❤✉➞♥ tr♦♥❣ Rn ✱ ❧➔ ♠ët ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣✳ ❚❤➟t

✈➟②✱ ①➨t ✷ ✤✐➸♠ x ✈➔ y ✱ ✈➔ α ∈ (0, 1) ❜➜t ❦➻✳ ❉♦ Z ❧➔ t➟♣ ✤â♥❣✱ ♥➯♥ tç♥
t↕✐ ❝→❝ ✤✐➸♠ v ∈ Z ✈➔ w ∈ Z s❛♦ ❝❤♦

f (x) = x − v
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t✱ ❦❤✐

f (y) = y − w

♦,

·

✺

♦

♦.

❧➔ ♠ët ❝❤✉➞♥ ❊✉❝❧✐❞❡✱ t❤❡♦ ❬✸✱


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❚❤❡♦r❡♠ ✷✳✶✵❪✱ v ✈➔ w ❧➔ ❝→❝ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ✈➔ y ❧➯♥ Z ✳ ❱➻ Z ❧➔ ♠ët
t➟♣ ❧ç✐✱ ♥➯♥ tê ❤ñ♣ ❧ç✐ ❝õ❛ ❝→❝ ✤✐➸♠ ♥➔② αv + (1 − α)w✱ ✈î✐ α ∈ (0, 1)✱
❝ô♥❣ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ Z ✳ ❉♦ ✤â✱

f (αx + (1 − α)y) = min αx + (1 − α)y − z
z∈Z

♦

≤ αx + (1 − α)y − [αv + (1 − α)w]
= α(x − v) + (1 − α)(y − w)

♦

≤α x−v

♦

♦

+ (1 − α) y − w

♦

= αf (x) + (1 − α)f (y).
❚r♦♥❣ ✈➼ ❞ö tr➯♥✱ t➼♥❤ ❧ç✐ ❝õ❛ t➟♣ Z ❧➔ ❝➛♥ t❤✐➳t✳ ❍➔♠ ❦❤♦↔♥❣ ❝→❝❤
✤➳♥ ♠ët t➟♣ ❦❤æ♥❣ ❧ç✐ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët ❤➔♠ ❧ç✐✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ▼ët ❤➔♠ f ✤÷ñ❝ ❣å✐ ❧➔

❧ç✐ ❝❤➦t ♥➳✉ ❜➜t ✤➥♥❣ t❤ù❝

✭✶✳✶✮ ❧➔ ❝❤➦t ✈î✐ ♠å✐ x1 = x2 ✈➔ 0 < α < 1✳

❇ê ✤➲ ✶✳✸✳

◆➳✉ f ❧ç✐ t❤➻ domf ❧➔ ♠ët t➟♣ ❧ç✐✳

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ x1 ∈ domf ✈➔ x2 ∈ domf ✱ t❤➻ t❤❡♦ ❇ê ✤➲ ✶✳✷✱ t❛ ❝â
f (αx1 + (1 − α)x2 ) < +∞.
❑❤✐ ✤â αx1 + (1 − α)x2 ∈ domf ✱ ♥➯♥ domf ❧➔ ♠ët t➟♣ ❧ç✐✳

❇ê ✤➲ ✶✳✹✳

◆➳✉ fi, i ∈ I ✱ ❧➔ ♠ët ❤å ❝→❝ ❤➔♠ ❧ç✐✱ t❤➻
f (x) = sup fi (x)
i∈I

❧➔ ♠ët ❤➔♠ ❧ç✐✳
✻

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â
epif =

epifi .
i∈I

❚❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ ♠➺♥❤ ✤➲ ✈➔ t❤❡♦ ❬✸✱ ▲❡♠♠❛ ✷✳✷❪✱ t➟♣ epif ❧ç✐✳ ❉♦ ✤â✱
❤➔♠ f ❧➔ ❧ç✐✳

❱➼ ❞ö ✶✳✹✳ ❱î✐ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣✱ t❛ ①→❝ ✤à♥❤ ❣✐→ trà r✐➯♥❣ ❧î♥ ♥❤➜t
❝õ❛ ♥â ❧➔ λmax (A)✳ ❉♦ ✤â✱ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Sn ❝õ❛ ❝→❝ ♠❛ tr➟♥ ✤è✐ ①ù♥❣
❝â ❦➼❝❤ t❤÷î❝ n × n t❛ ①➨t ❤➔♠

f (A) = λmax (A),
❱➻ λmax (A) = max y, Ay , ✈➔ ♠é✐ ❤➔♠ fy (A) = y, Ay t✉②➳♥ t➼♥❤✱ ♥➯♥
y =1

❤➔♠ λmax (·) ❧➔ ❤➔♠ ❧ç✐✳

◆➳✉ f ❧➔ ♠ët ❤➔♠ ❧ç✐✱ t❤➻ ✈î✐ ♠å✐ x1, x2, . . . , xn ✈➔ α1
0, α2 ≥ 0, . . . , αm ≥ 0 s❛♦ ❝❤♦ α1 + α2 + . . . + αm = 1✱ t❛ ❝â

❇ê ✤➲ ✶✳✺✳

≥

f (α1 x1 + α2 x2 + . . . + αm xm ) ≤ α1 f (x1 ) + α2 f (x2 ) + . . . + αm f (xm ).

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ❝→❝ ✤✐➸♠


i



x

 , i = 1, 2, . . . , m,
i
f (x )
t❤✉ë❝ epif. ❚❤❡♦ t➼♥❤ ❧ç✐ ❝õ❛ t➟♣ epif ✱ tê ❤ñ♣ ❧ç✐ ❝õ❛ ❝→❝ ✤✐➸♠ ♥➔②

α1 f (x1 ) + α2 f (x2 ) + . . . + αm f (xm ), ✈î✐ α1 + α2 + . . . + αm = 1✱ ❝ô♥❣
t❤✉ë❝ epif ✳ ❉♦ ✤â✱

f (α1 x1 + α2 x2 + . . . + αm xm ) ≤ α1 f (x1 ) + α2 f (x2 ) + . . . + αm f (xm ).

✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

◆➳✉ ❝→❝ ❤➔♠ fi, i = 1, 2, . . . , m, ❧➔ ❧ç✐✱ t❤➻ ✈î✐ ♠å✐ c1 ≥
0, c2 ≥ 0, . . . , cm ≥ 0 ❤➔♠ f (x) = c1 f1 (x) + c2 f2 (x) + . . . + cm fm (x) ❧ç✐✳

❇ê ✤➲ ✶✳✻✳

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ✭✶✳✶✮ ✤ó♥❣ ✈î✐ ♠é✐ fi✱ t❛ ❝â t❤➸ ♥❤➙♥ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝
❝õ❛ ❝❤ó♥❣ ✈î✐ ci ✈➔ ❝ë♥❣ ❧↕✐ t❛ ✤÷ñ❝ ❦➳t q✉↔ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳
▼ët ❤➔♠ f : Rn → R ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ ♥➳✉ ✈î✐ ♠é✐ ❝❤✉é✐
❤ë✐ tö ❝õ❛ ❝→❝ ✤✐➸♠ xk t❤➻ t❛ ❝â

f ( lim xk ) ≤ lim inf f (xk ).
k→∞

k→∞

▼ët ❤➔♠ f : Rn → R ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ t➟♣
❧➔ ♠ët t➟♣ ✤â♥❣✳

❇ê ✤➲ ✶✳✼✳
epif

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ♠ët ❞➣② ❝→❝ ✤✐➸♠ (xk , αk ) t❤✉ë❝ epif ✱ ✈➔ ❣✐↔ sû xk →
x ✈➔ αk → α, ❦❤✐ k → ∞. ◆➳✉ f ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ t❤➻
f (x) ≤ lim inf f (xk ) ≤ lim αk = α,
k→∞

k→∞

s✉② r❛ (x, α) ∈ epif ✳
●✐↔ sû t➟♣ epif ✤â♥❣✱ ♥❤÷♥❣ f ❦❤æ♥❣ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✳ ❑❤✐ ✤â tç♥
t↕✐ ♠ët ❞➣② xk ⊂ Rn ❤ë✐ tö ✤➳♥ ♠ët sè ✤✐➸♠ x ∈ Rn s❛♦ ❝❤♦

f (x) > lim f (xk ),

k→∞

ð ✤â ❣✐î✐ ❤↕♥ ❜➯♥ ♣❤↔✐ ❝â t❤➸ ❧➔ −∞✳ ❑❤✐ ✤â✱ ∃ε > 0 s❛♦ ❝❤♦ f (xk ) <

f (x) − ε ✈î✐ ♠å✐ k ✤õ ❧î♥✳ ❉♦ ✤â
(xk , f (x) − ε) ∈ epif

✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

✈î✐ ♠å✐ ❦ ✤õ ❧î♥✳ ❱➻ t➟♣ epif ✤â♥❣ ♥➯♥ ✤✐➸♠ ❣✐î✐ ❤↕♥ (x, f (x) − ε) ❧➔
♠ët ♣❤➛♥ tû ❝õ❛ t➟♣ epif. ❚ù❝ ❧➔ f (x) − ε ≥ f (x)✱ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â f
♣❤↔✐ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✳

❇ê ✤➲ ✶✳✽✳

◆➳✉ f : Rn → R ❧➔ ❤➔♠ ❧ç✐✱ t❤➻ ✈î✐ ♠é✐ β ∈ R t➟♣
✭✶✳✷✮

Mβ = {x : f (x) ≤ β}

❧➔ t➟♣ ❧ç✐✳ ❍ì♥ ♥ú❛✱ ♥➳✉ f ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ t❤➻ t➟♣ Mβ ❧➔ t➟♣ ✤â♥❣ ✈î✐
♠å✐ β ✳
❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ x ∈ Mβ ✈➔ y ∈ Mβ ✱ t❤➻ t❤❡♦ ❇ê ✤➲ ✶✳✷✱
f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y) ≤ β,
✈➻ ✈➟② αx + (1 − α)y ∈ Mβ ✳

◆➳✉ f ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ t➟♣ epif ✤â♥❣ ✭t❤❡♦ ❇ê ✤➲ ✶✳✼✮✳ ❳➨t t➟♣
tr♦♥❣ Rn × R✿

Mβ × {β} = epif ∩ {(x, β) : x ∈ Rn }.
❚➟♣ tr➯♥ ✤â♥❣ ✈➻ epif ✤â♥❣✳ ❉♦ ✤â t➟♣ Mβ ✤â♥❣✳
❚➟♣ Mβ ð tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔

t➟♣ ♠ù❝ ❞÷î✐ ❝õ❛ f ✳ ▼ët ❤➔♠ ❝â ❝→❝ t➟♣

♠ù❝ ❞÷î✐ ❧ç✐ t❤➻ ❝❤÷❛ ❝❤➢❝ ❧ç✐✱ ❝❤➥♥❣ ❤↕♥ f (x) =

|x|, x ∈ R.

❈❤♦ X ⊂ Rn ❧➔ ♠ët t➟♣ ❧ç✐ ✈➔ f : Rn → R ❧➔ ♠ët ❤➔♠ ❧ç✐✳
❑❤✐ ✤â t➟♣ X ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉
❇ê ✤➲ ✶✳✾✳

min f (x)
x∈X

✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❧➔ t➟♣ ❧ç✐✳
❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ ❜➔✐ t♦→♥ tè✐ ÷✉ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✱ t❤➻ t➟♣ X ré♥❣ ♥➯♥
t➟♣ X ❧ç✐✳ ❈❤♦ X = ∅ ✈➔ x ∈ X, β = f (x)✳ ❑❤✐ ✤â

X = X ∩ Mβ
✈î✐ Mβ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✷✮✳ ❚❤❡♦ ❬✸✱ ▲❡♠♠❛ ✷✳✷❪✱ t➟♣ X ❧➔ t➟♣
❧ç✐✳

✶✳✷ ❍➔♠ ❧ç✐ trì♥
❑➼ ❤✐➺✉ ∇f (x) ❝❤♦ ❣r❛❞✐❡♥t ❝õ❛ ❤➔♠ f t↕✐ x✱





∂f (x)
 ∂x1 
 ∂f (x) 
 ∂x 
 2 

∇f (x) =  ✳  .
 ✳✳ 


∂f (x)
∂xn

ð ✤➙② x1 , x2 , . . . , xn ❜✐➸✉ t❤à tå❛ ✤ë ❝õ❛ ✈❡❝t♦r x✳

●✐↔ sû r➡♥❣ ❤➔♠ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â
❧ç✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈î✐ ♠å✐ x ✈➔ y

✣à♥❤ ❧þ ✶✳✶✳
(i) f

f (y) ≥ f (x) + ∇f (x), y − x ;
(i) f

✭✶✳✸✮

❧ç✐ ❝❤➦t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈î✐ ♠å✐ x = y
f (y) > f (x) + ∇f (x), y − x .

✶✵

✭✶✳✹✮


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮●✐↔ sû f ❧ç✐✱ ✈➔ tç♥ t↕✐ x✱ y ✈î✐ ε > 0 s❛♦ ❝❤♦
f (y) ≤ f (x) + ∇f (x), y − x − ε.
❚❛ ①➨t z = αy + (1 − α)x ✈î✐ 0 < α < 1✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✱ t❛ ❝â

f (z) ≤ αf (y) + (1 − α)f (x) ≤ f (x) + α ∇f (x), y − x − αε.
❤❛②

f (z) − f (x) ≤ α ∇f (x), y − x − αε.
❈❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝❤♦ α t❛ ✤÷ñ❝

f (z) − f (x)
≤ ∇f (x), y − x − ε.
α

✭✶✳✺✮

❈â z = αy + (1 − α)x ♥➯♥ z = x + αd ✈î✐ d = y − x✳ ❈❤♦ α ↓ 0✱ ❦❤✐ ✤â

lim
α↓0

f (z) − f (x)
f (x + αd) − f (x)
= lim
= f (x; d) = ∇f (x), d .
α↓0
α
α

✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ✭✶✳✺✮✳ ❱➟② f ❧ç✐ t❤➻ ✈î✐ ♠å✐ x ✈➔ y

f (y) ≥ f (x) + ∇f (x), y − x .
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû ✈î✐ ♠å✐ x ✈➔ y ❝â f (y) ≥ f (x) + ∇f (x), y − x .
❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ f ❧➔ ❤➔♠ ❧ç✐✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû y, z ❧➔ ❝→❝ ✤✐➸♠ tò② þ✱

y = z ✱ ✈➔ x = αy + (1 − α)z ✈î✐ α ∈ (0, 1)✳ ❑❤✐ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â
f (y) ≥ f (x) + ∇f (x), y − x ,
f (z) ≥ f (x) + ∇f (x), z − x .

✶✶

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

◆❤➙♥ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❧➛♥ ❧÷ñt ✈î✐ α ✈➔ 1 − α✱ ✈➔ ❝ë♥❣ ❧↕✐ t❛ ✤÷ñ❝

αf (y) + (1 − α)f (z) ≥ f (x),
❚❤❛② x = αy + (1 − α)z t❛ ✤÷ñ❝

αf (y) + (1 − α)f (z) ≥ f (αy + (1 − α)z),
❙✉② r❛ f ❧➔ ❤➔♠ ❧ç✐✳
✭✐✐✮ ◆➳✉ f ❧ç✐ ❝❤➦t✱ t❤➻ f ❧ç✐ ✈➔ ✭✶✳✺✮ ✤ó♥❣✳ ❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❜➜t
✤➥♥❣ t❤ù❝ ✭✶✳✺✮ ❧➔ ❝❤➦t✱ ♥➳✉ y = x ✈➔ α ∈ (0, 1)✳ ●✐↔ sû f (y) = f (x) +

∇f (x), y − x ✳ ❈❤♦ z = 21 x + 21 y ✳ ❱➻ f ❧ç✐ ❝❤➦t ♥➯♥ f (αx + (1 − α)y) <
αf (x) + (1 − α)f (y)✳ ❑❤✐ ✤â✱ t❛ ❝â
1
1
1
1
1
f (z) = f ( x + y) < f (x) + f (y) = f (x) + ∇f (x), y − x . ✭✶✳✻✮
2
2
2
2
2
❈❤♦ v = βx + (1 − β)z ✈î✐ 0 < β < 1✳ ❚÷ì♥❣ tü t❛ ✤÷ñ❝

1
f (v) < βf (x) + (1 − β)f (z) < f (x) + (1 − β) ∇f (x), y − x .
2
❱➻ v − x = (1 − β)(z − x) = 21 (1 − β)(y − x)✱ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð
t❤➔♥❤

f (v) < f (x) + ∇f (x), v − x ,
♠➙✉ t❤✉➝♥ ✈î✐ ✭✶✳✺✮✱ ♥➯♥ ❣✐↔ sû s❛✐✳ ❱➟② f (y) > f (x) + ∇f (x), y − x
✈î✐ ♠å✐ y = x✳
◆❣÷ñ❝ ❧↕✐✱ t❛ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ✭✐✮✳
◆➳✉ f : Rn → R ❧ç✐ ✈➔ ❦❤↔ ✈✐ t↕✐ x t❤➻ f (y) ≥ f (x) + ∇f (x), y − x

✶✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

✈î✐ ♠å✐ y ∈ Rn ✳ ◆➳✉ f ❧ç✐ ❝❤➦t✱ t❤➻ f (y) > f (x) + ∇f (x), y − x ✈î✐ ♠å✐

y ∈ Rn ✳

❱➼ ❞ö ✶✳✺✳ ❳➨t ❤➔♠ f : Rn → R ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣✱
f (x) = x, Ax ,
ð ✤â A ❧➔ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳ ❍➔♠ f ❧ç✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A ❧➔ ♠❛ tr➟♥
♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✱ ✈➔ f ❧ç✐ ❝❤➦t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A ❧➔ ♠❛ tr➟♥ ①→❝ ✤à♥❤
❞÷ì♥❣✳ ❚❤➟t ✈➟②✱

∇f (x) = 2Ax,
✈➔ ✈î✐ ♠å✐ x ✈➔ y t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤

f (y) − f (x) − ∇f (x), y − x = y, Ay − x, Ax − 2 Ax, y − x
= y, Ay + x, Ax − 2 Ax, y
= y − x, A(y − x) .
❈→❝ ❜✐➸✉ t❤ù❝ ð ♣❤➼❛ ❜➯♥ ♣❤↔✐ ❦❤æ♥❣ ➙♠ ✈î✐ ♠å✐ x, y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A
♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ❇✐➸✉ t❤ù❝ ♥➔② ❧➔ ❞÷ì♥❣ ✈î✐ ♠å✐ y = x ♥➳✉ ✈➔ ❝❤➾
♥➳✉ A ①→❝ ✤à♥❤ ❞÷ì♥❣✳

✶✳✸ ✣↕♦ ❤➔♠ t❤❡♦ ❤÷î♥❣
❚r♦♥❣ r➜t ♥❤✐➲✉ ❝→❝ ù♥❣ ❞ö♥❣✱ ❝❤ó♥❣ t❛ t❤÷í♥❣ ❣➦♣ ❝→❝ ❤➔♠ ❦❤æ♥❣
trì♥✳ ❈❤➥♥❣ ❤↕♥✱ ❝❤✉➞♥ ❊✉❝❧✐❞❡ ❦❤æ♥❣ ❦❤↔ ✈✐ t↕✐ ✵
1/2

n

x2j

x =
j=1

✶✸

.


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

❚r♦♥❣ t❤ü❝ t➳✱ ❦❤æ♥❣ ❝❤✉➞♥ ♥➔♦ ❧➔ ❦❤↔ ✈✐ t↕✐ ✵✳ ▼ët sè ❝❤✉➞♥✱ ❝❤➥♥❣
❤↕♥

n

x

1

|xj | ❤♦➦❝ x

=

∞

= max |xj |

j=1

1≤j≤n

❦❤æ♥❣ ❦❤↔ ✈✐ t↕✐ r➜t ♥❤✐➲✉ ✤✐➸♠✳ ❈→❝ ❤➔♠ ❦❤æ♥❣ trì♥ r➜t ♣❤ê ❜✐➳♥ tr♦♥❣
❝→❝ ♠æ ❤➻♥❤ tè✐ ÷✉✳
❑❤→✐ ♥✐➺♠ ❣r❛❞✐❡♥t ❝õ❛ ♠ët ❤➔♠ trì♥ ❝â t❤➸ ✤÷ñ❝ tê♥❣ q✉→t ❝❤♦
tr÷í♥❣ ❤ñ♣ ❝→❝ ❤➔♠ ❦❤æ♥❣ trì♥✱ ♥â✐ r✐➯♥❣ ❝❤♦ ❝→❝ ❤➔♠ ❧ç✐ ❦❤æ♥❣ trì♥✳
✣➸ ❤✐➸✉ ❝→❝❤ ①➙② ❞ü♥❣ ♥➔②✱ tr÷î❝ ❤➳t ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t
q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ❤➔♠ ❧ç✐✳

❈❤♦ f : Rn → R ❧➔ ♠ët ❤➔♠ ❧ç✐✳ ❱î✐ ♠é✐ x ∈ int domf

tç♥ t↕✐ δ > 0 ✈➔ L s❛♦ ❝❤♦
❇ê ✤➲ ✶✳✶✵✳

|f (y) − f (x)| ≤ L y − x

❦❤✐

y − x < δ.

❈❤♦ f : Rn → R ❧➔ ♠ët ❤➔♠ ❧ç✐ ✈➔ ❝❤♦ x ∈ domf ✳ ❑❤✐ ✤â ✈î✐ ♠é✐

d ∈ Rn ✤↕✐ ❧÷ñ♥❣
f (x; d) = lim
τ ↓0

f (x + τ d) − f (x)
,
τ

✭✶✳✼✮

✤↕♦ ❤➔♠ t❤❡♦ ❤÷î♥❣ d ❝õ❛ f t↕✐ x✳
❇ê ✤➲ ✶✳✶✶✳ ❱î✐ ♠é✐ x ∈ domf ✈➔ ♠é✐ d ∈ Rn ❣✐î✐ ❤↕♥ tr♦♥❣ ✭✶✳✼✮ tç♥
t↕✐ ✭❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥✮✳ ◆➳✉ x ∈ int domf ✱ ❦❤✐ ✤â f (x; d) ❧➔ ❤ú✉ ❤↕♥
✈î✐ ♠å✐ d✳
✤÷ñ❝ ❣å✐ ❧➔

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t❤÷ì♥❣
Q(τ ) =

f (x + τ d − f (x))
.
τ

✶✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

◆➳✉ f (x + τ d) = +∞ ✈î✐ ♠å✐ τ > 0✱ t❤➻ Q(τ ) = +∞ ✈î✐ ♠å✐ τ > 0 ✈➔

f (x; d) = +∞✳ ◆➳✉ f (x + τ d) < +∞ ✈î✐ ♠ët sè τ0 > 0✱ t❤➻ t❤❡♦ t➼♥❤
❧ç✐ ❝õ❛ f s✉② r❛ f (x + τ d) < +∞ ✈î✐ ♠å✐ 0 < τ < τ0 ✈➔ Q(τ ) ✤÷ñ❝ ①→❝
✤à♥❤ ✈î✐ ❝→❝ τ ♥➔②✳ ❈❤♦ 0 < τ1 < τ2 < τ0 ✳ ❚❛ ❝â

x + τ1 d = (1 −

τ1
τ1
)x + (x + τ2 d).
τ2
τ2

❚❤❡♦ t➼♥❤ ❧ç✐ ❝õ❛ f t❛ ✤÷ñ❝

f (x + τ1 d) ≤ (1 −

τ1

τ1
)f (x) + f (x + τ2 d),
τ2
τ2

❝â t❤➸ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉

f (x + τ1 d) − f (x) ≤

τ1
[f (x + τ2 d) − f (x)].
τ2

❈❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝❤♦ τ1 t❛ t❤➜② ❝→❝ t❤÷ì♥❣ ❧➔ ✤ì♥ ✤✐➺✉✿

Q(τ1 ) ≤ Q(τ2 ) ✈î✐ ♠å✐ 0 ≤ τ1 ≤ τ2 .

✭✶✳✽✮

❉♦ ✤â ❣✐î✐ ❤↕♥ tr♦♥❣ ✭✶✳✼✮ tç♥ t↕✐ ✭❤ú✉ ❤↕♥ ❤♦➦❝ ❜➡♥❣ −∞✮✳ ❚❤❡♦ t➼♥❤
✤ì♥ ✤✐➺✉ ❝õ❛ Q(·) s✉② r❛

f (x; d) ≤ Q(τ ) ✈î✐ ♠å✐ τ > 0
◆➳✉ x ∈ int domf ✱ t❤❡♦ ❇ê ✤➲ ✶✳✶✵ s✉② r❛ ✈î✐ ♠å✐ τ ✤õ ♥❤ä

|Q(τ )| =

|f (x + τ d) − f (x)|
≤L d ,
τ

✈➔ ❞♦ ✤â ❣✐î✐ ❤↕♥ ❝õ❛ Q(τ ) ✈î✐ τ ↓ 0 ♣❤↔✐ ❤ú✉ ❤↕♥✳

✶✺

✭✶✳✾✮


ữỡ
t ữợ

ữợrt ữợ
f : Rn R ởt ỗ tữớ
x domf ởt tr g Rn tọ
f (y) f (x) + g, y x ợ ồ y Rn
ữủ ồ ởt



ữợrt srt ừ f t x

ừ tt ữợrt ừ f t x ữủ ồ ữợ

ừ

f t x ữủ f (x)
ữợrt õ ởt ỵ ồ ró r sỷ

g f (x) t tự õ tr ỗ t ừ f ổ
tr ỗ t ừ l(y) = f (x) + g, y x

ợ ộ (y, v) epif t õ

v f (y) f (x) + g, y x ,




õ tốt ồ

r



g, y x + (1)(v f (x)) 0.
õ (g, 1) ởt tỷ ừ õ t Nepif (x, f (x))
r ởt số ữủ ụ ú ởt tr

(u, ) Nepif (x, f (x)) = 0 t g = u/ ởt ữợrt ừ
f t x t ỗ




xln(x) x x > 0,



f (x) = 0
x = 0,

+
x < 0.
(0, 0) tt t ừ t tr ỗ t õ (u, 0)
tr õ u < 0 f ổ õ ữợrt t x = 0

sỷ f : Rn R ởt ỗ tữớ x
domf ởt tr g ởt ữợrt ừ f t x
ờ

f (x; d) g, d

ợ ồ

d Rn .



ự sỷ ú õ ợ ộ y tứ t ữủ
f (y) f (x) + f (x; y x) f (x) + g, y x ,
s r g ởt ữợrt
ữủ sỷ g f (x) õ ợ ộ d > 0 t
tự ữợrt tọ

f (x + d f (x))
g, d

= g, d .






❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

◗✉❛ ❣✐î✐ ❤↕♥ ✈î✐ τ ↓ 0 t❛ ✤÷ñ❝ ✭✷✳✷✮✳

❈❤♦ f : Rn → R ❧➔ ♠ët ❤➔♠ ❧ç✐✳ ●✐↔ sû x ∈ int domf ✳
❑❤✐ ✤â✱ ∂f (x) ❧➔ ♠ët t➟♣ ❧ç✐✱ ✤â♥❣✱ ❜à ❝❤➦♥ ✈➔ ❦❤→❝ ré♥❣✳ ❍ì♥ ♥ú❛✱ ✤è✐
✈î✐ ♠é✐ ❤÷î♥❣ d ∈ Rn t❛ ❝â✿
✣à♥❤ ❧þ ✷✳✶✳

f (x; d) = max g, d .
g∈∂f (x)

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈î✐ ♠é✐ d tç♥
t↕✐ g ∈ ∂f (x) s❛♦ ❝❤♦

f (x; d) = g, d .

✭✷✳✸✮

❳➨t ❤❛✐ t➟♣ tr♦♥❣ Rn+1 ✿

E = {(y, v) : v > f (y)},
✈➔

L = {(x + τ d, f (x) + τ f (x; d)) : τ ∈ R}.
❱➻

f (x + τ d) ≤ f (x) + τ f (x; d) ∀τ ∈ R,
♥➯♥ ❝→❝ t➟♣ ♥➔② ❧ç✐ ✈➔ ❦❤æ♥❣
  ❝â ✤✐➸♠ ❝❤✉♥❣✳ ❚❤❡♦ ✤à♥❤ ❧þ t→❝❤ ❬✸✱
u
❚❤❡♦r❡♠ ✷✳✶✺❪✱ tç♥ t↕✐ z =   ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ (y, v) ∈ E
γ
✈➔ ♠é✐ τ ∈ R t❛ ❝â

u, y + γv ≥ u, x + τ d + γ[f (x) + τ f (x; d)].
◆➳✉ γ < 0✱ ❝❤♦ v → ∞ t❤➻ ❞➝♥ ✤➳♥ ♠➙✉ t❤✉➝♥✳ ❱➻ ✈➟② γ ≥ 0.

✶✽

✭✷✳✹✮


õ tốt ồ

r

sỷ = 0 x ởt tr ừ ỳ ú t
õ t ồ y tứ ừ ọ B ừ x s tỗ t v > f (y) t

= 0 tứ t õ u, y u, x ợ ồ y B õ t

r u = 0 t ợ z = 0 õ > 0
ừ t g = u/ v f (y) t ữủ

f (y) g, y f (x) + f (x; d) g, x + d
ợ ồ x int domf R t tỷ t
ữủ

[f (x; d) g, d ] f (y) f (x) g, y x ,
ợ ồ R t tự r số = 0
ú õ g ởt ữợrt
t = 0 t s r

f (y) f (x) + g, y x ợ ồ x int domf.



t ỗ ợ ộ y domf t õ

f (y) f (x) 2[f ((x + y)/2) f (x)],
(x + y)/2 int domf ử f ((x + y)/2) t t
ú ợ ồ y domf ợ ồ y Rn õ g ởt ữợrt
ừ f t x t ữợ ổ rộ
sỷ g 1 f (x), g 2 f (x) õ ợ ồ y

f (y) f (x) + g 1 , y x ,




❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉

f (y) ≥ f (x) + g 2 , y − x .
◆❤➙♥ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈î✐ α ✈➔ 1 − α✱ ð ✤â α ∈ (0, 1)✱ t❛ ✤÷ñ❝

αg 1 + (1 − αg 2 ) ∈ ∂f (x)✳ ❉♦ ✤â ❞÷î✐ ✈✐ ♣❤➙♥ ❧➔ t➟♣ ❧ç✐✳
◆➳✉ g k ∈ ∂f (x) ✈➔ g k → g ✱ ❦❤✐ ✤â ❣✐î✐ ❤↕♥ q✉❛ ❜➜t ✤➥♥❣ t❤ù❝

f (y) ≥ f (x) + g k , y − x ,
t❛ ❦➳t ❧✉➟♥ r➡♥❣ g ∈ ∂f (x)✳ ❉♦ ✤â ❞÷î✐ ✈✐ ♣❤➙♥ ❧➔ t➟♣ ✤â♥❣✳
❈❤♦ g ∈ ∂f (x)✳ ❚❤❡♦ ❬✸✱ ▲❡♠♠❛ ✷✳✸✻❪ ❝❤♦ x + τ d ✤õ ❣➛♥ x t❛ ❝â

f (x + τ d) − f (x) ≤ τ L d .
❉♦ ✤â

f (x; d) ≤ L d , ✈î✐ ♠å✐ d.
❚❤❡♦ ❇ê ✤➲ ✷✳✶ t❛ ✤÷ñ❝

g, d ≤ f (x; d) ≤ L d , ✈î✐ ♠å✐ d,
tù❝ ❧➔ g ≤ L✳
❚ø ❝❤ù♥❣ ♠✐♥❤ t❛ t❤➜② ❞÷î✐ ✈✐ ♣❤➙♥ ∂f (x) ❧ç✐ ✈➔ ✤â♥❣ ✈î✐ ♠å✐ x t↕✐
✤â f (·) ❝â ➼t ♥❤➜t ♠ët ❞÷î✐✲❣r❛❞✐❡♥t ✭❧➔ ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥✮✳

❇ê ✤➲ ✷✳✷✳

◆➳✉ ♠ët ❤➔♠ ❧ç✐ f : Rn → R ❧➔ ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥ t↕✐ x✱ ❦❤✐

✤â ✈î✐ ♠é✐ d

f (x; d) = sup g, d .
g∈∂f (x)

❍ì♥ ♥ú❛✱ ♥➳✉ f (x; d) < ∞ t❤➻ ❝➟♥ tr➯♥ ✤ó♥❣ ð tr➯♥ ✤↕t ✤÷ñ❝✳
✷✵