Một số vấn đề về lý thuyết đối ngẫu liên hợp và lý thuyết đối ngẫu lagrange

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
-------------------------------

LÊ BÁ LONG NHẬT

MỘT SỐ VẤN ĐỀ VỀ LÝ THUYẾT ĐỐI NGẪU
LIÊN HỢP VÀ LÝ THUYẾT ĐỐI NGẪU
LAGRANGE
Chuyên ngành: Toán ứng dụng
Mã số
: 8 46 01 12

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
TS. Dương Thị Việt An

THÁI NGUYÊN - 2020


▼ö❝ ❧ö❝
❉❛♥❤ ♠ö❝ ❦þ ❤✐➺✉

✷

▼ð ✤➛✉

✹

▲í✐ ❝↔♠ ì♥

✼

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✽

✶✳✶
✶✳✷
✶✳✸
✶✳✹

❚➟♣ ❧ç✐ ✈➔ ❍➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❧✐➯♥ ❤ñ♣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t
❉÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳
▼ët sè ❦➳t q✉↔ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳
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✷ ▼ët sè ✈➜♥ ✤➲ ✈➲ ❧þ t❤✉②➳t ✤è✐ ♥❣➝✉
✷✳✶
✷✳✷
✷✳✸
✷✳✹
✷✳✺

P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✣è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✣è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❱➼ ❞ö ✈➔ →♣ ❞ö♥❣ sì ✤ç ✤è✐ ♥❣➝✉
⑩♣ ❞ö♥❣ t➼♥❤ t♦→♥ ❞÷î✐ ✈✐ ♣❤➙♥

❑➳t ❧✉➟♥

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✽

✶✶
✶✸
✶✹

✶✼

✶✼
✷✵
✷✹
✷✼
✸✶

✸✺
✶


ử ỵ
R
R
R+

X

x
x
M N
|x|
||x||
B X (0, 1)

t A
inf f (x)

xK

sup f (x)
xK

(ã)
epi f
dom f
x , x

trữớ số tỹ
t số tỹ s rở
t số tỹ ổ
ổ ủ ố ừ X
t rộ
ợ ồ x
tỗ t x
ừ t ủ M N
tr tt ố ừ x
ừ tỡ x
ỡ õ tr X
tr ừ t A
ừ t số tỹ {f (x) | x K}
sr ừ t số tỹ {f (x) | x K}
ừ t
tr ỗ t ừ f
ỳ ừ f

tr ừ x t x



∂f (x)
f∗
f ∗∗
l.s.c.
N (x)
val(P )

❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❤➔♠ ❧ç✐ f t↕✐ x
❤➔♠ ❧✐➯♥ ❤ñ♣ ❝õ❛ ❤➔♠ f
❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ ❤➔♠ f
♥û❛ ❧✐➯♥ tö❝ ❞÷î✐
❤å ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x
t➟♣ ❝→❝ ❣✐→ trà tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ P

✸



ỵ tt ố ởt ở q trồ ừ ỵ tt tố ữ
ữỡ ự ợ ộ t t t ỏ ồ
t ố õ ởt t ố t ố t ố õ
ố q ợ t t ừ t õ t ữủ
st tổ q t q tr t t t ữủ
t t t ố t ố tr ộ
q t ừ ú ỳ ủ tr qt
t s tứ tỹ t

t q t ồ tr ổ ổ
ữủ ự tứ ỳ t trữợ t ợ ổ t q
t t ổ t tố ữ tr ổ
õ trú ự t ữ t tố ữ t
õ t ữ t q t ồ tr ổ ổ

P t ởt tr ỳ t ỡ t ừ
t ờ r t ỗ ỵ tt tr ú
ớ ỳ t t t ừ t ỗ ỗ ữợ



♥✐➺♠ ♠ð rë♥❣ ❝❤♦ ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ❦❤✐ ❤➔♠ ❦❤æ♥❣ ❦❤↔ ✈✐✳ ✣✐➲✉ ♥➔② ❝❤♦
t❤➜② ✈❛✐ trá ❝õ❛ ❞÷î✐ ✈✐ ♣❤➙♥ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❤✐➺♥ ✤↕✐ ❝ô♥❣ ❝â t➛♠ q✉❛♥
trå♥❣ ♥❤÷ ✈❛✐ trá ❝õ❛ ✤↕♦ ❤➔♠ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❝ê ✤✐➸♥✳
❚r♦♥❣ ▲þ t❤✉②➳t tè✐ ÷✉ ♥â✐ ❝❤✉♥❣ ✈➔ ●✐↔✐ t➼❝❤ ❧ç✐ ♥â✐ r✐➯♥❣✱ ❝→❝ q✉②
t➢❝ t➼♥❤ tê♥❣ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣ ❝â ✈❛✐ trá ❤➳t sù❝
q✉❛♥ trå♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❦❤✐ t❛ ❧➔♠ ✈✐➺❝ ✈î✐ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â r➔♥❣ ❜✉ë❝✳
▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❧þ t❤✉②➳t ✤è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣ ✈➔
❧þ t❤✉②➳t ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ❝❤♦ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ❧ç✐ ❝â t❤❛♠ sè tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❚ø ✤â →♣ ❞ö♥❣ ❧÷ñ❝ ✤ç ✤è✐ ♥❣➝✉ ✤➸ ♥❣❤✐➯♥ ❝ù✉ q✉②
t➢❝ t➼♥❤ tê♥❣ ❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣ ❞÷î✐ ♥❤ú♥❣ ✤✐➲✉
❦✐➺♥ ❝❤➼♥❤ q✉② t❤➼❝❤ ❤ñ♣✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❞à❝❤ r❛ ❚✐➳♥❣ ❱✐➺t ♠ët sè ♥ë✐ ❞✉♥❣
tø ♠ö❝ ✷✳✺ ❉✉❛❧✐t② ❚❤❡♦r② tr♦♥❣ ❝✉è♥ s→❝❤ ❝❤✉②➯♥ ❦❤↔♦ ✧P❡rt✉r❜❛t✐♦♥
❆♥❛❧②s✐s ♦❢ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠s✧ ✭❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✵✮ ❝õ❛ ❝→❝
t→❝ ❣✐↔ ❏✳ ❋✳ ❇♦♥♥❛♥s ❛♥❞ ❆✳ ❙❤❛♣✐r♦ ❬✸❪✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ t→❝
❣✐↔ ❝ô♥❣ t➻♠ ❤✐➸✉✱ tê♥❣ ❤ñ♣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥ ✈➔ ❝è ❣➢♥❣ ❞✐➵♥
✤↕t ❝❤✐ t✐➳t ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❝→❝ ♠➺♥❤ ✤➲ ✈➔ ❝→❝ ✤à♥❤ ❧þ✳
▲✉➟♥ ✈➠♥ ❣ç♠ ♣❤➛♥ ♠ð ✤➛✉✱ ♣❤➛♥ ❦➳t ❧✉➟♥✱ ❞❛♥❤ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠

❦❤↔♦✱ ✈➔ ❤❛✐ ❝❤÷ì♥❣ ❝â ♥ë✐ ❞✉♥❣ ♥❤÷ s❛✉✿

❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦✐➳♥

t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ❧ç✐✱ ❤➔♠ ❧ç✐✱ ❤➔♠ ❧✐➯♥ ❤ñ♣ ❝ò♥❣ ♠ët sè ❦➳t q✉↔ ❜ê trñ
♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ð ❝❤÷ì♥❣ s❛✉✳
✺


❈❤÷ì♥❣ ✷✿ ▼ët sè ✈➜♥ ✤➲ ✈➲ ❧þ t❤✉②➳t ✤è✐ ♥❣➝✉ tr➻♥❤ ❜➔②

❤❛✐ ❝→❝❤ t✐➳♣ ❝➟♥ ✈➲ ❧þ t❤✉②➳t ✤è✐ ♥❣➝✉✿ ✣è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣ ✈➔ ✣è✐ ♥❣➝✉
▲❛❣r❛♥❣❡✳ ❱➼ ❞ö ✈➔ →♣ ❞ö♥❣ sì ✤ç ✤è✐ ♥❣➝✉ ❝ô♥❣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ ð ❝❤÷ì♥❣
♥➔②✳ ✣➦❝ ❜✐➺t✱ ð ♣❤➛♥ ❝✉è✐ ❝❤÷ì♥❣✱ ♠ët ❦➳t q✉↔ ✈➲ q✉② t➢❝ t➼♥❤ t♦→♥ ❞÷î✐
✈✐ ♣❤➙♥ ❝õ❛ tê♥❣ ❤❛✐ ❤➔♠ ❧ç✐✱ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✱ ❝❤➼♥❤ t❤÷í♥❣ t❤✉ ✤÷ñ❝
❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ sì ✤ç ✤è✐ ♥❣➝✉✳

✻


▲í✐ ❝↔♠ ì♥
▲✉➟♥ ✈➠♥ ♥➔② ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐
❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ ❚❙✳ ❉÷ì♥❣ ❚❤à ❱✐➺t ❆♥✳
❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ ❈æ ✤➣ ❤÷î♥❣ ❞➝♥ ❤✐➺✉ q✉↔ ✈➔
tr✉②➲♥ ❝❤♦ ❡♠ ♥❤ú♥❣ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣
✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳
❊♠ ❝ô♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱
tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ✤✐➲✉
❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ ð tr÷í♥❣✳
❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✻ t❤→♥❣ ✽ ♥➠♠ ✷✵✷✵

❍å❝ ✈✐➯♥

▲➯ ❇→ ▲♦♥❣ ◆❤➟t

✼


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦✐➳♥ t❤ù❝ ❝ì
❜↔♥ ✈➲ t➟♣ ❧ç✐✱ ❤➔♠ ❧ç✐✱ ❤➔♠ ❧✐➯♥ ❤ñ♣ ❝ò♥❣ ♠ët sè ❦➳t q✉↔ ❜ê trñ ♥❤➡♠
♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ s❛✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛
❝❤÷ì♥❣ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✸❪✳

✶✳✶ ❚➟♣ ❧ç✐ ✈➔ ❍➔♠ ❧ç✐
●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ t÷ì♥❣ ù♥❣ ❧➔ X ∗✱
D ⊂ X, f : D → R = R ∪ {±∞}. ❈→❝ t➟♣ ❤ñ♣ ❞÷î✐ ✤➙②✿
epif := {(x, α) ∈ D × R | f (x) ≤ α},
domf := {x ∈ D | f (x) < +∞},

❧➛♥ ❧÷ñt ✤÷ñ❝ ❣å✐ ❧➔ tr➯♥ ✤ç t❤à ✈➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ ❤➔♠ f. ❍➔♠ f ✤÷ñ❝
❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ domf = ∅ ✈➔ f (x) > −∞, ∀x ∈ D.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚➟♣ A ⊂ X ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ ♥➳✉
∀x, y ∈ A, ∀λ ∈ (0, 1) ⇒ λx + (1 − λ)y ∈ A.

◗✉② ÷î❝✿ ❚➟♣ ∅ ❧➔ t➟♣ ❧ç✐✳
✽

❱➼ ❞ö ✶✳✶✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ♠➦t ♣❤➥♥❣✱ ✤♦↕♥ t❤➥♥❣✱

✤÷í♥❣ t❤➥♥❣✱ t❛♠ ❣✐→❝✱ ❤➻♥❤ ❝➛✉ ❧➔ ❝→❝ t➟♣ ❧ç✐✳

▼➺♥❤ ✤➲ ✶✳✶✳ ✭①❡♠ ❬✶✱ tr❛♥❣ ✹❪✮ ●✐↔ sû Aα ⊂ X(α ∈ I) ❧➔ ❝→❝ t➟♣ ❧ç✐✱

✈î✐ I ❧➔ t➟♣ ❝❤➾ sè ❜➜t ❦ý✳ ❑❤✐ ✤â A =

Aα
α∈I

❝ô♥❣ ❧➔ t➟♣ ❧ç✐✳

▼➺♥❤ ✤➲ ✶✳✷✳ ✭①❡♠ ❬✶✱ tr❛♥❣ ✹❪✮ ●✐↔ sû t➟♣ Ai
λi ∈ R, 1, m✳

∈ X

❧➔ ❝→❝ t➟♣ ❧ç✐✱

❑❤✐ ✤â λ1A1 + · · · + λmAm ❧➔ t➟♣ ❧ç✐✳

▼➺♥❤ ✤➲ ✶✳✸✳ ✭①❡♠ ❬✶✱ tr❛♥❣ ✹❪✮ ●✐↔ sû Xi ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✱ t➟♣
Ai ⊂ Xi

❧ç✐ (i = 1, n)✳ ❑❤✐ ✤â✱ t➼❝❤ ✣➲ ❝→❝ A1 × A2 × ... × An ❧➔ t➟♣ ❧ç✐

tr♦♥❣ X1 × X2 × ... × Xn.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❍➔♠ f : D → R ✤÷ñ❝ ❣å✐ ❧➔ ❧ç✐ tr➯♥ D ♥➳✉ epif ❧➔ t➟♣
❧ç✐ tr♦♥❣ X × R✳

▼➺♥❤ ✤➲ ✶✳✹✳ ✭①❡♠ ❬✶✱ tr❛♥❣ ✹✵❪✮ ❈❤♦ f : X → (−∞, +∞]✳ ❑❤✐ ✤â f ❧➔
❤➔♠ ❧ç✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉

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

✭✶✳✶✮

❈❤ù♥❣ ♠✐♥❤✳ ⇒) ❱➻ f ❧➔ ❤➔♠ ❧ç✐ ♥➯♥ epif ❧➔ t➟♣ ❧ç✐✳ ❑❤✐ ✤â ✈î✐ ♠å✐
(x, r) ∈ epif ✱ (y, s) ∈ epif ✱ λ ∈ (0, 1)✱

t❛ ❝â

λ(x, r) + (1 − λ)(y, s) = (λx + (1 − λ)y, λr + (1 − λ)s) ∈ epif
⇔ f (λx + (1 − λ)y) ≤ λr + (1 − λ)s
⇔ f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), (❧➜② r = f (x), s = f (y)).

◆➳✉ x ❤♦➦❝ y ❦❤æ♥❣ t❤✉ë❝ domf t❤➻ f (x) = +∞ ❤♦➦❝ f (y) = +∞✳ ❑❤✐
✤â ✭✶✳✶✮ ✤ó♥❣✳
⇐) ◆❣÷ñ❝ ❧↕✐ ❣✐↔ sû ✭✶✳✶✮ ✤ó♥❣✳ ▲➜② (x, r) ∈ epif, (y, s) ∈ epif ✱ ✈î✐ ♠å✐
✾


(0, 1)

t ự
(x, r) + (1 )(y, s) epif.

t ợ t (x, r) epif, (y, s) epif s r f (x) r, f (y) s
õ
f (x + (1 )y) f (x) + (1 )f (y)
r + (1 )s.

ứ õ s r (x + (1 )y, r + (1 )s) epif, õ
(x, r) + (1 )(y, s) epif.

ử C ởt t ỗ ừ X ừ C ữủ


C (x) =




0

x C,



+

x C

ỗ
t t epi C = C ì [0, +) t ỗ tr X ì R.

ỵ t tự s sỷ f : X (, +]
õ

f

ỗ ợ ồ

i 0 (i = 1, m)

m

i = 1,
i=1

x1 , x2 , ..., xm X,
f (1 x1 + ... + m xm ) 1 f (x1 ) + ... + m f (xm ).






✶✳✷ ❍➔♠ ❧✐➯♥ ❤ñ♣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t
❚r♦♥❣ ♠ö❝ ♥➔② ❝❤ó♥❣ tæ✐ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❧✐➯♥ ❤ñ♣ ✈➔ ♠ët sè t➼♥❤
❝❤➜t✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ♠ö❝ ✤÷ñ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉ ❬✶❪✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ X ✳ ❍➔♠ f ∗ ①→❝ ✤à♥❤

tr➯♥ X ∗ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

f ∗ (x∗ ) = sup{ x∗ , x − f (x)}
x∈X

✭✶✳✸✮

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧✐➯♥ ❤ñ♣ ❝õ❛ ❤➔♠ f ✳
❚ø ✭✶✳✸✮ t❛ ❝â f ∗(x∗) ≥

x∗ , x − f (x)✱

✈î✐ ♠å✐ x ∈ X ✱ ❤❛②

f ∗ (x∗ ) + f (x) ≥ x∗ , x , ∀x ∈ X.

✭✶✳✹✮

❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ✤÷ñ❝ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲❋❡♥❝❤❡❧✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❍➔♠ f ∗∗ = (f ∗)∗✱ tù❝ ❧➔
f ∗∗ (x) = sup { x∗ , x − f ∗ (x∗ )}
x∗ ∈X ∗

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ f ✳

❱➼ ❞ö ✶✳✸✳ ❈❤♦ ❤➔♠ ❛❢❢✐♥❡ f (x) =

x∗ , x + α

✈î✐ x ∈ X ✳ ❑❤✐ ✤â

f ∗ (x∗ ) = sup{ x∗ , x − x∗0 , x − α}
x

= sup{ x∗ − x∗0 , x − α}
x


−α, ♥➳✉ x∗ = x∗0 ,
=


+∞, ♥➳✉ x∗ = x∗0 .

❱➼ ❞ö ✶✳✹✳ ❈❤♦ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣ f (x) = ex, x ∈ R✳ ❑❤✐ ✤â ❤➔♠ ❧✐➯♥
❤ñ♣ t❤ù ♥❤➜t ❝õ❛ f ❧➔

f ∗ (x∗ ) = sup{ x∗ , x − f (x)} = sup{x∗ x − ex }.
x

x
✶✶


◆➳✉ x∗ = 0 t❤➻ f ∗(x∗) = f ∗(0) = 0.
◆➳✉ x∗ < 0 t❤➻ f ∗(x∗) = +∞.
◆➳✉ x∗ > 0✳ ❳➨t ❤➔♠ sè ϕ(x) = x∗x − ex✳ ❚❛ ❝â ϕ (x) = x∗ − ex✱ ❣✐↔✐
♣❤÷ì♥❣ tr➻♥❤ ϕ (x) = 0 t❛ ✤÷ñ❝ x = ln x∗. ❍➔♠ sè ϕ(x) ✤↕t ❝ü❝ ✤↕✐ t↕✐
✤✐➸♠ ♥➔② ✈➻ ϕ”(x) = −ex < 0✱ ✈î✐ ♠å✐ x✳ ❱➟②




0




∗ ∗
f (x ) = +∞





x∗ ln x∗ − x∗

♥➳✉ x∗ = 0,
♥➳✉ x∗ < 0,

♥➳✉ x∗ > 0.
❚➼♥❤ t♦→♥ t÷ì♥❣ tü t❛ t❤✉ ✤÷ñ❝ ❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ f ♥❤÷ s❛✉
f ∗∗ (x) = sup{ x∗ , x − f ∗ (x∗ )} = sup{x∗ x − x∗ ln x∗ + x∗ } = ex , ∀x ∈ R.
x∗

x∗

❚✐➳♣ t❤❡♦✱ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧✐➯♥ ❤ñ♣✳ ❚❛ ♥â✐ f > g ❝â
♥❣❤➽❛ ❧➔ f (x) > g(x) ✈î✐ ♠å✐ x ∈ X ✳

▼➺♥❤ ✤➲ ✶✳✺✳ ❇➜t ✤➥♥❣ t❤ù❝ f ≥ f ∗∗ ✤ó♥❣ ✈î✐ ♠å✐ ❤➔♠ f ✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â f ∗∗(x) = supx{ x∗, x − f ∗(x∗)} ≤ f (x) ✭❞♦ ❜➜t

✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲❋❡♥❝❤❡❧✮✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❍➔♠ f ✤÷ñ❝ ❣å✐ ✤â♥❣ ♥➳✉ epi f ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ X × R✳
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐ ✭❧✳s✳❝✳✮ t↕✐ x ∈ X

♥➳✉ ♠å✐ ε > 0 tç♥ t↕✐ U ∈ N (x) s❛♦ ❝❤♦ f (x ) ≥ f (x) − ε ✈î✐ ♠å✐ x
◆➳✉ f ❧✳s✳❝✳ t↕✐ ♠å✐ ✤✐➸♠ x ∈ X t❤➻ f ✤÷ñ❝ ❣å✐ ❧➔ ❧✳s✳❝✳ tr➯♥ X ✳

∈ U.

◆❤➟♥ ①➨t ✶✳✶✳ ❍➔♠ f ❧✳s✳❝✳ tr➯♥ X ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ✤â♥❣ ✈➔ dom f ❝ô♥❣

✤â♥❣✳ ❱➼ ❞ö ✤ì♥ ❣✐↔♥ X = R, f (x) = x1 ✈î✐ x > 0 ✈➔ f (x) = +∞ ✈î✐ ♠å✐
x ❦❤æ♥❣ ➙♠✱ ❝â epi f ✤â♥❣✱ t✉② ♥❤✐➯♥ f ❦❤æ♥❣ ❧✳s✳❝✳ tr➯♥ X ✭❦❤æ♥❣ ❧✳s✳❝✳
t↕✐ x = 0✮✳
✶✷


f ởt tr X õ ủ
f

ỗ õ ợ tổổ tr ổ X

ự f tr ừ ồ
f ỗ ỡ ỳ f tr ừ tử x
x , x f (x) (x dom f ) f õ ợ tổổ t õ
tữỡ ữỡ ợ t ỷ tử ữợ

ỵ ỵ r f tr X

ổ tr
õ



õ

f = f



f

ỗ

ữợ ừ ỗ
sỷ f ởt ỗ tữớ tr ổ

ỗ ữỡ X x0 domf ởt x
ữợ rt ừ f t x0

X

ữủ ồ

f (x) f (x0 ) + x , x x0 , x X.

ủ tt ữợ rt ừ f t x0 ữủ ồ ữợ ừ
f t x0 ỵ f (x0 ) õ
f (x0 ) = {x X | f (x) f (x0 ) x , x x0 , x X}.

ữợ x0 / domf t f (x0) =
t ồ tứ ữợ t t r
(x) := f (x0 ) + x , x x0 , x X

õ ỗ t ởt s tỹ ừ epif t (x0, f (x0)).



❱➼ ❞ö ✶✳✺✳ ❳➨t ❤➔♠ ❝❤➾ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ❱➼ ❞ö ✶✳✷✳ ◆➳✉ C ❧➔

t➟♣ ❧ç✐ ✈➔ x0 ∈ C t❤➻

∂δC (x0 ) = {x∗ ∈ X ∗ | x∗ , x − x0 ≤ 0, ∀x ∈ C} = NC (x0 ),

ð ✤â NC (x0) = {x∗ ∈ X ∗ |
❝õ❛ t➟♣ ❧ç✐ C t↕✐ ✤✐➸♠ x0✳

x∗ , x − x0 ≤ 0, ∀x ∈ C}

❧➔ ♥â♥ ♣❤→♣ t✉②➳♥

❱➼ ❞ö ✶✳✻✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ϕ : X → R+ ❧➔ ❤➔♠ ❝❤✉➞♥
❝õ❛ X ✱ tù❝ ❧➔ ϕ(x) = ||x||✳ ❑❤✐ ✤â
∂ϕ(¯
x) =





{x∗ ∈ X ∗ | ||x∗ || ≤ 1} = B X ∗ (0, 1)

♥➳✉

x¯ = 0,



{x∗ ∈ X ∗ | ||x∗ || = 1, x∗ , x
¯ = ||¯
x||}

♥➳✉

x¯ = 0.

▼➺♥❤ ✤➲ ✶✳✼✳ ❉÷î✐ ✈✐ ♣❤➙♥ ∂f (x0) ❧➔ t➟♣ ❧ç✐✱ ✤â♥❣ ②➳✉∗✳
✣à♥❤ ❧þ ✶✳✸✳ ✭①❡♠ ❬✷✱ tr❛♥❣ ✶✶✾❪✮ ◆➳✉ f ❧➔ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣ ✈➔ ❧✐➯♥

tö❝ t↕✐ ♠ët ✤✐➸♠ ♥➔♦ ✤â✱ t❤➻ t↕✐ ♠å✐ ✤✐➸♠ x0 ∈ int(domf )✱ ∂f (x0) ❧➔ ❦❤→❝
ré♥❣✱ ❧ç✐✱ ✈➔ ❝♦♠♣❛❝t ②➳✉∗✳

✶✳✹ ▼ët sè ❦➳t q✉↔ ❜ê trñ
✣à♥❤ ❧þ ✶✳✹✳ ✭①❡♠ ❬✶✱ tr❛♥❣ ✶✶✶✲✶✶✷❪✮ ●✐↔ sû f ❧➔ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣

tr➯♥ X ✈➔ x ∈ dom f ✳ ❑❤✐ ✤â✿

x∗ ∈ ∂f (¯
x) ⇔ f (¯
x) + f ∗ (x∗ ) = x∗ , x¯ .

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ∈ ∂f (¯x)✳ ❑❤✐ ✤â✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❞÷î✐ ✈✐ ♣❤➙♥
t❛ ❝â

f (x) − f (¯
x) ≥ x∗ , x − x¯ , ∀ x ∈ X.

✶✹


✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈î✐
f (x) − f (¯
x) ≥ x∗ , x − x∗ , x¯ , ∀ x ∈ X
⇔ x∗ , x¯ − f (¯
x) ≥ x∗ , x − f (x), ∀ x ∈ X.

▲➜② ✧s✉♣✧ ❤❛✐ ✈➳ t❤❡♦ x ∈ X t❛ ✤÷ñ❝
x∗ , x¯ − f (¯
x) ≥ sup{ x∗ , x − f (x)}.
x∈X

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❧✐➯♥ ❤ñ♣ t❛ ❝â sup{ x∗, x
x∈X
✤â ✭✶✳✺✮ t÷ì♥❣ ✤÷ì♥❣

✭✶✳✺✮

− f (x)} = f ∗ (x∗ )✳

f ∗ (x∗ ) + f (¯

x) ≤ x∗ , x¯ .

❉♦

✭✶✳✻✮

▼➦t ❦❤→❝✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲❋❡♥❝❤❡❧ t❛ ❝â
f ∗ (x∗ ) + f (¯
x) ≥ x∗ , x¯ .

✭✶✳✼✮

❚ø ✭✶✳✻✮ ✈➔ ✭✶✳✼✮ s✉② r❛
f ∗ (x∗ ) + f (¯
x) = x∗ , x¯ .

◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû f ∗(x∗) + f (¯x)
❋❡♥❝❤❡❧✱ t❛ ❝â

= x∗ , x¯ .

❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲

f ∗ (x∗ ) + f (¯
x + td) ≥ x∗ , x¯ + td , ∀ t > 0, ∀ d ∈ X
⇔ f (¯
x + td) ≥ x∗ , x¯ − f ∗ (x∗ ) + t x∗ , d , ∀ t > 0, ∀ d ∈ X.

❈❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝❤♦ t > 0 t❛ ✤÷ñ❝
f (¯

x + td) − f (¯
x)
≥ x∗ , d , ∀ t > 0, ∀ d ∈ X.
t

❉♦ ✤â

f (¯
x + td) − f (¯
x)
≥ x∗ , d , ∀ t > 0, ∀ d ∈ X,
t>0
t

inf

✶✺


❤❛②
f (¯
x + td) ≥ x∗ , d , ∀ d ∈ X.

❉♦ ✤â x∗ ∈ ∂f (¯x)✳

▼➺♥❤ ✤➲ ✶✳✽✳ ✭①❡♠ ❬✸✱ ▼➺♥❤ ✤➲ ✷✳✶✶✽❪✮ ❈❤♦ f : X → R ❧➔ ❤➔♠ ❧ç✐✳ ❑❤✐
✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✳
✭✐✮ ❱î✐ ♠é✐ x ∈ X ♠➔ f ∗∗(x) ❤ú✉ ❤↕♥✱ ❦❤✐ ✤â

∂f ∗∗ (x) = argmax{ x∗ , x − f ∗ (x∗ ), x∗ ∈ X ∗ }.

✭✶✳✽✮

✭✐✐✮ ◆➳✉ f ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥ t↕✐ x t❤➻ f ∗∗(x) = f (x).
✭✐✐✐✮ ◆➳✉ f ∗∗(x) = f (x) ✈➔ ❣✐→ trà ♥➔② ❤ú✉ ❤↕♥✱ ❦❤✐ ✤â ∂f (x) = ∂f ∗∗(x).

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✶✳✹ ❝❤♦ tr÷í♥❣ ❤ñ♣ f ∗ ✈➔ f ∗∗ t❛ ✤÷ñ❝
x∗ ∈ ∂f ∗∗ (¯
x) ⇔ f ∗∗ (¯
x) + f ∗ (x∗ ) = x∗ , x .

✭✶✳✾✮

▼➦t ❦❤→❝✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ f ✱ t❛ ❝â
f ∗∗ (x) = sup { x∗ , x − f ∗ (x∗ )}.
x∗ ∈X ∗

❉♦ ✤â t❛ t❤✉ ✤÷ñ❝ ✭✶✳✽✮✳
✭✐✐✮ ❚❛ ❧✉æ♥ ❝â f (x) ≥ f ∗∗(x) ✈î✐ ♠å✐ x ∈ X ✳ ◆➳✉ f ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥✱
❦❤✐ ✤â
x∗ ∈ ∂f (x) ⇔ f (x) + f ∗ (x∗ ) = x∗ , x .

❉♦ ✤â f (x) ≤ f ∗∗(x). ❱➟② f ∗(x) = f ∗∗(x).
✭✐✐✐✮ ❚ø ✭✶✳✾✮ ✈➔ ✭✶✳✶✵✮ t❛ t❤✉ ✤÷ñ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

✶✻

✭✶✳✶✵✮

ữỡ

ởt số ỵ tt ố
ố ữỡ ộ t ỹ t t ố ữủ
t tữỡ ự ợ t ỹ t ố s ớ
t ố t õ t ữủ tr tố ữ
ừ t ố õ ỹ t ố ỵ tt
ố r ữỡ ú tổ tr t q
tở ố ủ ố r ũ ợ ử tr
t t ữợ ừ tờ ỗ ỷ tử ữợ
tữớ ở ừ ữỡ ữủ tờ ủ tt tứ
ử t r ừ ố s

Pt t
sỷ X, U, Y ổ tổổ ỗ ữỡ X , U , Y
ổ tổổ ố tữỡ ự t t tố ữ
min f (x),
xX



P


ð ✤â f : X → R ❧➔ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✳ ❚❛ ❣✐↔ sû r➡♥❣ ❜➔✐ t♦→♥ ✭P✮
✤÷ñ❝ ♥❤ó♥❣ ✈➔♦ ❤å ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â t❤❛♠ sè s❛✉✿
min ϕ(x, u),
x∈X

✭Pu✮

ð ✤â ϕ : X × U → R ❧➔ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✳ ❑❤✐ u = 0 ❜➔✐ t♦→♥ (P0)
trò♥❣ ✈î✐ ❜➔✐ t♦→♥ ✭P✮✱ tù❝ ❧➔ ϕ(·, 0) = f (·)✳
❍➔♠ ❣✐→ trà tè✐ ÷✉ t÷ì♥❣ ù♥❣ ✈î✐ ❜➔✐ t♦→♥ (Pu) ✤÷ñ❝ ❝❤♦ ❜ð✐
v(u) := inf ϕ(x, u) .
x∈X

❚❛ ♥â✐ r➡♥❣ ❜➔✐ t♦→♥ ✭P✮ ✭t÷ì♥❣ ù♥❣ (Pu)✮ ❧➔ t÷ì♥❣ t❤➼❝❤ ♥➳✉ ♠✐➲♥ ❤ú✉
❤✐➺✉ ❝õ❛ f (·) ✭t÷ì♥❣ ù♥❣ ϕ(·, u)✮ ❦❤→❝ ré♥❣✳ ❘ã r➔♥❣ (Pu) ❧➔ t÷ì♥❣ t❤➼❝❤
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ v(u) < +∞.
❙❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❣✐→ trà tè✐ ÷✉ v(·).

▼➺♥❤ ✤➲ ✷✳✶✳ ◆➳✉ ❤➔♠ ϕ(x, u) ❧ç✐✱ ❦❤✐ ✤â ❤➔♠ ❣✐→ trà tè✐ ÷✉ v(·) ❝ô♥❣
❧➔ ❤➔♠ ❧ç✐✳

❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x1, x2 ∈ X, u1, u2 ∈ U ❜➜t ❦➻✱ ✈î✐ ♠å✐ λ ∈ (0, 1)✱ ❞♦
t➼♥❤ ❧ç✐ ❝õ❛ ❤➔♠ ϕ(x, u) ♥➯♥ t❛ ❝â

λϕ(x1 , u1 ) + (1 − λ)ϕ(x2 , u2 ) ≥ ϕ(λx1 + (1 − λ)x2 , λu1 + (1 − λ)u2 )
≥ v(λu1 + (1 − λ)u2 ).

▲➜② inf ❤❛✐ ✈➳ t❤❡♦ x1, x2 t❛ ✤÷ñ❝
λv(u1 ) + (1 − λ)v(u2 ) ≥ v(λu1 + (1 − λ)u2 ).

❱➟② v(·) ❧➔ ❤➔♠ ❧ç✐✳

◆❤➟♥ ①➨t ✷✳✶✳ ❍➔♠ ❣✐→ trà tè✐ ÷✉ v(·) ❦❤æ♥❣ ❝❤➼♥❤ t❤÷í♥❣ ❤♦➦❝ ♥û❛ ❧✐➯♥

tö❝ ❞÷î✐ t❤➟♠ ❝❤➼ ❝↔ ❦❤✐ ϕ(x, u) ❧➔ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✱ ♥û❛ ❧✐➯♥ tö❝
✶✽

❞÷î✐✳ ❈❤➥♥❣ ❤↕♥✱ ❧➜②
ϕ(x, u) =

✈î✐ ♠å✐

x, u ∈ R✳




x,

♥➳✉ ex − u ≤ 0



+∞,

tr♦♥❣ ❝→❝ tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐,

❱➻

❧➔ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✱ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â
ϕ : R × R → R ∪ {+∞} ❧➔ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✱ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✳ ❚❛ ❝â
f (x) = x

v(u) = inf ϕ(x, u) =
x∈X




−∞,

♥➳✉ u > 0,



+∞,

♥➳✉ u ≤ 0.

❉♦ ✤â v(·) ❦❤æ♥❣ ❝❤➼♥❤ t❤÷í♥❣✱ ❦❤æ♥❣ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✳

▼➺♥❤ ✤➲ ✷✳✷✳ ❍➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ♥❤➜t ❝õ❛ v(·) ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝
v ∗ (u∗ ) = ϕ∗ (0, u∗ ),

tr♦♥❣ ✤â
ϕ∗ (x∗ , u∗ ) =

{ x∗ , x + u∗ , u − ϕ(x, u)} .

sup
(x,u)∈X×U

✭✷✳✶✮

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❧✐➯♥ ❤ñ♣✱ t❛ ❝â

v ∗ (u∗ ) = sup

u∗ , u −

=

sup

inf

ϕ(x, u)

(x,u)∈X×U

u∈U

u∗ , u − ϕ(x, u)

(x,u)∈X×U

= ϕ∗ (0, u∗ ).

❚❤➟t ✈➟②✱ ❞♦ ϕ(x, u) ≥ x∈X
inf ϕ(x, u)✱ ♥➯♥ −ϕ(x, u) ≤ − inf ϕ(x, u)✱ s✉② r❛
x∈X
sup
u∈U

u∗ , u −

inf

ϕ(x, u)

≤

(x,u)∈X×U

sup
(x,u)∈X×U

✶✾

u∗ , u − ϕ(x, u) .

✭✷✳✷✮


◆❣÷ñ❝ ❧↕✐
sup

{ u∗ , u − ϕ(x, u)}

(x,u)∈X×U

≥ sup { u∗ , u − ϕ(x, u)} , ∀u ∈ U
x∈X

= u∗ , u + sup(−ϕ(x, u)), ∀u ∈ U
x∈X

= u∗ , u − inf ϕ(x, u), ∀u ∈ U.
x∈X

❉♦ ✤â
sup
u∈U

u∗ , u −

ϕ(x, u) ≥

inf
(x,u)∈X×U

sup

u∗ , u − ϕ(x, u) .

(x,u)∈X×U

✭✷✳✸✮

❚ø ✭✷✳✷✮✱ ✭✷✳✸✮✱ ❦➳t ❤ñ♣ ✈î✐ ✭✷✳✶✮ t❛ t❤✉ ✤÷ñ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

▼➺♥❤ ✤➲ ✷✳✸✳ ❍➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ v(·) ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝
v ∗∗ (u) = sup { u∗ , u − ϕ∗ (0, u∗ )}.
u∗ ∈U ∗

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❧✐➯♥ ❤ñ♣ t❤ù ❤❛✐ ❝õ❛ v✱ t❛ ❝â

v ∗∗ (u) = sup { u∗ , u − v ∗ (u∗ )}.
u∗ ∈U ∗

❚ø ✤â✱ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✷ ✣è✐ ♥❣➝✉ ❧✐➯♥ ❤ñ♣
❇➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ (Pu) ❧➔
max
{ u∗ , u − ϕ∗ (0, u∗ )}.
∗
∗

u ∈U

✭Du✮

❚r÷í♥❣ ❤ñ♣ r✐➯♥❣ ❦❤✐ u = 0✱ ❜➔✐ t♦→♥ (D0) trð t❤➔♥❤ ❜➔✐ t♦→♥
max
{−ϕ∗ (0, u∗ )},
∗
∗

u ∈U

✷✵

✭❉✮


✤÷ñ❝ ①❡♠ ♥❤÷ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ❑þ ❤✐➺✉ t➟♣ t➜t ❝↔ ❝→❝

❣✐→ trà tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (Pu) ✭t÷ì♥❣ ù♥❣ ❝õ❛ ❜➔✐ t♦→♥ (Du)✮ ❧➔ val(Pu)
✭t÷ì♥❣ ù♥❣ val(Du)✮✳ ❑❤✐ ✤â t❛ ❝â val(Pu) = v(u) ✈➔ val(Du) = v∗∗(u)✳
❱➻ v(u) ≥ v∗∗(u)✱ ✈î✐ ♠å✐ u ✭❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲❋❡♥❝❤❡❧✮ ♥➯♥ t❛ ❧✉æ♥ ❝â
val(Pu ) ≥ val(Du ).

▼➺♥❤ ✤➲ ✷✳✹✳ ●✐↔ sû ϕ(x, u) ❧➔ ❤➔♠ ❧ç✐✱ ❝❤➼♥❤ t❤÷í♥❣✳ ◆➳✉ v∗∗(u) ❤ú✉
❤↕♥✱ ❦❤✐ ✤â t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du)✱ ❦þ ❤✐➺✉ ❧➔ S(Du) trò♥❣
✈î✐ ∂v∗∗(u).

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✽✱ t❛ ❝â
∂v ∗∗ (u) = argmax{ u∗ , u − v ∗ (u∗ ), u∗ ∈ U ∗ }.

❍ì♥ ♥ú❛✱ ✈➻ v∗(u∗) = ϕ∗(0, u∗). ❉♦ ✤â t❛ ❝â t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐
♥❣➝✉ trò♥❣ ✈î✐ ∂v∗∗(u).

✣à♥❤ ❧þ ✷✳✶✳ ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✿

✭✐✮ ❈❤♦ tr÷î❝ u ∈ U, ∂v(u) = ∅✱ ❦❤✐ ✤â val(Pu) = val(Du) ✈➔ t➟♣
♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du) trò♥❣ ✈î✐ ∂v(u)✳
✭✐✐✮ ◆➳✉ val(Pu) = val(Du) ✈➔ ❤ú✉ ❤↕♥✱ ❦❤✐ ✤â S(Du) = ∂v(u).
✭✐✐✐✮ ◆➳✉ val(Pu) = val(Du) ✈➔ x¯ ∈ X, u¯∗ ∈ U ∗ ❧➔ ♥❣❤✐➺♠ t÷ì♥❣ ù♥❣
❝õ❛ ❜➔✐ t♦→♥ (Pu) ✈➔ (Du)✳ ❑❤✐ ✤â ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✤ó♥❣✱
ϕ(¯
x, u) + ϕ∗ (0, u¯∗ ) = u¯∗ , u .

✭✷✳✹✮

◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮ t❤ä❛ ♠➣♥ ✈î✐ x¯ ✈➔ u¯∗ t÷ì♥❣ ù♥❣ ❧➔ ♥❣❤✐➺♠
tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ (Pu) ✈➔ (Du)✳ ❑❤✐ ✤â val(Pu) = val(Du).

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✲❋❡♥❝❤❡❧✱ ♥➳✉ ∂v(u) ❦❤→❝ ré♥❣
✷✶


t❤➻ t❛ ❝â
v(u) ≥ u∗ , u − v ∗ (u∗ ).

❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✱
v(u) = u∗ , u − v ∗ (u∗ ) ♥➳✉

✈➔ ❝❤➾ ♥➳✉ u∗ ∈ ∂v(u).

❱➻ v∗(u∗) = ϕ∗ (0, u∗)✱ ✤✐➲✉ ♥➔② s✉② r❛ r➡♥❣ u∗ ❧➔ ♠ët ♥❣❤✐➺♠ tè✐ ÷✉
❝õ❛ (Du) ♥➳✉ u∗ ∈ ∂v(u) ✈➔ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ✤â val (Pu) = val (Du)✱
✈➔ ♥❣÷ñ❝ ❧↕✐✳ ❉♦ ✤â ✭✐✮ ✈➔ ✭✐✐✮ ①↔② r❛✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹✱ t❛ ❝â
S(Du ) = ∂v ∗∗ (u)✳ ❱➻ v(·) ❧➔ ❦❤↔ ❞÷î✐ ✈✐ ♣❤➙♥ t↕✐ u✱ ♥➯♥ tø ▼➺♥❤ ✤➲ ✶✳✽
t❛ ❝ô♥❣ ❝â ∂v∗∗(u) = ∂v(u)✱ ✈➔ ❞♦ ✤â ✭✐✮ ①↔② r❛✳ ◆➳✉ v∗∗(u) = v(u)✱ t❤➻
♠ët ❧➛♥ ♥ú❛ ∂v∗∗(u) = ∂v(u)✱ ✈➔ ❦❤✐ ✤â ✭✐✐✮ ①↔② r❛✳
◆➳✉ val (Pu) = val (Du) ✈➔ ❣✐→ trà ♥➔② ❤ú✉ ❤↕♥✱ t❤➻ rã r➔♥❣ tø
♥❤ú♥❣ ❧➟♣ ❧✉➟♥ tr➯♥✱ x¯ ∈ X ✈➔ u¯∗ ∈ U ∗ ❧➛♥ ❧÷ñt ❧➔ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛
(Pu ) ✈➔ (Du ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮ ✤ó♥❣✳ ❍ì♥ ♥ú❛✱ t❛ ❝ô♥❣ t❤➜②
r➡♥❣✱ ♥➳✉ ✭✷✳✹✮ ✤ó♥❣ t❤➻ val (Pu) = val (Du)✱ ✈➔ ❞♦ ✤â ✭✐✐✐✮ ①↔② r❛ ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❚❛ ♥â✐ r➡♥❣ ❜➔✐ t♦→♥ (Pu) t➽♥❤ ♥➳✉ val(Pu) ❤ú✉ ❤↕♥ ✈➔

❤➔♠ ❣✐→ trà tè✐ ÷✉ v(·) ❦❤↔ ❞÷î✐ ✈✐ t↕✐ u✱ ♥❣❤➽❛ ❧➔ ∂v(u) = ∅✳
❚ø ✣à♥❤ ❧þ ✷✳✶✱ t❛ ❝â ❦➳t q✉↔ s❛✉✳

▼➺♥❤ ✤➲ ✷✳✺✳ ●✐↔ sû r➡♥❣ val(Pu) ❤ú✉ ❤↕♥✳ ◆➳✉ (Pu) t➽♥❤✳ ❑❤✐ ✤â ❦❤æ♥❣

❝â ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛ (Pu) ✈➔ (Du)✱ ✈➔ t➟♣ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐
t♦→♥ ✤è✐ ♥❣➝✉ (Du) ❦❤→❝ ré♥❣✳ ◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ ❣✐ú❛
(Pu ) ✈➔ (Du ) ❜➡♥❣ ❦❤æ♥❣✱ t❤➻ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ (Du ) ❝â ♠ët ♥❣❤✐➺♠ tè✐ ÷✉
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (Pu) t➽♥❤✳
✷✷


ỵ ỵ ố sỷ r (x, u) ỗ tữớ

tr tố ữ v(u) = val(Pu) ỳ tử t u U õ
val(Pu ) = val(Du ) S(Du ) = ỡ ỳ S(Du ) = v(
u)

ự v(ã) ỗ tử t u t ỵ t õ v(u)
rộ õ t ỵ t t ữủ ự

v(ã) tử t u ữủ ữ ởt q
õ t t ữợ tữỡ ữỡ ử v(ã)
ỗ val(Pu) ỳ t tữỡ ữỡ ợ val(Pu)
tr ởt ừ u ỡ ỳ U ổ ỳ
t v(ã) tử t u u int(dom v) X, U
ổ t t õ t q s

X, U ổ sỷ r
tữớ ỗ ỷ tử ữợ v(u) ỳ õ
v(ã) tử t u u int(dom v).
(x, u)

ử x = (x1, x2) R2 u R
(x, u) :=

x1 ,

x1 + ex2 + u


+,

0,

tr trữớ ủ .

ợ ồ u R tt ừ ỵ tọ v(u) = u ợ
ồ u R r
(0, u ) := v (u ) =



0,

u = 1,


+,

tr trữớ ủ

ổ õ ố ỳ (Pu) (Du) S(Du) = {1}

✷✳✸ ✣è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡
❚r♦♥❣ ♠ö❝ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝❤ t✐➳♣ ❝➟♥ ✤è✐ ♥❣➝✉ ❞ò♥❣ ❤➔♠
▲❛❣r❛♥❣❡✳ P❤➛♥ ❝✉è✐ ❝❤ó♥❣ tæ✐ ❝ô♥❣ ❝❤➾ r❛ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✤è✐ ♥❣➝✉ ❧✐➯♥
❤ñ♣ ✭▼ö❝ ✷✳✷✮ ✈➔ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡✳
❈❤♦ KX ⊂ X, KY ⊂ Y ❧➔ ❝→❝ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣ ❜➜t ❦➻✳ ❚❛ ①➨t ❝➦♣
❜➔✐ t♦→♥ ❣è❝ ✈➔ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ t❤æ♥❣ q✉❛ ❤➔♠ ▲ : KX × KY → R✱
✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
min sup

x∈KX y∈KY

▲(x, y),

✭P L✮

▲(x, y).

✭DL✮

max inf

y∈KY x∈KX

❚❛ ❣å✐ ▲ ❧➔ ❤➔♠ ▲❛❣r❛♥❣❡ t÷ì♥❣ ù♥❣ ✈î✐ ❝→❝ ❜➔✐ t♦→♥ ð tr➯♥✳ ❍✐➺✉ sè ❣✐ú❛
val(P L ) − val(DL ) ✭❦❤✐ val(P L ) ✈➔ val(DL ) ❦❤æ♥❣ ❝ò♥❣ ♥❤➟♥ ❣✐→ trà ✈æ
❤↕♥✮ ✤÷ñ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐ ♥❣➝✉ t÷ì♥❣ ù♥❣ ✈î✐ ❝➦♣ ❜➔✐ t♦→♥ ✤è✐
♥❣➝✉ ð tr➯♥✳ ❚❛ ♥â✐ r➡♥❣ (¯x, y¯) ∈ KX × KY ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ❝õ❛

❤➔♠ ▲(x, y) ♥➳✉ ▲(¯x, y¯) ∈ R ✈➔✿

▲(¯x, y) ≤ ▲(¯x, y¯) ≤ ▲(x, y¯),

∀(x, y) ∈ KX × KY .

✣à♥❤ ❧þ ✷✳✸✳ ✭✐✮ ❚❛ ❝â val(DL) ≤ val(P L)✳ ❍ì♥ ♥ú❛ ❦❤♦↔♥❣ ❝→❝❤ ✤è✐

♥❣➝✉ val(DL) − val(P L) ✭♥➳✉ ♥â ①→❝ ✤à♥❤✮ ❧➔ ❦❤æ♥❣ ➙♠✳
✭✐✐✮ ❍➔♠ ▲(x, y) ❝â ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ❜➔✐ t♦→♥ (DL)✈➔
(P L ) ❝â ❝ò♥❣ ❣✐→ trà tè✐ ÷✉ ✈➔ t➟♣ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ♠é✐ ❜➔✐ t♦→♥ ❧➔
❦❤→❝ ré♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ②➯♥ ♥❣ü❛ ✤÷ñ❝ ❦➼ ❤✐➺✉
S(P L ) × S(DL )✳
✷✹