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

VŨ VIỆT BÌNH

ĐIỀU KIỆN CẦN VÀ ĐỦ CHO TỰA NGHIỆM
HỮU HIỆU YẾU CỦA BÀI TỐN TỐI ƯU
ĐA MỤC TIÊU KHƠNG TRƠN

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2020


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

VŨ VIỆT BÌNH

ĐIỀU KIỆN CẦN VÀ ĐỦ CHO TỰA NGHIỆM
HỮU HIỆU YẾU CỦA BÀI TỐN TỐI ƯU
ĐA MỤC TIÊU KHƠNG TRƠN
Chun ngành: Tốn ứng dụng
Mã số: 8 46 01 12

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
GS.TS. Đỗ Văn Lưu

THÁI NGUYÊN - 2020


ử ử
ỵ

ởt số tự





ữợ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷✳ ◆â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉

✽

✷✳✶✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✷✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✸✳ ✣✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸ ✣è✐ ♥❣➝✉

✷✺

✸✳✶✳ ✣è✐ ♥❣➝✉ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✷✳ ✣è✐ ♥❣➝✉ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✾
✸✶


✐

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝ỉ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✤ë❝ ❧➟♣ ❝õ❛ r✐➯♥❣
❜↔♥ t❤➙♥ tæ✐ ữợ sỹ ữợ ồ ừ ộ ❱➠♥ ▲÷✉✳ ❈→❝ ♥ë✐
❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉✱ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ tr tỹ ữ tứ ổ
ố ữợ t ý tự trữợ
r tr ✈➠♥ tỉ✐ ❝â sû ❞ư♥❣ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝
✤➲✉ ❝â tr➼❝❤ ❞➝♥ ✈➔ ❝❤ó t❤➼❝❤ ỗ ố t t ý sỹ ♥➔♦
tæ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵
❚→❝ ❣✐↔

❱ô ❱✐➺t ❇➻♥❤


✐✐

▲í✐ ❝↔♠ ì♥
❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tỉ✐ ✤➣ ♥❤➟♥

✤÷đ❝ sü ❣✐ó♣ ù t t ừ ữớ ữợ ộ ❱➠♥ ▲÷✉✳
❚ỉ✐ ❝ơ♥❣ ♠✉è♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❑❤♦❛ ❚♦→♥✲❚✐♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱
✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤
tèt ❧✉➟♥ ✈➠♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ ❜↔♥ t❤➙♥ t→❝ ❣✐↔ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥
✈➠♥ ❝â t❤➸ ❝â ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♠✉è♥ ♥❤➟♥ ✤÷đ❝ ỵ ỗ
õ õ ỹ ừ t❤➛② ❝ỉ✱ ✈➔ ❝→❝ ❜↕♥✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵
❚→❝ ❣✐↔

❱ô ❱✐➺t ❇➻♥❤




ỵ
coM

ỗ ừ t M

coM
coneM

ỗ õ ừ t➟♣ M

M−

❝ü❝ ➙♠ ❝õ❛ M

Ms
X∗

❝ü❝ ➙♠ ❝❤➦t ❝õ❛ M

T (M, x)
TC (M, x)

♥â♥ t✐➳♣ ❧✐➯♥ ❝õ❛ M t↕✐ x

N (M, x)
f − (x, d)

♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ❝õ❛ M t↕✐ x

õ ỗ s r M

ổ ố tổ ♣æ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X
♥â♥ t✐➳♣ t✉②➳♥ ❈❧❛r❦❡ ❝õ❛ M t x
ữợ ừ f t x t ♣❤÷ì♥❣ d

f + (x, d)
f 0 (x, d)

✤↕♦ ❤➔♠ ❉✐♥✐ tr➯♥ ❝õ❛ f t↕✐ x t❤❡♦ ♣❤÷ì♥❣ d

∂C f (x)
∂f (x)

ữợ r ừ f t x

t ữ

tữỡ ự

KT
KT V CP

❑✉❤♥✲❚✉❝❦❡r

✤↕♦ ❤➔♠ s✉② rë♥❣ ❈❧❛r❦❡ ❝õ❛ f t↕✐ x t ữỡ d
ữợ ừ ỗ f t x

✤✐➸♠ tỵ✐ ❤↕♥ ✈❡❝tì ❑✉❤♥✲ ❚✉❝❦❡r


✷

▼ð ✤➛✉
✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥
❑❤✐ t➼♥❤ t♦→♥ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✱ s❛✉ ♠ët sè ❤ú✉ ❤↕♥ ữợ tt
t tố ữ t ❤ú✉ ❤✐➺✉ ①➜♣ ①➾✳ ❱➻ ✈➟② ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉
❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ①➜♣ ①➾ ❧➔ r➜t ❝➛♥ t❤✐➳t✳ ❚ø ✤â ❞➝♥ ✤➳♥ ✈✐➺❝ ♥❣❤✐➯♥
❝ù✉ ❝→❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ●♦❧❡st❛♥✐✕❙❛❞❡❣❤✐✕❚❛✈❛♥ ✭✷✵✶✼✮ ✤➣ ♥❣❤✐➯♥
❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❑✉❤♥✲ ❚✉❝❦❡r ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✭✇❡❛❦
q✉❛s✐ ❡❢❢✐❝✐❡♥t s♦❧✉t✐♦♥✮ ✈➔ tỹ ỳ qs t st
ỵ ✤è✐ ♥❣➝✉ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✳
▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ tü❛ ♥❣❤✐➺♠
❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ t tố ữ ử t ợ st
ữỡ q ữợ r ừ st ❍✳ ❙❛❞❡❣❤✐✱ ❨✳

❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥

✸✽✭✷✵✶✼✮✱ ✽✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐ ♥❣➝✉ ②➳✉✱
♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝✳

✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ t
ỗ ❜❛ ❝❤÷ì♥❣✱ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝ ❝→❝
t➔✐ ❧✐➺✉ t❤❛♠

ữỡ ợ t

tự tr ởt số tự

ỡ ữợ r õ t t õ t r
ữỡ ợ t ✤➲✿

✧✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉✧ tr➻♥❤

❜➔② ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♠ỵ✐ ✤➙② ❝õ❛ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳

❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥


✸

✸✽✭✷✵✶✼✮✱ ✻✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐
ố ữủ

ữỡ ợ t ố

tr ỵ ố

✤è✐ ♥❣➝✉ ♥❣÷đ❝ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉
▼♦♥❞✲❲❡✐r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✺ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵
❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

❱ô ❱✐➺t ❇➻♥❤


✹

❈❤÷ì♥❣ ✶

▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ởt số tự ỡ ữợ ❈❧❛r❦❡✱
♥â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❝➛♥ ❞ị♥❣
tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ❈→❝ ❦✐➳♥ t❤ù❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝
t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✶✱✷✱✹❪✳

✶✳✶✳ ữợ r
sỷ x = (x1 , . . . , x ) ✈➔ y = (y1 , . . . , y ) ❧➔ ❤❛✐ ✈❡❝tì tr♦♥❣ R ✳ ❈→❝ ❦➼
❤✐➺✉ s❛✉ ✤➙② s➩ ✤÷đ❝ sû ❞ư♥❣ s❛✉ ♥➔②✿

x = y,

♥➳✉ xi = yi ,

✈ỵ✐ ♠å✐ i,

x

y,

♥➳✉ xi ≤ yi ,

✈ỵ✐ ♠å✐ i,

x < y,

♥➳✉ xi < yi ,

✈ỵ✐ ♠å✐ i,

x ≤ y,

♥➳✉ x

y ✈➔ x = y.

●✐↔ sû M ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✳ ❚❤ỉ♥❣ t❤÷í♥❣✱ ❝❧ M ✱ ✐♥t M ✱ ❝♦(M ) ✈➔
❝♦♥❡ (M ) ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ❜❛♦ ✤â♥❣✱ ♣❤➛♥ tr ỗ õ s

M tữỡ ự ỹ ➙♠ ✈➔ ❝ü❝ ➙♠ ❝❤➦t ❝õ❛ M ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
M − := ξ ∈ R

ξ, ν ≤ 0,

∀ν ∈ M ,

M s := ξ ∈ R

ξ, ν < 0,

∀ν M ,

tr õ Ã, Ã t ổ ữợ tr♦♥❣ R ✳
❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦➼ ❤✐➺✉ t❤æ♥❣ t❤÷í♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥ ✭①❡♠
❬✷❪✮✳


✺

●✐↔ sû ϕ : R → R ❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶ ✣↕♦ ❤➔♠ t❤❡♦ ♣❤÷ì♥❣ s✉② rë♥❣ ✭❣❡♥❡r❛❧✐③❡❞ ❞✐r❡❝t✐♦♥❛❧
❞❡r✐✈❛t✐✈❡✮ ❝õ❛ ϕ t↕✐ x t❤❡♦ ♣❤÷ì♥❣ ν ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉

ϕ◦ (x; ν) = lim sup
y→x,t↓0

ϕ(y + t) (y)
.
t

ữợ r rs s✉❜❞✐❢❢❡r❡♥t✐❛❧✮ ❝õ❛ ϕ t↕✐
x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

∂C ϕ(x) = {ξ ∈ R | ξ, ν ≤ ϕ◦ (x; ν) ∀ν ∈ R }.
❈❤➥♥❣ ❤↕♥✱ ❤➔♠ f (x) = x x0 ổ t x0 ữợ ♣❤➙♥
❈❧❛r❦❡ ❝õ❛ ♥â t↕✐ x0 ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ B[0, 1] := B tr♦♥❣ R ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸ ữợ ừ ỗ : R

R t↕✐ x ∈ R ✤÷đ❝

①→❝ ✤à♥❤ ♥❤÷ s❛✉✿

∂ϕ(x) = {ξ ∈ R : ξ, x − x ≤ ϕ(x) − ϕ(x)}.
❚❛ ❜✐➳t r➡♥❣ →♥❤ ①↕ ν → ϕ◦ (x; ν) ởt ỗ ữợ ừ õ
t t ỗ t = 0 tỗ t↕✐ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔

∂ϕ◦ (x; ·)(0) ✈➔ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✿
∂C ϕ(x) = ∂ϕ◦ (x; ·)(0).
❙❛✉ ✤➙② ởt số t t ừ ữợ r ❝õ❛ ♠ët ❤➔♠ ▲✐♣s❝❤✐t③
✤à❛ ♣❤÷ì♥❣✳

❇ê ✤➲ ✶✳✶ ❬✷❪ ●✐↔ sû ϕ, ψ : R

❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣
♠ët ❧➙♥ ❝➟♥ ❝õ❛ x ∈ R ✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✿
✐✮ ∂C ϕ(x) ❧➔ t➟♣ ❝♦♥ ❦❤→❝ rộ t ỗ ừ R
ợ ồ ∈ R ✱ ϕ◦ (x; ν) = max{ ξ, ν |ξ ∈ ∂C ϕ(x)}✳
✐✐✐✮ ❱ỵ✐ ❜➜t ❦➻ sè λ, ∂C λϕ(x) = λ∂C ϕ(x).
✐✈✮ ❍➔♠ ν → ϕ◦ (x; ν) ỳ t t ữỡ ữợ t t
tr R
ỗ t❤➻ ❝â ∂C (ϕ + ψ)(x) = ∂C ϕ(x) + ∂C ψ(x).
→R

✻

✶✳✷✳ ◆â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡
❙❛✉ ✤➙② t❛ ✤÷❛ ✈➔♦ ♠ët ✈➔✐ ♥â♥ s➩ ✤÷đ❝ sû ❞ư♥❣ s❛✉ ♥➔②✳ ●✐↔ sû

M ⊂ R , x0 ∈ ❝❧M ✱
• ◆â♥ t✐➳♣ ❧✐➯♥ ✭❝♦♥t✐♥❣❡♥t ❝♦♥❡✮ ❝õ❛ M t↕✐ x0 ❧➔
T (M, x0 ) := ν ∈ R ∃tn ↓ 0, ∃νn → ν; x0 + tn νn ∈ M .
• ◆â♥ t✐➳♣ t✉②➳♥ ❈❧❛r❦❡ ✭❈❧❛r❦❡✬s t❛♥❣❡♥t ❝♦♥❡✮ ❝õ❛ M t↕✐ x0 ❧➔
TC (M, x0 )
:= ν ∈ R ∀tn ↓ 0, ∀xn → x0 ✈ỵ✐ xn ∈ M, ∃νn → ν; xn + tn νn ∈ M .
• ◆â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ✭❈❧❛r❦❡✬s ♥♦r♠❛❧ ❝♦♥❡✮ ❝õ❛ M t↕✐ x0 ❧➔
N (M, x0 ) := ν ∈ R

w, ν ≤ 0,

∀w ∈ TC (M, x0 ) .

❚❛ ❜✐➳t r➡♥❣

TC (M, x0 ) ⊂ T (M, x0 ).
●✐↔ sû Q ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ fi , i ∈ {1, . . . , m}, gj , j ∈

{1, . . . , n}, hk , k ∈ {1, . . . , p} ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥
◦
◦
❝õ❛ x0 ∈ ❝❧Q ✈➔ α ∈ ✐♥t(Rm
+ )✱ t❤❡♦ ❇ê ✤➲ ✶✳✶✱ fi (x0 ; ·) + αi · ✱ gj (x0 ; ·)

hk (x0 ; Ã) ỳ ữợ t t tr➯♥ R ✱ ∂C (−hk )(x0 ) = −∂C hk (x0 ) ✈➔
h◦k (x0 ; −ν) = (−hk )◦ (x0 ; ν)✳ ❍ì♥ ♥ú❛✱ TC (Q, x0 ) ❧➔ ♠ët t ỗ õ
ừ R 0 TC (Q, x0 )✳
❈→❝ ❦➳t q✉↔ s❛✉ ✤➙② ❧➔ ❧➔ ❝➛♥ t❤✐➳t ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐
÷✉ ❦✐➸✉ ❑✉❤♥✲❚✉❝❦❡r ✈➔ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳

❍➺ q✉↔ ✶✳✶ ❬✹❪ ●✐↔ sû f = (f1, . . . , fm)✱ g = (g1, . . . , gn) ✈➔ h = (h1, . . . , hp)

❧➔ ❝→❝ ❤➔♠ tỡ ợ t st ữỡ tr R ✳ ●✐↔ sû
r➡♥❣ Q ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✱ x0 ∈ clQ ✈➔ α ∈ int(Rm+ )✳ ❑❤✐ ✤â✱ ❝→❝ ♣❤→t
❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ ❍➺ s❛✉ ✤➙② ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✿
◦
f ◦ (x0 ; ν) = (f1◦ (x0 ; ν), . . . , fm
(x0 ; ν)) < −α ν ,

g ◦ (x0 ; ν) = (g1◦ (x0 ; ν), . . . , gn◦ (x0 ; ν))

0,


✼

h◦ (x0 ; ν) = (h◦1 (x0 ; ν), . . . , h◦p (x0 ; ν))

0,

h◦ (x0 ; −ν) = (h◦1 (x0 ; −ν), . . . , h◦p (x0 ; ))

0,

TC (Q, x0 ).


ỗ t (, µ, ν, ν˜) ∈ Rm+ × Rn+ × Rp+ × Rp+, λ = 0 s❛♦ ❝❤♦
m

p

n

0∈

λi ∂C fi (x0 ) +
i=1

i=1
p

k=1
m

+

❈❤ù♥❣ ♠✐♥❤✳

ν k ∂C hk (x0 )

µj ∂C gj (x0 ) +

ν˜k ∂C (−hk )(x0 ) +

✭✶✳✶✮

λi αi B + N (Q, x0 ).
i=1

k=1

❍➺ q✉↔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ờ ỵ

tr t❤❛② Q ❜➡♥❣ TC (Q, x0 ) ✈➔ ❝→❝ ❤➔♠ fi , gj t÷ì♥❣
ù♥❣ ❜ð✐ fi0 (x0 ; .) + αi ||.|| ✈➔ (gj0 (x0 , .), h0k (x0 ; .), (−hk )0 (x0 ; .))✳

❍➺ q✉↔ ✶✳✷ ❬✹❪ ●✐↔ sû f = (f1, . . . , fm)✱ g = (g1, . . . , gn) ✈➔ h = (h1, . . . , hp)

❧➔ ❝→❝ ❤➔♠ ✈❡❝tì ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ R ✈➔ Q
❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✈➔ x0 ∈ clQ✳ ●✐↔ sû r➡♥❣ α ∈ int(Rm+ ) ✈➔ ✈ỵ✐ ♠é✐
i0 ∈ I = {1, . . . , m}✱ ♥â♥




n

(∂C fi (x0 ) + αi B) + cone co

Di0 = cone co 

j=1

i∈I\{i0 }
p

+ cone co

∂C hk (x0 )
k=1

∂C gj (x0 )

p

+ cone co

∂C (−hk )(x0 )

+ N (Q, x0 )

k=1

✤â♥❣✱ t❤➻ ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ ❍➺ s❛✉ ✤➙② ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✿

◦
f ◦ (x0 ; ν) = (f1◦ (x0 ; ν), . . . , fm
(x0 ; ν)) < −α ν ,

g ◦ (x0 ; ν) = (g1◦ (x0 ; ν), . . . , gn◦ (x0 ; ν))
h◦ (x0 ; ν) = (h◦1 (x0 ; ν), . . . , h◦p (x0 ; ν))

0,
0,

h◦ (x0 ; −ν) = (h◦1 (x0 ; −ν), . . . , hp (x0 ; ))

0,

TC (Q, x0 ).


ỗ t (, à, , ) Rm++ ì Rn+ ì Rp+ × Rp+, λ = 0 s❛♦ ❝❤♦ ✭✶✳✶✮ ✤ó♥❣✳


✽

❈❤÷ì♥❣ ✷

✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐
÷✉
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ✈➔ ❑✉❤♥✲❚✉❝❦❡r
♠↕♥❤ ❝ò♥❣ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠
❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ❝â r➔♥❣ ❜✉ë❝✳
❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✸✕✽❪✳

✷✳✶✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ ①➨t ❜➔✐ t♦→♥ ❝â r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝✱
✤➥♥❣ t❤ù❝ ✈➔ r➔♥❣ ❜✉ë❝ t➟♣ s❛✉ ✤➙②✿

(MP)

min f (x) := (f1 (x), . . . , fm (x)),

✈ỵ✐ r➔♥❣ ❜✉ë❝✿

g(x) := (g1 (x), . . . , gn (x))

0,

h(x) := (h1 (x), . . . , hp (x)) = 0,
x ∈ Q,
tr♦♥❣ ✤â fi , i ∈ I = {1, . . . , m}✱ gj , j ∈ J = {1, . . . , n}✱ hk , k ∈ K =

{1, . . . , p} ❧➔ ❝→❝ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❣✐→ trà t❤ü❝ tr➯♥ R ✈➔ Q ⊆ R
❧➔ t➟♣ ❜➜t ❦➻✳
❑➼ ❤✐➺✉ S ❧➔ ♠✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✱ ❝ö t❤➸

S := x ∈ R g(x)

0, h(x) = 0, x ∈ Q .


✾

✣à♥❤ ♥❣❤➽❛ ✷✳✶ ✣✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x0 ✤÷đ❝ ❣å✐ ❧➔
❛✮ ◆❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ ❤ú✉ ❤✐➺✉✮ ♥➳✉ tỗ t ởt

U ừ x0 s ợ ❜➜t ❦ý x ∈ U ∩ S ✭t✳÷✱ x ∈ S ✮ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉
❦❤ỉ♥❣ ✤ó♥❣

f (x) ≤ f (x0 ).
❜✮ ◆❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ ❤ú✉ tỗ t ởt
U ừ x0 s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý x ∈ U ∩ S ✭t✳÷✳✱ x ∈ S ✮ ❜➜t ✤➥♥❣ t❤ù❝
s❛✉ ❦❤ỉ♥❣ ✤ó♥❣

f (x) < f (x0 ).

✣à♥❤ ♥❣❤➽❛ ✷✳✷ ❬✻❪ ✣✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x0 ✤÷đ❝ ❣å✐ ❧➔
❛✮ ❚ü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ữỡ tỹ ỳ P
tỗ t↕✐ α ∈ ✐♥t(Rm
+ ) ✈➔ ❧➙♥ ❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý x ∈ U ∩ S
✭t✳÷✳✱ x ∈ S ✮ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ✤ó♥❣

f (x) ≤ f (x0 ) − α x − x0 .

✭✷✳✶✮

❜✮ ❚ü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ tỹ ỳ
ừ P tỗ t ∈ ✐♥t(Rm
+ ) ✈➔ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x0 s ợ
t ý x U S tữ x ∈ S ✮ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ✤ó♥❣

f (x) < f (x0 ) − α x − x0 .

✭✷✳✷✮

❚❛ ♥â✐ r➡♥❣ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ ❧➔ tü❛ ♥❣❤✐➺♠
❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❝õ❛ ✭▼P✮ t❤❡♦ α ♥➳✉ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤ó♥❣ ✈ỵ✐

α ∈ ✐♥t(Rm

+ )✳
❍➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣
✭♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣
✭t✳÷✳✱ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❝õ❛ ✭▼P✮✳ ✣✐➲✉ ♥❣÷đ❝ ❧↕✐ ♥â✐
❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✱ ❝â t❤➸ t❤➜② tr♦♥❣ ✈➼ ❞ö s❛✉ ✤➙②✳


✶✵

❱➼ ❞ư ✷✳✶ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉ ✤➙②✿
min f (x) = (x2 − x, −x),

(P1)

✈ỵ✐ r➔♥❣ ❜✉ë❝

g(x) = −x ≤ 0,
h(x) = 0,
x ∈ Q,
tr♦♥❣ ✤â Q = {x ∈ R : |x| ≤ 1}. ợ = (1, 1) ổ tỗ t x ∈ S
s❛♦ ❝❤♦ ✭✷✳✶✮ ✭❤♦➦❝ ✭✷✳✷✮✮ ✤ó♥❣ t↕✐ x0 = 0✱ ♥❤÷ ✈➟② x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠
❤ú✉ ❤✐➺✉ ✭②➳✉✮ t❤❡♦ α = (1, 1)✳ ❉➵ ❞➔♥❣ ❦✐➸♠ ❝❤ù♥❣ r➡♥❣ x0 = 0 ❦❤æ♥❣ ❧➔
♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭②➳✉✮ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✶✮✳
✣➸ t❤✉➟♥ t✐➺♥ t❛ ✤÷❛ ✈➔♦ ♠ët ✈➔✐ ỵ s

S l := x R fi (x) ≤ fi (x0 ) − αi x − x0 , ∀i = l, g(x)
F :=

0, h(x) = 0, x ∈ Q

(∂C fi (x0 ) + αi B)
i∈I

F l :=

(∂C fi (x0 ) + αi B)
i∈I\{l}

G :=

∂C gj (x0 )
j∈J(x0 )

∂C hk (x0 ) ∪

H :=
k∈K

∂C (−hk )(x0 ),
k∈K

tr♦♥❣ ✤â J(x0 ) ỵ t số r ở t➼❝❤ ❝ü❝ t↕✐ ✤✐➸♠ ❝❤➜♣
♥❤➟♥ ✤÷đ❝ x0 ✈➔ α ∈ ✐♥t(Rm
+ ) ✈➔ B ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ tr♦♥❣ R ✳
▼ö❝ ✤➼❝❤ ❝õ❛ ♣❤➛♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❑✉❤♥✲
❚✉❝❦❡r ✈➔ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭②➳✉✮ ✈ỵ✐ ❝→❝ ✤✐➲✉
❦✐➺♥ ❝❤➼♥❤ q✉② s❛✉ ✤➙②✿

(F i )s

H−

Gs

TC (Q, x0 ) = ∅, ∀i ∈ I.

✭❈◗✶✮

m
−

−

−

T (S i , x0 ).

✭❈◗✷✮

TC (Q, x0 ) ⊆ T (S, x0 ).

F s G− H − T (Q, x ) ⊆ T (S, x )
C
0
0
(F i )s G− H − T (Q, x ) = ∅, ∀i ∈ I.
C
0

✭❈◗✸✮

F

G

H

TC (Q, x0 ) ⊆
i=1

Fs

G−

H−

✭❈◗✹✮

◆❤➟♥ ①➨t ✷✳✶ ▲✐ ❬✸❪ ✤➣ ❝❤➾ r❛ r➡♥❣✿ ♥➳✉ ❦❤æ♥❣ ❝â ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣


✶✶

B ✈➔ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝ ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ x0 ∈ ✐♥t Q✱ ✭❈◗✶✮ ❧➔ ♠ët
t÷ì♥❣ tü ❦❤ỉ♥❣ trì♥ ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✤÷đ❝ ❝❤♦ ❜ð✐ ▼❛❡❞❛ ❬✼❪ ❧➔
♠ët tê♥❣ q✉→t ❤â❛ ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❈♦tt❧❡ ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣
❦❤↔ ✈✐ ❧✐➯♥ tư❝ ❝õ❛ ✭▼P✮✱ ✭❈◗✶✮ q✉② ✈➲ ♠ët ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ②➳✉ ❤ì♥
✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❝õ❛ ❝õ❛ ▼❛❡❞❛ ✈➔ ♥❤÷ ❧➔ ♠ët tê♥❣ q✉→t ❤â❛ ❝õ❛ ✤✐➲✉
❦✐➺♥ ❝❤➼♥❤ q✉② ▼❛♥❣❛s❛r✐❛♥✲❋r♦♠♦✈✐t③✳
❇ð✐ ✈➻ ✭❈◗✷✮ ♣❤ö tở t(Rm

+ ) ú ỵ r ổ tỗ t
q ỳ q✉② s❛✉ ✤➣ ❝❤♦ tr♦♥❣ ❬✺❪✿
−

∂C fi (x0 )

m
−

G

H

−

T (X i , x0 ),

TC (Q, x0 ) ⊆

✭❈◗✮

i=1

i∈I

tr♦♥❣ ✤â

X i := x ∈ R fj (x) ≤ fj (x0 ), ∀j = i, g(x)

0, h(x) = 0, x ∈ Q .

✣➸ ♠✐♥❤ ❤å❛ ✤✐➲✉ ♥➔②✱ t❛ ①➨t ❱➼ ❞ö ✷✳✶✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ ✈ỵ✐

α = (2, 2)✱ ✭❈◗✷✮ ✤ó♥❣ t↕✐ x0 = 0 ♥❤÷♥❣ ✭❈◗✮ ❦❤ỉ♥❣ ✤ó♥❣ t↕✐ x0 = 0✳
❚✉② ♥❤✐➯♥✱ ♥➳✉ t❛ ①➨t Q = [0, 1] ✈➔ α = (1, 1) t❤➻ ✭❈◗✮ ✤ó♥❣ t↕✐ x0 = 0
♥❤÷♥❣ ✭❈◗✷✮ ❦❤ỉ♥❣ ✤ó♥❣ t↕✐ x0 = 0✳

❇ê ✤➲ ✷✳✶ ❈→❝ s✉② ❧✉➟♥ s❛✉ ✤➙② ✤ó♥❣✿
✭❈◗✶✮ ⇒ ✭❈◗✷✮ ⇒ ✭❈◗✸✮,
✭❈◗✶✮ ⇒ ✭❈◗✹✮

❈❤ù♥❣ ♠✐♥❤✳

✈➔ ✭❈◗✹✮ ⇒ ✭❈◗✸✮.

❇ð✐ ✈➻ ♥➳✉ t ỗ t t A + B

ụ ỗ t ự tữỡ tỹ ▼➺♥❤ ✤➲ ✼✳✶ tr♦♥❣ ❬✸❪ t❛
♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
ỵ B(x0 , ) t x0 ✱ ❜→♥ ❦➼♥❤ δ ✳ ✣➸ tr➻♥❤ ❜➔② ❦➳t q✉↔
❝❤➼♥❤ ừ t ỵ s

❧➼ ✷✳✶ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✳ ●✐↔
sû

✤ó♥❣ t↕✐ (x0, α)✳ ❑❤✐ ✤â tỗ t
B(x0 , ) S s ổ t❤ä❛ ♠➣♥✿
✭❈◗✷✮

δ > 0

s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐

x ∈


✶✷

f ◦ (x0 , x − x0 ) ≤ −α x − x0 ,
gj◦ (x0 , x − x0 ) ≤ 0 ∀j ∈ J(x0 ),
h◦ (x0 , x − x0 )

✭✷✳✸✮

0,

(−h)◦ (x0 , x − x0 )

0,

x − x0 TC (Q, x0 ).

ự

sỷ ữủ ợ ộ > 0 tỗ t x B(x0 , δ) ∩ S

s❛♦ ❝❤♦ ❤➺ ✭✷✳✸✮ ✤ó♥❣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② ❦➨♦ t❤❡♦

x − x0 ∈ F −

G−

H−

TC (Q, x0 ).



ỡ ỵ t t = x − x0 ✳ ❇ð✐ ✈➻ f ◦ (x0 , ν) õ
tỗ t i0 I s❛♦ ❝❤♦ fi◦0 (x0 ; ν) + αi0 ν < 0✳ ●✐↔ sû ε > 0 t❤ä❛ ♠➣♥

fi◦0 (x0 ; ν) + αi0 ν < −ε.
❇ð✐ ✈➻

fi+0 (x0 ; ν) fi0 (x0 ; ),
t s r tỗ t 0 > 0 s❛♦ ❝❤♦

fi0 (x0 + tν) − fi0 (x0 )
+ αi0 ν < −ε ∀t ∈ (0, δ0 ).
t

✭✷✳✺✮

▼➦t ❦❤→❝✱ ❜ð✐ ✈➻ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✱ tø ✭✷✳✹✮ t❛ s✉② r❛ ν ∈
m
i
i=1 T (S , x0 )

✈➔ ✈➻ ✈➟② ν ∈ T (S i0 , x0 ) õ tỗ t tn 0 νn → ν s❛♦

❝❤♦

x0 + tn νn ∈ S i0 .

✭✷✳✻✮

❇➙② ❣✐í ❣✐↔ sû ❤➡♥❣ sè ▲✐♣s❝❤✐t③✐❛♥ ❝õ❛ fi0 tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ x0 ❧➔ k ✳ ❉♦
✤â✱ ✈ỵ✐ n ✤õ ❧ỵ♥ t❛ ❝â

fi0 (x0 + tn νn ) ≤ fi0 (x0 + tn ν) + ktn νn − ν .
❱➻ ✈➟②

fi0 (x0 + tn νn ) − fi0 (x0 )
fi (x0 + tn ν) − fi0 (x0 )
≤ 0
+ k νn − ν .
tn
tn

✭✷✳✼✮


✶✸

❚÷ì♥❣ tü✱ tø ❜➜t ✤➥♥❣ t❤ù❝ νn ≤ νn − ν + ν t❛ ♥❤➟♥ ✤÷đ❝
✭✷✳✽✮

αi0 νn ≤ αi0 νn − ν + αi0 ν .
❇ð✐ ✈➻ k νn − ν → 0 ✈➔ αi0 νn − ν → 0 t❛ s✉② r❛ ✈ỵ✐ n ✤õ ❧ỵ♥

ε
k νn − ν < ,
4
ε
αi0 νn − ν < .
4

✭✷✳✾✮

❚ø ✭✷✳✺✮ ✈➔ ✭✷✳✼✮✲✭✷✳✾✮✱ ✈ỵ✐ n ✤õ ❧ỵ♥✱ t❛ ❝â

fi0 (x0 + tn νn ) − fi0 (x0 )
+ αi0 νn
tn
fi (x0 + tn ν) − fi0 (x0 )
≤ 0
+ α i 0 νn + k νn − ν
tn
fi (x0 + tn ν) − fi0 (x0 )
≤ 0
+ αi0 ν + αi0 νn − ν + k νn − ν
tn
fi (x0 + tn ν) − fi0 (x0 )
≤ 0
+ αi0 ν + (αi0 + k) νn − ν
tn
< −ε + (αi0 + k) νn − ν
✭✷✳✶✵✮

< 0.

◆❤÷ ✈➟②✱ tø ✭✷✳✻✮ ✈➔ ✭✷✳✶✵✮ t❛ ❦➳t ữủ r ợ ộ > 0 tỗ t

x0 + tn νn ∈ B(x0 , δ) ∩ S s❛♦ ❝❤♦
f (x0 + tn νn ) ≤ f (x0 ) − αtn νn .
✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦

α✳
▼ët ❝→❝❤ t÷ì♥❣ tü ❝❤ù♥❣ ♠✐♥❤ tr➯♥✱ t❛ ❝â t❤➸ ♣❤→t ❜✐➸✉ ự
ỵ s tỹ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳

✣à♥❤ ❧➼ ✷✳✷ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦

sỷ

ú t (x0, ) õ tỗ t
x B(x0 , δ) ∩ S ❤➺ s❛✉ ✤➙② ❦❤æ♥❣ t❤ä❛ ♠➣♥✿
✭❈◗✸✮

δ > 0

f ◦ (x0 , x − x0 ) < −α x − x0 ,
gj◦ (x0 , x − x0 ) ≤ 0 ∀j ∈ J(x0 ),

s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐


✶✹

h◦ (x0 , x − x0 )

✭✷✳✶✶✮

0,

(−h)◦ (x0 , x − x0 )

0,

x − x0 ∈ TC (Q, x0 ).

◆❤➟♥ ①➨t ✷✳✷ ❱ỵ✐ ✈➼ ❞ư ✷✳✶✱ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ ❦❤✐ Q = [0, 1] ✈➔

α = (2, 2)✱ x0 = 0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭②➳✉✮ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✱

✭❈◗✷✮ ✈➔ ✭❈◗✸✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , ) ổ ú ú
ỵ r ✭❈◗✮ ✤ó♥❣ t↕✐ x0 = 0 ♥❤÷♥❣ ♠å✐ x > 0 ởt ừ P
tr ỵ ừ ữ ỵ rở ỵ
tr trữớ ủ ỳ ✤à❛ ♣❤÷ì♥❣ s❛♥❣ tr÷í♥❣ ❤đ♣
tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣✳
❑❤✐ ❦❤æ♥❣ ❝â ❝→❝ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝ ✈➔ r➔♥❣ ❜✉ë❝ t➟♣✱ t❤➻ ❦❤→✐ ♥✐➺♠
✤✐➸♠ tỵ✐ ❤↕♥ ❑✉❤♥✲❚✉❝❦❡r ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉
❝õ❛ ✭▼P✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❬✽❪✳ ❙❛✉ ✤➙② ❦❤→✐ ♥✐➺♠ ♥➔② ✤÷đ❝ tê♥❣
q✉→t ❤â❛ t❤➔♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ✭▼P✮ ❝â tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❤♦➦❝ tü❛
♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉✳

✣à♥❤ ♥❣❤➽❛ ✷✳✸ ✣✐➸♠ ữủ x0 ữủ ồ tợ ✈❡❝tì
❑✉❤♥✲❚✉❝❦❡r ✭❑✉❤♥✲❚✉❝❦❡r ✈❡t♦r ❝r✐t✐❝❛❧ ♣♦✐♥t✮✱ ✈✐➳t t➢t ❧➔ ❑❚❱❈P✱ ❝❤♦

m

n
p
✭▼P✮ tỗ t t(Rm
+ ) (, à, ν) ∈ R+ × R+ × R ✱ λ = 0 s❛♦ ❝❤♦
m

0∈

λi ∂C fi (x0 ) +
i=1
m

+

p

n

µj ∂C gj (x0 ) +
j=1

νk ∂C hk (x0 )

✭✷✳✶✷✮

k=1

λi αi B + N (Q, x0 ),
i=1

µj gj (x0 ) = 0, ∀j ∈ J.
❚❛ ♥â✐ r➡♥❣ x0 ❧➔ ❑❚❱❈P t❤❡♦ α ❝❤♦ ✭▼P✮ ♥➳✉ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤ó♥❣
✈ỵ✐ α ∈ ✐♥t(Rm
+ )✳
✣à♥❤ ỵ s tr tố ữ ❑✉❤♥✲❚✉❝❦❡r ❝❤♦ tü❛
♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣✳

✣à♥❤ ❧➼ ✷✳✸ ✭✣✐➲✉ ❦✐➺♥ ❑❚✮ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣
t❤❡♦ α ✈➔

✭❈◗✷✮

t❤ä❛ ♠➣♥ t↕✐ (x0, α)✳ ❑❤✐ ✤â x0 ❧➔ ❑❚❱❈P t❤❡♦ α✳


✶✺

❈❤ù♥❣ ♠✐♥❤✳

❱➻ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ✈➔

✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✱ ỵ t tỗ t > 0 s❛♦ ❝❤♦ ✈ỵ✐
♠é✐ x ∈ B(x0 , δ) ∩ S ❤➺ ✭✷✳✸✮ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✱ ❝ơ♥❣ ♥❤÷ ❤➺ ✭✷✳✶✶✮ ❦❤ỉ♥❣
❝â àj = 0 ợ j
/ J(x0 ) ✈➔ sû ❞ö♥❣ ❤➺ q✉↔ ✶✳✶ t❛
p
p
n
s✉② r❛ tỗ t (, à, , ) Rm
+ ì R+ × R+ × R+ ✱ λ = 0 s❛♦ ❝❤♦ ✭✶✳✶✮ ✤ó♥❣✳

❇➡♥❣ ❝→❝❤ t❤❛② t❤➳ νk = ν k − ν˜k ∈ Rp ✭νk ❝â t❤➸ ❦❤ỉ♥❣ ➙♠ ❤♦➦❝ ❦❤ỉ♥❣
❞÷ì♥❣✮ ỵ ữủ ự
tữỡ tỹ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❑✉❤♥✲
❚✉❝❦❡r ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳

✣à♥❤ ❧➼ ✷✳✹ ✭✣✐➲✉ ❦✐➺♥ ❑❚✮ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛
♣❤÷ì♥❣ t❤❡♦ α ✈➔
α✳

✭❈◗✸✮

t❤ä❛ ♠➣♥ t↕✐ (x0, α)✳ ❑❤✐ ✤â x0 P t

t ỵ rở tờ qt õ ỵ
tr♦♥❣ ❬✻❪✳

✷✳✷✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤
❇➙② ❣✐í t❛ tr➻♥❤ tố ữ r q ữợ
♣❤➙♥ ❈❧❛r❦❡ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉
❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳

✣à♥❤ ❧➼ ✷✳✺ ✭✣✐➲✉ ❦✐➺♥ ❑❚ ♠↕♥❤✮ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉
✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✳ ●✐↔ sû

✭❈◗✷✮



t❤ä❛ ♠➣♥ t↕✐ (x0, α) ✈➔ ✈ỵ✐ ♠é✐ i0 ∈ I



Di0 = cone co 





(∂C fi (x0 ) + αi B) + cone co 

∂C gj (x0 )

i∈I\{i0 }
p

+ cone co

∂C hk (x0 )
k=1

j∈J(x0 )
p

+ cone co

∂C (−hk )(x0 )

+ N (Q, x0 )

k=1

✭✷✳✶✸✮

❧➔ ✤â♥❣✳ ❑❤✐ õ tỗ t (, à, ) Rm++ ì Rn+ × Rp s❛♦ ❝❤♦ ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮
✤ó♥❣✳


✶✻

❈❤ù♥❣ ♠✐♥❤✳

❇ð✐ ✈➻ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ✈➔

✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✱ ỵ t tỗ t > 0 s❛♦ ❝❤♦
✈ỵ✐ ♠é✐ x ∈ B(x0 , δ) ∩ S ❤➺ ✭✷✳✸✮ ❦❤ỉ♥❣ ✤ó♥❣✳ ❱➻ Di0 ✤â♥❣✱ ✈ỵ✐ ♠é✐ i0 I
tr àj = 0 ợ j
/ J(x0 ) sỷ ử q
p
p
n
tỗ t (, à, , ) Rm
++ ì R+ ì R+ × R+ s❛♦ ❝❤♦ ❜✐➸✉ ❞✐➵♥ ✭✶✳✶✮ t❤ä❛

♠➣♥✳ ❇➡♥❣ ❝→❝❤ t❤❛② t❤➳ νk = ν k − ν˜k ∈ Rp k õ t ổ
ổ ữỡ ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ❧➼ ✷✳✻ ✭✣✐➲✉ ❦✐➺♥ ❑❚ ♠↕♥❤✮ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉
②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦

t❤ä❛ t (x0, ) õ tỗ t

n
p
(, à, ) ∈ Rm
++ × R+ × R s❛♦ ❝❤♦ ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮ ✤ó♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ✭❈◗✹✮ ❦➨♦ t❤❡♦ ✭❈◗✸✮✱ t❤❡♦ ✣à♥❤ ỵ tỗ t (, à, )






n
p
Rm
+ ì R+ × R , λ = 0 s❛♦ ❝❤♦ ✭✷✳✶✷✮ ✤ó♥❣✳ ớ t sỷ r tỗ t
i I s i = 0 tỗ t i ∈ ∂C fi (x0 )✱ ηj ∈ ∂C gj (x0 )✱

ζk ∈ ∂C hk (x0 ), e ∈ B ✈➔ d ∈ N (Q, x0 ) s❛♦ ❝❤♦
m

p

n

λi ξi +
i=1

νk ζk +

µj η j +

j=1

m

λi αi e + d = 0.



i=1

k=1

t t ừ tỗ t ν ∈ TC (Q, x0 ) s❛♦ ❝❤♦

ξi , ν < −αi ν ,

∀i ∈ I \ {i },

ηj , ν ≤ 0,

∀j ∈ J,

ζk , ν = 0,

∀k ∈ K,



d, 0.
n

p
(, à, ) Rm
+ ì R+ ì R = 0 ợ ồ e ∈ B✱ e, ν ≤ ν , t❛ s✉②

r❛
m

λi ξi +
i=1

p

n

µj ηj +
j=1

m

νk ζk +
k=1

λi αi e + d,

< 0.



i=1

t ợ ữ Rm
++ ỵ ữủ ự


t ỵ rở tờ qt õ ỵ

tr tứ trữớ ủ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭②➳✉✮ s❛♥❣ tr÷í♥❣
❤đ♣ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭②➳✉✮ ✤à❛ ♣❤÷ì♥❣✳


✶✼

❙❛✉ ✤➙②✱ t❛ tr➻♥❤ ❜➔② ❤❛✐ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r
♠↕♥❤ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉✳

❱➼ ❞ö ✷✳✷ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿
(P 2)

min f (x) = (x2 − 2x, −2x)

✈ỵ✐ r➔♥❣ ❜✉ë❝✿


x
x≤0
g(x) =
,
−x x > 0



x x ≤ 0
h(x) =
,
0 x > 0

x ∈ Q = [0, 1].
❇ð✐ ✈➻ ✈ỵ✐ α = (2, 2) ❦❤ỉ♥❣ tỗ t x S s ú t x0 = 0✱
♥➯♥ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭✤à❛ ♣❤÷ì♥❣✮ t❤❡♦ α = (2, 2)✳ ❉➵ ❦✐➸♠
tr❛ r➡♥❣ x0 ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣✳ ❇➡♥❣ ♠ët t➼♥❤ t♦→♥
✤ì♥ ❣✐↔♥✱ t❛ ❝â

∂C f1 (0) = ∂C f2 (0) = {−2},
∂C g(0) = [−1, 1],
∂C h(0) = [0, 1],
∂C (−h)(0) = [−1, 0],
N (Q, 0) = (−∞, 0].
❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ✈ỵ✐ i = 1, 2✱ Di ✤â♥❣ ✈➔ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✳
❇➡♥❣ ❝→❝❤ ❧➜② λ1 = λ2 = µ = ν = e = 1✱ ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮ t❤ä❛ ♠➣♥✳

❱➼ ❞ö ✷✳✸ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿
(P 3)

min f (x) = (f1 (x), f2 (x))

✈ỵ✐ r➔♥❣ ❜✉ë❝✿

x ∈ S = {x ∈ R|g(x) ≤ 0, h(x) = 0, x ∈ Q},
tr♦♥❣ ✤â Q = {x ∈ R : |x| ≤ 2} ✈➔ g, h, fi : R → R✱ i = 1, 2 ✤÷đ❝ ❝❤♦ ❜ð✐



x2 − 1 (x − 1), x ≥ 1
− 1 (x − 1), x ≥ 1
2
2
f1 (x) =
, f2 (x) =
,
x,

x<1
−x,
x<1


✶✽


−(x − 1)2 , x ≥ 1
g(x) =
,
0,
x<1
❇ð✐ ✈➻ ✈ỵ✐ α =

1 1
2, 3


0,
x≥1

h(x) =
(x − 1)2 , x < 1.

ổ tỗ t x S s ú t↕✐ x0 = 1,

♥➯♥ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α =

∂C f1 (1) = 1,

1 1
2, 3

✳ ❚❛ ❝â

3
,
2

∂C f2 (1) = −1, −

1
,
2

∂C g(1) = ∂C h(1) = ∂C (−h)(1) = {0},
N (Q, 1) = {0}.
❚❛ ❝â ✭❈◗✹✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✳ ❇➡♥❣ ❝→❝❤ ❧➜② λ1 = λ2 = 1 ✈➔ µ = ν =

e = 0 ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮ t❤ä❛

ừ tố ữ
ỗ t❤ỉ♥❣ t❤÷í♥❣ ❝õ❛ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳
❍➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ : X R ữủ ồ ỗ t↕✐ x0 ∈ X ♥➳✉
✈ỵ✐ ♠å✐ x ∈ X ✱ t❛ ❝â

ϕ(x) ≥ ϕ(x0 ) + ξ, x − x0 , ∀ξ ∈ ∂C ϕ(x0 ).
✣➦❝ tr÷♥❣ s❛✉ ✤➙② ❝õ❛ ỗ ữủ tt tr t
ở sỹ

ỷ tử ữợ : X R ữủ ồ ỗ

✭❛♣♣r♦①✐♠❛t❡ ❝♦♥✈❡① ❢✉♥❝t✐♦♥✮ t↕✐ x0 ∈ X ♥➳✉ ✈ỵ✐ ♠å✐ > 0 tỗ t

> 0 s
(x) ϕ(x0 ) + ξ, x − x0 − α x − x0 , ∀x ∈ B(x0 , δ) ∩ X, ∀ξ ∈ ∂C ϕ(x0 ).
❑❤→✐ ♥✐➺♠ tr➯♥ ❞➝♥ ✤➳♥ ♠ët ❧ỵ♣ ❤➔♠ ♠ỵ✐ s❛✉ ✤➙②✳

✣à♥❤ ♥❣❤➽❛ ✷✳✺ ❍➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ϕ : X → R ✤÷đ❝ ❣å✐ ❧➔ ♠ët

❤➔♠ ỗ rt s t t x0 ✈ỵ✐




ồ > 0 tỗ t > 0 s ❝❤♦ ✈ỵ✐ ♠å✐ x ∈ B(x0 , δ) ∩ X ✱ t❛ ❝â

ξ, x − x0 + α x − x0 ≥ 0 ✈ỵ✐ ξ ∈ ∂C ϕ(x0 )
❦➨♦ t❤❡♦

ϕ(x) ≥ ϕ(x0 ) − α x − x0 .

✣✐➲✉ ♥➔② tữỡ ữỡ ợ

(x) < (x0 ) x x0
❦➨♦ t❤❡♦

ξ, x − x0 < −α x − x0 , ∀ξ ∈ ∂C ϕ(x0 ).
❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ♠å✐ ỗ ỗ t x0 ✳ ❚✉② ♥❤✐➯♥✱
❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥✱ t❛ ①➨t

ϕ(x) = −x2 − 2x,

x ∈ X = [1, 0].

õ ỗ ữ ổ ỗ t x0 = 0.
t ❤❛✐ ❧ỵ♣ ❤➔♠ ✈➔ s➩ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦
✤✐➸♠ ❑✉❤♥✲❚✉❝❦❡r ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❤♦➦❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉
❝õ❛ ✭▼P✮✳

✣à♥❤ ♥❣❤➽❛ ✷✳✻ ●✐↔ sû x0 ∈ R ✳ ❇➔✐ t♦→♥ ✭▼P✮ ✤÷đ❝ ❣å✐ ❧➔

❛✮ ❆❢❢✐♥❡ ❣✐↔ ỗ t strt rt s
t x0 ợ ồ t(Rm
+ ) tỗ t > 0 s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐

x ∈ B(x0 , δ) ✈ỵ✐ f (x) ≤ f (x0 ) − α x − x0 t ú
ỗ ①➾ ✭❑❚✲❛♣♣r♦①✐♠❛t❡ ♣s❡✉❞♦❝♦♥✈❡①✲❛❢❢✐♥❡✮ t↕✐ x0
♥➳✉ ✈ỵ✐ ♠å✐ α ∈ ✐♥t(Rm
+ ) tỗ t > 0 s ợ ộ x ∈ B(x0 , δ) ✈ỵ✐

f (x) < f (x0 ) − α x − x0 ❦➨♦ t❤❡♦ ❤➺ ✭✷✳✶✶✮ ú

õ r P ỗ ❝❤➦t tr➯♥ t➟♣ D ⊂ R
♥➳✉ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ B(x0 , δ) ∩ D✳ ❚❛ õ r P
ỗ t tr t➟♣ D ⊂ R ✱ ♥➳✉ (M P ) ❧➔ ỗ
t t x0 tr D ợ ồ x0 D ữỡ tỹ t
ỗ ụ ữủ
ú ỵ ợ t ổ r ở t ỗ
t ỗ t tữỡ ữỡ ợ t ỗ
tr




tr r ộ ỗ t ỗ
♥❣÷đ❝ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ✣➸ ♠✐♥❤ ❤å❛ ✤✐➲✉
♥➔② t❛ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿

(P 4)

min f (x) = (−x2 − 2x, −2x),

✈ỵ✐ r➔♥❣ ❜✉ë❝✿

g(x) = −x ≤ 0,
h(x) = 0,
x ∈ Q = [0, 1].
õ P ỗ t x0 = 0 ❜ð✐ ✈➻ ✈ỵ✐ ♠é✐ α =

(α1 , α2 ) ∈ ✐♥t(R2+ ) t❤➻ t❛ ❝❤➾ ❝➛♥ ①➨t δ s❛✉✿

α − 2, α > 2,

1
1
δ=
1,
α1 ≤ 2.
❚✉② ♥❤✐➯♥✱ P ổ ỗ t t x0 = 0 ❜ð✐
✈➻ ✈ỵ✐ α = (2, 2) ✈➔ ợ ộ > 0 tỗ t x B(x0 , δ) ∩ S s❛♦ ❝❤♦

f (x) ≤ f (x0 ) − α x − x0 ♥❤÷♥❣ ❤➺ ✭✷✳✶✶✮ ❦❤ỉ♥❣ ✤ó♥❣❀ ✈➼ ❞ư t↕✐ ✤✐➸♠
1
❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x = .
2

✣à♥❤ sỷ P ỗ ①➾ ❝❤➦t t↕✐ x0 tr➯♥ S ✳

●✐↔ sû x0 ❧➔ ❑❚❱❈P t❤❡♦ α✳ ❑❤✐ ✤â✱ x0 ❝ô♥❣ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉
t❤❡♦ α✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x0 ❧➔ ♠ët tỹ ỳ t ừ P
õ tỗ t↕✐ x ∈ S s❛♦ ❝❤♦ f (x) ≤ f (x0 ) − α x − x0 ✳ ❇ð✐ ✈➻ P
ỗ r t ừ ✭✷✳✶✶✮ t↕✐ x0 ✱ tø ✣à♥❤ ♥❣❤➽❛
✷✳✻ ❦➨♦ t❤❡♦ x − x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✶✮✳ ❇➡♥❣ ❝→❝❤ àj = 0 ợ

j
/ J(x0 ) sỷ ử ❍➺ q✉↔ ✶✳✶ s✉② r❛ x0 ❦❤æ♥❣ t❤➸ ❧➔ ❑❚❱❈P t❤❡♦ α✳
✣✐➲✉ ✤â ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✈➔ ✣à♥❤ ỵ ữủ ự
ự tữỡ tỹ t õ t ự ữủ ỵ s