Điều kiện cần tối ưu bậc nhất cho bài toán tối ưu đa mục tiêu lipschitz

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣

✣■➋❯ ❑■➏◆ ❈❺◆❚➮■ ×❯ ❇❾❈ ◆❍❻❚
❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ✣❆ ▼Ö❈ ❚■➊❯
▲■P❙❈❍■❚❩

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣

✣■➋❯ ❑■➏◆ ❈❺◆❚➮■ ×❯ ❇❾❈ ◆❍❻❚
❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ✣❆ ▼Ö❈ ❚■➊❯
▲■P❙❈❍■❚❩
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿

❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✾



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

t tọ ỏ t ỡ s s tợ t



ữớ t tr tử tự t t ú ù ữợ
tr sốt q tr ồ t ự t õ
r q tr ự ổ tr ọ ữ s sõt
ữủ sỹ õ õ ỵ ừ t ổ
t t ồ õ 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♦♥❣ ❦❤✐ ❧➔♠ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛
❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳

❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾

❙✐♥❤ ✈✐➯♥
❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣


▼ö❝ ❧ö❝
▲í✐ ♠ð ✤➛✉

✷

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✹

✶✳✶

❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

❉÷î✐ ✈✐ ♣❤➙♥ ❈❧❛r❦❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✸

◆â♥ t✐➳♣ t✉②➳♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✹

❈→❝ ✤à♥❤ ❧þ ❧✉➙♥ ♣❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r
✷✳✶

✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣

✷✳✷

✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ✶✻

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶
✶✶

✷✸

✶



ử ừ õ tr sỹ tỗ
t ừ ỳ ữỡ ỳ tỹ sỹ r
ữỡ t tố ữ ử t

(P ) ợ r ở tự

t tự ởt r ở t ủ tũ ỵ

Min f (x) = (f1 (x), f1 (x), . . . , f1 (x))
ợ

x C, gj 0, j = 1, 2, 3, . . . , m, hk (x) = 0, k {1, 2, . . . , n},

õ số

J = {1, 2, . . . , m}

x

fi : U R, i I = {1, 2, . . . , l}



st ữỡ tr t

X h : X R, k K = {1, 2, . . . , n}



P

gj : U R, j
U

ừ ổ

rt t ỳ

C U

rữợ t ú tổ tr tố ữ t

(P )

s õ sỷ ử ỵ ữ r tố ữ

tỷ r ố

ữ ú t t q õ ởt trỏ q
trồ tr tố ữ ử t ú ữủ sỷ ử sỹ t
ừ ử t tr tố ữ rt
ởt ỹ tr ỹ t ừ ởt ử t ữủ r




t ữỡ ừ tỷ r tữỡ ự ợ õ t t
ởt tr tỷ r tữỡ ự ợ ử t ữỡ
t õ tố ữ rsr
tt tỷ r tữỡ ự ợ ử t
ữỡ t õ tố ữ rsr
r tố ữ ởt ử t õ r ở
ữủ r ở ừ t ừ
tữớ ổ ừ ừ
t tố ữ ử t
q ữủ sỷ ử ữ r tố ữ
tr õ ữủ ỹ tr t q
ừ stt t q sỹ rở ừ t q tr
ừ r ừ r ở sỹ ừ r
ở sỹ
t t õ ỗ ữỡ
ữỡ tr ởt số t q ỡ ữủ sỷ ử
tr t ở õ ỹ ỵ tờ qt
ữỡ tr tố ữ t t
tố ữ ử t ử tr ởt tố ữ t
ởt ỳ ữỡ ừ t P ỵ
t tờ qt ú ú t ữ r tố ữ
t ử tr ởt số tố ữ t ởt

ỳ tỹ sỹ r ữỡ ừ P s õ ử
ỵ r tờ qt t ữủ tố ữ
P




❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠
❈❤♦

Rl ✱

Rl

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡

l

❝❤✐➲✉✳ ❱î✐

x = (x1 , . . . , xl ), y = (y1 , . . . , yl ) ∈

t❛ sû ❞ö♥❣ ❝→❝ q✉② ÷î❝ s❛✉✿

y,

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

xi ≥ yi , i = 1, . . . , l,

x ≥ y,

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

xi

x > y,

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

xi > yi , i = 1, . . . , l.

x

yi , x = y,

❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ö❝ t✐➯✉

▼✐♥ f (x)

ð ✤â✱ ❝→❝ ❤➔♠ sè✿

❇❛♥❛❝❤
❑❤✐

fi : U → R, i ∈ I = {1, 2, . . . , l}

✈➔

❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ t➟♣ ❝♦♥ ♠ð

X ✱ hk : X → R, k ∈ K = {1, 2, . . . , n}

C = X✱

✭P✮

x ∈ C, gj (x) ≤ 0, j = 1, . . . , m, hk (x) = 0, k = 1, . . . , n,

✈î✐ r➔♥❣ ❜✉ë❝

J = {1, 2, . . . , m}

= (f1 (x), f1 (x), . . . , f1 (x))

t❤➻ ❜➔✐ t♦→♥ ✭P✮ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔

✹

gj : U → R, j ∈
U

❝õ❛ ❦❤æ♥❣ ❣✐❛♥

❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✈➔

(P1 )✳

C ⊆ U✳


D = {x ∈ X : gj (x) ≤ 0, j ∈ J}

❑➼ ❤✐➺✉

✈➔

S = D ∩ C ∩ Ch ,

ð ✤â

Ch = {x ∈ X : h(x) = 0}✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳
(P )

▼ët ✤✐➸♠

x¯ ∈ S

x∈S

♥➳✉ ❦❤æ♥❣ tç♥ t↕✐

✤÷ñ❝ ❣å✐ ❧➔

t❤ä❛ ♠➣♥

♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉

fi (x) ≤ fi (¯
x)

✈î✐ ♠å✐

fi (x) < fi (¯
x)

✈î✐ ➼t ♥❤➜t ♠ët ❝❤➾ sè i✳ ❚❛ ❝ô♥❣ ♥â✐ r➡♥❣

❤ú✉ ❤✐➺✉ ❝õ❛

f

❝õ❛ ❜➔✐ t♦→♥

(P )

❤✐➺✉ ❝õ❛

f

❈❤♦
✈➔

lin B ✱

tr➯♥

tr➯♥

B

S✳

✣✐➸♠

x¯ ∈ S

x¯ ∈ S

i = 1, . . . , l

♥➳✉ tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥

❧➔ ♠ët ♥❣❤✐➺♠

V

❝õ❛

x¯

s❛♦ ❝❤♦

x¯

❧➔ ♥❣❤✐➺♠ ❤ú✉

S∩V✳

❧➔ t➟♣ ❝♦♥ ❝õ❛

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳

X ✳ ◆❤÷ t❤÷í♥❣ ❧➺✱ t❛ ❦➼ ❤✐➺✉ ❜➡♥❣ cone B ✱ conv B

▼ët ✤✐➸♠

x¯ ∈ S

✤÷ñ❝ ❣å✐ ❧➔ ♠ët

B✳

♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü

❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ♥➳✉ ♥â ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tç♥ t↕✐

❝❤♦✱ ✈î✐ ♠é✐

✈➔

✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣

t÷ì♥❣ ù♥❣✱ ❜❛♦ ♥â♥✱ ❜❛♦ ❧ç✐ ✈➔ ❜❛♦ t✉②➳♥ t➼♥❤ ❝õ❛

●❡♦❢❢r✐♦♥

❝õ❛ ❜➔✐ t♦→♥

M >0

s❛♦

i✱
fi (x) − fi (¯
x)
≤ M,
fj (¯
x) − fj (x)

❝❤♦ ♠ët sè

◆➳✉

j

t❤ä❛ ♠➣♥

x¯ ∈ S

fj (¯
x) < fj (x)

✈î✐ ♠å✐

x∈S

♥❤÷ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ t❤➻

♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛

f

tr➯♥

S✳

x¯

✈➔

fi (¯
x) > fi (x)✳

❝ô♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❧➔ ♠ët

❈❤ó♥❣ t❛ ♥â✐ r➡♥❣

x¯ ∈ S

❧➔

♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮ ♥➳✉ tç♥ t↕✐ ♠ët
❧➙♥ ❝➟♥

V

❝õ❛

x¯

♠➔

x¯

❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛

f

tr➯♥

S∩V✳
❚ò ✤à♥❤ ♥❣❤➽❛✱ t❛ t❤➜② r➡♥❣ ♠å✐ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❧➔
♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ▼ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤ú✉ ❤✐➺✉ t❤ü❝ sü
✤÷ñ❝ ❣å✐ ❧➔ ♠ët

♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤æ♥❣ t❤ü❝ sü

✭✐♠♣r♦♣❡r❧② ❡❢❢✐❝✐❡♥t✮✳ ◆❤÷ ✈➟②✱ ♠ët ✤✐➸♠

✺

x¯ ∈ X

✭t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●❡♦❢❢r✐♦♥✮

❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤æ♥❣


tỹ sỹ ừ t P ợ ồ

fi (
x) > fi (x)

M > 0

tỗ t

xX



iI

s



fi (x) fi (
x)
>M
fj (
x) fj (x)
ợ tt

j

tọ

fj (
x) < fj (x).

ữợ r

U



F

ởt st ữỡ tr ởt t

ừ ởt ổ



X



x U

t

s rở t r ừ

F

t

x

ữủ

F (x + tv) F (x)
, v X.
+
t
(x,t)(
x,0 )

F (
x, v) = lim sup



ữợ r ừ

F

t

x

C F (
x) = { Rp : , v F (
x, v), v Rp },

õ

X = Rp




, v

t t ổ ữợ ừ

v F (
x, v)



F (
x, .)(0)





ổ

Y

õ

F

v

tr

Rp

t t ữỡ ữợ ở t

ữợ ừ õ t t ỗ t




v=0

tỗ t ữủ

C F (
x) = F (
x, .)(0)

ởt tứ ổ

ú t õ r

ởt t tỷ t t tử tứ

X





F
Y

t t

x

ữủ ỵ

X

ởt

tỗ t

F (
x)

s


ợ ộ

v X

t õ

F (x + tv) F (x)
= F (
x)(v)
t
(x,t)(
x,0+ )
lim

sỹ ở tử t

v

tỹ ở tọ

tr t ủ t ố ũ

F

st

x

F t t x t F st x

õ t t

t

xS

1, 2, . . . ,

s

ởt ữợ

d

ữủ ồ

tỗ t

k 0

ữợ t t

xk S

ừ t

ổ ữợ

SX

k > 0, k =

xk x
.
k
k

d = lim

ứ t s r r

xk x

tr t ợ

tr ổ tỗ t

ờ S X x S TS (x) ừ tt ữợ t t
ừ S t x ởt õ õ
ự

sỷ

d TS (x)

ợ ồ

>0

t õ

xk x
,
d = lim
k0 (k /)


xk



k /

tọ ợ ữợ

d

õ

TS (x)

ởt õ


dj

k = 1, 2, . . . ,

ởt ữợ t t ừ

tọ



S

t

x



limj dj = d.



dj

xj,k



j,k

ữợ t


t ợ ồ

j

tỗ t

k(j)

s

xj,k(j) x
dj dj d .
j,k(j)
õ

xj,k(j) x
d 2 dj d .
j,k(j)
õ t r
ợ ữợ

d

xj,k(j)

õ



TS (x)

j,k(j) j = 1, 2, . . .

tọ

õ

õ t t õ trỏ q trồ tr tố ữ
t tố ữ t õ õ t t õ t ổ
ỗ tr ởt số trữớ ủ t õ ỵ tr ự
ử õ ỗ

ờ S X t ỗ x S õ
TS (x) = cl cone(S x).

ỵ
ỵ ỵ t tờ qt f1 , . . . , fl , g1 , . . . , gm
ữợ t t tứ Rp R h1 , . . . , hn t t
tứ Rp R ữủ hk (v) = ck , v , k K = {1, . . . , n} t
s


fi (x) < 0, i I = {1, . . . , l}, gj (x) 0, j J = {1, . . . , m}, hk (x) =

0, k K ổ õ x Rp






ỗ t (u, v, w) Rl ì Rm ì Rn s u 0, v
l

0

m

ui fi (0) +
i=1



n

w k ck .

vj gj (0) +
j=1

k=1

ỗ t (u, v, w) Rl ì Rm ì Rn s u 0, v
l

m

ui fi (d) +
i=1

0

0

n

wk hk (d) 0, d Rp

vj gj (d) +
j=1



k=1

õ (b) (c) tữỡ ữỡ ỏ (c) s r (a) ỡ ỳ õ
m

B = cone conv(

gj (0)) + lin{ck : k K}

j=1

õ t (a) s r (c) tr tữỡ ữỡ

ỵ ỵ r tờ qt f1 , . . . , fl , g1 , . . . , gm
ữợ t t tứ Rp R h1 , . . . , hn t t tứ

Rp R ữủ hk (v) = ck , v , k K = {1, . . . , n} sỷ r ợ
ộ i I = {1, . . . , l} õ
l

m

Bi = cone conv(

gj (0)) + lin{ck : k K}

fj (0)) + cone conv(
j=1

j=1,j=i

õ õ s tữỡ ữỡ


fi (x) 0, i I, fi (x) < 0 ợ t t ởt i gj (x) 0, j J, hk (x) =

0, k K ổ õ x Rp


ỗ t (u, v, w) Rl ì Rm ì Rn s u > 0, v
l

0

m

ui fi (0) +
i=1

n

vj gj (0) +
j=1



0
w k ck .

k=1


◆❤➟♥ ①➨t ✶✳✶✳ ▼➺♥❤ ✤➲ ✭❜✮ tr♦♥❣ ✣à♥❤ ❧þ ✶✳✷ t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ♠➺♥❤ ✤➲ s❛✉✿
✏❚ç♥ t↕✐

(u, v, w) ∈ Rl × Rm × Rn

l

m

ui fi (d) +
i=1

s❛♦ ❝❤♦

0

✈➔

n

wk hk (d) ≥ 0, ∀d ∈ Rp .

vj gj (d) +
j=1

u > 0, v

k=1

✶✵


❈❤÷ì♥❣ ✷
✣✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉
❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r

✷✳✶ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣
❚r♦♥❣ ♠ö❝ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝õ❛
♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ q✉↔ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ✈î✐ ❝→❝ r➔♥❣ ❜✉ë❝ ❜➜t
✤➥♥❣ t❤ù❝✱ ✤➥♥❣ t❤ù❝ ✈➔ ♠ët r➔♥❣ ❜✉ë❝ t➟♣ tò② þ✳
❇ê ✤➲ s❛✉ ✤➙② ❧➔ ❝➛♥ t❤✐➳t tr♦♥❣ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❦➳t q✉↔ ❝❤➼♥❤✳

❇ê ✤➲ ✷✳✶✳ ❈❤♦ F : U → R ❧➔ ♠ët ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ t➟♣ ❝♦♥
♠ð U ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✳ ●✐↔ sû r➡♥❣✿
✭✐✮

✭✐✐✮

✭✐✐✐✮

xn → x¯✱
F (xn ) ≥ F (¯
x)✱
1
(xn
n→+∞ tn

u = lim

− x¯) ✈î✐ tn → 0, tn > 0.

❑❤✐ ✤â✱ F ◦ (¯
x; u) ≥ 0.
❈❤ù♥❣ ♠✐♥❤✳

✣➦t

un =

1
tn (xn

− x¯)✱

❤❛② ❧➔

✶✶

xn = x¯ + tn un ✳

❉♦

F

❧➔ ▲✐♣s❝❤✐t③


x¯

✤à❛ ♣❤÷ì♥❣ t↕✐

✈➔

lim (¯
x + tn un ) = lim (¯
x + tn u) = x¯,

n→∞

tç♥ t↕✐

L

0

✈➔

n0 ∈ N

n→∞

s❛♦ ❝❤♦

|F (¯
x + tn un ) − F (¯
x + tn u)|

Ltn uk − u

❢♦r ❛❧❧

n

n0 .

❱➻ ✈➟②✱

F (¯
x + tn un ) − F (¯
x)

0

= [F (¯
x + tn un ) − F (¯
x + tn u)] + [F (¯
x + tn u) − F (¯
x)]

Ltk un − u + F (¯
x + tk u) − F (¯
x)
✈î✐ ♠å✐

n

n0 ✳

✣✐➲✉ ✤â ❦➨♦ t❤❡♦

0

lim L un − u + lim sup

n→∞

n→∞

lim sup
x→¯
x

F (¯
x + tn u) − F (¯
x)
tn

F (x + tu) − F (x)
.

t

t↓0

❉♦ ✈➟②✱ t❛ ❝â

F ◦ (¯
x, u)

❱î✐ ♠é✐ ✤✐➸♠

0✳

x¯ ∈ D✱

❣å✐

J(¯
x)

❧➔ t➟♣ ❤ñ♣ ❝→❝ r➔♥❣ ❜✉ë❝ ❤♦↕t t↕✐

✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

J(¯
x) = {j ∈ {1, 2, . . . , m} : gj (¯
x) = 0}.
❱î✐ ♠é✐

i ∈ I✱

t❛ ✤➦t

M i = {x ∈ X : gj (x) ≤ 0, ∀j ∈ J, fi (x) ≤ fj (¯
x)}
✶✷

x

✈➔


✈➔

M i.

M=
i∈I
◆â♥ t✉②➳♥ t➼♥❤ ❤â❛ ❝õ❛

Mi

t↕✐

x¯ ∈ M i

✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

L(M i ; x¯) = {d ∈ X : fi◦ (¯
x; d) ≤ 0, i = 1, 2, . . . , l, gj◦ (¯

x; d) ≤ 0, j ∈ J(¯
x)},
✈➔ ♥â♥ t✉②➳♥ t➼♥❤ ❤â❛ ❝õ❛ ▼ t↕✐

x¯ ∈ M

✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

L(M ; x¯) = {d ∈ X : fi◦ (¯
x; d) ≤ 0, i = 1, 2, . . . , l, gj◦ (¯
x; d) ≤ 0, j ∈ J(¯
x)}.
✣à♥❤ ❧þ s❛✉ ❝❤♦ t❛ ♠ët ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r
②➳✉ ✭❲❑❑❚✮ ❝❤♦ ♠ët ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳

✣à♥❤ ❧þ ✷✳✶✳ ❳➨t ❜➔✐ t♦→♥

✭P✮✳

❈❤♦ x¯ ∈ S = D ∩ C ∩ Ch ✳ ●✐↔ sû fi , i ∈ I ✱

✈➔ gj , j ∈ J ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U ✈➔ h : X → Rn ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t
t↕✐ x¯ ✈î✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t h (¯
x)✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② (GRC) s❛✉

L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯
x) ⊆ cl conv T (M i ∩ C ∩ Ch , x¯), ✈î✐ ➼t ♥❤➜t ♠ët i,
✭✷✳✶✮

✤ó♥❣ ✈î✐ i0 ∈ I ✱ ð ✤â Ker h (¯

x) = {d ∈ X : h (¯
x)(d) = 0}✳ ●✐↔ sû r➡♥❣ ❝→❝
❤➔♠ sè fi , i ∈ I \ {i0 } ❦❤↔ ✈✐ ❝❤➦t t↕✐ x¯✳ ◆➳✉ x¯ ∈ S ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛
♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥

✭P✮✱

t❤➻ ❤➺

fi◦ (¯
x; d) < 0, i = 1, 2, . . . , l,

✭✷✳✷✮

gj◦ (¯
x; d) ≤ 0, j ∈ J(¯
x),

✭✷✳✸✮

h (¯
x)(d) = 0,

✭✷✳✹✮

✶✸


d ∈ T (C, x¯),

✭✷✳✺✮

❦❤æ♥❣ ❝â ♥❣❤✐➺♠ d ∈ X ✳
❈❤ù♥❣ ♠✐♥❤✳

❉♦

x¯ ∈ S

❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✱ ♥➯♥

V

tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥

❝õ❛

x¯

x¯

s❛♦ ❝❤♦

f

tr➯♥

❉♦ ✤â✱

d ∈

❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛

V ∩ S✳
d ∈ X✳

●✐↔ sû ♥❣÷ñ❝ ❧↕✐ r➡♥❣ ❤➺ ✭✷✳✷✮✕✭✷✳✺✮ ❝â ♥❣❤✐➺♠

L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯
x)
Ch , x¯)✱

✈➔ tø ✭✷✳✶✮✱ t❛ ❝â

✈➔ ❞♦ ✤â✱ tç♥ t↕✐ ♠ët ❞➣②

d ∈ cl conv T (M i0 ∩ C ∩

{dim0 } ⊆ conv T (M i0 ∩ C ∩ Ch , x¯)

s❛♦ ❝❤♦

lim dim0 = d.

m→+∞

❱î✐ ♠é✐

dim0 ∈ conv T (M i0 ∩ C ∩ Ch , x¯), m = 1, 2, . . .✱

❝õ❛ ❜❛♦ ❧ç✐✱ tç♥ t↕✐

i0
i0 i0
, dm,k ∈ T (M i0 ∩ C ∩ Ch , x¯), k = 1, 2, . . . , Km
Km

i
Km0

i
Km0

i0
i0
k=1 λm,k dm,k ,

dim0 =

❚ø

i0
k=1 λm,k

✈➔

xnm,k,i0 ∈ M i0 ∩ C ∩ Ch ∩ V
✣✐➲✉ ✤â ❦➨♦ t❤❡♦✱

n✳

✈î✐ ♠å✐

✈î✐ ♠å✐

❦❤✐

n

p
fr (xm,k,i
) ≥ fr (¯
x)✱
0

{dnm,k,i0 }n

n → ∞

{tnm,k,i0 }n

✈➔

s❛♦ ❝❤♦

✈î✐

xnm,k,i0 = x¯ +

N0 ∈ N ✱

❑❤✐ ✤â✱ tç♥ t↕✐

s❛♦ ❝❤♦

n ≥ N0 ✳

fi0 (xnm,k,i0 ) − fi0 (¯
x) ≤ 0✱

❈â t❤➸ ❝❤➾ r❛ r➡♥❣ tç♥ t↕✐
s❛♦ ❝❤♦

tç♥ t↕✐ ❞➣②

tnm,k,i0 > 0, tnm,k,i0 → 0✱

tnm,k,i0 dnm,k,i0 ∈ M i0 ∩ C ∩ Ch

s❛♦ ❝❤♦

0
= 1, λim,k
≥ 0.

0
dim,k
∈ T (M i0 ∩ C ∩ Ch , x¯)✱

0

dnm,k,i0 → dim,k

t❤❡♦ ✤à♥❤ ♥❣❤➽❛

r = i0

✈î✐ ♠å✐

✈î✐ ♠å✐

✈➔ ♠ët ❞➣② ❝♦♥

np

n ≥ N0 ✳
n

p
{xm,k,i
}
0

❝õ❛

{xnm,k,i0 }

✤õ ❧î♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✱ t❛

sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ♣❤↔♥ ❝❤ù♥❣✳ ◆➳✉ ✤✐➲✉ ♥➔② ❦❤æ♥❣ ✤ó♥❣✱
t❤➻ ✈î✐ ♠å✐

✣➦t

r = i0 ✱

tç♥ t↕✐

Nr

s❛♦ ❝❤♦ ✈î✐ ♠å✐

˜ = max{N0 , Nr : r = i0 }✳
N

˜✱
fr (¯
x), ∀n ≥ N

✈➔ tø ✤â t❛ ❝â

✈î✐ ❣✐↔ t❤✐➳t ❧➔ ✤✐➸♠

x¯

n ≥ Nr , fr (xnm,k,i0 ) < fr (¯
x)✳

❑❤✐ ✤â✱ ✈î✐ ♠å✐

r = i0 ✱

fr (xnm,k,i0 ) <

t❛ ❝â

˜ ≥ N0 ✱
fi0 (xnm,k,i0 ) ≤ fi0 (¯
x), ∀n ≥ N

❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛

❉♦ ✤â✱ ♠➺♥❤ ✤➲ ❧➔ ✤ó♥❣ ✈➔ tç♥ t↕✐

r = i0 ✱
✶✹

f

tr➯♥

S ∩V

✭✈î✐

✈➔ ♠ët ❞➣② ❝♦♥

♠➙✉ t❤✉➝♥

˜

x = xN

m,k,i0 ✮✳
n

p
{xm,k,i
}
0

❝õ❛


{xnm,k,i0 }

s

n

p
fr (xm,k,i
) fr (
x)
0

ợ ồ

np

ừ ợ

õ tứ ờ t t ừ số

0
0
fr (
x)(dim,k
) = fr (
x; dim,k
) 0

fr

t

t t ừ số

x

t õ

fr x

t

õ
i

i

Km0

Km0

0
0
im,k
dim,k

fr (
x)(dim0 ) = fr (
x)
k=1
õ

fr (
x; d) = fr (
x)(d) 0,

0
0
im,k
fr (
x)(dim,k
)0

=
k=1

t ợ õ

ổ õ

t t ỹ tố ữ

t

(P1 )

ổ õ r ở t tự

(W KKT ) ố

C = X

ỵ t t (P1 ) ợ X = Rp , p 1 x S = D Ch
õ số h : Rp Rn rt t x S. fi , i I
gj , j J st ữỡ tr U sỷ r q

(GRC) ú ợ i0 I tự
L(M ; x) Ker h (
x) cl conv T (M i0 Ch , x).
sỷ r fi , i I \ {i0 } t t x õ B õ
õ

B = cone conv(m
x)) + lin{hk (
x) : k K}.
j=1 C gj (
x S ởt ỳ ữỡ ừ t (P1 ) t tỗ t
tỡ u = (u1 , . . . , ul ) Rl , v = (v1 , . . . , vm ) Rm w = (w1, . . . , wn) Rn
s u 0, v

0
vj gj (

x) = 0, j = 1, 2, . . . , m, ,





l

0

m

ui C fi (
x) +
i=1

n

wk hk (
x),

vj C gj (
x) +
j=1



k=1

tữỡ ữỡ

l

m

ui fi (
x, d)
i=1

ự

n

vj gj (
x, d)

+
j=1



x

wk hk (
x)(d) 0, d Rp ,

+
k=1

ởt ỳ ữỡ ừ t

t ỵ

ổ õ

t tờ qt ỵ tỗ t

d X

ỵ

u = (u1 , . . . , ul ) Rl , u 0

vj R, vj 0, j J(
x), w = (w1 , . . . , wn ) Rn



(P1 )

fi (
x; d) < 0, i = 1, . . . , l, gj (
x; d) 0, j J(
x),

hk (
x)(d) = 0, k = 1, 2, . . . , n

s ú

vj = 0, j
/ J(
x) gj (
x) = 0 ợ j J(
x) t õ

ỵ tữỡ ữỡ

ỳ tỹ sỹ
r ữỡ
r ử ú tổ s ữ r tố ữ t ởt
ỳ tỹ sỹ r ữỡ ừ t P ợ
r ở t tự tự r ở t r ú tổ
ụ tr ố

(SKKT )



ỳ tỹ sỹ r ữỡ ừ t P tr trữớ ủ
ổ õ r ở t

ỵ t t

P

x S = D C Ch sỷ

số fi , i I gj , j J st ữỡ tr U h



rt t x sỷ r q (GARC) s ú t x
l

T (M i C Ch , x),

L(M ; x) T (C, x) Ker h (
x)
i=1

x S ởt ỳ tỹ sỹ r ữỡ ừ t
P

õ

fi (
x; d) 0, i = 1, 2, . . . , l, i0 {1, 2, . . . , l} s fi0 (
x; d) < 0



gj (
x; d) 0, j J(
x),

h (
x)(d) = 0,



d T (C, x),



ổ õ d X
ự



x ởt ỳ tỹ sỹ r ữỡ ừ

t P tỗ t ởt

f

ỳ tỹ sỹ r ừ

tr

V

ừ

x

s

x

ởt ởt

V S

sỷ ự õ

d X

ổ t t

tờ qt ú t õ t sỷ

f1 (
x; d) < 0,
fi (
x; d) 0, i = 2, . . . , l.



ứ t ữủ

d L(M ; x) T (C, x) Ker h (
x)

d T (M i C Ch , x)

iI

{dni }n M i C Ch

ợ ồ



{tni }n



s



(GARC)

õ

ú ứ õ tỗ t

tni > 0, tni 0



dni d

ợ

n


✈➔

xni = x¯ + dni tni ∈ M i ∩ C ∩ Ch , n ≥ 1✳

❑❤✐ ✤â✱ tç♥ t↕✐

no ∈ N✱

s❛♦ ❝❤♦

xni ∈ M i ∩ C ∩ Ch ∩ V, n ≥ no ✳
i0 ∈ {2, 3, . . . , l}✳

❈è ✤à♥❤

Mi0 ✱

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛
❱î✐ ♠å✐

n ≥ n0 ✱

✈î✐

i0

❱î✐

i0

♥➔②✱ ❧➜② ❞➣② t÷ì♥❣ ù♥❣

✤â✱ t❛ ❝â

xni0

♥❤÷ tr➯♥✳

fi0 (xni0 ) − fi0 (¯
x) ≤ 0✳

t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣

An = {k ≥ 2 : fk (xni0 ) > fk (¯
x)} ⊂ {2, 3, . . . , l}.
❚❛ ❦❤➥♥❣ ✤à♥❤

An = ∅✱

An = ∅

t❤➻ ✈î✐ ♠å✐

✈î✐ ♠å✐

k≥2

n ≥ n0 ✳

❚❤➟t ✈➟②✱ ♥➳✉ tç♥ t↕✐

✭✤➦❝ ❜✐➺t✱ ✈î✐

k = i0 ✮

n ≥ n0

s❛♦ ❝❤♦

t❛ ❝â

fk (xni0 ) ≤ fk (¯
x).

✭✷✳✶✹✮

❚ø

f1 (¯
x + tni0 dni0 ) − f1 (¯
x)
f1 (¯
x + tni0 d) − f1 (¯
x)
lim sup
≤

lim
sup
n→∞
n→∞
tni0
tni0
f1 (¯
x + tni0 dni0 ) − f1 (¯
x + tni0 d)
+ lim sup
n→∞
tni0
≤ f1◦ (¯
x; d) + lim sup L1 dni0 − d
n→∞

= f1◦ (¯
x; d) < 0,
tç♥ t↕✐

ni0 ≥ n0

s❛♦ ❝❤♦

f1 (¯
x + tni0 dni0 ) − f1 (¯
x) < 0,
✈î✐ ♠å✐

n ≥ ni0 ✱

ð ✤â

L1 > 0

❧➔ ❤➺ sè ▲✐♣s❝❤✐t③ ❝õ❛

✭✷✳✶✺✮

f1 ✳

❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ✭✷✳✶✺✮ ♠➙✉ t❤✉➝♥ ✈î✐ t➼♥❤ ❤ú✉ ❤✐➺✉ ❝õ❛
tr➯♥

V ∩S

✈➔ ❞♦ ✤â

An = ∅

✈î✐ ♠å✐

n ≥ n0
✶✽

✳

x¯

❇➙② ❣✐í t❛ ❝è ✤à♥❤

{xni0r }nr

❝õ❛

{xni0 }n≥n0

k¯ ∈ {2, 3, . . . , l}

s❛♦ ❝❤♦

fk¯ (xni0r ) > fk¯ (¯
x) ❝❤♦ t➜t ❝↔ nr ✱ tù❝ ❧➔✱ k¯ ∈ Anr =

{k ∈ {2, 3, . . . , l} : fk (xni0r ) > fk (¯
x)}
❚❤❡♦ ❇ê ✤➲ ✷✳✶✱ t❛ ❝â

✳ ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ①➨t ❞➣② ❝♦♥

✱ ✈î✐ ♠å✐

fk¯◦ (¯
x, d) ≥ 0✳

nr ≥ n0 ✳

❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈➔ ✭✷✳✶✸✮ s✉② r❛

fk¯◦ (¯
x, d) = 0.
❚ø ❝→❝ ❤➔♠ sè

fi , i ∈ I

❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥

U✱

fi (¯
x + tni0 dni0 ) − fi (¯
x + tni0 d)
= 0, ✈î✐
lim
n→∞
tni0

t❛ ❝â

♠å✐

i∈I

✈➻

fi (¯
x + tni0 dni0 ) − fi (¯
x + tni0 d)
lim

≤ lim Li dni0 − d = 0
n
n→∞
n→∞
ti0
ð ✤â

Li > 0

❧➔ ❤➺ sè ▲✐♣s❝❤✐t③ ❝õ❛

fi ✳

❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ❝â

fi (¯
x + tni0 dni0 ) − fi (¯
x)
fi (¯
x + tni0 d) − fi (¯
x)
lim sup
≤
lim
sup
n→∞
n→∞
tni0
tni0
fi (¯

x + tni0 dni0 ) − fi (¯
x + tni0 d)
+ lim sup
n→∞
tni0
≤ fi◦ (¯
x; d), ∀i ∈ I.
❚ø

f1◦ (¯
x, d) < 0✱

t❛ t❤✉ ✤÷ñ❝

f1 (¯
x) − f1 (xni0 )
lim inf
> 0,
n→∞
tni0
✈➔ ❞♦ ✤â✱

✣è✐ ✈î✐ ❝❤➾ sè

f1 (¯
x) − f1 (xni0r )
> 0.
lim inf
nr →∞
tni0r

k¯

❝è ✤à♥❤ tr÷î❝ ✤â t❛ ❦✐➸♠ tr❛

✶✾

k¯ ∈ Anr ✱

✈î✐ ♠å✐

nr ≥ n0 ✱


❝❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t

fk¯ (xni0r ) − fk¯ (¯
fk¯ (xni0r ) − fk¯ (¯
x)
x)
0 ≤ lim inf
≤
lim
sup
≤ fk¯◦ (¯
x; d) = 0,
nr
nr
nr →∞
n
→∞

ti0
ti0
r
❞♦

fk¯ (xni0r ) > fk¯ (¯
x)

✈➔

tni0r > 0✳

❱➻ ✈➟②✱

fk¯ (xni0r ) − fk¯ (¯
x)
lim
= 0.
nr
nr →∞
ti0
✣✐➲✉ ✤â ❦➨♦ t❤❡♦

f1 (xni0r )

f1 (¯
x) −
= lim
n
nr →∞ fk¯ (x r ) − fk¯ (¯

x) nr →∞
i0
lim

♠➙✉ t❤✉➝♥ ✈î✐ ❣✐↔ t❤✐➳t r➡♥❣

f

tr➯♥

V ∩ S✳

f1 (¯
x)−f1 (xni0r )
tni0r
x)
fk¯ (xni0r )−fk¯ (¯
nr
ti0

= +∞,

x¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛

❉♦ ✤â✱ ❤➺ ✭✷✳✾✮✕ ✭✷✳✶✷✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✳

❱➼ ❞ö ✷✳✶✳ ❳➨t ❜➔✐ t♦→♥ Min f (x) = (|x1 |, x2 , x1 +x42 )✱ ð ✤â x = (x1 , x2 ) ∈ R2 ✳
❍➔♠

f = (f1 , f2 , f3 )

✈✐ ❧✐➯♥ tö❝ t↕✐

❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥

x¯ = (0, 0)✱

R2

✳ ❈→❝ ❤➔♠

✈➻ ✈➟② ❝❤ó♥❣ ❦❤↔ ✈✐ ❝❤➦t t↕✐

x¯ = (0, 0)✳

♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ❚❛ ❝â

M 1 = {(0, x2 ) : x2 ∈ R},
M 2 = {(x1 , x2 ) ∈ R2 : x2 ≤ 0},
M 3 = {(x1 , x2 ) ∈ R2 : x1 + x42 ≤ 0}.

✷✵

f2

✈➔

f3

✣✐➸♠

❦❤↔

x¯

❧➔


õ

T (M 1 , x) = {(0, d2 ) : d2 R},
T (M 2 , x) = {(d1 , d2 ) R2 : d2 0},
T (M3 , x) = {d = (d1 , d2 ) R2 : d1 0}


3

T (M i , x) = {(0, d2 ) : d2 0}.
i=1
õ s r r

(GRC)

i = 1

ú ợ

x; d) 0}, i = {1, 2, 3} = {(0, d2 ) : d2 0}
R2 : fi (

L(M ; x) = {d

õ

L(M, x)

cl conv T (M 1 , x)
t ừ ỵ ữủ tọ ú t õ t
t r

0, d1 < 0

x; d) < 0, i = 1, 2, 3,
fi (

ổ õ

d R2

L(M ; x) = 3i=1 T (M i , x)

ỗ t


x

d R2 (d = (0, d2 ), d2 < 0)

x; d) = 0
f3 (

|d1 | < 0, d2 <

õ tố ữ ừ ỵ

ữủ tọ t ỳ
ứ

ừ ỵ

(GARC)

s

ữủ tọ

x; d) < 0
x; d) = 0, f2 (
f1 (

ừ ỵ õ ởt

d R2

t ố ợ ỳ tỹ sỹ r ữỡ
ổ ữủ tọ õ

x

ổ ởt ỳ tỹ

sỹ r ữỡ ừ t t ụ õ t ữủ
tr trỹ t t

f2 (x)f2 (
x)
f3 (
x)f3 (x)

=



a
a4

x

=

1
a3 t tợ

+

x = (0, a), a > 0
ợ

t r

a 0, a > 0

ởt ỳ ữ ỳ

ữỡ ừ ỵ tr [] ổ tọ



fi (
x; d)