Điều kiện tối ưu cấp cao cho cực tiểu địa phương chặt và cực tiểu pareto địa phương chặt

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ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM

PHẠM QUỲNH TRANG

ĐIỀU KIỆN TỐI ƯU CẤP CAO
CHO CỰC TIỂU ĐỊA PHƯƠNG CHẶT
VÀ CỰC TIỂU PARETO ĐỊA PHƯƠNG CHẶT

LUẬN VĂN THẠC SĨ TOÁN HỌC

Thái Nguyên - Năm 2015
Số hóa bởi Trung tâm Học liệu - ĐHTN

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✐

▲ê✐ ❝❛♠ ➤♦❛♥
❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr✉♥❣
t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ❝➳❝ ➤Ị t➭✐ ❦❤➳❝✳ ❚➠✐ ❝ị♥❣ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ♠ä✐ sù
❣✐ó♣ ➤ì ❝❤♦ ✈✐Ư❝ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❝➯♠ ➡♥ ✈➭ ❝➳❝ t❤➠♥❣ t✐♥ trÝ❝❤
❞➱♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ ❝❤Ø râ ♥❣✉å♥ ❣è❝✳
❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺
◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥

P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣

✐✐

▲ê✐ ❝➯♠ ➡♥
▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ✈➭ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝
❚❤➳✐ ◆❣✉②➟♥ ớ sự ớ ọ ủ P ỗ ❱➝♥ ▲➢✉✳ ◗✉❛ ➤➞②✱
t➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬② ❣✐➳♦✱ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛
❤ä❝ ủ ì P ỗ ờ t t×♥❤ ❤➢í♥❣ ❞➱♥ tr♦♥❣ s✉èt
q✉➳ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ t➳❝ ❣✐➯✳ ➜å♥❣ t❤ê✐ t➳❝ ❣✐➯ ❝ò♥❣ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝
t❤➬② ❝➠ tr♦♥❣ ❦❤♦❛ ❚♦➳♥✱ ❦❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠✱ ➜➵✐ ❤ä❝
❚❤➳✐ ◆❣✉②➟♥✱ ➤➲ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ ➤Ĩ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➳❝
❣✐➯ ❝ị♥❣ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ➤Õ♥ ❣✐❛ ➤×♥❤ ✈➭ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✷✶❇✱
➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥✳
▲✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ó tr➳♥❤ ❦❤á✐ ữ tế sót t rt ợ
sự ỉ ❜➯♦ t❐♥ t×♥❤ ❝đ❛ ❝➳❝ t❤➬② ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ö♣✳
❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺
◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥

P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣


✐✐✐

▼ơ❝ ❧ơ❝

▲ê✐ ❝❛♠ ➤♦❛♥

✐

▲ê✐ ❝➯♠ ➡♥

✐✐

▼ơ❝ ❧ơ❝

✐✐✐

▼ë ➤➬✉

✶

✶

➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❲❛r❞

✸

✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✷ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

✻

✶✳✸ ❍➭♠

✷

C 1,1

m

✳ ✳ ✳ ✳ ✳ ✳ ✳

✈➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛
❘❛❤♠♦✲❙t✉❞♥✐❛rs❦✐

✷✷

✷✳✶ ❈➳❝ ❦Õt q✉➯ ❜ỉ trỵ
✷✳✷ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✷✳✸ ➜✐Ị✉ ❦✐Ư♥ ➤ñ tè✐ ➢✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✷✳✹

➜➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t

❑Õt ❧✉❐♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾


✶

▼ë ➤➬✉

✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❧✉❐♥ ✈➝♥

▲ý t❤✉②Õt ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❧➭ ♠ét ❜é ♣❤❐♥ q✉❛♥ trä♥❣ ❝đ❛ ❧ý t❤✉②Õt tè✐
➢✉ ❤ã❛✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ột é t ị ợ t ❝➳❝ ➤✐Ĩ♠
❞õ♥❣✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ é t tì r ợ ệ tố
tr t ❝➳❝ ➤✐Ĩ♠ ❞õ♥❣ ➤ã✳ ❑❤➳✐ ♥✐Ư♠ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ t

m

ợ

ị ĩ ở r ề ệ tố ➢✉ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣

m ➤➢ỵ❝ t❤✐Õt ❧❐♣ ❜ë✐ ❆✉s❧❡♥❞❡r ❬✶❪✱ ❙t✉❞♥✐❛rs❦✐ ❬✶✷❪✱ ❉✳❱✳ ▲✉✉ ❬✶✵❪✱ ❲❛r❞ ❬✶✹❪✳
❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛
♣❤➢➡♥❣ ❝❤➷t ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦✳ ❊✳❉✳ ❘❛❤♠♦ ✲ ▼✳
❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮ ➤➲ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ➤➵♦ ❤➭♠ ❙t✉❞♥✐❛rs❦✐ ➤➢❛ r❛ ✶✾✽✻ ❝❤♦
❤➭♠ ✈Ð❝t➡ ✈➭ ❞➱♥ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣
❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ tr♦♥❣ ữ ề ề
t ợ ề t➳❝ ❣✐➯ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ❈❤Ý♥❤ ✈× ✈❐②
❡♠ ❝❤ä♥ ➤Ị t➭✐✿ ✧➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝
t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳✧
✷✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉

❙➢✉ t➬♠ ✈➭ ➤ä❝ t➭✐ ❧✐Ö✉ tõ ❝➳❝ s➳❝❤✱ t➵♣ ❝❤Ý t♦➳♥ ❤ä❝ tr♦♥❣ ♥➢í❝ ✈➭ q✉è❝ tÕ
❧✐➟♥ q✉❛♥ ➤Õ♥ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦✳ ◗✉❛ ➤ã✱ t×♠ ❤✐Ĩ✉ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ✈✃♥
➤Ị ♥➭②✳
✸✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥

▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ t×♠ ❤✐Ĩ✉ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝


✷
t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳ ❈ơ t❤Ĩ✱ ❝❤ó♥❣ t➠✐ ➤ä❝
❤✐Ĩ✉ ✈➭ tr×♥❤ ❜➭② ❧➵✐ ♠ét ❝➳❝❤ t➢ê♥❣ ♠✐♥❤ ❤❛✐ ❜➭✐ ❜➳♦ s❛✉✿
✶✳ ❉✳❊✳ ❲❛r❞✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ str✐❝t ❧♦❝❛❧ ♠✐♥✐♠❛ ❛♥❞ ♥❡❝❡ss❛r② ❝♦♥❞✐✲
t✐♦♥s ❢♦r ✇❡❛❦ s❤❛r♣ ♠✐♥✐♠❛✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✵✭✶✾✾✹✮✱ ✺✺✶✲✺✼✶✳
✷✳ ❊✳❉✳ ❘❛❤♠♦✱ ▼✳ ❙t✉❞♥✐❛rs❦✐✱ ❍✐❣❤❡r ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r str✐❝t ❧♦❝❛❧ P❛r❡t♦
♠✐♥✐♠❛ ✐♥ t❡r♠s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏✳ ▼❛t❤✳
❆♥❛❧✳ ❆♣♣❧✳ ✸✾✸✭✷✵✶✷✮✱ ✷✶✷✲✷✷✶✳
✹✳ ◆é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥

▲✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ♣❤➬♥ ♠ë ➤➬✉✱ ✷ ❝❤➢➡♥❣✱ ❦Õt ❧✉❐♥ ✈➭ ❞❛♥❤ ♠ơ❝ ❝➳❝ t➭✐ ❧✐Ư✉
t❤❛♠ ❦❤➯♦
❈❤➢➡♥❣ ✶✿ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ự tể ị t ủ r

rì ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ tể ị t ủ r
ớ ữ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ❦❤➳❝ ♥❤❛✉✳ ❱í✐ ➤✐Ị✉ ❦✐Ư♥
❝❤Ý♥❤ q✉②✱ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝✃♣ ❝❛♦ trë t❤➭♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝
t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦✳
❈❤➢➡♥❣ ✷✿ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ự tể Prt ị t ủ
trs

rì ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣

m ❝❤♦ ❤➭♠

✈❡❝t➡ ✈➭ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣
❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐ ✭❬✶✵❪✱ ✷✵✶✷✮✳

m


✸

❈❤➢➡♥❣ ✶

➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ị
t ủ r

r ú t trì ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉
➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦ ❝đ❛ ❲❛r❞ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ➤➵♦ ❤➭♠ ❝✃♣ ❝❛♦ t❤❡♦
♣❤➢➡♥❣ ❦❤➳❝ ♥❤❛✉✳ ❱í✐ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ trë t❤➭♥❤ ➤✐Ị✉ ❦✐Ư♥
➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ó✉ ❝❤➷t ❝✃♣ ❝❛♦✳ ❈➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ❧➭
❝đ❛ ❲❛r❞ ❬✶✹❪✳

✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛

❳Ðt ❜➭✐ t♦➳♥ tè✐ ➢✉ s❛✉✿
✭✶✳✶✮

min {f (x) |x ∈ S } ,
tr♦♥❣ ➤ã

f ✿ Rn → R {+} S

ột t rỗ tr

Rn

ị ♥❣❤Ü❛ ✶✳✶✳✶

❈❤♦

·

❧➭ ❝❤✉➮♥ ➙❝❧✐t tr♦♥❣

Rn ✳ ❱í✐ ε > 0✱ ➤➷t

B (x, ε) := {y ∈ Rn | y − x ≤ ε} .
✭❛✮ ❚❛ ♥ã✐ r➺♥❣
tå♥ t➵✐

x¯ ∈ S

❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥

ε > 0 s❛♦ ❝❤♦
f (x) > f (¯
x) (∀x ∈ S ∩ B (¯

x, ε) \ {¯
x}) .

(1.1)

♥Õ✉


✹
✭❜✮ ❈❤♦

m ≥ 1 ❧➭ ♠ét sè ♥❣✉②➟♥✳ ❚❛ ♥ã✐ r➺♥❣ x¯ ∈ S

❝❤➷t ❝✃♣

❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣

m ❝ñ❛ (1.1) ♥Õ✉ tå♥ t➵✐ ε > 0✱ β > 0 s❛♦ ❝❤♦
m

f (x) − f (¯
x) ≥ β x − x¯

✭✶✳✷✮

(∀x ∈ S ∩ B (¯
x, ε)) .

◆❤❐♥ ①Ðt ✶✳✶✳✶

✭❛✮ ◆❤❐♥ t❤✃② r➺♥❣✱ ♥Õ✉

x¯

❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ t

ột ự tể ị

j

ớ ọ

m

tì ó ❝ị♥❣

j > m✳

✭❜✮ ❘â r➭♥❣ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❜✃t ❦ú ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛
t ỗ ự tể ị ❝❤➷t ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛
♣❤➢➡♥❣ ❝❤➷t ❝✃♣

m ✈í✐ m ♥➭♦ ➤ã✳ ❈❤➻♥❣ ❤➵♥✱ ❝❤♦ ❤➭♠ f : [0, +∞) → R
f (x) = x1/x ,

✈í✐

x > 0,

f (0) = 0,

✈➭

S := [0, +∞) . ❑❤✐ ➤ã✱ x = 0 ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ♠➭ ❦❤➠♥❣ ❧➭

♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

m ✈í✐ m ❜✃t ❦ú✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷

✭❛✮ ❈❤♦

S⊂ Rp . ◆ã♥ ❧ï✐ ①❛ ❝đ❛ S

➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐

0+ S := {y ∈ Rp |s + ty ∈ S, ∀s ∈ S, t ≥ 0} .
✭❜✮ ◆ã♥ t✐Õ♣ t✉②Õ♥ ❧➭ ♠ét ➳♥❤ ①➵
✈➭

p

p

A : 2R × Rp 2R

s ớ ỗ

S Rp

x Rp , A (S, x) ột ó ó tể rỗ ớ ỗ S Rp x S, t❛ ❝ã

0+ S ⊂ 0+ A (S, x) .
❈➳❝ ♥ã♥ t✐Õ♣ t✉②Õ♥ q✉❛♥ trä♥❣ ë ➤➞② ❧➭ ♥ã♥ t✐Õ♣ ❧✐➟♥✱ ♥ã♥ t✐Õ♣ t✉②Õ♥ ❯rs❡s❝✉
✈➭ ❝➳❝ ♥ã♥ ♣❤➬♥ tr♦♥❣ t➢➡♥❣ ø♥❣✳
◆ã♥ tế ợ ị ĩ ở

K (S, x) := y ∃ (tn , yn ) → 0+ , y

s❛♦ ❝❤♦

x + tn yn ∈ S, ∀n ;

◆ã♥ t✐Õ♣ t✉②Õ♥ ❯rs❡s❝✉ ợ ị ĩ ở

k (S, x) := y (tn ) → 0+ , ∃ (yn ) → y

✈í✐

x + tn yn ∈ S, ∀n ;


✺
✈➭ ❝➳❝ ♥ã♥ ♣❤➬♥ tr♦♥❣ t➢➡♥❣ ø♥❣ ❝đ❛ ❝❤ó♥❣ ❧➭

IK (S, x) := y ∃ (tn ) → 0+

s❛♦ ❝❤♦

∀ (yn ) → y, x + tn yn ∈ S, ∀n ➤đ ❧í♥

✈➭

Ik (S, x) := y ∀ (tn , yn ) → 0+ , y , x + tn yn ∈ S, ∀n ➤đ ❧í♥ .

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✸

: Rn → R ∪ {+∞} ❤÷✉ ❤➵♥ t➵✐ x ∈ Rn . ❑Ý

●✐➯ sö A ❧➭ ♠ét ♥ã♥ t✐Õ♣ t✉②Õ♥✱ ✈➭ f
❤✐Ư✉ tr➟♥ ➤å t❤Þ ❝đ❛

y

❧➭ ❡♣✐f.

f

A✲ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ủ f

t

x t

ợ ị ĩ ở

f A (x; y) := inf {r |(y, r) ∈ A ( ❡♣✐f, (x, f (x)))} .
ớ ó tế tế ợ ị ĩ ♥❤➢ ë tr➟♥✱ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣
t➢➡♥❣ ø♥❣ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ ❝➳❝ ❣✐í✐ ❤➵♥ s❛✉ ✭①❡♠ ❬✶✺❪✮

f K (x; y) = lim inf
(f (x + tv) − f (x)) /t,
+
(t,v)→(0 ,y)

f k (x; y) = lim sup inf (f (x + tv) − f (x)) /t
v→y

t→0+

(f (x + tv) − f (x)) /t,

:= sup lim sup inf
ε>0

t→0+

v∈B(y,ε)

f IK (x; y) = lim inf
sup (f (x + tv) − f (x)) /t
+
t→0

v→y

sup (f (x + tv) − f (x)) /t,
:= inf lim inf
+
ε>0 t→0

v∈B(y,ε)

f Ik (x; y) = lim sup (f (x + tv) − f (x)) /t.
(t,v)→(0+ ,y)

◆Õ✉

f

❧➭ ❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐

♣❤➢➡♥❣ tr➟♥ ❜➺♥❣
tr♦♥❣

x ✈í✐ ➤➵♦ ❤➭♠ ∇f (x) , t❤× ❝➯ ❜è♥ ➤➵♦ ❤➭♠ t❤❡♦

∇f (x) , y ,

tr♦♥❣ ➤ã

❧➭ ❦Ý ❤✐Ư✉ tÝ❝❤ ✈➠ ❤➢í♥❣ tr♦♥❣

Rn ế f st ị t x, tì f K (x; ·) = f IK (x; ·) ✈➭

f k (x; ·) = f Ik (x; ·) ✭①❡♠ ❬✶✻❪✮✳
♠➭

·, ·

f K (x; ·) > −∞

✈➭

▼ét ❧í♣ ❤➭♠ ➤➳♥❣ ❝❤ó ý ❜❛♦ ❣å♠ ❝➳❝ ❤➭♠

f K (x; ·) = f k (x; ·) .

tr➟♥ ❦❤➯ ✈✐ ❡♣✐❞✐❢❢❡r❡♥t✐❛❜❧❡ t➵✐

x✳

f

❈➳❝ ❤➭♠ sè ♥❤➢ t❤Õ ➤➢ỵ❝ ❣ä✐ ❧➭


✻
❈ị♥❣ ♥❤➢ tr♦♥❣ ❬✶✷❪✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ➤➵♦ ❤➭♠ ❝✃♣ ❝❛♦

dm f K (x; y) = lim inf
(f (x + tv) − f (x)) /tm
+
(t,v)→(0 ,y)

✈➭ ➤Þ♥❤ ♥❣❤Ü❛ t➢➡♥❣ tù ❝❤♦ dm f k (x; y) , dm f IK

✶✳✷

(x; y) ✈➭ dm f Ik (x; y) .

➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

●✐➯ sư

x¯ ∈ S

m

✈➭ ❦ý ❤✐Ö✉

K (¯
x) := K (S, x¯) ∩ y f K (x; y) ≤ 0 ;
is

❧➭ ❤➭♠ ❝❤Ø ❝ñ❛ t❐♣

S:

 0, ♥Õ✉ x ∈ S,
is (x) =
 +∞, ♥Õ✉ x ∈
/ S.

➜Þ♥❤ ❧ý ✶✳✷✳✶
❈❤♦
✭❛✮

[12]

m > 1✱ ❝➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❝đ❛ ❜➭✐ t♦➳♥ ✭✶✳✶✮❀

✭❜✮ ❱í✐ ♠ä✐

y ∈ Rn \ {0} ,
dm (f + iS )K (¯
x; y) > 0;

✭❝✮ ❇✃t ➤➻♥❣ t❤ø❝ ✭✶✳✸✮ ➤ó♥❣ ✈í✐ ♠ä✐
◆Õ✉

m = 1✱

✭✶✳✸✮

y ∈ K (¯
x) \ {0} .

t❤× ✭❛✮✱ ✭❜✮✱ ✭❝✮ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ♥Õ✉

K (¯
x)

tr♦♥❣ ✭❝✮✳
◆❤❐♥ ①Ðt ✶✳✷✳✶

➜✐Ị✉ ❦✐Ư♥ ✭❜✮ ủ ị ý ó tể ợ t ở

b ồ t➵✐

β > 0 s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ y ∈ Rn ,
dm (f + iS )K (¯
x; y) ≥ β y

m

.

➤➢ỵ❝ t❤❛② ❜ë✐

K (S, x¯)


✼
❚❤❐t ✈❐②✱ ♥Õ✉ ✭❛✮ ➤ó♥❣ t❤× ✭❜✮ ➤ó♥❣ ✈➭

dm (f + iS )K (¯
x; 0) = 0.
❱× t❤Õ ✭ˆ
b✮ t❤á❛ ♠➲♥ ✈í✐

β := min dm (f + iS )K (¯
x; y) | y = 1 .
❘â r➭♥❣ ✭ˆ
b✮ ❦Ð♦ t❤❡♦ ✭❜✮✱ ✈➭ ❞♦ ➤ã ✭❛✮✱ ✭ˆb✮✱ ✭❝✮ t➢➡♥❣ ➤➢➡♥❣✳
❇ỉ ➤Ị ✶✳✷✳✶
●✐➯ sư

g, h : Rp × Rn → R ∪ {+∞} ✈➭ (a, b) ∈ Rp × Rn

s❛♦ ❝❤♦

lim inf g (t, v) > −∞,
(t,v)→(a,b)

lim inf h (t, v) > −∞.
(t,v)→(a,b)
❑❤✐ ➤ã✱ ❝➳❝ ❦Õt q✉➯ s❛✉ ➤ó♥❣✿
✭✐✮

lim inf (g + h) (t, v) ≥ lim inf g (t, v) + lim inf h (t, v) ;
(t,v)→(a,b)

✭✐✐✮

✭✐✐✐✮

(t,v)→(a,b)

(t,v)→(a,b)

lim inf (g + h) (t, v) ≤ lim inf g (t, v) + lim sup h (t, v) ;
(t,v)→(a,b)

(t,v)→(a,b)

(t,v)→(a,b)

lim inf (g + h) (t, v) ≤ lim sup inf g (t, v) + lim inf sup h (t, v) .
(t,v)→(a,b)

t→a

v→b

t→a

v→b

❈❤ø♥❣ ♠✐♥❤

❈➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✐✮ ✈➭ ✭✐✐✮ ❤✐Ĩ♥ ♥❤✐➟♥ ➤ó♥❣✳ ❈❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ✭✐✐✐✮✳ ●ä✐

L := lim sup inf g (t, v) ,
t→a

v→b

M := lim inf sup h (t, v) ,
t→a

v→b

✈➭ ❣✐➯ sö

ε > 0, à > 0. ị ĩ ủ tå♥ t➵✐ δ ∈ (0, µ) s❛♦ ❝❤♦✱ ✈í✐
♠ä✐ λ > 0, tå♥ t➵✐ t¯ ∈ B (a, λ) s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ v ∈ B (b, δ) , t❛ ❝ã
h (t¯, v) ≤ M + ε/2.

✽
❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ▲✱ tå♥ t➵✐
tå♥ t➵✐

λ (δ) ∈ (0, δ) s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ t ∈ B (a, λ (δ)) ,

v (t) ∈ B (b, δ) ✈í✐
g (t, v (t)) ≤ L + ε/2.

❈❤ä♥

t¯ ∈ B (a, λ (δ)) s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ v ∈ B (b, δ) ,
h (t¯, v) ≤ M + ε/2.

❑❤✐ ➤ã✱

t¯ ∈ B (a, µ) , v (t¯) ∈ B (b, µ) , ✈➭
g (t¯, v (t¯)) + h (t¯, v (t¯)) ≤ L + M + .

ì

à ợ ọ tù ý✱ s✉② r❛
lim inf (g + h) (t, v) ≤ L+M +ε.
(t,v)→(a,b)

✷

❈➳❝ q✉② t➽❝ tÝ♥❤ t♦➳♥ ❝❤♦ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ s❛✉ ➤➞② ❧➭ ❝➳❝ ❤Ư q✉➯

trù❝ t✐Õ♣ ❝đ❛ ❜ỉ ➤Ị ✶✳✷✳✶✳
❍Ư q✉➯ ✶✳✷✳✶
●✐➯ sư

f1 , f2 : Rn → R ∪ {+∞} ❤÷✉ ❤➵♥ t➵✐ x¯, ✈➭ ❣✐➯ sö fiK (¯
x; y) > −∞, i =

1, 2. ●✐➯ sö m ≥ 1 ❧➭ ♠ét sè ♥❣✉②➟♥✳ ❑❤✐ ➤ã✿

✭✐✮

dm (f1 + f2 )K (¯
x; y) ≥ dm f1K (¯
x; y) + dm f2K (¯
x; y) ,

✭✐✐✮

✭✐✐✐✮

dm (f1 + f2 )K (¯
x; y) ≤ dm f1K (¯
x; y) + dm f2Ik (¯
x; y) ,
dm (f1 + f2 )K (¯
x; y) ≤ dm f1k (¯
x; y) + dm f2IK (¯
x; y) .

❑❤✐ ❝❤ó ý r➺♥❣

dm iA
x; ·) = iA(S,¯x) (·) ,
S (¯

✈í✐

A := K, k, IK, Ik,

❚❛ ❝ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤ñ tè✐ ➢✉ ➤➡♥ ❣✐➯♥ ❝❤♦ ❜➭✐ t♦➳♥ ✭✶✳✶✮✳


✾
❍Ư q✉➯ ✶✳✷✳✷
●✐➯ sư

[12]

x¯ ∈ S.

✭❛✮ ◆Õ✉

f K (¯
x; y) > 0

✈í✐ ♠ä✐

y ∈ K (S, x¯) \ {0}

t❤×

x¯

❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛

♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✶ ❝đ❛ ✭✶✳✶✮✳
✭❜✮ ◆Õ✉

m>1

✈➭

dm f K (¯
x; y) > 0

t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

✈í✐ ♠ä✐

y ∈ K (¯
x) \ {0}

t❤×

x¯

❧➭ ♠ét ❝ù❝

m ❝đ❛ ✭✶✳✶✮✳

❈❤ø♥❣ ♠✐♥❤

●✐➯ sư

y ∈ K (S, x¯) \ {0} . ❚õ ❤Ư q✉➯ ✶✳✷✳✶✭✐✮ ✈➭ ❣✐➯ t❤✐Õt ✭❛✮
(f + iS )K (¯
x; y) ≥ f K (¯
x; y) + iK(S,¯x) (y) > 0.

ị ý tì

x ột ự tể ị ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✶ ❝ñ❛ ✭✶✳✶✮✳ ❈❤ø♥❣

♠✐♥❤ ✭❜✮ t➢➡♥❣ tù ✭❛✮✳

✷

❚➢➡♥❣ tù✱ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ❞➢í✐ ➤➞② ❞Ơ ó ợ từ ị ý
✈➭ ✭✐✐✐✮ ❝đ❛ ❤Ư q✉➯ ✶✳✷✳✶✳
❍Ư q✉➯ ✶✳✷✳✸
●✐➯ sư

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ≥ 1 ❝ñ❛ ✭✶✳✶✮✳ ❑❤✐ ➤ã✿

✭❛✮

dm f K (¯
x; y) > 0, ∀y ∈ Ik (S, x¯) \ {0} ,

✭❜✮

dm f Ik (¯
x; y) > 0, ∀y ∈ K (S, x¯) \ {0} ,

✭❝✮

dm f k (¯
x; y) > 0, ∀y ∈ IK (S, x¯) \ {0} ,

✭❞✮

dm f IK (¯
x; y) > 0, ∀y ∈ k (S, x¯) \ {0} .

❈❤ø♥❣ ♠✐♥❤

❚❤❡♦ ➤Þ♥❤ ❧Ý ✶✳✷✳✶ ✈➭ ❤Ư q✉➯ ✶✳✷✳✶ ✭✐✐✮ t❛ ❝ã

0 < dm (f + iS )K (¯
x; y) ≤ dm f K (¯
x; y) + iIk(S,¯x) (y),
✈í✐ ♠ä✐

y ∈ Rn \ {0} . ❉♦ ➤ã✱ ✈í✐ ♠ä✐ y ∈ Ik (S, x¯) \ {0} , t❛ ❝ã
dm f K (¯
x; y) > 0


✶✵

✷

❈❤ø♥❣ ♠✐♥❤ ✭❜✮✱ ✭❝✮✱ ✭❞✮ t➢➡♥❣ tù ✭❛✮✳

◆❤❐♥ ①Ðt ✶✳✷✳✷

➜✐Ò✉ ❦✐Ư♥ ➤đ tr♦♥❣ ❤Ư q✉➯ ✶✳✷✳✷ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❝➬♥ ❝❤♦ tè✐ ➢✉ ✈➭ ➤✐Ị✉
❦✐Ư♥ ❝➬♥ ❝đ❛ ❤Ư q✉➯ ✶✳✷✳✸ ❦❤➠♥❣ ♣❤➯✐ ➤ñ ❝❤♦ tè✐ ➢✉✳ ❚✉② ♥❤✐➟♥ ❝ã ♥❤✐Ị✉ tr➢ê♥❣
❤ỵ♣ tr♦♥❣ ➤ã ❤Ư q✉➯ ✶✳✷✳✷ ✈➭ ✶✳✷✳✸ ❝ã t❤Ĩ ❦Õt ❤ỵ♣ ➤Ĩ ♥❤❐♥ ➤➢ỵ❝ tÝ♥❤ ❝❤✃t ➤➷❝
tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

m✳ ❈❤ó♥❣ t❛ ♠✐♥❤ ❤ä❛ ➤✐Ị✉ ♥➭② ❜➺♥❣ ❤❛✐

tr➢ê♥❣ ❤ỵ♣ s❛✉✳
▼Ư♥❤ ➤Ị ✶✳✷✳✶
❑Ý ❤✐Ư✉

intS

❧➭ ♣❤➬♥ tr♦♥❣ ❝đ❛

S✳

●✐➯ sư

x¯ ∈ intS ✳

❑❤✐ ➤ã✱

x¯

❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛

♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✶ ❝đ❛ ✭✶✳✶✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉

f K (¯
x; y) > 0, ∀y ∈ Rn \ {0} .
❱í✐

m ❃ ✶✱ x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❝ñ❛ ✭✶✳✶✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉
dm f K (¯
x; y) > 0, ∀y = 0

✈í✐

f K (¯
x; y) ≤ 0.

❈❤ø♥❣ ♠✐♥❤

❉♦

x¯ ∈ intS ✱ t❛ ❝ã
K (S, x¯) = Ik (S, x¯) = Rn .

➜✐Ị✉ ❦✐Ư♥ ➤đ ➤➢ỵ❝ s✉② r❛ trù❝ t✐Õ♣ tõ ❤Ư q✉➯ ✶✳✷✳✷ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ tõ ❤Ư q✉➯

✷

✶✳✷✳✸✭❛✮✳

▼Ư♥❤ ➤Ị ✶✳✷✳✷
●✐➯ sư

f : Rn → R ∪ {+∞} ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ t➵✐ x¯ ✈➭ k (S, x¯) = K (S, x¯) .

❑❤✐ ➤ã✱

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ❝đ❛ ✭✶✳✶✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉
f K (¯
x; y) > 0, ∀y ∈ K (S, x¯) \ {0} .

✭✶✳✹✮

❈❤ø♥❣ ♠✐♥❤

➜✐Ị✉ ❦✐Ư♥ ➤đ ❝đ❛ ✭✶✳✹✮ ❝❤Ý♥❤ ❧➭ ❤Ư q✉➯ ✶✳✷✳✷✭❛✮✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❝ñ❛ ✭✶✳✹✮ ❧➭ ♠ét


✶✶
❤Ư q✉➯ ❝đ❛ ❤Ư q✉➯ ✶✳✷✳✸✭❞✮ ✈➭ sù ❦✐Ư♥

f K (¯
x; ·) = f IK (¯
x; ·) ❦❤✐ f

▲✐♣s❝❤✐t③ ➤Þ❛

✷

♣❤➢➡♥❣ t➵✐ x
¯.

◆❤❐♥ ①Ðt ✶✳✷✳✸

●✐➯ t❤✐Õt

K (S, x¯) = k (S, x¯) t❤á❛ ♠➲♥ ❦❤✐ S

❧å✐✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✶

✭❛✮ ●✐➯ sư

S ⊂ Rp

✈➭

x ∈ S ✳ ◆ã♥ t✐Õ♣ t✉②Õ♥ ❈❧❛r❦❡ ❝ñ❛ S

T (S, x) := y ∀ (xn , tn ) → x, 0+
s❛♦ ❝❤♦
✭❜✮ ❈ù❝ ❝đ❛ ♥ã♥ ❧å✐

S ⊂ Rp

✈í✐

t➵✐

x ❧➭ t❐♣ ❤ỵ♣

{xn } ⊂ S, ∃ {yn } → y

xn + tn yn ∈ S, ∀n} .

❧➭ t❐♣ ❤ỵ♣

S 0 := {y ∈ Rp | y, x ≤ 0, ∀x ∈ S } .
✭❝✮ ▼✐Ị♥ ❤÷✉ ❤✐Ư✉ ❝đ❛ ❤➭♠

f : Rn → R ∪ {±∞} ❧➭ t❐♣ ❤ỵ♣

domf := {x ∈ Rn |f (x) < +∞} .
✭❞✮ ●✐➯ sö

A ❧➭ ♥ã♥ t✐Õ♣ t✉②Õ♥✳ A✲❞➢í✐ ❣r❛❞✐❡♥t ❝đ❛ f

t➵✐

x ∈ domf

❧➭ t❐♣ ❤ỵ♣

∂ A f (x) := x∗ ∈ Rn x∗ , y ≤ f A (x; y) , ∀y ∈ Rn .

▼Ư♥❤ ➤Ị ✶✳✷✳✸
●✐➯ sư

[13]

fi : Rn → R ∪ {+∞}✱ i❂✶✱✷✱

♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✱

x¯ ∈ domf1 ∩ domf2 ✱

✈➭

domf1T (¯
x; ·) − domf2T (¯
x; ·) = Rn .
❑❤✐ ➤ã✱

✭✶✳✺✮

∀y ∈ R, t❛ ❝ã
x; y) ≤ f1K (¯
x; y) + f2k (¯
x; y) .
(f1 + f2 )K (¯

✭✶✳✻✮


✶✷
❍➡♥ ♥÷❛✱ ♥Õ✉

f1K (¯

x; ·) ✈➭ f2k (¯
x; ·) ❧å✐ t❤×
∂ K (f1 + f2 ) (¯
x) ⊂ ∂ K f1 (¯
x) + ∂ k f2 (¯
x) .

❉✃✉ ❜➺♥❣ ①➯② r❛ tr♦♥❣ ✭✶✳✻✮ ✈➭ ✭✶✳✼✮ ♥Õ✉ t❤➟♠ ➤✐Ị✉ ❦✐Ư♥
✈✐ t➵✐

f1

❤♦➷❝

✭✶✳✼✮

f2

❧➭ tr➟♥ ❦❤➯

x¯.

❚õ ♠Ư♥❤ ➤Ị ✶✳✷✳✸ t❛ s✉② r❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ s❛✉ ➤➞② ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣
❝❤➷t ❝✃♣ ♠ét✳
❍Ư q✉➯ ✶✳✷✳✹

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ❝ñ❛ ✭✶✳✶✮✱ tr♦♥❣ ➤ã f : Rn →

●✐➯ sư

R ∪ {+∞} ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✱ S

➤ã♥❣✱ ✈➭

domf T (¯
x; ·) − T (S; x¯) = Rn .

✭✶✳✽✮

❑❤✐ ➤ã✿

✭❛✮

f K (¯
x; y) > 0, ∀y ∈ k (S, x¯) \ {0} ,

✭❜✮

f k (¯
x; y) > 0, ∀y ∈ K (S, x¯) \ {0} .

❈❤ø♥❣ ♠✐♥❤

➜Ó ❝❤ø♥❣ ♠✐♥❤ ✭❛✮ t❛ ❧✃② f1

:= f, f2 := iS

tr♦♥❣ ♠Ư♥❤ ➤Ị ✶✳✷✳✸✳ ➜✐Ị✉ ❦✐Ư♥ ✭✶✳✺✮

t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ✭✶✳✽✮ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✳ ❚õ ➤Þ♥❤ ❧Ý ✶✳✷✳✶ ✈➭ ✭✶✳✻✮ s✉② r❛ ✈í✐

♠ä✐

y = 0✱ t❛ ❝ã
0 < (f + iS )K (¯
x; y) ≤ f K (x; y) + ik(S;¯x) (y) .

❉♦ ➤ã✱

f K (¯
x; y) > 0, ∀y ∈ k (S; x¯) \ {0} .
➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ✭❜✮ t❛ ➤➯♦ ♥❣➢ỵ❝ ✈❛✐ trß ❝đ❛ f1 ✈➭ f2 .

✷

❈❤ó♥❣ t❛ ❝ị♥❣ ❝ã t❤Ĩ sư ❞ơ♥❣ ♠Ư♥❤ ➤Ị ✶✳✷✳✸ ➤Ĩ ♥❤❐♥ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❦❤➳❝
❝đ❛ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét✳


✶✸
➜Þ♥❤ ❧ý ✶✳✷✳✷
●✐➯ sư
sư

f

f : Rn → R ∪ {+∞}

tr➟♥ ❦❤➯ ✈✐ t➵✐

x¯

❤♦➷❝

♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✱

K (S, x¯) = k (S, x¯)✳

S

➤ã♥❣ ✈➭ ✭✶✳✽✮ ➤ó♥❣✳ ●✐➯

❑❤✐ ➤ã✱

x¯

❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛

♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ❝đ❛ ✭✶✳✶✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✭✶✳✹✮ ➤ó♥❣✳ ❍➡♥ ♥÷❛ ♥Õ✉
✈➭

f K (¯
x; ·)

K (S, x¯) ồ tì x ự tể ị t ♠ét ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐

β > 0 s❛♦ ❝❤♦
B (0, β) ⊂ ∂ K f (¯
x) + (K (S, x¯))0 .

✭✶✳✾✮

❈❤ø♥❣ ♠✐♥❤

●✐➯ sư

f

tr➟♥ ❦❤➯ ✈✐ t➵✐

x¯✳

❚õ ➤Þ♥❤ ❧ý ✶✳✷✳✶ ✈➭ ♠Ư♥❤ ➤Ị ✶✳✷✳✸✱

♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉✱ ✈í✐ ♠ä✐

x¯

❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛

y ∈ Rn \ {0} ,

0 < (f + iS )K (¯
x; y) = f k (¯
x; y) + iK
x; y)
S (¯
= f K (¯
x; y) + iK(S,¯x) .
❉♦ ➤ã✱

x¯

❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✭✶✳✹✮ ➤ó♥❣✳ ❇➞②

❣✐ê✱ ❣✐➯ sư

f K (¯
x; ·)

✈➭

K (S; x¯)

❧å✐✳ ◆❤➢ ➤➲ ➤Ị ❝❐♣ tr♦♥❣ ♥❤❐♥ ①Ðt ✶✳✷✳✶

❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐

x¯

❧➭

β > 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐

y ∈ Rn ,
β y ≤ (f + iS )K (¯
x; y) .

✭✶✳✶✵✮

❇ë✐ ✈×

(f + iS )K (¯
x; ·) = f K (¯
x; ·) + iK(S,¯x) (·) ,
tõ ♠Ư♥❤ ➤Ị ✶✳✷✳✸✱ t❛ ❝ã

(f + iS )K (¯
x; ·) ❧å✐✳ ❱× ✈❐②✱ ✭✶✳✶✵✮ t➢➡♥❣ ➤➢➡♥❣ ✈í✐
B (0, β) ⊂ ∂ K (f + iS ) (¯
x) ,

✈➭ tõ ♠Ö♥❤ ➤Ò ✶✳✷✳✸✱ t❛ ❝ã

∂ K (f + is ) (¯
x) = ∂ k f (¯
x) + ∂ K iS (¯
x)
= ∂ K f (¯
x) + (K (S, x¯))0 .
❉♦ ➤ã✱ x
¯ ❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✭✶✳✾✮ ➤ó♥❣✳ ◆Õ✉ t❤❛②
❜➺♥❣ ➤✐Ị✉ ❦✐Ư♥ K

(S, x¯) = k (S, x¯)✱ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳

✷


✶✹
◆❤❐♥ ①Ðt ✶✳✷✳✹

➜Þ♥❤ ❧Ý ✶✳✷✳✷ ❜❛♦ ❤➭♠ t❤ù❝ sù ♠Ư♥❤ ➤Ị ✶✳✷✳✷✱ ❜ë✐ ✈× ♥Õ✉
t➵✐

x¯

t❤× ❞♦♠f T

(¯
x; ·) = Rn

f

▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣

✈➭ ✭✶✳✽✮ ➤ó♥❣✳ ❙❛✉ ➤➞② ❧➭ ♠ét ✈Ý ❞ơ ➤ó♥❣ ✈í✐ ➤Þ♥❤

❧Ý ✶✳✷✳✷ ♥❤➢♥❣ ❦❤➠♥❣ ➤ó♥❣ ✈í✐ ♠Ư♥❤ ➤Ị ✶✳✷✳✷✳ ●✐➯ sö

S = R✱ x¯ = 0✱

✈➭ ❤➭♠

f : Rn → R

 0,
♥Õ✉ x = 0
f (x) =
 1/2n , ♥Õ✉ 1/2n+1 < |x| ≤ 1/2n , n = 0, ±1, ±2, ....
❑❤✐ ➤ã

♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ✈➭

f

f T (0; y) = i{0} (y) , f K (0; y) = |y| , ∀y ∈ R.

ë ➤➞②✱ f ❦❤➠♥❣ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ t➵✐ x¯ = 0, ♥❤➢♥❣ ✭✶✳✽✮ ➤ó♥❣ ✈➭ K (S, x¯) =
k (S, x¯)✳

❱× ✭✶✳✾✮ t❤á❛ ♠➲♥ ✈í✐

✶✳✷✳✷ ❦Ð♦ t❤❡♦

✶✳✸

❍➭♠

β =1

✈➭ ✈×

f K (0; ·)✱ K (S, 0)

❧➭ ❧å✐✱ ➤Þ♥❤ ❧ý

x¯ = 0 ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét✳

C 1,1

✈➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷

➜Ĩ t×♠ ❝➳❝ tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

m > 1, tr➢í❝

❤Õt t❛ ①➳❝ ➤Þ♥❤ ❝➳❝ ❧í♣ ❤➭♠ ♠➭ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ❜➺♥❣ ♥❤❛✉✳ ❚r♦♥❣
♣❤➬♥ ♥➭②✱ ❝❤ó♥❣ t❛ ①Ðt ♠ét ❧í♣ tế tr trờ ợ

m = 2

ị ĩ

sè

f : Rn → R ∪ {+∞}

❋rÐ❝❤❡t t➵✐

➤➢ỵ❝ ❣ä✐ ❧➭

C 1,1

C 1,1

❇ỉ ➤Ị ✶✳✸✳✶

✭✐✮

❧➭

♥Õ✉

f

❦❤➯ ✈✐ ❧✐➟♥ tơ❝

t❤➢ê♥❣ ①✉✃t ❤✐Ư♥ tr♦♥❣ tè✐ ➢✉✱ ❝❤➻♥❣ ❤➵♥ tr♦♥❣ ♣❤➢➡♥❣ ♣❤➳♣

❤➭♠ ♣❤➵t ➤Ó ❣✐➯✐ ❜➭✐ t♦➳♥ ♣❤✐ t✉②Õ♥ ✈í✐ ❞÷ ❧✐Ư✉

f

x

x ✈➭ ∇f (·) ▲✐♣s❝❤✐③ ➤Þ❛ ♣❤➢➡♥❣ t➵✐ x✳

❈➳❝ ❤➭♠

◆Õ✉

t➵✐

C 1,1

t➵✐

x ✈➭ ∇f (x) = 0 t❤×

d2 f K (x, ·) = d2 f IK (x, ·) ,

C 2.


✶✺
✭✐✐✮

d2 f k (x, ·) = d2 f Ik (x, ·) .

❈❤ø♥❣ ♠✐♥❤

❚õ ➤Þ♥❤ ♥❣❤Ü❛ t❛ ❝ã

d2 f K (x; ·) ≤ d2 f IK (x; ·) .
●✐➯ sö

y ∈ Rn

✈➭

d2 f K (x; y) ≤ r.
➜Ó ❝❤ø♥❣ ♠✐♥❤ ✭✐✮ t❛ ❝❤Ø ❝➬♥ ❝❤Ø r❛ r➺♥❣

d2 f IK (x; y) ≤ r.
❈❤♦

ε > 0✳

❱×

f

❧➭

C 1,1

t➵✐

x

♥➟♥ tå♥ t➵✐

L > 0, δ > 0

s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐

z✱

w ∈ B (x, δ) ,
∇f (z) − ∇f (ω) ≤ L z − ω .
❈❤ä♥

λ ∈ (0, min {ε, ε/4L ( y + ε)})
s❛♦ ❝❤♦

x + (0, λ) B (y, λ) ⊂ B (x, δ) .
▲✃② v, ω

∈ B (y, λ) , t ∈ (0, ) . ị ý trị tr ì tồ t➵✐ θ ∈ (0, 1)

s❛♦ ❝❤♦✱ ✈í✐

z := θv + (1 − θ) ω, t❛ ❝ã

f (x + tv) − f (x + tω) = ∇f (x + tz) , t (v − ω)
= ∇f (x + tz) − ∇f (x) , t (v − ω) ,
≤ Lt2 z

v−ω

≤ Lt2 ( y + ε) 2λ
≤ t2 ε/2.
❉♦ ➤ã✱

(f (x + tv) − f (x + tω)) /t2 ≤ ε/2.
❇➞② ❣✐ê✱ ❧✃②

v ∈ B (y, λ) , η > 0✳ ❱×
d2 f K (x; y) ≤ r

✈×

∇f (x) = 0,


✶✻
♥➟♥ tå♥ t➵✐

ω ∈ B (y, λ) , t ∈ (0, min {η, λ}) s❛♦ ❝❤♦
(f (x + tω) − f (x)) /t2 ≤ r + ε/2.

❉♦ ➤ã✱

(f (x + tv) − f (x)) /t2 = (f (x + tv) − f (x + tω)) /t2 + (f (x + tω) − f (x)) /t2
≤ ε/2 + r + ε/2 = r + ε.
❉♦

v, η, ε tï② ý✱ t❛ ❝ã
d2 f IK (x; y) ≤ r.
✷

❉♦ ➤ã ✭✐✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ ✭✐✐✮ t➢➡♥❣ tù✳

❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❝❤ó♥❣ t❛ ❧➭ ♠ét tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣
❝❤➷t ❝✃♣ ✷ ❝❤♦ ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ t♦➳♥ ❤ä❝✿

min {f (x) |gi (x) ≤ 0, i = 1, ..., m, hi (x) = 0, i = 1, ..., p} ,
tr♦♥❣ ➤ã

f, gi , hi

❧➭

✭✶✳✶✶✮

C 1,1 ✳ ❈❤ó♥❣ t❛ t t t ớ ỗ x ❧➭

➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭✶✳✶✶✮✱ t❛ ➤➷t

I (x) := {i |gi (x) = 0} .

●✐➯ sö

λi ≥ 0, i = 1, ...., m, µi ∈ R, i = 1, ..., p, r ợ ị

s

L (x) := f (x) +

λi gi (x) +
i=1

❚❛ ♣❤➞♥ ❤♦➵❝❤

p

m

µi hi (x).
i=1

I (x) t❤➭♥❤ ❝➳❝ t❐♣
J (x) := {i ∈ I (x) |λi > 0}

✈➭

M (x) := {i ∈ I (x) |λi = 0} .
❚r♦♥❣ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ✷✱ t❐♣ ❤ỵ♣ ❝➳❝ ♣❤➢➡♥❣✿

D (x) := {y ∈ Rn | ∇gi (x) , y ≤ 0, ∀i ∈ M (x) , ∇gi (x) , y = 0, ∀i ∈ J (x) ,

✶✼

∇hi (x) , y = 0, i = 1, ..., p}
sÏ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣✳
❈❤ó♥❣ t❛ sÏ t❤✐Õt ❧❐♣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝❤♦ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣ ❤❛✐✳ ❈➳❝ ➤✐Ị✉
❦✐Ư♥ ♥➭② tỉ♥❣ q✉➳t ❤ã❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝✃♣ ❤❛✐ t❤➠♥❣ t❤➢ê♥❣ ❝❤♦ ❜➭✐ t♦➳♥ ✭✶✳✶✶✮
✈í✐ ❞÷ ❧✐Ư✉

C2

❝❤♦ ❜➭✐ t♦➳♥ ❦❤➯ ✈✐ ✈í✐ ❤➭♠ ♠ơ❝ t✐➟✉ ✈➭ ❤➭♠ r➭♥❣ ❜✉é❝ ❦❤➯ ✈✐

❋rÐ❝❤❡t✳ ❚r♦♥❣ s✉èt ♣❤➬♥ ♥➭②✱

x¯ ❧➭ ột ể ợ ủ

ị ý
sử

f, gi , i = 1, ...., m, hi , i = 1, ...., p,

❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐

x¯✳

●✐➯ sö tå♥ t➵✐

λi ≥ 0, µi ∈ R s❛♦ ❝❤♦ ∇L (¯
x) = 0 ✈➭ λi gi (¯

x) = 0 ✈í✐ i = 1, ..., m ✈➭
2

d LK (¯
x; y) > 0, ∀y ∈ D (¯
x) \ {0} .
❑❤✐ ➤ã✱

✭✶✳✶✷✮

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷ ❝đ❛ ✭✶✳✶✶✮✳

❈❤ø♥❣ ♠✐♥❤

❈❤ó♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✳ ●✐➯ sö
♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ ❞➲②
t♦➳♥ ✭✶✳✶✶✮ s❛♦ ❝❤♦

{xn }

x¯ ❦❤➠♥❣ ột ự tể ị

ể ợ ❝ñ❛ ❜➭✐

{xn } → x¯ ✈➭
f (xn ) − f (¯
x) ≤ xn − x¯ 2 /n.

✭✶✳✶✸✮

➜➷t

tn := xn − x¯ , yn := (xn − x¯) /tn .
❑❤✐ ➤ã✱ tn

→ 0+ . ❑❤➠♥❣ ♠✃t tÝ♥❤ ❝❤✃t tæ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sư {yn } → y

✈í✐

y = 0 ♥➭♦ ➤ã✳ ❚❛ ❝ã y ∈ D (¯
x) ✈➭
(L (xn ) − L (¯
x)) /t2n ≤ (f (xn ) − f (¯
x)) /t2n ≤ 1/n.
❉♦ ➤ã✱

d2 LK (¯
x; y) ≤ lim inf ≤ (L (xn ) − L (¯
x)) /t2n 0.
n

ì tỏ ị ý ợ ❝❤ø♥❣ ♠✐♥❤✳

✷

❚r➢í❝ ❦❤✐ tr×♥❤ ❜➭② ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ❝✃♣ ❤❛✐ ❝❤♦ ✭✶✳✶✶✮✱ ❝❤ó♥❣ t❛ ①Ðt ♠è✐
q✉❛♥ ❤Ư ❣✐÷❛ ♥❣❤✐Ư♠ ❝đ❛ ✭✶✳✶✶✮ ✈➭ ❝ù❝ t✐Ĩ✉ ❝đ❛ ❤➭♠ ▲❛❣r❛♥❣❡ tr➟♥ t❐♣ ❤ỵ♣ r➭♥❣
❜✉é❝ ❝đ❛ ✭✶✳✶✶✮✳

✶✽
❇ỉ ➤Ị ✶✳✸✳✷
●✐➯ sư x
¯ ❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ ✭✶✳✶✶✮❀

λi ≥ 0, i = 1, ..., m, µi ∈

R, i = 1, ..., p. ◆Õ✉ λi gi (¯
x) = 0 ✈í✐ i = 1, ..., m tì x ột ự tể ị
t ❤❛✐ ❝đ❛

L tr➟♥ t❐♣ ❤ỵ♣

C := {x |gi (x) ≤ 0, i = 1, ..., m, gi (x) = 0, i ∈ J (¯
x) , hi (¯
x) = 0, i = 1, ..., p} .
❈❤ø♥❣ ♠✐♥❤

❑Ý ❤✐Ư✉

S

❧➭ t❐♣ ❤ỵ♣ r➭♥❣ ❜✉é❝ ❝đ❛ ✭✶✳✶✶✮✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ tå♥ t➵✐

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

ε > 0, β > 0

x ∈ S ∩ B (¯
x, ε)✱ t❛ ❝ã

f (x) − f (¯
x) > β x − x¯ 2 .

●✐➯ sö

x ∈ C ∩ B (¯
x, ε) . ❑❤✐ ➤ã✱
p

L (x) − L (¯
x) = f (x) − f (¯
x) +

λi (gi (x) − gi (¯
x)) +

ui (hi (x) − hi (¯
x))
i=1

J(¯
x)

= f (x) − f (¯
x) ≥ β x − x¯ 2 .
❉♦ ➤ã✱

✷

x¯ ❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝đ❛ L tr➟♥ C.

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✷

●✐➯ sư gi , i

= 1, ..., m, hi , i = 1, ..., p, ❧➭ C 1

t➵✐

x¯. ❚❛ ♥ã✐ r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤

q✉② ▼❛♥❣❛s❛r✐❛♥✲❋r♦♠♦✇✐t③ ❝❤➷t ✭❙▼❋❈◗✮ ➤ó♥❣ t➵✐
✭✐✮

x¯ ♥Õ✉

∇gi (¯
x) , i ∈ J (x) , ∇hi (¯
x) , i = 1, ..., p ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤❀

✭✐✐✮ ❚å♥ t➵✐

z ∈ Rn

s❛♦ ❝❤♦

∇gi (¯
x) , z < 0, ∀i ∈ M (¯
x) ,
∇gi (¯

x) , z = 0, ∀i ∈ J (¯
x) ,
∇hi (¯
x) , z = 0, i = 1, .., p.
➜✐Ị✉ ❦✐Ư♥ ❙▼❋❈◗ ❧➭ ➤đ ➤Ĩ s✉② r❛

k (C, x¯) = D (¯
x)

❝❤♦ t❐♣

C

tr♦♥❣ ❜ỉ ➤Ị

✶✳✸✳✷ ✭①❡♠ ❬✾❪✮✳ ❱í✐ sù ❦✐Ư♥ ♥➭②✱ ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❝✃♣ ❤❛✐ ❝❤♦
✭✶✳✶✶✮✳


✶✾
➜Þ♥❤ ❧ý ✶✳✸✳✷

x¯

●✐➯ sư

❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝đ❛ ✭✶✳✶✶✮✱ tr♦♥❣ ➤ã

❋rÐ❝❤❡t t➵✐
➤ó♥❣ t➵✐

x¯

✈➭

gi , i = 1, ..., m, hi , i = 1, ..., p,

❧➭

C1

t➵✐

x¯.

f

❦❤➯ ✈✐

●✐➯ sư ❙▼❋❈◗

x¯ ✈í✐ λi ≥ 0, µi ∈ R t❤á❛ ♠➲♥ ∇L (¯
x) = 0, λi gi (¯
x) = 0, i = 1, ..., m.

❑❤✐ ➤ã✱

d2 LIK (¯
x; y) > 0, ∀y ∈ D (¯
x) \ {0} .

✭✶✳✶✹✮

❈❤ø♥❣ ♠✐♥❤

❚❤❡♦ ❜ỉ ➤Ị ✶✳✸✳✷✱

L

❝ã ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ tr➟♥

C

t➵✐

x¯.

❱×

✈❐②✱ t❤❡♦ ❤Ư q✉➯ ✶✳✷✳✸✭❞✮ t❛ ❝ã

d2 LIK (¯
x; y) > 0, ∀y ∈ k (C, x¯) \ {0} .
➜✐Ị✉ ❦✐Ư♥ ✭✶✳✶✹✮ ➤➢ỵ❝ s✉② r❛ tõ sù ❦✐Ư♥ k (C, x
¯)

= D (¯
x) ✈í✐ ❣✐➯ t❤✐Õt ❙▼❋❈◗✳ ✷

❇➞② ❣✐ê ❝❤ó♥❣ t❛ sư ❞ơ♥❣ ❝➳❝ ➤Þ♥❤ ❧Ý ✶✳✸✳✶✱ ✶✳✸✳✷ ✈➭ ❜ỉ ➤Ị ✶✳✸✳✶ ➤Ĩ s✉② r❛ ♠ét

tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝❤♦ ❜➭✐ t♦➳♥

C 1,1 .

➜Þ♥❤ ❧ý ✶✳✸✳✸
●✐➯ sư

f, gi , i = 1, .., m, hi , i = 1, ..., p, ❧➭ C 1,1 t➵✐ x¯. ●✐➯ sö ❙▼❋❈◗ t❤á❛ ♠➲♥
λi ≥ 0, µi ∈ R

∇L (¯
x) = 0, λi gi (¯
x) = 0, ∀i = 1, ...m.

t➵✐

x¯

➤ã✱

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝đ❛ ✭✶✳✶✶✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✭✶✳✶✷✮

✈í✐

s❛♦ ❝❤♦

❑❤✐

➤ó♥❣✳

▼ét ❤Ư q✉➯ ❝đ❛ ➤Þ♥❤ ❧Ý ✶✳✸✳✸ ❧➭✿ tr♦♥❣

C 2 ✲ tè✐ ➢✉✱ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝✃♣ ❤❛✐ ❧➭ ➤➷❝

tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ♥Õ✉ ❙▼❋❈◗ t❤á❛ ♠➲♥✳
❈➳❝❤ t✐Õ♣ ❝❐♥ ❦❤➳❝ ❝❤♦

C 1,1 ✲ tè✐

❝ñ❛ ♠ét ❤➭♠ C 1,1 . ❙❛✉ ➤➞②✱ ∇2 f
❤❛✐ ❧➬♥ t➵✐

➢✉ ❧➭ ❞ù❛ tr➟♥ ❦❤➳✐ ♥✐Ö♠ ❍❡ss✐❛♥ s✉② ré♥❣

(x) sÏ ❦Ý ❤✐Ư✉ ❍❡ss✐❛♥ ❝đ❛ ❤➭♠ f

x.

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✸

●✐➯ sö

f : Rn → R ❧➭ C 1,1

t➵✐

x¯ ✈➭

Ef := x ∇2 f (¯
x)

tå♥ t➵✐

.

❦❤➯ ✈✐ ❋rÐ❝❤❡t


✷✵
❍❡ss✐❛♥ s✉② ré♥❣ ❝đ❛

f

t➵✐

x¯ ❧➭ t❐♣ ❝➳❝ n × n ♠❛ tr❐♥

∂ 2 f (¯
x) := conv A ∃ {xj } → x¯ ✈í✐ xj ∈ Ef

✈➭

∇2 f (xj ) → A ,

✭✶✳✶✺✮

tr♦♥❣ ➤ã ❝♦♥✈ ❦Ý ❤✐Ư✉ ❜❛♦ ❧å✐✳
❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ s❛✉ ➤➞② ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ủ t
ó ợ ị í ✶ ❝đ❛ ❬✼❪✳
➜Þ♥❤ ❧ý ✶✳✸✳✹ ✭❬✽❪✮

●✐➯ sư
✈➭

f, gi , hi , ❧➭ C 1,1 t➵✐ x¯. ●✐➯ sö tå♥ t➵✐ λi ≥ 0, µi ∈ R s❛♦ ❝❤♦ ∇L (¯
x) = 0

λi gi (¯
x) = 0 ✈í✐ i = 1, ..., m; ✈➭ ✈í✐ ♠ä✐ A ∈ ∂ 2 L (¯
x) , t❛ ❝ã
y, Ay > 0, ∀y ∈ D (¯
x) \ {0} .

❑❤✐ ➤ã✱

✭✶✳✶✻✮

x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝đ❛ ✭✶✳✶✶✮✳

❑❤➠♥❣ ❣✐è♥❣ ♥❤➢ ✭✶✳✶✷✮✱ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✶✻✮ ❦❤➠♥❣ ❝❤♦ ♠ét tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣
❝đ❛ ❝ù❝ t✐Ĩ✉ ➤➵✐ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐✳ ➜✐Ị✉ ♥➭② ➤➢ỵ❝ ♠✐♥❤ ❤ä❛ ❜➺♥❣ ✈Ý ❞ơ s❛✉
➤➞②✳
❱Ý ❞ơ ✶✳✸✳✶

❈❤♦

Z ❧➭ t❐♣ ❤ỵ♣ ❝➳❝ sè ♥❣✉②➟♥✱ ✈➭ ❤➭♠ f : R → R



 0, ♥Õ✉ x = 0,

f (x) = x2 + x cos 2n+2 x − 3π + 1 /2n , ♥Õ✉ π/2n+1 < x ≤ π/2n , n ∈ Z,



f (−x) , ♥Õ✉ x < 0.
❇ë✐ ✈×

f (x) ≥ x2 , ∀x ∈ R,
f (0) = 0,
t❛ ❝ã

x¯ = 0 ❧➭ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣ ❤❛✐ ❝đ❛ f

tr➟♥

R. ❚❛ ❝ã f ❧➭ C 1 tr➟♥ R ✈➭ ❦❤➯

✈✐ ❤❛✐ ❧➬♥ trõ r❛ t❐♣ ➤Õ♠ ➤➢ỵ❝ s❛✉✿

Ω := {0} ∪ {π/2n |n ∈ Z } ,

✈í✐

f (¯
x) ≤ 10 + 16π, ∀x ∈
/ Ω.

f



C 1,1

tr

R. ớ ỗ n,

f

3/2n+2 = 2 12π < 0.

❉♦ ➤ã✱

2 − 12π ∈ ∂ 2 f (0) ,
✈➭ ✭✶✳✶✻✮ ❦❤➠♥❣ t❤á❛ ♠➲♥ ✈í✐ ❜✃t ❦× t❐♣ r➭♥❣ ❜✉é❝ ♠➭

D (0) \ {0} = ∅.