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

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✐ ❝➳❝ ➤Ò t➭✐ ❦❤➳❝✳ ❚➠✐ ❝ò♥❣ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ♠ä✐ sù
❣✐ó♣ ➤ì ❝❤♦ ✈✐Ö❝ t❤ù❝ ❤✐Ö♥ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢î❝ ❝➯♠ ➡♥ ✈➭ ❝➳❝ t❤➠♥❣ t✐♥ trÝ❝❤
❞➱♥ tr♦♥❣ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢î❝ ❝❤Ø râ ♥❣✉å♥ ❣è❝✳
❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺
◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥

P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣


✐✐

▲ê✐ ❝➯♠ ➡♥
▲✉❐♥ ✈➝♥ ➤➢î❝ t❤ù❝ ❤✐Ö♥ ✈➭ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝
❚❤➳✐ ◆❣✉②➟♥ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❝ñ❛ P●❙✳ ❚❙✳ ➜ç ❱➝♥ ▲➢✉✳ ◗✉❛ ➤➞②✱
t➳❝ ❣✐➯ ①✐♥ ➤➢î❝ ❣ö✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬② ❣✐➳♦✱ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛
❤ä❝ ❝ñ❛ ♠×♥❤✱ P●❙✳ ❚❙✳ ➜ç ❱➝♥ ▲➢✉✱ ♥❣➢ê✐ ➤➲ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥ tr♦♥❣ s✉èt
q✉➳ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝ñ❛ t➳❝ ❣✐➯✳ ➜å♥❣ t❤ê✐ t➳❝ ❣✐➯ ❝ò♥❣ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝
t❤➬② ❝➠ tr♦♥❣ ❦❤♦❛ ❚♦➳♥✱ ❦❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠✱ ➜➵✐ ❤ä❝
❚❤➳✐ ◆❣✉②➟♥✱ ➤➲ t➵♦ ♠ä✐ ➤✐Ò✉ ❦✐Ö♥ ➤Ó t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➳❝
❣✐➯ ❝ò♥❣ ❣ö✐ ❧ê✐ ❝➯♠ ➡♥ ➤Õ♥ ❣✐❛ ➤×♥❤ ✈➭ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✷✶❇✱
➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥✳
▲✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ó tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ t➳❝ ❣✐➯ r✃t ♠♦♥❣ ♥❤❐♥ ➤➢î❝
sù ❝❤Ø ❜➯♦ t❐♥ t×♥❤ ❝ñ❛ ❝➳❝ t❤➬② ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ö♣✳
❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺
◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥

P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣


✐✐✐

▼ô❝ ❧ô❝

▲ê✐ ❝❛♠ ➤♦❛♥

✐

▲ê✐ ❝➯♠ ➡♥

✐✐

▼ô❝ ❧ô❝

✐✐✐

▼ë ➤➬✉

✶

✶

➜✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝ñ❛ ❲❛r❞

✸

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

✸

✶✳✷ ➜✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝❤♦ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

✻

✶✳✸ ❍➭♠

✷

C 1,1

m

✳ ✳ ✳ ✳ ✳ ✳ ✳

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

➜✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝ñ❛
❘❛❤♠♦✲❙t✉❞♥✐❛rs❦✐

✷✷

✷✳✶ ❈➳❝ ❦Õt q✉➯ ❜æ trî
✷✳✷ ➜✐Ò✉ ❦✐Ö♥ ❝➬♥ tè✐ ➢✉

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

✷✳✸ ➜✐Ò✉ ❦✐Ö♥ ➤ñ tè✐ ➢✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✷✳✹

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

❑Õt ❧✉❐♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

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

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

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

✶

▼ë ➤➬✉

✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❧✉❐♥ ✈➝♥

▲ý t❤✉②Õt ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❧➭ ♠ét ❜é ♣❤❐♥ q✉❛♥ trä♥❣ ❝ñ❛ ❧ý t❤✉②Õt tè✐
➢✉ ❤ã❛✳ ❈➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ♠ét ❝❤♦ ♣❤Ð♣ t❛ ①➳❝ ➤Þ♥❤ ➤➢î❝ t❐♣ ❝➳❝ ➤✐Ó♠
❞õ♥❣✳ ❈➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ♣❤Ð♣ t❛ t×♠ r❛ ➤➢î❝ ❝➳❝ ♥❣❤✐Ö♠ tè✐ ➢✉
tr♦♥❣ t❐♣ ❝➳❝ ➤✐Ó♠ ❞õ♥❣ ➤ã✳ ❑❤➳✐ ♥✐Ö♠ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

m

➤➢î❝

➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ ❈r♦♠♠❡ ❬✷❪✳ ❈➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ó✉ ❝❤➷t ❝✃♣

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

❙➢✉ t➬♠ ✈➭ ➤ä❝ t➭✐ ❧✐Ö✉ tõ ❝➳❝ s➳❝❤✱ t➵♣ ❝❤Ý t♦➳♥ ❤ä❝ tr♦♥❣ ♥➢í❝ ✈➭ q✉è❝ tÕ

❧✐➟♥ q✉❛♥ ➤Õ♥ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦✳ ◗✉❛ ➤ã✱ t×♠ ❤✐Ó✉ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ✈Ò ✈✃♥
➤Ò ♥➭②✳
✸✳ ▼ô❝ ➤Ý❝❤ ❝ñ❛ ❧✉❐♥ ✈➝♥

▼ô❝ ➤Ý❝❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ t×♠ ❤✐Ó✉ ✈Ò ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝


✷
t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ó✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳ ❈ô t❤Ó✱ ❝❤ó♥❣ t➠✐ ➤ä❝
❤✐Ó✉ ✈➭ 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 ❧✉❐♥ ✈➭ ❞❛♥❤ ♠ô❝ ❝➳❝ t➭✐ ❧✐Ö✉
t❤❛♠ ❦❤➯♦
❈❤➢➡♥❣ ✶✿ ➜✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝ñ❛ ❲❛r❞

❚r×♥❤ ❜➭② ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝ñ❛ ❲❛r❞
❬✶✸❪ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ❦❤➳❝ ♥❤❛✉✳ ❱í✐ ➤✐Ò✉ ❦✐Ö♥
❝❤Ý♥❤ q✉②✱ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤ñ ❝✃♣ ❝❛♦ trë t❤➭♥❤ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝
t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦✳
❈❤➢➡♥❣ ✷✿ ➜✐Ò✉ ❦✐Ö♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ó✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝ñ❛
❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐

❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣

m ❝❤♦ ❤➭♠

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

m






ề ệ tố ự tể ị
t ủ r

r ú t trì ề ệ tố ự tể
ị t ủ r ớ ữ t
ớ ề ệ í q ề ệ ủ trở t ề ệ
tr ự tể t ết q trì tr
ủ r



ệ ị ĩ

ét t tố s


min {f (x) |x S } ,
tr ó

f Rn R {+} S

ột t rỗ tr

Rn

ị ĩ



ã

t tr

Rn ớ > 0 t

B (x, ) := {y Rn | y x } .
ó r
tồ t

x S

ột ự tể ị t ủ t

> 0 s
f (x) > f (
x) (x S B (
x, ) \ {
x}) .

(1.1)

ế





m 1 ột số ó r x S

t

ột ự tể ị

m ủ (1.1) ế tồ t > 0 > 0 s
m

f (x) f (
x) x x



(x S B (
x, )) .

ét

t r ế

x

ột ự tể ị t

ột ự tể ị

j

ớ ọ

m

tì ó ũ

j > m

õ r ột ự tể ị t m t ỳ ột ự tể ị
t ỗ ự tể ị t ột ự tể ị
t

m ớ m ó f : [0, +) R
f (x) = x1/x ,

ớ

x > 0,

f (0) = 0,

S := [0, +) . ó x = 0 ột ự tể ị t

ột ự tể ị t

m ớ m t ỳ

ị ĩ



S Rp . ó ù ủ S

ợ ị ở

0+ S := {y Rp |s + ty S, s S, t 0} .
ó tế tế ột


p

p

A : 2R ì Rp 2R

s ớ ỗ

S Rp

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

0+ S 0+ A (S, x) .
ó tế tế q trọ ở ó tế ó tế tế rss
ó tr t ứ
ó tế ợ ị ĩ ở

K (S, x) := y (tn , yn ) 0+ , y

s

x + tn yn S, n ;

ó tế tế rss ợ ị ĩ ở

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

ớ

x + tn yn S, n ;



ó tr t ứ ủ ú

IK (S, x) := y (tn ) 0+

s

(yn ) y, x + tn yn S, n ủ ớ

Ik (S, x) := y (tn , yn ) 0+ , y , x + tn yn S, n ủ ớ .

ị ĩ

: Rn R {+} ữ t x Rn . í

sử A ột ó tế tế f
ệ tr ồ tị ủ

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 ở tr t
t ứ ó tể ể ễ ớ s

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
vy

t0+

(f (x + tv) f (x)) /t,

:= sup lim sup inf
>0

t0+

vB(y,)

f IK (x; y) = lim inf
sup (f (x + tv) f (x)) /t
+
t0

vy

sup (f (x + tv) f (x)) /t,
:= inf lim inf
+
>0 t0

vB(y,)

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

ế

f

rét t

tr
tr

x ớ f (x) , tì ố t

f (x) , y ,

tr ó

í ệ tí ớ tr

Rn ế f st ị t x, tì f K (x; ã) = f IK (x; ã)

f k (x; ã) = f Ik (x; ã)


ã, ã

f K (x; ã) >

ột ớ ú ý ồ

f K (x; ã) = f k (x; ã) .

tr rt t

x

f

số tế ợ ọ



ũ tr t ị ĩ

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

ị ĩ t tự dm f k (x; y) , dm f IK



(x; y) dm f Ik (x; y) .

ề ệ tố ự tể ị t

sử

x S

m

ý ệ

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

ỉ ủ t

S:

0, ế x S,
is (x) =
+, ế x
/ S.

ị ý



[12]

m > 1 t ể s t

x ột ự tể ị t m ủ t

ớ ọ

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

t tứ ú ớ ọ
ế

m = 1



y K (
x) \ {0} .

tì t ế

K (
x)

tr
ét

ề ệ ủ ị ý ó tể ợ t ở

b ồ t

> 0 s ớ ọ y Rn ,
dm (f + iS )K (
x; y) y

m

.

ợ t ở

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➢➡♥❣ ➤➢➡♥❣✳
❇æ ➤Ò ✶✳✷✳✶
●✐➯ sö

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❝ ✭✐✮ ✈➭ ✭✐✐✮ ❤✐Ó♥ ♥❤✐➟♥ ➤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, µ > 0. ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ ▼✱ 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.

ị ĩ ủ tồ t
tồ t

() (0, ) s ớ ọ t B (a, ()) ,

v (t) B (b, ) ớ
g (t, v (t)) L + /2.

ọ

t B (a, ()) s ớ ọ v B (b, ) ,
h (t, v) M + /2.

ó

t B (a, à) , v (t) B (b, à) ,
g (t, v (t)) + h (t, v (t)) L + M + .

ì

à ợ ọ tù ý s r
lim inf (g + h) (t, v) L+M +.
(t,v)(a,b)



q t tí t t s ệ q
trự tế ủ ổ ề
ệ q

sử

f1 , f2 : Rn R {+} ữ t x, sử fiK (
x; y) > , i =

1, 2. sử m 1 ột số ó



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) .

ú ý r

dm iA

x; ã) = iA(S,x) (ã) ,
S (

ớ

A := K, k, IK, Ik,

ó ề ệ ủ tố t


✾
❍Ö q✉➯ ✶✳✷✳✷
●✐➯ sö

[12]

x¯ ∈ S.

✭❛✮ ◆Õ✉

f K (¯
x; y) > 0

✈í✐ ♠ä✐

y ∈ K (S, x¯) \ {0}

t❤×

x¯

❧➭ ♠ét ❝ù❝ t✐Ó✉ ➤Þ❛

♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✶ ❝ñ❛ ✭✶✳✶✮✳
✭❜✮ ◆Õ✉

m>1

✈➭

dm f K (¯
x; y) > 0

t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

✈í✐ ♠ä✐

y ∈ K (¯
x) \ {0}

t❤×

x¯

❧➭ ♠ét ❝ù❝

m ❝ñ❛ ✭✶✳✶✮✳

❈❤ø♥❣ ♠✐♥❤

●✐➯ sö

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

❚❤❡♦ ➤Þ♥❤ ❧ý ✶✳✷✳✶ t❤×

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

♠✐♥❤ ✭❜✮ t➢➡♥❣ tù ✭❛✮✳

✷

❚➢➡♥❣ tù✱ ➤✐Ò✉ ❦✐Ö♥ ❝➬♥ tè✐ ➢✉ ❞➢í✐ ➤➞② ❞Ô ❞➭♥❣ ❝ã ➤➢î❝ tõ ➤Þ♥❤ ❧ý ✶✳✷✳✶ ✈➭ ❝➳❝
♣❤➬♥ ✭✐✐✮ ✈➭ ✭✐✐✐✮ ❝ñ❛ ❤Ö q✉➯ ✶✳✷✳✶✳
❍Ö q✉➯ ✶✳✷✳✸
●✐➯ sö

x¯ ❧➭ ♠ét ❝ù❝ t✐Ó✉ ➤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 ✶✳✷✳✶ ✈➭ ❤Ö 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✉ ❦✐Ö♥ ➤ñ tr♦♥❣ ❤Ö q✉➯ ✶✳✷✳✷ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❝➬♥ ❝❤♦ tè✐ ➢✉ ✈➭ ➤✐Ò✉
❦✐Ö♥ ❝➬♥ ❝ñ❛ ❤Ö q✉➯ ✶✳✷✳✸ ❦❤➠♥❣ ♣❤➯✐ ➤ñ ❝❤♦ tè✐ ➢✉✳ ❚✉② ♥❤✐➟♥ ❝ã ♥❤✐Ò✉ tr➢ê♥❣
❤î♣ tr♦♥❣ ➤ã ❤Ö q✉➯ ✶✳✷✳✷ ✈➭ ✶✳✷✳✸ ❝ã t❤Ó ❦Õt ❤î♣ ➤Ó ♥❤❐♥ ➤➢î❝ tÝ♥❤ ❝❤✃t ➤➷❝
tr➢♥❣ ❝ñ❛ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣

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

tr➢ê♥❣ ❤î♣ s❛✉✳
▼Ö♥❤ ➤Ò ✶✳✷✳✶
❑Ý ❤✐Ö✉

intS

❧➭ ♣❤➬♥ tr♦♥❣ ❝ñ❛

S✳

●✐➯ sö

x¯ ∈ intS ✳

❑❤✐ ➤ã✱

x¯

❧➭ ❝ù❝ t✐Ó✉ ➤Þ❛

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

f K (¯
x; y) > 0, ∀y ∈ Rn \ {0} .
❱í✐

m ❃ ✶✱ x¯ ❧➭ ♠ét ❝ù❝ t✐Ó✉ ➤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 .

➜✐Ò✉ ❦✐Ö♥ ➤ñ ➤➢î❝ s✉② r❛ trù❝ t✐Õ♣ tõ ❤Ö q✉➯ ✶✳✷✳✷ ✈➭ ➤✐Ò✉ ❦✐Ö♥ ❝➬♥ tõ ❤Ö q✉➯

✷

✶✳✷✳✸✭❛✮✳

▼Ö♥❤ ➤Ò ✶✳✷✳✷

●✐➯ sö

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

❑❤✐ ➤ã✱

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

✭✶✳✹✮

❈❤ø♥❣ ♠✐♥❤

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


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

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❛ ✶✳✷✳✶

✭❛✮ ●✐➯ sö

S ⊂ Rp

✈➭

x ∈ S ✳ ◆ã♥ t✐Õ♣ t✉②Õ♥ ❈❧❛r❦❡ ❝ñ❛ S

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

S ⊂ Rp

✈í✐

t➵✐

x ❧➭ t❐♣ ❤î♣

{xn } ⊂ S, ∃ {yn } → y

xn + tn yn ∈ S, ∀n} .

❧➭ t❐♣ ❤î♣

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

f : Rn → R ∪ {±∞} ❧➭ t❐♣ ❤î♣

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

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

t➵✐

x ∈ domf

❧➭ t❐♣ ❤î♣

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

▼Ö♥❤ ➤Ò ✶✳✷✳✸
●✐➯ sö

[13]

fi : Rn → R ∪ {+∞}✱ i❂✶✱✷✱

♥ö❛ ❧✐➟♥ tô❝ ❞➢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❤➟♠ ➤✐Ò✉ ❦✐Ö♥
✈✐ t➵✐

f1

❤♦➷❝

✭✶✳✼✮

f2

❧➭ tr➟♥ ❦❤➯

x¯.

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

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

●✐➯ sö

R ∪ {+∞} ♥ö❛ ❧✐➟♥ tô❝ ❞➢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♦♥❣ ♠Ö♥❤ ➤Ò ✶✳✷✳✸✳ ➜✐Ò✉ ❦✐Ö♥ ✭✶✳✺✮

t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ✭✶✳✽✮ tr♦♥❣ tr➢ê♥❣ ❤î♣ ♥➭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} .
➜Ó ❝❤ø♥❣ ♠✐♥❤ ✭❜✮ t❛ ➤➯♦ ♥❣➢î❝ ✈❛✐ trß ❝ñ❛ f1 ✈➭ f2 .

✷

❈❤ó♥❣ t❛ ❝ò♥❣ ❝ã t❤Ó sö ❞ô♥❣ ♠Ö♥❤ ➤Ò ✶✳✷✳✸ ➤Ó ♥❤❐♥ ➤➢î❝ ➤➷❝ tr➢♥❣ ❦❤➳❝
❝ñ❛ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét✳



ị ý
sử
sử

f

f : Rn R {+}

tr t

x

ử tụ ớ

K (S, x) = k (S, x)

S

ó ú

ó

x

ột ự tể ị

t ột ủ ế ỉ ế ú ữ ế


f K (
x; ã)

K (S, x) ồ tì x ự tể ị t ột ỉ tồ t

> 0 s
B (0, ) K f (
x) + (K (S, x))0 .



ứ

sử

f

tr t

x

ừ ị ý ệ ề

t ột ế ỉ ế ớ ọ

x

ự tể ị

y Rn \ {0} ,

0 < (f + iS )K (
x; y) = f k (
x; y) + iK
x; y)
S (
= f K (
x; y) + iK(S,x) .
ó

x

ự tể ị t ột ế ỉ ế ú

ờ sử

f K (
x; ã)



K (S; x)

ồ ề tr ét

ự tể ị t ột ế ỉ ế tồ t

x



> 0 s ớ ọ

y Rn ,
y (f + iS )K (
x; y) .



ở ì

(f + iS )K (

x; ã) = f K (
x; ã) + iK(S,x) (ã) ,
từ ệ ề t ó

(f + iS )K (
x; ã) ồ ì t ớ
B (0, ) K (f + iS ) (
x) ,

từ ệ ề t ó

K (f + is ) (
x) = k f (
x) + K iS (
x)
= K f (
x) + (K (S, x))0 .
ó x
ự tể ị t ột ế ỉ ế ú ế t
ề ệ K

(S, x) = k (S, x) ứ t tự




✶✹
◆❤❐♥ ①Ðt ✶✳✷✳✹

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

t➵✐

x¯

t❤× ❞♦♠f T

(¯
x; ·) = Rn

f

▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣

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

❧Ý ✶✳✷✳✷ ♥❤➢♥❣ ❦❤➠♥❣ ➤ó♥❣ ✈í✐ ♠Ö♥❤ ➤Ò ✶✳✷✳✷✳ ●✐➯ 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

♥ö❛ ❧✐➟♥ tô❝ ❞➢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✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ♠ét✳

C 1,1

✈➭ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷

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

m > 1, tr➢í❝

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

m = 2✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶

❍➭♠ sè

f : Rn → R ∪ {+∞}

❋rÐ❝❤❡t t➵✐

➤➢î❝ ❣ä✐ ❧➭

C 1,1

C 1,1

❇æ ➤Ò ✶✳✸✳✶

✭✐✮

❧➭

♥Õ✉

f

❦❤➯ ✈✐ ❧✐➟♥ tô❝

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

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

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, ã) = d2 f Ik (x, ã) .

ứ

ừ ị ĩ t ó

d2 f K (x; ã) d2 f IK (x; ã) .
sử

y Rn



d2 f K (x; y) r.
ể ứ t ỉ ỉ r r

d2 f IK (x; y) r.


> 0

ì

f

C 1,1

t

x

tồ t

L > 0, > 0

s ớ ọ

z

w B (x, ) ,
f (z) f () L z .
ọ

(0, min {, /4L ( y + )})
s

x + (0, ) B (y, ) B (x, ) .
v,

B (y, ) , t (0, ) . ị ý trị tr ì tồ t (0, 1)

s ớ

z := v + (1 ) , t ó

f (x + tv) f (x + t) = f (x + tz) , t (v )
= f (x + tz) f (x) , t (v ) ,
Lt2 z

v

Lt2 ( y + ) 2
t2 /2.
ó

(f (x + tv) f (x + t)) /t2 /2.
ờ

v B (y, ) , > 0 ì
d2 f K (x; y) r

ì

f (x) = 0,



tồ t

B (y, ) , t (0, min {, }) s
(f (x + t) f (x)) /t2 r + /2.

ó

(f (x + tv) f (x)) /t2 = (f (x + tv) f (x + t)) /t2 + (f (x + t) f (x)) /t2
/2 + r + /2 = r + .


v, , tù ý t ó
d2 f IK (x; y) r.


ó ợ ứ ứ t tự

ết q í ủ ú t ột tí t tr ủ ự tể ị
t t q t ọ

min {f (x) |gi (x) 0, i = 1, ..., m, hi (x) = 0, i = 1, ..., p} ,
tr ó

f, gi , hi





C 1,1 ú t t t t ớ ỗ x

ể ợ ủ t t t

I (x) := {i |gi (x) = 0} .
sử

i 0, i = 1, ...., m, ài R, i = 1, ..., p, r ợ ị

s

L (x) := f (x) +

i gi (x) +
i=1



p

m

ài hi (x).
i=1

I (x) t t
J (x) := {i I (x) |i > 0}



M (x) := {i I (x) |i = 0} .
r ề ệ tố t ợ

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 ❧❐♣ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤ñ ❝❤♦ ❝ù❝ t✐Ó✉ ❝❤➷t ❝✃♣ ❤❛✐✳ ❈➳❝ ➤✐Ò✉
❦✐Ö♥ ♥➭② tæ♥❣ q✉➳t ❤ã❛ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤ñ ❝✃♣ ❤❛✐ t❤➠♥❣ t❤➢ê♥❣ ❝❤♦ ❜➭✐ t♦➳♥ ✭✶✳✶✶✮
✈í✐ ❞÷ ❧✐Ö✉

C2

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

❋rÐ❝❤❡t✳ ❚r♦♥❣ s✉èt ♣❤➬♥ ♥➭②✱

x¯ ❧➭ ♠ét ➤✐Ó♠ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ❝ñ❛ ✭✶✳✶✶✮✳

➜Þ♥❤ ❧ý ✶✳✸✳✶
●✐➯ sö

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✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ✷ ❝ñ❛ ✭✶✳✶✶✮✳

❈❤ø♥❣ ♠✐♥❤

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

{xn }

x¯ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ❝ù❝ t✐Ó✉ ➤Þ❛

❝➳❝ ➤✐Ó♠ ❝❤✃♣ ♥❤❐♥ ➤➢î❝ ❝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❤Ó ❣✐➯ sö {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→∞

❱× ✈❐②✱ ✭✶✳✶✷✮ ❦❤➠♥❣ t❤á❛ ♠➲♥✳ ➜Þ♥❤ ❧ý ➤➢î❝ ❝❤ø♥❣ ♠✐♥❤✳

✷

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


✶✽
❇æ ➤Ò ✶✳✸✳✷

●✐➯ sö x
¯ ❧➭ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ ✭✶✳✶✶✮❀

λi ≥ 0, i = 1, ..., m, µi ∈

R, i = 1, ..., p. ◆Õ✉ λi gi (¯
x) = 0 ✈í✐ i = 1, ..., m t❤× x¯ ❧➭ ♠ét ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣
❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛

L tr➟♥ t❐♣ ❤î♣

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

❑Ý ❤✐Ö✉

S

❧➭ t❐♣ ❤î♣ r➭♥❣ ❜✉é❝ ❝ñ❛ ✭✶✳✶✶✮✳ ❚❤❡♦ ❣✐➯ 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✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ L tr➟♥ C.

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✷

●✐➯ sö gi , i

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

t➵✐

x¯. ❚❛ ♥ã✐ r➺♥❣ ➤✐Ò✉ ❦✐Ö♥ ❝❤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.
➜✐Ò✉ ❦✐Ö♥ ❙▼❋❈◗ ❧➭ ➤ñ ➤Ó s✉② r❛

k (C, x¯) = D (¯
x)

❝❤♦ t❐♣

C

tr♦♥❣ ❜æ ➤Ò

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



ị ý

x

sử

ự tể ị t ủ tr ó

rét t
ú t

x

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



C1

t

x.

f



sử

x ớ i 0, ài R tỏ L (
x) = 0, i gi (
x) = 0, i = 1, ..., m.

ó

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

ứ

ổ ề

L

ó ột ự tể ị t tr

C

t

x.

ì

t ệ q t ó

d2 LIK (
x; y) > 0, y k (C, x) \ {0} .
ề ệ ợ s r từ sự ệ k (C, x
)

= D (
x) ớ tết

ờ ú t sử ụ ị í ổ ề ể s r ột
tí t tr ủ ự tể ị t t

C 1,1 .

ị ý
sử

f, gi , i = 1, .., m, hi , i = 1, ..., p, C 1,1 t x. sử tỏ
i 0, ài R

L (
x) = 0, i gi (
x) = 0, i = 1, ...m.

t

x

ó

x ột ự tể ị t ủ ế ỉ ế

ớ

s



ú

ột ệ q ủ ị í tr

C 2 tố ề ệ ủ

tr ự tể ị t ế tỏ
tế

C 1,1 tố

ủ ột C 1,1 . 2 f
t

ự tr ệ ss s rộ

(x) sẽ í ệ ss ủ f

x.

ị ĩ

sử

f : Rn R C 1,1

t

x

Ef := x 2 f (
x)

tồ t

.

rét


✷✵
❍❡ss✐❛♥ s✉② ré♥❣ ❝ñ❛

f

t➵✐

x¯ ❧➭ t❐♣ ❝➳❝ n × n ♠❛ tr❐♥

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

✈➭

∇2 f (xj ) → A ,

✭✶✳✶✺✮

tr♦♥❣ ➤ã ❝♦♥✈ ❦Ý ❤✐Ö✉ ❜❛♦ ❧å✐✳
❈➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤ñ s❛✉ ➤➞② ❝❤♦ ❝ù❝ t✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ ❜➭✐ t♦➳♥
✭✶✳✶✶✮ ❝ã ➤➢î❝ ❞♦ ➤Þ♥❤ ❧Ý ✶ ❝ñ❛ ❬✼❪✳
➜Þ♥❤ ❧ý ✶✳✸✳✹ ✭❬✽❪✮
●✐➯ sö
✈➭

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✐Ó✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ ✭✶✳✶✶✮✳

❑❤➠♥❣ ❣✐è♥❣ ♥❤➢ ✭✶✳✶✷✮✱ ➤✐Ò✉ ❦✐Ö♥ ✭✶✳✶✻✮ ❦❤➠♥❣ ❝❤♦ ♠ét tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣
❝ñ❛ ❝ù❝ t✐Ó✉ ➤➵✐ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐✳ ➜✐Ò✉ ♥➭② ➤➢î❝ ♠✐♥❤ ❤ä❛ ❜➺♥❣ ✈Ý ❞ô s❛✉
➤➞②✳
❱Ý ❞ô ✶✳✸✳✶

❈❤♦

Z ❧➭ t❐♣ ❤î♣ ❝➳❝ 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✐Ó✉ ❝❤➷t ❝✃♣ ❤❛✐ ❝ñ❛ f

tr➟♥

R. ❚❛ ❝ã f ❧➭ C 1 tr➟♥ R ✈➭ ❦❤➯

✈✐ ❤❛✐ ❧➬♥ trõ r❛ t❐♣ ➤Õ♠ ➤➢î❝ s❛✉✿

Ω := {0} ∪ {π/2n |n ∈ Z } ,

✈í✐

f (¯
x) ≤ 10 + 16π, ∀x ∈
/ Ω.


✷✶
❍➭♠

f

❧➭

C 1,1

tr➟♥

R. ❚✉② ♥❤✐➟♥✱ ✈í✐ ♠ç✐ n,

f

3π/2n+2 = 2 − 12π < 0.

❉♦ ➤ã✱

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

D (0) \ {0} = ∅.