1. Trang chủ
  2. » Luận Văn - Báo Cáo

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

45 5 0

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

THÔNG TIN TÀI LIỆU

Thông tin cơ bản

Định dạng
Số trang 45
Dung lượng 498,87 KB

Nội dung

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 Trung tâm Học liệu - ĐHTN http://www.lrc-tnu.edu.vn/ ✐ ▲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❝ ❤✐Ö♥ ✈➭ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ọ ủ P ỗ t ❣✐➯ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬② ❣✐➳♦✱ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❝đ❛ ♠×♥❤✱ P●❙✳ ỗ t tì 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❤Ĩ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ t➳❝ ❣✐➯ r✃t ♠♦♥❣ ợ ỉ t tì ủ t ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ö♣✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺ ◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥ P❤➵♠ ◗✉ú♥❤ ❚r❛♥❣ ✐✐✐ ▼ô❝ ❧ô❝ ▲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♥ ➤đ 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♥ tố 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❝ tể ị t m ợ ị ĩ ❈r♦♠♠❡ ❬✷❪✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ➤➷❝ tr➢♥❣ ❝❤♦ ❝ù❝ t✐Ĩ✉ ❝❤➷t ❝✃♣ m ➤➢ỵ❝ t❤✐Õt ❧❐♣ ❜ë✐ ❆✉s❧❡♥❞❡r ❬✶❪✱ ❙t✉❞♥✐❛rs❦✐ ❬✶✷❪✱ ❉✳❱✳ ▲✉✉ ❬✶✵❪✱ ❲❛r❞ ❬✶✹❪✳ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ự tể ị t ữ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦✳ ❊✳❉✳ ❘❛❤♠♦ ✲ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮ ➤➲ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ➤➵♦ ❤➭♠ ❙t✉❞♥✐❛rs❦✐ ➤➢❛ r❛ ✶✾✽✻ ❝❤♦ ❤➭♠ ✈Ð❝t➡ ✈➭ ❞➱♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉✳ ➜➞② ❧➭ ➤Ị t➭✐ ➤➢ỵ❝ ♥❤✐Ị✉ 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 ➤✐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✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦✳ ❈❤➢➡♥❣ ✷✿ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐ ❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐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✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❲❛r❞ ❚r♦♥❣ ❝❤➢➡♥❣ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ➤✐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✐Ĩ✉ ❝❤➷t ❝✃♣ ❝❛♦✳ ❈➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ❧➭ ❝ñ❛ ❲❛r❞ ❬✶✹❪✳ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ❳Ðt ❜➭✐ t♦➳♥ tè✐ ➢✉ s❛✉✿ ✭✶✳✶✮ {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 > s❛♦ ❝❤♦ f (x) > f (¯ x) (∀x ∈ S ∩ B (¯ x, ε) \ {¯ x}) (1.1) ♥Õ✉ ✹ ✭❜✮ ❈❤♦ m ≥ ❧➭ ♠ét sè ♥❣✉②➟♥✳ ❚❛ ♥ã✐ r➺♥❣ x¯ ∈ S ❝❤➷t ❝✃♣ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ m ❝đ❛ (1.1) ♥Õ✉ tå♥ t➵✐ ε > 0✱ β > 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 ❝✃♣ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ j ✈í✐ ♠ä✐ m✱ t❤× ♥ã ❝ị♥❣ j > m✳ ✭❜✮ ❘â r➭♥❣ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❜✃t ❦ú ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t✳ ❚✉② ♥❤✐➟♥✱ ❦❤➠♥❣ ỗ ự tể ị t ột ự 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 = ❧➭ ♠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❛♦ ỗ S Rp x Rp , A (S, x) ột ó ó tể rỗ 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 ứ ó tế ợ ị ĩ K (S, x) := y ∃ (tn , yn ) → 0+ , y s❛♦ ❝❤♦ x + tn yn ∈ S, n ; ó tế tế rss ợ ị ĩ ❜ë✐ 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 ❝ñ❛ y ❧➭ ❡♣✐f f A✲ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ủ f t x t ợ ị ĩ f A (x; y) := inf {r |(y, r) ∈ A ( ❡♣✐f, (x, f (x)))} ❱í✐ ❝➳❝ ♥ã♥ tế tế ợ ị ĩ 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 ❝✃♣ ●✐➯ sö x¯ ∈ S m ✈➭ ❦ý ❤✐Ö✉ K (¯ x) := K (S, x¯) ∩ y f K (x; y) ≤ ; is ❧➭ ❤➭♠ ❝❤Ø ❝ñ❛ t❐♣ S:   0, ♥Õ✉ x ∈ S, is (x) =  +∞, ♥Õ✉ x ∈ / S ➜Þ♥❤ ❧ý ✶✳✷✳✶ ❈❤♦ ✭❛✮ [12] m > 1✱ ❝➳❝ ♣❤➳t ❜✐Ó✉ 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 ✭ˆ b✮ ❚å♥ t➵✐ β > s❛♦ ❝❤♦✱ ✈í✐ ♠ä✐ y ∈ Rn , dm (f + iS )K (¯ x; y) ≥ β y m ➤➢ỵ❝ t❤❛② ❜ë✐ K (S, x¯) ✷✼ ➤➢➡♥❣ ❧➭ t❤❡♦ ♥ã♥ ❞➢➡♥❣ Rp+ := [0, ∞)p ▼➷❝ ❞ï ë ➤➞② f ỉ trị t ữ í tết trì tr trớ ữ í tr ✈✐Ư❝ ①Ðt ♠ét sè ❧♦➵✐ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝đ❛ ❈❤ó♥❣ t❛ ❦Ý ❤✐Ư✉ S✳ N (x) ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ ❧➞♥ ❝❐♥ ❝đ❛ x ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✸ ●✐➯ sư ✈➭ ❤➭♠ ❝❤Ø ✈❡❝t➡ ❝đ❛ f ✭❬✺❪✮ m ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ✈➭ x¯ ∈ S ✭❛✮ ❚❛ ♥ã✐ r➺♥❣ x ¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ t m ệ ữ ệ ị ❝❤➷t ❝✃♣ ♥Õ✉ tå♥ t➵✐ m✮ ❝ñ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ ❦Ý ❤✐Ö✉ α > ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ m (f (x) + Rp+ ) ∩ B (f (¯ x) , α x − x¯ ✭❜✮ ❚❛ ♥ã✐ r➺♥❣ m ) = ∅, ∀x ∈ S ∩ U \ {¯ x} ✭✷✳✶✾✮ x¯ ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ s✐➟✉ ❝❤➷t ❝✃♣ m❀ s✉♣❡r✲str✐❝t ❧♦❝❛❧ P❛r❡t♦ ♠✐♥✐♠✐③❡r ♦❢ ♦r❞❡r ❝✃♣ x¯ ∈ StrL (m, f, S) m ệ ữ ệ ị s ❝❤➷t ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ ❦Ý ❤✐Ư✉ x¯ ∈ SStrL (m, f, S) ♥Õ✉ tå♥ t➵✐ α > 0✱ U ∈ N (¯ x) , ✈➭ t➵✐ ♥❤✐Ò✉ ♥❤✃t p ♥ã♥ ♠ë Ai ✭ ❦❤➠♥❣ ❝❤ø❛ ✵✮ i ∈ I ⊂ I, s❛♦ ❝❤♦ {Vi := x¯ + Ai : i ∈ I } ❧➭ ♠ét ♣❤ñ ❝ñ❛ S ∩ U \ {¯ x} ✈➭ m fi (x) > fi (¯ x) + α x − x¯ , ∀x ∈ S ∩ U ∩ Vi ✭✷✳✷✵✮ ▼Ư♥❤ ➤Ị ✷✳✶✳✺ ✭❬✹❪✮ x¯ ∈ ❙tr▲(m, f, S) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ η ∈ intRp+ f (x) tø❝ ❧➭✱ ✈í✐ ỗ f ( x) + x x m ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ , ∀x ∈ S ∩ U \ {¯ x} , ✭✷✳✷✶✮ x ∈ S ∩ U \ {¯ x} , ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❦❤➠♥❣ ➤ó♥❣ fi (x) fi (¯ x) + ηi x − x¯ fj (x) < fj (¯ x) + ηj x − x¯ m , m , ∀i ∈ I, ✈í✐ j ♥➭♦ ➤ã ∈ I ▼ë ré♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tõ ❬✶✷❪ ❝❤♦ ❝➳❝ ❤➭♠ ✈❡❝t➡✱ ❜➞② ❣✐ê ❝❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❝➳❝ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣ ❞ m m✿ f (¯ x + tv) − f (¯ x) , ,y) tm f (¯ x; y) := lim inf + (t,v)→(0 ✭✷✳✷✷✮ ✷✽ f (¯ x + tv) − f (¯ x) d¯m f (¯ x; y) := lim sup , tm (t,v)→(0+ ,y) ✭✷✳✷✸✮ tr ó ợ ị ĩ ị ♥❣❤Ü❛ ✷✳✶✳✷✳ ❈❤Ý♥❤ ①➳❝ ❤➡♥✱ t❛ ❝ã ❞ m f (¯ x; y) := sup inf t∈(0,δ) δ>0 f (¯ x + tv) − f (¯ x) , tm ✭✷✳✷✹✮ f (¯ x + tv) − f (¯ x) tm ✭✷✳✷✺✮ v∈B(y,δ) d¯m f (¯ x; y) := inf sup δ>0 t∈(0,δ) v∈B(y,δ) ➳♣ ❞ơ♥❣ ♠Ư♥❤ ➤Ị ✷✳✶✳✸✱ t❛ ❝ã f (¯ x; y) = (❞m f1 (¯ x; y) , , ❞m fp (¯ x; y)) , ✭✷✳✷✻✮ x; y) , , d¯m fp (¯ x; y) , d¯m f (x; y) = d¯m f1 (¯ ✭✷✳✷✼✮ ❞ m tr♦♥❣ ➤ã ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ✈Õ ♣❤➯✐ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ ♥❤➢ tr♦♥❣ ❬✶✶❪✳ ❈❤ó♥❣ ❝ã t❤Ĩ ❝ã ❣✐➳ trÞ ✈➠ ❤➵♥✱ ✈➭ ❞♦ ➤ã ❝➳❝ ➤➵♦ ❤➭♠ ✭✷✳✷✷✮✲✭✷✳✷✸✮ ♥ã✐ ❝❤✉♥❣ t❤✉é❝ ❱í✐ ¯ p✳ R ¯ (¯ m = 1, t❛ ❦ý ❦✐Ö✉ ❞f (¯ x; y) ✈➭ df x; y) t❤❛② ❝❤♦ ❞1 f (¯ x; y) ✈➭ d¯1 f (¯ x; y) ❚❛ ❦ý ❤✐Ö✉ dm f (¯ x; y) := lim+ (t,v)→(0 ❦❤✐ ❣✐í✐ ❤➵♥ ♥➭② tå♥ t➵✐ ❤÷✉ ❤➵♥ tr♦♥❣ t➢➡♥❣ tù ❝ã t❤Ĩ sư ❞ơ♥❣ ❝❤♦ ❤➭♠ ❑ý ❤✐Ư✉ g f (¯ x + tv) − f (¯ x) , ,y) tm ✭✷✳✷✽✮ ¯ p ✭tø❝ ❧➭✱ dm f (¯ R x; y) ∈ Rp ✮✳ ❈➳❝ ❦ý ❤✐Ö✉ ♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ t➢➡♥❣ ø♥❣ tr♦♥❣ K (S, x¯) ❧➭ ♥ã♥ t✐Õ♣ ❧✐➟♥ ❝ñ❛ S t➵✐ Y x¯ : K (S, x¯) := y ∈ X : ∃ (tn , yn ) → 0+ , y s❛♦ ❝❤♦ x + tn yn ∈ S, ∀n ✭✷✳✷✾✮ ➜➷t x; y) Kf (S, x¯) := K (S, x¯) ∩ {y ∈ X : ❞f (¯ 0} p = K (S, x¯) ∩ {y ∈ X : ❞fi (¯ x; y) 0} ✭✷✳✸✵✮ i=1 ❱× K (S, x¯) ❧➭ ♠ét ♥ã♥ ➤ã♥❣ ✈➭ ỗ số fi ( x; Ã) t t ❞➢➡♥❣ ✈➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✱ ♥➟♥ t❐♣ ❤ỵ♣ ✭✷✳✸✵✮ ❝ị♥❣ ❧➭ ♠ét ♥ã♥ ➤ã♥❣ ✭❝❤ø❛ ✵✮✳ ❚❛ sÏ sư ❞ơ♥❣ ❤➭♠ ❝❤Ø ✈Ð❝t➡ ❝đ❛ t❐♣ S✿ ¯p ∆ (x |S ) := (δ (x |S ) , , δ (x |S )) ∈ R ✈í✐ x ∈ X, ✭✷✳✸✶✮ ✷✾ tr♦♥❣ ➤ã   ♥Õ✉ x ∈ S, δ (x |S ) :=  ∞ ♥Õ✉ x ∈ / S ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ ¯p fS : X → R f S = f1S , , fpS , ✷✳✷ f S := f + ∆ (· |S ) ❑❤✐ ➤ã✱ ♥❤➢ s❛✉✿ S tr♦♥❣ ➤ã fi ✭✷✳✸✷✮ := fi + δ (· |S ) , i ∈ I ✭✷✳✸✸✮ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ➜Ĩ ♣❤➳t ❜✐Ĩ✉ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉✱ t❛ sÏ ❝➬♥ ➤Õ♥ ♥ã♥ ❞➢➡♥❣ ♠ë ré♥❣ ¯ p+ := R [0, ∞]p ➜Þ♥❤ ❧ý ✷✳✷✳✶ ●✐➯ sö x¯ ∈ ❙tr▲(m, f, S) ✭❛✮ ●✐➯ sö D = ∅ ✈➭ dg (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ❑❤✐ ➤ã tå♥ t➵✐ β > s❛♦ ❝❤♦ m d¯m f (¯ x; y) ∈ / B (0, β y ¯ p+ , )−R ∀y ∈ K (C, x¯) ∩ {u ∈ X : dg (¯ x; u) ∈ −intD} ✭❜✮ ●✐➯ sö Y = Rq ✈➭ ✭✷✳✸✹✮ D = Rq+ ❑❤✐ ➤ã tå♥ t➵✐ β > s❛♦ ❝❤♦ m d¯m f (¯ x; y) ∈ / B (0, β y ¯ p+ , )−R ¯ (¯ ∀y ∈ K (C, x¯) ∩ u ∈ X : dg x; u) < ✭✷✳✸✺✮ ❈❤ø♥❣ ♠✐♥❤ ✭❛✮ ✭❙ö ❞ơ♥❣ ý t➢ë♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ✸✳✶✭❛✮⇒ ✭❜✮ tr♦♥❣ ❬✼❪✮✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ tå♥ t➵✐ α > ✈➭ U ∈ N (¯ x) s❛♦ ❝❤♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✾✮ ➤ó♥❣✳ ➜✐Ị✉ ➤ã t➢➡♥❣ ➤➢➡♥❣ ✈í✐ f (x) − f (¯ x) ∈ / B 0, α x − x¯ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ ✭✷✳✸✹✮ ➤ó♥❣ ✈í✐ ❑❤✐ ➤ã tå♥ t➵✐ m ¯ p+ , ∀x ∈ S ∩ U \ {x} −R β = α/2m y ∈ K (C, x¯) , u ∈ B (0, β y ✭✷✳✸✻✮ ●✐➯ sư r➺♥❣ ➤✐Ị✉ ➤ã ❦❤➠♥❣ ➤ó♥❣✳ m ¯ p+ ) ✈➭ z ∈ R dg (¯ x; y) ∈ −intD s❛♦ ❝❤♦ ✭✷✳✸✼✮ ✸✵ ✈➭ d¯m f (¯ x; y) = u − z ✭❈❤ó ý r➺♥❣ d¯m f (¯ x; y) ❝ã t❤Ó ❝ã t❤➭♥❤ ♣❤➬♥ ❜➺♥❣ −∞✮✳ ▲✃② ε > s❛♦ ❝❤♦ u + εe ∈ B (0, β y tr♦♥❣ ➤ã ✭✷✳✸✽✮ u m ✭✷✳✸✾✮ ), e := (1, , 1)T ∈ Rp ❑❤✐ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✽✮ ❦Ð♦ t❤❡♦ d¯m f (¯ x; y) < u + εe ❧➭ ♠ét ♥ã♥ ✈➭ ✭✷✳✹✵✮ ❱× D ➤ã y = ✭❝➬♥ ❧➢✉ ý r➺♥❣ dg (¯ x; 0) = ❞♦ ➤Þ♥❤ ♥❣❤Ü❛ ✭✷✳✷✽✮ ❜ë✐ ✈× v = y ❧➭ ♠ét ❝➳❝❤ ❝❤ä♥ ❝đ❛ v ✮✳ D=Y ♥➟♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮ ❦Ð♦ t❤❡♦ dg (¯ x; y) = 0, ✈➭ ❞♦ ❉♦ ➤ã✱ t❤❡♦ ♠Ư♥❤ ➤Ị ✷✳✶✳✹ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮✱ t❛ t×♠ ➤➢ỵ❝ δ > s❛♦ ❝❤♦ δ ✭✷✳✹✶✮ y /2, ✭✷✳✹✷✮ x¯ + (0, δ) B (y, δ) ⊂ U, f (¯ x + tv) − f (¯ x) < u + εe, ∀t ∈ (0, δ) ✈➭ ∀v ∈ B (y, δ) tm g (¯ x + tv) − g (¯ x) ∈ −D, ∀t ∈ (0, δ) ✈➭ ∀v ∈ B (y, δ) t ❱× y ∈ K (C, x ¯) ♥➟♥ g (¯ x + tv) ∈ g (¯ x) − D ⊂ −D ✈í✐ ♠ä✐ D t❛ s✉② r❛ t ∈ (0, δ) ❚õ ✭✷✳✹✶✮ ✲ ✭✷✳✹✹✮ ✈➭ ✭✷✳✹✻✮ t❛ s✉② r❛ tå♥ t➵✐ ✭✷✳✹✹✮ ✭✷✳✹✺✮ (¯ x + (0, δ) B (y, δ)) ∩ C = ∅ ❍➡♥ ♥÷❛✱ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✹✮ ✈➭ tÝ♥❤ ❧å✐ ❝ñ❛ ✭✷✳✹✸✮ ✈➭ v ∈ B (y, δ) ✭✷✳✹✻✮ λ ∈ (0, δ) ✈➭ ω ∈ B (y, δ) s❛♦ ❝❤♦ x¯ + λω ∈ C ∩ U \ {x} , ✭✷✳✹✼✮ f (¯ x + λω) − f (¯ x) < u + εe, λm ✭✷✳✹✽✮ ✸✶ ✭✷✳✹✾✮ g (¯ x + λω) ∈ −D ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✾✮ ✈➭ ✭✷✳✹✽✮ ❦Ð♦ t❤❡♦ f (¯ x + λω) − f (¯ x) ∈ B (0, β y λm m ) − RP+ , ❤♦➷❝ t➢➡♥❣ ➤➢➡♥❣ f (¯ x + λω) − f (¯ x) ∈ B (0, β λy m ) − RP+ ✭✷✳✺✵✮ ✭❈❤ó ý r➺♥❣ f (¯ x + λω) − f (¯ x) ❦❤➠♥❣ t❤Ĩ ♥❤❐♥ ❣✐➳ trÞ ✈➠ ❤➵♥ ❜ë✐ ✈× f ❇➞② ❣✐ê✱ ❧✃② x := x¯ + λω, tõ ✭✷✳✹✼✮✱ ✭✷✳✹✾✮ ✈➭ ✭✷✳✺✵✮ t❛ ❝ã x ∈ C ∩ U \ {¯ x} , g (x) ∈ −D, f (x) − f (¯ x) ∈ B (0, β λy m ❤÷✉ ❤➵♥✮✳ ) − RP+ ✭✷✳✺✶✮ ❚õ ✭✷✳✹✶✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ω−y x4 , ✈➭ ❞♦ ➤ã (u1 , u2 ) = u21 + u22 > x2 xm ❚õ ➤ã s✉② r❛ x3 , x2 + R2+ ∩ B ((0, 0) , αxm ) = ∅ ❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝ã ✈í✐ dg (0; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y∈R ✈➭ dg (0; y) = −y < y > ❍➡♥ ♥÷❛✱ K (C, x¯) = R ❙ư ❞ơ♥❣ ✭✷✳✷✼✮✱ ✈í✐ y > 0, t❛ ❝ã     0, y , ♥Õ✉ m = 2, d¯m f (0; y) = y , ∞ , ♥Õ✉ m = 3,    (∞, ) , ế m > ó ỗ β ∈ (0, 1) , ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✺✮ ➤ó♥❣✱ tø❝ ❧➭ ¯ 2+ d¯m f (0; y) ∈ / B (0, βy m ) − R ✈í✐ ♠ä✐ ✈í✐ m ✈➭ y > ▼è✐ q✉❛♥ ❤Ư ♥➭② ➤ó♥❣ ✈í✐ m = ❜ë✐ ✈× β < 1, ✈➭ ✈× d¯m f (0; y) ❝❤ø❛ ❝➳❝ t❤➭♥❤ ♣❤➬♥ , ò t ủ m ỗ t tr♦♥❣ ¯ 2+ B (0, βy m ) − R ❤♦➷❝ ❤÷✉ ❤➵♥ ❤♦➷❝ ❜➺♥❣ −∞ ✸✸ ❱Ý ❞ơ ✷✳✷✳✷ ❈❤♦ f = (f1 , f2 ) : R2 → R2 ✈➭ g : R2 → R ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ f (x1 , x2 ) := x21 + x2 , x21 − x2 , g (x1 , x2 ) := − |x2 | ❳Ðt ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ tr♦♥❣ ➤ã S := (x1 , x2 ) ∈ R2 : −g (x1 , x2 ) = |x2 | ∈ R+ = R2 ë ➤➞② D ✭ := R+ C := R2 ✮✳ ✈➭ ➜✐Ó♠ x¯ = (0, 0) ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❤❛✐✳ ➜Ĩ t❤✃② ➤✐Ị✉ ♥➭②✱ t❛ ❧✃② ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ α = 1/2 ✈➭ U := B (¯ x, 1) ❑❤✐ ➤ã✱ S ∩ U \ {¯ x} = R2 ∩ U \ {¯ x} = (x1 , x2 ) : < x21 + x22 < ❇➞② ❣✐ê✱ ❧✃② (x1 , x2 ) ∈ S ∩ U \ {¯ x} ✈➭ (h1 , h2 ) ∈ R2+ ; t❛ ①Ðt trờ ợ ế x2 < tì x2 f (x1 , x2 ) + (h1 , h2 ) > ✭✐✐✮ ◆Õ✉ x22 ❉♦ ➤ã✱ x1 + x22 = α (x1 , x2 ) − x¯ ; −1 < x2 < t❤× −x2 f (x1 , x2 ) + (h1 , h2 ) > x21 + x22 |f1 (x1 , x2 ) + h1 | = x21 + x2 + h1 ✭✷✳✺✹✮ x22 ❉♦ ➤ã✱ x21 + x22 |f2 (x1 , x2 ) + h2 | = x21 − x2 + h2 x1 + x22 = α (x1 , x2 ) − x¯ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✹✮ ✈➭ ✭✷✳✺✺✮ ❦Ð♦ t❤❡♦ ✭✷✳✶✾✮ ➤ó♥❣ ✈í✐ ❈❤ó♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✺✮✳ ❚❛ ❝ã dg (¯ x; (y1 , y2 )) = − |y2 | < ✈í✐ y2 = ✭✷✳✺✺✮ m = K (C, x¯) = R2 ▲✃② ✈❡❝t➡ ❜✃t ❦ú ✈➭ y = (y1 , y2 ) y2 = ❑❤✐ ➤ã✱ f1 (tv1 , tv2 ) t2 v12 + tv2 ¯ d f1 (¯ x; (y1 , y2 )) = lim sup = lim sup t2 t2 (t,v)→(0+ ,y) (t,v)→(0+ ,y)   ∞, ♥Õ✉ y > 0, v2 2 = lim sup v1 + =  −∞, ♥Õ✉ y2 < 0, t (t,v)→(0+ ,y) ✈í✐ ✸✹ ✈➭ f2 (tv1 , tv2 ) t2 v12 − tv2 ¯ d f2 (¯ x; (y1 , y2 )) = lim sup = lim sup t2 t2 (t,v)→(0+ ,y) (t,v)→(0+ ,y)   −∞, ♥Õ✉ y > 0, v2 2 = lim sup v1 − =  ∞, ♥Õ✉ y2 < t (t,v)→(0+ ,y) ❉♦ ➤ã✱   (∞, −∞) , ♥Õ✉ y > 0, 2 ¯ d f (¯ x; (y1 , y2 )) =  (−∞, ∞) , ♥Õ✉ y2 < 0, tø❝ ❧➭ ✭✷✳✸✺✮ ➤ó♥❣ ❜ë✐ ✈× ❦❤➠♥❣ ❝ã ✈❡❝t➡ ♥➭♦ tr♦♥❣ ❝ã t❤➭♥❤ ♣❤➬♥ ❜➺♥❣ ✷✳✸ B (0, β y m ¯ p+ )−R ❝ã t❤Ĩ ∞ ➜✐Ị✉ ❦✐Ư♥ ➤đ tè✐ ➢✉ ➜Þ♥❤ ❧ý ➤➬✉ t✐➟♥ ✈Ị ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤➢ỵ❝ ♣❤➳t ❜✐Ĩ✉ ❝❤♦ t❐♣ ❤ỵ♣ r➭♥❣ ❜✉é❝ S tï② ý ✭❝❤ó♥❣ t❛ ❣✐➯ sư ❦❤➠♥❣ ❝ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✽✮✮✳ ➜Þ♥❤ ❧ý ✷✳✸✳✶ ●✐➯ sö ❞✐♠X ✭❛✮ ◆Õ✉ < ∞✱ ✈➭ x¯ ❧➭ ♠ét ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✳ m > ✈➭ ❞ t❤× ¯ p+ , ∀y ∈ Kf (S, x¯) \ {0} f (¯ x; y) ∈ / −R m S ✭✷✳✺✻✮ x¯ ∈ ❙❙tr▲(m, f, S) ✭❜✮ ◆Õ✉ ❞f t❤× S ¯ p+ , ∀y ∈ K (S, x¯) \ {0} (¯ x; y) ∈ / −R ✭✷✳✺✼✮ x¯ ∈ ❙❙tr▲(1, f, S) ❈❤ø♥❣ ♠✐♥❤ ✭❛✮ ❚õ ✭✷✳✷✻✮ ✈➭ ✭✷✳✸✸✮✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✻✮ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ max ❞m fiS (¯ x; y) > 0, ∀y ∈ Kf (S, x¯) \ {0} i∈I ✭✷✳✺✽✮ ✸✺ ❚õ s✉② ❧✉❐♥ ✭❝✮⇒ ✭❛✮ tr♦♥❣ ❬✻✱ ➤Þ♥❤ ❧Ý ✸✳✶❪ t❛ s✉② r❛ x¯ ∈ SStrL (m, f, S) ✷ ❈❤ø♥❣ ♠✐♥❤ ♣❤➬♥ ✭❜✮ t➢➡♥❣ tù✳ ❚r♦♥❣ ➤Þ♥❤ ❧Ý t✐Õ♣ t❤❡♦ t❛ sư ❞ơ♥❣ ❦Ý ❤✐Ư✉ s❛✉ ➤➞② ❝❤♦ ❜❛♦ ➤ã♥❣ ❝ñ❛ ♥ã♥ s✐♥❤ ❜ë✐ D + g (¯ x)✿ ✭✷✳✺✾✮ Dg(¯x) := cl cone (D + g (¯ x)) ❱× D ❧➭ t❐♣ ❧å✐ ♥➟♥ Dg(¯x) ❧➭ ♠ét ♥ã♥ ❧å✐ ➤ã♥❣✳ ➜Þ♥❤ ❧ý ✷✳✸✳✷ ●✐➯ sư ❞✐♠X < ∞, ✭✷✳✶✽✮✳ ●✐➯ sö dg (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ◆Õ✉ ❞ m ✈➭ x¯ ❧➭ ♠ét ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭✷✳✶✼✮ ✲ ¯ p+ , ∀y ∈ K (C, x¯) ∩ u ∈ X : dg (¯ f (¯ x; y) ∈ / −R x; u) ∈ −Dg(¯x) \ {0} ✭✷✳✻✵✮ t❤× x¯ ∈ ❙tr▲(m, f, S) ❈❤ø♥❣ ♠✐♥❤ ✭❙ư ❞ơ♥❣ ý t➢ë♥❣ ❝đ❛ ➤Þ♥❤ ❧Ý ✹✳✶ tr♦♥❣ ❬✶✵❪✮✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ x ¯ ∈ / StrL (m, f, S) ❑❤✐ ➤ã✱ tõ ➤Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✸✭❛✮ s✉② r ỗ số xn S B (¯ x, 1/n) \ {¯ x} ✈➭ dn = (dn,1 , , dn,p ) ∈ Rp+ f (xn ) − f (¯ x) + dn ∈ B 0, xn − x¯ n n, tå♥ t➵✐ s❛♦ ❝❤♦ m ➜✐Ị✉ ♥➭② t➢➡♥❣ ➤➢➡♥❣ ✈í✐ dn f (xn ) − f (¯ x) m + xn − x¯ xn − x¯ m ∈ B 0, ❑❤➠♥❣ ♠✃t tÝ♥❤ ❝❤✃t tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sư tơ ➤Õ♥ ♠ét ✈❡❝t➡ xn = x¯ + tn ❇ë✐ ✈× v ✈í✐ ✈í✐ ♠ä✐ v n = ❉♦ ❑Ý ❤✐Ö✉ tn xn ∈ S ⊂ C = n ✭✷✳✻✶✮ := (xn − x¯) / xn − x¯ xn − x¯ ♥➟♥ t❛ ❝ã t❤× tn v ∈ K (C, x¯) → 0+ ❤é✐ ✈➭ ❞♦ ✭✷✳✷✾✮✳ dg (¯ x; v) tå♥ t➵✐✱ ❝❤♦ ♥➟♥ g (¯ x + tn ) − g (¯ x) n→∞ tn dg (¯ x; v) := lim ✭✷✳✻✷✮ ✸✻ ❍➡♥ ♥÷❛✱ g (¯ x + tn ) = g (xn ) ∈ −D ❉♦ ➤ã✱ g (¯ x + tn ) − g (¯ x) ∈ cone (−D − g (¯ x)) ⊂ −Dg(¯x) , ∀n tn ❚õ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✻✷✮ ✈➭ ✭✷✳✻✸✮✱ Dg(¯x) v ∈ K (C, x¯) , v = 1, ✭✷✳✻✸✮ ✈➭ tÝ♥❤ ➤ã♥❣ ❝ñ❛ t❛ ❝ã v ∈ K (C, x¯) ∩ u ∈ X : dg (¯ x; u) ∈ −Dg(¯x) \ {0} ✭✷✳✻✹✮ ❚õ ✭✷✳✻✶✮ s✉② r❛ fi (xn ) − fi (¯ x) dn,i + m xn − x¯ xn − x¯ lim n→∞ = 0, ∀i ∈ I m ❇✐Ó✉ t❤ø❝ tr➟♥ ❝ã t❤Ó ✈✐Õt ❧➵✐ ♥❤➢ s❛✉ lim n→∞ fi (¯ x + tn ) − fi (¯ x) dn,i + m tm tn n ❚õ s r ỗ m fi (¯ x; v) = 0, ∀i ∈ I ✭✷✳✻✺✮ i, t❛ ❝ã lim inf n→∞ fi (¯ x + tn ) − fi (¯ x) tm n fi (¯ x + tn ) − fi (¯ x) dn,i dn,i + + lim inf − n→∞ n→∞ tm tm tm n n n dn,i = lim inf − m n→∞ tn p ➜✐Ị✉ ❦✐Ư♥ dn ∈ R+ ❦Ð♦ t❤❡♦ −dn,i /tm ❚õ ✭✷✳✻✻✮ s✉② r❛ n = lim ❞ m fi (¯ x; v) ✭✷✳✻✻✮ (∀i) ❉♦ ➤ã✱ ❞ m ¯ p+ f (¯ x; v) ∈ −R ✷ ❉♦ ✭✷✳✻✹✮✱ ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ✭✷✳✻✵✮✳ ❱Ý ❞ơ ✷✳✸✳✶ ❳Ðt ❜➭✐ t♦➳♥ ✭✷✳✶✼✮✱ tr♦♥❣ ➤ã f = (f1 , f2 ) : R → R2   x sin , x − sin x x f (x) :=  (0, 0) , , ợ ị ế ế x = 0, x = 0, ✸✼ ✈➭ t❐♣ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ➤➢ỵ❝ ❝❤♦ ❜ë✐✿ S := {x ∈ R : −g (x) = |x| ∈ R+ , x ∈ R+ } = R+ ë ➤➞② D := R+, C ✭ sử := R+ g:RR ợ ị g (x) := − |x|✳✮ x¯ = ❱í✐ ♠ä✐ y ∈ R, t❛ ❝ã dg (0; y) = − |y| ✈➭ dg (0; y) ∈ −Dg(0) = −R+ ❘â r➭♥❣ K (R+ , 0) = R+ ❉♦ ➤ã✱ K (R+ , 0) ∩ u ∈ R : dg (¯ x; u) ∈ −Dg(0) = R+ ❱í✐ ♠ä✐ y ∈ R+ , t❛ ❝ã ¯ (0; y) = y, ❞f1 (0; y) = −y, df ❞ f2 ¯ (0; y) = 3y (0; y) = y, df ❉♦ ➤ã✱ ❞f ¯ 2+ , ∀y ∈ R+ \ {0} (0; y) = (−y, y) ∈ / −R ❱× ✈❐② ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✻✵✮ t❤á❛ ♠➲♥✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ t❛ ❧✃② f (x) |f2 (x)| = x − sin ❉♦ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✾✮ ➤ó♥❣ ✈í✐ α = 1/2 t❤× > αx, ∀x ∈ S\ {0} = R+ \ {0} x m = ✈➭ U = R ➜✐Ò✉ ➤ã ❝ã ♥❣❤Ü❛ ❧➭ x¯ = ❧➭ ♠ét ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ❝❤➷t ✭t♦➭♥ ❝ơ❝✮ ❝✃♣ ♠ét✳ ▲➢✉ ý r➺♥❣ ✈Ý ❞ơ ♥➭② ❦❤➠♥❣ t❤Ĩ ❣✐➯✐ q✉②Õt ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤Þ♥❤ ❧Ý ✹✳✶ tr♦♥❣ ❬✶✵❪ ❜ë✐ ✈× ❞f ✈í✐ ♠ä✐ ✷✳✹ ¯ (0; y) (0; y) = df y = ❉♦ ➤ã df (0; ·) ❦❤➠♥❣ tå♥ t➵✐✳ ➜➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ Prt ị t ú t trì ột tí ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ m ❝❤♦ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ✭✷✳✶✼✮ ♠➭ ❦❤➠♥❣ ❝ã ❜✃t ❦× ❤➵♥ ❝❤Õ ❣× ✈Ị t❐♣ r➭♥❣ ❜✉é❝ S ♥❤➢ tr♦♥❣ ✭✷✳✶✽✮✳ ❑Õt q✉➯ ♥➭② t➢➡♥❣ tù ✈í✐ ❬✶✷✱ ➤Þ♥❤ ❧Ý ✸✳✶❪ ♥❤➢♥❣ ❜❛♦ ❣å♠ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ tï② ý✳ ➜Þ♥❤ ❧ý ✷✳✹✳✶ ●✐➯ sö ❞✐♠X < ∞, x¯ ∈ S, dm f (¯ x; y) tå♥ t➵✐ ✈í✐ ♠ä✐ y ∈ X ❑❤✐ ➤ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❞➢í✐ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✸✽ ✭❛✮ x¯ ∈ ❙tr▲(m, f, S) ✭❜✮ ❚å♥ t➵✐ β > s❛♦ ❝❤♦ dm f (¯ x; y) ∈ / B (0, β y ✭❝✮ m ¯ p+ , ∀y ∈ K (S, x¯) \ {0} )−R ✭✷✳✻✼✮ ¯ p+ , ∀y ∈ K (S, x¯) \ {0} dm f (¯ x; y) ∈ / −R ❈❤ø♥❣ ♠✐♥❤ ➜Ó ❝❤ø♥❣ ♠✐♥❤✭❛✮ C =S ⇒ ✈➭ ❦❤➠♥❣ ❝ã ✭❜✮✱ ❝❤ó♥❣ t❛ ❧➷♣ ❧➵✐ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➤Þ♥❤ ❧Ý ✷✳✷✳✶✭❛✮ ✈í✐ g ❈❤ó ý r➺♥❣ t❛ ♣❤➯✐ t❤➟♠ ❤➵♥ ❝❤Õ ➜✐Ị✉ ♥➭② ❝ã ➤➢ỵ❝ ❞♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮✳ ❚➢➡♥❣ tù ✭❝✮ ✷✳✸✳✷✳ ❈ß♥ s✉② ❧✉❐♥ ✭❜✮ ⇒ ✭❝✮ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳ ⇒ y =0 tr♦♥❣ ✭✷✳✻✼✮✳ ó ợ ị í ết ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝đ❛ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮ ✈➭ ❊✳❉✳ ❘❛❤♠♦ ✲ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✷✵✶✷✮✳ ❈➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❜❛♦ ❣å♠✿ • ❑❤➳✐ ♥✐Ư♠ A ✲ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝đ❛ ❲❛r❞❀ • ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ị t ủ r ã ệ t❤❡♦ ♣❤➢➡♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝✃♣ ❝❛♦ ❝❤♦ ❤➭♠ ✈❡❝t➡ ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐❀ • ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ❝✃♣ ❝❛♦ ❝đ❛ ❘❛❤♠♦ ✲ ❙t✉❞♥✐❛rs❦✐✳ ➜✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝✃♣ ❝❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤➷t ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ị t ề t ợ ♥❤✐Ị✉ t➳❝ ❣✐➯ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ✹✵ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❆✳ ❆✉s❧❡♥❞❡r ✭✶✾✽✹✮✱ ❙t❛❜✐❧✐t② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥♦♥❞✐❢✲ ❢❡r❡♥t✐❛❜❧❡ ❞❛t❛✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♥tr♦❧ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✷✷✱ ♣♣✳ ✷✸✾✲✷✺✹✳ ❬✷❪ ▲✳ ❈r♦♠♠❡ ✭✶✾✼✽✮✱ ❙tr♦♥❣ ✉♥✐q✉❡♥❡s✿ ❆ ❢❛r r❡❛❝❤✐♥❣ ❝r✐t❡r✐♦♥ ❢♦r t❤❡ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐✈❡ ♣r♦❝❡❞✉r❡s✱ ◆✉♠❡r✐s❝❤❡ ▼❛t❤❡♠❛t✐❦✱ ✈♦❧✳ ✷✾✱ ♣♣✳ ✶✼✾✲ ✶✾✸✳ ❬✸❪ ❆✳❱✳ ❋✐❛❝❝♦✱ ●✳P✳ ▼❝❈♦r♠✐❝❦ ✭✶✾✻✽✮✱ ◆♦♥❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✿ ❙❡q✉❡♥t✐❛❧ ✉♥❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ t❡❝❤♥✐q✉❡s✱ ❏♦❤♥ ❲✐❧❡②✱ ◆❡✇ ❨♦r❦✳ ❬✹❪ ❆✳ ●✉♣t❛✱ ❉✳ ❇❤❛t✐❛✱ ❆✳ ▼❡❤r❛ ✭✷✵✵✼✮✱ ❍✐❣❤❡r ♦r❞❡r ❡❢❢✐❝✐❡♥❝②✱ s❛❞❞❧❡ ♣♦✐♥t ♦♣t✐♠❛❧✐t②✱ ❛♥❞ ❞✉❛❧✐t② ❢♦r ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳ ✷✽ ✭✸✲✹✮✱ ✸✸✾✲✸✺✷✳ ❬✺❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✷✮✱ ❙tr✐❝t ❡❢❢✐❝✐❡♥❝② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✻✺✱ ✷✻✹✲✷✽✹✳ ❬✻❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✸✮✱ ❙tr✐❝t ♠✐♥✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♥♦♥❞✐❢❢❡r❡♥t✐❛❜❧❡ ♠✉❧t✐✲ ♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✶✻✱ ✾✾✲✶✶✻✳ ❬✼❪ ❇✳ ❏✐♠Ð♥❡③ ✭✷✵✵✾✮✱ ❱✳◆♦✈♦✱ ▼✳ ❙❛♠❛✱ ❙❝❛❧❛r✐③❛t✐♦♥ ❛♥❞ ♦♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s ❢♦r str✐❝t ♠✐♥✐♠✐③❡rs ✐♥ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ ❝♦♥t✐♥❣❡♥t ❡♣✐❞❡r✐✈❛t✐✈❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✺✷ ✼✽✽✲✼✾✽✳ ❬✽❪ ❉✳ ❑❧❛tt❡✱ ❑✳ ❚❛♠♠❡r ✭✶✾✽✽✮✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r s✉❢❢✐❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐✲ t✐♦♥s ❢♦r C 1,1 ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✶✾✱ ♣♣✳✶✻✾✲✶✼✾✳ ✹✶ ❬✾❪ ❏✳ ❑②♣❛r✐s✐s ✭✶✾✽✺✮✱ ❖♥ ✉♥✐q✉❡♥❡ss ♦❢ ❑✉❤♥✲❚✉❝❦❡r ♠✉❧t✐♣❧✐❡rs ✐♥ ♥♦♥❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣✱ ✈♦❧✳ ✸✷✱ ♣♣✳ ✷✹✷✲✷✹✻✳ ❬✶✵❪ ❉✳❱✳ ▲✉✉ ✭✷✵✵✽✮✱ ❍✐❣❤❡r✲♦r❞❡r ♥❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r str✐❝t P❛r❡t♦ ♠✐♥✐♠❛ ✐♥ t❡r♠s ♦❢ ❙t✉❞♥✐❛rs❦✐✬s ❞❡r✐✈❛t✐✈❡s✱ ❖♣t✐♠✐③❛t✐♦♥ ✺✼ ✭✹✮✱ ✺✾✸✲✻✵✺✳ ❬✶✶❪ ❊✳❉✳ ❘❛❤♠♦✱ ▼✳ ❙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❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✾✸✱ ✷✶✷✲✷✷✶✳ ❬✶✷❪ ▼✳ ❙t✉❞♥✐❛rs❦✐ ✭✶✾✽✻✮✱ ◆❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ✐s♦❧❛t❡❞ ❧♦❝❛❧ ♠✐♥✐♠❛ ♦❢ ♥♦♥s♠♦♦t❤ ❢✉♥❝t✐♦♥s✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✷✹✱ ✶✵✹✹✲✶✵✹✾✳ ❬✶✸❪ ❉✳❊✳ ❲❛r❞✱ ❏✳▼✳ ❇♦r✇❡✐♥ ✭✶✾✽✼✮✱ ◆♦♥s♠♦♦t❤ ❝❛❧❝✉❧✉s ✐♥ ❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♥tr♦❧ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✷✺✱ ♣♣✳ ✶✸✶✷✲✶✸✹✵✳ ❬✶✹❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✾✹✮✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ str✐❝t ❧♦❝❛❧ ♠✐♥✐♠❛ ❛♥❞ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ✇❡❛❦ s❤❛r♣ ♠✐♥✐♠❛✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✵✱ ✺✺✶✲✺✼✶✳ ❬✶✺❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✽✼✮✱ ■s♦t♦♥❡ t❛♥❣❡♥t ❝♦♥❡s ❛♥❞ ♥♦♥s♠♦♦t❤ ♦♣t✐♠✐③❛t✐♦♥✱ ❖♣✲ t✐♠✐③❛t✐♦♥✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✼✻✾✲✼✽✸✳ ❬✶✻❪ ❉✳❊✳ ❲❛r❞ ✭✶✾✽✽✮✱ ❚❤❡ q✉❛♥t✐❢✐❝❛t✐♦♥❛❧ t❛♥❣❡♥t ❝♦♥❡s✱ ❈❛♥❛❞✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✹✵✱ ♣♣✳ ✻✻✻✲✻✾✹✳

Ngày đăng: 24/03/2021, 17:41

Nguồn tham khảo

Tài liệu tham khảo Loại Chi tiết
[2] L. Cromme (1978), Strong uniquenes: A far reaching criterion for the con- vergence of iterative procedures, Numerische Mathematik, vol. 29, pp. 179- 193 Sách, tạp chí
Tiêu đề: Strong uniqueness: A far reaching criterion for the convergence of iterative procedures
Tác giả: L. Cromme
Nhà XB: Numerische Mathematik
Năm: 1978
[4] A. Gupta, D. Bhatia, A. Mehra (2007), Higher order efficiency, saddle point optimality, and duality for vector optimization problems, Numer. Funct.Anal. Optim. 28 (3-4), 339-352 Sách, tạp chí
Tiêu đề: Higher order efficiency, saddle point optimality, and duality for vector optimization problems
Tác giả: A. Gupta, D. Bhatia, A. Mehra
Nhà XB: Numerical Functional Analysis and Optimization
Năm: 2007
[5] B. JimÐnez (2002), Strict efficiency in vector optimization, J. Math. Anal.Appl. 265, 264-284 Sách, tạp chí
Tiêu đề: Strict efficiency in vector optimization
Tác giả: B. JimÐnez
Nhà XB: J. Math. Anal.Appl.
Năm: 2002
[7] B. JimÐnez (2009), V.Novo, M. Sama, Scalarization and optimality con- ditions for strict minimizers in multiobjective optimization via contingent epiderivatives, J. Math. Anal. Appl. 352 788-798 Sách, tạp chí
Tiêu đề: Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives
Tác giả: B. JimÐnez, V. Novo, M. Sama
Nhà XB: J. Math. Anal. Appl.
Năm: 2009
[8] D. Klatte, K. Tammer (1988), On second-order sufficient optimality condi- tions for C 1,1 optimization problems, Optimization, vol. 19, pp.169-179 Sách, tạp chí
Tiêu đề: On second-order sufficient optimality condi- tions for C 1,1 optimization problems
Tác giả: D. Klatte, K. Tammer
Nhà XB: Optimization
Năm: 1988
[10] D.V. Luu (2008), Higher-order necessary and sufficient conditions for strict Pareto minima in terms of Studniarski's derivatives, Optimization 57 (4), 593-605 Sách, tạp chí
Tiêu đề: Higher-order necessary and sufficient conditions for strict Pareto minima in terms of Studniarski's derivatives
Tác giả: D.V. Luu
Nhà XB: Optimization
Năm: 2008
[13] D.E. Ward, J.M. Borwein (1987), Nonsmooth calculus in finite dimensions, SIAM Journal on Control and Optimization, vol. 25, pp. 1312-1340 Sách, tạp chí
Tiêu đề: Nonsmooth calculus in finite dimensions
Tác giả: D.E. Ward, J.M. Borwein
Nhà XB: SIAM Journal on Control and Optimization
Năm: 1987
[1] A. Auslender (1984), Stability in mathematical programming with nondif- ferentiable data, SIAM Journal on Control and Optimization, vol. 22, pp.239-254 Khác
[3] A.V. Fiacco, G.P. McCormick (1968), Nonlinear programming: Sequential unconstrained minimization techniques, John Wiley, New York Khác
[6] B. JimÐnez (2003), Strict minimality conditions in nondifferentiable multi- objective programming, J. Optim. Theory Appl. 116, 99-116 Khác
[9] J. Kyparisis (1985), On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Mathematical Programming, vol. 32, pp. 242-246 Khác
[11] E.D. Rahmo, M. Studniarski (2012), Higher-order conditions for strict local Pareto minima in terms of generalized lower and upper directional deriva- tives, J. Math. Anal. Appl. 393, 212-221 Khác
[12] M. Studniarski (1986), Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim. 24, 1044-1049 Khác
[14] D.E. Ward (1994), Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl. 80, 551-571 Khác
[15] D.E. Ward (1987), Isotone tangent cones and nonsmooth optimization, Op- timization, vol. 18, pp. 769-783 Khác
[16] D.E. Ward (1988), The quantificational tangent cones, Canadian Journal of Mathematics, vol. 40, pp. 666-694 Khác

TÀI LIỆU CÙNG NGƯỜI DÙNG

TÀI LIỆU LIÊN QUAN