ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM TRẦN THỊ NHÀN ĐIỀU KIỆN CẦN VÀ ĐỦ CHO NGHIỆM HỮU HIỆU CỦA BÀI TOÁN TỐI ƯU ĐA MỤC TIÊU QUA DƯỚI VI PHÂN SUY RỘNG LUẬN VĂN THẠC SĨ TOÁN HỌC Thái Nguyên - Năm 2015 ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr✉♥❣ t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ❝➳❝ ➤Ị t➭✐ ❦❤➳❝✳ ❚➠✐ ❝ò♥❣ 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 ❧✉❐♥ ✈➝♥ ❚r➬♥ ❚❤Þ ◆❤➭♥ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ✈➭ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ❞➢í✐ sù ọ ủ P ỗ ◗✉❛ ➤➞②✱ t➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬② ❣✐➳♦✱ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ủ ì P ỗ 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✳ ❚➳❝ ❣✐➯ ❝ò♥❣ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ➤Õ♥ ❣✐❛ ➤×♥❤ ✈➭ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ❚♦➳♥ ❑✷✶❜✱ ➤➲ ➤é♥❣ ✈✐➟♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ▲✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ĩ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ t➳❝ rt ợ ỉ t tì ❝đ❛ ❝➳❝ t❤➬② ❝➠ ✈➭ ❜➵♥ ❜❒ ➤å♥❣ ♥❣❤✐Ư♣✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✺ ◆❣➢ê✐ ✈✐Õt ❧✉❐♥ ✈➝♥ ❚r➬♥ ❚❤Þ ◆❤➭♥ ✐✐✐ ▼ơ❝ ❧ơ❝ ▲ê✐ ❝❛♠ ➤♦❛♥ ✐ ▲ê✐ ❝➯♠ ➡♥ ✐✐ ▼ô❝ ❧ô❝ ✐✐✐ ▼ë ➤➬✉ ✶ ✶ ✸ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ✶✳✶ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❜ỉ trỵ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶✳ ❉➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ✶✳✶✳✷✳ ❈➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ ❈❧❛r❦❡✲❘♦❝❦❛❢❡❧❧❛r✱ ❈❧❛r❦❡✱ ▼✐❝❤❡❧✲P❡♥♦t ✶✳✶✳✸✳ ❉➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉②✱ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tè✐ t❤✐Ó✉ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ó✉ P❛r❡t♦ ②Õ✉ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ➜✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ✷✹ ✷✳✶ ➜✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ➜✐Ị✉ ❦✐Ư♥ ➤đ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ✳ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶ ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ❧✉❐♥ ✈➝♥ ◆➝♠ ✶✾✾✹✱ ❉❡♠②❛♥♦✈ ❬✺❪ ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝♦♠♣➝❝ ❧å✐✳ ❑❤➳✐ ♥✐Ư♠ ♥➭② ❧➭ ♠ét tỉ♥❣ q✉➳t ❤♦➳ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❧å✐ tr➟♥ ✈➭ ❧â♠ ❞➢í✐ ✭①❡♠ ❬✻❪✮✳ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ➤ã♥❣✱ ❦❤➠♥❣ ❧å✐ ✈➭ ❏❛❝♦❜✐❛♥ ①✃♣ ①Ø ➤➢ỵ❝ ➤Ị ①✉✃t ❜ë✐ ❏❡②❛❦✉♠❛r ✈➭ ▲✉❝ tr♦♥❣ ❬✾❪ ✈➭ ❬✶✵❪✳ ❑❤➳✐ ♥✐Ư♠ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❧➭ tỉ♥❣ q✉➳t ❤♦➳ ❝đ❛ ♠ét sè ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❞➢í✐ ✈✐ ♣❤➞♥ ➤➲ ❜✐Õt ❝đ❛ ❈❧❛r❦❡ ❬✹❪✱ ▼✐❝❤❡❧✲P❡♥♦t ❬✶✼❪✱ ▼♦r❞✉❦❤♦✈✐❝❤ ❬✶✽❪✳ ▼ét ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ❝đ❛ ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ ➤❛ ♠ơ❝ t✐➟✉ ữ ỉ ợ r ▲✉❝ ❬✶✷❪✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ❉✉tt❛✲ ❈❤❛♥❞r❛ ❬✼✱✽❪ ❝❤♦ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ ✈í✐ ❝➳❝ r➭♥❣ ❜✉é❝ ❜✃t ➤➻♥❣ t❤ø❝✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ✈➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ▲✉✉ ❬✶✺❪ ✈í✐ ❝➳❝ r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣✳ ❉ù❛ ✭✷✵✶✹✮ tr➟♥ ➤➲ ➤Þ♥❤ t❤✐Õt ❧Ý ❧❐♣ ▲❥✉st❡r♥✐❦ ❝➳❝ ➤✐Ị✉ ♠ë ❦✐Ư♥ ré♥❣ tè✐ ➢✉ ❝đ❛ ❝❤♦ ❏✐♠Ð♥❡③✲◆♦✈♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ✭✷✵✵✷✮✱ ②Õ✉ ❝đ❛ ❉✳❱✳▲✉✉ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ✭❝♦♥✈❡①✐❢✐❝❛t♦r✮✳ ➜➞② ❧➭ ➤Ị t➭✐ ➤❛♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ t➳❝ ❣✐➯ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ❈❤Ý♥❤ ✈× t❤Õ ❡♠ ❝❤ä♥ ➤Ị t➭✐ ✿ ➇➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ❝❤♦ ♥❣❤✐Ư♠ ❤÷✉ ❤✐Ư✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ q✉❛ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣➈✳ ✷✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ✷ ❙➢✉ t➬♠ ✈➭ ➤ä❝ t➭✐ ❧✐Ö✉ tõ ❝➳❝ s➳❝❤✱ t➵♣ ❝❤Ý t♦➳♥ ❤ä❝ tr♦♥❣ ♥➢í❝ ✈➭ q✉è❝ tÕ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ❜➭✐ t♦➳♥ tè✐ ➢✉ ✈Ð❝ t➡✳ ◗✉❛ ➤ã✱ t×♠ ❤✐Ĩ✉ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ✈✃♥ ➤Ị ♥➭②✳ ✸✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ❝❤♦ ♥❣❤✐Ư♠ ❤÷✉ ❤✐Ư✉ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr♦♥❣ ❜➭✐ ❜➳♦ ❝ñ❛ ❉✳ ❱✳ ▲➢✉ ➤➝♥❣ tr♦♥❣ t➵♣ ❝❤Ý ❏♦✉r♥❛❧ ♦❢ ❖♣t✐♠✐③❛t✐♦♥ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✶✻✵ ✭✷✵✶✹✮✱ ♣♣✳ ✺✶✵✲✺✷✻✳ ✹✳ ◆é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ♣❤➬♥ ♠ë ➤➬✉✱ ✷ ❝❤➢➡♥❣✱ ❦Õt ❧✉❐♥ ✈➭ ❞❛♥❤ ♠ơ❝ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❈❤➢➡♥❣ ✶✿ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ❚r×♥❤ ❜➭② ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ✈í✐ ❝➳❝ ❤➭♠ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣✳ ❈❤➢➡♥❣ ✷✿ ➜✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❚r×♥❤ ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❝❤♦ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ✈í✐ ❝➳❝ ❤➭♠ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ✈Ò tÝ♥❤ ❧å✐ s✉② ré♥❣✱ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ ❝➬♥ tè✐ ➢✉ trë t❤➭♥❤ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ tè✐ ➢✉✳ ✸ ❈❤➢➡♥❣ ✶ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ ②Õ✉ ❚r♦♥❣ ❝❤➢➡♥❣ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣✳ ❈➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ t❤❛♠ ❦❤➯♦ tr♦♥❣ ❬✾❪✱ ❬✶✹❪✳ ✶✳✶ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❜ỉ trỵ ✶✳✶✳✶✳ ❈❤♦ ❉➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ f trị tự rộ ợ ị tr➟♥ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❉✐♥✐ ❞➢í✐ ✈➭ tr➟♥ v ∈ Rn f− t➵✐ t➵✐ x¯ x¯ f ❝ñ❛ t➵✐ x¯ ∈ Rn f − (¯ x; v) := lim inf f (x + tv) − f (¯ x) , t f + (¯ x; v) := lim sup f (¯ x + tv) − f (¯ x) t t↓0 f f+ ✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ➤➵♦ t❤❡♦ ♣❤➢➡♥❣ ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ t↓0 ◆Õ✉ ✈➭ Rn f + (¯ x; v) = f − (¯ x; v) ✱ tì trị ó ợ ọ ❝đ❛ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ v ✈➭ ❦ý ❤✐Ư✉ ❧➭ f (¯ x; v) ✳ ❍➭♠ ♥Õ✉ tå♥ t➵✐ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ❝ñ❛ ♥ã t➵✐ ❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐ x¯ ✈í✐ ➤➵♦ ❤➭♠ ❋rÐ❝❤❡t ∇f (¯ x) f x¯ t❤× ❣ä✐ ❧➭ ❦❤➯ ✈✐ t❤❡♦ ♣❤➢➡♥❣ t❤❡♦ ♠ä✐ ♣❤➢➡♥❣✳ ◆Õ✉ f f (¯ x; v) = ∇f (¯ x, v) ❧➭ ✹ f ❚❤❡♦ ❬✾❪ ❤➭♠ ∂∗ f (¯ x) ✮ t➵✐ ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr➟♥ x¯ ∈ Rn ♥Õ✉ ∂ ∗ f (¯ x) ✭❤❛② (∂∗ f (¯ x)) ⊆ Rn f − (¯ x; v) ≤ sup inf (∀v ∈ Rn ), ξ, v (∀v ∈ Rn ) ξ∈∂∗ f (¯ x) ▼ét t❐♣ ➤ã♥❣ ♥Õ✉ ∂ ∗ f (¯ x) ❚❤❡♦ ∂ ∗ f (¯ x) ⊆ Rn ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝đ❛ ➤å♥❣ t❤ê✐ ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr➟♥ ✈➭ ❞➢í✐ ❝ñ❛ ❬✽❪ ❤➭♠ ∂ ∗ f (¯ x) ⊆ Rn t➵✐ f x¯ ➤➢ỵ❝ ♥Õ✉ ✭❤❛② ❞➢í✐ ✮ ❧➭ t❐♣ ➤ã♥❣ ✈➭ ξ, v ξ∈∂ ∗ f (¯ x) f + (¯ x; v) ≥ ∂ ∗ f (¯ x) ❣ä✐ ❧➭ ∂ ∗ f (¯ x) ❝ã ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❜➳♥ f t➵✐ x¯ ❝❤Ý♥❤ f t➵✐ x¯ ✳ q✉② tr➟♥ ❧➭ t❐♣ ➤ã♥❣ ✈➭ f + (¯ x; v) ≤ sup (∀v ∈ Rn ) ξ, v ξ∈∂ ∗ f (¯ x) ✭✶✳✶✮ ❱Ý ❞ô ✶✳✶✳✶ ❈❤♦ ❤➭♠ f :R→R x, f (x) := x4 − 4x3 + 4x2 , 0, ợ ị x Q ∩ [0; +∞[, x ∈ Q ∩ ]−∞; 0], ❦❤✐ , tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ❦❤➳❝ tr♦♥❣ ➤ã Q ❧➭ t❐♣ ❝➳❝ sè ❤÷✉ tû✳ ❑❤✐ ➤ã v, + f (0; v) = 0, ❦❤✐ v ≥ 0, ❦❤✐ v < 0, f − (0; v) = (∀v ∈ R) ❚❐♣ {0; 1} ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❜➳♥ ❝❤Ý♥❤ q✉② tr➟♥ ❝đ❛ ♥ã ❝ò♥❣ ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr➟♥ ❝đ❛ ré♥❣ ❞➢í✐ ❝ñ❛ ❚❤❡♦ ❬✾❪✱ f t➵✐ ♥Õ✉ f t➵✐ x¯ ✳ ❚❐♣ {0} f t➵✐ x¯ ✱ ❝❤♦ ♥➟♥ ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② x¯ ✳ ①➯② r❛ ➤➻♥❣ t❤ø❝ tr♦♥❣ tì f ( x) ợ ọ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥✳ ❱í✐ ♠ét ❤➭♠ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣✱ ❞➢í✐ ✈✐ ♣❤➞♥ ✺ ❈❧❛r❦❡ ✈➭ ❞➢í✐ ✈✐ ♣❤➞♥ ▼✐❝❤❡❧✲P❡♥♦t ❧➭ ♥❤÷♥❣ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝đ❛ x¯ ✭①❡♠ ❬✾❪✮✳ f t➵✐ ❍➡♥ ♥÷❛ ✈í✐ ♠ét ❤➭♠ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② tr♦♥❣ t❤❡♦ ♥❣❤Ü❛ ❈❧❛r❦❡ ❬✹❪✱ ❞➢í✐ ✈✐ ♣❤➞♥ ❈❧❛r❦❡ ❧➭ ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② f tr➟♥ ✭①❡♠ ❬✼❪✮✳ ❈❤ó ý r➺♥❣✱ ♥Õ✉ ❤➭♠ tr➟♥ t➵✐ x¯ ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② t❤× ♥ã ❝ò♥❣ ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❜➳♥ ❝❤Ý♥❤ q✉② tr➟♥ t➵✐ x¯ ➤ã ♥ã ➤➢ỵ❝ ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr➟♥ t➵✐ x¯ ✱ ✈➭ ❞♦ ✳ ❱Ý ❞ô ✶✳✶✳✷ ét f :RR ợ ị x2 cos π , x f (x) = 0, ❚❛ ❝ã f t➵✐ x¯ = t➢➡♥❣ ø♥❣ ❧➭ x = f t➵✐ f x¯ ∈ Q t❤❡♦ tÝ♥❤ t➵✐ x¯ ∈ Q t❤❡♦ Q Q ♥Õ✉ t➵✐ {0} ♣❤➞♥ ❈❧❛r❦❡ ✳ ❈➳❝ t❐♣ x¯ ✳ ❚❐♣ {0} ✈➭ ▼✐❝❤✐❧❡✲ {0} [−π; π] ✱ ✈➭ ❧➭ ❞➢í✐ s Q ế ỗ f (x) ≤ f (¯ x) ⇒ ∀t ∈ ]0, 1[ , ➤➢ỵ❝ ❣ä✐ ❧➭ tù❛ ❧å✐ tr➟♥ ✈➭ ✈✐ x¯ ❚❤❡♦ ❬✶✻❪ ♠ét ❤➭♠ ❣✐➳ trÞ t❤ù❝ ♠ë ré♥❣ ❣ä✐ ❧➭ tù❛ ❧å✐ t➵✐ ❉➢í✐ [−π; π] ❧➭ ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝ñ❛ ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥ ❝ñ❛ ♥Õ✉ ±f f ①➳❝ ➤Þ♥❤ tr➟♥ t❐♣ x∈Q Q f Q ⊆ Rn ➤➢ỵ❝ ✱ f (tx + (1 − t)¯ x) ≤ f (¯ x) ❧➭ tù❛ ❧å✐ t➵✐ s✉② ré♥❣ ❞➢í✐ ❧å✐ tr➟♥ ♠ét t❐♣ ❧å✐ f ❧➭ tù❛ t ỗ r ỉ r r ế x¯ t❤❡♦ Q x∈Q f ✳ ❣ä✐ ❧➭ tù❛ t✉②Õ♥ ✳ ❧➭ ❧✐➟♥ tơ❝✱ tù❛ ❧å✐ ✈➭ ❝ã ♠ét ❞➢í✐ tì ỗ f (x) f (y) ⇒ ∃ξ (n) ∈ ∂∗ f (y), f ❦❤✐ ✳ {−π; π} ◆Õ✉ x = 0, f + (0; v) = f − (0; v) = 0, (∀v ∈ R) P❡♥♦t ❝ñ❛ f ❦❤✐ x, y ∈ Q ✱ lim (ξ (n) , x − y) ≤ n→∞ ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥ t➵✐ x¯ t❤× t❛ ❝ã ♠Ư♥❤ ➤Ị s❛✉ ➤➞②✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✶ ●✐➯ sư f ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥ ∂ ∗ f (¯ x) t➵✐ x¯ ✈➭ f tù❛ ❧å✐ ✻ t➵✐ x¯ ∈ Q t❤❡♦ t❐♣ ❧å✐ Q✳ ❑❤✐ ➤ã✱ ∀x ∈ Q, f (x) ≤ f (¯ x) ⇒ ∀ξ ∈ ∂ ∗ f (¯ x), ξ, x − x¯ ≤ ❈❤ø♥❣ ♠✐♥❤ ❱× f x¯ ❧➭ tù❛ ❧å✐ t➵✐ t❤❡♦ Q ỗ xQ tỏ f (x) f (¯ x) ✱ t❛ ❝ã f + (¯ x; x − x¯) ≤ ❉♦ tÝ♥❤ ❝❤Ý♥❤ q✉② tr➟♥ ❝ñ❛ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ♠➲♥ ∂ ∗ f (¯ x) ỗ xQ tỏ f (x) f (¯ x) ✱ t❛ ❝ã ξ, x − x¯ = f + (¯ x; x − x¯) ≤ sup ξ∈∂ ∗ f (¯ x) ✷ ❚õ ➤ã✱ t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ❬✷✵❪✱ ❤➭♠ t❤ù❝ ♠ë ré♥❣ tr➟♥ Q f ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❞➢í✐ ❧å✐ ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐➯ ❧å✐ t✐Ư♠ ❝❐♥ ❞➢í✐ tr➟♥ ∃ξ (n) ∈ ∂∗ f (x), ❍➭♠ ❣✐➳ trÞ t❤ù❝ ♠ë ré♥❣ ❧➭ ❣✐➯ ❧å✐ t✐Ö♠ ❝❐♥ t➵✐ x¯ f Q ế ỗ x, y Q lim ξ (n) , y − x ≥ ⇒ f (y) ≥ f (x) n→∞ ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ t❤❡♦ ∃ξ (n) ∈ conv∂ ∗ f ( x), Q ế ỗ xQ f (¯ x) t➵✐ x¯ ➤➢ỵ❝ ❣ä✐ t❛ ❝ã lim ξ (n) , x − x¯ ≥ ⇒ f (x) ≥ f (¯ x) n→∞ tr♦♥❣ ➤ã ❝♦♥✈ ❦Ý ❤✐Ö✉ ❜❛♦ ❧å✐ ❱Ý ❞ô ✶✳✶✳✸ ❈❤♦ ∂∗ f (x) f, g : R → R x, x ≤ 0, f (x) := x, x > 0, x ∈ Q, x, g(x) := 2x, x ∈ (R\Q) ∩ ]−∞, 0] , x ∈ (R\Q) ∩ [0, ∞[ x, ✶✾ tr♦♥❣ ➤ã ➜✐Ị✉ λ = (λk )k∈J , µ = (µi )i∈I(¯x) , γ = (γj )j∈L ❦✐Ö♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ✭P✮ ❞➢í✐ ♥❣➠♥ ữ s rộ ợ t ể s❛✉✳ ➜Þ♥❤ ❧ý ✶✳✷✳✷ ●✐➯ sư r➺♥❣ x¯ ❧➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ✭P✮ ✈➭ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✶ t❤♦➯ ♠➲♥✳ ❑❤✐ ➤ã tå♥ t➵✐ ¯ µ (λ, ¯) = (0, 0) ✈➭ γ¯ ∈ Rl s❛♦ ❝❤♦ ¯ k conv∂ ∗ fk (¯ λ x) + ∈ cl ¯ k ≥ (∀k ∈ J) , µ λ ¯i ≥ (∀i ∈ I (¯ x))✱ k∈J µ ¯i conv∂ ∗ gi (¯ x) + γ¯j ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ✭✶✳✶✼✮ ❈❤ø♥❣ ♠✐♥❤ ❚❛ ❝❤Ø r❛ r➺♥❣ ∈ clD (¯ x) ✭✶✳✶✽✮ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ 0∈ / clD (¯ x) ❚❛ ❝ã t❐♣ D (¯ x) t❐♣ ❧å✐ rê✐ ♥❤❛✉ ❧➭ rỗ ó t ụ ị ❧ý t➳❝❤ ♠➵♥❤ ❝❤♦ ❝➳❝ D (¯ x) ✈➭ {0} ✭①❡♠ ❬✶✾❪✱ ❤Ö q✉➯ ✶✶✳✹✳✷✮✱ tå♥ t➵✐ v0 ∈ Rn , v0 = s❛♦ ❝❤♦ sup ξ, v0 < ✭✶✳✶✾✮ ξ∈D(¯ x) ❚❤❡♦ ➤Þ♥❤ ❧ý ✶✳✷✳✶✱ ❤Ư ✭✶✳✷✮ ✲ ✭✶✳✺✮ ❧➭ ❦❤➠♥❣ t➢➡♥❣ t❤Ý❝❤✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ λk = 1, λp = 0, (∀p ∈ J, p = k), µi = (∀i ∈ I(¯ x)), γ = ✈➭ = ζ ∈ N (C; x¯) ✱ tõ ✭✶✳✶✾✮ t❛ s✉② r❛ sup ξk , v0 < (∀k ∈ J) ✭✶✳✷✵✮ ηi , v0 < (∀i ∈ I(¯ x)) ✭✶✳✷✶✮ ξk ∈conv∂ ∗ fk (¯ x) ❚➢➡♥❣ tù ♥❤➢ tr➟♥✱ t❛ ❝ã sup ηi ∈conv∂ ∗ gi (¯ x) ✷✵ ❚❛ ❝❤Ø r❛ ∇hj (¯ x), v0 = ❚❤❐t ✈❐② ❝➳❝❤ ❧✃② ♥Õ✉ ✭✶✳✷✷✮ ❧➭ s❛✐✱ t❤× (∀j ∈ L) ∇hj0 (¯ x), v0 = ✈í✐ ✭✶✳✷✷✮ j0 ♥➭♦ ➤ã ∈ L ✳ ❇➺♥❣ ξs ∈ ∂ ∗ fs (¯ x) , λs = 1, λk = (∀k ∈ J, k = s) , µi = 0(∀i ∈ I(¯ x)), γj = (∀j ∈ L, j = j0 ) , = ζ ∈ N (C; x¯), tõ ✭✶✳✶✾✮ t❛ s✉② r❛ ξs , v0 + γj0 ∇hj0 (¯ x), v0 < ✭✶✳✷✸✮ ❚❛ ❝❤ó ý r➺♥❣ | ξs , v0 | < +∞ ❈❤♦ ♥Õ✉ γj0 ✈➭ ∇hj0 (¯ x), v0 > ➤đ ❧í♥ ♥Õ✉ ∇hj0 (¯ x), v0 < | ∇hj0 (¯ x), v0 | < +∞ ✱ ❝ß♥ γj0 < ✈í✐ ❣✐➳ trÞ t✉②Ưt ➤è✐ ➤đ ❧í♥ ✱ t❛ sÏ ➤✐ ➤Õ♥ ♠ét ♠➞✉ t❤✉➱♥ ✈í✐ ✭✶✳✷✸✮✳ ❉♦ ✈❐②✱ ✭✶✳✷✷✮ ❧➭ ➤ó♥❣✳ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤Ø r❛ r➺♥❣ v0 ∈ T (C; x¯) ❚❤❐t ✈❐②✱ ♥Õ✉ ✭✶✳✷✹✮ ❦❤➠♥❣ ➤ó♥❣ t❤× sÏ ❇➺♥❣ ❝➳❝❤ ❝❤♦ I(¯ x)), γ = ∃η0 ∈ N (C; x¯) ✭✶✳✷✹✮ s❛♦ ❝❤♦ α>0 t❛ ❝ã αη0 ∈ N (C; x¯) ✈➭ λs ξ0 , v0 + α η0 , v0 < η0 , v0 > ✳ λk = (∀k ∈ J, k = s) , λs > 0, ξs ∈ ∂ ∗ fs ( x) , ài = 0(i ì (η0 , v0 ) > ✈í✐ α ✭✶✳✷✺✮ ➤đ ❧í♥ t❛ ♥❤❐♥ ➤➢ỵ❝ ♠ét ♠➞✉ t❤✉➱♥ ✈í✐ ✭✶✳✷✺✮✳ ❉♦ ➤ã η0 , v0 ≤ 0, (∀η ∈ N (C; x¯)) ❈❤ó ý r➺♥❣ T (C; x¯) ❧➭ ♠ét ♥ã♥ ❧å✐ ➤ã♥❣✳ ❚õ ➤ã t❛ ❝ã v0 ∈ N (C; x¯) = T 00 (C; x¯) = T (C; x¯) ❑Õt ❤ỵ♣ ✭✶✳✷✵✮ ✲ ✭✶✳✷✷✮ ✈➭ ✭✶✳✷✹✮ t❛ s✉② r❛ ❤Ö ✭✶✳✻✮ ✲ ✭✶✳✶✵✮ ❝ã ♠ét ♥❣❤✐Ö♠ v0 , ✈➭ ❝ò♥❣ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❤Ư ✭✶✳✷✮ ✲ ✭✶✳✺✮✳ ➜✐Ị✉ t ị ý ì (n) ✈❐② ✭✶✳✶✽✮ ➤ó♥❣ ✈➭ ❞♦ ➤ã tå♥ t➵✐ (n) 0, ηi (n) (n) λk ≥ 0, ξk ∈ conv∂ ∗ fk (¯ x) (∀k ∈ J) , µi (n) ∈ conv∂ ∗ gi (¯ x) (∀i ∈ I (¯ x)) , γj ∈ R (∀j ∈ L) ✈➭ ζ (n) ∈ N (C; x¯) ≥ ✈í✐ (n) λ(n) , µI(¯x) , γ (n) = (0, 0, 0) (n) (n) (n) (n) = lim n→∞ s❛♦ ❝❤♦ µi ηi λk ξk + k∈J (n) γj ∇hj (¯ x) + ξ (n) , + ✭✶✳✷✻✮ j∈L i∈I(¯ x) tr♦♥❣ ➤ã (n) λ(n) = λk (n) k∈J (n) , µI(¯x) = µi (n) i∈I(¯ x) , γ (n) = γj j∈L ì (n) (n) , àI(x) , (n) = (0, 0, 0) , t❛ ❝ã t❤Ó ①❡♠ ♥❤➢ (n) (λ(n) , µI(¯x) , γ (n) ) = (∀n) ❑❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t ❝ã t❤Ĩ ❣✐➯ sư ✈í✐ ¯ ≥ 0, µ λ ¯I(¯x) ≥ 0, γ¯ ∈ Rl ✈➭ (n) ¯ µ λ(n) , µI(¯x) , γ (n) → λ, ¯I(¯x) , γ¯ ¯ µ (λ, ¯I(¯x) , γ¯ ) = ✳ ❇ë✐ ✈× clA + clB ⊆ cl(A + B), ✈➭ tõ ✭✶✳✷✻✮ t❛ s✉② r❛ ¯ k clconv∂ ∗ fk (¯ λ x) + 0∈ k∈J µ ¯i clconv∂ ∗ gi (¯ x) i∈I(¯ x) γ¯j ∇hj (¯ x) + N (C; x¯) + j∈L ¯ k conv∂ ∗ fk (¯ λ x) + ⊆ cl k∈J µ ¯i conv∂ ∗ gi (¯ x) + γ¯j ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ✭✶✳✷✼✮ ❚❛ ❝ã ✭✶✳✷✼✮ ➤ó♥❣ ✈í✐ ¯ µ λ, ¯ = (0, 0) ✳ ❚❤❐t ✈❐② ♥Õ✉ , = (0, 0) tì = ✳ ❚✉② ♥❤✐➟♥✱ ❦Õt ❤ỵ♣ ✈í✐ ✭✶✳✷✼✮ t❛ ➤✐ ➤Õ♥ ♠➞✉ t❤✉➱♥ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❘❈✮✳ ❉♦ ➤ã ¯ µ λ, ¯ = (0, 0) ✳ ❙✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷ ✷✷ ❍Ư q✉➯ ✶✳✷✳✶ C = Rn ●✐➯ sö r➺♥❣ ✈➭ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✷ t❤♦➯ ♠➲♥ tr♦♥❣ ➤ã ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❘❈✮ tr♦♥❣ ♠Ư♥❤ ➤Ị ✶✳✷✳✶ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ ➤✐Ị✉ ❦✐Ư♥✿ ❤Ư ¯ k ≥ (∀k ∈ J), µ ∇h1 (¯ x), , ∇hl (¯ x) ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ λ ¯i ≥ ¯ µ ¯ = (0, 0) ✈➭ γ¯j ∈ R (∀j ∈ L) s❛♦ ❝❤♦ ✭✶✳✶✼✮ ➤ó♥❣✳ (∀i ∈ I(¯ x)) ✈í✐ λ, ❈❤ø♥❣ ♠✐♥❤ C = Rn ❱í✐ ✱ t❛ ❝ã N (C; x¯) = {0} ✳ ❉♦ ➤ã✱ ♥Õ✉ ∇h1 (¯ x), , ∇hl (¯ x) ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤ t❤× ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❘❈✮ t❤á❛ ♠➲♥✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ✶✳✷✳✷ t❛ s✉② r❛ ✷ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ t❐♣ D(¯ x) ❧➭ t❐♣ ➤ã♥❣✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ❤Ư q✉➯ trù❝ t✐Õ♣ s❛✉ ➤➞② ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✶✱ tr♦♥❣ ➤ã ❜❛♦ ➤ã♥❣ tr♦♥❣ ✭✶✳✶✼✮ ❝ã t❤Ĩ ❜á ➤➢ỵ❝✳ ❍Ư q✉➯ ✶✳✷✳✷ C = Rn ✳ ●✐➯ sư ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✷ ➤ó♥❣ ✈➭ t❐♣ D(¯ x) ¯ k ≥ (∀k ∈ J), µ ¯ µ ❑❤✐ ➤ã ∃λ ¯i ≥ (∀i ∈ I(¯ x)) ✈í✐ λ, ¯ = (0, 0) ✈➭ ●✐➯ sö ➤ã♥❣✳ γ¯j ∈ R (∀j ∈ L) s❛♦ ❝❤♦ ¯ k conv∂ ∗ fk (¯ λ x) + 0∈ k∈J µ ¯i conv∂ ∗ gi (¯ x) + γ¯j ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ◆❤❐♥ ①Ðt ✶✳✷✳✷ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ (k ∈ J) ✈➭ C = Rn ✱ ❝ò♥❣ ♥❤➢ tr♦♥❣ ♥❤❐♥ ①Ðt ✸✳✶ tr♦♥❣ ❬✶✸❪✱ ♥Õ✉ ∂ ∗ gi (¯ x)(i ∈ (I(¯ x)) k∈J t❤× D(¯ x) ❜Þ ❝❤➷♥ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞②✿ conv∂ ∗ fk (¯ x) ∪ 0∈ / conv ❧➭ ∂ ∗ fk (¯ x) conv∂ ∗ gi (¯ x) + lin {∇hj (¯ x) : j ∈ L} , i∈I(¯ x) t❐♣ ➤ã♥❣✱ conv∂ ∗ fk (¯ x) (k ∈ J) tr♦♥❣ ✈➭ ➤ã ❧✐♥ ❦Ý ❤✐Ö✉ ❜❛♦ conv∂ ∗ gi (¯ x)(i ∈ (I(¯ x)) t✉②Õ♥ tÝ♥❤✳ conv∂ ∗ fk (¯ x) ∪ k∈J conv∂ ∗ gi (¯ x) i∈I(¯ x) ✈❐② t❛ ❝ã ❧➭ ❝♦♠♣➝❝ ✈➭ ❞♦ ➤ã t❐♣ s❛✉ ❝ò♥❣ ❝♦♠♣➝❝✿ conv ❚❤❐t ✷✸ ❉♦ ➤ã✱ conv∂ ∗ fk (¯ x) ∪ E(¯ x) := conv kJ t ó ữ ì conv gi (¯ x) +lin {∇hj (¯ x) : j ∈ L} i∈I(¯ x) 0∈ / E(¯ x) ✈➭ D(¯ x) = coneE(¯ x), ❝❤♦ ♥➟♥ D(¯ x) ❧➭ t❐♣ ➤ã♥❣✱ tr♦♥❣ ➤ã coneE(¯ x) ❧➭ ♥ã♥ s✐♥❤ ❜ë✐ E(¯ x) ét ề ệ t ợ tr ị ❧ý ✶✳✷✳✷✱ ❝➳❝ ❤Ö q✉➯ ✶✳✷✳✶✱ ✶✳✷✳✷ ✈➭ ♠ét ✈➭✐ ề ệ tố trì ữ ∂ ∗ f (¯ x) ❧➭ tèt ❤➡♥ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ tè✐ ➢✉ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ ♥❤➢ ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ ❈❧❛r❦❡✱ ▼♦r❞✉❦❤♦✈✐❝❤ ✈➭ ▼✐❝❤❡❧✲P❡♥♦t✳ ✷✹ ❈❤➢➡♥❣ ✷ ➜✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❚r♦♥❣ ❝❤➢➡♥❣ ✷ ❝❤ó♥❣ t❛ sÏ tr×♥❤ ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r✱ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ô❝ t✐➟✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ❞➢í✐ ♥❣➠♥ ữ s rộ ết q trì ❜➭② tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ t❤❛♠ ❦❤➯♦ tr♦♥❣ ❬✶✹❪✳ ✷✳✶ ➜✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✈➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲ ❚✉❝❦❡r ❈❤➢➡♥❣ ✶ ➤➲ tr×♥❤ ❜➭② ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ✭✶✳✶✼✮ ✈í✐ ❣✐➯ t❤✐Õt ✶✳✷✳✶ ✈➭ ❣✐➯ t❤✐Õt ❝đ❛ ♠Ư♥❤ ➤Ị ✶✳✷✳✶✱ tr♦♥❣ ➤ã ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❘❈✮ ➤ó♥❣✿ 0∈ γj ∇hj (¯ x) + N (C; x¯) ⇒ γ1 = = γl = j∈L ❚r♦♥❣ tr➢ê♥❣ ợ C = Rn tì ề ệ trở t ➤✐Ị✉ ❦✐Ư♥✿ ❤Ư ❧➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤✳ ❈❤ó ý r➺♥❣ tr♦♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✭✶✳✶✼✮ t❛ ❝ã ➜Ĩ ♥❤❐♥ ợ ề ệ ệ ữ ệ tr ó ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② s❛✉ ➤➞② ♠➭ t❛ ❦ý ❤✐Ư✉ ❧➭ ✭❈◗✶✮✿ ❝➳❝ sè ak > (k ∈ J, k = s) , bi > (i ∈ I (¯ x)) ∇h1 , , ∇hl ¯ µ λ, ¯ = (0, 0) ¯=0 λ ✱ t❛ ➤➢❛ ✈➭♦ ∃s ∈ J d0 ∈ T (C, x¯) s❛♦ ❝❤♦ ✱ ✳ ✱ ✈➭ ✷✺ ξk , d0 ≤ −ak (∀ξk ∈ conv∂ ∗ fk (¯ x) , ∀k ∈ J, k = s) ; ηi ; d0 ≤ −bi ✭✐✮ (∀ηi ∈ conv∂ ∗ gi (¯ x) , ∀i ∈ I (¯ x)) ; ✭✐✐✮ ∇hj (¯ x) , d0 = (∀j ∈ L) ❈❤ó♥❣ t❛ ❝ï♥❣ ➤➢❛ ✈➭♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❈◗✷✮✿ ❱í✐ ♠ä✐ ✈➭ λk ≥ (k ∈ J, k = s) ; µi ≥ (∀i ∈ I (¯ x)) ✱ ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵✱ γj ∈ R (∀j ∈ L) ✱ t❛ ❝ã µi conv∂ ∗ gi (¯ x) λk conv∂ ∗ fk (¯ x) + 0∈ / cl k∈J,k=s i∈I(¯ x) γj ∇hj (¯ x) + N (C; x¯) + j∈L ❚r♦♥❣ ♣❤➬♥ t✐Õ♣ t❤❡♦ t❛ tr×♥❤ ❜➭② ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ✭❈◗✶✮ ✈➭ ✭❈◗✷✮✳ ▼Ư♥❤ ➤Ị ✷✳✶✳✶ ●✐➯ sư r➺♥❣ fk ✈➭ gi ❝ã ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ tr➟♥ ∂ ∗ fk (¯ x) ✈➭ ∂ gi ( x) t x ỗ k J, k = s, i ∈ I (¯ x)✱ h ❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐ x¯ ✈➭ C ✭❈◗✶✮ ❦Ð♦ t❤❡♦ ✭❈◗✷✮ ❧➭ ❧å✐✳ ❑❤✐ ➤ã (∀s ∈ J)✳ ❈❤ø♥❣ ♠✐♥❤ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ✭❈◗✶✮ ➤ó♥❣✱ ♥❤➢♥❣ ✭❈◗✷✮ s❛✐✱ ❝ã ♥❣❤Ü❛ ❧➭ (∀i ∈ I (¯ x)) ✈í✐ (λ(s) , µ) = (0, 0), tr♦♥❣ ➤ã ∃λk ≥ (∀k ∈ J, k = s) ; µ (λ(s) = (λk )k∈J,k=s, µ = (µi )i∈I(¯x) ), ξk(n) ∈ conv∂ ∗ fk (¯ x) (∀k ∈ J, k = s) , (n) ηi ∈ conv∂ ∗ gi (¯ x) (∀i ∈ I (¯ x)) , γj ∈ R (∀j ∈ L) ✈➭ ζ (n) ∈ N (C; x¯) s❛♦ ❝❤♦ (n) lim n→∞ (n) λk ξk + k∈J,k=s ❚õ ✭❈◗✶✮ s✉② r❛ µi ηi s❛♦ ❝❤♦ (n) = lim n→∞ j∈L i∈I(¯ x) ∃d0 ∈ T (C; x¯) γj ∇hj (¯ x) + ζ (n) + (n) λk ξk , d0 + k∈J,k=s µi ηi , d0 i∈I(¯ x) γj ∇hj (¯ x), d0 + ζ (n) , d0 + j∈L = ✷✻ (n) ≤ lim n ì ((s) , à) = (0, 0) kJ,k=s ài ηi , d0 ✭✷✳✶✮ i∈I(¯ x) tõ ➤✐Ị✉ ❦✐Ư♥ ✭✐✮ tr♦♥❣ ✭❈◗✶✮ s✉② r❛ (n) (n) lim n→∞ (n) λk ξk , d0 + µi ηi , d0 λk ξk , d0 + k∈J,k=s i∈I(¯ x) λk ak − ≤− k∈J,k=s µi bi < i∈I(¯ x) ✷ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ✭✷✳✶✮✳ ❉♦ ➤ã t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ▼ét ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❝❤♦ ♥❣❤✐Ư♠ ữ ệ ó tể ợ t ể s ị ❧ý ✷✳✶✳✶ x¯ ●✐➯ sư ❧➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ✭P✮✳ ●✐➯ sư t✃t ❝➯ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✷ t❤♦➯ ♠➲♥ ✈➭ ❣✐➯ sư ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❈◗✶✮ ❤♦➷❝ ✭❈◗✷✮ ➤ó♥❣ ✭✈í✐ s ∈ J ✮✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ ¯ s > 0, λ ¯ k ≥ 0(∀k ∈ J, k = s), µ λ ¯i ≥ 0(∀i ∈ I(¯ x)), γ¯j ∈ R(∀j ∈ L) s❛♦ ❝❤♦ ¯ k conv∂ ∗ fk (¯ λ x) + ∈ cl k∈J µ ¯i conv∂ ∗ gi (¯ x) i∈I(¯ x) γ¯j ∇hj (¯ x) + N (C; x¯) + ✭✷✳✷✮ j∈L ❈❤ø♥❣ ♠✐♥❤ ➳ ♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ✶✳✷✳✷ t❛ s✉② r❛ ¯ k ≥ (∀k ∈ J) , µ ∃ λ ¯i ≥ (∀i ∈ I (¯ x)) ¯ µ ¯ µ ¯ k ) , (¯ (λ, ¯) = (0, 0) (λ, ¯) = (λ k∈J µi )i∈I(¯ x) ¯s = λ ◆Õ✉ ✱ t❤× tõ ✭❈◗✶✮ ❤♦➷❝ ✭❈◗✷✮ t❛ s✉② ✈➭ r❛ γ¯ ∈ Rl ♠➞✉ ✈í✐ s❛♦ ❝❤♦ ✭✷✳✷✮ ➤ó♥❣✳ t❤✉➱♥ ✈í✐ ✭✷✳✷✮✳ ❱× ✈❐② ¯s > λ ✳ ✷ ❚õ ➤ã✱ t❛ s✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ◆❤❐♥ ①Ðt ✷✳✶✳✶ ❚❛ ➤➢❛ ✈➭♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❈◗✸✮ ②Õ✉ ❤➡♥ ❤➵♥ ❝❤Õ ✭❈◗✶✮✿ tå♥ t➵✐ d0 ∈ ✷✼ T (C, x¯) ✈➭ ❝➳❝ sè bi > (i ∈ I (¯ x)) s❛♦ ❝❤♦ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✮ tr♦♥❣ ✭❈◗✶✮ ➤ó♥❣ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ✭✐✬✮ ηi , d0 ≤ −bi (∀ηi ∈ conv∂ ∗ gi (¯ x) , ∀i ∈ I(¯ x)) ❚❛ ❝ò♥❣ ➤➢❛ ✈➭♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❈◗✹✮ ②Õ✉ ❤➡♥ ❤➵♥ ❝❤Õ ✭❈◗✷✮✿ ✈í✐ ♠ä✐ µi ≥ (∀i ∈ I (¯ x)) ✱ ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵✱ ✈➭ µi conv∂ ∗ gi (¯ x) + 0∈ / cl γj ∈ R (∀j ∈ L) ✱ γj ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ❇➺♥❣ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ♠Ư♥❤ ➤Ị ✷✳✶✳✶✱ t❛ s✉② r❛ r é t ữ ế tr ị ❧ý ✷✳✶✳✶ ✭❈◗✶✮ ✈➭ ✭❈◗✷✮ ✭✈í✐ ♥➭♦ ➤ã ∈J s ✮ t➢➡♥❣ ø♥❣ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ ✭❈◗✸✮ ✈➭ ✭❈◗✹✮✱ t❛ t❤✉ ➤➢ỵ❝ ➤✐Ị✉ ❦✐Ư♥ ¯k > λ ✳ ➜✐Ị✉ ❦✐Ư♥ ♥➭② ②Õ✉ ❤➡♥ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✭✷✳✷✮ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ¯s > λ ✳ k∈J ❈➳❝ ❣✐➯ t❤✐Õt s❛✉ ❧➭ ❝➬♥ t❤✐Õt ➤Ĩ ❞➱♥ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈í✐ ❝➳❝ ❤Ư sè ▲❛❣r❛♥❣❡ t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ❤➭♠ ♠ơ❝ t✐➟✉ ❧➭ ❞➢➡♥❣✳ ●✐➯ t❤✐Õt ✷✳✶✳✶ ỗ kJ í q tr t i ∈ I(¯ x) ✱ ❝➳❝ ❤➭♠ ∂ ∗ fk (¯ x) tr×♥❤ ✈➭ ❜➭② ∂ ∗ gi (¯ x) ♠ét ➤✐Ị✉ fk t➵✐ x¯ ❦✐Ư♥ ✈➭ gi ❝ã ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❜➳♥ ❀ ❝➳❝ ❤➭♠ ❝➬♥ gi (i ∈ / I(¯ x)) ❧✐➟♥ tô❝ t➵✐ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❝❤♦ ❝ù❝ x¯ ✳ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❝➳❝ ♥❤➞♥ tư ▲❛❣r❛♥❣❡ ❞➢➡♥❣ t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ❤➭♠ ♠ơ❝ t✐➟✉✳ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ●✐➯ sư x¯ ❧➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ✭P✮✳ ●✐➯ sư ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý ✶✳✷✳✷ t❤♦➯ ♠➲♥ tr♦♥❣ ➤ã ❣✐➯ t❤✐Õt ✶✳✷✳✶ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ ❣✐➯ t❤✐Õt ✷✳✶✳✶✳ ●✐➯ sư r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉② ✭❈◗✶✮ ❤♦➷❝ ✭❈◗✷✮ ✭✈í✐ ♠ä✐ ➤ã✱ tå♥ t➵✐ s ∈ J ✮ ➤ó♥❣✳ ¯ k > 0(∀k ∈ J), µ λ ¯i ≥ (∀i ∈ I (¯ x)) , γ¯j ∈ R (∀j ∈ L) s❛♦ ❝❤♦ ¯ k conv∂ ∗ fk (¯ λ x) + ∈ cl k∈J µ ¯i conv∂ ∗ gi (¯ x) i∈I(¯ x) γ¯j ∇hj (¯ x) + N (C; x¯) + jL ứ sJ ỗ (s) ➳♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ✷✳✶✳✶ t❛ s✉② r❛ tå♥ t➵✐ (s) (s) (∀k ∈ J, k = s), µi ≥ (∀i ∈ I (¯ x)) , γj ∈ R (∀j ∈ L) (s) (s) k∈J s❛♦ ❝❤♦ (s) µi ∂ ∗ gi (¯ x) + λk ∂ ∗ fk (¯ x) + ∈ cl (s) λs > 0, λk ≥ γj ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ✭✷✳✸✮ ❈❤ó ý r➺♥❣ cl (A) + cl (B) ⊆ cl (A + B) ✳ ▲✃② s = 1, , r tr♦♥❣ ✭✷✳✸✮ ✈➭ ❝é♥❣ ❤❛✐ ✈Õ ❝đ❛ ❝➳❝ ❜❛♦ ❤➭♠ t❤ø❝ ♥❤❐♥ ➤➢ỵ❝✱ t❛ s✉② r❛ 0∈ s∈J ⊆ cl i∈I(¯ x) ¯ k ∂ ∗ fk (¯ λ x) + µ ¯i ∂ ∗ gi (¯ x) + ✈➭ γ¯j = s∈J j∈L γ¯j ∇hj (¯ x) + N (C; x¯) , (s) s∈J,s=k (s) γj γj ∇hj (¯ x) + N (C; x¯) j∈L i∈I(¯ x) ¯ k = λ(s) λ s + (s) µi ∂ ∗ gi (¯ x) + k∈J k∈J tr♦♥❣ ➤ã (s) (s) λk ∂ ∗ fk (¯ x) + cl λk > (∀k ∈ J), µ ¯i = (s) s∈J µi ≥ ( ∀i ∈ I (¯ x)) ∈ R (∀j ∈ L) ✷ ❚õ ➤ã✱ t❛ s✉② r❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷ ➜✐Ò✉ ❦✐Ư♥ ➤đ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ❚r♦♥❣ ♣❤➬♥ ♥➭②✱ t❛ ❝ã t❤Ĩ t❤✃② r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r sÏ trë t❤➭♥❤ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❦❤✐ t❛ ➤➢❛ ✈➭♦ ♠ét ✈➭✐ ❣✐➯ t❤✐Õt ❧å✐ s✉② ré♥❣✳ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ●✐➯ sư x¯ ❧➭ ♠ét ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✮✳ ●✐➯ sư r➺♥❣ fk ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥ ∂ ∗ fk (¯ x) t➵✐ x¯, ∀k ∈ J, gi ❝ã ♠ét ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ∂ ∗ gi (¯ x) t➵✐ x¯, ∀i ∈ I (¯ x)✱ ✈➭ h ❧➭ ❦❤➯ ✈✐ ❋rÐ❝❤❡t t➵✐ x¯✳ ❍➡♥ ♥÷❛✱ ❣✐➯ sư r➺♥❣ ✭✐✮ ¯ k ≥ (∀k ∈ J) ✈í✐ λ = (λk )k∈J = 0, µ ∃λ ¯i ≥ (∀i ∈ I (¯ x)) ✈➭ γ¯j ∈ R (∀j ∈ L) s❛♦ ❝❤♦ ¯ k conv∂ ∗ fk (¯ λ x) + ∈ cl k∈J µ ¯i conv∂ ∗ gi (¯ x) i∈I(¯ x) ✷✾ γ¯j ∇hj (¯ x) + N (C; x¯) , + ✭✷✳✹✮ j∈L ✭✐✐✮ f ❧➭ ❣✐➯ ❧å✐ t✉②Õ♥ tÝ♥❤ t➵✐ ❑❤✐ ➤ã Rn+ x¯ t❤❡♦ ❈❀ gi ✲ t✐Ư♠ ❝❐♥ ✈➠ ❤➢í♥❣ t➵✐ hj ❧➭ tù❛ ❧å✐ ✈➭ ❧➭ tù❛ x¯ t❤❡♦ ❈ (∀i ∈ I (¯ x) , j ∈ L)❀ ❈ ❧➭ t❐♣ ❧å✐✳ x¯ ❧➭ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ❝đ❛ ✭P✮✳ ❈❤ø♥❣ ♠✐♥❤ ❚õ ✭✷✳✹✮ s✉② r❛ tå♥ t➵✐ (∀i ∈ I (¯ x)) ✈➭ n→∞ gi ζ¯(n) ∈ N (C; x¯) k∈J s❛♦ ❝❤♦ (n) ¯ k ξ¯(n) + λ k lim ❇ë✐ ✈× (n) x) (∀k ∈ J) , η¯i ∈ conv∂ ∗ gi (¯ x) ξ¯k(n) ∈ conv∂ ∗ fk (¯ µ ¯i η¯i γ¯j ∇hj (¯ x) + ζ¯(n) + ✭✷✳✺✮ j∈L i∈I(¯ x) ❧➭ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝❤Ý♥❤ q✉② tr➟♥ ∀i ∈ I (¯ x) ✱ tõ ♠Ư♥❤ ➤Ị ✶✳✶✳✶ t❛ s✉② r❛ ✈í✐ ♠ä✐ ηi , x − x¯ ≤ ❞♦ = ∂ ∗ gi (¯ x) ∀i ∈ M t➵✐ x¯ t❤❡♦ M ✱ ✈í✐ ♠ä✐ ✱ (∀ηi ∈ ∂ ∗ gi (¯ x)) , gi (x) − gi (¯ x) ≤ 0(∀x ∈ M ) ✳ ➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ (∀ηi ∈ conv∂ ∗ gi (¯ x)) ηi , x − x¯ ≤ ❉♦ ➤ã✱ (n) η¯i , x − x¯ ≤ ❱× hj (j ∈ L) ❧➭ tù❛ t✉②Õ♥ tÝ♥❤ t➵✐ x¯ t❤❡♦ (∀x ∈ M ) C ∇hj (¯ x) , x − x¯ = ❱× C ❧å✐ ♥➟♥ t❛ ❝ã ♥➟♥ t❛ ❝ã (∀x ∈ M ) x − x¯ ∈ T (C; x¯) ∀x ∈ C ζ¯(n) , x − x¯ ≤ ✭✷✳✻✮ ✭✷✳✼✮ ✳ ❱× ✈❐②✱ (∀x ∈ M ) ✭✷✳✽✮ ❑Õt ❤ỵ♣ ✭✷✳✺✮ ✲ ✭✷✳✾✮ t❛ s✉② r❛ ¯ k ξ¯(n) , x − x¯ ≥ λ k lim n→∞ ❱× fk ❝ã ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ q✉② t➽❝ ✹✳✶✱ ✹✳✷ ❬✾❪ t❛ s✉② r❛ ✭✷✳✾✮ k∈J ¯T f λ ¯ k ≥ ∀k ∈ J ∂ ∗ fk (¯ x) x¯ λ ¯ k ∂ ∗ fk (¯ λ x) ❝ã t➵✐ k∈J ✈➭ ✱ t❤❡♦ ❝➳❝ ❧➭ ♠ét ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ✸✵ ré♥❣ t➵✐ ¯T f λ x¯ ✳ ❙ư ❞ơ♥❣ tÝ♥❤ ❣✐➯ ❧å✐ ❧➭ ❣✐➯ ❧å✐ t✐Ư♠ ❝❐♥ t➵✐ x¯ n R+ t❤❡♦ M ✲ t✐Ö♠ ❝❐♥ ✈➠ ❤➢í♥❣ ❝đ❛ x¯ ✱ t❛ s✉② r❛ r➺♥❣ ✳ ❉♦ ➤ã tõ ✭✷✳✾✮ t❛ ❝ã ¯ f (x) ≥ λ, ¯ f (¯ λ, x) ❚õ ➤ã s✉② r❛ f (∀x ∈ M ) ❧➭ ❝ù❝ t✐Ó✉ P❛r❡t♦ ②Õ✉ ❝đ❛ ✭P✮✳ ➜ã ❧➭ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷ ✸✶ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝đ❛ ❉✳ ❱✳ ▲➢✉ ✭✷✵✶✹✮ ❝❤♦ ♥❣❤✐Ư♠ ❤÷✉ ❤✐Ư✉ ②Õ✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ❝ã r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✈➭ r➭♥❣ ❜✉é❝ t❐♣ ✈í✐ ❝➳❝ ❤➭♠ ▲✐♣s❝❤✐t③ ➤Þ❛ ♣❤➢➡♥❣ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣✱ ❜❛♦ ❣å♠✿ ✲ ❈➳❝ ❦✐Õ♥ t❤ø❝ ✈Ị ❞➢í✐ ✈✐ ♣❤➞♥ s✉② ré♥❣ ❝đ❛ ❏❡②❛❦✉♠❛r✲▲✉❝ ✭✶✾✾✾✮❀ ✲ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❋r✐t③ ❏♦❤♥ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣❀ ✲ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝❤Ý♥❤ q✉②❀ ✲ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ❑❛r✉s❤✲❑✉❤♥✲❚✉❝❦❡r ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉ ➤Þ❛ ♣❤➢➡♥❣❀ ✲ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ➤đ ❝❤♦ ❝ù❝ t✐Ĩ✉ P❛r❡t♦ ②Õ✉✳ ▲ý t❤✉②Õt ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tè✐ ➢✉ ❝❤♦ ♥❣❤✐Ư♠ ❤÷✉ ❤✐Ư✉ ❝đ❛ ❜➭✐ t♦➳♥ tè✐ ➢✉ ➤❛ ♠ơ❝ t✐➟✉ ❦❤➠♥❣ tr➡♥ ❞➢í✐ ♥❣➠♥ ♥❣÷ ❝➳❝ ❞➢í✐ ✈✐ ♣❤➞♥ ❧➭ ➤Ị t➭✐ ➤➲ ✈➭ ➤❛♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ t➳❝ ❣✐➯ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ♣❤➳t tr✐Ĩ♥✳ ✸✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❚➭✐ ❧✐Ö✉ ❚✐Õ♥❣ ❱✐Öt❪ ❬ ❬✶❪ ỗ P tí ọ ĩ tt ộ ỗ ❱➝♥ ▲➢✉ ✭✶✾✾✾✮✱ ●✐➯✐ tÝ❝❤ ▲✐♣s❝❤✐t③✱ ◆❳❇ ❦❤♦❛ ❤ä❝ ✈➭ ❦Ü t❤✉❐t ❍➭ ◆é✐✳ ❚➭✐ ❧✐Ö✉ ❚✐Õ♥❣ ❆♥❤❪ ❬ ❬✸❪ ❆✉❜✐♥✱ ❏✳P✳✱ ❈❡❧❧✐♥❛✱ ❆✳ ✭✶✾✽✹✮✿ ❉✐❢❢❡r❡♥t✐❛❧ ■♥❝❧✉s✐♦♥s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✳ ❬✹❪ ❈❧❛r❦❡✱ ❋✳❍✳ ✭✶✾✽✸✮✿ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ◆♦♥s♠♦♦t❤ ❆♥❛❧②s✐s✱ ❲✐❧❡② ■♥t❡r✲ s❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦✳ ❬✺❪ ❉❡♠②❛♥♦✈✱ ❱✳❋✳ ✭✶✾✾✹✮✿ ❈♦♥✈❡①✐❢✐❝❛t✐♦♥ ❛♥❞ ❝♦♥❝❛✈✐❢✐❝❛t✐♦♥ ♦❢ ❛ ♣♦s✐t✐✈❡❧② ❤♦♠♦❣❡♥❡♦✉s ❢✉♥❝t✐♦♥ ❜② t❤❡ s❛♠❡ ❢❛♠✐❧② ♦❢ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ❯♥✐✈❡rs✐❛ ❞✐ P✐s❛✱ ❘❡♣♦rt ✸✱ ✷✵✽✱ ✽✵✷✳ ❬✻❪ ❉❡♠②❛♥♦✈✱ ❱✳❋✳✱ ❘✉❜✐♥♦✈✱ ❆✳▼✳ ✭✶✾✾✺✮✿ ❈♦♥str✉❝t✐✈❡ ◆♦♥s♠♦♦t❤ ❆♥❛❧②s✐s✱ ❱❡r❧❛❣ P❡t❡r ▲❛♥❣✱ ❋r❛♥❦❢✉rt✳ ❬✼❪ ❉✉tt❛✱ ❏✳✱ ❈❤❛♥❞r❛✱ ❙✳ ✭✷✵✵✹✮✿ ❈♦♥✈❡①✐❢✐❝❛t♦rs✱ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈❡①✐t② ❛♥❞ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✳ ❖♣t✐♠✐③❛t✐♦♥ ✺✸✱ ✼✼✲✾✹✳ ❬✽❪ ❉✉tt❛✱ ❏✳✱ ❈❤❛♥❞r❛✱ ❙✳ ✭✷✵✵✷✮✿ ❈♦♥✈❡①✐❢✐❝❛t♦rs✱ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✈❡①✐t② ❛♥❞ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✶✸✱ ✹✶✲✻✺✳ ❬✾❪ ❏❡②❛❦✉♠❛r✱ ❱✳✱ ▲✉❝✱ ❉✳❚✳ ✭✶✾✾✾✮✿ ◆♦♥s♠♦♦t❤ ❝❛❧❝✉❧✉s✱ ♠✐♥✐♠❛❧✐t②✱ ❛♥❞ ♠♦♥♦t♦♥✐❝✐t② ♦❢ ❝♦♥✈❡①✐❢✐❝❛t♦rs✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✵✶✱ ✺✾✾✲✻✷✶✳ ✸✸ ❬✶✵❪ ❏❡②❛❦✉♠❛r✱ ❱✳✱ ▲✉❝✱ ❉✳❚✳ ✭✶✾✾✽✮✿ ❆♣♣r♦①✐♠❛t❡ ❏❛❝♦❜✐❛♥ ♠❛tr✐❝❡s ❢♦r ♥♦♥✲ s♠♦♦t❤ ❝♦♥t✐♥✉♦✉s ♠❛♣s ❛♥❞ C1 ✲ ♦♣t✐♠✐③❛t✐♦♥✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✸✻✱ ✶✽✶✺✲✶✽✸✷✳ ❬✶✶❪ ❏✐♠Ð♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳ ✭✷✵✵✷✮✿ ❆ ❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❡①t❡♥s✐♦♥ ♦❢ ▲②✉st❡r♥✐❦ t❤❡♦r❡♠ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✼✵✱ ✸✹✵✲✸✺✻✳ ❬✶✷❪ ▲✉❝✱ ❉✳ ❚✳ ✭✷✵✵✷✮✿ ❆ ♠✉❧t✐♣❧✐❡r r✉❧❡ ❢♦r ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜✲ ❧❡♠s ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❞❛t❛✳ ❙■❆▼ ❏✳ ❖♣t✐♠✳ ✶✸✱ ✶✻✽✲✶✼✽✳ ❬✶✸❪ ▲✉✉✱ ❉✳ ❱✳ ✭✷✵✶✷✮✿ ◆❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ❡❢❢✐❝✐❡♥❝② ✐♥ t❡r♠s ♦❢ t❤❡ ▼✐❝❤❡❧✲P❡♥♦t s✉❜❞✐❢❢❡r❡♥t✐❛❧s✱ ❖♣t✐♠✐③❛t✐♦♥ ✻✶✱ ✶✵✾✾✲✶✶✶✼✳ ❬✶✹❪ ▲✉✉✱ ❉✳ ❱✳ ✭✷✵✶✹✮✿ ◆❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ❡❢❢✐❝✐❡♥❝② ✈✐❛ ❝♦♥✈❡①✐❢✐❝❛t♦rs✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✻✵✱ ✺✶✵✲✺✷✻✳ ❬✶✺❪ ▲✉✉✱ ❉✳ ❱✳ ✭✷✵✶✹✮✿ ❈♦♥✈❡①✐❢✐❝❛t♦rs ❛♥❞ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ❡❢❢✐❝✐❡♥❝②✱ ❖♣t✐♠✐③❛t✐♦♥ ✻✸✱ ✸✷✶✲✸✸✺✳ ❬✶✻❪ ▼❛♥❣❛ss❛r✐❛♥✱ ❖✳▲✳ ✭✶✾✻✾✮✿ ◆♦♥❧✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✳ ▼❝●r❛✇✲❍✐❧❧✱ ◆❡✇ ❨♦r❦✳ ❬✶✼❪ ▼✐❝❤❡❧✱ P✳✱ P❡♥♦t✱ ❏✳ ✲P✳ ✭✶✾✽✹✮✿ ❈❛❧❝✉❧ s♦✉s✲❞✐❢❢Ðr❡♥t✐❡❧ ♣♦✉r ❞❡s ❢♦♥❝t✐♦♥s ❧✐♣s❝❤✐t③✐❡♥♥❡s ❡t ♥♦♥❧✐♣s❝❤✐t③✐❡♥♥❡s✱ ❈✳❘✳ ▼❛t❤✳ ❆❝❛❞✳ ❙❝✐✳ ✶✷✱ ✷✻✾✲✷✼✷✳ ❬✶✽❪ ▼♦r❞✉❦❤♦✈✐❝❤✱ ❇✳❙✳✱ ❙❤❛♦✱ ❨✳ ✭✶✾✾✺✮✿ ❖♥ ♥♦♥❝♦♥✈❡① s✉❜❞✐❢❢❡r❡♥t✐❛❧ ❝❛❧❝✉✲ ❧✉s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳ ✷✱ ✷✶✶✲✷✷✽✳ ❬✶✾❪ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳❚✳ ✭✶✾✼✵✮ ✿ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✳ ❬✷✵❪ ❨❛♥❣✱ ❳✳◗✳ ✭✷✵✵✺✮✿ ❈♦♥t✐♥✉♦✉s ❣❡♥❡r❛❧✐③❡❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❖♣t✐♠✐③❛t✐♦♥ ✺✹✱ ✹✾✺✲✺✵✻✳ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r