Các điều kiện tối ưu cấp hai với hiện tượng envelope like trong các bài toán quy hoạch đa mục tiêu không trơn

i BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC KINH TẾ TP.HCM ĐỀ TÀI NGHIÊN CỨU KHOA HỌC CẤP TRƯỜNG CÁC ĐIỀU KIỆN TỐI ƯU CẤP HAI VỚI HIỆN TƯỢNG ENVELOPE-LIKE TRONG CÁC BÀI TỐN QUY HOẠCH ĐA MỤC TIÊU KHƠNG TRƠN Mã số: CS – 2013 - 37 Chủ nhiệm: TS Nguyễn Đình tuấn Tp Hồ Chí Minh 12 – 2013 ệ ệ ệ ệ ữỡ ỵ ❞♦ ❝❤å♥ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸ ✷✳ ▼ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ✹✳ ❑➳t ❝➜✉ ❝õ❛ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✹ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❝ỉ♥❣ ❝ư tr♦♥❣ ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ữỡ lờ ổ ữợ ❝ơ♥❣ ♥❤÷ ✈❡❝tì ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✼ ữỡ t ữợ tr ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣ ✳✳✳✳✶✶ ❈❤÷ì♥❣ tố ữ ợ ❤✐➺♥ t÷đ♥❣ ❡♥✈ ❡❧♦♣❡✲❧✐❦❡✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✶✺ ❈❤÷ì♥❣ ✺✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷✺ ❑➳t ❧✉➟♥ ✈➔ ❤÷ỵ♥❣ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✷✾ ❈→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ✤➲ t➔✐ ự ữỡ ỵ ✤➲ t➔✐✳ ❚r♦♥❣ q✉② ❤♦↕❝❤ t♦→♥ ❤å❝✱ ✈➔ tê♥❣ q✉→t ❤ì♥ tr♦♥❣ tè✐ ÷✉ ❤â❛✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝❤✐➳♠ ♠ët ✈à tr➼ q✉❛♥ trå♥❣✱ ✈➻ ♥â ❝✉♥❣ ❝➜♣ t❤æ♥❣ t✐♥ t❤➯♠ q✉❛♥ tr♦♥❣ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët✳ ✣➸ ✤→♣ ù♥❣ ❝❤♦ sü ♣❤➙♥ ❧♦↕✐ ù♥❣ ❞ö♥❣ t❤ü❝ t➳✱ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤÷đ❝ ①❡♠ ①➨t✱ ✈➔ ❞♦ ✤â ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t ♥❣❤✐➯♥ ❝ù✉✱ ♥❣➔② ❝➔♥❣ trð ♥➯♥ ♣❤ù❝ t↕♣ ❤ì♥✳ ❚✉② ♥❤✐➯♥ ❝❤ó♥❣ t❛ ❝â t❤➸ ♥❤➟♥ t❤➜② r➡♥❣✱ tr♦♥❣ ✤❛ sè ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❧✐➯♥ q✉❛♥ ✤➣ ❝â✱ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝â t❤➸ ✤÷đ❝ ❦❤➥♥❣ ✤à♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❦➳t q✉↔ ❝ê ✤✐➸♥ ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ❝→❝ ❤➔♠ ♠ö❝ t✐➯✉ ✭❤♦➦❝ ❝→❝ ❤➔♠ ▲❛❣r❛♥❣❡ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝â r➔♥❣ ❜✉ë❝✮ t↕✐ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉ ❧➔ ❦❤æ♥❣ ➙♠✳ ❑❛✇❛s❛❦✐ ❬✶✼❪ ❧➔ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❝❤♦ t r t ữợ ❤➔♠ ▲❛❣r❛♥❣❡ ❝â t❤➸ ➙♠ t↕✐ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉✱ t ữợ t ủ ❜ð✐ ❤➔♠ ♠ö❝ t✐➯✉ ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝ ♥➡♠ tr➯♥ ♣❤➛♥ ✤➦❝ ❜✐➺t ❝õ❛ ❜✐➯♥ ❝õ❛ ♥â♥ ❤ñ♣ ➙♠ tr♦♥❣ t➼❝❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ↔♥❤✳ ➷♥❣ ➜② ❣å✐ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❑❛✇❛s❛❦✐ ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ♥❣ ❤✐➯♥ ❝ù✉ ♣❤→t tr✐➸♥ tr♦♥❣ ❬✻✱ ✽✱ ✷✺✱ ✷✻❪✱ ❧✉æ♥ ❧✉æ♥ ①❡♠ ①➨t ❝→❝ ❜➔✐ t q ổ ữợ tở ợ C2 ố ữ tr♦♥❣ ❬✶✼❪✳ ❚r♦♥❣ q✉② ❤♦↕❝❤ ✤❛ ♠✉❝ t✐➯✉✱ ❝→❝ ❦➳t q✉↔ ✤➛✉ t✐➯♥ t❤✉ë❝ ❦✐➸✉ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✶✹✱ ✶✺❪ ❝ơ♥❣ ①➨t ❝❤♦ ❝→❝ tr÷í♥❣ ❤đ♣ trì♥✳ ✣è✐ ✈ỵ✐ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✱ ●✉t✐➨rr❡③✲❏✐♠➨♥❡③✲◆♦✈♦ ❬✶✸❪ ũ t ữợ ✈➔ ♣❛r❛❜♦❧✐❝ ✤❛ trà ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tố ữ ợ tữủ ①➨t ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♠➔ ✤↕♦ ❤➔♠ ❝õ❛ ♥â ❧➔ ❧✐➯♥ tö❝ ❤♦➦❝ ê♥ ✤à♥❤ t↕✐ ✤✐➸♠ ♥❣❤✐➯♥ ❝ù✉✳ ❚✉② ♥❤✐➯♥✱ ✈➝♥ ❝á♥ ♥❤✐➲✉ t→❝ ❣✐↔ ❝❤÷❛ ♥❤➟♥ r❛ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐✳ ✣✐➲✉ ♥➔② ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ët sè s❛✐ ❧➛♠ ❦❤ỉ♥❣ ❜✐➳t✳ ❍ì♥ ♥ú❛✱ tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ♥â✐ tr➯♥✱ ✤ỉ✐ ❦❤✐ ❦❤ỉ♥❣ ①→❝ ✤à♥❤ ✤÷đ❝ ❦❤✐ ♥➔♦ t❤➻ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ✈➔ ❦❤✐ t ổ q st tr ỗ ❝↔♠ ❤ù♥❣ ❝❤♦ ♠ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ ❧➔♠ rã ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐✳ ▼➦t ❦❤→❝✱ ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ❝❤➼♥❤ ❝❤♦ tè✐ ÷✉ ❦❤ỉ♥❣ trì♥ ❧➔ ✤➲ ①✉➜t ✈➔ →♣ ❞ư♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ t❤➼❝❤ ❤ñ♣ ✤➸ t❤❛② t❤➳ t rt ổ tỗ t ❦❤✐ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉✳ ◆❤✐➲✉ ❧♦↕✐ ✤↕♦ ❤➔♠ ✤➣ ✤÷đ❝ ❞ị♥❣✱ ♠é✐ ❧♦↕✐ ✤➲✉ ❝â t❤✉➟♥ ❧đ✐ r✐➯♥❣ tr♦♥❣ ♠ët sè t➼♥❤ ❤✉è♥❣ ❝ư t❤➸ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤✉➟♥ ❧đ✐ ❝❤♦ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣✳ ●➛♥ ✤➙②✱ ❝→❝ ✤↕♦ ❤➔♠ ✤❛ trà ❝❤♦ ❤➔♠ ✈❡❝tì ✤ì♥ trà ✤➣ ✤÷đ❝ sû ❞ư♥❣ ❤✐➺✉ q✉↔ ✤➸ ❝✉♥❣ ❝➜♣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû tr♦♥❣ ❝→❝ q✉② ❤♦↕❝❤ ❦❤æ♥❣ trì♥✱ ①❡♠ ❬✺✱ ✶✵✱ ✶✶✱ ✶✸✱ ✶✾✱ ✷✸❪ ✭♥❤÷♥❣ tr♦♥❣ ❬✺✱ ✶✵✱ ✶✶✱ ✶✾✱ ✷✸❪✱ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❦❤ỉ♥❣ ①↔② r q st ỗ ự t t❤❡♦ ❝❤♦ ♠ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ t❤ù ❤❛✐ ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ →♣ ❞ư♥❣ t ữợ r ữủ ①✉➜t tr♦♥❣ ❬✶✾❪✮ ❝ị♥❣ ✈ỵ✐ t➼♥❤ ❝❤➜t l✲ê♥ ✤à♥❤ ✭✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✷✱ ✸✱ ✹✱ ✶✷❪✮ ✤➸ ✤↕t ữủ tố ữ ợ t❤✐➺♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❣➛♥ tr t ữợ ❤❛✐ ❍❛❞❛♠❛r❞ t↕✐ ♠ët ✤✐➸♠ t❤➻ ❧ỵ♥ ❤ì♥ ❣✐→ trà t ữợ ♣❛r❛❜♦❧✐❝✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝õ❛ ❝❤ó♥❣ tỉ✐ ♠↕♥❤ ❤ì♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tr♦♥❣ ❬✶✸❪✳ ❍ì♥ ♥ú❛✱ ❝❤ó♥❣ tỉ✐ ♥ỵ✐ ❧ä♥❣ ✸ ❝→❝ ❣✐↔ t❤✐➳t ❝❤➼♥❤ ✤➦t r❛ tr♦♥❣ ❬✶✸❪✿ t❤❛② t❤➳ ❧➛♥ ❧÷đt t➼♥❤ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ ✈➔ ê♥ ✤à♥❤ ❜ð✐ t➼♥❤ ❦❤↔ ✈✐ ❝❤➦t ✈➔ l✲ê♥ ✤à♥❤✳ ✷✳ ▼ư❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ s❛✉ ✤➙②✳ ❈❤♦ ❝→❝ ❤➔♠ f : Rn → R m✱ g : Rn → Rp ✈➔ h : Rn Rr C õ ỗ õ tr Rm K t ỗ tr Rp t ữợ sỹ t ú tổ P f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K✱ h(x) = 0✳ n ◆➳✉ K = R +✱ t❤➻ r➔♥❣ ❜✉ë❝ g(x) ∈ −K trð t❤➔♥❤ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤æ♥❣ tữớ ú tổ ũ t ữợ tr r ữợ tt t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② l✲ê♥ ✤à♥❤ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮ ✤➸ t❤✐➳t ❧➟♣ tố ữ ợ ợ t ❝❤➜t ❡♥✈❡❧♦♣❡✲❧✐❦❡ ✤÷đ❝ ❧➔♠ rã ❤ì♥ ❝❤♦ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ✭P✮✳ ❈ư t❤➸✱ ✤➲ t➔✐ t❤ü❝ ❤✐➺♥ ❝→❝ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤➙②✳ lờ ổ ữợ ụ ữ ✈❡❝tì ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✰ ❑❤→✐ t ữợ tr ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tố ữ ợ tữủ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✳ ❈→❝ ❦➳t q ✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ✤➣ ✤÷đ❝ t→❝ ❣✐↔ ✈➔ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ◗✉è❝ t➳✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❚♣✳ ❍❈▼ ❝æ♥❣ ❜è tr➯♥ ♠ët t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ q✉ è❝ t➳ tr♦♥❣ ❤➺ t❤è♥❣ ■❙■✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❞ị♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ❣ ✐↔✐ t➼❝❤ ✤❛ trà ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✳ t t t ỗ ữỡ ã ữỡ ỵ tỹ ❤✐➺♥ ✤➲ t➔✐✱ ♠ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ t ã ữỡ ởt số ổ tr♦♥❣ ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ã ữỡ lờ ổ ữợ ụ ữ tỡ ởt số t t ú ã ữỡ t ữợ tr t t ú ã ữỡ tố ữ ợ tữủ ã ữỡ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐✳ ✹ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❝ỉ♥❣ ❝ư tr♦♥❣ ❣✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ trì♥ ✈➔ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ N ✈➔ R ❧➛♥ ❧÷đt ❧➔ ❝→❝ t➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥ ✈➔ sè t❤ü❝✳ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X✱ X∗ ❧➔ ✤è✐ ♥❣➝✉ t♦♣♦ ❝õ❛ ♦❢ X❀ (., ) ❧➔ t➼❝❤ ✤è✐ ♥❣➝✉✳ "." ❧➔ ❝❤✉➞♥ tr♦♥❣ ❦❤æ♥❣ ❣ ✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❜➜t ❦ý ✈➔ d(y, S) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø ✤✐➸♠ y ✤➳♥ t➟♣ S✳ Bn(x, r) = {y ∈ Rn : "x − y" < r}❀ Sn = {y ∈ Rn : "y" = 1}❀n S∗ = {y ∈ (Rn) : "y" = 1} L(X, Y ) ỵ ❤✐➺✉ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥Xtø ✈➔♦ Y ✱ tr♦♥❣ ✤â ❧➔ ❝→❝ ❦❤æ♥❣ X ✈➔ ợ õ C Rn ỵ CY∗ = {c∗ ∈ (Rn)∗ : (c∗, c) ≥ 0, ∀c ∈ C} ❧➔ ♥â♥ ✤è✐ ❝ü❝ ❝õ❛ C✳ ❱ỵ✐ A Rn ỵ rA tA A A ❝♦♥✈A✱ ❝♦♥❡A ✈➔ ▲✐♥A ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ố tr õ ỗ õ ❝õ❛ A ✈➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ s✐♥❤ ❜ð✐ A✳ ❱ỵ✐ t > ✈➔ r ∈ N✱ o(tr ) ỵ ởt tở t s❛♦ ❝❤♦ o(tr )/tr → ❦❤✐ t → 0+✳ ❈❤ó♥❣ t❛ ❤➣② ♥❤ỵ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ s❛✉ ✤➙②✳ ⑩♥❤ ①↕ f : Rn → X✱ tr♦♥❣ ✤â X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x ∈ Rn ♥➳✉ ♥â ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t f t (x) t↕✐ x ✈➔ limy→x,t→0+ suph∈Sn " t t (f (y + th) − f (y)) − f (x)h" = 0✳ ❱ỵ✐ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ f : Rn → Rm✱ ❏❛❝♦❜✐❛♥ s✉② rë♥❣ ❈❧❛r❦❡ ❝õ❛ f t↕✐ x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ∂f (x) = conv{limf t (xk ) : xk ∈ Ω, xk → x}✱ tr♦♥❣ ✤â f ❦❤↔ ✈✐ tr♦♥❣ Ω✱ ợ t trũ t ỵ r ▼ët ✈➔✐ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❏❛❝♦❜✐❛♥ s✉② rë♥❣ ❈❧❛r❦❡ ✤÷đ❝ ❧✐➺t ❦➯ tr♦♥❣ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✶ ✭❬✼❪✮✳ ❈❤♦ f : Rn → Rm ❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ x✳ ❑❤✐ ✤â✱ ✭✐✮ ∂f (x) t ỗ rộ tr L(Rn, Rm); ✭✐✐✮ ∂f (x) ❧➔ t➟♣ ♠ët ✤✐➸♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x✿ ∂f (x) = {f t (x)}; ✭✐✐✐✮ ∂f (x) = {limk→∞vk : vk ∈ ∂f (xk ), xk → x}✱ ♥â✐ ❝→❝❤ ❦❤→❝ ✭✈➻ ∂f (x) ❧➔ ❝♦♠♣➠❝✮✱ →♥❤ ①↕ ∂f (.) ỳ tử tr t x; ỵ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ▲❡❜♦✉r❣✮ ♥➳✉ f ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ữỡ tr ởt ỗ U x a, b ∈ U ✱ t❤➻ f (b) − f (a) ∈ conv(∂f ([a, b])(b − a)) ✈➔ ❦❤✐ m = tỗ t ởt c (a, b) s❛♦ ❝❤♦ f (b) − f (a) ∈ ∂f (c)(b − a) ●✐í ✤➙②✱ t❛ ♥❤➢❝ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♣❤➛♥ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ x0, u ∈ Rn ✈➔ M ⊂ Rn✳ ✭❛✮ ◆â♥ ❝♦♥t✐♥❣❡♥t ❝õ❛ M t↕✐ x0 ❧➔ ✺ T (M, x0) = {v ∈ Rn : ∃tk → 0+, ∃vk → v, ∀k ∈ N, x0 + tkvk ∈ M} ✳ ✭❜✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❝õ❛ M t↕✐ x0 ❧➔ IT (M, x0) = {v ∈ Rn : ∀tk → 0+, ∀vk → v, ∀k ✤õ ❧ỵ♥, x0 + tkvk ∈ M} ✳ ✭❝✮ ❚➟♣ ❝♦♥t✐♥❣❡♥t ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0, u) ❧➔ T 2(M, x0, u) = {w ∈ Rn : ∃tk → 0+, ∃wk → w, ∀k ∈ N, x0 + tku + t2 wk ∈ M} ✳ k ✭❞✮ ◆â♥ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ M t↕✐ (x0, u) ❧➔ T tt (M, x0, u) = {w ∈ Rn : ∃(tk, rk ) → (0+, 0+) : tk/rk → 0, ∃wk → w✱ ∀k ∈ N, x0 + tku +2 tkrkwk ∈ M} ✳ ✭❡✮ ❚➟♣ ❦➲ ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0, u) ❧➔ A2(M, x0, u) = {w ∈ Rn : ∀tk → 0+, ∃wk → w, ∀k ∈ N, x0 + tku + t2 wk ∈ M} ✳ ✭❢✮ ❚➟♣ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ ❝õ❛ M t↕✐ (x0, u) ❧➔ k IT 2(M, x0, u) = {w ∈ Rn : ∀tk → 0+, ∀wk → w, ∀k ✤õ ❧ỵ♥, x0 + tku + t2 wk ∈ M} ✳ k ▼➺♥❤ ✤➲ s❛✉ ✤➙② tâ♠ t➢t ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ tr➯♥✳ ▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ M ⊂ Rn✱ x0 ∈ Rn ✈➔ u ∈ Rn✳ ❑❤✐ ✤â✱ ✭✐✮ IT 2(M, x0, u) ⊂ A2(M, x0, u) ⊂ T 2(M, x0, u) ⊂ clcone[cone(M − x0) − u]; ✭✐✐✮ ♥➳✉ u ƒ∈ T (M, x0)✱ t❤➻ T 2(M, x0, u) = ∅✳ ◆➳✉✱ t❤➯♠ ỳ M ỗ tM = u T (M, x0)✱ t❤➻ ✭①❡♠ ❬✶✺✱ ✷✹✱ ✷✽❪✮ ✭✐✐✐✮ ✐♥t❝♦♥❡(M − x0) = IT (intM, x0); ✭✐✈✮ ♥➳✉ A2(M, x0, u) ƒ= ∅✱ t❤➻ IT 2(M, x0, u) = intA2(M, x0, u), clIT 2(M, x0, u) = A2(M, x0, u); ✭✈✮ ♥➳✉ u ∈ cone(M − x0)✱ t❤➻ ✭❛✮ IT 2(M, x0, u) = intcone[cone(M − x0) − u]❀ ✭❜✮ A2(M, x0, u) = clcone[cone(M − x0) − u]✳ ✻ ữỡ lờ ổ ữợ ụ ♥❤÷ ✈❡❝tì ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣ ❍➔♠ h : Rn → Rm ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ t x Rn tỗ t ởt U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✱ "h(y) − h(x)" ≤ ϑ"y − x"✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶ ✭❬✷✱ ✶✷❪✮ ✭✐✮ ✣↕♦ ❤➔♠ t ữợ ữợ tữỡ ự tr : Rn R t x t ữợ u ữủ ♥❣❤➽❛ ❜ð✐ ϕl(x, u) = lim inft →0+ (ϕ(x + tu) − ϕ(x)) t + u →0 ✭t÷ì♥❣ ù♥❣✱ ϕ (x, u) = (ϕ(x + tu) − ϕ(x))✮✳ t lim supt ✭✐✐✮ ❍➔♠ ϕ ✤÷đ❝ ❣å✐ ❧➔ l✲ê♥ tữỡ ự uờ t x tỗ t ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x ✈➔ ϑ > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ y ∈ U ✈➔ u ∈ Sn✱ ✭✷✳✶✮ ✭t÷ì♥❣ ù♥❣✱ |ϕu(y, u) − ϕu(x, u)| ≤ ϑ"y − x"✮✳ ✭✷✳✷✮ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ l✲ê♥ ✤à♥❤ ϕ : Rn → R ✤÷đ❝ tâ♠ t➢t tr♦♥❣ ♠➺♥❤ ✤➲ s❛✉✳ |ϕl(y, u)−ϕl(x, u)| ≤ ϑ"y−x" ▼➺♥❤ ✤➲ ✷✳✷ ✭✐✮ ✭❬✷✱ ✹❪✮ ❍➔♠ l✲ê♥ ✤à♥❤ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❦❤↔ ✈✐ ❝❤➦t✳ ✭✐✐✮ ✭❬✶✷❪✮ ϕ ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ϕ ❧➔ rt t x tỗ t ởt ❝➟♥ U ❝õ❛ x s❛♦ ❝❤♦ ϕ ❧➔ ▲✐♣s❝❤✐t③ tr➯♥ U tỗ t > s "ϕ (y) −t ϕ (x)" ≤ tϑ"y − x" ❤➛✉ ❤➳t tr♦♥❣ U ✭t❤❡♦ ♥❣❤➽❛ ✤ë ✤♦ ▲❡❜❡s❣✉❡✮✳ ✭✐✐✐✮ ✭❬✶✷❪✮ ❈→❝ ❦❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ ✈➔ u✲ê♥ ✤à♥❤ ❧➔ t÷ì♥❣ ✤÷ì♥❣❀ ❤ì♥ ♥ú❛ ❧➙♥ ❝➟♥ U ✈➔ ❤➡♥❣ sè ϑ ✤÷đ❝ →♣ ❞ư♥❣ ❣✐è♥❣ ♥❤❛✉ tr♦♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✮ ❤♦➦❝ ✭✷✳✷✮✳ ❇ð✐ ▼➺♥❤ ✤➲ ✷✳✷ ✭✐✐✐✮✱ tr♦♥❣ ♣❤➛♥ s t sỷ t ữợ ữợ lờ ỏ t ữợ tr u✲ê♥ ✤à♥❤ ✤÷đ❝ ♥❤➢❝ ✤➳♥ ❦❤✐ ❝➛♥ t❤✐➳t✳ ❑❤→✐ ♥✐➺♠ l✲ê♥ ✤à♥❤ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ❝→❝ ❤➔♠ ✈❡❝tì ♥❤÷ s t ữợ ữợ ✭t÷ì♥❣ ù♥❣✱ tr➯♥✮ ❝õ❛ ❤➔♠ Φ : Rn → Rm ❘ã   f t (x0) = (0, 1)✱ gt (x0) ❂ 0 0✱ g(x0) = (0, 1, 0) K r K õ ỗ ✤â♥❣ ✈ỵ✐ ✐♥tK ƒ= ∅✳ ❇➙② ❣✐í t❛ t➼♥❤ K(g(x0)) = cone(K + g(x0))✳ ❱ỵ✐ (t1, t2, t3) ∈ K(g(x0))✱ (t1, t2, t3) = α[(k1, k2, k3) + (0, 1, 0)] ⇔ t1 = αk1, t2 = α(k2 + 1), t3 = αk3 ✈ỵ✐ α ≥ ✈➔ (k1, k2, k3) ∈ K✳ ◆➳✉ α = 0✱ t❤➻ (t1, t2, t3) = (0, 0, 0)✳ ◆➳✉ α > 0✱ t❤➻ k1 = t1/α, k2 = (t2/α) − 1, k3 = t3/α✳ k2 ≤ ⇔ t2 ≤ α✱ k3 ≤ ⇔ t23 ≤ 0✱ k2k3 ≥ 2k ⇔ (t2 − α)t3 ≥ 2t2✳ ✭✹✳✶✹✮ ❚❛ ❝â ❚❛ ❝â ❤❛✐ tr÷í♥❣ ❤đ♣✳ ◆➳✉ t3 = 0✱ t❤➻ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ t1 = 0✳ ❱➻ t2 ≤ α✱ t ữủ (t1, t2, t3) = (0, 2, 0) ợ β2 ∈ R✳ ✭✹✳✶✺✮ ◆➳✉ t3 < 0✱ t❤➻ t2 ≤ α + (2t1 /t3)✳ ❱➻ t❤➳ (t1, t2, t3) = (β1, β2, β3) ✈ỵ✐ β1, β2 ∈ R, β3 < 0✳ ✭✹✳✶✻✮ ❚ø ✭✹✳✶✹✮ ✲ ✭✹✳✶✻✮✱ t❛ ✤↕t ✤÷đ❝ ❍ì♥ ♥ú❛✱ K(g(x0)) = {(k1, k2, k3) ∈ R3 : k3 < 0} ∪ {(0, k2, 0) : k2 ∈ R} T (−K, g(x0)) = {(k1, k2, k3) ∈ R3 : k3 ≥ 0}, N (−K, g(x0)) = {λ(0, 0, −1) : λ ∈ R}, Λ1(x0) = {(c∗, k∗) ∈ R × R3 : c∗ = α > 0, k∗ = α(0, 0, −1)} ❚÷ì♥❣ tü ♥❤ư tr♦♥❣ ❱➼ ❞ö ✹✳✶✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ ❦❤æ♥❣ ❦❤↔ ✈✐ tr♦♥❣ U \ {x0}✱ ✈ỵ✐ ❜➜t ❝ù ❧➙♥ ❝➟♥ U ❝õ❛ x0✳ ❈❤♦ u = (u1, u2) ∈ R2 s❛♦ ❝❤♦ (f, g)t (x0)u ∈ −[C×❝❧ K(g(x0))\ ✐♥t(C × K(g(x0)))]✳ ❑❤✐ ✤â✱ u = (u1, 0) ✈ỵ✐ u1 ∈ R ✈➔ gt (x0)u = (u1, 0, 0)✳ ◆➳✉ u ƒ= 0✱ tù❝ ❧➔✱ u1 ƒ= 0✱ t❤➻ gt (x0)u ƒ∈ −K(g(x0))✳ ◆➳✉ u = 0✱ t❤➻ gt (x0)u ∈ −K(g(x0)) ✈➔ {(β, γ, 0, β) : β, γ ∈ R}✳ D2(f, g)(x0, u) = ❱➻ t❤➳✱ ✈ỵ✐ ♠å✐ (y0, z0) ∈ D2(f, g)(x0, u) tỗ t (c , k ) = (1, 0, 0, −1) ∈ C × K(g(x0)) \ {(0, 0)} s❛♦ ❝❤♦ c∗ ◦ f t (x0) + k∗ ◦ gt (x0) = ✈➔ (c∗, y0) + (k, z0) = õ ỵ ỵ ổ t ❜ä x0✳ ▼➦❝ ❦❤→❝✱ ❝❤å♥ u = (1, 0)✱ t❛ ❝â ∗ ∗ A2(−K, g(x0), gt (x0)u) = {(k1, k2, k3) ∈ R3 : k3 ≥ 4}✱ D2(f, g)(x0, u) = {(−6 + β, γ, 2, β) : β, γ ∈ R}✳ ❱ỵ✐ α > 0✱ ❝❤♦ (c∗, k∗) = (α, 0, 0, −α) ∈ Λ1(x0)✱ t❛ ❝â supk∈A2(−K,g(x0),gt (x0)u)(k∗, k) = −4α✱ ✈➔✱ ✈ỵ✐ (y0, z0) ∈ D2(f, g)(x0, u)✱ (c∗, y0) + (k∗, z0) = α(−6 + β) − αβ = −6α < −4α✳ ❇ð✐ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮✱ x0 ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ✷✸ ❱➼ ❞ö t✐➳♣ t❤❡♦ ♠✐♥❤ ❤å❛ ✈✐➺❝ →♣ ❞ö♥❣ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✐✮ ❝õ❛ ❍➺ q✉↔ ✹✳✺ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♠➔ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✮ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ö ✹✳✸✳ ❈❤♦ C = R+, K = {0}✱ x0 = (0, 0)✱ f ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✹✳✶ ✈ ➔ g : R2 → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ g(x1, x2) = x3 + x2✳ ❑❤✐ ✤â✱ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ f t (x0) = (0, 1)✳ ❱➻ ✐♥tK = ∅✱ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮ ❦❤ỉ♥❣ t❤➸ →♣ ❞ư♥❣ ✤÷đ❝✳ ❚❤û ❦✐➸♠ tr❛ ❦❤➥♥❣ ✤à♥❤ ✭✐✐✐✮✱ t❛ ❝â M = {(x1, x2) ∈ R2 : x3 + x2 = 0}✱ T (M, x0) = {(u1, u2) ∈ R2 : u1 ≤ 0, u2 = 0}✳ ❱ỵ✐ u = (−1, 0) ∈ T (M, x0)✱ T tt (M, x0, u) = R2✳ ▲➜② w = (w1, w2) ✈ỵ✐ w2 < 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ c∗ ∈ C∗ \ {0} ✈ỵ✐ (c∗, f t (x0)u) = 0✱ tù❝ ❧➔✱ c∗ > 0✱ t❛ ❝â (c∗, f t (x0)w) = c∗w < 0✳ ❇ð✐ ✭✐✐✐✮✱ x0 ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳ ▲÷✉ þ r➡♥❣ ❝→❝ ①➜♣ ①➾ ✭✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❬✶✱ ✶✻❪✮ ❧➔ ❝→❝ ❧♦↕✐ ✤↕♦ ❤➔♠ r➜t tê♥❣ q✉→t tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✭①❡♠ ❬✶✽✱ ✷✵✱ ✷✶❪✮✳ ❱➼ ❞ö s❛✉ ✤➙② ❝❤ù♥❣ tä r➡♥❣ ❍➺ q✉↔ ✹✳✺ ❝â t❤➸ →♣ ❞ư♥❣ ✤÷đ❝ tr♦♥❣ ❦❤✐ ❝→❝ t tr ỵ ✹✳✶ ❝õ❛ ❬✷✵❪ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❱➼ ❞ư ✹✳✹✳ ❈❤♦ C = R+, K = R✱ x0 = 0✱ ✈➔ f, g : R → R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ g = ✈➔ −x2/2 ♥➳✉ x ≥ 0, √ f (x) = x /2 − (2/3)x −x ♥➳✉ x < 0, ✭f ✤÷đ❝ ❧➜② tø ❱➼ ❞ö ✹✳✶ ❝õ❛ ❬✶✾❪✮✳ ❑❤✐ ✤â✱ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝ t↕✐ x0 ✈➔ f t (x0) = 0✳ ❚➟♣ ❤ñ♣ ❝→❝ ♥❤➙♥ tû ❋r✐t③ ❏♦❤♥ ❝õ❛ ✭P✮ ❧➔ Λ1(x0) = {(c∗, k∗) ∈ R2 : c∗ > 0, k∗ = 0}✳ ❱ỵ✐ u = 1✱ A2(−K, g(x0), gt (x0)u) = R✱ D2(f, g)(x0, u) = {(−1, 0)}✱ ✈➔ ✈ỵ✐ ♠å✐ (c∗, k∗) ∈ Λ1(x0)✱ t❛ ❝â (c∗, −1) + (k∗, 0) = −c∗ < supk∈A2(−K,g(x0),gt (x0)u)(k∗, k) = 0✳ ❇ð✐ ❍➺ q✉↔ ✹✳✺ ✭✐✐✮✱ x0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ❇➙② ❣✐í t❛ t❤û ũ ỵ r ợ ♠å✐ u ∈ Rn, T (M, x0, u) = T tt (M, x0, u) = R✳ ❚❛ t➼♥❤ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✈➔ ❝→❝ t➟♣ ❧✐➯♥ q ✉❛♥ ♥❤÷ s❛✉✿ Bf (x0) = {−1/2} ∪ (α, +∞) ✈ỵ✐ α > 0✱ ❝❧Bf (x0) = {−1/2} ∪ [α, +∞) ✈➔ Bf (x0)∞ = [0, +∞)✳ ❱➻ t❤➳✱ x0 t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tr♦♥❣ ✤à♥❤ ỵ ổ ọ ❈❤÷ì♥❣ ✺✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ ①➨t ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = C K õ t ổ ỗ ỵ✐ ♣❤➛♥ tr♦♥❣ ré♥❣✳ ❇ê ✤➲ s❛✉ ✤➙② ✤÷đ❝ sû ❞ö♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝➜♣ ❤❛✐✳ ❇ê ✤➲ ✺✳✶✳ ●✐↔ sû x0 ∈ M ⊆ Rn✳ ◆➳✉ xk ∈ M \ {x0} ❤ë✐ tö ✤➳♥ x0✱ t tỗ t u T (M, x0) \ {0} ởt ũ ỵ xk s ❝❤♦ ✭✐✮ (xk − x0)/tk → u✱ tr♦♥❣ ✤â tk = "xk − x0"; ✭✐✐✮ ✭❬✶✺❪✱ ❇ê ✤➲ ✸✳✹✮ ❤♦➦❝ z T 2(M, x0, u)u tỗ t s (xk −x0 −tku)/ t2 → z k ❤♦➦❝ z ∈ T tt (M, x0, u) ∩ u⊥ \ {0} rk 0+ tỗ t s tk/rk → 0+ ✈➔ (xk − x0 − tku)/ 12 tkrk → z✱ tr♦♥❣ ✤â u⊥ ❧➔ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ u Rn ỵ t t ✭P✮ ✈ỵ✐ h = 0✱ ❝❤♦ f ✈➔ g ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ∈ M ✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ✤➸ x0 ❧➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝➜♣ ❤❛✐✳ ✭✐✮ ∀u ∈ Sn✱ ∃(c∗, k∗)∗∈ C∗ × K(g(x0))∗∗ s❛♦ ❝❤♦ (c , f (x (x0)u) > 0✳ 0)u) + (k , g t t ✭✐✐✮ ∀u ∈ Sn ✈ỵ✐ u ∈ T (M, x0) ✈➔ f t (x0)u ∈ −C✱ t❛ ❝â ✭❛✮ ∀w ∈ T 2(M, x0, u)∗ ∩ ∗u⊥✱ ∀(y0, z0) ∈ d2(f, g)(x0, u) ✈ỵ✐ gt (x0)w + z0 ∈ T (−K, g(x0), gt (x0)u)✱ ∃(c , k ) ∈ Λ1(x0) ✭✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❍➺ q✉↔ ✹✳✺✮ t❤ä❛ ♠➣♥ (c∗, y0) + (k∗, z0) > (k∗, gt (x0)w + z0); ✭✺✳✶✮ ✭❜✮ ∀w ∈ T tt (M, x0, u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C∗ \ {0} ✈ỵ✐ (c∗, f t (x0)u) = t❤ä❛ ♠➣♥ (c∗, f t (x0)w) > 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❳❡♠ ỵ tr sỷ ự r tỗ t xk Mk Bn(x0, ) \ {x0} ✈➔ ck ∈ C s❛♦ ❝❤♦ f (xk ) − f (x0) + ck ∈ Bm(0, t2 )✱ ✭✺✳✷✮ k k tr♦♥❣ ✤â tk = "xk − x0"✳ ❚❛ ❝â t❤➸ ❣✐↔ sû (xk − x0)/tk → u ∈ T (M, x0) ∩ Sn✳ ❈❤✐❛ ✭✺✳✷✮ tk q ợ t ữủ f t (x0)u ∈ −C✳ ❇ð✐ ❇ê ✤➲ ✺✳✶✱ ❝❤➾ ❝➛♥ ①➨t trữớ ủ s rữớ ủ ởt ỗ t↕✐ w ∈ T 2(M, x0, u)∩u⊥ s❛♦ ❝❤♦ wk := (xk −x0 −tku)/ t2 → w✳ k ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✮✱ t❛ ❝â t (yk, zk ) := (f, g)(xk ) − (f, g)(x0) − tk (f, g ) ( x 0) u tk/2 ✈ỵ✐ (y, z) ∈ Dp(f,2 g)(x0, u, w)✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✺✱ (y, z) = (f, g)t (x0)w + (y0, z0)✳ → (y, z)✱ ✈ỵ✐ (y0, z0) ∈ d2(f, g)(x0, u)✳ ❱➻ g(xk ) ∈ −K 1✈➔ zk = (g(xk ) − g(x0) − tkgt (x0)u)/ t → z = tg (x0)w + z0✱ k s✉② r❛ r➡♥❣ gt (x0)w + z0 ∈ T 2(−K, g(x0), gt (x0)u)✱ ✈➔ ❞♦ ✤â gt (x0)u ∈ T (−K, g(x0))✳ ❇ð✐ ❣✐↔ tt tỗ t (c, k) 1(x0) tọ ♠➣♥ ✭✺✳✶✮✳ ❈ë♥❣ t❤➯♠ (c∗, f t (x0)w) ✷✺ ✈➔♦ ❤❛✐ ✈➳ ❝õ❛ ✭✺✳✶✮ ✈➔ ❞ò♥❣ ✤➥♥❣ t❤ù❝ c∗ ◦ f t (x0) + k∗ ◦ gt (x0) = 0✱ t tự tữỡ ữỡ ợ t tứ ✭✺✳✷✮ t❛ s✉② r❛ (c∗, f t (x0)w + y0) > 0✳ (f (xk ) − f (x0) −t tkf (x0)u)/ ✭✺✳✸✮ t + (ct k + tkf (x0)u)/ k t → 0✳ 2 k ❱➻ sè ❤↕♥❣ ✤➛✉ t✐➯♥ ð ✤➙② ❧➔ yk ✱ sü ❤ë✐ tö tr➯♥ ❞➝♥ ✤➳♥ y = f t (x0)w + y0 ∈ −❝❧❝♦♥❡(C + f t (x0)u)✳ ❱➻ f t (x0)u ∈ −C✱ gt (x0)u ∈ T (−K, g(x0)) ✈➔ c∗ ◦ f t (x0) + k∗ ◦ gt (x0) = 0✱ t❛ ∗s✉② r❛ (c , f (x = 0✱ ✈➔ ❞♦ ✤â c∗ ∈ ❬❝❧❝♦♥❡✭C + f (x0)u)]∗✳ ❱➻ t❤➳✱ (c∗, f (x0)w +0y)u) 0) ≤ t t t t ợ rữớ ủ ỗ t rk 0+ s tk/rk ✈➔ wk := (xk − x0 − tku)/ t2krk → w ∈ T tt (M, x0, u) ∩ u \ {0} tt tỗ t c∗ ∈ C∗ \ {0} ✈ỵ✐ (c∗, f t (x0)u) = ✈➔ (c∗, f t (x0)w) > 0✳ ❇ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ t❛ ❝â (f (xk ) − f (x0) − tkft (x0)u)/ tkrk →t f (x0)w✳ ▼➦t ❦❤→❝✱ tø ✭✺✳✷✮ t❛ s✉② r❛ (f (xk ) − f (x0) −t tkf (x0)u)/ tkrk + (ct k + tkf (x0)u)/ t k r k → 0✳ ❱➻ t❤➳✱ f t (x0)w ∈ −❝❧❝♦♥❡(C + f t (x0)u)✳ ❉♦ ✤â✱ (c∗, f t (x0)w) t Q ữ ỵ tr ỵ ✤÷đ❝ s✉② r❛ ❜ð✐ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✭❛t ✮ ∀w ∈ Rn✱ ∀(y0, z0) ∈ d2(f, g)(x0, u) ✈ỵ✐ gt (x0)w + z0 ∈ T 2(−K, g(x0), gt (x0)u)✱ ∗ ∗ ∃(c , k ) ∈ Λ1(x0) t❤ä❛ ♠➣♥ ✭✺✳✶✮✳ ✭❛tt ✮ ∀(y0, z0) ∈ d2(f, g)(x0, u)✱ ∃(c∗, k∗) ∈ Λ1(x0)✱ (c∗, y0) + (k∗, z0) > supk∈T 2(−K,g(x0),gt (x0)u)(k∗, k)✳ ✭✺✳✹✮ ✭✐✐✮ ✣✐➲✉ ❦✐➺♥ tr ỵ ữủ s r ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✭❜t ✮ ∀w ∈ u⊥ \ {0} ✈ỵ✐ g0 t (x )w ∈ T tt (−K, g(x ), gt (x )u)✱ ∃(c∗, k∗) ∈ Λ (x )✱ 0 (k∗, gt (x0)w) < 0✳ ✭❜tt ✮ ∀w ∈∗ u⊥ \ {0} ✈ỵ✐ gt (x )w ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x )) − gt (x ) ∗ u]✱ ∃(c , k ) ∈ 0 Λ1(x0)✱ (k∗, gt (x0)w) < 0✳ ❚❤ü❝ ✈➟②✱ ❜ð✐ ▼➺♥❤ ✤➲ ✸✳✹ ✭✐✐✐✮✱ ♥➳✉ w ∈ T tt (M, x0, u)✱ t❤➻ gt (x0)w ∈ T tt (−K, g(x0), gt (x0)u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0)) − gt (x0)u] õ ỵ t ỵ ✸ ❝õ❛ ❬✶✸❪✱ tr♦♥❣ ✤â f t ✈➔ gt ✤÷đ❝ ❣✐↔ sû ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ✈➔ ❝→❝ ✤✐➲✉ ữủ ũ ữ ỵ r t ✈➻ D2(f, g)(x0, u) ♥❤÷ tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ tr ỵ t ũ t ọ ỡ d2(f, g)(x0, u) ✤➸ ✤÷đ❝ ❦➳t ❧✉➟♥ ♠↕♥❤ ❤ì♥✳ ✭✐✐✐✮ ❇➙② ❣✐í t❛ ♥â✐ ✤➳♥ ❧ê ❤ê♥❣ ❣✐ú❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tr ỵ tr ỵ t ữ ỵ r u T (M, x0)✱ t❤➻ gt (x0)u ∈ T (−K, g(x0)) = K(g(x0)) t ỵ ụ ú ữợ u Sn s (f, g)t (x0)u [CìK(g(x0))] ữủ t ữ tr ỵ ỡ ỳ ỵ tữỡ ự ỵ ụ ú tỹ sỹ ỡ t t❤❛② D2 ✭t÷ì♥❣ ù♥❣✱ d2✮ ❜ð✐ d2 ✭t÷ì♥❣ ù♥❣✱ D2✮✳2◆❣❛② ợ ữ tr pK tọ ♠➣♥ t➼♥❤ ❝❤➜t s❛✉ t↕✐ g(x0)✿ T (−K, g(x0), u) = A (−K, g(x0), u) ✈ỵ✐ ♠å✐ u ∈ R ✱ ❧ê ❤ê♥❣ ♥â✐ tr➯♥ ❧➔ ♥❤ä✿ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤÷đ❝ t❤❛② ❜ð✐ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤➦t tr♦♥❣ ✤✐➲ ✉ ❦✐➺♥ ✤õ✳ ❚r♦♥❣ ✈➼ s ỵ ❝❤➢♥✱ tr♦♥❣ ❦❤✐ ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② t❤➻ ❦❤æ♥❣✳ t ữ ợ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭θ ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✹✳✶✮ −x2 + x2 ♥➳✉ x1 ≥ 0, f (x1, x2) = θ(x1) + x2 ♥➳✉ x1 < ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✹✳✷✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f ❧➔ l✲ê♥ ✤à♥❤ t↕✐ x0 ✈➔ f t (x0) = (0, 1)✳ ❑❤✐ ✤â✱ ✈ỵ✐ u = (1, 0) ∈ S2 ✈➔ (c∗, k∗) ∈ C∗ × K(g(x0))∗✱ t❛ ❝â (c∗, ft (x0)u) + (k∗,t g (x0)u) = 0✳ ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭✐✮ tr♦♥❣ ✣à♥❤ ỵ ổ tọ u = (u1, u2) ∈ S2 s❛♦ ❝❤♦ (f, g)t (x0)u ∈ −[C×❝❧ K(g(x0))]✳ ❑❤✐ ✤â✱ u = (u1, 0) ✈ỵ✐ u1 = ±1✳ ❚❛ ❝â T 2(−K, g(x0), gt (x0)u) = A2(−K, g(x0), gt (x0)u)✱ ✈➔ ❞♦ ✤â✱ ✈ỵ✐ k∗ = (0, 0, −1) ∈ N (−K, g(x0))✱ supk∈T 2(−K,g(x0),gt (x0)u)(k∗, k) = −4✳ ◆➳✉ u1 = 1✱ t❤➻ ✈ỵ✐ (y0, z0) ∈ d2(f, g)(x0, u) = {(2, 0, 2, 0)} tỗ t (c∗, k∗) = (1, 0, 0, −1) ∈ Λ1(x0) t❤ä❛ ♠➣♥ (c∗, y0) + (k∗, z0) = −2 > supk∈T 2(−K,g(x0),gt (x0)u)(k∗, k)✳ ◆➳✉ u1 = −1✱ t❤➻ ✈ỵ✐ (y0, z0) d2(f, g)(x0, u) tỗ t tk 0+ s❛♦ →+∞ ❝❤♦ y0 = limk ✈➔ z0 = (0, 2, 0)✳ ❱➻ t❤➳✱ ✈ỵ✐ (c∗, k∗) = (1, 0, 0, −1) ∈ Λ1(x0)✱ t❛ ❝â θ(−tk ) tk/2 θ (−tk ) (c∗, y0) + (k∗, z0) = limk→+∞ t k/ 22 ≥ > supk∈T 2(−K,g(x0),gt (x0)u)(k∗, k)✳ ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭❛tt ✮ t❤ä❛ ♠➣♥✱ ✈➔ ❞♦ õ ỵ ụ tọ ♠➣♥✳ ✷✼ ❈❤♦ w = (w1, w2) ∈ v⊥ \ {(0, 0)}✱ tù❝ ❧➔✱ w1 = ✈➔ w2 ƒ= 0✱ ♥➳✉ gt (x0)w = (0, 0, w2) ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0)) − gt (x0)u] = {(k1, k2, k3) ∈ R3 : k3 ≥ 0}✱ t❤➻ w2 > 0✳ ❈❤å♥ t t (c∗, k∗) = (1, 0, 0, −1) ∈ Λ1(x0)✱ t❛ ❝â (k∗, gt (x0)w) = −w2 < 0✳ ❱➻ t❤➳✱ ✤✐➲✉ ❦✐➺♥ ✭❜ ✮ t❤ä❛ ♠➣♥✱ ✈➔ ❞♦ õ ỵ ụ tọ t q ỵ x0 ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝➜♣ ❤❛✐ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✳ ❱➻ f t ❦❤æ♥❣ ê♥ ✤à♥❤ t↕✐ x0✱ ✣à♥❤ ỵ ỵ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❍ì♥ ♥ú❛✱ ✈➻ ✈➳ tr→✐ ❝õ❛ õ t ỵ ổ ữủ t ữợ ự ♠ð rë♥❣ ✤➲ t➔✐ ❚r♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ✤➛✉ t✐➯♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ❤➔♠ l✲ê♥ ổ ữợ ụ ữ tỡ st ởt sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ ✤➲ t t ữợ tr ❤❛✐ ✈➔ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❈✉è✐ ũ ũ t ữợ tr r ữợ tt t tr ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② l✲ê♥ ✤à♥❤ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ❝→❝ tố ữ ợ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❤❛② ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣✱ ợ t t ữủ ró ỡ tr t♦→♥ q✉② ❤♦↕❝❤ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ✭P✮✳ ❚r♦♥❣ ❦➳ ❤♦↕❝❤ ♥❣❤✐➯♥ ❝ù✉ t÷ì♥❣ ❧❛✐✱ ❝❤ó♥❣ tỉ✐ s➩ ♠ð rở ữợ ự t t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ ❦❤→ tê♥❣ q✉→t s❛✉ ✤➙②✿ ✭P✶✮ ♠✐♥ f (x)✱ s✳t✳ x ∈ S✱ g(x) ∈ −K, h(x) = 0✱ tr♦♥❣ ✤â f : X → Y ✱ g : X → Z✱ ✈➔ h : X → W ❧➔ ❝→❝ →♥❤ ①↕✱ X ✈➔ W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ✈➔ Z ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ S ⊂ X✱ C Y õ ỗ õ K Z t ỗ ú tổ s tt ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ✈➔ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✈➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ❝õ❛ ❜➔✐ t♦→♥ ✭P✶✮ ❜➡♥❣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû ❋r✐t③✲ r ũ t ữợ tr ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐✳ ❈❤ó♥❣ tỉ✐ ✤➦t r❛ ❝→❝ ❣✐↔ t❤✐➳t ❣✐↔♠ ♥❤➭✿ st❡❛❞✐♥❡ss ✈➔ ❦❤↔ ✈✐ ❝❤➦t ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✱ t÷ì♥❣ ù♥❣❀ ê♥ ✤à♥❤ ✈➔ l✲ê♥ ✤à♥❤ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✱ t÷ì♥❣ ù♥❣✳ ✷✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❑✳ ❆❧❧❛❧✐ ❛♥❞ ❚✳ ❆♠❛❤r♦q✱ ❙❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ♥❡❝❡s✲ s❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❖♣t✐♠✐③❛t✐♦♥ ✹✵ ✭✶✾✾✼✮ ✷✷✾✲✷✹✻✳ ❬✷❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ✐♥ ✉♥❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛✲ t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❆✮ ✶✶✸ ✭✷✵✵✽✮ ✷✽✸✲✷✾✽✳ ❬✸❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❉❡❝r❡❛s❡ ♦❢ C 1,1 ♣r♦♣❡rt② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❘❆■❘❖✲ ❖♣❡r✳ ❘❡s✳ ✹✸ ✭✷✵✵✾✮ ✸✺✾✲✸✼✷✳ ❬✹❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ l✲st❛❜❧❡ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✵ ✭✷✵✵✾✮ ✷✸✶✼✲✷✸✷✹✳ ❬✺❪ ❉✳ ❇❡❞♥❛✞r➼❦ ❛♥❞ ❑✳ P❛st♦r✱ ❖♥ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❝♦♥str❛✐♥❡❞ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✹ ✭✷✵✶✶✮ ✶✸✼✷✲✶✸✽✷✳ ❬✻❪ ❏✳❋✳ ❇♦♥♥❛♥s ❛♥❞ ❆✳ ❙❤❛♣✐r♦✱ P❡rt✉r❜❛t✐♦♥ ❆♥❛❧②s✐s ♦❢ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦ ✭✷✵✵✵✮✳ ❬✼❪ ❋✳❍✳ ❈❧❛r❦❡✱ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ◆♦♥s♠♦♦t❤ ❆♥❛❧②s✐s✱ ❲✐❧❡② ■♥t❡rs❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✸✮✳ ❬✽❪ ❘✳ ❈♦♠✐♥❡tt✐✱ ▼❡tr✐❝ r❡❣✉❧❛r✐t②✱ t❛♥❣❡♥t s❡ts ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✷✶ ✭✶✾✾✵✮ ✷✻✺✲✷✽✼✳ ❬✾❪ ❆✳▲✳ ❉♦♥❝❤❡✈✱ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ■♠♣❧✐❝✐t ❋✉♥❝t✐♦♥s ❛♥❞ ❙♦❧✉t✐♦♥ ▼❛♣♣✐♥❣s✱ ❙♣r✐♥❣❡r✱ ❉♦r❞r❡❝❤t ✭✷✵✵✾✮✳ ❬✶✵❪ ■✳ ●✐♥❝❤❡✈✱ ❆✳ ●✉❡rr❛❣❣✐♦ ❛♥❞ ▼✳ ❘♦❝❝❛✱ ❙❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢ ♦r C1,1 ❝♦♥✲ str❛✐♥❡❞ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✶✵✹ ✭✷✵✵✺✮ ✸✽✾✲✹✵✺✳ ❬✶✶❪ ■✳ ●✐♥❝❤❡✈✱ ❆✳ ●✉❡rr❛❣❣✐♦ ❛♥❞ ▼✳ ❘♦❝❝❛✱ ❋r♦♠ s❝❛❧❛r t♦ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❆♣♣❧✳ ▼❛t❤✳ ✺✶ ✭✷✵✵✻✮ ✺✲✸✻✳ ❬✶✷❪ ■✳ ●✐♥❝❤❡✈✱ ❖♥ s❝❛❧❛r ❛♥❞ ✈❡❝t♦r l✲st❛❜❧❡ ❢✉♥❝t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✹ ✭✷✵✶✶✮ ✶✽✷✲ ✶✾✹✳ ❬✶✸❪ ❈✳ ●✉t✐➨rr❡③✱ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❖♥ s❡❝♦♥❞ ♦r❞❡r ❋r✐t③ ❏♦❤♥ t②♣❡ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✶✷✸ ✭✷✵✶✵✮ ✶✾✾✲✷✷✸✳ ❬✶✹❪ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✐♥ s❡t ❝♦♥str❛✐♥❡❞ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ ▼❡t❤♦❞s ❖♣❡r✳ ❘❡s✳ ✺✽ ✭✷✵✵✸✮ ✷✾✾✲✸✶✼✳ ❬✶✺❪ ❇✳ ❏✐♠➨♥❡③ ❛♥❞ ❱✳ ◆♦✈♦✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ s❡❝♦♥❞✲♦r❞❡r t❛♥❣❡♥t s❡ts✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✹✾ ✭✷✵✵✹✮ ✶✷✸✲✶✹✹✳ ❬✶✻❪ ❆✳ ❏♦✉r❛♥✐ ❛♥❞ ▲✳ ❚❤✐❜❛✉❧t✱ ❆♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♠❡tr✐❝ r❡❣✉❧❛r✐t② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✳ ✶✽ ✭✶✾✾✷✮ ✸✾✵✲✹✵✵✳ ❬✶✼❪ ❍✳ ❑❛✇❛s❛❦✐✱ ❆♥ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ♦❢ ✐♥❢✐♥✐t❡❧② ♠❛♥② ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts ♦♥ s❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✹✶ ✭✶✾✽✽✮ ✼✸✲✾✻✳ ❬✶✽❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❋✐rst ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉ s✐♥❣ ✷✾ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✸✵ ✭✷✵✵✻✮ ✷✽✾✲✸✵✽✳ ❬✶✾❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ♠✉ ❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ✉s✐♥❣ ❍❛❞❛♠❛r❞ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✸✸ ✭✷✵✵✼✮ ✸✹✶✲✸✺✼✳ ❬✷✵❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❋✐rst ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛s ❞❡r✐✈❛t✐✈❡s ♦❢ ♠❛♣♣✐♥❣s ✐♥ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✺✽ ✭✷✵✵✽✮ ✶✹✼✲✶✻✻✳ ❬✷✶❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛✲ t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ✐♥❝❧✉ s✐♦♥ ❝♦♥str❛✐♥ts✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✹ ✭✷✵✶✶✮ ✹✸✸✽✲✹✸✺✶✳ ❬✷✷❪ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈ ❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✼ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ❬✷✸❪ ▲✳ ▲✐✉✱ P✳ ◆❡✐tt❛❛♥♠☎❛❦✐ ❛♥❞ ▼✳ ❑✞r➼✞③❡❦✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥❞♦♠✐♥❛t❡❞ s♦❧✉t✐♦♥s ♦❢ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ C1,1 ❞❛t❛✱ ❆♣♣❧✳ ▼❛t❤✳ ✹✺ ✭✷✵✵✵✮ ✸✽✶✲✸✾✼✳ ❬✷✹❪ ❨✳ ▼❛r✉②❛♠❛✱ ❙❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥❧✐♥❡❛r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜✲ ❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s t♦ ❛♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✳ ✶✺ ✭✶✾✾✵✮ ✹✻✼✲✹✽✷✳ ❬✷✺❪ ❏✳P✳ P❡♥♦t✱ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❝♦♠♣♦s✐t❡ ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✻✼ ✭✶✾✾✹✮ ✷✷✺✲✷✹✺✳ ❬✷✻❪ ❏✳P✳ P❡♥♦t✱ ❙❡❝♦♥❞ ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❝♦♥str❛✐♥ts✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✸✼ ✭✶✾✾✾✮ ✸✵✸✲✸✶✽✳ ❬✷✼❪ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡r✲ s❡② ✭✶✾✼✵✮✳ ❬✷✽❪ ❉✳ ❊✳ ❲❛r❞✱ ❈❛❧❝✉❧✉s ❢♦r ♣❛r❛❜♦❧✐❝ s❡❝♦♥❞✲♦r❞❡r ❞❡r✐✈❛t✐✈❡s✱ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳ ✶ ✭✶✾✾✸✮ ✷✶✸✲✷✹✻✳ ✸✵ ❈→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❬✶❪ P✳ ◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳ ❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥ ✈❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■✿ l✲st❛❜✐❧✐t② ❛♥❞ s❡t✲✈❛❧✉❡❞ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✹✵✸ ✭✷✵✶✸✮ ✻✾✺✲✼✵✷✳ ❬✷❪ P✳ ◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳ ❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥ ✈❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■■✿ ❖♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✹✵✸ ✭✷✵✶✸✮ ✼✵✸✲✼✶✹✳ ✸✶