Các điều kiện tối ưu cấp hai với hiện tượng envelope like cho các bài toán tối ưu vectơ không trơn trong các không gian vô hạn chiều

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 CHO CÁC BÀI TỐN TỐI ƯU VECTƠ KHƠNG TRƠN TRONG CÁC KHÔNG GIAN VÔ HẠN CHIỀU Mã số: CS – 2014 - 43 Chủ nhiệm: TS Nguyễn Đình Tuấn Tp Hồ Chí Minh - 2014 ▼Ư❈ ▲Ư❈ ❈❤÷ì♥❣ ♠ð ✤➛✉✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸ ữỡ ợ t t ự ởt sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✺ ❈❤÷ì♥❣ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✾ ❈❤÷ì♥❣ ✸✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✶✸ ❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ tố ữ t ữợ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✸ ữỡ ỵ t➔✐✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ ✈➻ ♥â ❧➔♠ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ❤♦➔♥ t❤✐➺♥ ❤ì♥ ❜➡♥❣ ♥❤ú♥❣ t❤ỉ♥❣ t✐♥ ❝➜♣ ❤❛✐ ❣✐ó♣ ➼❝❤ r➜t ♥❤✐➲✉ tr♦♥❣ ✈ ✐➺❝ ♥❤➟♥ r❛ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝ơ♥❣ ♥❤÷ ✤÷❛ r❛ ❝→❝ t❤✉➟t t♦→♥ sè ✤➸ t➼♥❤ ❝→❝ ♥❣❤✐➺♠ ♥➔②✳ ❇↔♥ ❝❤➜t ❝õ❛ t❤ỉ♥❣ t✐♥ ❝➜♣ ❤❛✐ ♥➔② ❧➔ ♥❤÷ s❛✉✳ ◆â✐ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ❦❤➥♥❣ ✤à♥❤ r➡♥❣ t↕✐ ✤✐➸♠ ❝ü❝ trà✱ ✤↕♦ t ữợ ủ t✐➯✉ ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝✱ ❦❤æ♥❣ t❤✉ë❝ ✈➲ ♣❤➛♥ tr♦♥❣ ❝õ❛ ♥â♥ ✭❤đ♣✮ ➙♠ tr♦♥❣ t➼❝❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ↔♥❤✳ t ữợ õ t tr ❝õ❛ ♥â♥ ♥â✐ tr➯♥✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝✉♥❣ ❝➜♣ t❤ỉ♥❣ t✐♥ t❤➯♠✿ õ t ữợ ▲❛❣r❛♥❣❡ ❧➔ ❦❤æ♥❣ ➙♠✳ ❚✉② ♥❤✐➯♥✱ ✈➔♦ ♥➠♠ ✶✾✽✽✱ ❑❛✇❛s❛❦✐ ❬✶✹❪ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ✤➣ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ❦❤✐ t❛ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ ❤➔♠ ▲❛❣r❛♥❣ ❡ ❝â t❤➸ t ữợ ởt ①↕ ❤ñ♣ ♥â✐ tr➯♥ ♥➡♠ tr➯♥ ♣❤➛♥ ✤➦❝ ❜✐➺t ❝õ❛ ❜✐➯♥ ❝õ❛ ♥â♥ ➙♠✳ ➷♥❣ ❣å✐ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ◆❤✐➲✉ ♥❤➔ ♥❣ ❤✐➯♥ ❝ù✉ ✈➝♥ ❦❤ỉ♥❣ ú ỵ tữủ s ❧➛♠ ✤→♥❣ t✐➳❝✳ ◆❤✐➲✉ t→❝ ❣✐↔ ❦❤→❝ ❝❤➾ ①➨t ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ❦❤æ♥❣ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ♥➔②✱ ✈➔ ✈➻ t❤➳ ❦❤ỉ♥❣ ❝â ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛✳ ✣➣ ❝â ♥❤✐➲✉ ✤â♥❣ ❣â♣ q✉❛♥ trå♥❣ ❝❤♦ ❤✐➺♥ t÷đ♥❣ t❤ó ✈à ♥➔②✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❑❛✇❛s❛❦✐ ✤÷đ❝ ♠ð rë♥❣ ✈➔ ♣❤→t tr✐➸♥ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ổ ữợ tr ✷✺❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ❦❤↔ ✈✐ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✶✵✱ ✶✶❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ✭❤ú✉ ❤↕♥ ❝❤✐➲✉✮ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ ♠ët tr♦♥❣ ❬✼❪ ✈➔ ❝❤♦ q✉② ❤♦↕❝❤ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❝❤➦t tr♦♥❣ ❬✷✵✱ ✷✶❪✳ ❈❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② r➡♥❣ ❝➛♥ ♣❤↔✐ ❣✐↔✐ t❤➼❝❤ rã r➔♥❣ ❤ì♥ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛ ✈➔ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ♥➔② ❦❤ỉ♥❣ ①↔② r❛✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❝❤ó♥❣ tỉ✐ s ró ỡ ố ợ ỳ ữợ r ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ❍ì♥ ♥ú❛✱ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ ♠ù❝ ✤ë ❦❤ỉ♥❣ trì♥ ❝➜♣ ❝❛♦ ❤ì♥ ❧✉ỉ♥ ❧✉ æ♥ ❧➔ ♠ët ♥❤✉ ❝➛✉ t❤ü❝ t➳✳ ❉♦ ✤â✱ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ❝→❝ ①➜♣ ①➾ ✤➣ ✤÷đ❝ ✤➲ ①✉➜t tr♦♥❣ ❬✶✱ ✶✸❪ ❞ị♥❣ ❧➔♠ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❞ị♥❣ ❝→❝ ①➜♣ ①➾ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✶❪ ✈ỵ✐ ❣✐↔ t❤✐➳t r➡♥❣ t➜t ❝↔ ❝→❝ ①➜♣ ①➾ ✤÷đ❝ sû ❞ư♥❣ ❧➔ ❝♦♠♣❛❝t✳ ❈→❝ ①➜♣ ①➾ ❝â t❤➸ ❦❤ỉ♥❣ ❜à ❝❤➦♥ ✤➣ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✶✺✱ ✶✼✲✶✾❪ ❝❤♦ ♥❤✐➲✉ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤→❝ ♥❤❛✉✳ ✣↕♦ ❤➔♠ s✉② rë♥❣ t❤✉ë❝ ❧♦↕✐ ♥➔② t✐➺♥ ❧đ✐ ð ❝❤ê ❧➔ ♥❣❛② ❝↔ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❧✐➯♥ tö❝ t↕✐ ♠ët ✤✐➸♠ ❝â t❤➸ ❝â ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ t↕✐ ✤✐➸♠ ♥➔②✳ ❚✉② ♥❤✐➯♥✱ ✤➸ t➟♣ tr✉♥❣ tr➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➜♣ ❤❛✐ ✈➔ ró ữợ ữủ ❣➙② r❛ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✱ ❝❤ó♥❣ tỉ✐ ❝❤õ ②➳✉ ①➨t ❝→❝ →♥❤ ①↕ ❦❤↔ ✈✐ ❝➜♣ ♠ët✳ ❈→❝ q✉❛♥ s→t tr ỗ ự ự ❝õ❛ ❝❤ó♥❣ tỉ✐ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ →♣ ❞ö♥❣ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲ ✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ợ ợ tữủ t tố ÷✉ ✈ ❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ →♥❤ ①↕ tr♦♥❣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❦❤↔ ✈✐ ❝❤➦t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮ ✈➔ ❦❤ỉ♥❣ ❝➛♥ ❦❤↔ ✈✐ ❧✐➯♥ tư❝✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❝↔✐ t❤✐➺♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù ✉ ✸ ❣➛♥ ✤➙②✳ ✷✳ ▼ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ s❛✉ ✤➙②✳ ❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ C ⊂ Y õ ỗ õ K Z t ỗ f : X Y g : X → Z✱ ✈➔ h : X → W ❧➔ t ữợ sỹ t ❝❤ó♥❣ tỉ✐ ❧➔ ✭P✮ ♠✐♥Cf (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = 0✳ ❈❤ó♥❣ tỉ✐ ❞ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ①➜♣ ①➾ ✈ỵ✐ ♠ù❝ ✤ë ❦❤ỉ♥❣ trỡ ữợ tt t tr ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲ ✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ tr→♥❤ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❧✐➯♥ tö❝✱ ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tố ữ ợ t t t♦→♥ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔♠ rã ❤ì♥ ✈➜♥ ✤➲ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛ ✈➔ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳ ❈ö t❤➸✱ ✤➲ t➔✐ t❤ü❝ ❤✐➺♥ ❝→❝ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤➙②✳ ✰ ❑❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳ ✰ ❑❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐ ✈ỵ✐ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳ ✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣ ✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ✤➣ ✤÷đ❝ t→❝ ❣✐↔ ✈➔ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ◗✉è❝ t➳✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❚♣✳ ❍❈▼ ❝æ♥❣ ❜è tr♦♥❣ ♠ët ❜➔✐ ❜→♦ tr➯♥ t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ q✉è❝ t➳ tr♦♥❣ ❤➺ t❤è♥❣ ■❙■ ❬✷✷❪✿ P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈ ❡❧♦♣❡✲❧✐❦❡ ❡❢❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✼ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❞ị♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ❣✐↔✐ t➼❝❤ ✤❛ trà ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ✹✳ ❑➳t ❝➜✉ ❝õ❛ ✤➲ t➔✐✳ ✣➲ t➔✐ ỗ ữỡ ã ữỡ ỵ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✱ ♠ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ự t ã ữỡ ợ t t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t ú ởt ã ữỡ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ã ữỡ tố ữ ã ữỡ tố ÷✉ ✤õ ❝➜♣ ❤❛✐✳ ✹ ❈❤÷ì♥❣ ✶✿ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➯♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ C Y õ ỗ õ K Z t ỗ f : X → Y ✱ g : X → Z✱ ✈➔ h : X → W ❧➔ ❝→❝ →♥❤ ①↕✳ ❈❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì s❛✉ ✤➙②✿ ✭P✮ ♠✐♥Cf (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = 0✳ ❈❤ó♥❣ tổ ũ ỵ ỡ N = {1, 2, , n, } ✈➔ R ❧➔ t➟♣ ❤ñ♣ ❝→❝ 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}❀ BX (x, r) = {y ∈ X : "x − y" < r}✱ SX = {y ∈ X : "y" = 1} ✈➔ ✤è✐ ✈ỵ✐ BX (0, 1) t❛ ✈✐➳t ✤ì♥ ❣✐↔♥ ❧➔ BX ✳ L(X, Y ) ỵ ổ ①↕ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø X ✈➔♦ Y ✈➔ B(X, X, Y ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ s♦♥❣ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø X×X ✈➔♦ Y ✱ tr♦♥❣ ✤â X ✈➔ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❱ỵ✐ Pn✱ P tr♦♥❣ L(X,♣Y )✱ t❛ ✈✐➳t Pn −→ P ❤❛② P = ♣✲lim Pn ♥➳✉ Pn ❤ë✐ tử P ỵ tữỡ tỹữủũ Mn, M ∈ B(X, X, Y )✳ ❱ỵ✐ ♥â♥ C X ỵ C = {c X : (c , c) ≥ 0, ∀c ∈ C} ❧➔ õ ố ỹ ữỡ C ợ A X ỵ rA tA A A A A A(x) ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐✱ ♣❤➛♥ tr♦♥❣✱ õ õ ỗ A ❜❛♦ ♥â♥ ❝õ❛ ♣❤➛♥ ❞à❝❤ ❝❤✉②➸♥ A + x✳ ❱ỵ✐ t > ✈➔ r ∈ N✱ o(tr ) ❧➔ ỵ ởt tở t s❛♦ ❝❤♦ o(tr )/tr → ❦❤✐ t → 0+✳ C1,1 ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ ❦❤↔ ✈✐ ❋r➨❝❤❡t s❛♦ ❝❤♦ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ ①➨t X✱ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ h : X → Y ❧➔ →♥❤ ①↕✳ ❚❛ ♥â✐ h ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ♥➳✉ tỗ t ởt U x0 > s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ x∈U✱ "h(x) − h(x0)" ≤ κ"x − x0"✳ h ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ∈ X ♥➳✉ ♥â ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ht (x0) t↕✐ x0 ✈➔ t limy→x0,yt →x0 t t "h(y ) − h(y ) − h (x0 )(y − = 0✳ y )" t "y − y " ❍✐➸♥ ♥❤✐➯♥ r➡♥❣ ♥➳✉ h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0✱ t❤➻ h ❧➔ ▲✐♣s❝❤✐t③ ❣➛♥ x0✳ ❑➳t q✉↔ s❛✉ ✤➙② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü ♥❤÷ ❇ê ✤➲ ✸ ❝õ❛ ❬✼❪✳ ▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ h ❧➔ →♥❤ ①↕ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x0 ∈ X ✈ỵ✐ ht ❧➔ ê♥ ✤à♥❤ t↕✐ x0✱ ✈➔ u, w ∈ X✳ ◆➳✉ (tn, rn) → (0+, 0+)✱ tn/rn → 0+✱ ✈➔ wn := (xn − x0 2− tnu)/ t n rn → w ✱ t❤➻ h(xn) − h(x0) − tnht (x0)u yn := t n rn /2 t → h (x0)w✳ ❚❛ ♥❤ỵ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ s❛✉ ✤➙②✳ ✺ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ x0, u ∈ X ✈➔ S ⊂ X✳ ✭❛✮ ◆â♥ ❝♦♥t✐♥❣❡♥t ✭❤❛② ❇♦✉❧✐❣❛♥❞✮ ❝õ❛ S t↕✐ x0 ❧➔ T (S, x0) = {v ∈ X | ∃tn → 0+, ∃vn → v, ∀n ∈ N, x0 + tnvn ∈ S}✳ ✭❜✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ✭♥â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❈❧❛r❦❡✱ t÷ì♥❣ ù♥❣✮ ❝õ❛ S t↕✐ x0 ❧➔ IT (S, x0) = {v ∈ X | ∀tn → 0+, ∀vn → v, ∀n ✤õ ❧ỵ♥, x0 + tnvn ∈ S} ✭ ITC (S, x0) = {v ∈ X | ∀xn →S x0, ∀tn → 0+, ∀vn → v, ∀n ✤õ ❧ỵ♥, xn + tnvn ∈ S}✮✳ ✭❝✮ ❚➟♣ ❝♦♥t✐♥❣❡♥t ✭t➟♣ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0, u) ❧➔ T 2(S, x0, u) = {w ∈ X | ∃tn → 0+, ∃wn → w, ∀n ∈ N, x0 + tnu + t2 wn ∈ S} n ✭A2(S, x0, u) = {w ∈ X | ∀tn → 0+, ∃wn → w, ∀n ∈ N, x0 + tnu + t2 wn ∈ S}✮✳ n ✭❞✮ ◆â♥ t✐➳♣ ①ó❝ ✭♥â♥ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0, u) ❧➔ T tt (S, x0, u) = {w ∈ X | ∃(tn, rn) → (0+, + ): → 0, ∃wn → w✱ r n tn ∀n ∈ N, x0 + tnu +2 tnrnwn ∈ S} ✭Att (S, x0, u) = {w ∈ X | ∀(tn, rn) → (0+, +) : → 0, ∃wn → w, r n tn ∀n ∈ N, x0 + tnu + 21 tnrnwn ∈ S}✮✳ ✭❡✮ ❚➟♣ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0, u) ❧➔ IT 2(S, x0, u) = {w ∈ X | ∀tn → 0+, ∀wn → w, ∀n ✤õ ❧ỵ♥, x0 + tnu + t2 wn ∈ S}✳ n ✭❢✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0, u) ❧➔ IT tt (S, x0, u) = {w ∈ X | ∀(tn, rn) → (0+, 0+ ) : r → 0, ∀wn → w, n tn ∀n ✤õ ❧ỵ♥, x0 + tnu 2+ tnrnwn ∈ S}✳ ❈→❝ ♥â♥ T (S, x0)✱ IT (S, x0) ✈➔ ITC (S, x0) ✈➔ ❝→❝ t➟♣ T 2(S, x0, u)✱ A2(S, x0, u) ✈ ➔ IT 2(S, x0, u) ✤÷đ❝ ❜✐➳t rã✳ ❈→❝ ♥â♥ Att (S, x0, u) ✈➔ T tt (S, x0, u) ✤÷đ❝ P❡♥♦t ❬✷✺✱ ✷✻❪ sû ❞ư♥❣✳ ❈❤ó♥❣ tỉ✐ ✤à♥❤ ♥❣❤➽❛ ♥â♥ IT tt (S, x0, u) ởt tỹ ữ ỵ r➡♥❣ ♥➳✉ x0 ƒ∈ clS✱ t❤➻ t➜t ❝↔ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ð tr➯♥ ❧➔ ré♥❣✳ ❱➻ t❤➳✱ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ①➨t ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝❤➾ t↕✐ ♥❤ú♥❣ ✤✐➸♠ t❤✉ë❝ ❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ ✤❛♥❣ ①➨t✳ ❈❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ð tr➯♥ tr♦♥❣ ❜❛ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ S ⊂ X ✈➔ x0, u ∈ X✳ ❑❤✐ ✤â✱ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤÷đ❝ ❜✐➳t rã ✭✐✮ IT 2(S, x0, u) ⊂ A2(S, x0, u) ⊂ T 2(S, x0, u) ⊂ clcone[cone(S − x0) − u]❀ ✭✐✐✮ IT 2(S, x0, u) = IT 2(intS, x0, u) ✈➔ ♥➳✉ u ∈ bd[cone(S −x0 )]✱ t❤➻ ƒ∈ IT (S, x0, u)❀ ✭✐✐✐✮ ♥➳✉ u ƒ∈ T (S, x0)✱ t❤➻ T 2(S, x0, u) = ∅✳ t nr n t n rn → ✈➔ ❞♦ ✤â f t (x0)w ∈ −❝❧❝♦♥❡(C + f t (x0)u)✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✭✐✐✮ ợ M = ã "( r2tn )Mn" a > 0✳ ❑❤✐ ✤â✱ "Mn" → ∞ ✈➔ tn"Mn" → 0✳ ❉♦ ✤â✱ Mn/"Mn" ♣ n −→ M ∈ ♣✲Bf (x0)∞ \ {0}✱ ✈➔ t❛ ❝â ✷✽ a(f (xn) − f (x0) − tnf t (x0)u) tn2"Mn" → f t (x0)w + aM (u, u)✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥✱ ✭✷✶✮ ❞➝♥ ✤➳♥ ✤✐➲✉ ♠➙✉ t❤✉➝♥ f t (x0)w + aM (u, u) ∈ −❝❧❝♦♥❡(C + f t (x0)u)✳ Mn • "( 2tn )Mn" → ∞✳ ❑❤✐ ✤â✱ "Mn" → −→ M ∈ ♣✲ f B ∞ \ {0}✱ ✈➔ t❛ ✤÷đ❝ r ∞✱ n "Mn (x ) ♣ " t f (xn) − f (x0) − tnf (x0)u tn2"Mn" M (u, u) ữỡ tỹ ữ trữợ t ✤✐ ✤➳♥ ✤✐➲✉ ❦❤ỉ♥❣ t❤➸ ✤÷đ❝ M (u, u) ∈ −❝❧❝♦♥❡(C + f t (x0)u)✳ Q ◆❤➟♥ ①➨t ✹✳✷✳ ✭✐✮ tr ỵ ró r ữủ s✉② r❛ ❜ð✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭❛t ✮ ∀(M, N ) ∈ ♣✲❝❧B(f,g)(x0)✱ ∃(c∗, k∗) ∈ Λ1(x0)✱ ∈T ∗ t (c∗, M (u, u)) + (k∗, N (u, u)) (−K,g(x0),g (x0)u) (k , k)✱ > supk t ✈➔ ∀(M, N ) ∈ ♣✲B(f,g)(x0)∞ \ {0}✱ ∃c∗ ∈ C ∗ \ {0}✿ (c∗, f (x0)u) = 0✱ (c∗, M (u, u)) > 0✳ ✭✐✐✮ ●✐↔ sû r➡♥❣ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x0 ✈ỵ✐ gt ❧➔ ê♥ t x0 õ ỵ ụ ú ♥➳✉ t❛ t❤❛② w ∈ T tt (S, x0, u)∩u⊥ \{0} tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ✭❜✮ ❜ð✐ w ∈ u⊥ \{0} ✈ỵ✐ gt (x0)w ∈ T tt (−K, g(x0), gt (x0)u) ✭❤❛② gt (x0)w ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K−g(x0)) −gt (x0)u]✮✳ ❚❤➟t ✈➟②✱ ♥➳✉ w ∈ T tt (S, x0, u)✱ t❤➻ ❜ð✐ ▼➺♥❤ ✤➲ ✶✳✶✱ gt (x0)w ∈ T tt (−K, g(x0), gt (x0)u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0)) − gt (x0)u]✳ ❍➺ q✉↔ s❛✉ ✤➙② ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø ✣à♥❤ ỵ ợ (f, g) rt ❤❛✐ t↕✐ x0✳ ❍➺ q✉↔ ✹✳✸✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ f ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ∈ S✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ❝❤♦ x0 ∈ ▲❋❊(2, f, S) ợ u SX tỗ t (c∗, k∗∗) ∈ C ∗ × K(g(x0))∗ s❛♦ ❝❤♦ (c , f (x )u) + (k , gt (x0)u) > 0✳ t ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0) ✈ỵ✐ f t (x0)u ∈ −C✱ t❛ ❝â ✭❛✮ ∀w ∈ T 2(S, x0, u)∩u⊥✿ gt (x0)w+gtt (x0)(u, u) ∈ T 2(−K, g(x0), gt (x0)u)✱ ∃(c∗, k∗) ∈ Λ1(x0)✱ t (c∗, f (x0)w tt (x0)(u, u)) > 0❀ +f ✭❜✮ ∀w ∈ T tt (S, x0, u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C∗ \ {0}✿ (c∗, f t (x0)u) = 0✱ t (c∗, f (x0)w) > q rở ỵ ❝õ❛ ❬✷✺❪✱ tr♦♥❣ ✤â Y = R✱ ✈➔ ❍➺ q✉↔ ✼ ❝õ❛ ❬✼❪✱ tr♦♥❣ ✤â Y ✈➔ Z ❧➔ ❤ú✉ q ữợ ữủ s r tự tứ ỵ ❞ị♥❣ ❍❡ss✐❛♥ s✉② rë♥❣ ❈❧❛r❦❡ ✈➔ tü❛ ❍❡ss✐❛♥ ❏❡②❛❦✉♠❛r✲▲✉❝✱ t÷ì♥❣ ù♥❣✳ ❍➺ q✉↔ ✹✳✹✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X, Y, Z ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ (f, g) t❤✉ë❝ ❧ỵ♣ C1,1 t↕✐ x0 ∈ S✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ ✤õ ❝❤♦ x0 ∈ ✷✾ ▲❋❊(2, f, S)✳ ✭✐✮ ợ u SX tỗ t (c, k ) ∗∈ C ∗ × K(g(x0))∗ s❛♦ ❝❤♦ ∗ (c , ft (x0)u) + (k t , g (x0)u) > 0✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0) ✈ỵ✐ f t (x0)u ∈ −C✱ t❛ ❝â ✭❛✮ ∀w ∈ T 2(S, x0, u)∩u⊥✱ ∀(M, NC ) ∈ ∂2 (f, g)(x0)✿ gt (x0)w+N (u, u) ∈ T 2(−K, g(x0), gt (x0)u)✱ ∃(c∗, k∗) ∈ t ∗ (c∗, M (u, u)) + (k∗, N (u, u)) > (k , g (x0)w + N (u, u)); Λ1(x0)✱ ✭❜✮ ∀w ∈ T tt (S, x0, u) ∩ u⊥ \ {0}✱ ∃c∗ ∈ C∗ \ {0}✿ (c∗, f t (x0)u) = 0✱ (c∗, f t (x0)w) > 0✳ ❍➺ q✉↔ ✹✳✹ ✭✐✐✮ ♠ð rë♥❣ ❍➺ q✉↔ ✽ ❝õ❛ ❬✼❪✳ ❍➺ q✉↔ ✹✳✺✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ h = 0✱ ❣✐↔ sû r➡♥❣ X, Y, Z ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲ ✉ ✈➔ (f, g) t❤✉ë❝ ❧ỵ♣ C t↕✐ x0 ∈ S✳ ●✐↔ sû ❤ì♥ ♥ú❛ f ✈➔ g ❝â ❝→❝ →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ∂ 2f (.) ✈➔ ∂ g(.)✱ t÷ì♥❣ ù♥❣✱ ❧➔ ♥ú❛ ❧✐➯♥ tư❝ tr➯♥ t↕✐ x0✳ ❑❤✐ ✤â✱ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ ✤õ ❝❤♦ x0 ∈ ▲❋❊(2, f, S)✳ ✭✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ✱∗∃(c∗, k∗) ∈ C∗ ×∗ K(g(x0))∗✱ (c , f (x )u) + (k , gt (x0)u) > 0✳ t ✭✐✐✮ ❱ỵ✐ ♠å✐ u ∈ SX ∩ T (S, x0) ✈ỵ✐ f t (x0)u ∈ −C✱ t❛ ❝â ✭❛✮ ∀w ∈ T 2(S, x0, u)∩u⊥✱ ∀(M, N ) ∈ ❝❧❝♦∂2(f, g)(x0)✿ gt (x0)w+N (u, u) ∈ T 2(−K, g(x0), gt (x0)u)✱ ∃(c∗, k∗) ∈ Λ1(x0)✱ t (c∗, M (u, u)) + (k∗, N (u, u)) > (k∗, g (x0)w + N (u, u)), ✈➔ ∀(M, N ) ∈ ❝♦ ∂2(f, g) \ {0}✿ N (u, u) ∈ T tt (−K, g(x ), gt (x )u)✱ ∃c∗ ∈ C∗ \ {0}✿ (x0) ∞ u)) > 0❀ 0 (c , f (x0)u) = 0✱ (c∗, M (u, ∗ t ✭❜✮ ∀w ∈ T tt (S, x0, u) ∩u⊥ \ {0}✱ ∀M ∈ ❝♦∂2f (x0) t ✈➔ ∀M ∈ ❝♦∂2f (x0) ∞ ∞ ✱ ∃c∗ ∈ C∗ \ {0}✿ (c∗, f t (x0)u) = 0✱ (c∗, f (x0)w + M (u, u)) > \ {0}✱ ∃c∗ ∈ C∗ \ {0}✿ (c∗, f t (x0)u) = 0✱ (c∗, M (u, u)) > r s ỵ ✹✳✶ t➻♠ r❛ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥✱ tr♦♥❣ ❦❤✐ ✤â ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② t❤➻ ❦❤ỉ♥❣✳ ❱➼ ❞ư ✹✳✶✳ ❈❤♦ C = R+✱ K = {(k1, k2, k3) ∈ R3|k2k31 ≥ 2k2, k2 ≤ 0, k3 ≤ 0}✱ (x0, y0) = (0, 0)✱ ✈➔ f : R2 → R ✈➔ g : R2 → R3 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✐❢ x ≥ 0, y ≥ 0,  +y f (x, y) =  x−x2 + y ✐❢ x ≥ 0, y < 0,  3θ(x) + y ✐❢ x < 0, g(x, y) = (x, x2 + 1, y)✱ tr♦♥❣ ✤â θ ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ❱➼ ❞ö ✸✳✷✳ ❑❤✐ ✤â✱ t↕✐ (0, 0)✱ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥❤÷♥❣ ❦❤ỉ♥❣ ❦❤↔ ✈✐ ❝❤➦t✱ ✈➔ t❛ ❝â ✸✵   f t (0, 0) = (0, 1)✱ gt (0, 0) ❂ 0 0✱ g(0, 0) = (0, 1, 0) ∈ −K✱ T (−K, g(0, 0)) = {(k1, k2, k3) ∈ R3| k3 ≥ 0}, N (−K, g(0, 0)) = {λ(0, 0, −1)| λ ∈ R}, Λ1(0, 0) = {(c∗, k∗) ∈ R × R3| c∗ = α > 0, k∗ = α(0, 0, −1)} ❚❛ ❝â t❤➸ ❧➜② ❝→❝ ①➜♣ ①➾ ❝➜♣ ❤❛✐ Bf (0, 0) = {Mβ| β ∈ {−1} ∪ (1, ∞)}✱ B(f,g)(0, 0) = {(Mβ, N )| β ∈ {−1} ∪ ∞)} 3✱ (1, tr♦♥❣ ✤â Mβ ❂ β ✈➔ N : R2 × R → R ❧➔ ♠❛ tr➟♥ × × 0 0 0 N ❂ 0 0 0✱ tù❝ ❧➔✱ N (u, v) = (0, u1v1, 0)✱ ✈ỵ✐ u, v ∈ R2✳ ❱➻ t❤➳✱ ❝❧Bf (0, 0) = {Mβ| β ∈ {−1} ∪ [1, ∞)}✱ ❝❧B(f,g)(0, 0) = {(Mβ, N )| β ∈ {−1} ∪ [1, ∞)}✱ Bf (0, 0)∞ = {Mβ| β ≥ 0}✱ B(f,g)(0, 0)∞ = {(Mβ, 02×2×3)| β ≥ 0}✳ ∗ ∗ ∗ ∗ ❈❤å♥ u = (1, 0) ∈ S2✳ ∗❚❛ ❝â✱ ✈ỵ✐ ♠å✐ ∗ (c , k ) ∈ C × K(g(0, 0)) ✱ (c , f (0, 0)u) + (k , g (0, 0)u) = ✳ t t ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ✭✐✮ tr♦♥❣ ỵ ổ tọ u = (u1, u2) ∈ S2 s❛♦ ❝❤♦ (f, g)t (0, 0)u ∈ −[C×❝❧K(g(0, 0))]✳ ❑❤✐ ✤â✱ u = (u1, 0) ✈ỵ✐ u1 = ±1✳ ❚❛ ❝â T 2(−K, g(0, 0), gt (0, 0)u) = A2(−K, g(0, 0), gt (0, 0)u)✱ ✈➔ ❞♦ ✤â✱ ✈ỵ✐ k∗ = (0, 0, −1) ∈ N (−K, g(0, 0))✱ supk∈T 2(−K,g(0,0),gt (0,0)u)(k∗, k) = −4 ✭q✉❛♥ s→t r➡♥❣ tữủ r ợ (M, N ) B(f,g)(0, 0) tỗ t (c, k) = (1, 0, 0, −1) ∈ Λ1(0, 0) t❤ä❛ (c∗, 2Mβ (u, u)) + (k∗, 2N (u, u)) = 2β > supk∈T 2(−K,g(0,0),gt (0,0)u)(k∗, k) t ✈➔✱ ✈ỵ✐ ♠å✐ (Mβ, N ) B(f,g)(0, 0) \ {0} tỗ t c = ∈ C∗ \ {0} ✈ỵ✐ (c∗, f (0, 0)u) = t❤ä❛ (c∗, Mβ (u, u)) = β > 0✳ ❱➻ t❤➳✱ ✭❛t ✮ ❝õ❛ ◆❤➟♥ ①➨t ✹✳✷ ✈➔ õ tr ỵ tọ ỡ ♥ú❛✱ ❝❤♦ w = (w1, w2) ∈ v⊥ \ {(0, 0)}✱ tù❝ ❧➔✱ w1 = ✈➔ w2 ƒ= 0✱ ♥➳✉ gt (0, 0)w = (0, 0, w2) ∈ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(0, 0)) − gt (0, 0)u] = {(k1, k2, k3) ∈ R3| k3 ≥ 0}✱ t❤➻ w2 > 0✳ ❱➻ t❤➳✱ t ✈ỵ✐ ♠å✐ Mβ ∈ Bf (0, 0)∞✱ tỗ t c = C \ {0} ợ (c∗, f (0, 0)u) = t❤ä❛ (c∗, f t (0, 0)w + Mβ (u, u)) = w2 + β > 0✱ ✈➔✱ ✈ỵ✐ ♠å✐ Mβ ∗∈ Bf (0, 0)∞ \ {0} tỗ t c = C \ {0} ✈ỵ✐ (c∗, f (0, t 0)u) = t❤ä❛ (c , Mβ (u, u)) = β > 0✳ ❱➻ t❤➳✱ ❜ð✐ ◆❤➟♥ ①➨t ✹✳✷ ✭✐✐✮✱ ✤✐➲✉ ❦✐➺♥ ✭✐✐✮ ỵ tọ q (0, 0) ∈ ▲❋❊(2, f, S)✳ ❱➻ f ƒ∈ C1 t↕✐ (0, 0)✱ ❝→❝ ❍➺ q✉↔ ✼✱ ✽ ❝õ❛ ❬✼❪✱ ✣à♥❤ ỵ q ✹✳✺ ð tr➯♥ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ❍ì♥ ♥ú❛✱ ✈➻ d2(f, g)((0, 0), u) = ỵ ❬✼❪ ❝ơ♥❣ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝✳ ✸✶ ❑➳t ❧✉➟♥ ✈➔ ữợ ự rở t r t ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ✤➛✉ t✐➯♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ❦❤↔♦ s→t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❈✉è✐ ❝ị♥❣✱ ❞ò♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ♥➔② ữợ tt t tr tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tố ữ ợ ữỡ ữỡ ợ t ❝❤➜t ❡♥✈❡❧♦♣❡✲❧✐❦❡ ✤÷đ❝ ❧➔♠ rã ❤ì♥✱ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ ✭P✮✳ ❚r♦♥❣ ❦➳ ❤♦↕❝❤ ♥❣❤✐➯♥ ❝ù✉ t÷ì♥❣ ❧❛✐✱ ❝❤ó♥❣ tổ s rở ữợ ự t ❜➡♥❣ ❝→❝❤ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❜❛♦ ❤➔♠ t❤ù❝ ❦❤→ tê♥❣ q✉→t s❛✉ ✤➙②✿ ✭P✶✮ ♠✐♥Cf (x)✱ s❛♦ ❝❤♦ x ∈ S✱ ∈ F (x)✱ tr♦♥❣ ✤â f : X → Y ❧➔ →♥❤ ①↕ ✤ì♥ trà ✈➔ F : X → 2Z ❧➔ →♥❤ ①↕ ✤❛ trà✱ X ✈ ➔ Z ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ S ⊂ X✱ ✈➔ C ⊂ Y ❧➔ ♥â♥ ỗ õ ú tổ s tt tè✐ ÷✉ ❝➛♥ ✈➔ ✤õ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✈➔ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ❝õ❛ ❜➔✐ t♦→♥ ✭P✶✮ ❜➡♥❣ ❝→❝ q✉② t➢❝ ♥❤➙♥ tû ❋r✐t③✲ ❏♦❤♥✲▲❛❣r❛♥❣❡✳ ❈❤ó♥❣ tỉ✐ ❞ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ f t ữợ tr ❝❤♦ F ✱ ✈➔ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t ú ởt ữợ t❤✐➳t ✤÷đ❝ ❣✐↔♠ ♥❤➭✳ ✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆❧❧❛❧✐✱ ❑✳✱ ❆♠❛❤r♦q✱ ❚✳✿ ❙❡❝♦♥❞✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ♣r✐♠❛❧ ❛♥❞ ❞✉❛❧ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ❖♣t✐♠✐③❛t✐♦♥ ✹✵ ✭✶✾✾✼✮ ✷✷✾✲✷✹✻✳ ❬✷❪ ❇❡❞♥❛✞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✐♠✳ ✷✶ ✭✶✾✾✵✮ ✷✻✺✲✷✽✼✳ ❬✻❪ ❉♦♥t❝❤❡✈✱ ❆✳ ▲✳✱ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳ ❚✳✿ ❘❡❣✉❧❛r✐t② ❛♥❞ ❝♦♥❞✐t✐♦♥✐♥❣ ♦❢ s♦❧✉ t✐♦♥ ♠❛♣♣✐♥❣s ✐♥ ✈❛r✐❛t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❙❡t✲✈❛❧✉❡❞ ❆♥❛❧✳ ✶✷ ✭✷✵✵✹✮ ✼✾✲✶✵✾✳ ❬✼❪ ●✉t✐➨rr❡③✱ ❈✳✱ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❖♥ s❡❝♦♥❞ ♦r❞❡r ❋r✐t③ ❏♦❤♥ t②♣❡ ♦♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✭❙❡r✳ ❇✮ ✶✷✸ ✭✷✵✶✵✮ ✶✾✾✲✷✷✸✳ ❬✽❪ ❍✐r✐❛rt✲❯rr✉t②✱ ❏✳ ❇✳✱ ❙tr♦❞✐♦t✱ ❏✳ ❏✳✱ ◆❣✉②❡♥✱ ❱✳ ❍✳✿ ●❡♥❡r❛❧✐③❡❞ ❍❡ss✐❛♥ ♠❛tr✐① ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♣r♦❜❧❡♠s ✇✐t❤ C1,1 ❞❛t❛✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✶✶ ✭✶✾✽✹✮ ✹✸✲✺✻✳ ❬✾❪ ❏❡②❛❦✉♠❛r✱ ❱✳✱ ▲✉❝✱ ❉✳ ❚✳✿ ◆♦♥s♠♦♦t❤ ❱❡❝t♦r ❋✉♥❝t✐♦♥s ❛♥❞ ❈♦♥t✐♥✉♦✉s ❖♣t✐♠✐③❛✲ t✐♦♥✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ✭✷✵✵✽✮✳ ❬✶✵❪ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✐♥ s❡t ❝♦♥str❛✐♥❡❞ ❞✐❢❢❡r✲ ❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ ▼❡t❤✳ ❖♣❡r✳ ❘❡s✳ ✺✽ ✭✷✵✵✸✮ ✷✾✾✲✸✶✼✳ ❬✶✶❪ ❏✐♠➨♥❡③✱ ❇✳✱ ◆♦✈♦✱ ❱✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ❞✐❢❢❡r❡♥t✐❛❜❧❡ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✈✐❛ s❡❝♦♥❞✲♦r❞❡r t❛♥❣❡♥t s❡ts✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✹✾ ✭✷✵✵✹✮ ✶✷✸✲✶✹✹✳ ❬✶✷❪ ❏♦✉r❛♥✐✱ ❆✳✿ ▼❡tr✐❝ r❡❣✉❧❛r✐t② ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ✐♥❝❧✉s✐♦♥ ❝♦♥str❛✐♥ts✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✶ ✭✶✾✾✹✮ ✾✼✲✶✷✵✳ ❬✶✸❪ ❏♦✉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✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❢ ♦r ♥♦♥✲ s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ❣❡♥❡r❛❧ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳ ✶✻ ✭✷✵✵✾✮ ✶✻✾✲✶✽✻✳ ❬✶✾❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❈♦rr✐❣❡♥❞✉♠ t♦ ✏❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ ❛♣♣r♦①✐♠❛✲ t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✉♥❞❡r ❣❡♥❡r❛❧ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✧✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳ ✶✽ ✭✷✵✶✶✮ ✽✾✼✲✾✵✶✳ ❬✷✵❪ ❑❤❛♥❤✱ 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✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✹✵✸ ✭✷✵✶✸✮ ✻✾✺✲✼✵✷✳ ❬✷✶❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ t❤❡ ❡♥✈ ❡❧♦♣❡✲ ❧✐❦❡ ❡❢❢❡❝t ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣✱ ■■✿ ❖♣t✐♠❛❧✐t② ❝♦♥✲ ❞✐t✐♦♥s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✹✵✸ ✭✷✵✶✸✮ ✼✵✸✲✼✶✹✳ ❬✷✷❪ ❑❤❛♥❤✱ P✳ ◗✳✱ ❚✉❛♥✱ ◆✳ ❉✳✿ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈ ❡❧♦♣❡✲❧✐❦❡ ❡❢✲ ❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✼✼ ✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳ ❬✷✸❪ ▼❛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❡♥♦t✱ ❏✳ P✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❝♦♠♣♦s✐t❡ ♦♣t✐♠✐③❛t✐♦♥✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✻✼ ✭✶✾✾✹✮ ✷✷✺✲✷✹✺✳ ❬✷✺❪ P❡♥♦t✱ ❏✳ P✳✿ ❙❡❝♦♥❞ ♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❝♦♥str❛✐♥ts✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✸✼ ✭✶✾✾✽✮ ✸✵✸✲✸✶✽✳ ❬✷✻❪ P❡♥♦t✱ ❏✳ P✳✿ ❘❡❝❡♥t ❛❞✈❛♥❝❡s ♦♥ s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✱ ✐♥ ❖♣t✐♠✐③❛✲ t✐♦♥✱ ❱✳ ❍✳ ◆❣✉②❡♥✱ ❏✳ ❏✳ ❙tr♦❞✐♦t✱ P✳ ❚♦ss✐♥❣s ❡❞s✳✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✭✷✵✵✵✮ ✸✺✼✲✸✽✵✳ ❬✷✼❪ ❘♦❝❦❛❢❡❧❧❛r✱ ❘✳ ❚✳✿ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡② ✭✶✾✼✵✮✳ ❬✷✽❪ ❚❛❛✱ ❆✳✿ ❙❡❝♦♥❞✲♦r❞❡r ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜✲ ❧❡♠s ✇✐t❤ ✐♥❝❧✉s✐♦♥ ❝♦♥str❛✐♥ts✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳ ✺✵ ✭✷✵✶✶✮ ✷✼✶✲✷✾✶✳ ❬✷✾❪ ❲❛r❞✱ ❉✳ ❊✳✿ ❈❛❧❝✉❧✉s ❢♦r ♣❛r❛❜♦❧✐❝ s❡❝♦♥❞✲♦r❞❡r ❞❡r✐✈❛t✐✈❡s✱ ❙❡t ❱❛❧✉❡❞ ❆♥❛❧✳ ✶ ✭✶✾✾✸✮ ✷✶✸✲✷✹✻✳ ✸✹ t