Về Bài Toán Tối Ưu Trong Học Độ Tương Tự

ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - TRẦN VĂN PHƢỢNG VỀ BÀI TOÁN TỐI ƢU TRONG HỌC ĐỘ TƢƠNG TỰ LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - TRẦN VĂN PHƢỢNG VỀ BÀI TOÁN TỐI ƢU TRONG HỌC ĐỘ TƢƠNG TỰ Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Nguyễn Thanh Sơn THÁI NGUYÊN - 2019 ✐✐✐ ▼ö❝ ❧ö❝ ỵ ữỡ t♦→♥ tè✐ ÷✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✻ ✶✳✶ ❙ì ❧÷đ❝ ✈➲ ❜➔✐ t♦→♥ tè✐ ÷✉ ✻ ✶✳✶✳✶ ❇➔✐ t♦→♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✷ ❑❤→✐ q✉→t ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â r➔♥❣ ❜✉ë❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✸ ❚è✐ ÷✉ ❤➔♠ ♠ư❝ t✐➯✉ ❜➟❝ ❤❛✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ tè✐ ÷✉ ✶✳✷✳✶ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❣✐↔♠ s➙✉ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ❝❤➢♥ ❧♦❣❛r✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✹ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ❈❤÷ì♥❣ ✷ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✷✳✶ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✶ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✷ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✹ ❑❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡ ❝â trå♥❣ sè ✳ ✳ ✳ ✳ ✷✽ ✐✈ ✷✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ❝❤♦ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✸ ❱➼ ❞ö sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ỵ H ổ rt t❤ü❝ ∇f ❣r❛❞✐❡♥t ❝õ❛ ❤➔♠ sè✱ ❣r❛❞ ∇2 f ❍❡ss✐❛♥ ❝õ❛ ❤➔♠ sè A f f ✈➔ ❧➔ ♠❛ tr➟♥ ❝ï ❝❤✉➞♥ ❊✉❝❧✐❞ ❝õ❛ ♠❛ tr➟♥ A λ(A) ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A A≥0 ♠❛ tr➟♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ A>0 ♠❛ tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣ x∗ ✤✐➸♠ ❝ü❝ t✐➸✉ ❤❛② ❝ü❝ t✐➸✉ f (x∗ ) ❣✐→ trà ❝ü❝ t nìn ợ ữợ tự tữ ợ ✭❆rt✐❢✐❝❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ✲ ❚r➼ t✉➺ ♥❤➙♥ t↕♦✮ ✈➔ ■♦❚ ✭■♥t❡r♥❡t ♦❢ ❚❤✐♥❣s ✲ ■♥t❡r♥❡t ✈↕♥ ✈➟t✮ ✤❡♠ ✤➳♥ ♥❤ú♥❣ ✤ët ♣❤→ ❜➜t ♥❣í ✈➲ ❝ỉ♥❣ ♥❣❤➺✳ ❱❛✐ trá ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ♠ët q✉è❝ ❣✐❛✱ ♠ët ✈ò♥❣ ❧➣♥❤ t❤ê ❧ỵ♥ ✤➳♥ ♠ù❝ ✤÷đ❝ ♥❤➟♥ ✤à♥❤✱ r➡♥❣ ❛✐ ❞➝♥ ✤➛✉ ✈➲ ❝æ♥❣ ♥❣❤➺ ♥➔② s➩ ❝❤✐➳♥ t❤➢♥❣ tr♦♥❣ ❝✉ë❝ ❝↕♥❤ tr❛♥❤ ✈➲ ❝æ♥❣ ♥❣❤➺✱ ❦✐♥❤ t➳✳✳✳ ❚r♦♥❣ t❤ü❝ t➳✱ ❚r➼ t✉➺ ♥❤➙♥ t↕♦ ♠➔ ❝ư t❤➸ ❤ì♥ ♥ú❛ ❧➔ ❍å❝ ♠→② ✭▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✮ ✤➣ ❧➔ ♠ët ♥❣➔♥❤ ❝♦♥ ♣❤→t tr✐➸♥ tø ❧➙✉ ❝õ❛ ❦❤♦❛ ❤å❝ ♠→② t➼♥❤✳ ▲✉➟♥ ✈➠♥ ♥➔② s➩ ①➨t ♠ët ❜➔✐ t♦→♥ ♥❤ä tr♦♥❣ ❧➽♥❤ ✈ü❝ rë♥❣ ❧ỵ♥ ữợ õ t õ t÷ì♥❣ tü✳ ❚r♦♥❣ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡✱ ❤❛② tê♥❣ q✉→t ❤ì♥ ❧➔ tr♦♥❣ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ ❦❤→✐ ♥✐➺♠ ❦❤♦↔♥❣ ❝→❝❤ ✤÷đ❝ ❞ò♥❣ ✤➸ ✤♦ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ✤è✐ t÷đ♥❣✳ ✣è✐ t÷đ♥❣ ❧➔ trò♥❣ ♥❤❛✉ ♥➳✉ ❦❤♦↔♥❣ ❝→❝❤ ❜➡♥❣ ✵✱ ð ❣➛♥ ♥❤❛✉ ♥➳✉ ❦❤♦↔♥❣ ❝→❝❤ ♥❤ä ✈➔ ①❛ ♥❤❛✉ ♥➳✉ ❦❤♦↔♥❣ ❝→❝❤ ❧ỵ♥✳ ❇➙② ❣✐í✱ t❛ ①➨t ♠ët t➟♣ ❤đ♣ tê♥❣ q✉→t ❤ì♥ ♥❤÷ t➟♣ ❝→❝ ❤➻♥❤ ❝❤ư♣ ♠➦t ♥❣÷í✐✱ ❤❛② t➟♣ ❝→❝ ✈➠♥ ❜↔♥✳ ●✐↔ sû t❛ ❝â t❤➸ ❜✐➳♥ ✤ê✐ ❝→❝ ✤è✐ t÷đ♥❣ ✤â t❤➔♥❤ ❝→❝ ✤è✐ t÷đ♥❣ t♦→♥ ❤å❝ ♥❤÷ ✈➨❝ tì ❤♦➦❝ ♠❛ tr➟♥✳ ❚❛ ❝➛♥ ♣❤↔✐ ①➙② ❞ü♥❣ ♠ët ♣❤➨♣ ✤♦ ✤➸ ❝â t❤➸ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ❤❛✐ ♥❤â♠ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ♥❤❛✉ ✈➔ ❦❤→❝ ♥❤❛✉✳ ❱➲ ♠➦t ✤à♥❤ ❧÷đ♥❣✱ ❤❛✐ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ♥❤❛✉ ♥➳✉ ❝â ❦❤♦↔♥❣ ❝→❝❤ ✭t❤❡♦ ♣❤➨♣ ✤♦ ✈ø❛ ✤÷đ❝ ①➙② ❞ü♥❣✮ ♥❤ä ✈➔ ❤❛✐ ✤è✐ t÷đ♥❣ ❦❤→❝ ♥❤❛✉ s➩ ❝â ❦❤♦↔♥❣ ❝→❝❤ ❧ỵ♥✳ ❈➙✉ ❤ä✐ t✐➳♣ t❤❡♦ ❧➔ ①➙② ❞ü♥❣ ♣❤➨♣ ✤♦ ✤â ♥❤÷ t❤➳ ♥➔♦❄ Þ t÷ð♥❣ tü ✸ ♥❤✐➯♥ ❧➔ ❦❤→✐ q✉→t ❤â❛ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡✳ ❚❛ ❝â ✈ỵ✐ x−y tr♦♥❣ ✤â E I = x−y (x − y)T (x − y) = = x−y A A = t❤➻ (x − y)T I(x − y), I ❜➡♥❣ ♠ët ♠❛ tr➟♥ (x − y)T A(x − y) ✭✵✳✵✳✶✮ ❧➔ ♠ët ♠❛ tr➟♥ ✤ì♥ ✈à✳ ❇➙② ❣✐í✱ t❛ t❤❛② ✤è✐ ①ù♥❣ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ x, y Rn ữ ỵ r ❦❤✐ ✤â ✭✵✳✵✳✶✮ ❝❤➾ ❧➔ ♠ët ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤✱ tù❝ ❧➔ ❤❛✐ ✤✐➸♠ 0✳ ◆❤÷ ✈➟②✱ ✈✐➺❝ ①➙② ❞ü♥❣ ❦❤♦↔♥❣ ❦❤→❝ ♥❤❛✉ ❝â t❤➸ ❝â ❦❤♦↔♥❣ ❝→❝❤ ❜➡♥❣ ❝→❝❤ ✤÷đ❝ q✉② ✈➲ ✈✐➺❝ t➻♠ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥û❛ ữỡ ổ õ ủ ỵ t ✤➙② ❧➔ ♠ët ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❝â ❧í✐ ❣✐↔✐✳ Ð ❣â❝ ✤ë ❍å❝ ♠→②✱ ♠✉è♥ ♠→② ♣❤➙♥ ❜✐➺t ✤÷đ❝ t❤➳ ♥➔♦ ❧➔ ❤❛✐ ✤è✐ t÷đ♥❣ ❧➔ t÷ì♥❣ tü✱ t❤➳ ♥➔♦ ❧➔ ❦❤ỉ♥❣ t÷ì♥❣ tü t❤➻ t❛ ♣❤↔✐ ❞↕② ♥â✳ ❚❤ỉ♥❣ t S ủ ỵ trữợ ❤❛✐ t➟♣ ❝♦♥ ✤è✐ t÷đ♥❣ ✭❣✐↔ sû ❧➔ ♥❤❛✉✱ ❝á♥ D Rn ✮ ♠➔ tr♦♥❣ ✤â S ✈➔ D ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❝❤ù❛ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ❝❤ù❛ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ ❦❤ỉ♥❣ t÷ì♥❣ tü✳ ▼ët ♣❤➨♣ ✤♦ tèt✱ A✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ✤➦❝ tr÷♥❣ ❜ð✐ ♠❛ tr➟♥ ✐✮ ❑❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤è✐ t÷đ♥❣ t❤✉ë❝ S ♣❤↔✐ t❤ä❛ ♠➣♥ ❜❛ ✤✐➲✉✿ t❤❡♦ ♠❛ tr➟♥ A ❝➔♥❣ ♥❤ä ❝➔♥❣ tèt❀ ✐✐✮ ❑❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤è✐ t÷đ♥❣ t❤✉ë❝ D t tr A tữỡ ố ợ ▼❛ tr➟♥ A ♣❤↔✐ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤✱ tù❝ ❧➔ A ♣❤↔✐ ố ự ỷ ữỡ ỳ ủ ỵ tr➯♥ ✤➣ ❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ tè✐ ÷✉ s❛✉✿ x−y ❛r❣ A A ✭✵✳✵✳✷✮ (xi ,xj )∈S s❛♦ ❝❤♦ x−y (xi ,xj )∈D A ≥ 1, A ≥ ✭✵✳✵✳✸✮ ✹ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ tr➻♥❤ ❜➔② ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ tè✐ ÷✉ ♥↔② s✐♥❤ tr♦♥❣ ❤å❝ ✤ë t÷ì♥❣ tü ✭✵✳✵✳✷✮✱ ✭✵✳✵✳✸✮✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ❇➔✐ t♦→♥ tè✐ ÷✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ tè✐ ÷✉ ❤â❛✳ ❈❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët ❝→❝❤ ❧÷đ❝ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝✳ ❚r♦♥❣ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤✐ s➙✉ ✈➔♦ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t s➩ ❞ò♥❣ ð ❈❤÷ì♥❣ ✷✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ ❬✷❪✱ ❬✹❪✳ ❈❤÷ì♥❣ ✷✳ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❝❤õ ✤➲ ❦❤→✐ q✉→t ✈➲ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü✳ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤✐ ✈➔♦ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❜➔✐ t♦→♥ ❍å❝ ✤ë t÷ì♥❣ tü t❤❡♦ ❧♦↕t✳ ❈á♥ ♠ët sè ❝❤õ ✤➲ r➜t t❤ó ✈à ❦❤→❝ ♥❤÷ ❍å❝ ✤ë t÷ì♥❣ tü ♦♥❧✐♥❡✱ ❍å❝ ✤ë t÷ì♥❣ tỹ ỹ tr ỵ tt tổ t ổ t ✤÷đ❝ tr➻♥❤ ❜➔② ❞♦ ❦❤✉ỉ♥ ❦❤ê ❝â ❤↕♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❝ơ♥❣ ♥❤÷ sü ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝✳ ❙❛✉ ❝ò♥❣✱ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ❝❤♦ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✈➔ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ✤â✳ ❈❤÷ì♥❣ ♥➔② t❤❛♠ ❦❤↔♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✸❪✱ ❬✺❪✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ tr♦♥❣ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❚❙✳ ◆❣✉②➵♥ ỡ ữớ t t ữợ t ❣✐↔ ❤♦➔♥ ✺ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ô♥❣ ữủ ỷ ỡ tợ tr÷í♥❣ ❚❍P❚ ◆❣✉②➵♥ ✣➠♥❣ ✣↕♦ ✈➔ t➟♣ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ tê ❚♦→♥✲❚✐♥ ❝õ❛ ❚r÷í♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ t→❝ ❣✐↔ t❤❛♠ ❣✐❛ ❤å❝ ❝❛♦ ❤å❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❚r➛♥ ❱➠♥ P❤÷đ♥❣ ✻ ❈❤÷ì♥❣ ✶ ❇➔✐ t♦→♥ tè✐ ÷✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ tè✐ ÷✉ ❤â❛✳ ❈❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët ❝→❝❤ ❧÷đ❝ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝❤♦ ❜➔✐ t tố ữ ỗ t ổ r ✈➔ ❜➔✐ t♦→♥ ❝â r➔♥❣ ❜✉ë❝✳ ❚r♦♥❣ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤✐ s➙✉ ✈➔♦ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t s➩ ❞ò♥❣ ð ❈❤÷ì♥❣ ✷✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ ❬✷❪✱ ❬✹❪✳ ✶✳✶ ❙ì ❧÷đ❝ ✈➲ ❜➔✐ t♦→♥ tè✐ ÷✉ ▼ư❝ ♥➔② s➩ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ❦➳t q✉↔ ❝ì ❜↔♥ ✤➸ ❝â ❝→✐ ♥❤➻♥ ❦❤→✐ q✉→t ✈➲ ❜➔✐ t♦→♥ tè✐ ÷✉✳ ✶✳✶✳✶ ❇➔✐ t♦→♥ tè✐ ÷✉ ❈❤♦ f : Rn → R✳ ❚➻♠ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ x∗ f (x∗ ) ≤ f (x), ∀x ∈ Ux∗ , tr♦♥❣ ✤â✱ U x∗ ❧➔ ❧➙♥ ❝➟♥ ✤à❛ ♣❤÷ì♥❣ ♥➔♦ ✤â ❝õ❛ f (x) x ❝õ❛ f✱ ♥❣❤➽❛ ❧➔✱ ✭✶✳✶✳✶✮ x∗ ✳ ✣➸ ♥❣➢♥ ❣å♥✱ t❛ ✈✐➳t ✭✶✳✶✳✷✮ ✷✸ ❝❤➜t ❧÷đ♥❣ ↔♥❤ ❦❤ỉ♥❣ tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ t❤ü❝ t➳✳ ❱➲ ♠➦t ❞ú ❧✐➺✉✱ ✤➙② ❧➔ ♠ët ♠❛ tr➟♥ ❝ï m×n ♠➔ ❝→❝ ♣❤➛♥ tû ❝â ❣✐→ trà ❤♦➦❝ 0✳ ✐✐✮ ❷♥❤ ①→♠ ✭❣r❛② ✐♠❛❣❡s✮✳ ❑❤➢❝ ♣❤ư❝ ♥❤÷đ❝ ✤✐➸♠ ✈➲ ✤ë tr✉♥❣ t❤ü❝ ❝õ❛ ↔♥❤ ♥❤à ♣❤➙♥✱ ↔♥❤ ①→♠ ❝❤♦ ♣❤➨♣ ❝→❝ ❣✐→ trà ♣✐①❡❧ t❤✉ë❝ ✈ỵ✐ ✤ë ✤➟♠ ❝õ❛ ♠➔✉ ❣✐↔♠ ❞➛♥ tø ✤❡♥ [0, 1] (0) ✤➳♥ tr➢♥❣ (1)✳ ❑❤✐ ✤â✱ ❣✐→ trà ♣✐①❡❧ t❤✉ë❝ ❧ỵ♣ ❞♦✉❜❧❡ tr♦♥❣ ▼❆❚▲❆❇ ✈➔ s➩ tè♥ ❜②t❡s ✭= 64 ❜✐t✮ ✤➸ ❧÷✉ ♠ët ♣❤➛♥ tû ↔♥❤✳ ❚➜t ♥❤✐➯♥ ▼❆❚▲❆❇ ❝ơ♥❣ ❝❤♦ ♣❤➨♣ ❧÷✉ ↔♥❤ ①→♠ ❜➡♥❣ ❞ú ❧✐➺✉ ❧ỵ♣ ✉♥✐t✽ ✭✤á✐ ❤ä✐ ✶ ❜②t❡✮ ❜➡♥❣ ❝→❝❤ ❝❤✐❛ ✤ë ✤➟♠ tø ✤❡♥ ✤➳♥ tr➢♥❣ t❤➔♥❤ 256 ❣✐↔✐ ❝â ❣✐→ trà tø ✤➳♥ 255✳ ✐✐✐✮ ❷♥❤ ♠➔✉ ✭❝♦❧♦r ✐♠❛❣❡s✮✳ ❈ì ❝❤➳ ❧÷✉ ❞ú ❧✐➺✉ ❝õ❛ ↔♥❤ ♠➔✉ ự t ỡ rữợ t t t ỵ ỡ ổ ✏✤ë❝ ❧➟♣✑ ✈ỵ✐ ♥❤❛✉ ♠➔ t❛ ❝â t❤➸ t↕♦ r❛ ♠ët ♠➔✉ ♥❤➜t ✤à♥❤ ❜➡♥❣ ❝→❝❤ ♣❤❛ ❝❤➳ ❤❛✐ ❤❛② ♥❤✐➲✉ ♠➔✉ ❦❤→❝ ♥❤❛✉ ❧↕✐ ✈ỵ✐ ♥❤❛✉ t❤❡♦ t➾ ❧➺ ♥❤➜t ✤à♥❤✳ ❚r♦♥❣ t❤ü❝ t➳✱ ✈ỵ✐ ❜❛ ♠➔✉ ✤ä✱ ❧ư❝✭①❛♥❤ ❧→ ❝➙②✮✱ ❞÷ì♥❣ ✭①❛♥❤ ❞÷ì♥❣✮✱ ♥❣÷í✐ t❛ ❝â t❤➸ t↕♦ r ởt t ỹ tr ỵ ↔♥❤ ♠➔✉ ✭❤❛② t➯♥ ❣å✐ ❦❤→❝ ❧➔ ↔♥❤ ❘●❇ ✭r❡❞✱ r ữủ ữ ữ s ợ ộ ✈à tr➼ ♥❣÷í✐ t❛ ❝➛♥ ❣✐→ trà sè t÷ì♥❣ ù♥❣ ✈ỵ✐ ❜❛ ♠➔✉ ❝ì ❜↔♥✱ ❝→❝ ❣✐→ trà ♥➔② ❝â t❤➸ t❤✉ë❝ [0, 1] ♥➳✉ t❤✉ë❝ ❧ỵ♣ ❞♦✉❜❧❡ ❤♦➦❝ [0, 255] N tở ợ t ữ ♠➦t ❞ú ❧✐➺✉✱ ♠ët ↔♥❤ ♠➔✉ ❧➔ ♠ët ♠↔♥❣ (i, j) ❝❤✐➲✉ ❝ï m × n × 3✳ ✭❛✮ ❷♥❤ ♠➔✉✳ ♥➳✉ m × n ♣✐①❡❧ ✷✹ ✭❜✮ ❷♥❤ ①→♠✳ ✭❝✮ ❷♥❤ ♥❤à ♣❤➙♥✳ ❍➻♥❤ ✷✳✶✳✷✿ ▼ët sè ✤à♥❤ ❞↕♥❣ ↔♥❤ t❤ỉ♥❣ ❞ư♥❣✳ ❚r♦♥❣ ❝→❝ ❍➻♥❤ ✷✳✶✳✷✱ ❝❤ó♥❣ tỉ✐ ❧➛♥ ❧÷đt ♠✐♥❤ ❤å❛ ❝→❝ ❦✐➸✉ ❞ú ❧✐➺✉ ↔♥❤ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❝ò♥❣ ♠ët ↔♥❤✳ ❞✮ ❚r➼❝❤ rót ✤➦❝ tr÷♥❣ ◆❤÷ ð ❝❤÷ì♥❣ ■ ✤➣ ❝❤➾ r❛✱ ❞ú ❧✐➺✉ t❤÷í♥❣ ✤÷đ❝ ❝❤♦ ð ❞↕♥❣ ✈❡❝tì tr♦♥❣ ❦❤✐ ♠ët ↔♥❤ ✤÷đ❝ ▼❆❚▲❆❇ ❧÷✉ ð ❞↕♥❣ ♠↔♥❣ sè ❤❛✐ ❤♦➦❝ ❜❛ ❝❤✐➲✉✳ ✣÷ì♥❣ ♥❤✐➯♥ ♠→② ❞ú ❧✐➺✉ sè ❝â t❤➸ ❞➵ ❞➔♥❣ ✤÷đ❝ ❝❤✉②➸♥ ✤ê✐ s❛♥❣ ❞↕♥❣ ✈❡❝tì sè tr♦♥❣ ▼❆❚▲❆❇✳ ❈❤➥♥❣ ❤↕♥✱ ♠ët ♠❛ tr➟♥ ❝ï ❝❤✉②➸♥ t❤➔♥❤ ♠ët ✈❡❝tì m×n mìn õ t ởt ữ t ❜ë ❞ú ❧✐➺✉ ❝õ❛ ♠ët t➜♠ ↔♥❤ ♥❤÷ ✈➟② s➩ r➜t tè♥ ❜ë ♥❤ỵ✳ ❈❤➥♥❣ ❤↕♥✱ ❜ù❝ ↔♥❤ ♠➔✉ ❝õ❛ ❡♠ ❜➨ tr♦♥❣ ❤➻♥❤ ✷✳✶✳✷✳✭❛✮ ♥➳✉ ✤÷đ❝ ❧÷✉ ð ❞↕♥❣ ❞ú ❧✐➺✉ sè ❧➔ ♠ët ❜✐➳♥ ❝â ❦➼❝❤ ❝ï ✶✳✸✺▼❜✳ õ ữớ t t tt ợ ❜➡♥❣ ❝→❝❤ ❣✐ú ❧↕✐ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ q✉❛♥ trå♥❣ ♥❤➜t ❝õ❛ ❜ù❝ ↔♥❤✳ ❈ỉ♥❣ ✈✐➺❝ ✤â ❣å✐ ❧➔ tr➼♥❤ rót ✤➦❝ tr÷♥❣ ✭❢❡❛t✉r❡ ❡①tr❛❝t✐♦♥✮✳ ❙è ❧÷đ♥❣ ✤➦❝ tr÷♥❣ ✈➔ ❧♦↕✐ ✤➦❝ tr÷♥❣ ♥➔♦ ✤÷đ❝ ❣✐ú ❧↕✐ ♣❤ư t❤✉ë❝ ✈➔♦ ♥❤✉ ữớ ỷ ỵ r õ ổ ❝ö ❈♦♠♣✉t❡r ✈✐s✐♦♥ s②st❡♠s ❚♦♦❧❜♦① ✈➔ ❙t❛t✐st✐❝s ❛♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❚♦♦❧❜♦① ❝â s➤♥ ❝→❝ ❧➺♥❤ ❤é trđ ✤➸ tr➼❝❤ rót ✤➦❝ tr÷♥❣✳ ❚❛ s➩ ❦❤ỉ♥❣ ✤✐ s➙✉ ✈➔♦ ❝❤õ ✤➲ ♥➔② ✈➻ tr♦♥❣ ✈➼ ❞ö sè✱ t❛ ❝❤➾ sû ❞ö♥❣ ❝→❝ ❞ú ❧✐➺✉ ✤ì♥ ❣✐↔♥ ❝â s➤♥✳ ✷✳✶✳✷ ❇➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ❇➙② ❣✐í✱ t❛ ①➨t ♠ët t➟♣ ❤đ♣ tê♥❣ q✉→t ❤ì♥ ♥❤÷ t➟♣ ❝→❝ ❤➻♥❤ ❝❤ư♣ ♠➦t ♥❣÷í✐✱ ❤❛② t➟♣ ❝→❝ ✈➠♥ ❜↔♥✳ ●✐↔ sû t❛ ❝â t❤➸ ❜✐➳♥ ✤ê✐ ❝→❝ ✤è✐ t÷đ♥❣ ✤â t❤➔♥❤ ❝→❝ ✤è✐ t÷đ♥❣ t♦→♥ ❤å❝ ♥❤÷ ✈❡❝tì ❤♦➦❝ ♠❛ tr➟♥✳ ❚❛ ❝➛♥ ♣❤↔✐ ①➙② ❞ü♥❣ ♠ët ♣❤➨♣ ✤♦ ✤➸ ❝â t❤➸ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ❤❛✐ ♥❤â♠ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ♥❤❛✉ ✈➔ ❦❤→❝ ♥❤❛✉✳ ❱➲ ♠➦t ✤à♥❤ ❧÷đ♥❣✱ ❤❛✐ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ♥❤❛✉ ♥➳✉ ❝â ❦❤♦↔♥❣ ❝→❝❤ ✭t❤❡♦ ♣❤➨♣ ✤♦ ✈ø❛ ✤÷đ❝ ①➙② ❞ü♥❣✮ ♥❤ä ✈➔ ❤❛✐ ✤è✐ t÷đ♥❣ ❦❤→❝ ♥❤❛✉ s➩ ❝â ❦❤♦↔♥❣ ❝→❝❤ ❧ỵ♥✳ ❈➙✉ ❤ä✐ t✐➳♣ t❤❡♦ ❧➔ ỹ õ ữ t ị tữ tü ♥❤✐➯♥ ❧➔ ❦❤→✐ q✉→t ❤â❛ x, y ∈ Rn ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡✳ ❚❛ ❝â ✈ỵ✐ x−y tr♦♥❣ ✤â E I = x−y t❤➻ (x − y)T (x − y) = = I ❜➡♥❣ ♠ët ♠❛ tr➟♥ (x − y)T A(x − y) ✭✷✳✶✳✶✮ ❧➔ ♠ët ♠❛ tr➟♥ ✤ì♥ ✈à✳ ❇➙② ❣✐í✱ t❛ t❤❛② ✤è✐ ①ù♥❣ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ x−y A A (x − y)T I(x − y), = ữ ỵ r õ ❝❤➾ ❧➔ ♠ët ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤✱ tù❝ ❧➔ ❤❛✐ ✤✐➸♠ ❦❤→❝ ♥❤❛✉ ❝â t❤➸ ❝â ❦❤♦↔♥❣ ❝→❝❤ ❜➡♥❣ 0✳ ◆❤÷ ✈➟②✱ ✈✐➺❝ ①➙② ❞ü♥❣ ❦❤♦↔♥❣ ❝→❝❤ ✤÷đ❝ q✉② ✈➲ ✈✐➺❝ t➻♠ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ổ õ ủ ỵ t ởt ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❝â ❧í✐ ❣✐↔✐✳ Ð ❣â❝ ✤ë ❍å❝ ♠→②✱ ♠✉è♥ ♠→② ♣❤➙♥ ❜✐➺t ✤÷đ❝ t❤➳ ♥➔♦ ❧➔ ❤❛✐ ✤è✐ t÷đ♥❣ ❧➔ t÷ì♥❣ tü✱ t❤➳ ♥➔♦ ❧➔ ❦❤ỉ♥❣ t÷ì♥❣ tỹ t t õ ổ t ủ ỵ trữợ t sỷ ❧➔ Rn ✮ ♠➔ tr♦♥❣ ✤â S S ✈➔ D ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ✤è✐ t÷đ♥❣ ✭❣✐↔ ❝❤ù❛ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ t÷ì♥❣ tü ♥❤❛✉✱ ❝á♥ D ✷✻ ❝❤ù❛ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ ❦❤ỉ♥❣ t÷ì♥❣ tü✳ ▼ët ♣❤➨♣ ✤♦ tèt✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ✤➦❝ tr÷♥❣ ❜ð✐ ♠❛ tr➟♥ A✱ ♣❤↔✐ t❤ä❛ ♠➣♥ ❜❛ ✤✐➲✉✿ ✐✮ ❑❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤è✐ t÷đ♥❣ t❤✉ë❝ S t❤❡♦ ♠❛ tr➟♥ A ❝➔♥❣ ♥❤ä ❝➔♥❣ tèt❀ ✐✐✮ ❑❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❝→❝ ✤è✐ t÷đ♥❣ t❤✉ë❝ D t❤❡♦ tr A tữỡ ố ợ tr A ♣❤↔✐ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ❣✐↔ ❦❤♦↔♥❣ ❝→❝❤✱ tù❝ ❧➔ A ♣❤↔✐ ✤è✐ ①ù♥❣ ỷ ữỡ ỳ ợ ỵ tr ❞➝♥ ✤➳♥ ❜➔✐ t♦→♥ tè✐ ÷✉ s❛✉✿ x−y ❛r❣ A A ✭✷✳✶✳✷✮ (xi ,xj )∈S s❛♦ ❝❤♦ x−y A ≥ 1, A ≥ ✭✷✳✶✳✸✮ (xi ,xj )∈D ỗ t s s ♥➯✉ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➸ ❜➔✐ t♦→♥ tè✐ ÷✉ ỗ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿ ✐✮ ❚➟♣ ❝→❝ ♠❛ tr➟♥ ✤è✐ ự ỷ ữỡ P(n) ởt t ỗ ✐✐✮ ❍➔♠ ♠ö❝ t✐➯✉ x(i) − x(j) f (A) = (x(i) ,x(j) )S ỗ ố ợ A ✷✼ ✐✐✐✮ ❍➔♠ x(i) − x(j) g(A) = A (x(i) ,x(j) )∈D ❧➔ ♠ët ❤➔♠ ❧ã♠✳ ❑❤✐ ✤â ❜➔✐ t tố ữ ỗ ự x ∈ Rn ❚❤➟t ✈➟②✱ ❝❤♦ A1 , A2 ∈ ❙P❙❉(n) ✈ỵ✐ ♠å✐ λ ∈ (0; 1) ✈➔ t❛ ❝â xT (λA1 + (1 − λ)A2 )x = λxT A1 x + (1 − λ)xT A2 x ≥ ❉♦ A1 , A2 ∈ ❙P❙❉(n)✳ ❚ø ✤â λA1 + (1 )A2 P(n) P(n) ởt t ỗ ❑❤➥♥❣ ✤à♥❤ ✭✐✐✮ ✈➔ ✭✐✐✐✮ ✤÷đ❝ ❞➵ ❞➔♥❣ s✉② r❛ ✈➻ ❝↔ t✉②➳♥ t➼♥❤ ✤è✐ ✈ỵ✐ ❝♦♥ ❜➜t ❦ý ❝õ❛ (x(i) ,x(j) )∈M A f (A) ✈ỵ✐ ❝→❝ ❤➺ sè ❞÷ì♥❣✳ ❚❤➟t ✈➟②✱ ❝❤♦ M ✈➔ g(A) ❧➔ ❧➔ ♠ët t➟♣ Rn ✱ x(i) − x(j) λA1 +µA2 (x(i) − x(j) )T (λA1 + µA2 )(x(i) − x(j) ) = (x(i) ,x(j) )∈M = (x(i) − x(j) )T A1 (x(i) − x(j) ) λ (x(i) ,x(j) )∈M + (x(i) − x(j) )T A2 (x(i) − x(j) ) µ (x(i) ,x(j) )∈M = x(i) − x(j) λ A1 (x(i) ,x(j) )∈M x(i) − x(j) +µ (x(i) ,x(j) )∈M A2 ◆❤÷ ✈➟② ✤è✐ t÷đ♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ởt t tố ữ ỗ õ r ✷✳✶✳✹ ❑❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ❑❤→✐ ♥✐➺♠ ❦❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ①✉➜t ♣❤→t tø t❤è♥❣ ❦➯ ❤å❝✳ ◆â ❞ü❛ tr➯♥ ♠❛ tr➟♥ ❤✐➺♣ ♣❤÷ì♥❣ s❛✐ Σ ✭❤❛② ♠❛ tr➟♥ ❤✐➺♣ ❜✐➳♥✮ ❝õ❛ ❞ú ✷✽ ❧✐➺✉✳ ◆â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ❧➔ ♠ët ♠ð rë♥❣ ❝õ❛ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡ dM ahal (x(i) , x(j) ) := ((x(i) − x(j) )T Σ−1 (x(i) − x(j) ))1/2 ❚r♦♥❣ ❤å❝ ✤ë t÷ì♥❣ tü✱ ♥❣÷í✐ t❛ sû ❞ư♥❣ ❦❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ✈ỵ✐ ♥ë✐ ❤➔♠ rë♥❣ ❤ì♥✳ ❈ö t❤➸✱ ❝❤♦ ❆ ❧➔ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣✱ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ❜➜t ❦ý✳ ❑❤✐ ✤â✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❦❤♦↔♥❣ ❝→❝❤ ù♥❣ ✈ỵ✐ ❆ ❧➔ dA (x, y) = ((x − y)T A(x − y))1/2 ✭✷✳✶✳✹✮ ❑❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ❝â ♠è✐ ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡ t❤ỉ♥❣ q✉❛ ♣❤➨♣ ❝❤✐➳✉ ♥❤÷ s❛✉✳ ●✐↔ sû r(A) õ tỗ t tr LA Rnìi õ ❤↕♥❣ ❦ s❛♦ ❝❤♦ = k ≤ n✳ A = LA LTA ✳ ❑❤✐ ❑❤✐ ✤â✱ ✭✷✳✶✳✹✮ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ❧↕✐ t❤➔♥❤ dA (x, y) = ((x − y)T LA LTA (x − y))1/2 = ((LTA (x − y))T (LTA (x − y)))1/2 = LA x − LA y = d(LA x, LA y) ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ ❦❤♦↔♥❣ ❝→❝❤ ❊✉❝❧✐❞❡ ❝â trå♥❣ sè ❚❛ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ♠ö❝ ♥➔② ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ❦❤♦↔♥❣ ❝→❝❤ ▼❛❤❛❧❛♥♦❜✐s ❦❤✐ ❆ ❧➔ ♠ët ♠❛ tr➟♥ ❝❤➨♦ A = diag(a11 , , ann ), aii ≥ ❚❛ t❤➜② ❦❤✐ ✤â 1/2 n aii (xi − yi ) dA (x, y) = , ✭✷✳✷✳✶✮ i=1 ❦❤♦↔♥❣ ❝→❝❤ ✭✷✳✷✳✶✮ ✤➦t ❝→❝ trå♥❣ sè ❝õ❛ ✈➨❝tì x−y aii ❦❤→❝ ♥❤❛✉ ✤è✐ ✈ỵ✐ t❤➔♥❤ ♣❤➛♥ ❦❤✐ t➼♥❤ ✤ë ❞➔✐✳ ✣✐➲✉ ❦✐➺♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ❝õ❛ trð t❤➔♥❤ ✤✐➲✉ ❦✐➺♥ aii ≥ 0, i = 1, , n✳ A ✷✾ ❚✐➳♣ t❤❡♦✱ t❛ s➩ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ❝❤➢♥ ❧♦❣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔②✳ ❍➔♠ ❝❤➢♥ ❧♦❣ t÷ì♥❣ ù♥❣ ✈ỵ✐ ❇➔✐ t♦→♥ ✭✷✳✶✳✷✮✱ ✭✷✳✶✳✸✮ ❝â ❞↕♥❣ x(i) − x(j) P(a; µ) = (x(i) ,x(j) )∈S x(i) − x(j) − µ log A A (x(i) ,x(j) )∈D ✭✷✳✷✳✷✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ✤â t❛ ✤➣ ❦➼ ❤✐➺✉ a ≥ 0, ✭✷✳✷✳✸✮ a = (a1 , , an ) ❧➔ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ ❆ ✤➸ t❤➸ ❤✐➺♥ rã ❜✐➳♥ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ ♠ët ✈➨❝tì ❝ï n✳ ❚r♦♥❣ ✭✷✳✷✳✷✮ µ ❧➔ t❤❛♠ sè ❝❤➢♥✳ ❚❤❛② ✈➻ ❣✐↔✐ ♠ët ❤å ❝→❝ ❇➔✐ t♦→♥ ✭✷✳✷✳✷✮✱ ✭✷✳✷✳✸✮ ✈ỵ✐ ❝→❝ ❣✐→ trà ♥❤ä ❞➛♥✱ ❝→❝ t→❝ ❣✐↔ tr♦♥❣ ❬✺❪ ✤➣ ❝è ✤à♥❤ µ µ = 1✳ ▼➦❝ ❞ò ✤➙② ởt ữợ ỡ õ t ♥❤÷♥❣ ❝→❝ ✈➼ ❞ư sè ✭①❡♠ ▼ư❝ ✷✳✷✳✸✮ ❝❤➾ r❛ r➡♥❣ ♥â ✤õ ❤ú✉ ❤✐➺✉ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ q✉→ ♣❤ù❝ t↕♣✳ ❚❤❡♦ ✤â✱ t❛ s➩ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✤➸ tè✐ ÷✉ ❤â❛ ❤➔♠ ♠ư❝ t✐➯✉ x(i) − x(j) P(a; 1) = A (x(i) ,x(j) )∈S x(i) − x(j) − log (x(i) ,x(j) )∈D A ợ a0 õ ởt ữ ỵ ❧➔ tr♦♥❣ t➼♥❤ t♦→♥✱ t❛ ❦❤æ♥❣ ❝❤å♥ ❝❤➼♥❤ ①→❝ ✤ë ❞➔✐ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ◆❡✇t♦♥ ❦❤✐ ❝➟♣ ♥❤➟t sû ❞ö♥❣ ♠ët t❤❛♠ sè ∇2 P (ak )−1 ∇P (ak ) ♠➔ α ❞÷ì♥❣✿ α∇2 P (ak )−1 ∇P (ak ) ữợ s ✤✐➸♠ ❝➟♣ ♥❤➟t t✐➳♣ t❤❡♦ ❧✉æ♥ t❤ä❛ ♠➣♥ r➔♥❣ ❜✉ë❝ ✭✷✳✷✳✺✮✳ ✷✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ❝❤♦ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü ❚❛ ①➨t tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❦❤✐ ❆ ❧➔ ♠❛ tr➟♥ ✤➛② ✤õ✳ ❑❤✐ ✤â✱ ❜✐➳♥ ❆ ❝â t❤➸ ❝♦✐ ♥❤÷ ♠ët ✈➨❝tì ❝❤ù❛ n2 ❜✐➳♥✳ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ♥❤÷ ✸✵ tr➻♥❤ ❜➔② ð ▼ư❝ ✶✳✷✳✶ s➩ tr ổ ũ ủ tr trữớ ủ n ợ ❱➻ t❤➳ t❛ t➻♠ ♠ët ♣❤÷ì♥❣ ♣❤→♣ r➫ ❤ì♥✳ Ð t s sỷ ỵ tữ ữỡ rt tr ữ ỵ r➡♥❣ ð ▼ư❝ ✶✳✷✳✹✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣r❛❞✐❡♥t ✤÷đ❝ tr➻♥❤ ❜➔② ❝❤♦ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ ✤ì♥ ❣✐↔♥✿ < xi < bi ✳ ❚❛ ❝➛♥ ♣❤↔✐ ❦❤→✐ qt ỵ tữ õ õ t ữủ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ❇➔✐ t♦→♥ tè✐ ÷✉ ✭✷✳✶✳✷✮✱ ✭✷✳✶✳✸✮✳ ❈ö t❤➸✱ ♥➳✉ q✉❛♥ s→t ❦ÿ ❇➔✐ t♦→♥ ✭✷✳✶✳✷✮✱ ✭✷✳✶✳✸✮ ❝â tỵ✐ ❤❛✐ r➔♥❣ ❜✉ë❝✿ ✣✐➲✉ ❦✐➺♥ ✤è✐ ①ù♥❣ ❦❤ỉ♥❣ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ ✤✐➲✉ D✳ ❦✐➺♥ t→❝❤ tr➯♥ t➟♣ ữ ợ t t tr õ ộ ữợ t t❤ü❝ ❤✐➺♥ ♠ët ♣❤➨♣ ❝❤✐➳✉✱ ð ✤➙② t❛ ♣❤↔✐ t❤ü❝ ộ ữợ ởt ❦✐➺♥ ♥ú❛ ✤➸ ❣✐ó♣ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t❤➔♥❤ ❝ỉ♥❣ ❧➔ ✈✐➺❝ t➼♥❤ t♦→♥ ♣❤➨♣ ❝❤✐➳✉ ♣❤↔✐ t÷ì♥❣ ✤è✐ ✤ì♥ ❣✐↔♥✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ✭✷✳✶✳✸✮ ❧↕✐ t÷ì♥❣ ✤è✐ ♣❤ù❝ t↕♣✱ ✤➦❝ ❜✐➺t ❧➔ s♦ s→♥❤ ✈ỵ✐ ❤➔♠ ♠ư❝ t✐➯✉✳ ❱➻ t❤➳ t❛ ✤➦t ❧↕✐ ❇➔✐ t♦→♥ ✭✷✳✶✳✷✮✱ ✭✷✳✶✳✸✮ ð ♠ët ❣â❝ ♥❤➻♥ ❦❤→❝✳ ❈ö t❤➸✱ t❤❛② ✈➻ t❛ t➻♠ ♠ët ♠❡tr✐❝ tè✐ t❤✐➸✉ ❤â❛ ❦❤♦↔♥❣ ❝→❝❤ ❝→❝ ✤✐➸♠ t❤✉ë❝ S ✈➔ ✤↔♠ ❜↔♦ ❦❤♦↔♥❣ ❝→❝❤ ❝→❝ ✤✐➸♠ ❝õ❛ ❦❤♦↔♥❣ ❝→❝❤ ❝→❝ ✤✐➸♠ t❤✉ë❝ S ♥❤ä ❤ì♥ 1✳ D D ❧ỵ♥ ❤ì♥ 1✱ t❛ s➩ tè✐ ✤❛ ❤â❛ ✈➔ ❣✐ú ❝❤♦ ❦❤♦↔♥❣ ❝→❝❤ ❝→❝ ✤✐➸♠ tở õ t õ t tố ữ ợ tữỡ ữỡ ợ t max g(A) = A ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ x(i) − x(j) A, ✭✷✳✷✳✻✮ x(i) − x(j) A ✭✷✳✷✳✼✮ (x(i) ,x(j) )∈D f (A) = ≤ 1, (x(i) ,x(j) )∈S A ≥ ✭✷✳✷✳✽✮ ✣✐➸♠ ❦❤→❝ ❜✐➺t q✉❛♥ trå♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✷✳✻✮✲✭✷✳✷✳✽✮ s♦ ✈ỵ✐ ❜➔✐ t♦→♥ ✭✷✳✶✳✷✮✱ ✭✷✳✶✳✸✮ ❧➔ t❛ ✤➣ t❤❛② t❤➳ r➔♥❣ ❜✉ë❝ ✭✷✳✶✳✸✮ ❜ð✐ r➔♥❣ ❜✉ë❝ ❞↕♥❣ ❜➟❝ ❤❛✐ ✭✷✳✷✳✼✮✳ ✣✐➲✉ ♥➔② s➩ ❣✐ó♣ ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐➳✉ ✤ì♥ ❣✐↔♥ ❤ì♥ ✈➔ ❞♦ ✤â t➠♥❣ ❤✐➺✉ q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❦❤✐ →♣ ❞ư♥❣ ✈➔♦ ❤å❝ ✤ë ✸✶ t÷ì♥❣ tü✳ ❙❛✉ ✤➙②✱ t❛ s➩ ❜➔♥ t❤➯♠ ♠ët sè ❦❤➼❛ ❝↕♥❤ ❝ö t❤➸ ❝õ❛ ❜➔✐ t♦→♥✳ ❚❤❡♦ ❝→❝❤ ♣❤→t ❜✐➸✉ ♠ỵ✐ ✭✷✳✷✳✻✮✲✭✷✳✷✳✽✮✱ t❛ ❦➼ ❤✐➺✉✿ C1 = A: x (i) −x (j) ≤1 , A (x(i) ,x(j) )∈S C2 = A : A ≥ ❚❛ s➩ tâ♠ t➢t ❝→❝ t❤❛♦ t→❝ ❝❤➼♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✭t❤ü❝ tr ởt ữợ tr t t rt ✹ ❚❤✉➟t t♦→♥ ❝❤✐➳✉ ❣r❛❞✐❡♥t ❝❤♦ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü✳ ▲➦♣ ▲➦♣ A := ❛r❣ minA { A − A F : A ∈ C1 } A := ❛r❣ minA { A − A F : A ∈ C2 } ✤➳♥ ❦❤✐ A ❤ë✐ tö A := A + α(∇A g(A))⊥∇A f ✤➳♥ ❦❤✐ ❤ë✐ tö❀ ❚r♦♥❣ ❚❤✉➟t t♦→♥ ✹✱ ♣❤➨♣ ❝❤✐➳✉ t❤ù ♥❤➜t ✤÷đ❝ t❤ü❝ ❤✐➺♥ t❤ỉ♥❣ q✉❛ ✈✐➺❝ tè✐ ÷✉ ❤â❛ ♠ët ❤➔♠ ♠ư❝ t✐➯✉ ❜➟❝ ❤❛✐ ❛r❣ A − A A ∈C1 F ❱ỵ✐ r➔♥❣ ❜✉ë❝ t✉②➳♥ t➼♥❤ ✭✷✳✷✳✼✮✳ ❇➔✐ t♦→♥ ♥➔② ❝â t❤➸ ✤÷đ❝ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ tr➻♥❤ ❜➔② ð ▼ư❝ ✶✳✶✳✸✳ P❤➨♣ ❝❤✐➳✉ t❤ù ❤❛✐ t❤ü❝ ❝❤➜t ❧➔ t➻♠ ♠❛ tr ố ự ỷ ữỡ ợ ❆ ♥❤➜t t❤❡♦ ❝❤✉➞♥ ❋r♦❜❡♥✐✉s✳ ❚❛ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ✈✐➺❝ ♥➔② t❤æ♥❣ q✉❛ t❤✉➟t t♦→♥ ♥❣➢♥ s❛✉ ✤➙② ❞ü❛ tr➯♥ ♠ët ❦➳t q✉↔ ❝õ❛ ◆✳ ❏✳ ❍✐❣❤❛♠ ❬✶❪✳ ✸✷ ❆❧❣♦r✐t❤♠ ✺ ❚❤✉➟t t♦→♥ ❝❤✐➳✉ ❧➯♥ t➟♣ ❝→❝ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✣➛✉ ✈➔♦✿ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ❆ ✣➛✉ r❛✿ ♠❛ tr➟♥ S ✤è✐ ①ù♥❣ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ❣➛♥ A ♥❤➜t ✶✿ P❤➙♥ t➼❝❤ ❣✐→ trà r✐➯♥❣ A = X T ΛX ✷✿ ❣→♥ Λ = max(0, Λ) ✸✿ S = X T ΛX ✳ ✷✳✷✳✸ ❱➼ ❞ư sè ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ①➨t ♠ët ✈➼ ❞ö sè ✤➸ ♠✐♥❤ ❤å❛ ❝❤♦ t❤✉➟t t♦→♥ ✤÷đ❝ tr➻♥❤ ❜➔② ð ▼ư❝ ✷✳✷✳✶✳ ❈→❝ ❞ú ❧✐➺✉ ❞♦ ❝❤ó♥❣ tỉ✐ tü t↕♦ r❛ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ợ ỵ tữ ỹ tr tr t ❧✐➺✉ ❬✺❪✳ ✣➸ t↕♦ ❞ú ❧✐➺✉✱ ❝❤ó♥❣ tỉ✐ ❝❤♦ ✷ ✤✐➸♠ R3 ✳ ✈➔ ❚↕✐ ♠é✐ ✤✐➸♠ Q P✱ ❝❤ó♥❣ tỉ✐ t↕♦ r❛ ✈ỵ✐ ✤ë ❧ỵ♥ ❝õ❛ ♥❤✐➵✉ ❜➡♥❣ 1/5 100 P (1, 4, 3) ✈➔ Q(3.5, 2, 4) tr♦♥❣ ✤✐➸♠ ❜➡♥❣ ❝→❝❤ ♥❤✐➵✉ ✤✐➸♠ ❦❤♦↔♥❣ ❝→❝❤ tø P tỵ✐ Q✳ ✣✐➸♠ P P, Q ✈➔ ❝→❝ ♥❤✐➵✉ ❝õ❛ ❝❤ó♥❣ ✤÷đ❝ ❜✐➸✉ t❤à tr♦♥❣ ❍➻♥❤ ✷✳✷✳✶✳ ❚❛ ❝â t❤➸ t❤➜② ❝❤ó♥❣ ❧➟♣ t❤➔♥❤ ❤❛✐ ♥❤â♠✱ t↕♠ ❣å✐ ❧➔ ♥❤â♠ P ✈➔ ♥❤â♠ ◗✳ ❇➙② ❣✐í✱ t❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ✤✐➸♠ t❤✉ë❝ tø♥❣ ♥❤â♠ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ tü ✈➔ q✉❛♥ ❤➺ ❝õ❛ ❝→❝ ✤✐➸♠ ❦❤→❝ ♥❤â♠ ❧➔ q✉❛♥ ❤➺ ❦❤ỉ♥❣ t÷ì♥❣ tü✳ ❚✐➳♣ ✤â✱ ❝❤ó♥❣ tỉ✐ t❤ü❝ 10 ữợ t ợ ỹ ✤➛✉ ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à ✭ù♥❣ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ ữ ỵ r t sỷ từ tử tr ữủ ữợ s ữợ t ak+1 = ak + P (ak )−1 ∇P (ak ) ❧✉æ♥ t❤ä❛ ♠➣♥ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝õ❛ 10 ak+1 ❧➔ ❦❤ỉ♥❣ ➙♠✳ ❈❤ó♥❣ tỉ✐ sû ❞ư♥❣ ữợ ữ tr t ữủ a10 = [1.1606 1.0180 0.0000]T ✸✸ 4.5 z 3.5 2.5 4 3.5 3 2.5 y x ❍➻♥❤ ✷✳✷✳✶✿ ❍❛✐ ♥❤â♠ ✤✐➸♠ ❞ú ❧✐➺✉✳ ❚è❝ ✤ë ❣✐↔♠ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ❣r❛❞✐❡♥t ✈➔ ❣✐→ trà ❤➔♠ ♠ư❝ t✐➯✉ ✤÷đ❝ t❤➸ ❤✐➺♥ tr♦♥❣ ❍➻♥❤ ✷✳✷✳✷✭❛✮ ✈➔ ❍➻♥❤ ✷✳✷✳✷✭❜✮✳ 1201.2162 1201.216 1201.2158 chuan gradient 1201.2156 1201.2154 1201.2152 1201.215 1201.2148 1201.2146 1201.2144 buoc lap ✭❛✮ 10 ✸✹ 2100 2000 gia tri 1900 1800 1700 1600 1500 1400 10 buoc lap ✭❜✮ ❍➻♥❤ ✷✳✷✳✷✿ ❈❤✉➞♥ ❝õ❛ ❣r❛❞✐❡♥t ✭❛✮ ✈➔ ❣✐→ trà ❤➔♠ t tr ữợ ụ ữ þ r➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ ♠ư❝ t✐➯✉ ❤➛✉ ♥❤÷ ❦❤ỉ♥❣ ❣✐↔♠ ✤÷đ❝ ♥ú❛ ✈➔ ❦❤✐ ❝❤↕② t❤➯♠ t❤➻ tå❛ ✤ë t❤ù ❜❛ ❝õ❛ ✱ s➩ ❞➛♥ ✈➲ 0✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➣ t❤û ♥❤✐➲✉ t➻♥❤ ❤✉è♥❣ ❦❤→❝ ♥❤❛✉ ✈➔ t❤➜② r➡♥❣✱ t❤➔♥❤ ♣❤➛♥ ♥➔♦ ❝õ❛ P ✈➔ ù♥❣ s➩ t✐➳♥ ✈➲ Q 0✳ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ ❤✐➺✉ ♥❤ä ♥❤➜t t❤➻ tå❛ ✤ë t÷ì♥❣ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ✤➣ ①➨t✱ ❝â t❤➸ t❤➜② t❤➔♥❤ ♣❤➛♥ t❤ù ❜❛ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ ❤✐➺✉ ❧➔ ♥❤ä ♥❤➜t✳ ❈✉è✐ ❝ò♥❣✱ t❛ s♦ s→♥❤ ❧↕✐ ✈à tr➼ ❝õ❛ √ s❛✉ ❦❤✐ ✤➣ ❝❤✐➳✉ q✉❛ →♥❤ ①↕ A P ✈➔ Q ❝ò♥❣ ❝→❝ ♥❤✐➵✉ ❝õ❛ ❝❤ó♥❣ ♥❤÷ t❤↔♦ ❧✉➟♥ ð ❝✉è✐ ▼ö❝ ✷✳✶✳✹ tr♦♥❣ ❍➻♥❤ ✷✳✷✳✸✳ ❈â t❤➸ t❤➜② ❝→❝ ♥❤â♠ ✤✐➸♠ tư ❧↕✐ tèt ❤ì♥ s♦ ✈ỵ✐ ♣❤➙♥ ❜è ❜❛♥ ✤➛✉✳ ✸✺ 10-3 2.8 z 2.6 2.4 2.2 1.8 4 3 y x ❍➻♥❤ ✷✳✷✳✸✿ ▼✐♥❤ ❤å❛ ❤❛✐ ♥❤â♠ ✤✐➸♠ ❞ú ❧✐➺✉ tr♦♥❣ ❦❤♦↔♥❣ ❝→❝❤ ♠ỵ✐✳ ✸✻ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ tè✐ ÷✉ ♥↔② s✐♥❤ tr♦♥❣ ❤å❝ ✤ë t÷ì♥❣ tü✱ ❈ư t❤➸✿ ✭✶✮ ❚r➻♥❤ ❜➔② ❧÷đ❝ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ t❤❡♦ ✤÷í♥❣ t❤➥♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝✳ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ qt ỵ tt tờ qt t tố ữ ❝â r➔♥❣ ❜✉ë❝✳ ✭✷✮ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝ s➩ ❞ò♥❣ ð ❈❤÷ì♥❣ ✷✳ ✭✸✮ ❚r➻♥❤ ❜➔② ❦❤→✐ q✉→t ✈➲ ❜➔✐ t♦→♥ ❤å❝ ✤ë t÷ì♥❣ tü✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈➔ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥✳ ❙❛✉ ✤â✱ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥✳ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆✳ ❏✳ ❍✐❣❤❛♠✳ ✧❈♦♠♣✉t✐♥❣ ❛ ♥❡❛r❡st s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ s❡♠✐❞❡❢✐✲ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✵✸✲✶✶✽✳ ❑❡❧❧❡② ✭✷✵✶✷✮✳ ■t❡r❛t✐✈❡ ▼❡t❤♦❞s ❢♦r ❖♣t✐♠✐③❛t✐♦♥✱ ♥✐t❡ ♠❛tr✐①✧✱ ❬✷❪ ❈✳ ❚✳ ❙■❆▼✱ P❤✐❧❛❞❡❧♣❤✐❛✳ ❬✸❪ ❇✳ ❑✉❧✐s ✭✷✵✶✷✮✳ ▼❡tr✐❝ ❧❡❛r♥✐♥❣✿ ❛ s✉r✈❡②✳ ❋♦✉♥❞❛t✐♦♥ ❛♥❞ ❚r❡♥❞s ✐♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣✱ ✺✭✹✮✱ ✷✽✼✲✸✻✹✳ ❬✹❪ ❏✳ ◆♦❝❡❞❛❧✱ ❙✳ ❏✳ ❲r✐❣❤t ✭✶✾✾✾✮✳ ◆✉♠❡r✐❝❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣ ◆❡✇ ❨♦r❦✱ ■♥❝✳ ❬✺❪ ❊✳ P✳ ❳✐♥❣✱ ❆✳ ❨✳ ◆❣✱ ■✳ ❏♦r❞❛♥ ❛♥❞ ❙✳ ❘✉ss❡❧❧✳ ✭✷✵✵✷✮✳ ✧❉✐s✲ t❛♥❝❡ ♠❡tr✐❝ ❧❡❛r♥✐♥❣✱ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❝❧✉st❡r✐♥❣ ✇✐t❤ s✐❞❡✲ Pr♦❝❡❡❞✐♥❣✳ ◆■P❙✬✵✷ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✶✺t❤ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ◆❡✉r❛❧ ■♥❢♦r♠❛t✐♦♥ Pr♦❝❡ss✐♥❣ ❙②st❡♠s✱ ✺✷✶✲ ✐♥❢♦r♠❛t✐♦♥✧✳ ✺✷✽✳ ...ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - TRẦN VĂN PHƢỢNG VỀ BÀI TOÁN TỐI ƢU TRONG HỌC ĐỘ TƢƠNG TỰ Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC... TỰ Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Nguyễn Thanh Sơn THÁI NGUYÊN - 2019 ✐✐✐ ử ỵ ữỡ ✶ ❇➔✐ t♦→♥ tè✐ ÷✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉... A>0 ♠❛ tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣ x∗ ✤✐➸♠ ❝ü❝ t✐➸✉ ❤❛② ❝ü❝ t✐➸✉ f (x∗ ) ❣✐→ tr ỹ t nìn ợ ữợ tự tữ ✈ỵ✐ ❆■ ✭❆rt✐❢✐❝❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ✲ ❚r➼ t✉➺ ♥❤➙♥ t↕♦✮ ✈➔ ■♦❚ ✭■♥t❡r♥❡t ♦❢ ❚❤✐♥❣s ✲ ■♥t❡r♥❡t ✈↕♥