(LUẬN văn THẠC sĩ) về toán tử chiếu metric lên tập lồi đóng và ứng dụng vào bài toán bất đẳng thức biến phân

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VÀNG VĂN HÀ VỀ TOÁN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐÓNG VÀ ỨNG DỤNG VÀO BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2020 download by : skknchat@gmail.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VÀNG VĂN HÀ VỀ TỐN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐĨNG VÀ ỨNG DỤNG VÀO BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC GS.TSKH Lê Dũng Mưu THÁI NGUYÊN - 2020 download by : skknchat@gmail.com ử ỵ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♥â✐ ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✐ ✐✐ ✶ ✸ ✶✳✶ ❚➟♣ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚♦→♥ tû ❝❤✐➳✉ ❦❤♦↔♥❣ ❝→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❈❤÷ì♥❣ ✷✳ Ù♥❣ ❞ư♥❣ ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✷✷ ✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ▼ët t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ download by : skknchat@gmail.com ✷✷ ỵ R t số tỹ Rn ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ∅ t➟♣ ré♥❣ ∀x ✈ỵ✐ ♠å✐ x tỗ t n x x x tỡ x x, y t ổ ữợ tỡ x ❝❤✉➞♥ ❝õ❛ ✈➨❝✲tì x ✈➔ y x V IP (F ; C) ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ S(F ; C) t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C) download by : skknchat@gmail.com ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ữợ sỹ ữợ ❝õ❛ ●❙✳❚❙❑❍✳ ▲➯ ❉ơ♥❣ ▼÷✉✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ỏ t ỡ s s tợ ữớ tớ t t ữợ ú ù t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ ❚❤➛② ❈ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✲❚✐♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ sốt tớ t t rữớ ỗ tớ tổ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ỗ t t ủ ♥❤➜t ❝❤♦ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✷✵✳ ❚→❝ ❣✐↔ ❱➔♥❣ ❱➠♥ ❍➔ download by : skknchat@gmail.com ✶ ▲í✐ ♥â✐ ✤➛✉ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ♣❤ê t❤ỉ♥❣✱ ❝❤ó♥❣ t❛ ✤➣ ❧➔♠ q✉❡♥ ✈ỵ✐ ♣❤➨♣ ❝❤✐➳✉ ✈✉ỉ♥❣ ❣â❝ ①✉è♥❣ ♠ët ♠➦t ♣❤➥♥❣ tr♦♥❣ ❦❤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ✈➔ ❧÷đ♥❣ ❣✐→❝✳ ❑❤→✐ ♥✐➺♠ ♥➔② ✤➣ ✤÷đ❝ ♠ð rë♥❣ ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♥❤✐➲✉ ❝❤✐➲✉✱ t❤➟♠ ❝❤➼ ✈ỉ ❤↕♥ ❝❤✐➲✉ ❝ị♥❣ ✈ỵ✐ t t ởt t ỗ õ ✈ỵ✐ ♠ët ❦❤♦↔♥❣ ❝→❝❤ ✭♠❡tr✐❝✮ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❒✲❝ì✲❧✐t✳ ⑩♥❤ ①↕ ❝❤✉②➸♥ ♠ët ✤✐➸♠ ❜➜t ❦ý ❝❤♦ trữợ tr ổ ởt tr ởt t trữợ ợ ọ t ữủ t♦→♥ tû ❝❤✐➳✉ ❧➯♥ t➟♣ ✤â✳ ◆❣÷í✐ t❛ ✤➣ ❝❤➾ r❛ r➡♥❣✱ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ t♦→♥ tû ❝❤✐➳✉ ởt t ỗ õ ữủ t tỷ t ỗ õ õ ✤➦❝ tr÷♥❣ t❤ó ✈à✱ ❞♦ ✤â ♥â ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ t♦→♥ ❤å❝ ✈➔ tỹ t ữ tr ỵ tt tố ữ ❤â❛✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❝➙♥ ❜➡♥❣ ✈➔ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ❜❛♦ ỗ tự ỡ t t ỗ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❒✲❝ì✲❧✐t Rn ✱ ❝→❝ ❦➳t q✉↔ ✈➲ t tỷ t ỗ õ t✐➳♣ t❤❡♦ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ →♣ ❞ö♥❣ t♦→♥ tû ❝❤✐➳✉ ✈➔♦ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ download by : skknchat@gmail.com ✷ ❝ù✉ tr♦♥❣ ữủ tr t ữỡ ợ t✐➯✉ ✤➲✿ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ✷✿ Ù♥❣ ❞ư♥❣ ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝→❝ ❝❤÷ì♥❣ ♥❤÷ s❛✉✿ r ữỡ tổ tr t ỗ ởt số t t ỡ t ỗ ỗ t tr ỵ t t ỗ ởt ữỡ tr ♥❣❤➽❛ t♦→♥ tû ❝❤✐➳✉✱ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉✳ ❈❤÷ì♥❣ ✷ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝õ❛ t♦→♥ tû ❝❤✐➳✉ ♠❡tr✐❝ ❧➯♥ ♠ët t ỗ õ t t t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧➔ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ q✉❛♥ trå♥❣ ❝õ❛ ●✐↔✐ t➼❝❤ ù♥❣ ❞ư♥❣✳ ❇➔✐ t♦→♥ ♥➔② ❧➔ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ tê♥❣ q✉→t t q ỗ ỡ ỳ t♦→♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤➲✉ õ t ổ t ữợ t t tự ❜✐➳♥ ♣❤➙♥✳ download by : skknchat@gmail.com ✸ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥✱ ỵ q t ỗ ỗ ởt ữỡ tr ự sỹ tỗ t t t ởt t ỗ õ ✈➔ ❦❤↔♦ s→t ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t♦→♥ tû ❝❤✐➳✉✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✸❪✳ ỗ ỗ rữợ t ú tổ ợ t t ỗ ởt số t➼♥❤ ❝❤➜t ❝➛♥ t❤✐➳t✳ ◆❤➢❝ ❧↕✐ r➡♥❣✱ ♠ët Rn ✤÷í♥❣ t❤➥♥❣ ❧➔ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✈➨❝✲tì ♥è✐ ❤❛✐ ✤✐➸♠ ✭❤❛✐ ✈➨❝✲tì✮ x ∈ Rn a, b tr♦♥❣ ❝â ❞↕♥❣ {x ∈ Rn |x = αa + βb, α, β ∈ Rn , α + β = 1} ✣♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ a ✈➔ b tr♦♥❣ Rn ❧➔ t➟♣ ❤đ♣ ❝→❝ ✈➨❝✲tì x ❝â ❞↕♥❣ {x ∈ Rn |x = αa + βb, α ≥ 0, β ≥ 0, α + β = 1} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët t➟♣ C ⊆ Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠å✐ ✤♦↕♥ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♥â✳ ❚ù❝ t ỗ C ự C ỗ ❝❤➾ ❦❤✐ download by : skknchat@gmail.com ✹ ∀x, y ∈ C, ∀λ ∈ [0, 1] ⇒ λx + (1 − λ)y ∈ C ❱➼ ❞ö ✶✳✶✳ ❛✮ ❚➟♣ ∅ ✈➔ t Rn t ỗ AB Rn ởt t ỗ trỏ ỗ ởt t ỗ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ X, Y tr♦♥❣ ❤➻♥❤ trá♥ ♥➡♠ trồ tr trỏ ỗ ữợ t ổ ỗ ✤÷í♥❣ ♥➨t ✤ùt ❝❤ù❛ ♥❤✐➲✉ ✤✐➸♠ ❦❤ỉ♥❣ ♥➡♠ tr♦♥❣ ❝→❝ t õ ổ ỗ õ x tờ ủ ỗ x1, , xk ♥➳✉ k k j λj x , λj > ∀j = 1, , k, x= j=1 λj = j=1 download by : skknchat@gmail.com ✺ ▼➺♥❤ ✤➲ ✶✳✶✳ ❚➟♣ ❤ñ♣ C ỗ õ ự tờ ủ ỗ õ ự C ỗ k k k ∈ N, ∀λ1 , , λk > : k λj = 1, ∀x , , x ∈ C ⇒ j=1 λj xj ∈ C j=1 ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✤õ ❧➔ ❤✐➸♥ ♥❤✐➯♥ tø ✤à♥❤ ♥❣❤➽❛✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ sè ✤✐➸♠✳ ❱ỵ✐ k = 2✱ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ s✉② r❛ ♥❣❛② tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t ỗ tờ ủ ỗ sỷ ✤ó♥❣ ✈ỵ✐ ●✐↔ sû x k−1 ✤✐➸♠✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ợ tờ ủ ỗ k k ✤✐➸♠✳ x1 , , x k ∈ C ✳ k k j λj x , λj > ∀j = 1, , k, x= ❚ù❝ ❧➔ j=1 λj = j=1 ✣➦t k−1 λj ξ= j=1 ❑❤✐ ✤â 0 x∗ = x∗ ∈ (1; 2) ✭✷✳✷✮ t❤➻ ✭✷✳✷✮ ❝❤➾ t❤➻ ✭✷✳✷✮ ❝❤➾ t❤ä❛ ♠➣♥ ✈ỵ✐ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trồ t ỹ t ỗ tr t ỗ t ũ t C ởt t ỗ õ rộ tr R n ởt ỗ F ❧➔ ♠ët →♥❤ ①↕ ✤✐ tø t➟♣ C ✈➔♦ Rn ✈➔ F (x) = f (x)✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (F ; C) tữỡ ữỡ ợ t ỹ tr P x ∈ C t❤ä❛ ♠➣♥ f (x∗) ≤ f (y) ∀y ∈ C ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✱ tù❝ ❧➔ f :C→R F (x∗ ), y − x∗ ≥ ∀y ∈ C download by : skknchat@gmail.com ✷✹ ❚❤❡♦ ❇ê f ỗ t õ f (y) − f (x∗ ) ≥ ▼➔ F (x) = f (x) ♥➯♥ f (x∗ ), y − x∗ , ∀y ∈ C f (x∗ ) ≤ f (y) ∀y ∈ C ✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❖P✮✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❖P✮✱ t❤❡♦ ✤✐➲✉ ❦✐➺♥ tè✐ ữ ỗ t õ ứ õ t s✉② r❛ − f (x∗ ) + NC (x∗ ) f (x∗ ) ∈ NC (x∗ ) ❤❛② −F (x∗ ) ∈ NC (x∗ )✳ ❚ù❝ ❧➔ −F (x∗ ), y − x∗ ≤ 0, ∀y ∈ C ⇔ F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❱➟② x∗ ❑❤✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ C V IP (F ; C) ởt õ ỗ tr ❦❤æ♥❣ ❣✐❛♥ Rn t❤➻ ❜➔✐ t♦→♥ V IP (F ; C) trð t❤➔♥❤ ❜➔✐ t♦→♥ ❜ò✿ ✭❈P✮ ❚➻♠ tr♦♥❣ ✤â x∗ ∈ C ✱ F (x∗ ) ∈ C s❛♦ ❝❤♦ C := {y ∈ C : x, y ≥ 0, ∀x ∈ C} F (x∗ ), x∗ = ❧➔ ♥â♥ ✤è✐ ♥❣➝✉ ❝õ❛ C✳ ❚❛ ❝â ♠➺♥❤ ✤➲ s C ởt õ ỗ ❝♦♠♣❛❝t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ R n t❤➻ ❜➔✐ t♦→♥ ❜ị P tữỡ ữỡ ợ t t tự ♣❤➙♥ V IP (F ; C)✱ ♥❣❤➽❛ ❧➔ t➟♣ ♥❣❤✐➺♠ ❤❛✐ ❜➔✐ t♦→♥ ♥➔② trò♥❣ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ V IP (F ; C)✱ tù❝ ❧➔ F (x∗ ), y − x∗ > download by : skknchat@gmail.com C õ ỗ x C ♥➯♥ y + x∗ ∈ C ✱ ∀y ∈ C ✳ ❚❤❛② y = y + x∗ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✮ t❛ ✤÷đ❝ F (x∗ ), y + x∗ − x∗ ≥ 0, ∀y ∈ C ⇔ F (x∗ ), y ≥ 0, ∀y ∈ C F (x∗ ) t❤✉ë❝ ♥â♥ ✤è✐ ♥❣➝✉ C ✳ y = x∗ ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✸✮ ❙✉② r❛ ❚❤❛② t❛ ✤÷đ❝ F (x∗ ), x∗ ≤ 0, ∀y ∈ C ❙✉② r❛ F (x∗ ), x∗ = ❤❛② x∗ ∈ C ✱ F (x∗ ) ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜ị ♣❤✐ t✉②➳♥ ✭❈P✮✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ x∗ ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜ò ✭❈P✮ t❤➻ F (x∗ ), x∗ = 0, F (x∗ ) ∈ C ❱➻ F (x∗ ) ∈ C ♥➯♥ F (x∗ ), y ≥ 0, ∀y ∈ C ✳ ❚❛ ❝â F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❤❛② x∗ ∈ C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✳ ✷✳✷ ▼ët t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ C R ❈❤♦ F : C Rn ởt t ỗ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❑❤✐ ✤â✱ →♥❤ ①↕ F ❧➔ ✣ì♥ tr C tỗ t ởt số γ > s❛♦ ❝❤♦ F (x) − F (y), x − y ≥ γ x − y , ∀x, y ∈ C; ✭✐✐✮ ✣ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ F (x) − F (y), x − y ≥ 0, ∀x, y ∈ C; download by : skknchat@gmail.com n ✈➔ ✷✻ ✭✐✐✐✮ ●✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ F (x), y − x ≥ ⇒ F (y), y − x ≥ 0, ∀x, y ∈ C ✭✐✈✮ P❛r❛✲✤ì♥ ✤✐➺✉ tr➯♥ C ♥➳✉ x∗ ∈ S(F, C), x ∈ C, F (x), x∗ −x = 0, F (x∗ ), x−x∗ = ⇒ x ∈ S(F, C) ◆❤➟♥ ①➨t ✷✳✶✳ ❱➼ ❞ö ✷✳✷✳ ❚❛ ❝â ✭✐✮ F ❛✮ ❈❤♦ →♥❤ ①↕ ⇒ ✭✐✐✮✱ ✭✐✐✮ ⇒ ✤ì♥ trà ①→❝ ✤à♥❤ tr➯♥ ✭✐✐✐✮ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ R ♥❤÷ s❛✉✿ F (x) = 2x, ∀x ∈ R ✈ỵ✐ F (x) ❧➔ ✤↕♦ ❤➔♠ ỗ t r F ỡ ✤✐➺✉ tr➯♥ tr➟♥ ✈✉ỉ♥❣ ❝ï n × n✳ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤✐ Q Q R✳ ❜✮ ❈❤♦ f ①→❝ ✤à♥❤ tr➯♥ F (x) = Qx✱ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ t❛ t❤➜② F R✳ ❑❤✐ ✤â ❞➵ tr♦♥❣ ✤â Q ❧➔ ♠❛ ❧➔ ✤ì♥ ✤✐➺✉ tr➯♥ t♦➔♥ ❧➔ ♠❛ tr➟♥ ✈✉ỉ♥❣✱ ✤è✐ ①ù♥❣✱ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ◆➳✉ ❧➔ ✤è✐ ①ù♥❣✱ ①→❝ ✤à♥❤ ữỡ t x2 ởt ỗ tr C t❤➻ f F ✤ì♥ ✤✐➺✉ ♠↕♥❤✳ ❚ê♥❣ q✉→t ❤ì♥ ❧➔ ỡ tr C ú ỵ ổ t tỷ ỡ ỗ ❚❛ ❝â ❝→❝ ❜ê ✤➲ s❛✉ s➩ ❝➛♥ ✤➸ sü ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ❝õ❛ t❤✉➟t t♦→♥✳ ❇ê ✤➲ ✷✳✶✳ ●✐↔ sû {νk } ✈➔ {δk } ❧➔ ❤❛✐ ❞➣② sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ νk+1 ≤ νk + δk ✈ỵ✐ +∞ k=1 δk < +∞✳ ❑❤✐ ✤â ❞➣② {νk } ❤ë✐ tö✳ ❇ê ✤➲ ✷✳✷✳ ❬✹❪ ●✐↔ sû θ✱ β ✈➔ ξ ❧➔ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ θ2 − βθ − ξ ≤ 0✱ ❦❤✐ ✤â ❬✹❪ βθ ≤ β + ξ download by : skknchat@gmail.com ✭✷✳✹✮ ✷✼ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè ❜➟❝ ❤❛✐ s(θ) = θ2 − βθ − ξ ✱ ❦❤✐ ✤â s(θ) ≤ s✉② r❛ θ≤ ✈➻ β+ β + 4ξ , θ > β ◆❤➙♥ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ ✈➔ →♣ ❞ư♥❣ t➼♥❤ ❝❤➜t a2 + b ab ≤ t❛ ✤÷đ❝ βθ ≤ 2−1 β + β ≤2 −1 β + 4ξ β + β + 4ξ β + 2 = 2−1 β + β + 2ξ = β + ξ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö t❛ s➩ ❣✐↔ sû t➟♣ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✮ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ s❛✉ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ F (y), x∗ − y ≤ 0, ∀y ∈ C ❚➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ✤÷đ❝ ỵ Sd (F ; C) t t ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✳ ❈❤♦ t❤❛♠ sè ❞÷ì♥❣ ρ ✈➔ ❝→❝ ❞➣② sè t❤ü❝ {ρk } ✈➔ {βk } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ρk > ρ, βk = +∞, ρk βk > 0, ∀k ∈ N, ✭✷✳✻✮ βk2 < +∞, ✭✷✳✼✮ download by : skknchat@gmail.com ✷✽ ❱➼ ❞ư t❛ ❧➜② ρk = ✈ỵ✐ ♠å✐ k ✱ k = Pữỡ ữợ ữợ ữợ ✷✿ ♥❣❤➽❛ ❧➔ ❈❤å♥ x0 ∈ C ✳ ●✐↔ sû xk ∈ C ✳ βk γk ✈ỵ✐ m > 0✳ k = 0✳ ▲➜② g k = F (xk )✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ γk = max{ρk , g k } ✭✷✳✽✮ αk g k + xk+1 − xk , x − xk+1 ≥ 0, ∀x ∈ C, ✭✷✳✾✮ αk = ❚➼♥❤ ✣➦t m k+1 xk+1 ∈ C tr♦♥❣ ✤â s❛♦ ❝❤♦ xk+1 = PC (xk − αk g k )✳ ứ t t s ứ t ữợ k xk+1 = xk õ sỡ ỗ tt t s ỡ ỗ tt t download by : skknchat@gmail.com gk = ❤❛② ✷✾ ▼➺♥❤ ✤➲ tt t ữợ s r ♠ët ❞➣② ❤ú✉ ❤↕♥ t❤➻ ✤✐➸♠ ❝✉è✐ ❝ò♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ V IP (F ; C)✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ gk ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ =0 F (xk ), y − xk = t❤➻ ✈ỵ✐ ♠å✐ xk+1 = PC (xk − αk F (xk ))✱ t❤➻ tø ✈➟② xk ❧➔ V IP (F ; C)✳ ❇➙② ❣✐í sỷ tt t t tú t ữợ ❧➔ xk = xk+1 y✱ xk = xk+1 ✳ ◆➳✉ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉✱ t❛ ❝â xk+1 − (xk − αk F (xk )), y − xk ≤ 0, ∀y ∈ C ❉♦ xk = xk+1 ✈➔ αk > 0✱ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ trð t❤➔♥❤ F (xk )), y − xk ≥ ∀y ∈ C, ♥❣❤➽❛ ❧➔ xk ❧➔ ♠ët ♥❣❤✐➺♠✳ ❚ø ❣✐í trð ✤✐✱ ❝❤ó♥❣ t❛ ❣✐↔ sû t❤✉➟t t♦→♥ s✐♥❤ r❛ ♠ët ❞➣② ổ ữủ ỵ {xk } õ t➼♥❤ ❝❤➜t s❛✉✳ ❇ê ✤➲ ✷✳✸✳ ❱ỵ✐ ♠é✐ k✱ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤ó♥❣ ✭✐✮ ✭✐✐✮ αk g k ≤ βk ❀ βk xk+1 − xk ≤ βk2 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚ø ✭✷✳✽✮ t❛ ❝â αk g ✭✐✐✮ ❇➡♥❣ ❝→❝❤ ❧➜② x = xk k βk g k = ≤ βk max{ρk , g k } tr♦♥❣ ✭✷✳✾✮ t❛ ✤÷đ❝ xk+1 − xk ≤ αk g k , xk − xk+1 download by : skknchat@gmail.com ✭✷✳✶✵✮ ✸✵ ≤ αk g k xk+1 − xk ✭✷✳✶✶✮ ≤ βk xk+1 − xk ❉♦ ✤â✱ tø ❇ê ✤➲ ✷✳✷ ✈ỵ✐ k∈N θ = xk+1 − xk , β = βk ξ = 0✱ ✈➔ ✈ỵ✐ ♠é✐ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ t❤✐➳t t✐➳♣ t❤❡♦ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ s❛✉ ♥➔②✳ ❆✶✳ ❚➟♣ ♥❣❤✐➺♠ S(F ; C) ❦❤→❝ ré♥❣❀ ▼➺♥❤ ✤➲ ✷✳✹✳ ●✐↔ sû ❆✶ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∗ ∈ S(F ; C) ✈ỵ✐ ♠é✐ k✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ xk+1 − x∗ ≤ xk − x∗ + 2αk F (xk ), x∗ − xk + δk , ✈➔ ✭✷✳✶✷✮ tr♦♥❣ ✤â δk = 2βk2✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✤ì♥ ❣✐↔♥✱ t❛ ❝â xk+1 − x∗ = xk − x∗ − xk+1 − xk ≤ xk − x∗ + xk − xk+1 , x∗ − xk+1 ❑➳t ❤ñ♣ ✭✷✳✶✸✮ ✈➔ ✭✷✳✾✮ ✈ỵ✐ xk+1 − x∗ x = x∗ + xk − xk+1 , x∗ − xk+1 ✭✷✳✶✸✮ t❛ s✉② r❛ ≤ xk − x∗ + αk g k , x∗ − xk+1 = xk − x∗ + αk g k , x∗ − xk ✭✷✳✶✹✮ + αk g k , xk − xk+1 ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③ ✈➔ ❇ê ✤➲ ✷✳✸ ✭✐✮✱ s✉② r❛ xk+1 − x∗ ≤ xk − x∗ + 2αk g k , x∗ − xk + 2βk xk − xk+1 (2.15) ❚❤❡♦ ✭✷✳✶✺✮ ✈➔ ❇ê ✤➲ ✷✳✸ ✭✐✐✮✱ t❛ ❝â xk+1 − x∗ ≤ xk − x∗ + 2αk F (xk ), x∗ − xk + 2βk2 download by : skknchat@gmail.com ✭✷✳✶✻✮ ✸✶ ❉♦ ✤â✱ ✈➻ αk > ♥➯♥ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ❞➣② {xk } ✤÷đ❝ s✐♥❤ ❜ð✐ t❤✉➟t t♦→♥✳ ❆✷✳ S(F ; C) ⊆ Sd (F ; C) ú ỵ r F tử ỡ t tt ú ỵ ●✐↔ sû ❆✶ ✈➔ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ ❧➔ ❞➣② ❤ë✐ tư ✈ỵ✐ ♠å✐ x∗ ∈ S(F ; C)❀ k ✭✐✐✮ {x } ❧➔ ❞➣② ❜à ❝❤➦♥✳ ✭✐✮ { xk − x∗ } ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû x∗ ∈ S(F ; C) ✈➔ k ∈ N✳ ❚❤❡♦ ❆✷ t❛ ❝â xk ≤ F (xk ), x∗ − ❝ị♥❣ ✈ỵ✐ ▼➺♥❤ ✤➲ ✷✳✹ s✉② r❛ xk+1 − x∗ tr♦♥❣ ✤â ≤ xk − x∗ + δk , ✭✷✳✶✼✮ δk = 2βk2 ❉♦ ✤â✱ t❤❡♦ ✭✷✳✼✮ ✈➔ ✭✷✳✽✮ t❛ ❝â +∞ δk < +∞ ✭✷✳✶✽✮ k=0 ❉♦ ✤â✱ tø ✭✷✳✶✼✮✱ ✭✷✳✶✽✮ ✈➔ ❇ê ✤➲ ✷✳✸ s✉② r❛ { xk − x∗ } ❧➔ ♠ët ❞➣② ❤ë✐ tö✳ ✭✐✐✮ ❙✉② r❛ tø ✭✐✮✳ ✣à♥❤ ỵ sỷ F tử t❤✐➳t ❆✶✱ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ t❛ ❝â lim sup F (xk ), x∗ − xk = ∀x∗ ∈ S(F ; C) k→+∞ download by : skknchat@gmail.com ✸✷ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x∗ ∈ S(F ; C)✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✹ ✈➔ ❆✷ s✉② r❛ ≤ 2αk [− F (xk ), x∗ − xk ] ≤ xk − x∗ − xk+1 − x∗ + δk ✭✷✳✶✾✮ ❉♦ ✤â✱ m αk [− F (xk ), x∗ − xk ] 0≤2 k=0 m ≤ x −x ∗ − x m+1 −x ∗ δk + ✭✷✳✷✵✮ k=0 m ≤ x0 − x∗ δk + k=0 ❑❤✐ m → +∞ t❛ ❝â +∞ +∞ k ∗ k ∗ αk [− F (x ), x − x ] ≤ x − x 0≤2 + δk , ✭✷✳✷✶✮ k=0 k=0 ❦➳t ❤ñ♣ ✈ỵ✐ ✭✷✳✶✽✮ s✉② r❛ +∞ αk [− F (xk ), x∗ − xk ] < +∞ 0≤ ✭✷✳✷✷✮ k=0 ▼➦t ❦❤→❝✱ t❛ ❝â { gk } ❧➔ ❞➣② ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ γk L = max{1, ρ−1 gk } ≤ k ρk ρ ∀k ∈ N ❉♦ ✤â αk = βk ρ βk ≥ γk L ρk ∀k ∈ N ✭✷✳✷✸✮ ❚ø ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮✱ t❛ ❝â +∞ k=0 βk [− F (xk ), x∗ − xk ] < +∞ ρk ❱➟②✱ tø ✭✷✳✷✹✮ ✈➔ ✭✷✳✼✮ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ download by : skknchat@gmail.com ✭✷✳✷✹✮ ✸✸ ✣➸ ❝â ✤÷đ❝ sü ❤ë✐ tư ❝õ❛ ❝↔ ❞➣② ❝❤ó♥❣ t❛ ✤÷❛ r❛ ❣✐↔ t❤✐➳t s❛✉✳ ❆✸✳ ●✐↔ sû x∗ ∈ S(F ; C) ✈➔ x¯ ∈ C ✳ ◆➳✉ F (¯ x), x∗ − x¯ = F (x∗ ), x¯ − x∗ = t❤➻ x¯ ∈ S(F ; C)❀ ✣à♥❤ ỵ sỷ tọ ♠➣♥✳ ❑❤✐ ✤â✱ ❞➣② {x } ❤ë✐ k tö ✤➳♥ ♠ët ♥❣❤✐➺♠ ❝õ❛ V IP (F ; C)✳ ❈❤ù♥❣ ♠✐♥❤✳ sỷ x S(F ; C) ỵ tỗ t ởt {xkj } {xk } s❛♦ ❝❤♦ lim sup F (xk ), x∗ − xk = lim F (xkj ), x∗ − xkj r ỵ t õ {xkj } {xkj } x¯ ∈ C ❧➔ ❞➣② ❜à ❝❤➦♥✳ ❱➟②✱ ❝â ❦❤ỉ♥❣ ♠➜t tê♥❣ q✉→t✱ ❝ư t❤➸ ❧➔ {xkj } lim xkj = x¯ F ✈➔ ♠ët ❞➣② s❛♦ ❝❤♦ j→+∞ ❉♦ ✭✷✳✷✺✮ j→+∞ k→+∞ ✭✷✳✷✻✮ ❧✐➯♥ tö❝✱ ♥➯♥ F (¯ x), x∗ − x¯ = lim F (xkj ), x∗ − xkj j→+∞ ✭✷✳✷✼✮ = ❚ø ❣✐↔ t❤✐➳t ❆✷ t❛ ❝â F (¯ x), x∗ − x¯ ≤ 0✱ ❞â ✤â t❛ ❝â F (¯ x), x∗ − x¯ = ❉♦ ✤â✱ t❛ s✉② r❛ { xk − x¯ } x¯ ∈ S(F ; C)✳ ⑩♣ ỵ ỳ t ữủ tư✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✷✻✮ s✉② r❛ lim xk = x¯, k→+∞ ✭✷✳✷✽✮ x¯ ∈ S(F ; C) download by : skknchat@gmail.com ✸✹ ❱➼ ❞ö ✷✳✸✳ ❈❤♦  ❈❤♦ F : R2 → R2 ①→❝ ✤à♥❤ ❜ð✐✿ F (x) = Ax ✈ỵ✐ A =  0  ✳ C = {x = (x1 , x2 ) ∈ R2 : x ≤ 1}✳ ❱ỵ✐ ♠å✐ x = (x1 , x2 ) ∈ C ✱ ✈ỵ✐ ♠å✐ y = (y1 , y2 ) ∈ C t❛ ❝â✿ F (x) − F (y), x − y = A(x − y), x − y = 2(x1 − y1 )(x1 − y1 ) + 2(x2 − y2 )(x2 − y2 ) = 2(x1 − y1 )2 + 2(x2 − y2 )2 = x − y ❉♦ ✤â F ❧➔ ✤ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈ỵ✐ γ = 2✳ ❚❛ ❝â F (x) − F (y) = A(x − y) = x − y ❉♦ ✤â F ❧➔ ✷ ✲ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ tr➯♥ ❚ø t➼♥❤ ✤ì♥ ✤✐➺✉ ♠↕♥❤ ❝õ❛ ❚➻♠ F C✳ s✉② r❛ ❜➔✐ t♦→♥✿ x∗ ∈ C ✿ F (x∗ ), y − x∗ ≥ 0, ∀y ∈ C ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ❉➵ t❤➜② r➡♥❣ x∗ = (0, 0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥✳ ❈❤å♥ ❝→❝ t❤❛♠ sè t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t ρk = 1, βk = , ∀k ≥ k+1 Pữỡ ữợ õ   x∈C    g k = A(xk ) = (2xk1 2xk2 )      xk+1 = PC (xk − αk g k ) download by : skknchat@gmail.com ✸✺ tr♦♥❣ ✤â αk = βk γk ✈ỵ✐ PC (x) = γk = max{ρk , g k }✳ P❤➨♣ ❝❤✐➳✉ tr➯♥    x   0 + (x − 0) x−0 ♥➳✉ x ≤1 ♥➳✉ x > C ❝â ❞↕♥❣ ❍❛② PC (x) = ❱ỵ✐ ✤✐➸♠ ❜❛♥ ✤➛✉ x0 =    x x    x 1 , 2 ♥➳✉ x ≤1 ♥➳✉ x > ✳ ▲➟♣ tr➻♥❤ tr➯♥ ▼❛t❧❛❜ t❛ ❝â ❜↔♥❣ ❦➳t q✉↔ s❛✉✿ k xk1 xk2 xk − x∗ ✶ ✲✵✳✷✵✼✶✵✻✼✽✶ ✲✵✳✷✵✼✶✵✻✼✽✶ ✵✳✷✾✷✽✾✸✷✶✽ ✷ ✵✳✵✻✾✵✸✺✺✾✸ ✵✳✵✻✾✵✸✸✺✺✾✸ ✵✳✵✾✼✻✸✶✵✼✷✾ ✸ ✵ ✵ ✵ download by : skknchat@gmail.com ✸✻ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t♦→♥ tỷ tr t ỗ õ ự ✈➔♦ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉✳ ❈ö t❤➸ ❧➔✿ ✶✳ ◆❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ t t ỡ t ỗ ỗ ỵ t t ỗ ợ t ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ❧➯♥ ởt t ỗ õ ổ tự t ❝õ❛ ♠ët ✤✐➸♠ ❧➯♥ ❝→❝ t➟♣ ✤➦❝ ❜✐➺t ♥❤÷ ♥û❛ ❦❤ỉ♥❣ ❣✐❛♥✱ ❤➻♥❤ ❝➛✉ ✤â♥❣ ❤❛② s✐➯✉ ❤ë♣✱✳✳✳ ✸✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♥➯✉ ♠è✐ ❧✐➯♥ q✉❛♥ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ✈ỵ✐ ❜➔✐ t♦→♥ ỹ t ỗ tr t ỗ t ❜ị ♣❤✐ t✉②➳♥✳ ✹✳ ❙û ❞ư♥❣ ♣❤➨♣ ❝❤✐➳✉ ✤➸ ①➙② ỹ tt t ữợ t ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ♣❛r❛✲✤ì♥ ✤✐➺✉✳ download by : skknchat@gmail.com ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❍✐➲♥✱ ▲➯ ❉ơ♥❣ ▼÷✉✱ ◆❣✉②➵♥ ❍ú✉ ✣✐➸♥ ✭✷✵✶✺✮✱ tr➻♥❤ t ỗ ự ố ❍➔ ◆ë✐✳ ●✐→♦ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ■✳ ❑♦♥♥♦✈ ✭✷✵✶✶✮✱ ❈♦♠❜✐♥❡❞ ❘❡❧❛①❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❢♦r ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s✱ ❙♣r✐♥❣❡r✳ ❬✸❪ ❍♦❛♥❣ ❚✉② ✭✷✵✶✸✮✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❙♣r✐♥❣❡r✳ ❬✹❪ P✳ ❙❛♥t♦s ❛♥❞ ❙✳ ❙❝❤❡✐♠❜❡r❣ ✭✷✵✶✶✮✱ ✏❆♥ ✐♥❡①❛❝t s✉❜❣r❛❞✐❡♥t ❛❧❣♦✲ r✐t❤♠ ❢♦r ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s✑✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤✲ ❡♠❛t✐❝s✱ ✸✵✱ ♣♣✳ ✾✶✲✶✵✼✳ download by : skknchat@gmail.com ...  - VÀNG VĂN HÀ VỀ TOÁN TỬ CHIẾU METRIC LÊN TẬP LỒI ĐÓNG VÀ ỨNG DỤNG VÀO BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI... tø ✭✷✳✶✼✮✱ ✭✷✳✶✽✮ ✈➔ ❇ê ✤➲ ✷✳✸ s✉② r❛ { xk − x∗ } ❧➔ ♠ët ❞➣② ❤ë✐ tö✳ ✭✐✐✮ ❙✉② r❛ tứ ỵ sỷ F tử ✈➔ ❝→❝ ❣✐↔ t❤✐➳t ❆✶✱ ❆✷ ✤➲✉ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â✱ t❛ ❝â lim sup F (xk ), x∗ − xk = ∀x∗ ∈ S(F ; C)... x¯ ≤ 0✱ ❞â ✤â t❛ ❝â F (¯ x), x∗ − x¯ = ❉♦ ✤â✱ t❛ s✉② r❛ { xk − x¯ } x¯ ∈ S(F ; C) ỵ ỳ t ữủ tử t ủ ợ s r lim xk = x¯, k→+∞ ✭✷✳✷✽✮ x¯ ∈ S(F ; C) download by : skknchat@gmail.com ✸✹ ❱➼ ❞ö ✷✳✸✳