ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - VŨ VIỆT BÌNH ĐIỀU KIỆN CẦN VÀ ĐỦ CHO TỰA NGHIỆM HỮU HIỆU YẾU CỦA BÀI TOÁN TỐI ƯU ĐA MỤC TIÊU KHÔNG TRƠN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2020 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - VŨ VIỆT BÌNH ĐIỀU KIỆN CẦN VÀ ĐỦ CHO TỰA NGHIỆM HỮU HIỆU YẾU CỦA BÀI TOÁN TỐI ƯU ĐA MỤC TIÊU KHƠNG TRƠN Chun ngành: Tốn ứng dụng Mã số: 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC GS.TS Đỗ Văn Lưu THI NGUYấN - 2020 ử ỵ ✤➛✉ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✷ ữợ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳ ◆â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉ ✽ ✷✳✶✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸✳ ✣✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸ ✣è✐ ♥❣➝✉ ✷✺ ✸✳✶✳ ✣è✐ ♥❣➝✉ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷✳ ✣è✐ ♥❣➝✉ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✾ ✸✶ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ r t tổ ữợ sỹ ữợ ❞➝♥ ❦❤♦❛ ❤å❝ ❝õ❛ ●❙✳ ❚❙✳ ✣é ❱➠♥ ▲÷✉✳ ❈→❝ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉✱ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② tr tỹ ữ tứ ổ ố ữợ t ý tự trữợ r tr ✈➠♥ tỉ✐ ❝â sû ❞ư♥❣ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ✤➲✉ ❝â tr➼❝❤ ❞➝♥ ✈➔ ❝❤ó t ỗ ố t t ý sỹ ❧➟♥ ♥➔♦ tæ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ❱ơ ❱✐➺t ❇➻♥❤ ✐✐ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ù t t ữớ ữợ ộ ❱➠♥ ▲÷✉✳ ❚ỉ✐ ❝ơ♥❣ ♠✉è♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❑❤♦❛ ❚♦→♥✲❚✐♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ tèt ❧✉➟♥ ✈➠♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ ❜↔♥ t❤➙♥ t→❝ ❣✐↔ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝â t❤➸ ❝â ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ố ữủ ỵ ỗ õ õ ✈➔ ①➙② ❞ü♥❣ ❝õ❛ ❝→❝ t❤➛② ❝æ✱ ✈➔ ❝→❝ ❜↕♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✵ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ❱ô ❱✐➺t ❇➻♥❤ ✶ ỵ coM ỗ t M coM coneM ỗ õ t M M ỹ ❝õ❛ M Ms X∗ ❝ü❝ ➙♠ ❝❤➦t ❝õ❛ M T (M, x) TC (M, x) ♥â♥ t✐➳♣ ❧✐➯♥ ❝õ❛ M t↕✐ x N (M, x) f − (x, d) ♥â♥ t r M t x õ ỗ s r❛ ❜ð✐ M ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ tæ ♣æ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ X ♥â♥ t✐➳♣ t✉②➳♥ ❈❧❛r❦❡ ❝õ❛ M t↕✐ x ữợ f t x t ♣❤÷ì♥❣ d f + (x, d) f (x, d) ✤↕♦ ❤➔♠ ❉✐♥✐ tr➯♥ ❝õ❛ f t↕✐ x t❤❡♦ ♣❤÷ì♥❣ d C f (x) f (x) ữợ r ❝õ❛ f t↕✐ x t✳ ÷✳ t÷ì♥❣ ù♥❣ KT KT V CP ❑✉❤♥✲❚✉❝❦❡r ✤↕♦ ❤➔♠ s✉② rë♥❣ ❈❧❛r❦❡ ❝õ❛ f t x t ữỡ d ữợ ỗ f t x tợ tỡ r ✷ ▼ð ✤➛✉ ✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❑❤✐ t➼♥❤ t♦→♥ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✱ s❛✉ ♠ët số ỳ ữợ tt t tố ữ ❝❤♦ t❛ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ①➜♣ ①➾✳ ❱➻ ✈➟② ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ①➜♣ ①➾ ❧➔ r➜t ❝➛♥ t❤✐➳t✳ ❚ø ✤â ❞➝♥ ✤➳♥ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ●♦❧❡st❛♥✐✕❙❛❞❡❣❤✐✕❚❛✈❛♥ ✭✷✵✶✼✮ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❑✉❤♥✲ ❚✉❝❦❡r ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✭✇❡❛❦ q✉❛s✐ ❡❢❢✐❝✐❡♥t s♦❧✉t✐♦♥✮ ✈➔ tü❛ ♥❣❤✐➺♠ ỳ qs t st ỵ ố ♥❣➝✉ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t tố ữ t ợ st ữỡ q ữợ r ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ✸✽✭✷✵✶✼✮✱ ✽✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐ ♥❣➝✉ ②➳✉✱ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝✳ ✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ỗ ữỡ t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ ✈ỵ✐ t✐➯✉ ✤➲✿✧❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✧ tr➻♥❤ ❜➔② ♠ët số tự ỡ ữợ r ♥â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡✳ ❈❤÷ì♥❣ ✷ ✈ỵ✐ t✐➯✉ ✤➲✿ ✧✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tố ữ tr t q ự ợ ✤➙② ❝õ❛ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ✸ ✸✽✭✷✵✶✼✮✱ ✻✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐ ♥❣➝✉ ②➳✉✱ ố ữủ ữỡ ợ t ố tr ỵ ố ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✺ t❤→♥❣ ✸ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❱ơ ❱✐➺t ❇➻♥❤ ✹ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè tự ỡ ữợ r õ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ❈→❝ ❦✐➳♥ t❤ù❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ữợ r sỷ x = (x1 , , x ) ✈➔ y = (y1 , , y ) ❧➔ ❤❛✐ ✈❡❝tì tr♦♥❣ R ✳ ❈→❝ ❦➼ ❤✐➺✉ s❛✉ ✤➙② s➩ ✤÷đ❝ sû ❞ư♥❣ s❛✉ ♥➔②✿ x = y, ♥➳✉ xi = yi , ✈ỵ✐ ♠å✐ i, x y, ♥➳✉ xi ≤ yi , ✈ỵ✐ ♠å✐ i, x < y, ♥➳✉ xi < yi , ✈ỵ✐ ♠å✐ i, x ≤ y, ♥➳✉ x y ✈➔ x = y ●✐↔ sû M ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ R ✳ ❚❤ỉ♥❣ t❤÷í♥❣✱ ❝❧ M ✱ ✐♥t M ✱ ❝♦(M ) ✈➔ ❝♦♥❡ (M ) ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ❜❛♦ ✤â♥❣✱ ♣❤➛♥ tr♦♥❣✱ ỗ õ s M tữỡ ự ỹ ➙♠ ✈➔ ❝ü❝ ➙♠ ❝❤➦t ❝õ❛ M ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ M − := ξ ∈ R ξ, ν ≤ 0, ∀ν ∈ M , M s := ξ ∈ R ξ, ν < 0, ∀ν ∈ M , tr♦♥❣ õ Ã, à t ổ ữợ tr R ❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❦➼ ❤✐➺✉ t❤ỉ♥❣ t❤÷í♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥ ✭①❡♠ ❬✷❪✮✳ ✺ ●✐↔ sû ϕ : R → R ❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶ ✣↕♦ ❤➔♠ t❤❡♦ ♣❤÷ì♥❣ s✉② rë♥❣ ✭❣❡♥❡r❛❧✐③❡❞ ❞✐r❡❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡✮ ❝õ❛ ϕ t↕✐ x t❤❡♦ ♣❤÷ì♥❣ ν ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ϕ◦ (x; ν) = lim sup y→x,t↓0 ϕ(y + tν) − ϕ(y) t ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ữợ r rs srt t x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ∂C ϕ(x) = {ξ ∈ R | ξ, ν ≤ ϕ◦ (x; ν) ∀ν ∈ R } ❈❤➥♥❣ ❤↕♥✱ ❤➔♠ f (x) = x − x0 ổ t x0 ữợ r ❝õ❛ ♥â t↕✐ x0 ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ B[0, 1] := B tr♦♥❣ R ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸ ữợ ỗ : R R t↕✐ x ∈ R ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ ∂ϕ(x) = {ξ ∈ R : ξ, x − x ≤ ϕ(x) − ϕ(x)} ❚❛ ❜✐➳t r➡♥❣ →♥❤ ①↕ ν (x; ) ởt ỗ ữợ õ t t ỗ t = tỗ t ữủ ❧➔ ∂ϕ◦ (x; ·)(0) ✈➔ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✿ ∂C ϕ(x) = ∂ϕ◦ (x; ·)(0) ❙❛✉ ✤➙② ❧➔ ♠ët số t t ữợ r ởt ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳ ❇ê ✤➲ ✶✳✶ ❬✷❪ ●✐↔ sû ϕ, ψ : R ❧➔ ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x ∈ R ✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✿ ✐✮ ∂C ϕ(x) ❧➔ t rộ t ỗ R ✐✐✮ ❱ỵ✐ ♠å✐ ν ∈ R ✱ ϕ◦ (x; ν) = max{ ξ, ν |ξ ∈ ∂C ϕ(x)}✳ ✐✐✐✮ ❱ỵ✐ ❜➜t ❦➻ sè λ, ∂C λϕ(x) = λ∂C ϕ(x) ✐✈✮ ❍➔♠ ν → ϕ◦ (x; ν) ❧➔ ❤ú✉ ❤↕♥✱ t❤✉➛♥ t ữỡ ữợ t t tr R ỗ t õ ∂C (ϕ + ψ)(x) = ∂C ϕ(x) + ∂C ψ(x) →R ✶✼ ❙❛✉ ✤➙②✱ t❛ tr➻♥❤ ❜➔② ❤❛✐ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ♠↕♥❤ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉✳ ❱➼ ❞ư ✷✳✷ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿ (P 2) f (x) = (x2 − 2x, −2x) ✈ỵ✐ r➔♥❣ ❜✉ë❝✿ x x≤0 g(x) = , −x x > x x ≤ h(x) = , 0 x > x ∈ Q = [0, 1] ❇ð✐ ✈➻ ✈ỵ✐ α = (2, 2) ❦❤ỉ♥❣ tỗ t x S s ú t x0 = 0✱ ♥➯♥ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭✤à❛ ♣❤÷ì♥❣✮ t❤❡♦ α = (2, 2)✳ ❉➵ ❦✐➸♠ tr❛ r➡♥❣ x0 ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣✳ ❇➡♥❣ ♠ët t➼♥❤ t♦→♥ ✤ì♥ ❣✐↔♥✱ t❛ ❝â ∂C f1 (0) = ∂C f2 (0) = {−2}, ∂C g(0) = [−1, 1], ∂C h(0) = [0, 1], ∂C (−h)(0) = [−1, 0], N (Q, 0) = (−∞, 0] ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ✈ỵ✐ i = 1, 2✱ Di ✤â♥❣ ✈➔ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α)✳ ❇➡♥❣ ❝→❝❤ ❧➜② λ1 = λ2 = µ = ν = e = 1✱ ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✷✮ t❤ä❛ ♠➣♥✳ ❱➼ ❞ö ✷✳✸ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿ (P 3) f (x) = (f1 (x), f2 (x)) ✈ỵ✐ r➔♥❣ ❜✉ë❝✿ x ∈ S = {x ∈ R|g(x) ≤ 0, h(x) = 0, x ∈ Q}, tr♦♥❣ ✤â Q = {x ∈ R : |x| ≤ 2} ✈➔ g, h, fi : R → R✱ i = 1, ✤÷đ❝ ❝❤♦ ❜ð✐ x2 − (x − 1), x ≥ − (x − 1), x ≥ 2 f1 (x) = , f2 (x) = , x, x tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x ∈ B(x0 , δ) ∩ X ✱ t❛ ❝â ξ, x − x0 + α x − x0 ≥ ✈ỵ✐ ξ ∈ ∂C ϕ(x0 ) ❦➨♦ t❤❡♦ ϕ(x) ≥ ϕ(x0 ) − x x0 tữỡ ữỡ ợ ϕ(x) < ϕ(x0 ) − α x − x0 ❦➨♦ t❤❡♦ ξ, x − x0 < −α x − x0 , ∀ξ ∈ ∂C ϕ(x0 ) ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ỗ ỗ t x0 ✳ ❚✉② ♥❤✐➯♥✱ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥✱ t❛ ①➨t ϕ(x) = −x2 − 2x, x ∈ X = [−1, 0] ❑❤✐ ✤â ϕ ❧➔ ỗ ữ ổ ỗ t x0 = ❙❛✉ ✤➙②✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❤❛✐ ❧ỵ♣ ❤➔♠ ✈➔ s➩ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ✤✐➸♠ ❑✉❤♥✲❚✉❝❦❡r ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❤♦➦❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ✭▼P✮✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✻ ●✐↔ sû x0 ∈ R ✳ ❇➔✐ t♦→♥ ✭▼P✮ ✤÷đ❝ ❣å✐ ❧➔ ❛✮ ❆❢❢✐♥❡ ❣✐↔ ỗ t strt rt s t x0 ợ t(Rm + ) tỗ t δ > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ B(x0 , δ) ✈ỵ✐ f (x) ≤ f (x0 ) − α x − x0 ❦➨♦ t❤❡♦ ❤➺ ✭✷✳✶✶✮ ✤ó♥❣✳ ❜✮ ỗ rt s t x0 ợ t(Rm + ) tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ B(x0 , δ) ✈ỵ✐ f (x) < f (x0 ) − α x − x0 ❦➨♦ t❤❡♦ ❤➺ ✭✷✳✶✶✮ ✤ó♥❣✳ ❚❛ ♥â✐ r P ỗ t tr t➟♣ D ⊂ R ♥➳✉ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ B(x0 , δ) ∩ D✳ ❚❛ ♥â✐ r P ỗ t tr t➟♣ D ⊂ R ✱ ♥➳✉ (M P ) ❧➔ ỗ t t x0 tr D ợ x0 D ữỡ tỹ t ỗ ụ ữủ ú ỵ ❧➔ ✈ỵ✐ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝✱ t➼♥❤ ❛❢❢✐♥❡ ❣✐↔ ỗ t ỗ t tữỡ ữỡ ợ t ỗ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ✷✵ ❉➵ ❦✐➸♠ tr❛ r➡♥❣ ♠é✐ ỗ t ỗ ữủ õ ❦❤ỉ♥❣ ✤ó♥❣✳ ✣➸ ♠✐♥❤ ❤å❛ ✤✐➲✉ ♥➔② t❛ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ s❛✉✿ (P 4) f (x) = (−x2 − 2x, −2x), ✈ỵ✐ r➔♥❣ ❜✉ë❝✿ g(x) = −x ≤ 0, h(x) = 0, x ∈ Q = [0, 1] ❚❛ ❝â ✭P✹✮ ❧➔ ❛❢❢✐♥❡ ❣✐↔ ỗ t x0 = ợ ♠é✐ α = (α1 , α2 ) ∈ ✐♥t(R2+ ) t❤➻ t❛ ❝❤➾ ❝➛♥ ①➨t δ s❛✉✿ α − 2, α > 2, 1 δ= 1, α1 ≤ P ổ ỗ ①➾ ❝❤➦t t↕✐ x0 = ❜ð✐ ✈➻ ✈ỵ✐ α = (2, 2) ợ ộ > tỗ t↕✐ x ∈ B(x0 , δ) ∩ S s❛♦ ❝❤♦ f (x) ≤ f (x0 ) − α x − x0 ♥❤÷♥❣ ❤➺ ✭✷✳✶✶✮ ❦❤ỉ♥❣ ✤ó♥❣❀ ✈➼ ❞ư t↕✐ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x = ✣à♥❤ ❧➼ sỷ P ỗ ❝❤➦t t↕✐ x0 tr➯♥ S ✳ ●✐↔ sû x0 ❧➔ ❑❚❱❈P t❤❡♦ α✳ ❑❤✐ ✤â✱ x0 ❝ô♥❣ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α ❝õ❛ ✭▼P✮✳ õ tỗ t x S s f (x) ≤ f (x0 ) − α x − x0 P ỗ ❑✉❤♥✲❚✉❝❦❡r ❝❤➦t ❝õ❛ ❤➺ ✭✷✳✶✶✮ t↕✐ x0 ✱ tø ✣à♥❤ ♥❣❤➽❛ ✷✳✻ ❦➨♦ t❤❡♦ x − x0 ❧➔ ♠ët ♥❣❤✐➺♠ àj = ợ j∈ / J(x0 ) ✈➔ sû ❞ö♥❣ ❍➺ q✉↔ ✶✳✶ s✉② r❛ x0 ❦❤æ♥❣ t❤➸ ❧➔ ❑❚❱❈P t❤❡♦ α✳ ✣✐➲✉ õ t ợ tt ỵ ữủ ❝❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â t ự ữủ ỵ s sỷ P ỗ ①➜♣ ①➾ t↕✐ x0 tr➯♥ S ✳ ●✐↔ sû x0 ❧➔ ❑❚❱❈P t❤❡♦ α✳ ❑❤✐ ✤â✱ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳ ❘ã r➔♥❣ ❧➔ ♠å✐ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉✮ ❝õ❛ ✭▼P✮ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✭tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❝õ❛ ✭▼P✮✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❑➳t q✉↔ s❛✉ ✤➙② ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➸ ✤↔♠ ❜↔♦ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❝ơ♥❣ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ✭❤♦➦❝ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉✮ ❝õ❛ ✭▼P✮✳ ✣à♥❤ ❧➼ ✷✳✾ ●✐↔ sû x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮ ✈➔ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0, ) sỷ P ỗ ①➾ ❑❚✲❝❤➦t t↕✐ x0 tr➯♥ S ✳ ❑❤✐ ✤â✱ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α ❝õ❛ ✭▼P✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x0 ❦❤æ♥❣ ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ỳ t õ tỗ t x S s❛♦ ❝❤♦ f (x) ≤ f (x0 ) − α x − x0 ✳ ❙û ❞ö♥❣ t➼♥❤ ❛❢❢✐♥❡ ❣✐↔ ỗ r t P t s r x − x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❝ô♥❣ ♥❤÷ ❤➺ ✭✷✳✸✮✳ ❇ð✐ ✈➻ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , ) tỗ t tn n x − x0 s❛♦ ❝❤♦ x0 + tn νn ∈ S t t ữủ r ợ ộ > ✈➔ n ✤õ ❧ỵ♥ xn = x0 + tn νn ∈ B(x0 , δ) ∩ S ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✮✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮✳ ❇➡♥❣ ♠ët ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ỵ t ữủ ỵ s ✣à♥❤ ❧➼ ✷✳✶✵ ●✐↔ sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ✈➔ t❤ä❛ ♠➣♥ t↕✐ (x0, ) sỷ P ỗ ①➜♣ ①➾ t↕✐ x0 tr➯♥ S ✳ ❑❤✐ ✤â x0 ❝ô♥❣ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α ❝õ❛ ✭▼P✮✳ ✭❈◗✸✮ ❘ã r➔♥❣ ❧➔ ♠å✐ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❝ơ♥❣ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❑➳t q✉↔ s❛✉ ✤➙② ❝❤♦ t❛ ✤✐➲✉ ❦✐➺♥ ✤↔♠ ❜↔♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ❝ơ♥❣ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣✳ ✷✷ ✣à♥❤ ❧➼ ✷✳✶✶ ●✐↔ sû x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ ✈➔ ✭❈◗✸✮ ✤ó♥❣ t↕✐ (x0, α)✳ ●✐↔ sû ✭▼P✮ ỗ t t x0 tr S ✳ ❑❤✐ ✤â x0 ❝ô♥❣ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ x0 ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ α ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮✳ ❑❤✐ ✤â ợ ộ > tỗ t x B(x0 , δ)∩S s❛♦ ❝❤♦ f (x) ≤ f (x0 )−α xx0 ỷ t ỗ ❝❤➦t ❝õ❛ ✭▼P✮✱ s✉② r❛ x − x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✶✮✳ ❇ð✐ ✈➻ ✭❈◗✸✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , ) sỷ ỵ tữỡ tỹ ữ tr ự ỵ t s r❛ ✈ỵ✐ ♠é✐ δ > ✈➔ n ✤õ ❧ỵ♥ xn = x0 +tn νn ∈ B(x0 , δ)∩S ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✮✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✳ ❉♦ ✤â ♠➙✉ t❤✉➝♥ ♥➔② ❝❤♦ t❛ ❝❤ù♥❣ ♠✐♥❤ ữủ ỵ ú t tõ tt t q tr➯♥ tr♦♥❣ ❤➺ q✉↔ s❛✉ ✤➙②✳ ❍➺ q✉↔ ✷✳✶ ●✐↔ sỷ P ỗ t t x0 tr➯♥ S ✈➔ ✤ó♥❣ t↕✐ (x0, α)✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭❛✮ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✱ ✭❜✮ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α✱ ✭❝✮ x0 ❧➔ ❑❚❱❈P t❤❡♦ α✱ ✭❞✮ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α✱ ✭❡✮ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝→❝ ❦➳t q ữủ s r tứ ỵ ✷✳✼✱ ✭❈◗✷✮ ✷✳✽✱ ✷✳✾✱ ✷✳✶✵ ✈➔ ✷✳✶✶✳ ✣➸ ❦➳t t❤ó❝ ♣❤➛♥ ♥➔②✱ t❛ ♠✐♥❤ ❤å❛ ♠ët ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ ❦❤↔ ✈✐ ✈➔ ❝❤➾ r❛ r➡♥❣ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❑❚❱❈P✱ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ❱➼ ❞ư ✷✳✹ ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ (P 5) f (x) = ✈ỵ✐ r➔♥❣ ❜✉ë❝✿ |x|, − 1 + |x| ✷✸ x ∈ S = {x ∈ R|g(x) 0, h(x) = 0, x ∈ Q} tr♦♥❣ ✤â Q = {x ∈ R : |x| ≤ 1} ✈➔ gj , hk : R → R✱ j = 1, 2; k = 1, ✤÷đ❝ ❝❤♦ ❜ð✐ x, g1 (x) = x2 , 0, h1 (x) = x2 , x≤0 , x>0 x≤0 x>0 , x, g2 (x) = x2 , 0, h2 (x) = x2 , x≤0 , x>0 x≤0 x > ❇ð✐ ✈➻ ✈ỵ✐ ♠é✐ t(R2+ ) > ổ tỗ t↕✐ x ∈ B(x0 , δ) ∩ S s❛♦ ❝❤♦ f (x) ≤ f (0) − α x − ✈➔ f (x) < f (0) − α x − t P ỗ ❑❚✲①➜♣ ①➾ ❝❤➦t t↕✐ x = tr➯♥ S ✳ ❚❛ t➻♠ ❝→❝ ✤✐➸♠ ❑❚❱❈P✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✿ ✭❛✮ x = 0✳ ❇➡♥❣ ♠ët t➼♥❤ t♦→♥ ✤ì♥ ❣✐↔♥ t❛ ❝â ∂C f1 (0) = ∂C f2 (0) = [−1, 1], ∂C g1 (0) = [0, ], ∂C g2 (0) = [0, ], ∂C h1 (0) = ∂C h2 (0) = ∂C (−h1 )(0) = ∂C (−h2 )(0) = {0}, N (Q, 0) = {0} ❈â t❤➸ t❤➜② r➡♥❣ ✈ỵ✐ λ = (1, 1), µ = (0, 0), ν = (0, 0)✱ α = (1, 1) ✈➔ e = t❤➻ ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣ ∈ [−2, 2] = λT ∂C f (0) + µT ∂C g(0) + ν T ∂C h(0) + λT αB + N (Q, 0), µT g(0) = ❇ð✐ ✈➻ x = ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➯♥ x = ❧➔ ❑❚❱❈P t❤❡♦ α = (1, 1)✳ ✭❜✮ −1 ≤ x < 0✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ g(x) = ( x2 , x3 ) < (0, 0) ✈➔ h(x) = (0, 0)✳ ❉♦ ✤â x ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ◆❤÷ ✈➟② f, g ✈➔ h ❝ô♥❣ ❦❤↔ ✈✐ t↕✐ x ♥➯♥ ∂C f (x), C g(x) C h(x) tữỡ ữỡ ợ f (x), ∇g(x) ✷✹ ✈➔ −1 1 h(x)✳ ◆❤÷ ✈➟②✱ t❛ ❝â ∂C f (x) = {(−1, (1−x) )}✱ ∂C g(x) = {( , )} ✈➔ ∂C h(x) = {(0, 0)} ✈➔ x ❧➔ ❑❚❱❈P ♥➳✉ tỗ t = (1 , ) = (µ1 , µ2 )✱ ν = (ν1 , ν2 )✱ α = (α1 , α2 ) ✈➔ e ∈ B s❛♦ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✷✮ ✤ó♥❣✱ tù❝ ❧➔ ∈λ1 · −1 + λ2 · 1 −1 + µ · + µ · + ν1 · + ν2 · (1 − x)2 + λ1 α1 · e + λ2 α2 · e + N (Q, x), x x µ1 · = 0, µ2 · = ✣✐➲✉ ❦✐➺♥ t❤ù ❤❛✐ ð tr➯♥ ❦➨♦ t❤❡♦ µ = (µ1 , µ2 ) = (0, 0) ✈➔ ❞♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ♠ët t❤ä❛ ♠➣♥ ✈ỵ✐ λ1 = λ2 = 1✱ ν1 = ν2 = 0✱ α = (1, (1−x) 2) ✈➔ e = 1✳ ❇ð✐ ✈➻ x ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➯♥ x ❧➔ ❑❚❱❈P t❤❡♦ α = (1, (1−x) )✳ ✭❝✮ x > 0, x < −1✳ ❉➵ ❦✐➸♠ tr❛ r➡♥❣ x ❦❤ỉ♥❣ ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ◆❤÷ ✈➟② x ❦❤æ♥❣ ❧➔ ❑❚❱❈P✳ ❉♦ ✤â x ❧➔ ❑❚❱❈P t❤❡♦ α = (1, (1−x) ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x ∈ S = [−1, 0]✳ ❇➙② ❣✐í✱ t❛ ①❡♠ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❤♦➦❝ tü❛ ♥❣❤✐➺♠ ❤ú✉ ỷ ỵ t rót r❛ ♠å✐ x ∈ S ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α = (1, (1−x) ) ✷✺ ❈❤÷ì♥❣ ✸ ✣è✐ ữỡ tr ỵ ố ②➳✉✱ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ❝õ❛ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥ ❬✹❪ ❝❤♦ ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳ ✸✳✶✳ ✣è✐ ♥❣➝✉ ②➳✉ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ❜➔✐ t♦→♥ ❦❤æ♥❣ ❝â ❝→❝ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝ ✈➔ r➔♥❣ ❜✉ë❝ t➟♣✱ ●✉♣t❛ ❬✻❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ✤è✐ ♥❣➝✉ ❲♦❧❢❡ t P ợ tt ỗ ①➾ s✉② rë♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐r ✭▼❲❉✮ ❝❤♦ ❜➔✐ t♦→♥ ✭▼P✮ ✤÷đ❝ tr➻♥❤ ❜➔② ❝ị♥❣ ợ ỵ ố ợ tt t ỗ t t ỗ t ố ♥❣➝✉ ▼♦♥❞✲❲❡✐r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮ ✤÷đ❝ ❝❤♦ ❜ð✐ (MWD) max f (u), ✈ỵ✐ r➔♥❣ ❜✉ë❝✿ m 0∈ p n λi ∂C fi (u) + i=1 µj ∂C gj (u) + j=1 m νk ∂C hk (u) + k=1 n p m (, à, ) Rm + ì R+ × R , λ = 0, α ∈ ✐♥t(R+ ), p n νk hk (u) ≥ µj gj (u) + j=1 k=1 λi αi B + N (Q, u), i=1 ✷✻ ●✐↔ sû SD ❧➔ t➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ ◆❤➢❝ ❧↕✐ S ❧➔ t➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼P✮✳ ❚❛ ❜➢t ✤➛✉ ❜➡♥❣ ✤à♥❤ ♥❣❤➽❛ ✤è✐ ♥❣➝✉ ②➳✉✳ ✣à♥❤ ❧➼ ✸✳✶ ✭✣è✐ ♥❣➝✉ ②➳✉✮ ●✐↔ sû (u, λ, µ, ν, α) ∈ SD ✈➔ ✭▼P✮ ❧➔ ❛❢❢✐♥❡ ỗ t t u tr Q õ ợ ộ tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ B(u, δ) ∩ S ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ✤ó♥❣✿ ✭✸✳✶✮ f (x) ≤ f (u) − γ x − u ❈❤ù♥❣ ♠✐♥❤✳ ❇ð✐ ✈➻ (u, λ, µ, ν, α) ❧➔ ♠ët ữủ tỗ t i ∈ ∂C fi (u), ηj ∈ ∂C gj (u)✱ ζk ∈ ∂C hk (u), e ∈ B ✈➔ d ∈ N (Q, u) s❛♦ ❝❤♦ m p n λi ξi + i=1 νk ζk + µj ηj + j=1 m ✭✸✳✷✮ λi αi e + d = i=1 k=1 sỷ ữủ tỗ t s ợ ộ > tỗ t x ∈ B(u, δ) ∩ S s❛♦ ❝❤♦ ✭✸✳✶✮ ✤ó♥❣✳ ❙û t ỗ t P t↕✐ u✱ t❛ ❦➳t ❧✉➟♥ r➡♥❣ x − u ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✶✮✳ ✣✐➲✉ ✤â ❦➨♦ t❤❡♦ ξi , x − u < −γi x − u ✈ỵ✐ ♠å✐ i ∈ I, ηj , x − u ≤ ✈ỵ✐ ♠å✐ j ∈ J(u), ζk , x − u = ✈ỵ✐ ♠å✐ k ∈ K, d, x − u ≤ n p ❚❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ (λ, µ, ν) ∈ Rm / J(u), + × R+ × R = àj = ợ j ∈ α ✈➔ sü ❦✐➺♥ ✈ỵ✐ ♠å✐ e ∈ B, e, x − u ≤ x − u ✱ t❛ ❦➳t ❧✉➟♥ ✤÷đ❝ γ r➡♥❣ m λi ξi + i=1 p n µj ηj + j=1 m λi αi e + d, x − u νk ζk + k=1 < i=1 ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✸✳✷✮ ✈➔ õ ự tọ ỵ ú ởt ự tữỡ tỹ ỵ t õ ỵ s❛✉✳ ✷✼ ✣à♥❤ ❧➼ ✸✳✷ ✭✣è✐ ♥❣➝✉ ②➳✉✮ ●✐↔ sû (u, λ, µ, ν, α) ∈ SD ✈➔ ✭▼P✮ ❧➔ ỗ t u tr Q õ ợ ộ tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ B(u, δ) ∩ S ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ✤ó♥❣✿ f (x) < f (u) − γ x − u ✭✸✳✸✮ ❚÷ì♥❣ tü ✣à♥❤ ♥❣❤➽❛ ✾ tr♦♥❣ ❬✻❪✱ t❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉ ✤➙②✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶ (u0, λ0, µ0, ν0, α0) ∈ SD ✤÷đ❝ ❣å✐ ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✭❤♦➦❝ ❧➔ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣✮ ❝õ❛ ✭▼❲❉✮ ♥➳✉ tỗ t t(RM + ) U ❝õ❛ (u0 , λ0 , µ0 , ν0 , ) s ợ ộ (u, , à, , α) ∈ U ∩ SD ❜➜t ✤➥♥❣ t❤ù❝ f (u0 ) + η u − u0 ≤ f (u) ✭t✳÷✳✱ f (u0 ) + η u − u0 < f (u)✮ ❦❤ỉ♥❣ ✤ó♥❣✳ ✸✳✷✳ ✣è✐ ♥❣➝✉ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ❈❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ q✉❛♥ trồ ỵ ố t q ự t q trồ ố tr ỵ t❤✉②➳t tè✐ ÷✉✳ ✣à♥❤ ❧➼ ✸✳✸ ✭✣è✐ ♥❣➝✉ ♠↕♥❤✮ ●✐↔ sû x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮ ✈➔ ✭❈◗✷✮ ✤ó♥❣ t↕✐ (x0, α0)✳ õ tỗ t n p (0 , à0 , ν0 ) ∈ Rm + × R+ × R s❛♦ ❝❤♦ (x0 , λ0 , µ0 , ν0 , α0 ) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ ✭▼❲❉✮✳ P ỗ t t↕✐ ♠é✐ u tr➯♥ Q✱ tr♦♥❣ ✤â (u, λ, µ, , ) SD ợ (, à, ) Rm+ × Rn+ × Rp ✈➔ α ♥➔♦ ✤â ∈ int(Rm+ ) t ợ ộ (x0, 0, à0, 0, α0) ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ γ ❝õ❛ ✭▼❲❉✮ ✈➔ ❣✐→ trà ❝õ❛ ❤❛✐ ❜➔✐ t♦→♥ ✭▼P✮ ✈➔ ✭▼❲❉✮ ❜➡♥❣ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇ð✐ ✈➻ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α0 ❝õ❛ ✭▼P✮ ✈➔ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , α0 ) sỷ ỵ s r tỗ n p t↕✐ (λ0 , µ0 , ν0 ) ∈ Rm + × R+ × R , λ0 = s❛♦ ❝❤♦ ✭✷✳✶✷✮ ✤ó♥❣✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ (x0 , λ0 , µ0 , ν0 , α0 ) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ ❇ð✐ ✈➻ ✭▼P✮ ❧➔ ❛❢❢✐♥❡ ❣✐↔ ỗ t t ộ u tr Q tr ✤â (u, λ, µ, ν, α) ∈ SD n p m ợ (, à, ) õ tở Rm + × R+ × R ✈➔ α ♥➔♦ ✤â ∈ ✐♥t(R+ ) ữ ỵ ố ú ợ ộ tỗ t ởt ❧➙♥ ❝➟♥ U ❝õ❛ (x0 , λ0 , µ0 , ν0 , α0 ) s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ (u, λ, µ, ν, α) ∈ U ∩ SD ❜➜t ✤➥♥❣ t❤ù❝ ✈❡❝tì f (x0 ) + γ u − x0 ≤ f (u) ❦❤æ♥❣ t❤ä❛ ♠➣♥✳ ❉♦ ✤â✱ tø ✣à♥❤ ♥❣❤➽❛ ✸✳✶ t❛ s✉② r❛ (x0 , λ0 , µ0 , ν0 , α0 ) ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ γ ❝õ❛ ✭▼❲❉✮✳ ❍ì♥ ♥ú❛✱ ❣✐→ trà ♠ư❝ t✐➯✉ ❝õ❛ ✭▼P✮ ✈➔ ✭▼❲❉✮ ❜➡♥❣ ♥❤❛✉ ❧➔ ❜➡♥❣ f (x0 ) ữỡ tỹ ữ ỵ t õ ỵ s ố sû x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α0 ❝õ❛ ✭▼P✮ ✈➔ ✭❈◗✸✮ ✤ó♥❣ t↕✐ (x0, α)✳ õ tỗ t (0, à0, 0) Rm+ ì Rn+ ì Rp s (x0, 0, à0, 0, 0) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ ◆➳✉ ✭▼P✮ ❝ơ♥❣ ỗ t n p u tr➯♥ Q✱ tr♦♥❣ ✤â (u, λ, µ, ν, α) ∈ SD ợ (, à, ) Rm + ì R+ × R ✈➔ α ♥➔♦ ✤â ∈ int(Rm α✱ (x0 , λ0 , µ0 , ν0 , α0 ) ❧➔ tü❛ ♥❣❤✐➺♠ + ) t❤➻ ✈ỵ✐ ♠é✐ γ ❤ú✉ ❤✐➺✉ ②➳✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ γ ❝õ❛ ✭▼❲❉✮ ✈➔ ❣✐→ trà ♠ö❝ t✐➯✉ ❝õ❛ ✭▼P✮ ✈➔ ✭▼❲❉✮ ❜➡♥❣ ♥❤❛✉✳ ❈→❝ ❦➳t q ố ữủ ữủ ự ữợ ✣à♥❤ ❧➼ ✸✳✺ ✭✣è✐ ♥❣➝✉ ♥❣÷đ❝✮ ●✐↔ sû x0 ❧➔ ữủ P (x0, 0, à0, ν0, α0) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ ◆➳✉ P ỗ t t x0 tr➯♥ S t❤➻ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α0 ❝õ❛ ✭▼P✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❣✐↔ sû r➡♥❣ x0 ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼P✮ ✈➔ (x0 , λ0 ✱ µ0 , ν0 , α0 ) ❧➔ ✤✐➸♠ ữủ õ tỗ t p m n (λ0 , µ0 , ν0 ) ∈ Rm + × R+ × R ✱ λ0 = ✈➔ α0 ∈ ✐♥t(R+ ) s❛♦ ❝❤♦ ✭✷✳✶✷✮ ✤ó♥❣ ✈➔ ✈➻ ✈➟② x0 ❧➔ ❑❚❱❈P t❤❡♦ α0 ❝õ❛ ✭▼P✮✳ ❇ð✐ ✈➻ P ỗ t t x0 tứ ỵ t s r x0 tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤❡♦ α0 ❝õ❛ ✭▼P✮✳ ❚÷ì♥❣ tü ỵ t ữủ ỵ s ✣à♥❤ ❧➼ ✸✳✻ ✭✣è✐ ♥❣➝✉ ♥❣÷đ❝✮ ●✐↔ sû x0 ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✷✾ ✭▼P✮ ✈➔ (x0, λ0, µ0, ν0, α0) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ P ỗ t x0 tr➯♥ S t❤➻ x0 ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α0 ❝õ❛ ✭▼P✮✳ ✸✵ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ tr➻♥❤ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ▼✳●♦❧❡st❛♥✐ ✈➔ ❝ë♥❣ sü ❬✹❪ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ổ trỡ q ữợ r ũ ợ ỵ ố t ố r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ỗ tự ỡ ữợ ✈✐ ♣❤➙♥ ❈❧❛r❦❡❀ ✲ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❑✉❤♥✲❚✉❝❦❡r ✈➔ r ữợ ổ ỳ ữợ r tố ữ ỵ ✤è✐ ♥❣➝✉ ♠↕♥❤✱ ②➳✉ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝ ❝❤♦ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐r ❝õ❛ ❜➔✐ t♦→♥ ✭▼P✮✳ ❚è✐ ÷✉ ✈➔ ✤è✐ ♥❣➝✉ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥ ❧➔ ✤➲ t➔✐ ❝â t➼♥❤ t❤í✐ sü✱ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ✸✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ✣é ❱➠♥ ▲÷✉✱ P❤❛♥ ❍✉② ❑❤↔✐ t ỗ tt ❍➔ ◆ë✐✳ ❬✷❪ ✣é ❱➠♥ ▲÷✉ ✭✶✾✾✾✮✱ ●✐↔✐ t➼❝❤ ▲✐♣s❝❤✐t③✱ ◆❳❇ ❑❤♦❛ ❤å❝ ✈➔ ❑➽ t❤✉➟t✱ ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❳✳ ❋✳ ▲✐ ✭✷✵✵✵✮✳ ❈♦♥str❛✐♥t q✉❛❧✐❢✐❝❛t✐♦♥s ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✵✻✿✸✼✸✲✸✾✽✳ ❬✹❪ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥✭✷✵✶✼✮✱ ✧◆❛❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❡❢❢✐❝✐❡♥❝② ✐♥ ♥♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❜❧❡♠s✧✱ ◆✉♠❡r✲ ✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✸✽✭✷✵✶✼✮✱ ◆♦✻✱ ✻✽✸✲✼✵✹✳ ❬✺❪ ▼✳ ●♦❧❡st❛♥✐ ❛♥❞ ❙✳ ◆♦❜❛❦❤t✐❛♥ ✭✷✵✶✸✮✳✧ ◆♦♥s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠♠✐♥❣✿ ❙tr♦♥❣ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s✧✳ P♦s✐t✐✈✐t② ✶✼✿✼✶✶✲ ✼✸✷✳ ❬✻❪ ❆✳ ●✉♣t❛✱ ❆✳ ▼❡❤r❛✱ ❛♥❞ ❉✳ ❇❤❛t✐❛ ✭✷✵✵✻✮✳ ❆♣♣r♦①✐♠❛t❡ ❝♦♥✈❡①✐t② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐s❛t✐♦♥✳ ❇✉❧❧✳ ❆✉st✳ ▼❛t❤✳ ❙♦❝✳ ✼✹✿✷✵✼✲✷✶✽✳ ❬✼❪ ❚✳ ▼❛❡❞❛ ✭✶✾✾✹✮✳ ❈♦♥str❛✐♥t q✉❛❧✐❢✐❝❛t✐♦♥s ✐♥ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣✲ t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✿ ❉✐❢❢❡r❡♥t✐❛❜❧❡ ❝❛s❡✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✽✵✿✹✽✸✲✺✵✵✳ ❬✽❪ ▼✳ ❆r❛♥❛✲❏✐♠➨♥❡③✱ ❆✳ ❘✉❢✐→♥✲▲✐③❛♥❛✱ ❘✳ ❖s✉♥❛✲●â♠❡③✱ ❛♥❞ ●✳ ❘✉✐③✲ ●❛r③â♦♥ ✭✷✵✵✽✮✳ Ps❡✉❞♦✐♥✈❡①✐t②✱ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❛♥❞ ❡❢❢✐❝✐❡♥❝② ✐♥ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♣r♦❜❧❡♠s❀ ❞✉❛❧✐t②✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✲❚❤❡♦r✳ ✻✽✿✷✹✲✸✹✳ ... - VŨ VIỆT BÌNH ĐIỀU KIỆN CẦN VÀ ĐỦ CHO TỰA NGHIỆM HỮU HIỆU YẾU CỦA BÀI TOÁN TỐI ƯU ĐA MỤC TIÊU KHƠNG TRƠN Chun ngành: Tốn ứng dụng Mã số: 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG... ngành: Tốn ứng dụng Mã số: 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC GS.TS Đỗ Văn Lưu THÁI NGUYÊN - 2020 ▼ö❝ ❧ö❝ ❇↔♥❣ ỵ ởt số tự ữợ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳