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 TỐN TỐI ƯU ĐA MỤC TIÊU KHƠNG 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 Lu THI NGUYấN - 2020 ử ỵ ▼ð ✤➛✉ ✶ ▼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✉ ❦❤♦❛ r t tổ ữợ sỹ ữợ ộ ữ ❈→❝ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉✱ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ 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➼❝❤ ❞➝♥ ✈➔ ú 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 ú ù t t ữớ ữợ ✣é ❱➠♥ ▲÷✉✳ ❚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 ❣â♣ ✈➔ 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) õ 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❤❡♦ ữỡ 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❛✉ ởt 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 ✤è✐ ♥❣➝✉ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ❦❤ỉ♥❣ trì♥✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t tố ữ t ợ st ữỡ q ữợ r ❝õ❛ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ✸✽✭✷✵✶✼✮✱ ✽✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐ ♥❣➝✉ ②➳✉✱ ♠↕♥❤ ✈➔ ✤è✐ ♥❣➝✉ ♥❣÷đ❝✳ ✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ỗ ữỡ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ö❝ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ữỡ ợ t tự tr ởt số tự ỡ ữợ ❈❧❛r❦❡✱ ♥â♥ t✐➳♣ t✉②➳♥ ✈➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❈❧❛r❦❡✳ ❈❤÷ì♥❣ ✷ ✈ỵ✐ t✐➯✉ ✤➲✿ ✧✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉✧ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♠ỵ✐ ✤➙② ❝õ❛ ▼✳ ●♦❧❡st❛♥✐✱ ❍✳ ❙❛❞❡❣❤✐✱ ❨✳ ❚❛✈❛♥ ✤➠♥❣ tr♦♥❣ t↕♣ ❝❤➼ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ✸ ✸✽✭✷✵✶✼✮✱ ✻✽✸✲✼✵✹ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❑✉❤♥✲❚✉❝❦❡r✱ ✤è✐ ♥❣➝✉ ố ữủ ữỡ ợ t ố tr ỵ ố ②➳✉✱ ♠↕♥❤ ✈➔ ✤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 số tự ỡ ữợ r ♥â♥ 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 ✳ ❚❤æ♥❣ t❤÷í♥❣✱ ❝❧ M ✱ ✐♥t M ✱ ❝♦(M ) ✈➔ ❝♦♥❡ (M ) ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ❜❛♦ ✤â♥❣✱ ♣❤➛♥ tr ỗ õ s M tữỡ ự ❈ü❝ ➙♠ ✈➔ ❝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↕ (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 ❧➔ ởt số t t ữợ r ♠ët ❤➔♠ ▲✐♣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 ❝â ∂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) ổ tỗ t x S s ú t↕✐ x0 = 0✱ ♥➯♥ x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✭✤à❛ ♣❤÷ì♥❣✮ t❤❡♦ α = (2, 2)✳ ❉➵ ❦✐➸♠ tr❛ r➡♥❣ x0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❤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 δ > 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 ) − α x − x0 ✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐✿ 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 δ > s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ B(x0 , δ) ✈ỵ✐ f (x) < f (x0 ) − α x − x0 ❦➨♦ t❤❡♦ ❤➺ ✭✷✳✶✶✮ ✤ó♥❣✳ ❚❛ õ r P ỗ t tr➯♥ t➟♣ D ⊂ R ♥➳✉ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ B(x0 , δ) ∩ D✳ ❚❛ õ 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 þ ❧➔ ✈ỵ✐ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝✱ t➼♥❤ ❛❢❢✐♥❡ ỗ t ỗ t tữỡ ữỡ ợ t ỗ ①➾ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ✷✵ ❉➵ ❦✐➸♠ tr❛ r➡♥❣ ộ 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 = ✈ỵ✐ ♠é✐ α = (α1 , α2 ) ∈ ✐♥t(R2+ ) t❤➻ t❛ ❝❤➾ ❝➛♥ ①➨t δ s❛✉✿ α − 2, α > 2, 1 δ= 1, α1 P ổ ỗ ❑❚✲①➜♣ ①➾ ❝❤➦t t↕✐ x0 = ❜ð✐ ✈➻ ✈ỵ✐ α = (2, 2) ✈➔ ✈ỵ✐ ♠é✐ δ > tỗ t x B(x0 , ) S s ❝❤♦ f (x) ≤ f (x0 ) − α x − x0 ♥❤÷♥❣ ❤➺ ✭✷✳✶✶✮ ❦❤ỉ♥❣ ✤ó♥❣❀ ✈➼ ❞ư t↕✐ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x = ✣à♥❤ sỷ P ỗ ①➾ ❝❤➦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 ①➾ ❑✉❤♥✲❚✉❝❦❡r ❝❤➦t ❝õ❛ ❤➺ ✭✷✳✶✶✮ t↕✐ x0 ✱ tø ✣à♥❤ ♥❣❤➽❛ ✷✳✻ ❦➨♦ t❤❡♦ x − x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✶✮✳ ❇➡♥❣ ❝→❝❤ ❧➜② µj = ✈ỵ✐ j∈ / J(x0 ) ✈➔ sû ❞ư♥❣ ❍➺ q✉↔ ✶✳✶ s✉② r❛ x0 ❦❤æ♥❣ t❤➸ ❧➔ ❑❚❱❈P t❤❡♦ α✳ õ t ợ tt ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ õ t ự ữủ ỵ s ✣à♥❤ ❧➼ ✷✳✽ ●✐↔ sû ✭▼P✮ ❧➔ ❛❢❢✐♥❡ ❑❚✲ ❣✐↔ ỗ t x0 tr S sỷ 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➜♣ 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❛ ❤➺ ✭✷✳✶✶✮ ❝ơ♥❣ ♥❤÷ ❤➺ ✭✷✳✸✮✳ ❇ð✐ ✈➻ ✭❈◗✷✮ t❤ä❛ ♠➣♥ t↕✐ (x0 , ) tỗ t tn n → x − x0 s❛♦ ❝❤♦ x0 + tn νn S t t ữủ r ợ ộ δ > ✈➔ 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♥❣ tỹ 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✮✳ ❑❤✐ õ ợ ộ > tỗ t x B(x0 , δ)∩S s❛♦ ❝❤♦ f (x) ≤ f (x0 ) xx0 ỷ t ỗ ①➾ ❝❤➦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✉② 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✐ α ∈ ✐♥t(R2+ ) ✈➔ δ > ❦❤æ♥❣ tỗ t x B(x0 , ) S s ❝❤♦ f (x) ≤ f (0) − α x − ✈➔ f (x) < f (0) − α x − ✱ ♥➯♥ ❜➔✐ t♦→♥ ✭P✺✮ ❧➔ ❛❢❢✐♥❡ ❣✐↔ ỗ t t x = tr S ✳ ❚❛ t➻♠ ❝→❝ ✤✐➸♠ ❑❚❱❈P✳ ❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤ñ♣ 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) ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ 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 ❦❤ỉ♥❣ ❧➔ ❑❚❱❈P✳ ❉♦ ✤â x ❧➔ ❑❚❱❈P t❤❡♦ α = (1, (1−x) ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x ∈ S = [−1, 0]✳ ❇➙② ❣✐í✱ t❛ ①❡♠ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❤♦➦❝ tü❛ ♥❣❤✐➺♠ ỳ ỷ ỵ t❛ rót r❛ ♠å✐ x ∈ S ❧➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✈➔ tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ②➳✉ t❤❡♦ α = (1, (1−x) ) ✷✺ ❈❤÷ì♥❣ ✸ ố 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➜♣ ①➾ s✉② rë♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐r ✭▼❲❉✮ ❝❤♦ ❜➔✐ t♦→♥ ✭▼P✮ ✤÷đ❝ tr➻♥❤ ❜➔② ũ ợ ỵ ố ợ tt t ỗ t t ỗ t ✤è✐ ♥❣➝✉ ▼♦♥❞✲❲❡✐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) ❧➔ ởt ữủ tỗ t ξi ∈ ∂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♥❣✳ ỷ 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❛ (, à, ) Rm / J(u), + ì R+ × R ✱ λ = ✈➔ µj = ✈ỵ✐ 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 ố 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, ν, 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 ♥➔② ❝❤ù♥❣ ♠✐♥❤ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ✤è✐ ♥❣➝✉ tr♦♥❣ ỵ tt tố ữ ố ●✐↔ sû x0 ❧➔ ♠ët tü❛ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t❤❡♦ α ❝õ❛ ✭▼P✮ ✈➔ ✭❈◗✷✮ ✤ó♥❣ t↕✐ (x0, 0) 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, ν, α) ∈ SD ✈ỵ✐ (λ, µ, ν) ∈ Rm+ × Rn+ × Rp ✈➔ α ♥➔♦ ✤â ∈ int(Rm+ ) t ợ ộ (x0, 0, à0, ν0, 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 , ) 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 ♠ët ❧➙♥ ❝➟♥ 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 ♠ö❝ 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, ) õ tỗ t (0, à0, 0) Rm+ × Rn+ × Rp s❛♦ ❝❤♦ (x0, λ0, µ0, ν0, α0) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼❲❉✮✳ ◆➳✉ ✭▼P✮ ụ 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 ✤➙②✳ ✣à♥❤ ❧➼ ✸✳✺ ✭✣è✐ ♥❣➝✉ ♥❣÷đ❝✮ ●✐↔ sû x0 ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭▼P✮ ✈➔ (x0, λ0, µ0, 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♥❣ tỹ ỵ t ữủ ỵ s ✤➙②✳ ✣à♥❤ ❧➼ ✸✳✻ ✭✣è✐ ♥❣➝✉ ♥❣÷đ❝✮ ●✐↔ sû x0 ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✷✾ ✭▼P✮ ✈➔ (x0, 0, à0, 0, 0) ữủ 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 ✲ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✤õ tè✐ ÷✉❀ ✲ ❈→❝ ✤à♥❤ ỵ ố ố ữủ ❜➔✐ 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 t❤✉➟t✱ ❍➔ ◆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♦♥ 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