Bài Toán Tối Ưu Với Ràng Buộc Là Bài Toán Bù Tổng Quát

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆●❯❨➍◆ ❚❍❆◆❍ ❚❹▼ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❱❰■ ❘⑨◆● ❇❯❐❈ ▲⑨ ❇⑨■ ❚❖⑩◆ ❇Ị ❚✃◆● ◗❯⑩❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ◆●❯❨➍◆ ❚❍❆◆❍ ❚❹▼ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❱❰■ ❘⑨◆● ❇❯❐❈ ▲⑨ ❇⑨■ ❚❖⑩◆ ❇Ò ❚✃◆● ◗❯⑩❚ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ò ữớ ữợ ❦❤♦❛ ❤å❝✿ ●❙✳❚❙❑❍✳ ◆●❯❨➍◆ ❳❯❹◆ ❚❻◆ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❈→❝ sè ❧✐➺✉✱ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tỉ✐ t➻♠ ✤å❝ ✈➔ tr➼❝❤ ❞➝♥ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✶✶❪✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② t❤→♥❣ ♥➠♠ ✷✵✶✼ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤❛♥❤ ❚➙♠ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t ữợ sỹ ữợ t t ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ♥❣÷í✐ t❤➛② ❝õ❛ ♠➻♥❤✱ tr♦♥❣ ♠ët tớ tứ ữợ t t q ợ ổ ỵ tt tố ữ tr✉②➲♥ ❝❤♦ t→❝ ❣✐↔ ♥❤ú♥❣ ❦✐♥❤ ♥❣❤✐➺♠ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱ ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺ t→❝ ❣✐↔ ✈÷đt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥ tr♦♥❣ ❝❤✉②➯♥ ♠æ♥ ✈➔ ❝✉ë❝ sè♥❣✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tợ sữ t ổ ❚♦→♥ ❤å❝ ✈➔ tr÷í♥❣ ❙÷ P❤↕♠ ❚❤→✐ ◆❣✉②➯♥✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ò♥❣✱ t→❝ ❣✐↔ ♠✉è♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ❚♦→♥ ❣✐↔✐ t➼❝❤ ❦✷✸✱ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ❝õ❛ ♠➻♥❤ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❝❤✐❛ s➫ ✈➔ ❦❤➼❝❤ ❧➺ ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② t❤→♥❣ ♥➠♠ ✷✵✶✼ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣✉②➵♥ ❚❤❛♥❤ ❚➙♠ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ư❝ ❧ư❝ ✐✐✐ ▼ð ✤➛✉ ✶ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✼ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❇➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ✳ ✳ ✳ ✶✳✷✳✶ ❑❤→✐ ♥✐➺♠ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ ❝ü❝ trà tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❇➔✐ t♦→♥ ❜ò ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❇➔✐ t♦→♥ ❜ò t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❇➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ❇➔✐ t♦→♥ ❜ò tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✸ ✶✸ ✷✵ ✸✷ ✷ ❇➔✐ t♦→♥ ❝ü❝ trà ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❧➔ ❜➔✐ t♦→♥ ❜ị tê♥❣ q✉→t ✸✻ ✷✳✶ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✾ ✾ ✶✵ ✸✻ ✐✈ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ ✤➦t r❛ ✳ ✳ ✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✹✼ ✺✶ ✶ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ❜ị ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝✿ ❦✐♥❤ t t tt t ỵ s t ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉✱✳✳✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❜ị ❤✐➺♥ ♥❛② ✈➝♥ ✤❛♥❣ ❧➔ ✈➜♥ ✤➲ t❤í✐ sü✱ ✤➦❝ ❜✐➺t ❧➔ ✈✐➺❝ t➻♠ r❛ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜ị ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ✳ ❇➔✐ t♦→♥ ❜ị ♥❣✉②➯♥ ❣è❝ ✤÷đ❝ ♣❤→t ❜✐➸✉ ✿ ❈❤♦ f : Rn → Rn✱ ❚➻♠ x¯ ∈ Rn+ s❛♦ ❝❤♦ f (¯ x) ∈ Rn+ ✈➔ < x¯, f (¯ x) >= 0, (0.1) tr♦♥❣ ✤â Rn ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ ♥ ✲ ❝❤✐➲✉ ✈➔ Rn+ = {x = (x1 , x2 , , xn ) ∈ Rn+ , xi ≥ 0, i = 1, 2, n} ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙② ♥❣÷í✐ t❛ tê♥❣ q✉→t t❤➔♥❤ ❜➔✐ t♦→♥✿ ❚➻♠ x¯ ∈ Rn− s❛♦ ❝❤♦ g(¯ x) ≥ 0, h(¯ x) ≥ ✈➔ < g(¯ x), h(¯ x) >= 0✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ✈✐➳t ♠ët ❝→❝❤ tê♥❣ q✉❛♥ ✈➲ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈ỵ✐ r➔♥❣ ❜✉ë❝ tê♥❣ q✉→t ♥❤÷ s❛✉✿ ❈❤♦ C ⊆ Rn✱ t➟♣ ✤â♥❣ S1, S2 ⊆ Rn✱ ❜➔✐ t♦→♥ t➻♠ x˜ ∈ C ∩ S˜ s❛♦ ❝❤♦ ˜ ϕ(˜ x) = ϕ(y), C˜ = C ∩ S, (0.2) yC tr õ t õ ỗ tr ổ ❣✐❛♥ ❊✉❝❧✐❞ Rn✱ S˜ = S˜1 ∩ S˜2 ✈➔ ˜ S˜1 = x ∈ Rn : g˜(x) ≤ 0, h(x) =0 , S˜2 = {x ∈ Rn : g(x) ≤ 0, h(x) ≤ 0, g(x), h(x) (0.3) Rq = 0} ✷ ❝→❝ ❤➔♠ t❤ü❝ ϕ : Rn → R, g˜ : Rn → Rm, h˜ : Rn → Rp✱ ❣ Rn Rq tử ỵ y = (y1, y2, , ym) ≤ ❝â ♥❣❤➽❛ ❧➔ yi ≤ 0, ∀i = 1, 2, n✳ ❚❛ ❣✐↔ t❤✐➳t ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ✭✵✳✶✮✱ ✭✵✳✷✮ ✈➔ ✭✵✳✸✮ ❧➔ ❦❤→❝ ré♥❣✳ ❚r÷í♥❣ ❤đ♣ ❦❤✐ m = n, g(x) = −x, h(x) = −F (x)✱ ✈ỵ✐ F : Rn → Rn ❧➔ →♥❤ ①↕ ❛❢❢✐♥✱ ♥❣❤➽❛ ❧➔ F (x) = M x + q, M ∈ Rn×n , q ∈ Rn , ❜➔✐ t♦→♥ ✭✵✶✮ ✤÷đ❝ ❣å✐ ❧➔ t ũ t t ỵ Pq ❤✐➸✉ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ tr➯♥ t❛ ✤➣ t❤✉ ữủ t q s ỵ M Rnìn P tr ợ tt ❝→❝ ✤à♥❤ t❤ù❝ ❝♦♥ ❝❤➼♥❤ ❝õ❛ ▼ ❞÷ì♥❣ t❤➻ ▲❈P✭q✱▼✮ õ ởt ợ q Rn ỵ ✵✳✷✳ ❬✽❪ ◆➳✉ q ❦❤ỉ♥❣ ➙♠ t❤➻ ❜➔✐ t♦→♥ ❜ị t✉②➳♥ t➼♥❤ LCP (q, M ) ❧✉ỉ♥ ❣✐↔✐ ✤÷đ❝ ✈➔ x = ❧➔ ♠ët ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❝õ❛ ♥â✳ ◆❣❤✐➯♥ ❝ù✉ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜ò t✉②➳♥ t t t tự ỵ ❤✐➺✉ ❜ð✐ ❱■✭❑✱❋✮✱ ❧➔ ❜➔✐ t♦→♥ t➻♠ ♠ët ✈❡❝tì x ∈ K ⊂ Rn s❛♦ ❝❤♦ y − x, F (x) ≥ 0, ∀y ∈ K ð ✤➙② F : K → Rn ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ K ❧➔ t õ ỗ t õ t ữủ t q s ỵ F = M x + q, M ∈ Rn×n , q ∈ Rn , x ∈ Rn+ t❤➻ VI (F, Rn+ ) ✈➔ ❜➔✐ t♦→♥ ❜ò t✉②➳♥ t➼♥❤ ▲❈P✭q✱▼✮ ❝â ♥❣❤✐➺♠ ❤♦➔♥ t♦➔♥ trò♥❣ ♥❤❛✉✳ ✸ ❚r÷í♥❣ ❤đ♣ ❦❤✐ n = m, g(x) = −x, h(x) = −F (x) ✈ỵ✐ F ❧➔ →♥❤ ①↕ ♣❤✐ t✉②➳♥ tø Rn ✈➔♦ Rn✱ ❜➔✐ t♦→♥ ✭✵✳✶✮ ✤÷đ❝ ❣å✐ t ũ t ỵ P ✤â ❧➔ ❜➔✐ t♦→♥ t➻♠ ✈❡❝tì x ∈ Rn s❛♦ ❝❤♦ x ≥ 0, F (x) ≥ 0, x, F (x) = 0, (0.4) ❤✐➺♥ ♥❛② ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✤➣ t➻♠ r❛ r➜t ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝❤♦ ❝→❝ ❧♦↕✐ ❜➔✐ t♦→♥ ♥➔②✳ ❚➜t ❝↔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤÷❛ r❛ ✤➲✉ ❞➝♥ tỵ✐ ❣✐↔✐ ♠ët ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❤♦➦❝ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣✳ ◆❤✐➺♠ ✈ư ❝õ❛ t tố ữ ợ r ❜✉ë❝ ❧➔ ❜➔✐ t♦→♥ ❜ị tê♥❣ q✉→t ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ rữợ t t ởt số ữỡ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜ị tê♥❣ q✉→t tr➯♥✳ ✶✳ P❤÷ì♥❣ sỷ ị tữ ữỡ ♥➔② ❧➔ ❜✐➳♥ ✤ê✐ ❜➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥ ◆❈P✭❋✮ ✈➲ ❜➔✐ t♦→♥ tè✐ ÷✉ q✉❛ ✈✐➺❝ sû ❞ư♥❣ ❝→❝ ❤➔♠ ❦❤♦↔♥❣✳ ❈ỉ♥❣ ❝ư t❤✉➟♥ t✐➺♥ ✤➸ t❤✐➳t ❧➟♣ ❤➔♠ ❦❤♦↔♥❣ ❧➔ ❈ ✲ ❤➔♠✱ ✤â ❧➔ ❤➔♠ φ : R2 → R t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t✿ φ(a, b) = ⇔ ab = 0, a ≥ 0, b ≥ ❚❛ ❝â ♠ët sè ❈ ✲ ❤➔♠ s❛✉✿ ✶✳ φN R (a, b) = {a, b} ; ✷✳ φM S (a, b) = ab + 2α1 (max{0, a − αb}2 − a2 +max {0, b − αa}2 − b2), α > 1; ✸✳ φF B (a, b) = √ a2 + b2 − a − b ❍➔♠ ❦❤♦↔♥❣ ✤÷đ❝ ①➙② ❞ü♥❣ tr➯♥ ❤➔♠ φN R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ sè ❞÷ tü ♥❤✐➯♥✳ ❍➔♠ φF B ❦❤ỉ♥❣ ➙♠ tr➯♥ R2 ✈➔ ❤➔♠ ❦❤♦↔♥❣ ✤÷đ❝ ①➙② ❞ü♥❣ tr➯♥ ♥â ❣å✐ ❧➔ ✹ ❤➔♠ ▲❛❣r❛♥❣❡ ➞♥ ✤÷đ❝ ✤÷❛ ✈➔♦ ❜ð✐ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ♥❤÷ ▼❛♥❣❛s❛r✐❛♥ ✈➔ ❙♦❧♦❞♦✈✳ ❍➔♠ φF B ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❋✐s❝❤❡r✳ ●➛♥ ✤➙②✱ ❞ü❛ tr➯♥ ❤➔♠ φF B ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ ✤➣ ♠ð rở ự ữ r ởt số ợ ❝â t➼♥❤ ❝❤➜t tèt ❤ì♥✳ ▲✉♦ ✈➔ ❚s❡♥❣ ✤➣ ✤÷❛ r❛ ♠ët ❧ỵ♣ ❝→❝ ❤➔♠ ❦❤♦↔♥❣ ♠ỵ✐ f˜ : Rn → R ①→❝ ✤à♥❤ ❜ð✐ n f˜(x) = ψ0 ( x, F (x) Rn ) ψi (−xi , −Fi ), + i=1 ð ✤➙② ψ0 :→ [0, ∞) ✈➔ ψi : R2 → [0, ∞) , i = 1, 2, , n tử ị tữ ợ ♥➔② ✤÷đ❝ ❑❛♥③♦✇ ❈✳✱ ❨❛♠❛s❤✐t❛ ◆✳ ✈➔ ❋✉❦✉s❤✐♠❛ ▼✳ ❬✶✵❪ sỷ ỹ ợ Pữỡ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❜➡♥❣ ❝→❝❤ ♥❤✐➵✉ ❤➔♠ ❜❛♥ ✤➛✉ t❤➔♥❤ ♠ët ❞➣② t t ữủ ỗ tr ố ợ t ũ ỗ ✈✐➺❝ ❣✐↔✐ ❞➣② ❝→❝ ❜➔✐ t♦→♥✿ x ≥ 0, Fε (x) ≥ 0, x, Fε (x) Rn = 0, (0.5) ð ✤➙②✱ Fε(x) = F (x) + εx ✈➔ ε t số ữỡ tử tợ Pữỡ ♣❤→♣ ❦➳t ❤ñ♣ ❤➔♠ ❦❤♦↔♥❣ ✈➔ ❤✐➺✉ ❝❤➾♥❤ ✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ t❛ ❞ü❛ tr➯♥ ❤➔♠ H ❧➔ ❤➔♠ ✤✐ tứ ổ Rn+1 tợ Rn+1 ữủ ỹ ❜ð✐ H (ε, z) = ⇔ ε = 0, x ∈ S , tr♦♥❣ ✤â S ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ✭✵✳✹✮✱ z := (ε, x) ∈ R × Rn , H(ε, z) := ε, G(ε, z) (0.6) ✸✽ ❉➵ t❤➜② ♠è✐ q✉❛♥ ❤➺ LICQ ⇒ ACQ ⇒ GCQ ●✐↔ sû X ❧➔ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭✷✳✶✮ ✈➔ ①→❝ ✤à♥❤ ❝→❝ t➟♣ ❝❤➾ sè ✤è✐ ✈ỵ✐ x∗ ∈ X ♥❤÷ s❛✉ Ig˜ (x∗ ) := {i |˜ gi (x∗ ) = 0} , ˆ i (x∗ ) > , I0+ (x∗ ) := i gˆi (x∗ ) = 0, h ˆ i (x∗ ) = , I00 (x∗ ) := i gˆi (x∗ ) = 0, h ˆ i (x∗ ) = I+0 (x∗ ) := i gˆi (x∗ ) = 0, h ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❈❤♦ x∗ ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ ▼P❊❈ ✭✷✳✶✮✱ x∗ ✤÷đ❝ ❣å✐ ❧➔✿ ✐✮✳ ❉ø♥❣ ②➳✉ ♥➳✉ ♥â ❝â ❜ë λ ∈ Rm, µ ∈ Rp, λ, ν ∈ Rq s❛♦ ❝❤♦ p m ∗ ∇f (x ) + λi ∇˜ gi (x ) + i=1 i=1 q q ˆ i (x∗ ) = νi ∇h ∗ − γi ∇ˆ gi (x ) − i=1 ˜ i (x∗ ) µi ∇h ∗ i=1 ✈➔ λi ≥ 0, λi g˜i (x∗ ) = (i = 1, 2, , m) , γi = (i ∈ I+0 (x∗ )) , νi = (i ∈ I0+ (x∗ )) ; ✐✐✮✳ C ✲ ❞ø♥❣ ♥➳✉ ♥â ❧➔ ❞ø♥❣ ②➳✉ ✈➔ λiνi ≥ ∀i ∈ I00 (x∗) ❀ ✐✐✐✮✳ M ✲ ❞ø♥❣ ♥➳✉ ♥â ❧➔ ❞ø♥❣ ②➳✉ ✈➔ λi > 0, νi > ❤♦➦❝ λiνi = ✈ỵ✐ ♠å✐ i ∈ I00 (x∗ ) ; ✐✈✮✳ ❉ø♥❣ ♠↕♥❤ ♥➳✉ ♥â ❧➔ ✤✐➸♠ ❞ø♥❣ ②➳✉ ✈➔ λi > 0, νi > ✈ỵ✐ ♠å✐ i ∈ I00 (x∗ ) ; ✈✮✳ B ✲ ❞ø♥❣ ♥➳✉ ∇f (x∗)T d ≥ ✈ỵ✐ ♠å✐ d ∈ TX (x∗)✳ ✸✾ ❈→❝ ❦❤→✐ ♥✐➺♠ tr➯♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥❤❛✉✱ ❞➵ t❤➜② ❉ø♥❣ ♠↕♥❤ ⇒ M ✲ ❞ø♥❣ ⇒ C ✲ ❞ø♥❣ ⇒ ❞ø♥❣ ②➳✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳ ▼ët ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x∗ ❝õ❛ ▼P❊❈ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✐✮✳ ▼P❊❈ ✲ ▲■❈◗ ♥➳✉ ❣r❛❞✐❡♥ts ˜ i (x∗ ) |i = 1, 2, , p {∇˜ gi (x∗ ) |i ∈ Ig˜ (x∗ )} ∪ ∇h ˆ i (x∗ ) |i ∈ I00 (x∗ ) ∪ I+0 (x∗ ) ∪ {∇ˆ gi (x∗ ) |i ∈ I00 (x∗ ) ∪ I0+ (x∗ )} ∪ ∇h ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ ✐✐✮✳ ▼P❊❈ ✲ ●❈◗ ♥➳✉ TX (x∗)o = LM P EC (x∗)o, ð ✤➙② ||LM P EC (x∗ ) := {d ∈ Rn ∇˜ gi (x∗ )T d ≤ 0, ∀i ∈ Ig˜ (x∗ ) , ˜ i (x∗ )T d = 0, ∀i = 1, 2, , p, ∇h ∇ˆ gi (x∗ )T d = 0, ∀i ∈ I0+ (x∗ ) , ˆ i (x∗ )T d = 0, ∀i ∈ I+0 (x∗ ) , ∇h ˆ i (x∗ )T d ≥ 0, ∀i ∈ I00 (x∗ ) , ∇ˆ gi (x∗ )T d ≥ 0, ∇h ∇ˆ gi (x∗ )T d ˆ i (x∗ )T d = 0, ∀i ∈ I00 (x∗ )} ∇h ✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ ▼P❊❈ ❑❛❞r❛♥✐✱ ❉✉ss❛✉❧t ✈➔ ❇❡♥❝❤❛❦r♦✉♥ ❬✾❪ ✤➣ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❜➡♥❣ ❝→❝❤ t❤❛② t❤➳ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ❜ò ˆ i (x) ≥ 0, gˆi (x) , h ˆ i (x) = 0, ∀i = 1, 2, , q, gˆi (x) ≥ 0, h ✭✷✳✸✮ ❜ð✐ ˆ i (x) ≥ −t, gˆi (x) − t, h ˆ i (x) − t ≤ 0, ∀i = 1, 2, , q gˆi (x) ≥ −t, h ✈ỵ✐ số t > t ỵ tt ữỡ tốt ỡ s ợ ữỡ trữợ õ t t ữủ t♦→♥ ❤✐➺✉ ❝❤➾♥❤ ❤➛✉ ❤➳t ❧➔ rí✐ r↕❝ ✈➻ t❤➳ ♠ët sè ❜➔✐ t♦→♥ ❣➦♣ ❦❤â ❦❤➠♥ ❦❤✐ ❣✐↔✐✳ ✣➸ ❧♦↕✐ ❜ä ♥❤ú♥❣ ♥❤÷đ❝ ✤✐➸♠ ♥➔②✱ ❑❛♥③♦✇ ✈➔ ❙❝❤✇❛rt③ ❬✶✶❪ ỹ ữỡ ợ tử tợ M ứ ữ s ❈❤♦ ❤➔♠ ϕ : Rn → R ①→❝ ✤à♥❤ ❜ð✐ ab ♥➳✉ a + b ≥ 0, ϕ (a, b) := ♥➳✉ a + b < − a2 + b ❉ü❛ ✈➔♦ ❤➔♠ ϕ✱ ❤å ✤➣ ①→❝ ✤à♥❤ →♥❤ ①↕ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ φ : Rn → Rq s❛♦ ❝❤♦ ˆ i (x) − t φi (x; t) : = ϕ gˆi (x) − t, h  ˆ i (x) − t ♥➳✉ gˆi (x) + h ˆ i (x) ≥ 2t,  gˆi (x) − t, h = 2  − (ˆ ˆ i (x) − t ♥➳✉ gˆi (x) + h ˆ i (x) < 2t, g (x) − t) + h i ð ✤➙② t ≥ ❧➔ ♠ët t số tũ ỵ õ õ t ỹ ❜➔✐ t♦→♥ ❤✐➺✉ ❝❤➾♥❤ ◆▲P✭t✮✱ t ≥ ♥❤÷ s❛✉ f (x) ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ g˜i (x) ≤ ∀i = 1, 2, , m, gˆi (x) ≥ ∀i = 1, 2, , q, ˆ i (x) ≥ ∀i = 1, 2, , q, h φi (x; t) ≤ ∀i = 1, 2, , q ❚❤❛② ✤✐➲✉ ❦✐➺♥ ❜ò ✭✷✳✸✮ ❜ð✐ ˆ i (x) ≥ 0, φi (x; t) ≤ ∀i = 1, 2, , q gˆi (x) ≥ 0, h ✈➔ ①→❝ ✤à♥❤ ❝→❝ ❝➦♣ ❝❤➾ sè✿ Ig˜(x) := {i |˜ gi (x) = 0} , Igˆ(x) := {i |ˆ gi (x) = 0} , ˆ i (x) = , Ihˆ (x) := i h Iφ (x; t) := {i |φi (x; t) = 0} , ✹✶ ✈ỵ✐ t ≥ x ữủ ố ợ Pt P❤➙♥ ❤♦↕❝❤ t➟♣ ❝❤➾ sè Iφ (x; t) s❛♦ ❝❤♦ ˆ i (x) − t = , Iφ00 (x; t) := i ∈ Iφ (x; t) gˆi (x) − t = 0, h ˆ i (x) − t > , Iφ0+ (x; t) := i ∈ Iφ (x; t) gˆi (x) − t = 0, h ˆ i (x) − t = Iφ+0 (x; t) := i ∈ Iφ (x; t) gˆi (x) − t > 0, h ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ φ t❛ s✉② r❛ ˆ i (x) − t ≥ 0, (ˆ ˆ i (x) − t = φi (x; t) = ⇔ gˆi (x) − t ≥ 0, h gi (x) − t) h ❚❤❛② t ❜➡♥❣ {tk }✱ ❤å ✤➣ t ữủ t q s ỵ {tk } ↓ ✈➔ xk , λk , µk , γ k , ν k , δ k ❧➔ ♠ët ❞➣② ✤✐➸♠ ❑❑❚ ❝õ❛ N LP (tk ) ✈ỵ✐ xk → x∗ ✳ ◆➳✉ ▼P❊❈ ✲ ▲■❈◗ ✤ó♥❣ t↕✐ x∗ t❤➻ x∗ ❧➔ ♠ët ✤✐➸♠ ▼ ✲ ❞ø♥❣ ❝õ❛ t P ỵ {tk } ↓ ✈➔ xk , λk , µk , γ k , ν k , δ k ❧➔ ♠ët ❞➣② ✤✐➸♠ ❑❑❚ ❝õ❛ N LP (tk ) ✈ỵ✐ xk → x∗ ✳ ◆➳✉ ▼P❊❈ ✲ ▲■❈◗ ✤ó♥❣ t↕✐ x∗ ✈➔ ❝â K ⊆ N s❛♦ ❝❤♦ gi xk ≤ tk , hi xk ≤ tk ✈ỵ✐ ♠å✐ k ∈ K ✈ỵ✐ ♠å✐ i ∈ I00 (x∗ ) ✤ó♥❣ t❤➻ x∗ ❧➔ ♠ët ✤✐➸♠ ❞ø♥❣ ♠↕♥❤ ❝õ❛ ❜➔✐ t♦→♥ ▼P❊❈ ✭✷✳✶✮✳ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ ✤➦t r❛ ✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❝ü❝ trà ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❧➔ ❜➔✐ t♦→♥ ❜ò tê♥❣ q✉→t✱ t❛ ✤➦t ˆ (x) ✈➔ ①➨t ❜➔✐ t♦→♥ ❝ü❝ trà✿ x˜ ∈ C ∩ S˜✱ t❤ä❛ g (x) = −ˆ g (x) , h (x) = −h ♠➣♥ ✤✐➲✉ ❦✐➺♥ ϕ (˜ x) = ϕ (y) , y∈C∩S˜ ✭✷✳✹✮ ð ✤➙② C t õ ỗ tr ổ Rn S = S˜1 ∩ S˜2✱ tr♦♥❣ ✤â ˜ (x) = , S˜1 = x ∈ Rn : g˜ (x) ≤ 0, h ✹✷ S˜2 = {x ∈ Rn : g (x) ≤ 0, h (x) ≤ 0, g (x) , h (x) Rm = 0} , ✭✷✳✺✮ ❝→❝ ❤➔♠ t❤ü❝ ϕ : Rn → R, g˜ : Rn → Rm, h˜ : Rn → Rp✱ g ✈➔ h : Rn → Rq ❧➔ ❧✐➯♥ tö❝✳ ❚❛ ❣✐↔ t❤✐➳t t➟♣ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✹✮✲✭✷✳✺✮ ❦❤→❝ ré♥❣✳ ❑❤✐ S˜2 = Rn ✭✷✳✹✮✲✭✷✳✺✮ ❧➔ ❜➔✐ t♦→♥ ❝ü❝ trà ♣❤✐ t✉②➳♥ ❝â r➔♥❣ ❜✉ë❝ tê♥❣ q✉→t✳ ❇➔✐ t♦→♥ ♥➔② ✤÷đ❝ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝ ♥❤÷ s❛✉ F˜α (xα ) = F˜α (x) , α > 0, n x∈ F˜α (xα ) = F˜ (x) + α x − x∗ Rn m Rp αµj ψj (x) + αµm+1 ϕ (x) + ✭✷✳✻✮ j=1 , ≤ µ1 ≤ µ2 < < µm+1 < 1, ð ✤➙② F˜ (x) = f˜1 (x) , f˜2 (x) , , f˜p (x) T ˜ i (x) , f˜i = h ✈➔ ψj (x) = max {0, g˜j (x)}✳ ❚✐➳♣ t❤❡♦✱ ✤➸ ❝↔✐ t✐➳♥ ✭✷✳✻✮✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ♠ỵ✐ ❦✐➸✉ ❚✐❦❤♦♥♦✈ ❝❤♦ ▼P❊❈ ✭✷✳✹✮✲✭✷✳✺✮✳ ✣➸ ❧➔♠ ✤✐➲✉ ♥➔②✱ t❛ ✤➦t✿ ϕj (x) = max {0, g˜j (x)} , j = 1, 2, , m, ϕm+j (x) = max {0, gj (x)} , j = 1, 2, , q, ϕm+q+j (x) = max {0, hj (x)} , j = 1, 2, , q, ˜ i (x) , i = 1, 2, , p, fi (x) = h fp+i (x) = gi (x)hi (x) , i = 1, 2, , q ❚❤❡♦ ❣✐↔ t❤✐➳t✱ ❧✐➯♥ tö❝✱ ❤➔♠ ϕj ✈➔ fj ữ tr ợ j = 1, 2, , N (:= m + 2q) ✈➔ i = 1, 2, , M (:= p + q) ❧➔ ❧✐➯♥ tö❝✳ ❉➵ ❝❤➾ r❛ r➡♥❣ ˜ g, h g˜, h, Sj := {x ∈ Rn : g˜j (x) ≤ 0} = {x ∈ Rn : ϕj (x) = 0} , j = 1, m, Sm+j := {x ∈ Rn : gj (x) ≤ 0} = {x ∈ Rn : ϕm+j (x) = 0} , j = 1, q, ✹✸ Sm+q+j := {x ∈ Rn : hj (x) ≤ 0} = {x ∈ Rn : ϕm+q+j (x) = 0} , j = 1, m, ❞♦ ϕj (x) ≥ ✈ỵ✐ ♠å✐ x ∈ Rn, j = 1, 2, , N ✳ ❚❛ ①➨t t➟♣ ˜ (x) = 0, g (x) , h (x) S0 := x ∈ Rn : h Rn =0 , ❘ã r➔♥❣ S0 := {x ∈ Rn : fi (x) = 0, i = 1, 2, , M } ❚ø fi ✈➔ ϕj ❧✐➯♥ tö❝✱ Sj ✱ j = 1, 2, , N ✱ ✤â♥❣✳ ❍ì♥ ♥ú❛✱ t❛ ❝â S˜ = Nj=0 Sj ✈➔ C = {x ∈ Rn : x PC (x) = 0} ỵ ❤✐➺✉ PC (x) ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝õ❛ ♣❤➛♥ tû x ∈ Rn ❧➯♥ C ✳ ❇ð✐ ✈➟②✱ ❜➔✐ t♦→♥ P tữỡ ữỡ ợ t tố ữ õ r ❜✉ë❝ ✤➥♥❣ t❤ù❝ s❛✉✿ t➻♠ ♠ët ♣❤➛♥ tû x˜ ∈ C ∩ ∩Nj=0Sj s❛♦ ❝❤♦ ϕ (˜ x) = ✭✷✳✼✮ ϕ (x) x∈C∩(∩N j=0 Sj ) ✣➸ ❣✐↔✐ ✭✷✳✼✮✱ ❝➛♥ ♣❤↔✐ ❣✐↔✐ ❤❛✐ ❜➔✐ t♦→♥ ❝♦♥ ✐✮✳ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ ϕj (x) = 0, fi (x) = ✈ỵ✐ j = 1, 2, , N, i = 1, 2, , M ✈➔ x − Pc (x) = 0; ✐✐✮✳ ❝ü❝ t✐➸✉ ❤➔♠ ϕ (x) tr➯♥ t➟♣ C ∩ ∩Nj=0Sj ❍✐➸♥ ♥❤✐➯♥✱ ♠é✐ ❜➔✐ t♦→♥ ổ ỗ tớ ❜➔✐ t♦→♥ ❝♦♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝ü❝ t✐➸✉ ❤➔♠ trì♥ ❚✐❦❤♦♥♦✈ tr➯♥ t♦➔♥ ❜ë ❦❤ỉ♥❣ ❣✐❛♥ Rn✳ ❈ư t❤➸✱ ❝❤ó♥❣ tæ✐ t➻♠ ♠ët ♣❤➛♥ tû xα ∈ Rn s❛♦ ❝❤♦ Fα (xα ) = minn Fα (x) , x∈R Fα (x) = x − Pc (x) RM ✭✷✳✽✮ N ϕj (x) + αµ ϕ (x) + α x − x∗ + Rn , j=1 ❱ỵ✐ ♠é✐ t❤❛♠ sè ❝è ✤à♥❤ α > 0✱ ð ✤➙② F (x) = (f1 (x) , f2 (x) , .fM (x))T , µ ∈ (0, 1) ✹✹ ❧➔ ♠ët sè ❝è ✤à♥❤ ✈➔ x∗ ❧➔ ♣❤➛♥ tû ♥➔♦ ✤â tr♦♥❣ Rn✳ ●✐↔ sû ϕ (x) ≥ ✈ỵ✐ ♠å✐ x ∈ Rn ✈➔ ♥â ❝â t➟♣ ♠ù❝ ❣✐ỵ✐ ♥ë✐✱ ♥❣❤➽❛ ❧➔ t➟♣ {x ∈ Rn : ϕ (x) ≤ c} ❣✐ỵ✐ ♥ë✐ ✈ỵ✐ c > ❜➜t ❦ý✳ ◆❤÷ ✤➣ ❜✐➳t tr♦♥❣ ❬✶✼❪✱ ❜➔✐ t♦→♥ ✭✷✳✽✮ ❝â ♠ët ˜ g, h ❝â ♥❣❤✐➺♠ xα ✈ỵ✐ ♠å✐ α > 0✳ ❍ì♥ ♥ú❛✱ ♥❣❤✐➺♠ ✈➝♥ ê♥ ✤à♥❤ ❦❤✐ ϕ, g˜, h, ♥❤✐➵✉ ❬✶✸❪✳ ❈❤ó♥❣ tỉ✐ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ sü ❤ë✐ tư ❝õ❛ ♥❣❤✐➺♠ ❤✐➺✉ q ỵ s ỵ k → 0, ❦❤✐ k → ∞ t❤➻ ♠å✐ ❞➣② xk ✱ ð ✤➙② xk := xαk ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✽✮ ✈ỵ✐ α t❤❛② ❜ð✐ αk ✱ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tư✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ♠å✐ ❞➣② ❝♦♥ ❤ë✐ tö ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✹✮✲✭✷✳✺✮✳ ◆➳✉ t❤➯♠ ✤✐➲✉ ❦✐➺♥ õ ởt t ỵ x t❤➻ lim xk = x˜ k→∞ ❉♦ xk ❧➔ ♠ët ❝ü❝ t✐➸✉ ❝õ❛ Fα (x)✱ tø ✭✷✳✽✮ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ k k x − PC x Rn k + F x RM N ϕj (xk ) + αkµ ϕ xk + j=1 + α k xk − x∗ Rn ≤ y − PC (y) Rn + F (y) RM ✭✷✳✾✮ N ϕj (y) + αkµ ϕ (y) + αk y − x∗ + Rn , j=1 ✈ỵ✐ ♣❤➛♥ tû ❝è ✤à♥❤ y ∈ Rn ❜➜t ❦ý✳ ▲➜② y ∈ C ∩ ∩Nj=0Sj ✱ t❛ ❝â y −PC (y) = = F (y) ✈➔ ϕj xk ≥ 0, j = 1, 2, , N ✳ ❚ø ✭✷✳✾✮ s✉② r❛ ϕ xk ≤ ϕ (y) + αk1−µ y − x∗ n R ✭✷✳✶✵✮ ❚ø ϕ ❧➔ ❝❤➦♥ ✤➲✉✱ t➟♣ xk ❜à ❝❤➦♥✳ ▲➜② xl ⊂ xk s❛♦ ❝❤♦ xl → x¯ ❦❤✐ l → ∞ ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ x¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✹✮✲✭✷✳✺✮✳ ❚ø ✭✷✳✾✮ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ϕ ✈➔ ϕj ✱ t❛ ❝â ≤ x k − PC x k Rn , F xl RM , ϕj xl ≤ αlµ ϕ (y) + αl y − x∗ Rn ✹✺ ❈❤♦ l → ∞ ð ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐✱ ♥❤í t➼♥❤ ❝❤➜t ❧✐➯♥ tư❝ ❝õ❛ PC ([17])✱ F ✈➔ ϕj ✤↔♠ ❜↔♦ r➡♥❣ x¯ = PC (¯ x) , F (¯ x) = = ϕj (¯ x)✱ ♥❣❤➽❛ ❧➔ x¯ ∈ C∩ ∩N j=0 Sj ✳ ❇➙② ❣✐í✱ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ x¯ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✹✮✲ ✭✷✳✺✮✳ ❚ø ✭✷✳✶✵✮ t❤❛② k ❜ð✐ l✱ t ữủ ( x) (y) ợ y ❜➜t ❦ý t❤✉ë❝ C ∩ ∩N j=0 Sj ✳ ❉♦ ✤â x¯ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✼✮✳ ❘ã r➔♥❣✱ ♥➳✉ x¯ ❧➔ ❞✉② ♥❤➜t t❤➻ x¯ = x˜ ✈➔ t➜t ❝↔ ❞➣② xk ❤ë✐ tư tỵ✐ x˜ ❦❤✐ k → ∞✳ ●✐↔ sû t❤❛② {ϕ, C, fi, ϕj } ❜ð✐ ❝→❝ ♥❤✐➵✉ ❧➔ ϕδ , Cδ , ϕδj s❛♦ ❝❤♦ ϕ (x) − ϕδ (x) ≤ δ, H (Cδ , C) ≤ δ, ✭✷✳✶✶✮ fi (x) − fi δ (x) ≤ δ, i = 1, 2, , M, ϕj (x) − ϕδj (x) ≤ δ, j = 1, 2, , N, ∀x ∈ Rn , δ → 0, ˜ g, h ✤÷đ❝ ❝❤♦ ①➜♣ ①➾ ð ✤➙② fiδ ✈➔ ϕδj ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ fi ✈➔ ϕj ð tr➯♥ g˜, h, ✈➔ H (Cδ , C) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❍❛✉s❞♦r❢❢ ỳ t õ ỗ C C t t♦→♥ tè✐ ÷✉ ❦❤ỉ♥❣ r➔♥❣ ❜✉ë❝ s❛✉ ❚➻♠ ♠ët ♣❤➛♥ tû xδα tr♦♥❣ Rn s❛♦ ❝❤♦ Fδα xδα = minn Fαδ (x) , α > 0, δ ≥ 0, x∈R Fδα (x) = x − PCδ (x) Rn δ + F (x) RM N ϕδj (x) + ✭✷✳✶✷✮ j=1 µ δ + α ϕ (x) + α x − x∗ 2Rn Ð ✤➙② F (x) = f1δ (x) , f2δ (x) , , fMδ (x) T ✳ ❚❤❡♦ ❬✶✼❪✱ ✈ỵ✐ ♠é✐ α > ❜➔✐ t♦→♥ ✭✷✳✶✷✮ õ ỵ x ú tổ t t ữủ t q s ỵ αk , δk → s❛♦ ❝❤♦ αδ k k → ❦❤✐ k → ∞ t❤➻ ♠å✐ ❞➣② xk ✱ ð ✤➙② xk := xδαkk ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ợ ữủt t k ✈➔ δk ✱ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tư✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ♠å✐ ❞➣② ❝♦♥ ❤ë✐ tö ❧➔ ♠ët ♥❣❤✐➺♠ õ ởt t ỵ x˜ t❤➻ lim xk = x˜ k→∞ ✹✻ ❈❤ù♥❣ ♠✐♥❤✳ k ❚ø ✭✷✳✶✷✮ t❛ ❝â k x − PCδk x + F n δk x R + F (y) RM N ϕδjk xk + j=1 + αkδ ϕδk xk + αk xk − x∗ δk RM k Rn ≤ y − PCδk (y) ✭✷✳✶✸✮ Rn N ϕδjk (y) + αkµ ϕδk (y) + αk y − x∗ + Rn , j=1 ✈ỵ✐ ♣❤➛♥ tû y ∈ Rn ❜➜t ❦ý✳ ▲➜② y ∈ C ∩ ∩Nj=0Sj t❛ ❝â y − PC (y) 0, F (y) = ✈➔ ϕj xk ≥ ϕj (y) = 0, j = 1, 2, , N ❚ø ✭✷✳✶✸✮ s✉② r❛ k k x − PCδk x + F δk xk n R ≤ PC (y) − PCδk (y) RM Rn + F δk (y) + αkµ ϕδk xk RM ✭✷✳✶✹✮ N ϕδjk (y) − ϕj (y) + αkµ ϕδk (y) + αk y − x∗ + = Rn j=1 ❚ø ✭✷✳✶✶✮ ✈➔ ✭✷✳✶✹✮ s✉② r❛ ϕ δk k x δk (1 + M ) δk2 1−µ y − x∗ + N ≤ ϕ (y) + µ µ + αk αk αk δk Rn , ✈➻ t❤➳ δk (1 + M ) δk2 + N µ + αk1−µ y − x∗ µ αk αk ϕ xk ≤ ϕ (y) + 2δk + Rn ✭✷✳✶✺✮ ❚ø δk , αk , αδ → ❦❤✐ k → ∞ ✈➔ ϕ ❧➔ ♠ù❝ ❝❤➦♥✱ t➟♣ xk ❜à ❝❤➦♥✳ ▲➜② xl ❧➔ ❞➣② ❝♦♥ ❝õ❛ ❞➣② xk s❛♦ ❝❤♦ xl → x¯ ❦❤✐ l → ∞✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ x¯ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✹✮✲✭✷✳✺✮✳ ❚❤➟t ✈➟②✱ tø ✭✷✳✶✹✮ ✈➔ ϕ (x) ≥ ✈ỵ✐ x ∈ Rn t❛ ❝â k k xl − PCδl xl ≤ (1 + M ) δl2 2 Rn 2δl αlµ + l δl + F x n R + αlµ ϕ (y) + N δ1 + αl y − x∗ Rn ❈❤♦ l → ∞ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐✱ ♥❤í t➼♥❤ ❝❤➜t ❧✐➯♥ tư❝ ❝õ❛ PC , F, αl , δl → ✈➔ ✭✷✳✶✶✮✱ t❛ ❝â x¯ − PC (¯ x) = = F (¯ x)✱ ♥❣❤➽❛ ❧➔ x¯ ∈ C ∩ S0 ✹✼ ❇➙② ❣✐í✱ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ x¯ ∈ Sj ✈ỵ✐ j = 1, 2, , N ❱ỵ✐ ♣❤➛♥ tû ❜➜t ❦ý y ∈ C ∩ ∩Nj=0Sj ϕj (y) = fi (y) = tø ✭✷✳✶✸✮ ✈➔ ϕj xl ≥ 0, t❛ ❝â ϕδi l xl + αlµ ϕδl xl ≤ PC (y) − PCδl (y) + F δl (y) − F (y) n R RM + ϕδi l (y) N + αlµ ϕδl (y) + αl y − x∗ ϕδjl (y) − ϕδjl xl + Rn j=i ≤ PC (y) − PCδl (y) δl + F (y) − F (y) n R RM + ϕδi l (y) N ϕδjl (y) − ϕj (y) + ϕj xl − ϕδjl xl + + αlµ ϕδl (y) + αl y − x∗ Rn j=i ❚✐➳♣ t❤❡♦✱ tø ϕ xl ≥ t❛ ❝â✿ ≤ ϕi xl ≤ (1 + M ) δl2 + 2δl + (N − 1) δl + ϕδi l (y) + αlµ ϕδl (y) + ϕ(xl ) − ϕδl xl + αl y − x∗ Rn ≤ (1 + M ) δl2 + 2δl + (N − 1) δl + ϕδi l (y) + αlµ ϕδl (y) + αlµ δl + αl y − x∗ Rn ❈❤♦ l → ∞ ð ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐✱ t❛ ❝â ≤ ϕi (¯x) ≤ ϕi (y) = 0✳ ◆❣❤➽❛ ❧➔ x¯ ∈ Si ❈✉è✐ ❝ò♥❣✱ ❝➛♥ ♣❤↔✐ ự x (2.7) ữ tr ợ ❜➜t ❦ý ♣❤➛♥ tû y ∈ C ∩ ∩Nj=0Sj ✱ tø ✭✷✳✶✺✮ s✉② r❛ ϕ (¯x) ≤ ϕ (y) ∀y ∈ C ∩ ∩N ˜ ❧➔ ❞✉② ♥❤➜t t❤➻ ♠å✐ ❞➣② xk ❤ë✐ tư tỵ✐ x˜ j=0 Sj ✳ ❘ã r x k ỵ ữủ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ❳➨t ❤➔♠✿ ϕ (x) = (x1 − 3)2 + (x2 − 2)2 + x23 , ✹✽ ˜ ❤➔♠ g˜ (x) ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ g˜ (x) ≤ ✈ỵ✐ ♠å✐ x ∈ R3, h(x) = Ax − b, ð ✤➙② A ❧➔ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ ✸ ❝â a22 = 1, aij = ✈ỵ✐ i, j = ✈➔ b = (b1 , b2 , b3 )T = (0, 1, 0)T , ❝→❝ ❤➔♠ g, h : R3 → R2 ❝â ❞↕♥❣ ♥❤÷ s❛✉✿ g (x1 , x2 , x3 ) = x21 + x22 + x23 − 25, −x1 + , h (x1 , x2 , x3 ) = (x1 − 3)2 + x22 + x23 − 4, x1 − , √ ✈➔ t➟♣ C = x ∈ R3 : (x1 − 3)2 + x22 + x3 − ≤ ✳ ❉➵ t❤➜② ❜➔✐ √ t♦→♥ ✭✷✳✹✮✲✭✷✳✺✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t x˜ = 3, 1, ✳ ❈❤å♥ µ = 12 ✈ỵ✐ x∗ = (0, 0, 0)✱ t❤❛♠ sè ❤✐➺✉ ❝❤➾♥❤ αk = (1 + k)−1 ✈➔ ✤✐➸♠ ❜❛♥ ✤➛✉ ❝❤♦ tr÷í♥❣ ❤đ♣ k = ❧➔ x0 = (20, 45, 15) ✤➸ ❝ü❝ t✐➸✉ ❤➔♠ trì♥ ❚✐❦❤♦♥♦✈ Fαk (x) = x − PC (x) R3 + F (x) R3 ϕ˜j (x) + αk2 ϕ (x) + αk x + R3 j=1 ð ✤➙② (x1 − 3)2 + x22 + x23 − , (−x1 + 3) (x1 − 4) F (x) = x2 − 1, x21 + x22 + x23 − 25 ✈➔ x21 + x22 + x23 − 25 ϕ˜1 (x) = ϕ1 (x) = x21 + x22 + x23 − 25 > x21 + x22 + x23 − 25 ≤ 0, ✈➔ x2, x3 ❜➜t ❦ý ✈➔ x2, x3 ❜➜t ❦ý , (x1 − 3)2 + x22 + x23 − ♥➳✉ (x1 − 3)2 + x22 + x23 − > ϕ˜3 (x) = ϕ3 (x) = ♥➳✉ (x1 − 3)2 + x22 + x23 − ≤ 0, (x1 − 4)2 ♥➳✉ x1 > ✈➔ x2 , x3 ❜➜t ❦ý ϕ˜4 (x) = ϕ4 (x) = ♥➳✉ x1 ≤ ✈➔ x2, x3 ❜➜t ❦ý ❚r♦♥❣ tr÷í♥❣ ❤đ♣ {C, fi, ϕi, ϕ} ❝â ♥❤✐➵✉ δk ✱ ❝❤ó♥❣ tỉ✐ t➼♥❤ Cδ , fiδ , ϕδ ✈ỵ✐ ϕ˜2 (x) = ϕ22 (x) (x1 − 3)2 = ♥➳✉ ♥➳✉ ♥➳✉ ♥➳✉ x1 > x1 ≤ k k Cδk = x ∈ R : (x1 − 3) + x22 + x3 − √ ≤ (1 + δk )2 , k ✹✾ ϕδk (x) = ϕ (x) + δk , g δk (x) = g (x) , bδk = b + δk , ✈➔ g δk (x1 , x2 , x3 ) = x21 + x22 + x23 − 25 + δk , −x1 + + δk , hδk (x1 , x2 , x3 ) = (x1 − 3)2 + x22 + x23 − + δk , x1 − + δk ❇➔✐ t♦→♥ ❜❛♥ ✤➛✉ ❧➔ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ✈➔ t❤➯♠ ♥➳✉ bδ1 ❤♦➦❝ bδ3 ❦❤→❝ ✵ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ Ax=bδ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✳ ❉♦ ✤â✱ ❜➔✐ t♦→♥ ♥❤✐➵✉ ❝ơ♥❣ s ổ ữủ ợ ữỡ ♠ỵ✐ ♥➔② t❤➻ xk ❧➔ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ s❛✉✿ k k k Fαδkk (x) = x − PCδk (x) + F δk (x) R ϕ˜δjk (x) + αk2 ϕδk (x) + αk x + R3 R3 , j=1 ữủ t t ợ k = (1 + k)−2 ✈➔ αk = (1 + k)−1, ð ✤➙② F δk (x) = δk , x2 − + δk , δk , f4δk , f5δk , f4δk = x21 + x22 + x23 − 25 + δk , (x1 − 3)2 + x22 + x23 − + δk , f5δk = (−x1 + + δk ) (x1 − + δk ) , ✈➔ ϕ˜δj k (x) , ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ ϕ˜j (x) tr t r ữỡ ợ t ữỡ t ỹ tr ợ r➔♥❣ ❜✉ë❝ ❧➔ ❜➔✐ t♦→♥ ❜ò tê♥❣ q✉→t✳ ✺✵ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ❚â♠ ❧↕✐✱ ❜➔✐ t♦→♥ ❜ò ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳✳ ❉♦ ✤â✱ ✈✐➺❝ t➻♠ r❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ r❛ ♥❣❤✐➺♠ ❝õ❛ ♥â ❧➔ r➜t q✉❛♥ trå♥❣✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❝ü❝ trà ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❧➔ ❜➔✐ t♦→♥ ❜ị tê♥❣ q✉→t✱ ✈➔ ❝→❝❤ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥ ✤â✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ♥❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✳ ✶✳ ◆❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ♥❤÷ P ✲ ❤➔♠✱ P0 ✲ ❤➔♠✱ P ✲ ❤➔♠ ✤➲✉✱ ❤➔♠ ✤ì♥ ✤✐➺✉✱ ❤➔♠ ✤ì♥ ✤✐➺✉ ♠↕♥❤✳✳✳✳ ✷✳ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✈➲ ❜➔✐ t♦→♥ ❜ò ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ tr♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ❜➔✐ t♦→♥ ❜ò t✉②➳♥ t➼♥❤ ✈➔ ❜➔✐ t♦→♥ ❜ò ♣❤✐ t✉②➳♥✳ ✸✳ ❚r➻♥❤ ❜➔② ✈➲ ❜➔✐ t♦→♥ ❜ò tê♥❣ q✉→t ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝õ❛ ♥â ✹✳ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ ❜ò tê♥❣ q✉→t✳ ✺✳ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❤✐➺✉ ❝❤➾♥❤ ❚✐❦❤♦♥♦✈ ❝❤♦ ❜➔✐ t♦→♥ tố ữ ợ r t ũ tờ q✉→t✳ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆✳ ❇✉♦♥❣ ✭✷✵✵✻✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ✉♥❝♦♥str❛✐♥❡❞ ✈❡❝t♦r ♦♣t✐✲ ♠✐③❛t✐♦♥ ♦❢ ❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ▼❛t❤✳ P❤②s✱ ✹✻✭✸✮✱ ♣♣✳ ✸✼✷✲✸✼✽✳ ❬✷❪ ◆✳ ❇✉♦♥❣ ❛♥❞ ◆✳ ❚✳ ❚✳ ❍♦❛ ✭✷✵✶✺✮✱ ✧❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠s ✇✐t❤ ❣❡♥❡r❛❧✐③❡❞ ❝♦♠♣❧❡♠❡♥t❛r✐t② ❝♦♥✲ str❛✐♥ts✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ▼❛t❤✳ ♣❤②s✱ ✺✺✭✹✮✱ ♣♣✳ ✺✻✹✲✺✼✶✳ ❬✸❪ ❇✉r❦❡ ❏✳ ❛♥❞ ❳✉ ❙✳ ✭✷✵✵✵✮✱ ✧❆ ♥♦♥✲✐♥t❡r✐♦r ♣r❡❞✐❝t♦r✲❝♦rr❡❝t♦r ♣❛t❤ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ♠♦♥♦t♦♥❡ ❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠ ✧✱ ▼❛t❤✳ Pr♦❣r❛♠✳✱ ✽✼✭✶✮✱♣♣✳✶✶✸✲✶✸✵✳ ❬✹❪ ❈❤❡♥ ✲ ❙✳ ❏ ❛♥❞ P❛♥ ❙✳ ✭✷✵✵✽✮✱ ✧❆ ❢❛♠✐❧② ♦❢ ◆❈P ❢✉♥❝t✐♦♥s ❛♥❞ ❛ ❞❡s❝❡♥t ♠❡t❤♦❞s ❢♦r t❤❡ ♥♦♥❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠✧✱ ❈♦♠♣✉t✳ ❖♣t✐♠✳ ❆♣♣❧✳✱ ✹✵✭✸✮✱ ♣♣✳ ✸✽✾✲✹✵✹✳ ❬✺❪ ❋❛❝❝❤✐♥❡✐ ❋✳ ❛♥❞ ❑❛♥③♦✇ ❈✳ ✭✶✾✾✼✮✱ ✧❜❡♦♥❞ ♠♦♥♦t♦♥✐❝✐t② ✐♥ r❡❣✲ ✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠s✧✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✱ ✾✭✷✮✱♣♣✳ ✶✶✺✵✲✶✶✻✶✳ ❬✻❪ ❋❛♥❣ ▲✳ ✭✷✵✶✵✮✱ ✧❆ ♥❡✇ ♦♥❡✲st❡♣ s♠♦♦t❤✐♥❣ ◆❡✇t♦♥ ♠❡t❤♦❞s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠s ✇✐t❤ P0✲ ❢✉♥❝t✐♦♥ ✧✱ ❆♣♣❧✳ ▼❛t❤✳ ❝♦♠♣✉t✳✱ ✷✶✻✱ ♣♣✳ ✶✵✽✼✲✶✵✾✺✳ ❬✼❪ ❋✐s❝❤❡r ❆✳ ✭✶✾✾✷✮✱ ✧❙♦❧✉t✐♦♥ ♦❢ ♠♦♥♦t♦♥❡ ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜✲ ❧❡♠s ✇✐t❤ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③✐❛♥ ❢✉♥❝t✐♦♥s✧✱ ▼❛t❤✳ Pr♦❣r❛♠✳✱ ✼✻✭✸✮✱ ♣♣✳ ✺✶✸✲✺✸✷✳ ✺✷ ❬✽❪ ●❡✐❣❡r ❈✳ ❛♥❞ ❑❛♥③♦✇ ❈✳ ✭✶✾✾✻✮✱ ✧❖♥ t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ ♠♦♥♦t♦♥❡ ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠s✧✱ ❈♦♠♣✉t✳ ❖♣t✐♠✳ ❆♣♣❧✳✱ ✺✭✷✮✱ ♣♣✳ ✶✺✺✲ ✶✼✸✳ ❬✾❪ ❑❛❞r❛♥✐ ❆✳✱ ❉✉ss❛✉❧t P✳ ❏✳ ❛♥❞ ❇❡♥❝❤❛❦r♦✉♥ ❆✳ ✭✷✵✵✾✮✱ ✧❆ ♥❡✇ r❡❣✉❧❛r✐③❛t✐♦♥ s❝❤❡♠❡ ❢♦r ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠s ✇✐t❤ ❝♦♠♣❧❡✲ ♠❡♥t❛r✐t② ❝♦♥str❛✐♥ts✧✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱✱ ✷✵✭✶✮✱ ♣♣✳ ✼✽✲✶✵✸✳ ❬✶✵❪ ❑❛♥③♦✇ ❈✳ ✭✶✾✾✼✮✱ ✧❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❝♦♥t✐♥✉❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❣r❛♠s ✇✐t❤ ✉♥✐❢♦r♠ P ✲ ❢✉♥❝t✐♦♥s✧✱ ❖♣❡r✳ ❘❡s✳ ▲❡tt✳✱ ✷✵✭✷✮✱ ♣♣✳ ✽✺✲✾✷✳ ❬✶✶❪ ❑❛♥③♦✇ ❈✳ ❛♥❞ ❙❝❤✇❛rt③ ❆✳ ✭✷✵✶✸✮✱ ✧❆ ♥❡✇ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠s ✇✐t❤ ❝♦♠♣❧❡♠❡♥t❛r✐t② ❝♦♥str❛✐♥ts ✇✐t❤ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♣r♦♣❡rt✐❡s✧✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱✱ ♣♣✳ ✼✼✵✲✼✾✽✳ ❬✶✷❪ ❑♦❥✐♠❛ ▼✳✱ ▼❡❣✐❞❞♦ ◆✳ ❛♥❞ ◆♦♠❛ ❚✳ ✭✶✾✾✶✮✱ ✧❍♦♠♦t♦♣② ❝♦♥t✐♥✲ ✉❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠s✧✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✳✱ ✶✻✭✹✮✱ ♣♣✳ ✼✺✹✲✼✼✹✳ ❬✶✸❪ ▲✐s❦♦✈❡ts ❆✳ ❖✳ ✭✶✾✽✶✮✱ ✧❱❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ ✉♥st❛❜❧❡ ♣r♦❜❧❡♠s✧✱ ◆❛✉❦❛✐ ❚❡❦❤♥✐❦❛✱ ▼✐♥s❦✳ ❬✶✹❪ ▼❛♥❣❛s❛r✐❛♥ ▲✳ ❖✳ ❛♥❞ ❙♦❧♦❞♦✈ ❱✳ ▼✳ ✭✶✾✾✾✮✱ ✧ ❆ ❧✐♥❡❛r❧② ❝♦♥✲ ✈❡r❣❡♥t ❞❡r✐✈❛t✐✈❡ ✲ ❢r❡❡ ❞❡s❝❡♥t ♠❡t❤♦❞s ❢♦r str♦♥❣❧② ♠♦♥♦t♦♥❡ ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦♣❡rt✐❡s✧✱ ❈♦♠♣✉t ✳ ❖♣t✐♠✳ ❆♣♣❧✳✱ ✶✹✱ ♣♣✳ ✶✺ ✲ ✶✻✳ ❬✶✺❪ ❖❧ss♦♥ ❉✳✭✷✵✶✵✮✱ ✧❚❤❡ ❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠✿ ▼❡t❤♦❞s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❙♣r✐♥❣❡r✱ ❙❋✷✽✷✼ ❚♦♣✐❝s ✐♥ ❖♣t✐♠✳ ❬✶✻❪ ❙✉♥ ❉✳ ✭✶✾✾✾✮✱ ✧❆ r❡❣✉❧❛r✐③❛t✐♦♥ ◆❡✇t♦♥ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ♥♦♥✲ ❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❣r❛♠s✧✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳✱ ✹✵✭✸✮✱ ♣♣✳ ✸✺✶✲✸✸✾✳ ❬✶✼❪ ❱❛s✐❧❡✈ P✳ ❋✳ ✭✶✾✽✵✮✱ ✧◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ❡①tr❡♠❡ ♣r♦❜❧❡♠s✧✱ ✧◆❛✉❦❛✧✱ ▼♦s❝♦✇✳