Cơ sở lý thuyết cho bài toán tối ưu có điều kiện

ĐẠI HỌC ĐÀ NẴNG TRƯỜNG ĐẠI HỌC SƯ PHẠM KHOA TỐN LÊ HỒNG NHUẬN CƠ SỞ LÝ THUYẾT CHO BÀI TỐN TỐI ƯU CĨ ĐIỀU KIỆN CHUN NGÀNH: SƯ PHẠM TOÁN HỌC Giảng viên hướng dẫn: T.S Phạm Quý Mười ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❑❍❖❆ ❚❖⑩◆ ✖✖✖ ✯ ✖✖✖ ▲➊ ❍❖⑨◆● ◆❍❯❾◆ ❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❈➶ ✣■➋❯ ❑■➏◆ ❈❍❯❨➊◆ ◆●⑨◆❍✿ ❙× P❍❸▼ ❚❖⑩◆ ❍➴❈ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ●✐↔♥❣ ữợ P ỵ ữớ ✶✷ ♥➠♠ ✷✵✶✾ ▼Ö❈ ▲Ö❈ ▲❮■ ◆➶■ ✣❺❯ ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●✐→ trà r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ●✐→ trà r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❉↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❚æ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ sè ❧✐➯♥ tö❝ tr➯♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ r tr tr t ỗ ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t s s✐➯✉ ♣❤➥♥❣ tü❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❈➶ ✣■➋❯ ❑■➏◆ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ❇➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ✳ ❇➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➌❚ ▲❯❾◆ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹ ✹ ✺ ✽ ✽ ✶✵ ✶✶ ✶✷ ✶✺ ✶✺ ✶✺ ✶✼ ✷✵ ✷✸ ✷✸ ✷✺ ✸✹ ✸✾ ✹✶ ✹✶ ✶ ỵ tt tố ữ ởt ✈ü❝ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ ✈➲ ❝ü❝ trà ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ✤í✐ sè♥❣ t❤ü❝ t➳ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝✳ ❚✉② ♥❤✐➯♥ ♣❤↔✐ ✤➳♥ ♥❤ú♥❣ ♥➠♠ ✸✵✱ ✹✵ t tố ữ ợ ữủ t ợ tữ ởt ỵ tt ợ ữợ ự ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ ❤ì♥ tr♦♥❣ ❝✉ë❝ sè♥❣ ♥❤í sü ♣❤→t tr✐➸♥ ❝õ❛ ❝æ♥❣ ♥❣❤➺ t❤æ♥❣ t✐♥✱ ✤➦❝ ❜✐➺t ❧➔ t õ ự ỡ s ỵ tt ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ trì♥ ❝â ✤✐➲✉ ❦✐➺♥✳ ❈ư t❤➸✱ ❦❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✱ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ tè✐ ÷✉ õ õ ỗ ữỡ q✉→t ♥❤ú♥❣ ✈➜♥ ✤➲ ❝❤✉♥❣ ♥❤➜t ❦❤✐ t✐➳♣ ❝➟♥ ✈ỵ✐ ỵ tt t tố ữ õ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ✿ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ❝→❝ ❦➼ ❤✐➺✉ ữủ sỷ tr ỗ ởt số tr r t ỗ ỗ số tử tr Rn ỵ r ỵ tr tr ụ ữ ỵ t t ỡ s t t ❝➟♥ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ð ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✷✿ ❈ì sð ❧➼ t❤✉②➳t ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ tè✐ ù✉ ♣❤✐ t✉②➳♥ ❝â ✤✐➲✉ ❦✐➺♥✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ù✉ ❝â ✤✐➲✉ ❦✐➺♥✳ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ t➼❝❤ ❝ü❝ ❤å❝ t➟♣ ✈➔ ự ữợ sỹ t t t ữợ t t ụ ữủ ❝❤♦ ♠➻♥❤ r➜t ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ❝ô♥❣ ữ ố ợ ởt ỵ tt ợ tr♦♥❣ ❚♦→♥ ❤å❝ ✈➔ ✤➳♥ ♥❛② ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤✳ ❊♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t tợ trữớ rữớ ✣❍❙P✱ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣ ✤➣ t↕♦ ❝ì ❤ë✐ ❝❤♦ ❡♠ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳ ❊♠ ❝ơ♥❣ ữủ tọ ỏ t ỡ tợ t P ỵ ữớ t t ữợ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❧➔♠ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ❝❤ó♥❣ ❡♠ tr♦♥❣ s✉èt ❜è♥ ♥➠♠ ❤å❝ ✈ø❛ q✉❛✳ ❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ✷ ✣➔ ◆➤♥❣✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ▲➯ ❍♦➔♥❣ ◆❤✉➟♥ ✸ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð P❤➛♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♠❛ tr➟♥✱ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✱ ❣✐→ trà r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❚ỉ♣ỉ✱ ❤➔♠ sè ❧✐➯♥ tư❝✱ ❝→❝ ✤à♥❤ tr tr t ỗ ỗ ỵ t ỹ ữ ởt ổ ✤➸ t✐➳♣ ❝➟♥ ❝→❝ ❦✐➳♥ t❤ù❝ ð ❝❤÷ì♥❣ s❛✉✳ ✶✳✶ ▼❛ tr➟♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ♠❛ tr➟♥ ❝ï m ì n ởt số ỳ t ỗ m ❤➔♥❣ ✈➔ n ❝ët ❣å✐ ❧➔ ♠❛ tr➟♥ m × n✱ ❦➼ ❤✐➺✉ ❧➔ Am×n✱ ❝â ❞↕♥❣✿  Am×n a11 a12 a1n  a21 a22 a2n  =  ✳✳  ✳ am1 am2 amn      tr ù m ì n ữủ m × n✲♠❛ tr➟♥✳ ◆➳✉ m ✈➔ n ✤➣ rã t❤➻ tr Amìn ữủ A (aij )m×n✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭✐✮ ▼ët m × n✲♠❛ tr➟♥ ✤÷đ❝ ❣å✐ ❧➔ ♠❛ ∀i ∈ {1, 2, , m}, j ∈ {1, 2, , n} tr➟♥ ❦❤æ♥❣✱ ❦➼ ❤✐➺✉ ❧➔ 0✱ ♥➳✉ aij = ✭✐✐✮ ▼ët ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ✭n × n✲♠❛ tr➟♥✮ ♠➔ ❝→❝ ♣❤➛♥ tû aij = ✈ỵ✐ i = j ✈➔ aii = 1, ∀i ∈ {1, 2, , n} ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n✱ ❦➼ ❤✐➺✉ ❧➔ In ❤♦➦❝ I✳ ❑❤✐ ✤â✿      In =  0  ✳✳  ✳ 0 0       ❈→❝ ♣❤➨♣ t♦→♥ ✈➲ ♠❛ tr➟♥ t❤æ♥❣ t❤÷í♥❣ ✤÷đ❝ ❤✐➸✉ ✈➔ t❤ü❝ ❤✐➺♥ ♥❤÷ ❜➻♥❤ t❤÷í♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ✹ ✭❛✮ ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ m × n✲♠❛ tr➟♥ A✱ ❦➼ ❤✐➺✉ ❧➔ AT ❧➔ ♠ët n ì m tr ợ aTij = aji tr➟♥ ✈✉ỉ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ♥➳✉ AT = A✳ ✭❝✮ ▼❛ tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ s✉② ❜✐➳♥ ♥➳✉ ❝â ♠❛ tr➟♥ A−1✳ ❑❤✐ ✤â ♠❛ tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❦❤↔ ♥❣❤à❝❤✱ t❤ä❛ ♠➣♥ A−1A = I = AA−1✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈✉æ♥❣ A ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â ❦❤ỉ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔②✱ ♥❣÷í✐ ✤å❝ ❝â t❤➸ t❤❛♠ ❦❤↔♦ t❤➯♠ tr♦♥❣ ❝→❝ ❣✐→♦ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✳ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ R n ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n✲❝❤✐➲✉ ❧➔ Rn ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët ✤✐➸♠ x tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ R ❧➔ ♠ët ❜ë n sè t❤ü❝ ✤÷đ❝ n s t tự tỹ ữủ t ữợ ❞↕♥❣ ♥✲❝ët  x1  x2    x =  ✳✳  ,  ✳  xn  ✈ỵ✐ ♠é✐ sè xi ∈ R✱ i ∈ {1, 2, , n} ✤÷đ❝ ❣å✐ ❧➔ tå❛ ✤ë t❤ù i ❝õ❛ ✤✐➸♠ x✳ ✣➲ t❤✉➟♥ t✐➺♥ t❛ q✉② ÷ỵ❝    x = (x1 , x2 , , xn )T =   x1 x2 ✳✳ ✳      xn ❑➼ ❤✐➺✉ = (0, 0, , 0)T ∈ Rn ❧➔ ❣è❝ tå❛ ✤ë ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Rn ✳ ▼é✐ ✤✐➸♠ x t❤✉ë❝ Rn ①→❝ ✤à♥❤ ♠ët ✈➨❝tì tr♦♥❣ Rn ✈ỵ✐ ✤✐➸♠ ❣è❝ ❧➔ ✈➔ ✤✐➸♠ ♥❣å♥ ❧➔ x✳ ❱➨❝tì ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ ✈➟② ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ x✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ✣♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ✭✈➨❝tì✮ x ✈➔ y tr♦♥❣ R ✱ ❦➼ ❤✐➺✉ ❧➔ [x, y]✱ n ❧➔ t➟♣ ❤đ♣ ❝→❝ ✤✐➸♠ ✭✈➨❝tì✮ ❝â ❞↕♥❣ αx + (1 − α) y, ∀ ≤ α ≤ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ ❤❛✐ ✈➨❝tì x = (x , x , , x ) ✈➔ y = (y , y , , y ) tr♦♥❣ T Rn ✳ T n õ ổ ữợ ❤❛✐ ✈➨❝tì✱ ❦➼ ❤✐➺✉ ❧➔ xT y✱ yT x ❤♦➦❝ n xT y = yT x = x, y = xi y i i=1 ✺ x, y n ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✐✐✮ ❍❛✐ ✈➨❝tì x ✈➔ y trü❝ ❣✐❛♦ ♥➳✉ xT y = yT x = x, y = 0✳ ✭✐✐✐✮ ✣ë ❞➔✐ ❤❛② ❝❤✉➞♥ ❝õ❛ ✈➨❝tì x✱ ❦➼ ❤✐➺✉ ❧➔ x ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ n T x = x x 1/2 x2i = i=1 ❍➻♥❤ ✶✳✶✿ ✣♦↕♥ t❤➥♥❣ ◆❤➟♥ ①➨t ✶✳✶✳ r ổ tỡ C t ổ ữợ ❤❛✐ ✈➨❝tì x, y ❝â ❝→❝ n t➼♥❤ ❝❤➜t s❛✉✿ ổ ữợ tỡ x, y ✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ n xi y i x, y = i=1 ✭✐✐✮ ✭✐✐✐✮ ✭✐✈✮ x ≥0 ✈➔ x, y = y, x x =0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳ ✳ ✈ỵ✐ λ ∈ C✳ x, λy = λ x, y ✈ỵ✐ λ ❧➔ ❧✐➯♥ ❤ñ♣ ❝õ❛ λ✳ ❚➼♥❤ ❝❤➜t ✶✳✶✳ ▼ët sè t➼♥❤ ❝❤➜t ✤➣ ❜✐➳t ✈➲ ❝❤✉➞♥ ❝õ❛ ♠ët ✈➨❝tì✿ ✭❛✮ x ≥ ✈➔ x = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳ ✭❜✮ αx =| α | x ✱ ✈ỵ✐ ♠å✐ α ∈ R✳ ✭❝✮ x + y ≤ x + y ✱ ✈ỵ✐ ♠å✐ ✈➨❝tì x, y tr♦♥❣ Rn✳ ✭❞✮ ❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ λx, y = λ x, y |xT y| ≤ x ✈ỵ✐ ♠å✐ ✈➨❝tì x, y tr♦♥❣ Rn✳ ✻ y , ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❈→❝ ✈➨❝tì a , a , , a ✤÷đ❝ ❣å✐ ❧➔✿ k ✭✐✮ P❤ư t❤✉ë❝ t t tỗ t số tỹ 1, 2, , k ổ ỗ tớ k s❛♦ ❝❤♦ i=1 λi = 0✳ ✭✐✐✮ ✣ë❝ t tk ổ tỗ t số tỹ 1, 2, , k ổ ỗ tớ ❜➡♥❣ s❛♦ ❝❤♦ i=1 λiai = 0✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ✭✐✮ ▼ët tê ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ✈➨❝tì a1, a2, , ak ❧➔ ♠ët ✈➨❝tì ❝â ❞↕♥❣ ki=1 λiai ✈ỵ✐ λ1, λ2, , λk ∈ R✳ ❑❤✐ ✤â t➟♣ ❤đ♣ ❝→❝ ✈➨❝tì ❞↕♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❤đ♣ ❝→❝ ✈➨❝tì s✐♥❤ ❜ð✐ a1, a2, , ak ✳ ✭✐✐✮ ❚➟♣ ❤đ♣ ❝→❝ ✈➨❝tì ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ s✐♥❤ r❛ Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝ì sð ❝õ❛ Rn✳ ▼é✐ ❝ì sð ❝õ❛ Rn ❝❤ù❛ ✤ó♥❣ n ✈➨❝tì✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❍↕♥❣ ❝õ❛ m × n✲♠❛ tr➟♥ A ❜➡♥❣ sè ❝ët ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥ ♥❤➜t ❝õ❛ ♠❛ tr➟♥ A ✈➔ ❝ô♥❣ ❜➡♥❣ sè ❤➔♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥ ♥❤➜t ❝õ❛ ♠❛ tr➟♥ A✳ ❍ì♥ ♥ú❛✱ ♥➳✉ ❤↕♥❣ ❝õ❛ m × n✲♠❛ tr➟♥ A ❜➡♥❣ min{m, n} t❤➻ ♠❛ tr➟♥ A ✤÷đ❝ ❣å✐ ❧➔ ❝â ❝➜♣ ✤➛② ✤õ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❝õ❛ E n ❧➔ ♠ët t➟♣ ❝♦♥ ✤â♥❣ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ổ ữợ tự a + àb ∈ M, ∀a, b ∈ M, ∀λ, µ ∈ R✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ❙è ❝❤✐➲✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ M ❧➔ sè ✈➨❝tì ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❧ỵ♥ ♥❤➜t tr♦♥❣ M ✳ ✣à♥❤ ♥❣❤➽❛⊥✶✳✶✷✳ ❱ỵ✐ M ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ E n✱ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ M M t ủ ỗ ❝→❝ ✈➨❝tì trü❝ ❣✐❛♦ ✈ỵ✐ ❝→❝ ✈➨❝tì t❤✉ë❝ M ✳ ❉ü❛ ✈➔♦ ❦➳t q✉↔ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♣❤➛♥ ❜ị trü❝ ❣✐❛♦ ❝õ❛ M ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ E n ✱ ❤ì♥ ♥ú❛ M ✈➔ M ⊥ ❝↔♠ s✐♥❤ ❦❤æ♥❣ ❣✐❛♥ E n ✳ ◆â✐ ❝→❝❤ ❦❤→❝ ✈ỵ✐ ♠é✐ ✈➨❝tì x ∈ E n ữủ t ữợ x = a + b ✈ỵ✐ a ∈ M, b ∈ M ⊥ ✳ ▲ó❝ ✤â✱ a, b ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝❤✐➳✉ trü❝ ❣✐❛♦ ❝õ❛ ✈➨❝tì x ❧➛♥ ❧÷đt ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ M ✈➔ M ⊥ ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳ ▼ët q✉❛♥ ❤➺ t÷ì♥❣ ù♥❣ A ❣→♥ ♠é✐ ✤✐➸♠ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ X ✈ỵ✐ ♠ët ✤✐➸♠ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Y ✤÷đ❝ ❣å✐ ❧➔ ♠ët →♥❤ ①↕ tø X ✤➳♥ Y ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ A:X →Y✳ ⑩♥❤ ①↕ A ❝â t❤➸ ❧➔ t✉②➳♥ t➼♥❤ ❤♦➦❝ ♣❤✐ t✉②➳♥ t➼♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✹✳ ❈❤✉➞♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ A✱ ❦➼ ❤✐➺✉ A = max x ≤1 Ax x ❚➼♥❤ ❝❤➜t ✶✳✷✳ ❱ỵ✐ ♠å✐ ✈➨❝tì x✱ t❛ ❝â✿ Ax ≤ A ✼ A ✱ ✤÷đ❝ ✤✐♥❤ ♥❣❤➽❛ ❧➔✿ ✶✳✸ ●✐→ trà r✐➯♥❣ ✈➔ ❞↕♥❣ t♦➔♥ ♣❤÷ì♥❣ ✶✳✸✳✶ ●✐→ trà r✐➯♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✺✳ ❈❤♦ ♠ët n ì n tr ổ A tỗ t ởt ổ ữợ R ởt tỡ x = t❤ä❛ ♠➣♥ Ax = λx t❤➻ λ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ù♥❣ ✈ỵ✐ ✈➨❝tì r✐➯♥❣ x ❝õ❛ A✳ ◆❤➟♥ ①➨t ✶✳✷✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ λ ∈ R ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A ❧➔ ♠❛ tr➟♥ A − λI s✉② ❜✐➳♥✱ tù❝ ❧➔ ❞❡t (A − λI) = ❚ø ♥❤➟♥ ①➨t tr➯♥ t❛ ❝â t❤➸ ♥❤➟♥ t❤➜② ♠ët ❣✐→ trà r✐➯♥❣ λ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ det (λI − A) = λn + an−1 λ + + a1 λ + a0 = ✣❛ t❤ù❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣ A✳ ỵ sỷ ữỡ tr trữ t (λI − A) = ❝â n ♥❣❤✐➺♠ t❤ü❝ ♣❤➙♥ ❜✐➺t λ1, λ2, , λn✳ ❑❤✐ ✤â tỗ t n tỡ t t x1, x2, , xn ❧➔ ❝→❝ ✈➨❝tì r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A✱ tù❝ ❧➔ ❝→❝ ✈➨❝tì ♥➔② t❤ä❛ ♠➣♥✿ Axi = λi xi , ∀i ∈ {1, 2, , n} ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â✿ ❞❡t (λxi A) = tỗ t tỡ xi = s❛♦ ❝❤♦ Axi = λi xi ✈ỵ✐ i ∈ {1, 2, , n}✳ ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❤➺ n ✈➨❝tì x1 , x2 , , xn ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❚❤➟t ✈➟②✿ ●å✐ c1 , c2 , , cn ổ ữợ s ni=1 ci xi = 0✳ ❳➨t ♠❛ tr➟♥ B = (λ2 I − A)(λ3 I − A) (λn I − A)✳ ❑❤✐ ✤â✿ Bxn = (λ2 I − A)(λ3 I − A) (λn I − A)xn = (λ2 I − A)(λ3 I − A) (λn xn − Axn ) Bxn = ✈➻ λn xn − Axn = 0✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿ Bxi = 0, ∀i = 2, 3, , n✳ ✽ ✣➸ ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ♠ët ❝→❝❤ ❞➵ ❞➔♥❣✱ t❛ ❣✐ỵ✐ t❤✐➺✉ ❤➔♠ ▲❛❣r❛♥❣✐❛♥✿ m L (x, λ) = f (x) − λi ci (x), ✭✷✳✸✹✮ i=1 ✈ỵ✐ λ = (λ1 , , λm )T ∈ Rm ❧➔ ♠ët ✈❡❝t♦r ♥❤➙♥ tû ▲❛❣r❛♥❣❡✳ ❇➙② ❣✐í t❛ s➩ ♣❤→t ❜✐➸✉ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ✶ ❝õ❛ ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❇ê rs ỵ ỵ rs ✕ ❑✉❤♥ ✕ ❚✉❝❦❡r✮✳ ❈❤♦ x ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ∗ ✭✷✳✶✮✲✭✷✳✸✮✳ ◆➳✉ ✤✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ SF D (x∗ , X) = LF D (x∗ , X) ú t tỗ t tỷ ▲❛❣r❛♥❣❡ λ∗i ✤➸ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥ t↕✐ (x∗, λ∗) : m ∗ ✭✷✳✸✻✮ λ∗i ∇ci (x∗ ) = 0, ∇f (x ) − i=1 ci (x∗ ) = 0, ci (x∗ ) ≥ 0, λ∗i ≥ 0, λ∗i ci (x∗ ) = 0, ∀i ∈ E, ∀i ∈ I, ∀i ∈ I, ∀i ∈ I ✭✷✳✸✼✮ ✭✷✳✸✽✮ ✭✷✳✸✾✮ ✭✷✳✹✵✮ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ♥➯♥ x∗ ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❉♦ ✤â ✭✷✳✸✼✮ ✈➔ ✭✷✳✸✽✮ t❤ä❛ ♠➣♥✳ ◆➳✉ d ∈ SF D (x∗ , X) t❤➻ ✈➻ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ữỡ t ỵ t õ dT f (x∗ ) ≥ ❱➟② ❤➺✿ dT ∇ci (x∗ ) = 0, i ∈ E, ✭✷✳✹✶✮ dT ∇ci (x∗ ) ≥ 0, i ∈ I(x∗ ), ✭✷✳✹✷✮ ✭✷✳✹✸✮ ∗ T d ∇f (x ) < ✈æ ♥❣❤✐➺♠✳ ◆➳✉ d ∈ / SF D(x∗ , X) t❤➻ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ ✭✷✳✸✺✮ t❛ ❝â d ∈ / LF D(x∗ , X) ỗ dT ci (x ) = 0, i ∈ E ✈➔ dT ∇ci (x∗ ) ≥ 0, i ∈ I(x∗ ) ✈æ ♥❣❤✐➺♠✳ ❱➟② tr♦♥❣ ♠å✐ tr÷í♥❣ ❤đ♣✱ ❤➺ ✭✷✳✹✶✮✲✭✷✳✹✸✮ ✈ỉ ♥❣❤✐➺♠✳ ❚❤❡♦ ❇ê ✤➲ ❋❛r❦❛s✱ t❛ ❝â ∇f (x∗ ) = λ∗i ∇ci (x∗ ) + λ∗i ∇ci (x∗ ) , ✭✷✳✹✹✮ i∈I(x∗ ) i∈E ✈ỵ✐ λ∗i ∈ R (i ∈ E) ✈➔ λ∗i ≥ (i ∈ I (x∗ )) ✣➦t λ∗i = 0(i ∈ I\I (x∗ )), t❛ ❝â m ∗ λ∗i ∇ci (x∗ ) , ∇f (x ) = i=1 tù❝ ❧➔ ✭✷✳✸✻✮✱ ✈➔ t❛ ❝â λ∗i ≥ 0, ∀i ∈ I ố ũ ú ỵ r i I (x ) , ci (x∗ ) = ✈➔ λ∗i ≥ ❉♦ ✤â λ∗i ci (x∗ ) = 0; ❦❤✐ i ∈ I\I (x∗ ) , ci (x∗ ) > ♥❤÷♥❣ λ∗i = ❉♦ ✤â λ∗i ci (x∗ ) = ❱➟② t❛ ❝â λ∗i ci (x∗ ) = 0, ∀ i ∈ I ✷✽ ✣✐➲✉ ❦✐➺♥ ✭✷✳✸✻✮✲✭✷✳✹✵✮ t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❑❛r✉s❤ ✕ ❑✉❤♥ ✕ ❚✉❝❦❡r✱ ❤❛② ✈✐➳t t➢t ❧➔ ✤✐➲✉ ❦✐➺♥ ❑❑❚✱ ✭✷✳✸✻✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤✐➸♠ ❞ø♥❣✱ ❜ð✐ ✈➻ ♥â ❝â t❤➸ ✈✐➳t✿ m ∗ ∗ ∗ ✭✷✳✹✺✮ λ∗i ∇ci (x∗ ) = ∇x L (x , λ ) = ∇f (x ) − i=1 ✣✐➲✉ ❦✐➺♥ ✭✷✳✸✼✮ ✈➔ ✭✷✳✸✽✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✱ ✭✷✳✸✾✮ ❧➔ ✤✐➲✉ ❦✐➺♥ ❦❤æ♥❣ ➙♠ ❝❤♦ ❝→❝ ♥❤➙♥ tû✱ ✈➔ ✭✷✳✹✵✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣✱ tù❝ ❧➔ λ∗i ✈➔ ci (x∗ ) ❦❤ỉ♥❣ t❤➸ ❝ị♥❣ ❦❤→❝ ❦❤ỉ♥❣✱ ❤❛② ♥â✐ ♠ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣ ❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ t÷ì♥❣ ù♥❣ ✈ỵ✐ ♥❤ú♥❣ r➔♥❣ ❜✉ë❝ ❦❤ỉ♥❣ ❤♦↕t ✤ë♥❣ ✤➲✉ ❜➡♥❣ ❦❤ỉ♥❣✳ ❚❛ ♥â✐ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❜ị ❝❤➦t ✤ó♥❣ ♥➳✉ ❝â ❝❤➼♥❤ ①→❝ ♠ët tr♦♥❣ ❤❛✐ sè λ∗i ✈➔ ci (x∗ ) ❜➡♥❣ ❦❤ỉ♥❣ ✈ỵ✐ ♠é✐ i ∈ I, ♥❣❤➽❛ ❧➔ t❛ ❝â λ∗i > ✈ỵ✐ ♠é✐ i ∈ I ∩ A (x∗ ) ▼ët ✤✐➲✉ ❦✐➺♥ ❜➜t ✤➥♥❣ t❤ù❝ ci ✤÷đ❝ ❣å✐ ❧➔ ❤♦↕t ✤ë♥❣ ♠↕♥❤ ♥➳✉ i ∈ I ∩ A (x∗ ) ✈➔ λ∗i > 0, ♥❣❤➽❛ ❧➔ λ∗i > ✈➔ ci (x∗ ) = ▼ët ✤✐➲✉ ❦✐➺♥ ❜➜t ✤➥♥❣ t❤ù❝ ci ✤÷đ❝ ❣å✐ ❧➔ ❤♦↕t ✤ë♥❣ ②➳✉ ♥➳✉ i ∈ I ∩ A (x∗ ) ✈➔ λ∗i = 0, ♥❣❤➽❛ ❧➔ λ∗i = ✈➔ ci (x∗ ) = ✣✐➲✉ ❦✐➺♥ ✭✷✳✸✺✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ ✭❈◗✮✳ ✣✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ ❧➔ q✉❛♥ trå♥❣ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ❑❑❚✳ ◆❤÷ ♠ët ✈➼ ❞ư ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❋❧❡t❝❤❡r ❝❤➾ r❛ r➡♥❣ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❈◗ ❦❤ỉ♥❣ ✤ó♥❣✱ t❤➻ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮ ❝â t❤➸ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠ët ✤✐➸♠ ❑❑❚✳ ❱➼ ❞ö ✷✳✷✳✶✳ (x1 ,x2 )∈R2 ✭✷✳✹✻✮ x1 ✭✷✳✹✼✮ ✭✷✳✹✽✮ s.t x31 − x2 ≥ 0, x2 ≥ ❚❛ t❤➜② r➡♥❣ x∗ = (0, 0) ❧➔ ❝ü❝ t✐➸✉ t♦➔♥ ❝ö❝ ❝õ❛ ❜➔✐ t♦→♥✳ ❚↕✐ x∗ , t❛ ❝â SF D (x∗ , X) = d|d = α , α≥0 ✭✷✳✹✾✮ LF D (x∗ , X) = d|d = α , α∈R ✭✷✳✺✵✮ ✈➔ ❉♦ ✤â ❈◗ ❦❤ỉ♥❣ ✤ó♥❣✳ ❇➡♥❣ t➼♥❤ t♦→♥ trü❝ t✐➳♣ t❛ ❝â ∇f (x∗ ) = , ∇c1 (x∗ ) = 0 , ∇c2 (x∗ ) = −1 ✭✷✳✺✶✮ ✣✐➲✉ ♥➔② ❝❤♦ t ổ tỗ t f (x∗ ) = λ∗1 ∇c1 (x∗ ) + λ∗2 ∇c2 (x∗ ) ✭✷✳✺✷✮ ❱➼ ❞ư ✤ì♥ ❣✐↔♥ ♥➔② ❝❤♦ t❛ t❤➜② t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❈◗✳ ❚✉② ♥❤✐➯♥ ❦❤ỉ♥❣ ❞➵ ✤➸ ❜✐➳t ✤✐➲✉ ❦✐➺♥ ❈◗ ✤ó♥❣ ❤❛② ❦❤ỉ♥❣✳ ❙❛✉ ✤➙②✱ t❛ s➩ ✤÷❛ r❛ ♠ët sè r➔♥❣ ❜✉ë❝ ❝ư t❤➸ ❞➵ ❦✐➸♠ tr❛ ✈➔ ✤÷đ❝ sû ❞ư♥❣ t❤÷í♥❣ ①✉②➯♥✳ ✣✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ✤ì♥ ❣✐↔♥ ✈➔ rã r➔♥❣ ♥❤➜t ❧➔ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ❤➔♠ t✉②➳♥ t➼♥❤✳ ✷✾ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✶✳ ◆➳✉ t➜t ❝↔ ❝→❝ ❤➔♠ ✤✐➲✉ ❦✐➺♥ c (x) (i ∈ A (x ) = E ∪ I(x )) ✤➲✉ i ∗ ∗ ❧➔ t✉②➳♥ t➼♥❤✱ t❤➻ t❛ ♥â✐ r➡♥❣ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ❤➔♠ t✉②➳♥ t➼♥❤ ✭▲❋❈◗✮ ❧➔ ✤ó♥❣✳ ❚ø ✤à♥❤ ♥❣❤➽❛✱ ♥➳✉ ci (x) (i ∈ A (x∗ )) ❧➔ ♥❤ú♥❣ ❤➔♠ t✉②➳♥ t➼♥❤✱ t❤➻ ✤✐➲✉ ❦✐➺♥ ❈◗ ✭✷✳✸✺✮ ✤ó♥❣ ✈➔ t❛ ❝â ❤➺ q✉↔ s❛✉ ✤➙②✿ ❍➺ q✉↔ ✷✳✷✳✶✳ ❈❤♦ x ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ ◆➳✉ ✤✐➲✉ ❦✐➺♥ ∗ r➔♥❣ ❜✉ë❝ ❤➔♠ t✉②➳♥ t➼♥❤ ✭▲❋❈◗✮ ❧➔ ✤ó♥❣ t↕✐ x∗, t❤➻ x∗ ❧➔ ♠ët ✤✐➸♠ ❑❑❚✳ ✣✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ q✉❛♥ trå♥❣ ♥❤➜t ✈➔ t❤÷í♥❣ ①✉②➯♥ ✤÷đ❝ sû ❞ư♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✭▲■❈◗✮ s❛✉ ✤➙②✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✷✳ ◆➳✉ ❝→❝ ❣r❛❞✐❡♥t ❝õ❛ ❝→❝ r➔♥❣ ❜✉ë❝ ❤♦↕t ✤ë♥❣ ∇c (x ) , i ∈ i ∗ A (x∗ ) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ t❤➻ t❛ ♥â✐ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ✤ë❝ ❧➟♣ t✉②➳♥ t ú ỵ x ♠ët∗ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥∗ ✤÷đ❝ ✈➔ A∗(x∗) ❧➔ t➟♣ ❤đ♣ ❝❤➾ sè ❝õ❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❤♦↕t ✤ë♥❣ t↕✐ x ◆➳✉ ❝→❝ ∇ci(x ), i ∈ A (x ) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ t❤➻ ✤✐➲✉ ❦✐➺♥ ❤↕♥ ❝❤➳ ✭✷✳✸✺✮ ✭❈◗✮ ✤ó♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ SF D(x∗, X) ⊆ LF D(x∗, X), t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ LF D (x∗, X) ⊆ SF D (x∗ , X) ❈❤♦ d LF D (x , X) tũ ỵ t A (x∗ ) = E ∪ I (x∗ ) = {1, , l} , me ≤ l ≤ n ❱➻ ∇c1 (x∗ ) , , ∇cl (x∗ ) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ♥➯♥ t❛ ❝â t❤➸ ❜ê s✉♥❣ bl+1 , , bn ✤➸ ∇c1 (x∗ ) , , ∇cl (x∗ ) , bl+1 , , bn ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❳➨t ❤➺ ♣❤✐ t✉②➳♥ r (x, θ) = 0, ✭✷✳✺✸✮ ✈ỵ✐ ❝→❝ t❤➔♥❤ ♣❤➛♥ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ri (x, θ) = ci (x) − θdT ∇ci (x∗ ) , i = 1, , l, ✭✷✳✺✹✮ ri (x, θ) = (x − x∗ )T bi − θ dT bi , i = l + 1, , n ✭✷✳✺✺✮ ❑❤✐ θ = ❤➺ ✭✷✳✺✸✮ ✤÷đ❝ ❣✐↔✐ ❜ð✐ x∗ , ❦❤✐ θ ≥ ✤õ ♥❤ä✱ ❜➜t ❝ù ♥❣❤✐➺♠ x ❝ơ♥❣ ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ ❚❛ ✈✐➳t A = [∇c1 (x) , , ∇cl (x)] , B = [bl+1 , , bn ] ❚❤➻ ♠❛ tr➟♥ ❏❛❝♦❜✐❛♥ J(x, θ) = ∇x rT (x, θ) = [A : B] ❚❛ ❝â J(x∗ ) = [A(x∗ ) : B] ❧➔ ♠❛ tr➟♥ ❦❤æ♥❣ s✉② ❜✐➳♥✳ ❈❤♦ ♥➯♥ t❤❡♦ ỵ tỗ t ởt ♠ð Ωx ❝õ❛ x∗ ✈➔ Ωθ ❝õ❛ θ = s ợ tỗ t ❞✉② ♥❤➜t ♥❣❤✐➺♠ x(θ) ∈ Ωx , ✈➔ x(θ) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✈➔ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ t❤❡♦ θ ❚ø ✭✷✳✺✸✮ ✈➔ sû ❞ö♥❣ ✤↕♦ ❤➔♠ ❤➔♠ sè ❤ñ♣ t❛ ❝â✿ 0= dri = dθ j ∂ri dxj ∂r + i , ; i = 1, , n, ∂xj dθ ∂θ ✸✵ ❤❛② ∇ci (x)T dx − ∇ci (x∗ )T d = 0, i = 1, , l, dθ dx bTi − bTi d = 0, i = l + 1, , n dθ ✭✷✳✺✻✮ ✭✷✳✺✼✮ ❍➺ tr➯♥ ❧➔ dx − J (x∗ )T d = dθ ∗ ❱➻ x = x t↕✐ θ = ❱➟② ❤➺ tr➯♥ trð t❤➔♥❤ JT J (x∗ ) dx |θ=0 − d = dθ ❱➻ ♠❛ tr➟♥ ❤➺ sè ❦❤æ♥❣ s✉② ❜✐➳♥✱ t❛ ♥❤➟♥ ✤÷đ❝ dx = d t↕✐ θ = dθ ❱➟② ♥➳✉ θk ↓ t❤➻ x(θk ) ởt ữủ ợ ữợ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ d, ♥❣❤➽❛ ❧➔ x (θk ) − x∗ → d θk ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② r➡♥❣ d ∈ SF D (x∗ , X) ❱➻ d ∈ LF D (x , X) tũ ỵ t õ LF D (x∗ , X) ⊆ SF D (x∗ , X) ứ ỵ tr ỵ t õ ỵ s ỵ x ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ ◆➳✉ ▲■❈◗ ✤ó♥❣✱ ∗ ♥❣❤➽❛ ❧➔ ∇ci (x∗) , i ∈ A (x) t t t tỗ t ❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ λ∗i (i = 1, , m) ✤➸ ✭✷✳✸✻✮✲✭✷✳✹✵✮ ✤ó♥❣✳ ❚❤➾♥❤ t❤♦↔♥❣ ❝❤ó♥❣ t❛ sû ❞ö♥❣ ❣✐↔ t❤✐➳t SF D (x∗ , X) ∩ D (x∗ ) = LF D (x∗ , X) ∩ D (x∗ ) ✭✷✳✺✽✮ ●✐↔ t❤✐➳t ♥➔② ❝â t❤➸ s✉② r❛ trü❝ t✐➳♣ tø ✤✐➲✉ ❦✐➺♥ ❈◗✭✷✳✸✺✮✳ ❚✉② ♥❤✐➯♥✱ ữủ ổ ú ợ tt ỵ SF D (x , X) ∩ D (x∗ ) = ∅✮ trð t❤➔♥❤ LF D (x∗ , X) ∩ D (x∗ ) = ∅, ổ õ ữợ t t t x ❍ì♥ ♥ú❛✱ ♥❤÷ ❧➔ ♠ët ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ỵ t õ ỵ x X ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ SF D (x , X) ∩ D (x ) = LF D (x∗ , X) ∩ D (x∗ ) , t❤➻ x∗ ❧➔ ∗ ∗ ∗ ❇➔✐ t♦→♥ ✭✷✳✶✮ ✲ ✭✷✳✸✮✳ ◆➳✉ ♠ët ✤✐➸♠ ❑❑❚✳ ❚✐➳♣ t❤❡♦✱ t❛ ❜➔♥ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤õ tố ữ ởt ỵ ✷✳✶✻✳ ❈❤♦ x ∗ ❈❤♦ f (x) ✈➔ ci(x), ∈ X (i = 1, , m) ❦❤↔ ✈✐ t↕✐ x∗ ◆➳✉ ✭✷✳✺✾✮ dT ∇f (x∗ ) > 0, ∀0 = d ∈ SF D (x∗ , X) , t❤➻ x∗ ❧➔ ♠ët ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮ ✲ ✭✷✳✸✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ x∗ ❦❤ỉ♥❣ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ t t tỗ t ởt xk X (k = 1, 2, ) s❛♦ ❝❤♦ ✭✷✳✻✵✮ f (xk ) ≤ f (x∗ ) , ✈➔ xk → x∗ , xk = x∗ (k = 1, 2, ) ❑❤æ♥❣ ♠➜t tê♥❣ q✉→t✱ t❛ ❣✐↔ sû r➡♥❣ xk − x∗ xk − x∗ ✣➦t dk = xk −x∗ xk −x∗ ✭✷✳✻✶✮ → d , δk = xk − x∗ ❚ø ✣à♥❤ ♥❣❤➽❛ ✷✳✽ t❛ ❝â ✭✷✳✻✷✮ d ∈ SF D (x∗ , X) ❚ø ✭✷✳✻✵✮ ✈➔ ✭✷✳✻✶✮ ✈➔ f (xk ) = f (x∗ ) + (xk − x∗ )T ∇f (x∗ ) + 0( xk − x∗ , ❜➡♥❣ ❝→❝❤ ❝❤✐❛ ❤❛✐ ✈➳ ❝❤♦ xk − x∗ s❛✉ ✤â ❧➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ k → ∞, t❛ ♥❤➟♥ ✤÷đ❝ ✭✷✳✻✸✮ dT ∇f (x∗ ) ≤ 0, ✣✐➲✉ ũ t ợ ỵ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❱➻ SF D(x∗ , X) ⊆ LF D(x∗ , X), t❛ ❝â ❤➺ q✉↔ s❛✉ ✤➙②✳ ❍➺ q✉↔ ✷✳✷✳✷✳ ❈❤♦ x ∗ ∈ X ❈❤♦ f (x) ✈➔ ci(x), (i = 1, , m) ❦❤↔ ✈✐ t↕✐ x∗ ◆➳✉ dT ∇f (x∗ ) > 0, ∀0 = d ∈ LF D (x∗ , X) , ✭✷✳✻✹✮ t❤➻ x∗ ❧➔ ♠ët ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮ ✲ ✭✷✳✸✮✳ ▼ët ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ q✉❛♥ trå♥❣ ❦❤→❝✱ ✤÷đ❝ ❣❤✐ ♥❤➟♥ ❜ð✐ ❋r✐t③ ❏♦❤♥✱ ❧➔ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝õ❛ ❋r✐t③ ❏♦❤♥✳ ỵ f (x) c (x) (i = 1, , m) ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ ♠ët t➟♣ ♠ð ❦❤→❝ i ré♥❣ ❝❤ù❛ t➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ X ◆➳✉ x∗ ❧➔ ♠ët ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮ ✲ ✭✷✳✸✮✱ t❤➻ tỗ t ởt số ởt tr λ∗ ✤➸ m λ∗0 ∇ ∗ λi ∇ci (x∗ ) = f (x ) − ✭✷✳✻✺✮ i=1 ci (x∗ ) = 0, i ∈ E, ci (x∗ ) ≥ 0, i ∈ I, λ∗i ≥ 0, i ∈ I, λ∗i ci (x∗ ) = 0, ∀i, ✭✷✳✻✻✮ ✭✷✳✻✼✮ ✭✷✳✻✽✮ ✭✷✳✻✾✮ m (λ∗i )2 > ✭✷✳✼✵✮ i=0 ✸✷ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ ❝→❝ ∇ci (x∗ ) (i ∈ A(x∗ )) tở t t t tỗ t i (i A (x )) ổ ỗ tớ ổ s❛♦ ❝❤♦ ✭✷✳✼✶✮ λ∗i ∇ci (x∗ ) = i∈A(x∗ ) ❈❤♦ λ∗0 = ✈➔ λ∗i = 0, (i ∈ I\I(x∗ )), t❛ ♥❤➟♥ ✤÷đ❝ ✭✷✳✻✺✮✲✭✷✳✼✵✮✳ ◆➳✉ ❝→❝ ∇ci (x∗ ) (i ∈ A(x∗ )) ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t t t ữủ ợ = 1, t ỵ tọ ữủ ❧➔ ✤✐➸♠ ❋r✐t③ ❏♦❤♥ ✈➔ ❤➔♠ ▲❛❣r❛♥❣❡ ❝â trå♥❣ m L (x, λ0 , λ) = λ0 f (x) − ✭✷✳✼✷✮ λi ci (x) i=1 ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❋r✐t③ ❏♦❤♥✳ ❈❤ó♥❣ t❛ ❦➳t t❤ó❝ ♣❤➛♥ ♥➔② ✈ỵ✐ ♠ët ✤✐➲✉ tố ữ ữỡ tr ỗ ữ ú t t t ỹ t ởt ỗ tr t ỗ ữủ t q ỗ ởt t q ỗ õ ❞↕♥❣ f (x) s.tx ∈ Ω ✭✷✳✼✸✮ ✈ỵ✐ f (x) ởt ỗ tr t ỗ ổ t❤÷í♥❣✱ tr♦♥❣ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ♣❤✐ t✉②➳♥ f (x) s.tci (x) = 0, i ∈ E, ci (x) 0, i I, f (x) ỗ ci (x), (i ∈ E) ❧➔ ❝→❝ ❤➔♠ t✉②➳♥ t➼♥❤✱ ✈➔ ci (x), (i ∈ I) ❧➔ ♥❤ú♥❣ ❤➔♠ ❧ã♠✱ t❤➻ t➟♣ ❤ñ♣ ✤✐➲✉ ❦✐➺♥ Ω = {x| ci (x) = 0, i ∈ E; ci (x) ≥ 0, i I} ởt t ỗ ởt t q ỗ ỵ s ❝❤➾ r❛ r➡♥❣ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ q ỗ ụ ỹ t t õ ỵ ộ ỹ t ữỡ t q ỗ ụ ỹ t t õ t S ỗ ỹ t t ởt t ỗ ự sỷ ♥❣÷đ❝ ❧↕✐ r➡♥❣ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ♠➔ ổ ỹ t t õ tỗ t↕✐ x1 ∈ Ω s❛♦ ❝❤♦ f (x1 ) < f (x∗ ) ❳➨t xθ = (1 − θ) x∗ + θx1 , θ ∈ [0, 1] ❚ø t➼♥❤ ỗ f t õ x Ω ✈➔ f (xθ ) ≤ (1 − θ) f (x∗ ) + θ f (x1 ) = f (x∗ ) + θ(f (x1 ) − f (x∗ )) < f (x∗ ) ❱ỵ✐ θ ✤õ ♥❤ä xθ ∈ B(x∗ , ε) ∩ Ω ❱➻ x∗ ❧➔ ❝ü❝ t✐➸✉ ữỡ t õ ợ ọ t f (xθ ) ≥ f (x∗ ) ❚❛ ❣➦♣ ♠➙✉ t❤✉➝♥✳ ❱➟② ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝ơ♥❣ ❧➔ ❝ü❝ t✐➸✉ t♦➔♥ ❝ö❝✳ ❈❤♦ x0 , x1 ∈ S ✣➦t xθ = (1 − θ) x0 + θ x1 , θ ∈ [0, 1] ❱➻ x0 , x1 ❧➔ ❝ü❝ t✐➸✉ t♦➔♥ ❝ö❝ ♥➯♥ f (xθ ) ≥ f (x0 ) = f (x1 ) f ỗ t❛ ❝â f (xθ ) ≤ (1 − θ) f (x0 ) + θ f (x1 ) = f (x0 ) = f (x1 ) ❉♦ ✤â✱ f (xθ ) = f (x0 ) = f (x1 ) ❱➟② xθ S, tự S ỗ ỵ t q ỗ ỹ t✐➸✉ ❝õ❛ ♥â✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲❛❣r❛♥❣✐❛♥ ❈❤♦ (x∗ , λ∗ ) ❧➔ ✤✐➸♠ ❑❑❚ ❜➜t ❦ý ❝õ❛ ❜➔✐ t♦→♥ q✉② ỗ õ L (x, ) = f (x) − λ∗i ci (x) − i∈E λ∗i ci (x) iI ỗ ố ợ x sỷ t ỗ số ỳ ❦✐➺♥ ❑❑❚✱ t❛ ❝â ✈ỵ✐ ❜➜② ❦➻ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ x✱ L (x, λ∗ ) ≥ L (x∗ , λ∗ ) + (x − x∗ )T ∇x L(x∗ , λ∗ ) = L (x∗ , λ∗ ) m ∗ λ∗i ci (x∗ ) = f (x ) − i=1 = f (x ) ú ỵ r x ❧➔ ♠ët ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✈➔ λ∗i ≥ 0, ∀ i ∈ I ♥➯♥ t❛ ❝â λ∗i ci (x) = 0, i ∈ E; λ∗i ci (x) ≥ 0, i ∈ I ❱➻ ✈➟② L (x, λ∗ ) ≤ f (x) ✭✷✳✼✼✮ ❚ø ✭✷✳✼✻✮ ✈➔ ✭✷✳✼✼✮ t❛ ♥❤➟♥ ✤÷đ❝ f (x) ≥ f (x∗ ) ✭✷✳✼✽✮ ❱➟② x ởt ỹ t ỵ ởt t q ỗ ợ t ỗ t õ t ởt ỹ t ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❈❤♦ x∗ ∈ X, ♥➳✉ dT ∇f (x∗ ) > 0, ∀0 = d ∈ SF D (x∗ , X) , ✭✷✳✼✾✮ t❤➻ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ tỗ t d SF D (x , X) s❛♦ ❝❤♦ dT ∇f (x∗ ) < 0, ✭✷✳✽✵✮ t❤➻ t ỵ x ổ t ỹ t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ ❑➳t q✉↔ ♥➔② ♥â✐ ✈ỵ✐ ❝❤ó♥❣ t❛ r➡♥❣ ♠✐➵♥ ❧➔ ✭✷✳✼✾✮ ❤♦➦❝ ✭✷✳✽✵✮ ✤ó♥❣ t❤➻ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ♠ët s➩ ✤÷đ❝ sû ❞ö♥❣ ✤➸ ①→❝ ✤à♥❤ ①❡♠ x∗ ❝â ♣❤↔✐ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❤❛② ✸✹ ❦❤ỉ♥❣✳ ❚✉② ♥❤✐➯♥✱ ❝❤ó♥❣ t❛ ❦❤ỉ♥❣ t❤➸ ❜✐➳t x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❤❛② ❦❤ỉ♥❣ ♥➳✉ ❝❤➾ ❞ị♥❣ ✤↕♦ ❤➔♠ ❝➜♣ ♠ët ❦❤✐ ❝↔ ✭✷✳✼✾✮ ✈➔ ✭✷✳✽✵✮ ✤➲✉ s❛✐✱ ♥❣❤➽❛ ❧➔ ❦❤✐ dT ∇f (x∗ ) ≥ 0, ∀d ∈ SF D (x∗ , X) ; ✭✷✳✽✶✮ dT ∇f (x∗ ) = 0, ∃0 = d ∈ SF D (x∗ , X) ✭✷✳✽✷✮ ❚r♦♥❣ ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❤ỉ♥❣ t✐♥ ✈➲ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❧➔ ❝➛♥ t❤✐➳t✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ỉ❝ ❈◗ ✭✷✳✸✺✮ ✤ó♥❣✳ ❚❤❡♦ ✭✷✳✽✶✮✱ ✭✷✳✸✺✮ ✈➔ ❇ê ✤➲ ❋❛r❦❛s ✭✷✳✷✶✮ t❤➻ x∗ ❧➔ ♠ët ✤✐➸♠ ❑❑❚✳ ứ tỷ r tỗ t↕✐ ∃0 = d ∈ SF D (x∗ , X) s❛♦ ❝❤♦ m T ∗ ✭✷✳✽✸✮ λ∗i dT ∇ci (x∗ ) = d ∇f (x ) = i=1 ❱➻ SF D(x∗ , X) ⊆ LF D(x∗ , X), ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ✣à♥❤ ♥❣❤➽❛ ✷✳✼✱ t❛ ❝â ✭✷✳✽✸✮ t÷ì♥❣ ữỡ ợ i dT ci (x ) = 0, ∀ i ∈ I (x∗ ) ❈❤ó♥❣ t❛ ✤÷❛ r❛ ♥❤ú♥❣ ✤à♥❤ ♥❣❤➽❛ s❛✉✳ ❈❤♦ x∗ ❧➔ ♠ët ✤✐➸♠ ❑❑❚ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮ ✈➔ λ∗ ❧➔ ✈❡❝t♦r ♥❤➙♥ tû ▲❛❣r❛♥❣❡ t÷ì♥❣ ù♥❣✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ ❤đ♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❤♦↕t ✤ë♥❣ ♠↕♥❤ ♥❤÷ s↕✉✿ I+ (x∗ ) = i|i ∈ I (x∗ ) ✈ỵ✐ λ∗i > ✭✷✳✽✺✮ ❚❛ ❝â I+ (x∗ ) ⊆ I (x∗ ) ✣à♥❤ ♥❣❤➽❛ ✷✳✷✶✳ ❈❤♦ x ❧➔ ♠ët ✤✐➸♠ ❑❑❚ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✱ ✈➔ λ ❧➔ ✈❡❝t♦r ∗ tỷ r tữỡ ự tỗ t dk (k = 1, 2, ) ✈➔ ❞➣② δk (k = 1, 2, ) ✤➸ ❝❤♦ x ∗ + δk d k ∈ X ✭✷✳✽✻✮ ci (xk ) = 0, i ∈ E ∪ I+ (x∗ ) , ci (xk ) ≥ 0, i ∈ I (x∗ ) \I+ (x∗ ) , ✭✷✳✽✼✮ ✭✷✳✽✽✮ t❤ä❛ ♠➣♥ ✈➔ dk → d, δk → 0, t❤➻ d ✤÷đ❝ ởt ữợ r ổ t x ủ tt ữợ r ổ ❤✐➺✉ ❞➣② t↕✐ x∗ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ S (x∗, λ∗) ,   ∗ ∗ S (x , λ ) = d   xk = x∗ + δk dk ∈ X, δk > 0, δk → 0, dk → d,  ci (xk ) = 0, i ∈ E ∪ I+ (x∗ ),  ci (xk ) ≥ 0, i ∈ I(x∗ ) − I+ (x∗ ) ❇ê ✤➲ ✷✳✸✳ ✣➦t H = {d|d ∈ SF D(x , X); m i=1 ∗ H ✸✺ ✭✷✳✽✾✮ λ∗i ci (xk ) = 0} ❑❤✐ ✤â✱ t❛ ❝â S (x∗ , λ∗ ) = ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû d ∈ S (x∗ , λ∗ ) ⇒ xk = x∗ + δk dk ∈ X, δk > 0, δk → 0, dk → d ⇒ d ∈ SF D (x∗ , X) ❚❛ t❤➜② r➡♥❣ ✭✷✳✽✼✮✲✭✷✳✽✽✮ ❦➨♦ t❤❡♦ r➡♥❣ m ✭✷✳✾✵✮ λ∗i ci (x∗ + δk dk ) = i=1 ❚❤➟t ✈➟② m m λ∗i ci (x∗ + δk dk ) = i=1 i=1 λ∗i ci = λ∗i ci (xk ) = (xk ) + i∈I(x∗ ) i∈I i∈E λ∗i ci λ∗i ci (xk ) λ∗i ci (xk ) = λ∗i ci (xk ) + i∈I (xk ) i∈I\I(x∗ ) λ∗i ci (xk ) = i∈I(x∗ ) λ∗i ci (xk ) + = i∈I+ (x∗ ) λ∗i ci (xk ) = i∈I(x∗ )\I+ (x∗ ) ❱➟② d ∈ H ❇➙② ❣✐í ❣✐↔ sû d ∈ H ⇒ d ∈ SF D (x∗ , X) ⇒ xk = x∗ + δk dk ∈ X, δk > 0, δk → ∗ ∗ ∗ 0, dk → d ❍ì♥ ♥ú❛✱ m i=1 λi ci (xk ) = ⇒ λi ci (xk ) = 0, i ∈ I ❱ỵ✐ i ∈ I+ (x ) ∗ ∗ t❤➻ t❛ ❝â λi > ♥➯♥ ci (xk ) = ❱➟② ci (xk ) = 0, i ∈ E ∪ I+ (x ) ◆❣♦➔✐ r❛ t❛ ❝â ci (xk ) ≥ 0, i ∈ I (x∗ )−I+ (x∗ ) ✭✈➻ I (x∗ )−I+ (x∗ ) ⊆ I) ❱➻ ✈➟②✱ t❛ ❝â d ∈ S (x∗ , λ∗ ) ❚ø ❇ê ✤➲ ✷✳✸✱ t❛ ❝â✿ m ∗ ∗ S (x , λ ) = ∗ ✭✷✳✾✶✮ λ∗i ci (xk ) = d, d ∈ SF D(x , X); i=1 ❚ø ✤➙② s✉② r❛✿ S (x∗ , λ∗ ) ⊆ SF D(x∗ , X) ✣à♥❤ ♥❣❤➽❛ ✷✳✷✷✳ ❈❤♦ x ❧➔ ♠ët ✤✐➸♠ ❑❑❚ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮ ✈➔ λ ❧➔ ✈❡❝t♦r ∗ ∗ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ t÷ì♥❣ ù♥❣✳ ◆➳✉ d ❧➔ ởt ữợ t t ữủ t x ú t d ữủ ởt ữợ r ❜✉ë❝ ✈æ ❤✐➺✉ t✉②➳♥ t➼♥❤ ❤â❛ t↕✐ x∗ ❚➟♣ ủ tt ữợ r ổ t t➼♥❤ ❤â❛ t↕✐ x∗ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ G (x∗, λ∗) ,   d = 0,   ∗ ∗ T ∗ ∗ G (x , λ ) = d d ∇ci (x ) = 0, i ∈ E ∪ I+ (x ),   dT ∇ci (x∗ ) ≥ 0, i ∈ I(x∗ )\I+ (x∗ ) ✭✷✳✾✷✮ ◆➳✉ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ t↕✐ x∗ ❧➔ ❞✉② ♥❤➜t✱ G (x∗, λ∗) ❝â t❤➸ ✈✐➳t ❧➔ G (x∗) ❇ê ✤➲ ✷✳✹✳ ✣➦t F = d|d ∈ LF D(x∗, X; dT ∇ci(x∗) = 0, i ∈ I+(x∗) ❑❤✐ ✤â✱ t❛ ❝â G (x∗ , λ∗ ) = F ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû d ∈ G (x∗ , λ∗ ) ⇒ dT ∇ci (x∗ ) = 0, i ∈ E ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ dT ∇ci (x∗ ) ≥ 0, i ∈ I (x∗ ) ❚❤➟t ✈➟②✱ t❛ ❝â I (x∗ ) = I+ (x∗ ) ∪ (I (x∗ ) \I+ (x∗ )) ❱➻ dT ∇ci (x∗ ) = i ∈ I+ (x∗ ) ✈➔ dT ∇ci (x∗ ) ≥ 0, i ∈ I (x∗ ) \I+ (x∗ ) ♥➯♥ dT ∇ci (x∗ ) ≥ ✸✻ 0, i ∈ I (x∗ ) ❱➟② d ∈ LF D (x∗ , X) ◆❣♦➔✐ r❛✱ t❛ ❝â dT ∇ci (x∗ ) = 0, i ∈ I+ (x∗ ) ✭✈➻ d ∈ G (x∗ , λ∗ )✮✳ ❱➟② d ∈ LF D(x∗ , X), dT ∇ci (x∗ ) = 0, i ∈ I+ (x∗ ) ♥➯♥ d ∈ F ❇➙② ❣✐í ❣✐↔ sû d ∈ F ⇒ d ∈ LF D (x∗ , X) ⇒ dT ∇ci (x∗ ) = 0, i ∈ E ❚❛ ❝ô♥❣ ❝â dT ∇ci (x∗ ) = 0, i ∈ I+ (x∗ ) ♥➯♥ dT ∇ci (x∗ ) = 0, i ∈ E ∪ I+ (x∗ ) ◆❣♦➔✐ r❛✱ t❛ ❝â dT ∇ci (x∗ ) ≥ 0, i ∈ I (x∗ ) ♥➯♥ dT ∇ci (x∗ ) ≥ 0, i ∈ I (x∗ ) \I+ (x∗ ) ❱➟② d ∈ G (x∗ , λ∗ ) ❚ø ❇ê ✤➲ ✷✳✹✱ t❛ ❝â✿ G (x∗ , λ∗ ) = d d ∈ LF D(x∗ , λ∗ ); d ∇ci (x∗ ) = 0, i ∈ I+ (x∗ ) T ✭✷✳✾✸✮ ❚ø ❝→❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ ❝â S (x∗ , λ∗ ) ⊆ SF D(x∗ , X), G (x∗ , λ∗ ) ⊆ LF D(x∗ , X) ✭✷✳✾✹✮ ✭✷✳✾✺✮ ❚÷ì♥❣ tü ♥❤÷ SF D(x∗ , X) ⊆ LF D(x∗ , X), t❛ ❝â S (x∗ , λ∗ ) ⊆ G (x∗ , λ∗ ) ✭✷✳✾✻✮ ❇➙② ❣✐í t❛ ♣❤→t ❜✐➸✉ ♥❤ú♥❣ ❦➳t q ỵ ❦✐➺♥ ❝➛♥ ❝➜♣ ✷✮✳ ❈❤♦ x ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇➔✐ t♦→♥ ∗ ✭✷✳✶✮✲✭✷✳✸✮✳ ◆➳✉ ✤✐➲✉ ❦✐➺♥ r➔♥❣ ❜✉ë❝ ❈◗ ✭✷✳✸✺✮ ✤ó♥❣ t❤➻ t❛ ❝â dT ∇2xx L (x∗ , λ∗ ) d ≥ 0, ∀ d ∈ S (x∗ , λ∗ ) , ✭✷✳✾✼✮ ✈ỵ✐ L (x, λ) ❧➔ ❤➔♠ ▲❛❣r❛♥❣✐❛♥✳ ❍ì♥ ♥ú❛✱ ♥➳✉ S (x∗ , λ∗ ) = G (x∗ , λ∗ ) , ✭✷✳✾✽✮ dT ∇2xx L (x∗ , λ∗ ) d ≥ 0, ∀ d ∈ G (x∗ , λ∗ ) ✭✷✳✾✾✮ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ d ∈ S (x∗ , λ∗ ) , ♥➳✉ d = t❤➻ dT ∇2xx L (x∗ , λ∗ ) d = ❚❛ ①➨t d = ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ S (x∗ , ) tỗ t {dk } {k } s ❝❤♦ ✭✷✳✽✻✮✲✭✷✳✾✵✮ ✤ó♥❣✳ ❉♦ ✤â✱ tø ✭✷✳✾✵✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ❑❑❚ t❛ ❝â f (x∗ + δk dk ) = L(x∗ + δk dk , λ∗ ) = L (x∗ , λ∗ ) + δk2 dTk ∇2xx L (x∗ , λ∗ ) dk + 0(δk2 ) = f (x∗ ) + δk2 dTk ∇2xx L (x∗ , λ∗ ) dk + δk2 ✭✷✳✶✵✵✮ ❱➻ x ỹ t ữỡ ợ k ✤õ ❧ỵ♥ t❛ ❝â f (x∗ + δk dk ) ≥ f (x∗ ) ✭✷✳✶✵✶✮ ❙û ❞ö♥❣ ✭✷✳✶✵✵✮✲✭✷✳✶✵✶✮ ✈➔ ❧➜② ❣✐ỵ✐ ❤↕♥ t❛ ❝â dT ∇2xx L (x∗ , λ∗ ) d ≥ ❱➻ d ∈ S (x∗ , ) tũ ỵ t õ ứ t ữủ tứ ỵ ✭ ✣✐➲✉ ❦✐➺♥ ✤õ ❝➜♣ ❤❛✐✮✳ ❈❤♦ x ❧➔ ♠ët ✤✐➸♠ ❑❑❚ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲ ∗ ✭✷✳✸✮✳ ◆➳✉ dT ∇2xx L (x∗ , λ∗ ) d > 0, ∀d ∈ G (x∗ , λ∗ ) , ✭✷✳✶✵✷✮ t❤➻ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ x∗ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ữỡ t t tỗ t ởt xk X (k = 1, 2, ) s❛♦ ❝❤♦ f (xk ) ≤ f (x∗ ), ✭✷✳✶✵✸✮ ✈➔ xk → x∗ , xk = x∗ (k = 1, 2, ) ❑❤æ♥❣ ♠➜t tê♥❣ q✉→t✱ t❛ ❣✐↔ sû r➡♥❣ xk − x∗ xk − x∗ → d ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ ✭✷✳✻✶✮✲✭✷✳✻✸✮✱ t❛ ❝â dT ∇f (x∗ ) ≤ ✭✷✳✶✵✹✮ d ∈ SF D (x∗ , X) ⊆ LF D (x∗ , X) ✭✷✳✶✵✺✮ ✈➔ ❚ø ✤✐➲✉ ❦✐➺♥ ❑❑❚ ✈➔ ✭✷✳✶✾✮ t❛ ❝â m ∗ T λi dT ∇ci (x∗ ) ≥ d ∇f (x ) = ✭✷✳✶✵✻✮ i=1 ❚ø ✭✷✳✶✵✹✮ ✈➔ ✭✷✳✶✵✻✮ t❛ s✉② r❛ dT ∇f (x∗ ) = ✭✷✳✶✵✼✮ ❚ø ✭✷✳✶✵✻✮ ✈➔ ✣à♥❤ ♥❣❤➽❛ ✷✳✼✱ t❛ ❝â λi dT ∇ci (x∗ ) = 0, ∀i ∈ I(x∗ ) ✭✷✳✶✵✽✮ ❱➻ t❤➳✱ tø ✭✷✳✶✵✺✮✱ ✭✷✳✶✵✽✮ ✈➔ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✷ t❛ ❝â d ∈ G (x∗ , λ∗ ) ✭✷✳✶✵✾✮ ❚ø ✭✷✳✶✵✸✮ t❛ ❝â L (x∗ , λ∗ ) ≥ L (xk , λ∗ ) = L (x∗ , λ∗ ) + δk2 dTk ∇2xx L (x∗ , λ∗ ) dk + δk2 ✭✷✳✶✶✵✮ ❈❤✐❛ ❤❛✐ ✈➳ ❝❤♦ δk2 ✈➔ ❧➜② ❣✐ỵ✐ ❤↕♥ t❛ ❝â dT ∇2xx L (x∗ , λ∗ ) d ≤ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✵✷✮✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✸✽ ✭✷✳✶✶✶✮ ❚❛ ✤à♥❤ ♥❣❤➽❛ A+ (x∗ , λ∗ ) = E ∪ {i | i ∈ I (x∗ ) , λ∗i > 0} , ✭✷✳✶✶✷✮ ❚➟♣ ❤đ♣ A+ (x∗ , λ∗ ) ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝❤➾ sè ❝→❝ ✤✐➲✉ ❦✐➺♥ ❤♦↕t ✤ë♥❣ ♠↕♥❤✳ ❚❛ ❝â ❤➺ q✉↔ s❛✉ ✤➙②✿ ❍➺ q✉↔ ✷✳✸✳✶✳ ❈❤♦ x ❧➔ ♠ët ✤✐➸♠ ❑❑❚ ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✮✲✭✷✳✸✮✳ ◆➳✉ ∗ dT ∇2xx L (x∗ , λ∗ ) d > ✭✷✳✶✶✸✮ dT ∇ci (x∗ ) = 0, ∀ i ∈ A+ (x∗ , λ∗ ) , ✭✷✳✶✶✹✮ ✈ỵ✐ ♠å✐ d t❤ä❛ ♠➣♥ t❤➻ x∗ ❧➔ ♠ët ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ d ∈ G (x∗ , λ∗ ) t❤➻ dT ∇ci (x∗ ) = 0, ∀i ∈ A+ (x∗ , λ∗ ) ❚❤➟t ✈➟②✱ ❣✐↔ sû d ∈ G (x∗ , λ∗ ) t❤➻ d ∈ LF D (x∗ , X) , t❛ ❝â A+ (x∗ , λ∗ ) = E ∪ I+ (x∗ ) ❱ỵ✐ i ∈ E t❤➻ dT ∇ci (x∗ ) = ✭❞♦ d ∈ LF D (x∗ , X)) ❱ỵ✐ i ∈ I+ (x∗ ) t❤➻ dT ∇ci (x∗ ) = ✭❞♦ d ∈ G (x∗ , λ∗ )✮✳ ❱➟② dT ∇ci (x∗ ) = 0, ∀i ∈ A+ (x∗ , λ∗ ) ❑❤✐ ✤â dT ∇2xx L (x∗ , λ∗ ) d > 0, ∀d ∈ G (x∗ , λ∗ ) ❱➻ ✈➟②✱ x∗ ❧➔ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❝❤➦t✳ ✷✳✹ ❇➔✐ t♦→♥ ố ỵ x ởt ỹ t t ố ỗ P f (x) x s.t ci (x) ≥ 0, i = 1, , m ✭✷✳✶✶✺✮ ◆➳✉ f (x) ✈➔ ci (x) , (i = 1, , m) ❧➔ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤➲✉ ✤ó♥❣ ✱ t❤➻ x∗ ✈➔ λ∗ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✿ ✭✷✳✺✽✮ max L(x, λ) x,λ s.t ∇x L (x, λ) = 0, λ ≥ ✭✷✳✶✶✻✮ ❍ì♥ ♥ú❛✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❜➡♥❣ ♥❤❛✉✱ ♥❣❤➽❛ ❧➔✿ f (x∗ ) = L (x∗ , λ∗ ) ❈❤ù♥❣ ♠✐♥❤✳ ✭✷✳✶✶✼✮ ❚ø ❣✐↔ tt ỵ tỗ t tû ▲❛❣r❛♥❣❡ λ∗ ≥ s❛♦ ❝❤♦ ∇x L (x∗ , λ∗ ) = ✈➔ λ∗i ci (x∗ ) = 0, i = 1, , m ✸✾ ❈❤♦ x, λ ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ t ố ỷ 0, t ỗ ❝õ❛ L ✈➔ ∇x L (x, λ) = 0, t❛ ❝â m ∗ ∗ ∗ ∗ λi ci (x∗ ) L (x , λ ) = f (x ) ≥ f (x ) − i=1 ∗ = L(x , λ) T ≥ L (x, λ) + (x∗ − x) ∇x L (x, λ) = L (x, λ) ✭✷✳✶✶✽✮ ❱➻ ✈➟② (x∗ , λ∗ ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ỵ s ữủ ỵ ✤è✐ ♥❣➝✉ ②➳✉✱ ❝❤♦ t❤➜② r➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ ♠ư❝ t✐➯✉ t↕✐ ♠å✐ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ ♠ư❝ t✐➯✉ t↕✐ ♠å✐ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t ố ỵ x ởt ữủ tũ ỵ t ố ✭✷✳✶✶✺✮✳ ❈❤♦ (x, λ) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ tị② þ ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✭✷✳✶✶✻✮✳ ❑❤✐ ✤â✱ f (x ) ≥ L (x, λ) ✭✷✳✶✶✾✮ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ x ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ✈➔ (x, λ) ❧➔ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ t ố ỷ t ỗ f, t➼♥❤ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❧✐➯♥ ❤đ♣✱ t➼♥❤ ❧ã♠ ❝õ❛ ci , t➼♥❤ ❦❤æ♥❣ ➙♠ ❝õ❛ ci (x ) ✈➔ λi , t❛ ❝â f (x ) − f (x) ≥ ∇f (x)T (x − x) m λi ∇ci (x)T (x − x) = i=1 m ≥ λi (ci (x ) − ci (x)) i=1 m ≥− λi ci (x) i=1 ❱➻ ✈➟②✱ m λi ci (x) = L (x, λ) f (x ) ≥ f (x) − i=1 ứ ỵ tr t õ inf f (x) sup L (x, λ) x x,λ ✹✵ ✭✷✳✶✷✵✮ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ✧❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ❈➶ ✣■➋❯ ❑■➏◆✧ ✤➣ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ tố ỵ ❝➛♥ ✈➔ ✤õ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ❝â ✤✐➲✉ ❦✐➺♥ tê♥❣ q✉→t✳ ❚➜t ❝↔ ❝→❝ ✤à♥❤ ❧➼ ✤➲✉ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t✳ ✹✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❆♥❞r③❡❥ ❘✉s③❝③②♥s❦✐✱ ◆♦♥❧✐♥❡❛r Pr❡ss✱ ✶st ❡❞✐t✐♦♥✱ ✷✵✵✻✳ ❖♣t✐♠✐③❛t✐♦♥✳ Pr✐♥❝❡t♦♥✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② ❬✷❪ ❉❛✈✐❞ ●✳▲✉❡♥❜❡r❣❡r✱ ❨✐♥②✉ ❨❡✱ ▲✐♥❡❛r ❛♥❞ ◆♦♥❧✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✳ ❙t❛♥❞❢♦r❞ ❯♥✐✈❡rs✐t②✱ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ✹t❤ ❡❞✐t✐♦♥✱ ✷✵✶✻✳ ❬✸❪ ❊❞✇✐♥ ❑✳P✳❈❤♦♥❣✱ ❙t❛♥✐s❧❛✇ ❍✳❩❛❦✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ ❧❡② ❛♥❞ ❙♦♥s✱■♥❝✳✱ P✉❜❧✐s❤❝❛t✐♦♥✱ ✹t❤ ❡❞✐t✐♦♥✱ ✷✵✶✸✳ ❬✹❪ ❍♦➔♥❣ ❚✉✢✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❡rs✱ ✶st ❡❞✐t✐♦♥✱ ✶✾✾✽✳ ♦❢ ❖♣t✐♠✐③❛t✐♦♥✱ ❆ ❏♦❤♥ ❲✐✲ ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤✲ ❬✺❪ ■❣♦r ●✐r✈❛✱ ❙t❡♣❤❡♥ ●✳◆❛s❤✱ ❆r✐❡❧❛ ❙♦❢❡r✱ ▲✐♥❡❛r ❛♥❞ ◆♦♥❧✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥✱ ❚❤❡ ❙♦❝✐❡t② ❛♥❞ ■♥❞✉str✐❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✷♥❞ ❡❞✐t✐♦♥✱ ✷✵✵✾✳ ❬✻❪ ❘✳❚②rr❡❧❧ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥✈❡① ❡❞✐t✐♦♥✱ ✶✾✼✷✳ ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷♥❞ ❬✼❪ ●✐♦r❣✐ ●✳✱ ●✉❡rr❛❣❣✐♦♥ ❆✳❛♥❞ ❚❤✐❡r❢❡❧❞❡r✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❖♣t✐♠✐③❛t✐♦♥✿ ❙♠♦♦t❤ ❛♥❞ ◆♦♥s♠♦♦t❤ ❈❛s❡✱ ❆♠st❡r❞❛♠✲ ❇♦st♦♥ ❊❧s❡r✈✐❡r P✉❜❧✐s❤✐♥❣✱ ✷♥❞ ❡❞✐t✐♦♥✱ ✷✵✵✹✳ ❬✽❪ ❲❡♥②✉ ❙✉♥✱ ❨❛✲❳✐❛♥❣ ❨✉❛♥✱ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❇✉s✐♥❡ss ▼❡❞✐❛✱ ✶st ❡❞✐t✐♦♥✱ ✷✵✵✻✳ ❚❤❡♦r② ❛♥❞ ▼❡t❤♦❞s✱ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ❬✾❪ ❙✉♥✱ ❲❡♥②✉ ❛♥❞ ❨✉❛♥✱ ❨❛✲❳✐❛♥❣✱ ❖♣t✐♠✐③❛t✐♦♥ t❤❡♦r② ❣r❛♠♠✐♥❣✱ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡❞✐❛✱ ✷✵✵✻✳ ✹✷ ❛♥❞ ♠❡t❤♦❞s✿ ♥♦♥❧✐♥❡❛r ♣r♦✲