Header Page of 54 ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ◆●❯❨➍◆ ▲➊ ◗❯❹◆ ✣■➋❯ ❑■➏◆ ❚➮■ ×❯ ❚❖⑨◆ ❈Ư❈ ❚❘❖◆● ◗❯❨ ❍❖❸❈❍ ❚❖⑨◆ P❍×❒◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✶✽ Footer Page of 54 Header Page of 54 ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ◆●❯❨➍◆ ▲➊ ◗❯❹◆ ✣■➋❯ ❑■➏◆ ❚➮■ ×❯ ❚❖⑨◆ ❈Ư❈ ❚❘❖◆● ◗❯❨ ❍❖❸❈❍ ❚❖⑨◆ P❍×❒◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣÷í✐ ữợ P ◆ë✐ ✲ ✷✵✶✽ Footer Page of 54 Header Page of 54 ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ữợ sỹ ữợ P ❚➙♠✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tợ ữớ t t ữợ ú ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❊♠ ❝ô♥❣ ❜➔② tọ ỏ t ỡ t tợ qỵ t ổ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ỗ rữớ P ỗ ❍✉②➺♥ ✣❛♥ P❤÷đ♥❣ ✲ ❍➔ ◆ë✐✱ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ✈➲ ♠å✐ ♠➦t tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷✽ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✽ ◆❣✉②➵♥ ▲➯ ◗✉➙♥ ✶ Footer Page of 54 Header Page of 54 ổ ữợ sỹ ữợ P ❈❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤ ✈ỵ✐ ✤➲ t➔✐✿ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣ ❞♦ tỉ✐ tü ❧➔♠✳ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ t♦➔♥ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ t❤➔♥❤ q✉↔ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❈→❝ ❦➳t q✉↔ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ tr tỹ ữủ ró ỗ ố ◆ë✐✱ ♥❣➔② ✷✽ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✽ ◆❣✉②➵♥ ▲➯ ◗✉➙♥ ✷ Footer Page of 54 Header Page of 54 ▼ö❝ ❧ö❝ ▲❮■ ❈❷▼ ❒◆ ▲❮■ ❈❆▼ ✣❖❆◆ ▲❮■ ▼Ð ✣❺❯ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶ ✷ ✹ ✻ ✶✳✶ ▼ët sè ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ❝õ❛ t ỗ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❇➔✐ t♦→♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝ ✈➔ ✤➦❝ tr÷♥❣ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷ ✣■➋❯ ❑■➏◆ ❚➮■ ×❯ ❚❖⑨◆ ❈Ư❈ ❚❘❖◆● ◗❯❨ ❍❖❸❈❍ ❚❖⑨◆ P❍×❒◆● ố ữ t ữỡ ợ r t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ✣➦❝ tr÷♥❣ ❝õ❛ ❇ê ✤➲ S ✈➔ ❝❤➼♥❤ q✉② ❤â❛ ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ tè✐ ÷✉ t♦➔♥ ❝ư❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❑➌❚ ▲❯❾◆ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✷ ✹✸ ✸ Footer Page of 54 Header Page of 54 ▲❮■ ▼Ð ✣❺❯ ❚è✐ ÷✉ t♦➔♥ ♣❤÷ì♥❣ ❧➔ ♠ët ❜ë ♣❤➟♥ ❝õ❛ q✉② t õ ự tr ỵ tt ❝ơ♥❣ ♥❤÷ tr♦♥❣ ✤í✐ sè♥❣ t❤ü❝ t➳✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤✉➟t t♦→♥ ❣✐↔✐ ❤ú✉ ❤✐➺✉ ❝→❝ ❜➔✐ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣ ❧➔ ♠ët ❝❤õ ✤➲ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ tr ữợ q t s ✤÷đ❝ ❤å❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ t♦→♥ ❣✐↔✐ t➼❝❤✱ ✈ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✤➣ ❤å❝ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ✿ ✏ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ t♦➔♥ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣ ✑✳ ▲✉➟♥ ✈➠♥ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ✤✐➲✉ ❦✐➺♥ tố ữ t ỳ ợ t tr q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣✳ ◗✉❛ ✤â t❤➜② ✤÷đ❝ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✤➣ ❤å❝ ✈➔ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣✳ ❱ỵ✐ ♥ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ♥❣♦➔✐ ♣❤➛♥ ❧í✐ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✳ ❑➳t q✉↔ ❝❤➼♥❤ t➟♣ tr✉♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈ư t❤➸ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ ♣❤➛♥ ✤➛✉ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t t ỗ ỵ tt tố ữ sỷ ữỡ t t ữ t ỗ ỗ ữợ õ tr t tố ữ ũ ợ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✱ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣✱ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝✳ ❈❤÷ì♥❣ ✷✳ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ t♦➔♥ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣✳ ❚r♦♥❣ ữỡ trữợ t s tr tố ữ t ữỡ ợ r t ữỡ ◗✉❛ ✤â✱ t❤✉ ✤÷đ❝ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝ơ♥❣ ♥❤÷ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ❝→❝ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝✳ ❚✐➳♣ ✤â✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❇ê ✤➲ S ✈➔ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ❝❤♦ ✹ Footer Page of 54 Header Page of 54 tè✐ ÷✉ t♦➔♥ ♣❤÷ì♥❣ tr➯♥ ♠ët r➔♥❣ ❜✉ë❝ t♦➔♥ ♣❤÷ì♥❣ t❤ỉ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r✳ ❱➔ s❛✉ ✤â ❧➔ ❝❤➼♥❤ q✉② ❤â❛ ❇ê ✤➲ S ❦❤æ♥❣ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r✳ ❈✉è✐ ❝❤÷ì♥❣✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ tè✐ ÷✉ t♦➔♥ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣✳ ✺ Footer Page of 54 Header Page of 54 ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ s➩ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ ❦➳t q✉↔ ❝➛♥ t❤✐➳t t ỗ ỵ tt tố ữ ♣❤ư❝ ✈ư ❝❤♦ ❈❤÷ì♥❣ ✷✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ❬✶❪✱ ❬✶✺❪✱ ❬✶✻❪✱ ❬✶✽❪ ✈➔ ❬✷✻❪✳ ✶✳✶ ▼ët sè ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ỗ r t t ỵ Rn ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ n✲❝❤✐➲✉ tr➯♥ tr÷í♥❣ sè t❤ü❝ R✳ ộ tỡ x Rn s ỗ n tồ ✤ë ❧➔ ❝→❝ sè t❤ü❝✳ ❱ỵ✐ ❤❛✐ ✈➨❝ tì x = (x1 , , xn )T ✈➔ y = (y1 , , yn )T t❤✉ë❝ Rn ✱ t❛ ♥❤➢❝ ❧↕✐ r➡♥❣ n x, y := xi yi i=1 t ổ ữợ ❤❛✐ ✈➨❝ tì x ✈➔ y✳ ❈❤✉➞♥ ❊✉❝❧✐❞❡ ❝õ❛ ♣❤➛♥ tû x ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ x, x x := ỵ Rn+ t tt ✈➨❝ tì ❦❤ỉ♥❣ ➙♠ ❝õ❛ Rn ✳ ❱ỵ✐ ❝→❝ ✈➨❝ tì x, y ∈ Rn , x ≥ y ✤÷đ❝ ❤✐➸✉ ❧➔ xi ≥ yi ✱ ✈ỵ✐ i = 1, , n✳ ▼❛ tr➟♥ ✤ì♥ ✈à (n ì n) ữủ ỵ In A ❞÷ì♥❣✳ ❝â ♥❣❤➽❛ ❧➔ ♠❛ tr➟♥ A ❧➔ ♥û❛ tr ợ tỷ ữớ 1, , n ữủ ỵ diag(α1 , , αn )✳ ❑❤æ♥❣ ❣✐❛♥ ❝õ❛ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ (n × n) ữủ ỵ S n t A B ✈➔ A B t÷ì♥❣ ù♥❣ ✤÷đ❝ ❤✐➸✉ ❧➔ ♠❛ tr➟♥ A − B ✻ Footer Page of 54 Header Page of 54 ❧➔ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✈➔ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ◆â♥ ♥û❛ ①→❝ ✤à♥❤ ❞÷ì♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ S+n := {M ∈ S n : M 0} ❚➼❝❤ tr♦♥❣ ✭✈➳t✮ ❝õ❛ A ✈➔ B ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ n n A · B = Tr[AB] = aij bji , i=1 j=1 tr♦♥❣ ✤â aij ❧➔ ♣❤➛♥ tû (i, j) ❝õ❛ A ✈➔ bji ❧➔ ♣❤➛♥ tû (j, i) ❝õ❛ B ✳ ❈❤♦ K ❧➔ ♠ët ♥â♥ tr♦♥❣ S n ✳ ❈❤✉➞♥ ❝õ❛ A ∈ S n ✈➔ ❦❤♦↔♥❣ ❝→❝❤ tø A ✤➳♥ ♥â♥ K ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➛♥ ❧÷đt ❧➔ A = (A · A)1/2 ✈➔ d(A, K) = inf A − B B∈K ◆❤➙♥ ✭❦❡r♥❡❧ ✮ ❝õ❛ ♠ët ♠❛ tr➟♥ A ∈ S n ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ KerA := {x ∈ Rn : Ax = 0} ❜❛♦ ✤â♥❣ ❝õ❛ D ữủ ỵ D ởt t ởt õ λK ⊆ K ✈ỵ✐ ♠å✐ λ ≥ 0✳ ❈ü❝ ➙♠ ❱ỵ✐ ♠ët t➟♣ ❝♦♥ D ⊂ Rn ✱ K ⊂ Rn ữủ t r K ỵ ❤✐➺✉ ❧➔ K ◦ := {d : dT x ≤ ∀x ∈ K} ❱ỵ✐ ❝→❝ ♥â♥ K1 , K2 ∈ Rn ✱ ❝ỉ♥❣ t❤ù❝ ❝ü❝ ✭❬✶✻❪✮ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ (K1◦ ∩ K2◦ )◦ = K1 + K2 n ✣➦❝ ❜✐➺t✱ ❦❤✐ K1 = −S+ ✈➔ K2 = t≥0 tH2 ✱ tr♦♥❣ ✤â H2 ❧➔ ♠❛ tr➟♥ ♥➔♦ ✤â tr♦♥❣ S n ✱ t❛ t❤✉ ✤÷đ❝✿ {X ∈ S+n : H2 · X ≤ 0}◦ = (K1◦ ∩ K2◦ )◦ = K1 + K2 = −S+n + tH2 t≥0 ✼ Footer Page of 54 ✭✶✳✶✮ Header Page 10 of 54 ▼ët ✤÷í♥❣ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ✭❤❛✐ ✈➨❝ tì✮ a, b tr♦♥❣ Rn ❧➔ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✈➨❝ tì x ∈ Rn ❝â ❞↕♥❣ {x ∈ Rn : x = αa + βb, α, β ∈ R, α + β = 1} ✣♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ a, b tr♦♥❣ Rn ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✈➨❝ tì x ∈ Rn ❝â ❞↕♥❣ {x ∈ Rn : x = αa + βb, α ≥ 0, β ≥ 0, α + β = 1} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ▼ët t➟♣ C ⊆ Rn ✤÷đ❝ ❣å✐ ởt t ỗ C ự t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♥â✳ ◆â✐ t C ỗ ∀x, y ∈ C, ∀λ ∈ [0, 1] =⇒ λx + (1 − λ)y ∈ C ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭❬✶✻❪✮ ❈❤♦ h : Rn −→ R ∪ {−∞, +∞}✳ ✭✐✮ dom h := {x ∈ Rn | h(x) < +∞} ❧➔ ✭✐✐✮ ❍➔♠ h ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ❤ú✉ ❞ư♥❣ ❝õ❛ h❀ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ h ❦❤ỉ♥❣ ❧➜② tr➯♥ ❣✐→ tr dom h = r ỗ t ✭❡♣✐❣r❛♣❤✮ ❝õ❛ h ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ ✭✐✐✐✮ epi h := {(x, r) ∈ Rn × R | x ∈ dom h, h(x) ≤ r}; ❍➔♠ ❧✐➯♥ ❤ñ♣ ✭❝♦♥❥✉❣❛t❡ ❢✉♥❝t✐♦♥✮ ❝õ❛ h✱ h∗ : Rn −→ R ∪ {+∞}✱ ✭✐✈✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ h∗ (v) := sup{v(x) − h(x) | x ∈ dom h}, tr♦♥❣ ✤â v(x) := v T x✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸ ✭❬✶❪✮ ❈❤♦ C ⊆ Rn ởt t ỗ rộ h : C −→ R✳ ◆➳✉ ❡♣✐ h ❧➔ ♠ët t ỗ tr Rn+1 t t õ h ởt ỗ tr C Footer Page 10 of 54 Header Page 33 of 54 ❈✉è✐ ❝ò♥❣✱ q✉❛ ❣✐ỵ✐ ❤↕♥ ❦❤✐ k −→ ∞✱ t❛ ❝â ♥❣❛② ữủ ự ỵ ✷✳✷✳✺ ✭❈❤➼♥❤ q✉② ❤â❛ ❇ê ✤➲ ❙ ✮ ❈❤♦ f, g : Rn −→ R ❧➔ ❝→❝ ❤➔♠ t♦➔♥ ♣❤÷ì♥❣ ợ [g ữỡ 0] = õ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ (a) g(x) ≤ =⇒ f (x) ≥ (b) (∀ε > 0)(∃λε ≥ 0)(∀x ∈ Rn )f (x) + λε g(x) + ε( x ❈❤ù♥❣ ♠✐♥❤ ✳ ❬✭❜✮ =⇒ ✭❛✮❪ + 1) ≥ ▲➜② x ∈ [g ≤ 0] ✈➔ ε > õ tỗ t s f (x) ≥ λε g(x) − ε( x + 1) ≥ −ε( x + 1) ❈❤♦ ε −→ 0✱ t❛ t❤✉ ✤÷đ❝ f (x) ≥ 0✳ ❉♦ ✤â✱ t❛ ❝â ❦❤➥♥❣ ✤à♥❤ ✭❛✮✳ ❬✭❛✮ =⇒ ✭❜✮❪ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✶✳ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ t❤✉➛♥ ♥❤➜t ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x) = xT Ax ✈➔ g(x) = xT Bx, tr♦♥❣ ✤â A, B ∈ S n ✳ ◆➳✉ n = 2✱ t❤➻ t❛ ❝â (0, 0) ∈ / {(f (x), g(x)) : x ∈ R2 } + intR+ × R+ ổ t tỗ t x0 s f (x0 ) < ✈➔ g(x0 ) ≤ 0✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭❛✮✮✳ ❑❤✐ ✤â✱ t❤❡♦ ❇ê ✤➲ ✶✳✶✳✾✱ t❛ ❝â (0, 0) ∈ / {(A · X, B · X) : X ∈ S+2 } + intR+ × R+ ▼➦t ❦❤→❝✱ t❛ ❝â X ∈ S+2 , B · X ≤ =⇒ A · X ≥ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ −A ∈ {X ∈ S+2 : B · X ≤ 0}◦ ✸✶ Footer Page 33 of 54 Header Page 34 of 54 ❚ø ú ỵ r {X S+2 : B ã X ≤ 0}◦ = −S+2 + λB λ≥0 s❛♦ ợ ộ > tỗ t λε ≥ ✈➔ Pε ∈ S+ (A + λε B) − Pε ≤ ε ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ Rn ✱ t❛ ❝â f (x) + λε g(x) = xT (A + λε B)x = xT Pε x + xT ((A + λε B) − Pε )x ≥ −ε x ❙✉② r❛ ✭❜✮ t❤ä❛ ♠➣♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ◆➳✉ n = 2✱ t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët t➟♣ V ⊆ R2 ♥❤÷ s❛✉✿ V := {(f (x), g(x)) | x = 1} + R2+ õ V ởt t ỗ õ t ♥➳✉ n = t❤➻ {(f (x), g(x)) | x = 1} ❧➔ ♠ët t➟♣ ✤ì♥ ♣❤➛♥ tû✳ ❍✐➸♥ ♥❤✐➯♥✱ õ V t õ ỗ n ≥ 3✱ tø ❇ê ✤➲ ✶✳✶✳✽ t❛ s✉② r❛ V ụ t õ ỗ tờ s ởt t ỗ t ởt t ỗ õ õ ỗ t t ố > 0✳ ❑❤✐ ✤â (−ε, 0) ∈ / V ✳ ◆➳✉ ổ t tỗ t x0 = s (x0 ) ≤ −ε < ✈➔ g(x0 ) ≤ 0, ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭❛✮✳ ❑❤✐ ✤â✱ t❤❡♦ ✤à♥❤ ỵ t t tỗ t r R, (1 , λ2 ) ∈ R2 \ {(0, 0)} s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ (a1 , a2 ) ∈ V t❛ ❝â −λ1 ε < r < λ1 a1 + λ2 a2 ✣➦❝ ❜✐➺t✱ t❛ ❝â (λ1 , λ2 ) ∈ R2+ \ {(0, 0)} ✈➔ ✈ỵ✐ ♠å✐ x ∈ Rn ♠➔ x = 1✱ t❛ ❝â −λ1 ε < r < λ1 f (x) + λ2 g(x) ✸✷ Footer Page 34 of 54 ✭✷✳✺✮ Header Page 35 of 54 ❚✐➳♣ ✤➳♥✱ t s r r tỗ t s❛♦ ❝❤♦ A + λε B + εIn ✭✷✳✻✮ ▼➔ ✤✐➲✉ ♥➔② ❝❤♦ t❛ f (x) + λε g(x) + ε x ≥ 0, ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❙✉② r❛ ✭❜✮ t❤ä❛ ♠➣♥✳ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣✿ ❚r÷í♥❣ ❤đ♣ ✶✳✶✳ ◆➳✉ λ1 = 0✱ t❤➻ λ2 > 0✳ ❚ø ✭✷✳✺✮✱ ✈ỵ✐ ♠é✐ x ∈ Rn ♠➔ x = 1✱ t❛ ❝â xT Bx = g(x) > r > λ2 ❙✉② r❛ B ❧➔ ♠❛ tr ữỡ õ tỗ t ✭✤õ ❧ỵ♥✮ s❛♦ ❝❤♦ A + λ0 B ❉♦ ✤â✱ ❦➳t ❧✉➟♥ t❤ä❛ ♠➣♥ ❦❤✐ t❛ ❝❤å♥ λε = λ0 ✳ ❚r÷í♥❣ ❤đ♣ ✶✳✷✳ ●✐↔ sû r➡♥❣ λ1 > 0✳ ❚ø ✭✷✳✺✮✱ ✈ỵ✐ ♠å✐ x ∈ Rn ♠➔ x = 1✱ t❛ ❝â ≤ λ1 f (x) + λ2 g(x) + λ1 ε = λ1 f (x) + λ2 g(x) + λ1 ε x ❙✉② r❛ f (x) + ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ✣➦t λε = λ2 g(x) + ε x λ1 ≥ 0, λ2 t❛ t❤✉ ✤÷đ❝ λ1 A + λε B + εIn ❚r÷í♥❣ ❤đ♣ ✷✳ ●✐↔ sû r➡♥❣ f (x) = xT Ax + aT x + α, g(x) = xT Bx + bT x + β ✸✸ Footer Page 35 of 54 Header Page 36 of 54 t t t ữợ tr Rn+1 t÷ì♥❣ ù♥❣ ✤÷đ❝ s✐♥❤ r❛ tø f ✈➔ g ✿ f (x, ρ) = xT Ax + ρaT x + ρ2 α = x ρ g(x, ρ) = xT Bx + ρbT x + ρ2 β = x ρ T Hf x , ρ ✭✷✳✼✮ Hg x ρ ✭✷✳✽✮ T ❚✐➳♣ t❤❡♦✱ t❛ ❦➳t ❧✉➟♥ r➡♥❣ g(x, ρ) ≤ =⇒ f (x, ρ) ≥ ❚❤❡♦ ❚r÷í♥❣ ❤đ♣ ✶✱ t❛ t❤✉ ✤÷đ❝ (∀ε > 0), (∃λε ≥ 0)(∀(x, ρ) ∈ Rn+1 )f (x, ρ) + λε g(x, ρ)+ ε + ( (x, ρ) + 1) ≥ ❈❤♦ ρ = 1✱ t❛ ✤÷đ❝ ε (∀ε > 0), (∃λε ≥ 0)(∀x ∈ Rn )f (x) + λε g(x) ≥ − ( x + 2) ≥ −ε( x + 1) ❙✉② r❛ ✭❜✮ t❤ä❛ ♠➣♥✳ sỷ r tỗ t (x0 , ) Rn × R s❛♦ ❝❤♦ f (x0 , ρ0 ) < ✈➔ g(x0 , ρ0 ) ≤ ◆➳✉ ρ = 0✱ t❤➻ f x0 ρ0 = ρ−2 f (x0 , ρ0 ) < g x0 ρ0 = ρ−2 g(x0 , ρ0 ) ≤ 0, ✈➔ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✭❛✮✳ ◆➳✉ ρ = 0✱ t❤➻ xT0 Ax0 < ✈➔ xT0 Bx0 ≤ ✸✹ Footer Page 36 of 54 Header Page 37 of 54 ❚✐➳♣ ✤➳♥✱ ❝è ✤à♥❤ x s❛♦ ❝❤♦ g(x) ≤ 0✳ ❱ỵ✐ ♠å✐ t ∈ R✱ t❛ ❝â f (tx0 + x) = t2 xT0 Ax0 + 2t(a + Ax)T x0 + f (x) ✈➔ g(tx0 + x) = t2 xT0 Bx0 + 2t(b + Bx)T x0 + g(x) ❇➙② ❣✐í✱ t❛ ❧↕✐ ❝❤✐❛ t❤➔♥❤ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ❚r÷í♥❣ ❤đ♣ ✷✳✶✳ (b + Bx)T x0 ≤ 0✳ ❚ø xT0 Ax0 < 0✱ ✈ỵ✐ ♠å✐ t > ✤õ ❧ỵ♥✱ t❛ ❝â f (tx0 + x) < ▼➦t ❦❤→❝✱ tø xT0 Bx0 ≤ 0, (b + Bx)T x0 ≤ ✈➔ g(x) ≤ 0✱ t❛ ❝â g(tx0 + x) = t2 xT0 Bx0 + 2t(b + Bx)T x0 + g(x) ≤ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭❛✮✳ ❚r÷í♥❣ ❤đ♣ ✷✳✷✳ (b + Bx)T x0 ≥ 0✳ ❑❤✐ ✤â✱ t÷ì♥❣ tü t❛ ①➨t ✤✐➸♠ −tx0 + x ợ t > ợ t ữủ f (−tx0 + x) < ✈➔ g(−tx0 + x) ≤ 0, ✤✐➲✉ ♥➔② ❝ơ♥❣ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❦❤➥♥❣ ✤à♥❤ ✭❛✮✳ ỵ ữủ ự t ữ t s t tr q ữợ tr ỵ ✷✳✷✳✺ t❛ ❣✐↔ t❤✐➳t t❤➯♠ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r t❤ä❛ ♠➣♥✱ t❤➻ ❝❤➼♥❤ q✉② ❤â❛ ❇ê ✤➲ S t❤✉ ❣å♥ ❧↕✐ t❤➔♥❤ ❇ê ✤➲ S ❝❤✉➞♥✳ ❍➺ q✉↔ ✷✳✷✳✻ ✭❇ê ✤➲ S ✮ ❈❤♦ f, g : Rn −→ R ❧➔ ❝→❝ ❤➔♠ t♦➔♥ ♣❤÷ì♥❣✱ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x) = xT Ax + aT x + α ✈➔ g(x) = xT Bx + bT x + β, A, B ∈ S n , a, b ∈ Rn ✈➔ α, β ∈ R✳ ❈❤♦ Hf , Hg ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr♦♥❣ ✭✶✳✼✮✳ ●✐↔ sû r➡♥❣ [g ≤ 0] = tỗ t x0 Rn s g(x0) < 0✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ (a) g(x) ≤ =⇒ f (x) ≥ (b) (∃λ ≥ 0) f (x) + λg(x) ≥ (c) (∃λ ≥ 0) Hf + λHg (d) (∀ε > 0)(∃λε ≥ 0)(∀x ∈ Rn )f (x) + λε g(x) + ε( x ✸✺ Footer Page 37 of 54 + 1) ≥ Header Page 38 of 54 ❈❤ù♥❣ ♠✐♥❤ ✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✹✱ t❛ ❝â ✭❜✮ ⇐⇒ ✭❝✮ ✈➔ ✭❝✮ =⇒ ✭❞✮✳ ❍ì♥ ♥ú❛✱ t ỵ t õ ✤â✱ t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ ✭❞✮ =⇒ ✭❝✮✳ ❬✭❞✮ =⇒ ✭❝✮❪ ●✐↔ sû t❛ ❝â ❦❤➥♥❣ ✤à♥❤ ✭❞✮✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✹✱ t❤➻ ∃{λk } ⊆ R+ , d(Hf + λk Hg , S+n+1 ) −→ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ Hf ∈ S+n+1 + λ(−Hg ) λ≥0 n+1 ❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤➾ r❛ r➡♥❣ S+ + λ≥0 λ(−Hg ) ❧➔ t➟♣ ✤â♥❣✳ ❚❤➟t ✈➟②✱ t❛ ✤➦t Zk = Pk + λk (−Hg ) n+1 ❱ỵ✐ Zk −→ Z ✱ tr♦♥❣ ✤â Pk ∈ S+ ✈➔ λk ≥ 0✳ ❱ỵ✐ x0 ∈ Rn ✱ t❛ ❧➜② X0 = x0 x0 T ❑❤✐ ✤â 0≤ x0 T Pk x0 = Pk · X = (Zk + λk Hg ) · X0 = Zk · X0 + λk g(x0 ) ❚ø g(x0 ) < 0✱ s✉② r❛ ≤ λk ≤ Zk · X −g(x0 ) ❉♦ Zk −→ Z ✈➔ ✈➻ t❤➳ ♥â ❜à ❝❤➦♥✱ ♥➯♥ λk ❜à ❝❤➦♥✳ ❇➡♥❣ ❝→❝❤ ❝❤✉②➸♥ q✉❛ ❞➣② ❝♦♥✱ t❛ ❝â λk −→ λ0 ≥ ❚ø ✤â Pk = Zk + λk Hg −→ Z + λ0 Hg ∈ S+n+1 , ✸✻ Footer Page 38 of 54 Header Page 39 of 54 ✈➔ ❞♦ ✤â t❛ ❝â Z = (Z + λ0 Hg ) + λ0 (−Hg ) ∈ S+n+1 + λ(−Hg ) λ≥0 n+1 ❙✉② r❛ t➟♣ S+ + λ≥0 λ(−Hg ) ❧➔ ✤â♥❣✳ ❚❤➔♥❤ t❤û Hf ∈ S+n+1 + λ(−Hg ) õ tỗ t s Hf + λHg ❍➺ q✉↔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✸ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ tè✐ ÷✉ t♦➔♥ ❝ư❝ ❳➨t ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ t♦➔♥ ♣❤÷ì♥❣ ✭◗▼P✮✿ x∈Rn s❛♦ ❝❤♦ ✭◗▼P✮ f (x) g(x) ≤ 0, tr♦♥❣ ✤â f, g ❧➔ ❝→❝ ❤➔♠ t♦➔♥ ♣❤÷ì♥❣ ❧➛♥ ❧÷đt ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ f (x) = xT Ax + aT x + α, g(x) = xT Bx + bT x + β, tr♦♥❣ ✤â A, B ∈ S n , a, b ∈ Rn , α, β ∈ R✳ ❇ê s õ ố ỳ ữợ ữợ s ữủ sû ❞ư♥❣ ✤➸ ✤➦❝ tr÷♥❣ ❞➣② tè✐ ÷✉ t♦➔♥ ❝ư❝ ❝õ❛ ❜➔✐ t♦→♥ ✭◗▼P✮ ❦❤æ♥❣ ❝â r➔♥❣ ❜✉ë❝ ✤à♥❤ t➼♥❤✳ ❇ê ✤➲ ✷✳✸✳✶ ●✐↔ sû r➡♥❣ f ❧➔ ♠ët ❤➔♠ ỗ tử tr Rn > 0, ≥ ✈➔ x∗0 ∈ ∂ε f (x0 )✳ ❑❤✐ õ tỗ t x , x Rn s ✭❬✶✻❪✮ x∗ε ∈ ∂f (xε ), xε − x0 ≤ ✸✼ Footer Page 39 of 54 √ ε, Header Page 40 of 54 |f (xε ) − f (x0 )| ≤ ✈➔ x∗ε − x∗0 ≤ √ √ √ ε( ε + β −1 ), ε(1 + β x∗0 ) ỵ tố ữ t ợ t P x ∈ [g ≤ 0]✳ ❑❤✐ ✤â✱ ❤❛✐ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✭❛✮ ✭❜✮ x ❧➔ ♠ët ❝ü❝ t t t P ỗ t ❞➣② {xk } ⊂ Rn ✈➔ {λk } ⊆ R+ s❛♦ ❝❤♦ ❦❤✐ k −→ ∞✱ t❛ ❝â xk − x −→ 0, λk g(xk ) −→ 0, ∇(f + λk g)(xk ) −→ 0, ✈➔ d(A + λk B, S+n ) −→ ❈❤ù♥❣ ♠✐♥❤ ✳ ❬✭❛✮ =⇒ ✭❜✮❪ ●✐↔ sû r➡♥❣ x ❧➔ ♠ët ❝ü❝ t✐➸✉ t♦➔♥ ❝ö❝ ❝õ❛ ❜➔✐ t♦→♥ ✭◗▼P✮✳ ❑❤✐ ✤â✱ g(x) ≤ =⇒ f (x) − f (x) ≥ , k N õ t ỵ tỗ t {λk } ⊆ R+ s❛♦ k ❝❤♦ ✈ỵ✐ ♠å✐ x ∈ Rn ✱ t❛ ❝â ▲➜② εk = h(x) := f (x) − f (x) + λk g(x) + εk ( x + 1) ≥ ✭✷✳✾✮ ✣➦❝ ❜✐➺t✱ t❛ ❝â ≥ λk g(x) ≥ −εk x k t ỵ r ợ ♠å✐ x ∈ Rn ✱ t❛ ❝â h(x) ≥ 0✳ ❉♦ ✤â ∇2 h(x) = 2(A + λk B + εk In ) ✸✽ Footer Page 40 of 54 0, Header Page 41 of 54 ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(A + λk B, S+n ) ≤ εk In −→ 0, ❦❤✐ k −→ ∞✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠å✐ x ∈ Rn ✱ t❤❡♦ ✭✷✳✶✵✮ ✈➔ ✭✷✳✾✮✱ t❛ ❝â h(x) − h(x) = (f (x) − f (x) + λk g(x) + εk x ) − (λk g(x) + εk x ) ≥ −εk − εk x ❚❛ ✤à♥❤ ♥❣❤➽❛ ηk = εk + εk x > ❑❤✐ ✤â ∈ ∂ηk h(x) ✈➔ ηk −→ ❦❤✐ k −→ ∞ ⑩♣ ❞✉♥❣ ❇ê ✤➲ ✷✳✸✳✶ ✭❧➜② β = 1, ε = ηk ✈➔ x∗0 = tỗ t xk s xk x ≤ √ ηk , |h(x) − h(xk )| ≤ ηk + √ ηk , ✈➔ ∈ ∂h(xk ) + √ ηk B = ∇(f + λk g)(xk ) + 2εk xk + tr♦♥❣ ✤â B ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à ✤â♥❣ tr♦♥❣ Rn ✳ ❚ø εk −→ ✈➔ ηk −→ 0, ❦❤✐ k −→ ∞, t❛ ❝â xk −→ x, ∇(f + λk g)(xk ) −→ ❈✉è✐ ❝ò♥❣✱ ✤➸ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤✱ t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ lim λk g(xk ) = k−→∞ ❚❤➟t ✈➟②✱ tø |h(x) − h(xk )| ≤ ηk + ✸✾ Footer Page 41 of 54 √ ηk , √ η k B, Header Page 42 of 54 t❛ ❝â |(λk g(x) + εk x ) − (f (xk ) − f (x) + λk g(xk ) + εk xk )| ≤ ηk + √ ηk ❑➳t ❤đ♣ ✈ỵ✐ xk −→ x, εk −→ ✈➔ ηk −→ 0, ❦➨♦ t❤❡♦ lim λk (g(x) − g(xk )) = k−→∞ ❚ø ✭✷✳✶✵✮✱ t❛ s✉② r❛ lim λk g(xk ) = lim λk g(x) = k−→∞ ❬✭❜✮ =⇒ ✭❛✮❪ k sỷ r tỗ t x0 s g(x0 ) ≤ ✈➔ f (x0 ) < f (x) ứ tỗ t xk x, {k } ⊆ R+ ✈ỵ✐ λk g(xk ) −→ s❛♦ ❝❤♦ ∇(f + λk g)(xk ) −→ ✈➔ d(A + λk B, S+n ) −→ ❚❛ ❝â f (x0 ) − f (x) = (f (x0 ) + λk g(x0 )) − (f (xk ) + λk g(xk )) + (f (xk ) − f (x)) + λk g(xk ) − λk g(x0 ) ≥ (f (x0 ) + λk g(x0 )) − (f (xk ) + λk g(xk )) + (f (xk ) − f (x)) + λk g(xk ) = (∇(f + λk g)(xk ))T (x0 − xk ) + (x0 − xk )T (A + λk B)(x0 − xk ) + (f (xk ) − f (x)) + λk g(xk ), ✹✵ Footer Page 42 of 54 Header Page 43 of 54 ð ✤➙② ❜➜t ✤➥♥❣ t❤ù❝ t❤ä❛ ♠➣♥ ✈➻ λk ≥ ✈➔ g(x0 ) ≤ 0✳ ❍ì♥ ♥ú❛✱ tø xk −→ x, ∇(f + λk g)(xk ) −→ 0, λk g(xk ) −→ ✈➔ f (x0 ) < f (x), ❜➡♥❣ ❝→❝❤ ❝❤♦ q✉❛ ❣✐ỵ✐ ❤↕♥ tr➯♥✱ t❛ ❝â lim sup (x0 − xk )T (A + λk B)(x0 − xk ) ≤ f (x0 ) − f (x) < k−→∞ ✭✷✳✶✶✮ ❈✉è✐ ❝ò♥❣✱ tø d(A + k B, S+n ) 0, n tỗ t {Pk } ⊆ S+ s❛♦ ❝❤♦ (A + λk B) − Pk −→ ❚ø ✤â✱ t❛ ❝â lim sup (x0 − xk )T (A + λk B)(x0 − xk ) k−→∞ 1 = lim sup (x0 − xk )T (A + λk B − Pk )(x0 − xk ) + (x0 − xk )T Pk (x0 − xk ) 2 k−→∞ ≥ lim sup (x0 − xk )T (A + λk B − Pk )(x0 − xk ) = 0, k−→∞ ✤✐➲✉ ♥➔② t ợ ỵ ữủ ự t♦➔♥✳ ✹✶ Footer Page 43 of 54 Header Page 44 of 54 ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✏ ✣✐➲✉ ❦✐➺♥ tè✐ ÷✉ t♦➔♥ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣ ✑ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ♠ët sè ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❍➺ t❤è♥❣ ❧↕✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tr♦♥❣ ❣✐↔✐ t ỗ ỵ tt tố ữ r õ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ✤✐➸♠ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✱ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣✱ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝✳ ✷✳ ❚r➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝ ✈➔ ✤➦❝ tr÷♥❣ ♥❤➙♥ tû ▲❛❣r❛♥❣❡✳ ✸✳ ❚r➻♥❤ ❜➔② tố ữ t ữỡ ợ r t ♣❤÷ì♥❣✳ ◗✉❛ ✤â✱ t❤✉ ✤÷đ❝ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝ơ♥❣ ♥❤÷ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ❝→❝ ❝ü❝ t✐➸✉ t♦➔♥ ❝ư❝✳ ✹✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❇ê ✤➲ S ✱ ✤è✐ ♥❣➝✉ ▲❛❣r❛♥❣❡ ❝❤♦ tè✐ ÷✉ t♦➔♥ ♣❤÷ì♥❣ tr➯♥ ♠ët r➔♥❣ ❜✉ë❝ t♦➔♥ ♣❤÷ì♥❣ t❤ỉ♥❣ q✉❛ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r ✈➔ t➼♥❤ ❝❤➼♥❤ q✉② ❤â❛ ❇ê ✤➲ S ❦❤æ♥❣ ✤✐➲✉ ❦✐➺♥ ❙❧❛t❡r✳ ◆❣♦➔✐ r❛✱ ❧✉➟♥ ✈➠♥ ❝á♥ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ tè✐ ÷✉ t♦➔♥ ❝ư❝ tr♦♥❣ q✉② ❤♦↕❝❤ t♦➔♥ ♣❤÷ì♥❣✳ ✹✷ Footer Page 44 of 54 Header Page 45 of 54 ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❍✐➲♥✱ ▲➯ ❉ơ♥❣ ▼÷✉✱ ◆❣✉②➵♥ ❍ú✉ ✣✐➸♥ ✭✷✵✶✺✮✱ ♠ỉ♥ t ỗ ự ố ❍➔ ◆ë✐✳ ❬❇❪ ◆❤➟♣ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❇❡❝❦ ❆✳✱ ❚❡❜♦✉❧❧❡ ▼✳ ✭✷✵✵✵✮✱ ✏●❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❜✐♥❛r② ❝♦♥str❛✐♥ts✑✱ ❖♣t✐♠✳✱ ✶✶✱ ♣♣✳ ✶✼✾✲✶✽✽✳ ❙■❆▼ ❏✳ ▲❡❝t✉r❡s ♦♥ ▼♦❞❡r♥ ❈♦✈❡① ❖♣t✐✲ ♠✐③❛t✐♦♥✿ ❆♥❛❧②s✐s✱ ❆❧❣♦r✐t❤♠s ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙■❆▼✲ ❬✸❪ ❇❡♥✲❚❛❧ ❆✳✱ ◆❡♠✐r♦✈s❦✐ ❆✳ ✭✷✵✵✵✮✱ ▼P❙✱ P❤✐❧❛❞❡❧♣❤✐❛✳ ❬✹❪ ❉❛❤❧ ●✳ ✭✷✵✵✵✮✱ ✏❆ ♥♦t❡ ♦♥ ❞✐❣♦♥❛❧❧② ❞♦♠✐♥❛t ♠❛tr✐❝❡s✑✱ ▲✐♥❡❛r ❜r❛ ❆♣♣❧✳✱ ✸✶✼✱ ♣♣✳ ✷✶✼✲✷✷✹✳ ❬✺❪ ❉❡r✐♥❦✉②✉ ❑✳✱ P✐♥❛r ▼✳ ❈✳✱ ❛♥ts✱ ❖♥ t❤❡ S−♣r♦❝❡❞✉r❡ ❆❧❣❡✲ ❛♥❞ s♦♠❡ ✈❛r✐✲ ▼❛t❤✳ ▼❡t❤♦❞s ❖♣❡r✳ ❘❡s✳ ✭s❡❡ ❤tt♣✿✴✴✇✇✇✳✐❡✳❜✐❧❦❡♥t✳❡❞✉✳tr✴ ♠✉st❛❢❛♣✴♣✉❜s✴✮✳ ❬✻❪ ❉☎ ✉r ▼✳✱ ❍♦rst ❘✳✱ ▲♦❝❛t❡❧❧✐ ▼✳ ✭✶✾✾✽✮✱ ✏◆❡❝❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❝♦♥✈❡① ♠❛①✐♠✐③❛t✐♦♥ r❡✈✐s✐t❡❞✑✱ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✶✼ ✭✷✮✱ ♣♣✳ ✻✸✼✲✻✹✾✳ ❏✳ ▼❛t❤✳ ❬✼❪ ●❧♦✈❡r ❇✳ ▼✳✱ ❏❡②❛❦✉♠❛r ❱✳✱ ❘✉❜✐♥♦✈ ❆✳ ▼✳ ✭✶✾✾✾✮✱ ✏❉✉❛❧ ❝♦♥❞✐t✐♦♥s ❝❤❛r❛❝t❡r✐③✐♥❣ ♦♣t✐♠❛❧✐t② ❢♦r ❝♦♥✈❡① ♠✉❧t✐✲♦❜❥❡❝t✐✈❡ ♣r♦❣r❛♠s✑✱ ▼❛t❤✳ Pr♦❣r✳✱ ✽✹✱ ♣♣✳ ✷✵✶✲✷✶✼✳ ✹✸ Footer Page 45 of 54 Header Page 46 of 54 ❬✽❪ ●❧♦✈❡r ❇✳ ▼✳✱ ■s❤✐③✉❦❛ ❨✳✱ ❏❡②❛❦✉♠❛r ❱✳✱ ❚✉❛♥ ❍✳ ❉✳ ✭✶✾✾✻✮✱ ✏❈♦♠✲ ♣❧❡t❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❢♦r ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ t❤❡ ♣♦✐♥t✇✐s❡ ♠✐♥✐♠✉♠ ♦❢ s✉❜❧✐♥❡❛r ❢✉♥❝t✐♦♥s✑✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱ ✻✱ ♣♣✳ ✸✻✷✲✸✼✷✳ ❬✾❪ ❍✐r✐❛rt✲❯rr✉t② ❏✳ ❇✳ ✭✷✵✵✶✮✱ ✏●❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✐♥ ♠❛①✐♠✐③✲ ✐♥❣ ❛ ❝♦♥✈❡① q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ✉♥❞❡r ❝♦♥✈❡① q✉❛❞r❛t✐❝ ❝♦♥str❛✐♥ts✑✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✷✶✱ ♣♣✳ ✹✹✺✕✹✺✺✳ ❬✶✵❪ ❍✐r✐❛rt✲❯rr✉t② ❏✳ ❇✳ ✭✶✾✾✽✮ ✏❈♦♥❞✐t✐♦♥s ❢♦r ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ✷✑✱ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✶✸✱ ♣♣✳ ✸✹✾✕✸✻✼✳ ❬✶✶❪ ❍✐r✐❛rt✲❯rr✉t② ❏✳ ❇✳✱ ▲❡♠❛r❡❝❤❛❧ ❈✳ ✭✶✾✾✸✮ ❈♦♥✈❡① ❏✳ ❆♥❛❧②s✐s ❛♥❞ ▼✐♥✲ ✐♠✐③❛t✐♦♥ ❆❧❣♦r✐t❤♠s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ◆❡✇ ❨♦r❦✳ ❍❛♥❞❜♦♦❦ ♦❢ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ◆♦♥✲ ❝♦♥✈❡① ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❑❧✉✇❡r ❉♦r❞r❡❝❤t✳ ❬✶✷❪ ❍♦rst ❘✳✱ P❛r❞❛❧♦s P✳ ✭✶✾✾✺✮ ❬✶✸❪ ❏❡②❛❦✉♠❛r ❱✳ ✭✷✵✵✵✮ ❋❛r❦❛s ▲❡♠♠❛✿ ●❡♥❡r❛❧✐③❛t✐♦♥s✱ ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ ❖♣t✐♠✐③❛t✐♦♥✱ ❱♦❧✳ ✷✱ ♣♣✳ ✽✼✕✾✶✱ ❑❧✉✇❡r ❇♦st♦♥✱ ❯❙❆✳ ❬✶✹❪ ❏❡②❛❦✉♠❛r ❱✳ ✭✷✵✵✻✮ ✏❚❤❡ str♦♥❣ ❝♦♥✐❝❛❧ ❤✉❧❧ ✐♥t❡rs❡❝t✐♦♥ ♣r♦♣❡rt② ❢♦r ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣✑✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ❙❡r ❆✱ ✶✵✻✱ ♣♣✳ ✽✶✕✾✷✳ ❬✶✺❪ ❏❡②❛❦✉♠❛r ❱✳✱ ❘✉❜✐♥♦✈ ❆✳ ▼✳✱ ❲✉ ❩✳ ❨✳ ✭✷✵✵✼✮ ✏◆♦♥❝♦♥✈❡① q✉❛❞r❛t✐❝ ♠✐♥✐♠✐③❛t✐♦♥ ✇✐t❤ q✉❛❞r❛t✐❝ ❝♦♥str❛✐♥ts✿ ●❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐✲ t✐♦♥s✑✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ❙❡r✳ ❆✱ ✶✶✵✱ ♣♣✳ ✺✷✶✕✺✹✶✳ ❬✶✻❪ ❏❡②❛❦✉♠❛r ❱❛✐t❤✐❧✐♥❣❛♠✱ ◗✉❛♥❣ ❍✉② ◆❣✉②❡♥✱ ▲✐ ●✉♦②✐♥ ✭✷✵✵✾✮✱ ✏◆❡❝✲ ❡ss❛r② ❛♥❞ s✉❢❢✐❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r S ✲❧❡♠♠❛ ❛♥❞ ♥♦♥❝♦♥✈❡① q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛t✐♦♥✑ ❖♣t✐♠ ❊♥❣✱ ✶✵✱ ♣♣✳ ✹✾✶✕✺✵✸✳ ❬✶✼❪ ▼♦r➨ ❏✳ ✭✶✾✾✸✮ ✏●❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ tr✉st r❡❣✐♦♥ ♣r♦❜❧❡♠✑ ▼❡t❤✳ ❙♦❢t✳✱ ✷✱ ♣♣✳ ✶✽✾✕✷✵✾✳ ❖♣t✐♠✳ ❋♦✉♥❞❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❖♣t✐✲ ♠✐③❛t✐♦♥✿ ❈♦♥✈❡① ❆♥❛❧②s✐s ✇✐t❤♦✉t ▲✐♥❡❛r✐t②✱ ❉♦r❞r❡❝❤❡t✳ ❬✶✽❪ P❛❧❧❛s❝❤❦❡ ❉✳✱ ❘♦❧❡✇✐❝③ ❙✳ ✭✶✾✾✼✮ ✹✹ Footer Page 46 of 54 Header Page 47 of 54 ❬✶✾❪ P❛r❞❛❧♦s P✳ ▼✳ ✭✶✾✾✶✮ ✏❈♦♥str✉❝t✐♦♥ ♦❢ t❡st ♣r♦❜❧❡♠s ✐♥ q✉❛❞r❛t✐❝ ❜✐✲ ✈❛❧❡♥t ♣r♦❣r❛♠♠✐♥❣✑✱ ❆❈▼ ❚r❛♥s✳ ▼❛t❤✳ ❙♦❢t✇❛r❡✱ ✶✼ ✭✶✮✱ ♣♣✳ ✼✹✕✽✼✳ ❬✷✵❪ P❡♥❣ ❏✳ ▼✳✱ ❨✉❛♥ ❨✳ ✭✶✾✾✼✮ ✏❖♣t✐♠✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ♠✐♥✲ ✐♠✐③❛t✐♦♥ ♦❢ ❛ q✉❛❞r❛t✐❝ ✇✐t❤ t✇♦ q✉❛❞r❛t✐❝ ❝♦♥str❛✐♥ts✑✱ ❖♣t✐♠✳✱ ✼ ✭✸✮✱ ♣♣✳ ✺✼✾✕✺✾✹✳ ❙■❆▼ ❏✳ ❬✷✶❪ P✙♥❛r ▼✳ ❈✳ ✭✷✵✵✹✮ ✏❙✉❢❢✐❝✐❡♥t ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❜✐✈❛❧❡♥t q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛t✐♦♥✑✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r✳ ❆♣♣❧✳✱ ✶✷✷ ✭✷✮✱ ♣♣✳ ✹✸✸✕✹✹✵✳ ❬✷✷❪ P♦❧②❛❦ ❇✳ ❚✳ ✭✶✾✾✽✮ ✏❈♦♥✈❡①✐t② ♦❢ q✉❛❞r❛t✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ❛♥❞ ✐ts ✉s❡ ✐♥ ❝♦♥tr♦❧ ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥✑✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r✳ ❆♣♣❧✳✱ ✾✾✱ ♣♣✳ ✺✺✸✕✺✽✸✳ ❬✷✸❪ ❘✉❜✐♥♦✈ ❆✳ ▼✳ ✭✷✵✵✵✮ ❆❜str❛❝t ❈♦♥✈❡①✐t② ❛♥❞ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❑❧✉✇❡r✱ ◆❡t❤❡r❧❛♥❞s✳ ❬✷✹❪ ❙t❡r♥ ❘✳✱ ❲♦❧❦♦✇✐❝③ ❍✳ ✭✶✾✾✺✮ ✏■♥❞❡❢✐♥✐t❡ tr✉st r❡❣✐♦♥ s✉❜♣r♦❜❧❡♠s ❛♥❞ ♥♦♥s②♠♠❡tr✐❝ ❡✐❣❡♥✈❛❧✉❡ ♣❡rt✉r❜❛t✐♦♥s✑✱ ❙■❆▼ ❏✳ ❖♣t✐♠✳✱ ✺✱ ♣♣✳ ✷✽✻✕✸✶✸✳ ❬✷✺❪ ❙tr❡❦❛❧♦✈s❦② ❆✳ ✭✶✾✾✽✮ ✏●❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ♥♦♥❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✑✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳✱ ✶✷ ✭✹✮✱ ♣♣✳ ✹✶✺✕✹✸✹✳ ❬✷✻❪ ❙✉♥ ❲✳✱ ❨✉❛♥ ❨✳ ✭✷✵✵✻✮✱ ❖♣t✐♠✐③❛t✐♦♥ ❚❤❡♦r② ❛♥❞ ▼❡t❤♦❞s✿ ◆♦♥❧✐♥✲ ❡❛r Pr♦❣r❛♠♠✐♥❣✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✷✼❪ ❨❛❦✉❜♦✈✐❝❤ ❱✳ ❆✳ ✭✶✾✼✼✮ ✏❚❤❡ S ✲♣r♦❝❡❞✉r❡ ✐♥ ♥♦♥❧✐♥❡❛r ❝♦♥tr♦❧ t❤❡✲ ♦r②✑✱ ❱❡st♥✐❦ ▲❡♥✐♥❣r✳ ❯♥✐✈✳✱ ✹✱ ♣♣✳ ✼✸✕✾✸✳ ✹✺ Footer Page 47 of 54