Các phép toán dưới vi phân của hàm lồi

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❈⑩❈ ◗❯❨ ❚➁❈ ❚➑◆❍ ❚❖⑩◆ ❉×❰■ ❱■ P❍❹◆ ❈Õ❆ ❍⑨▼ ▲➬■ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❈⑩❈ ◗❯❨ ❚➁❈ ❚➑◆❍ ❚❖⑩◆ ❉×❰■ ❱■ P❍❹◆ ❈Õ❆ ❍⑨▼ ▲➬■ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍➶❆ ▲❯❾◆ ✿ ❚❙✳ ◆●❯❨➍◆ ❱❿◆ ❚❯❨➊◆ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽ ▲❮■ ữủ ỷ ỡ tợ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ❝❤➳✳ ❑➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✵✼ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ữợ sỹ ữợ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ữủ t ổ trũ ợ t t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✼ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ỗ ỡ ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ trỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ❚➼♥❤ t ữợ ữợrt ữợ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ❈→❝ q✉② t t t ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à t ỗ ởt ❜ë ♠æ♥ q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥ ❤✐➺♥ t ỗ ự ỳ t t ỗ ỗ ữợ ♠ët ♠ð rë♥❣ ❝❤♦ ✤↕♦ ❤➔♠ ❦❤✐ ❤➔♠ ❦❤æ♥❣ ❦❤↔ ✈✐✱ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ỗ st q t t t ữợ ỗ õ trỏ q trồ tr ỵ tt tố ữ t q ợ ố ữủ t s ỡ ỗ t ữợ ỗ tổ ự t q t t t ữợ ỗ õ tr ởt ❝→❝❤ ❝â ❤➺ t❤è♥❣✱ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➔ q trồ t ỗ q t t t ữợ ỗ t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ tr➯♥ ố tr õ ỗ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ởt số tự ỡ t ỗ ỗ ữỡ tr q t t t ữợ ởt số t t ỡ ữợrt ữợ ♣❤➙♥✳ ▼ö❝ ✷✳✷ tr➻♥❤ ❜➔② ♠ët sè q✉② t➢❝ t➼♥❤ t ữợ tr ữợ ữỡ ỗ ✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❑➼ ❤✐➺✉ R := R ∪ {±∞} ✈➔ ❣å✐ ❧➔ t➟♣ sè t❤ü❝ ♠ð rë♥❣✳ ❈❤♦ f : Rn → R ❧➔ ♠ët ❤➔♠ số ỳ t tr ỗ t f t÷ì♥❣ ù♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐✿ domf = {x ∈ Rn : f (x) < +∞} , epif = {(x, v) ∈ Rn × R : v ≥ f (x)} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët t➟♣ X ∈ Rn ữủ ỗ ợ x1 , x2 ∈ X ✈➔ α ∈ [0, 1]✱ t❛ ❝â (1 − α)x1 + αx2 ∈ X ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ỗ ởt t X ữủ conv X tt t ỗ ❝❤ù❛ X ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ X ❧➔ ♠ët t ỗ õ tr Rn x Rn ▼ët ✤✐➸♠ t❤✉ë❝ X ❣➛♥ x ♥❤➜t ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ X ✈➔ ❦➼ ❤✐➺✉ ❧➔ ΠX (x)✳ ❚❤❡♦ ❬✸✱ ❚❤❡♦r❡♠ ✷✳✶✵❪✱ t❛ ❝â ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ ♠ët ✤✐➸♠ ❧➯♥ ♠ët t➟♣ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ r ỗ õ ổ tỗ t↕✐ ✈➔ ❞✉② ♥❤➜t✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët t➟♣ K ⊂ Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥â♥ ♥➳✉ αx ∈ K ✈ỵ✐ ♠å✐ α > ✈➔ x ∈ K ✳ ❇ê ✤➲ ✶✳✶✳ ●✐↔ sû X ❧➔ ♠ët t➟♣ ỗ õ t cone(X) = {x : x X, 0} ởt õ ỗ ✶✳✺✳ ❈❤♦ K ❧➔ ♠ët ♥â♥✳ ❚➟♣ ❤ñ♣ K ◦ := {y ∈ Rn : y, x ≤ 0, ∀x ∈ K} ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝ü❝ ❝õ❛ K ✳ X ởt t ỗ õ ✈➔ x ∈ X ✳ ❚➟♣ ❤ñ♣ NX (x) = {v ∈ Rn : ΠX (x + v) = x} ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ❝õ❛ X t↕✐ x✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ NX (x) = [cone(X − x)]◦ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ▼ët số f ữủ ỗ epif ởt t ỗ õ tốt ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❱➼ ❞ö ✶✳✶✳ ▼ët ✈➼ ❞ö ỗ x ln(x) x ♥➳✉ x > 0,    f (x) = ♥➳✉ x = 0,      +∞ ♥➳✉ x < ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ▼ët f ữủ ó f ỗ ♥❣❤➽❛ ✶✳✾✳ ▼ët ❤➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ f (x) > −∞ ✈ỵ✐ ♠å✐ x ✈➔ f (x) < +∞ ✈ỵ✐ ➼t ♥❤➜t ♠ët x✳ ❇ê ✤➲ ởt f t õ ỗ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ x1✱ x2 ✈➔ ≤ α ≤ f (αx1 + (1 − α)x2 ) ≤ αf (x1 ) + (1 − α)f (x2 ) ✭✶✳✶✮ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ x1 ∈/ domf ❤♦➦❝ x2 ∈/ domf ✱ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ t➛♠ t❤÷í♥❣✳ ◆➳✉ x1 ∈ domf ✈➔ x2 ∈ domf ✳ ❑❤✐ ✤â ❝→❝ ✤✐➸♠     x x  ∈ epif  ✈➔   f (x2 ) f (x1 ) f ỗ t αx + (1 − α)x   ∈ epif αf (x ) + (1 − α)f (x ) t tr ỗ t t õ ✭✶✳✶✮✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû t❛ ❝â ✭✶✳✶✮✱ (xi , v i ) ∈ epif ✱ i = 1, 2✱ ✈➔ α ∈ [0, 1]✳ ✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❑❤✐ ✤â✱ t❤❡♦ ✭✶✳✶✮✱ t❛ ❝â f (αx1 + (1 − α)x2 ) ≤ αf (x1 ) + (1 − α)f (x2 ) ≤ αv + (1 − α)v ❉♦ ✤â✱ (αx1 + (1 − α)x2 , αv + (1 − α)v ) ∈ epif ✳ ✣✐➲✉ ✤â ❝â epif ởt t ỗ t tự ✭✶✳✶✮ ❝â t❤➸ ✤÷đ❝ sû ❞ư♥❣ ♥❤÷ ♠ët ✤à♥❤ ♥❣❤➽❛ ỗ tữớ ❍➔♠ f (x) = x ð ✤â · ♦ ♦, ởt tr Rn ởt ỗ tữớ t ợ x, y Rn ✈➔ α ∈ [0, 1]✱ t❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✱ t❛ ❝â αx + (1 − α)y ♦ ≤ αx ♦ + (1 − α)y ♦ =α x ♦ + (1 − α) y ♦ ❱➼ ❞ö ✶✳✸✳ ●✐↔ sỷ Z ởt t ỗ õ tr Rn ❑❤♦↔♥❣ ❝→❝❤ tỵ✐ Z ✱ f (x) = x − z z∈Z ð ✤â · ♦ ♦, ❧➔ ♠ët tr Rn ởt ỗ tữớ ❚❤➟t ✈➟②✱ ①➨t ✷ ✤✐➸♠ x ✈➔ y ✱ ✈➔ α ∈ (0, 1) ❜➜t ❦➻✳ ❉♦ Z ❧➔ t➟♣ õ tỗ t v Z w ∈ Z s❛♦ ❝❤♦ f (x) = x − v ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ❦❤✐ f (y) = y − w ♦, · ✺ ♦ ♦ ❧➔ ♠ët ❝❤✉➞♥ ❊✉❝❧✐❞❡✱ t❤❡♦ ❬✸✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❙✉② r❛ ∂f (x) = NZ (z) x z ú ỵ r➡♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✤✐➸♠ z ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ❞✉② ♥❤➜t✱ ✈➻ ✈➟② ❝❤ó♥❣ t❛ ❝ơ♥❣ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ t➟♣ ❜➯♥ ♣❤↔✐ ❧➔ ♥❤÷ ♥❤❛✉ ❝❤♦ t➜t ❝↔ ❤➻♥❤ ❝❤✐➳✉ z ✳ ❚❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ộ z t ộ ữợ rt ởt tỷ ❝õ❛ t➟♣ ❜➯♥ ♣❤↔✐ ✭✷✳✼✮✳ ❱➻ g ∗ ≤ 1✱ ♠é✐ y t❛ ❝â f (y) = inf y − z z∈Z ♦ ≥ inf g, y − z z∈Z = g, y − x + g, x − z + inf g, z − z z∈Z ❱➻ g ∈ ∂ x − z t❛ ❝â g, x − z = x − z ♦ ♦ = f (x)✳ ❚ø ✭✷✳✼✮ t❛ ❝ô♥❣ ❝â g ∈ NZ (z)✱ ✈➻ ✈➟② g, z − z ≥ ✈ỵ✐ ♠å✐ z ∈ Z ✳ ❉♦ ✤â✱ ❝ỉ♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ❦➨♦ t❤❡♦ f (y) ≥ f (x) + g, y − x ợ y, g ởt ữợrt t❤➸✱ ♥➳✉ · ♦ ❧➔ ❝❤✉➞♥ ❊✉❝❧✐❞❡ ✈➔ x ∈ / Z ✱ ❦❤✐ ✤â ✤✐➸♠ z ❧➔ ❤➻♥❤ ❝❤✐➳✉ trü❝ ❣✐❛♦ ❝õ❛ x tr➯♥ Z ✳ ❉♦ ✤â ∂f (x) = NZ (ΠZ (x)) ∩ ∂ x − ΠZ (x) ữ ữợ ự ♠ët ♣❤➛♥ tû✿ ∂ x − ΠZ (x) = {x − ΠZ (x)}, ✈➔ x − ΠZ (x) ∈ NZ (ΠZ (x)) ✷✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❉♦ ✤â✱ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ❧➔ ❦❤↔ ✈✐ t↕✐ ♠é✐ x ∈ Z ✈➔ ♠é✐ ❣r❛❞✐❡♥t ❝õ❛ ♥â ❝❤♦ ❜ð✐ ∇f (x) = x − ΠZ (x) ❱➼ ❞ö ✷✳✸✳ ❈❤♦ f (x) = max fi (x), i∈I tr♦♥❣ ✤â I ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✈➔ fi ỗ ộ i ∈ I ✳ ✣à♥❤ ♥❣❤➽❛ I(x) = {i ∈ I : fi (x) = f (x)} ❑❤✐ ✤â ∂f (x) = conv{∇fi (x) : i ∈ I(x)} ❚❤➟t ✈➟②✱ ♥➳✉ s ∈ I(x) t❛ ❝â f (y) ≥ fs (y) ≥ fs (x) + ∇fs (x), y − x = f (x) + ∇fs (x), y − x ❉♦ õ fs (x) f (x) ữợ t ỗ õ t õ conv{fi (x) : i ∈ I(x)} ⊂ ∂f (x) ●✐↔ sû g ∈ f (x) tỗ t s g / conv{fi (x) : i ∈ I(x)} ❚❤❡♦ ❬✸✱ ❚❤❡♦r❡♠ ✷✳✶✹❪✱ t❛ ❝â t t t g ỗ tr tỗ t↕✐ d = ✈➔ ε > s❛♦ ❝❤♦ g, d ≥ max ∇fi (x), d + ε i∈I(x) ✷✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ τ > f (x + τ d) ≥ f (x) + τ g, d ≥ f (x) + τ max ∇fi (x), d + τ ε ✭✷✳✽✮ i∈I(x) ❈❤♦ τ > ✤õ ♥❤ä✱ tỗ t r I(x) s f (x + τ d) = fr (x + τ d)✳ ❱➻ t➼♥❤ fi tỗ t sè oi (τ ) s❛♦ ❝❤♦ fi (x + τ d) = fi (x) + τ ∇fi (x), d + oi (τ ) ✈➔ oi (τ )/τ ↓ ❦❤✐ τ ↓ 0✳ ❉♦ ✤â✱ ❝❤♦ τ > ♥❤ä f (x + τ d) ≤ f (x) + τ max ∇fi (x), d + max oi (τ ) i∈I(x) i∈I(x) ❑➳t ❤đ♣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈ỵ✐ ✭✷✳✽✮ t❛ ✤÷đ❝ f (x) + τ max ∇fi (x), d + max oi (τ ) ≥ f (x) + τ max ∇fi (x), d + τ ε i∈I(x) i∈I(x) i∈I(x) ❈❤✐❛ ❝❤♦ τ ✈➔ ❝❤♦ τ ↓ t❛ ✤÷đ❝ ≥ ε✱ ♠➙✉ t❤✉➝♥✳ ❱➼ ❞ö ✷✳✹✳ ❈❤♦ C ❧➔ t ỗ õ Rn t t C (x) =   0 ❝❤➾ ❝õ❛ C ♥➳✉ x ∈ C  +∞ ♥➳✉ x ∈ / C ❍➔♠ tr ỗ t ữợ δC t↕✐ ♠ët ✤✐➸♠ x ∈ C ✳ ❚❛ ❝â g ∈ ∂δC (x) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ δC (y)−δC (x) ≥ g, y −x ✈ỵ✐ ♠å✐ y ✳ ❑❤✐ y∈ / C t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ t➛♠ t❤÷í♥❣✳ g ởt ữợrt õ tốt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❝õ❛ δC (·) t↕✐ x ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ g, y − x ≤ 0, ✈ỵ✐ ♠å✐ y ∈ C ❉♦ ✤â g, d ≤ ✈ỵ✐ ♠å✐ d ∈ cone(C − x) t ữủ ữợ ❤➔♠ ❝❤➾ ♥❤÷ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥✿ ∂δC (x) = [cone(C − x)]◦ = NC (x) ❈ö t❤➸✱ ♥➳✉ K õ ỗ õ õ K (0) = K q t t t ữợ ♣❤➙♥ ❇ê ✤➲ ✷✳✺✳ ●✐↔ sû f : Rn → R ởt ỗ > h(x) = f (x) õ h ỗ h(x) = α∂f (x)✱ ✈ỵ✐ ♠å✐ x✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ h(x) = f (x) f ỗ h ụ ỗ õ g f (x) ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ y h(y) = αf (y) ≥ α[f (x) + g, y − x ] = h(x) + g, y x , tữỡ ữỡ ợ αg ∈ ∂h(x)✳ ●✐↔ sû f : Rn → R ởt ỗ A tr õ tữợ m ì n h(x) = f (Ax) ✤â ∂h(x) = AT ∂f (Ax), ✈ỵ✐ ♠å✐ x✳ ❇ê ✤➲ ✷✳✻✳ ✸✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â g ∈ ∂f (Ax) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ h(y) = f (Ay) ≥ f (Ax) + g, Ay − Ax = h(x) + AT g, y x , tữỡ ữỡ ợ AT g ∈ ∂h(x) ❱➼ ❞ö ✷✳✺✳ ❳➨t ❤➔♠ h(x) = x, Ax , tr♦♥❣ ✤â A ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ ◆❤➙♥ tû ❤â❛ A = U T U ✱ ✈ỵ✐ U ❦❤ỉ♥❣ s✉② ❜✐➳♥✱ t❛ t❤➜② r➡♥❣ h(x) = U x ❉♦ ✤â✱ tø ❱➼ ❞ư ✷✳✶ ✈➔ ❇ê ✤➲ ✷✳✻ t❛ ✤÷đ❝✿ ∂h(0) = {U T g : g ≤ 1} = {v : (U T )−1 v ≤ 1} = {v : v, A−1 v ≤ 1} ▼➔ h(x) ❧➔ ♠ët ❝❤✉➞♥ ❝â ❝❤✉➞♥ ✤è✐ ♥❣➝✉ ❧➔ v A−1 = v, A−1 v ✳ ●✐↔ sû f = f1 + f2 tr♦♥❣ ✤â f1 : Rn → R ✈➔ f2 : Rn → R ỗ tữớ tỗ t ♠ët ✤✐➸♠ x0 ∈ domf s❛♦ ❝❤♦ fi ❧✐➯♥ tö❝ t x0 t ỵ f (x) = ∂f1 (x) + ∂f2 (x), ✈ỵ✐ ♠å✐ x ∈ domf ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû g1 ∈ ∂f1(x) ✈➔ g2 ∈ ∂f2(x)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ y f (y) = f1 (y) + f2 (y) ≥ f1 (x) + g , y − x + f2 (x) + g , y − x = f (x) + g + g , y − x ✸✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❙✉② r❛ ∂f (x) ⊃ ∂f1 (x) + ∂f2 (x) ●✐↔ sỷ ởt ữợrt g f (x) tỗ t s ❝❤♦ g ∈ / ∂f1 (x) + ∂f2 (x)✳ ✣➸ ỵ t r t ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ∂f1 (x) + ∂f2 (x) ❧➔ ♠ët t ỗ õ ứ t ỗ ữợ f1 (x) f2 (x) ỵ ♥❤➟♥ ①➨t s❛✉ ❦❤✐ ❝❤ù♥❣ ♠✐♥❤ ♥â✮ ✈➔ ❬✸✱ ▲❡♠♠❛ ✷✳✸❪ t❤➻ ∂f1 (x) + ∂f2 (x) ❧➔ ♠ët t➟♣ ỗ ữợ õ tờ ∂f1 (x) + ∂f2 (x) ✤â♥❣✳ ❈❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ t t tt s ỵ ự ◆➳✉ ∂f1 (x) + ∂f2 (x) ✤â♥❣✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❦➳t ❤ñ♣ ❬✸✱ ❚❤❡♦r❡♠ ✷✳✶✹❪ ✈➔ t→❝❤ g tø tờ f1 (x) + f2 (x) õ tỗ t ởt ữợ d Rn > s ❝❤♦ g, d ≥ g + g , d + ε, ✭✷✳✾✮ ✈ỵ✐ ♠å✐ g ∈ ∂f1 (x) ✈➔ g2 ∈ ∂f2 (x)✳ ❙✉② r❛ g , d ❜à ❝❤➦♥ ✈ỵ✐ ♠å✐ g ∈ ∂f1 (x)✳ ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✷✳✷ t❛ ✤÷đ❝ f1 (x; d) = 1max g , d < ∞ f2 (x; d) = 2max g , d < ∞ g ∈∂f1 (x) ❚÷ì♥❣ tü g ∈∂f2 (x) ▲➜② ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ ❜➯♥ ♣❤↔✐ ✭✷✳✾✮ tr➯♥ g ∈ ∂f1 (x) ✈➔ g ∈ ∂f2 (x) t❛ ✤÷đ❝ g, d ≥ 1max g ∈∂f1 (x) g , d + 2max g ∈∂f2 (x) g , d + ε = f1 (x; d) + f2 (x; d) + ε ✸✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ▼➦t ❦❤→❝✱ g, d ≤ f (x; d) ♥➯♥ f (x; d) ≥ f1 (x; d) + f2 (x; d) + ε, ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â ởt ữợrt g f (x)\(f1 (x) + f2 (x)) ổ t tỗ t ởt tr f1 , f2 tử t x t ữợ ❝õ❛ ♥â ❧➔ ❝♦♠♣❛❝t✱ ✈➔ tê♥❣ ∂f1(x) + ∂f2(x) ✤â♥❣✳ ữợ ổ ❳➨t ❤❛✐ ❞➣② g1k ∈ ∂f1 (x) ✈➔ g2k ∈ ∂f2 (x) s❛♦ ❝❤♦ g1k + g2k → s, ❦❤✐ k → ∞ ●✐↔ sû g ∈ / ∂f1 (x) + ∂f2 (x)✳ ❱➻ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ ❞➣② ♥➔② ♣❤↔✐ ❧➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ tữỡ ự ữợ g1k ∞ ✈➔ g2k → ∞✳ ❳➨t ❞➣② g1k z = k g1 k ●✐↔ sû z k ❝â ❣✐ỵ✐ ❤↕♥ z ✳ ❚❛ ❝â f1 (x0 ) − f1 (x) ≥ g1k , x0 − x ❈❤✐❛ ❝❤♦ g1k ✈➔ q✉❛ ❣✐ỵ✐ ❤↕♥ ✈ỵ✐ k → ∞ t❛ ❦➳t ❧✉➟♥ r➡♥❣ z, x0 −x ≤ 0✳ ❚❛ ❝ô♥❣ ❝â g2k = s − g1k ✈➔ ❞♦ ✤â f2 (x0 ) − f2 (x) ≥ s − g1k , x0 − x ❈❤✐❛ ❝❤♦ g1k ✈➔ q✉❛ ❣✐ỵ✐ ❤↕♥ t❛ ✤÷đ❝ z, x0 − x ✳ ❉♦ ✤â z, x0 − x = ✸✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❱➻ f1 ❧✐➯♥ tử t x0 tỗ t > s ❝❤♦ f1 (x0 + εz) < ∞ t❛ ✤÷đ❝ f1 (x0 + εz) − f1 (x) ≥ g1k , x0 + εz − x ❈❤✐❛ ❝❤♦ g1k ✈➔ q✉❛ ❣✐ỵ✐ ❤↕♥ t❛ ❦➳t ❧✉➟♥ r➡♥❣ ≥ z, x0 − x + εz = ε z = ε, ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â✱ tê♥❣ ∂f1 (x) + ∂f2 (x) õ ỵ ữủ ỵ rr ❱➼ ❞ö ✷✳✻✳ ❳➨t t➟♣ K = K1 ∩ K2 ∩ ∩ Km , tr♦♥❣ ✤â ♠é✐ Ki , i = 1, 2, , m ởt õ ỗ ứ t ❜✐➳t r➡♥❣ ∂δK (0) = K ◦ ▼➦t ❦❤→❝✱ δK (x) = δK1 (x) + δK2 (x) + + δKm (x) ◆➳✉ K1 ∩ intK2 ∩ ∩ intKm = ∅✱ sû ❞ö♥❣ ✣à♥❤ ỵ t ữủ K (0) = K1 (0) + ∂δK2 (0) + + ∂δKm (0) = K1◦ + K2◦ + + Km ❚ê♥❣ q✉→t ❤ì♥✱ ❝❤♦ Xi , i = 1, 2, , m t ỗ ✤â♥❣✱ ✈➔ ❝❤♦ X = X1 ∩ X2 ∩ ∩ Xm ✸✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❚❛ ❝â ∂δX (x) = NX (x) ◆➳✉ X1 ∩ intX2 ∩ intXm = ỵ s r NX (x) = NX1 (x) + NX2 (x) + + NXm (x) ữợ ❤➔♠ ♠❛① ❳➨t ❤➔♠ F (x) = sup f (x, y) y∈Y ●✐↔ sû f : Rn × Y → R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ f (·, y) ỗ ợ y Y f (x, ·) ❧➔ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈ỵ✐ ♠å✐ x tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ①→❝ ✤à♥❤ ❝õ❛ ♠ët ✤✐➸♠ x0 ❀ ✭✐✐✐✮ ❚➟♣ Y ⊂ Rm ❝♦♠♣❛❝t✳ ❍➔♠ F ❧➔ ỗ t ổ tự õ ❝❤➼♥❤ t❤÷í♥❣ ❞♦ ✭✐✐✮✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤ó♥❣ t❛ tr♦♥❣ ữ r ổ tự t ữợ ♣❤➙♥ ❝õ❛ ❤➔♠ F t↕✐ x0 ✳ ❑➼ ❤✐➺✉ Y (x) ❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû y ∈ Y ♠➔ f (x, y) = F (x)✳ ❱➻ f (x, ·) ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ✈➔ Y ❧➔ ❝♦♠♣❛❝t✱ ♥➯♥ t➟♣ Y (x) ❦❤ỉ♥❣ ré♥❣ ✈➔ ❝♦♠♣❛❝t✱ ✈ỵ✐ ♠é✐ x tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ①→❝ ✤à♥❤ ❝õ❛ x0 ✳ ❑➼ x f (x0 , y) ữợ f (ã, y) t x0 ỵ ✷✳✸✳ ●✐↔ sû ❝â ✤✐➲✉ ❦✐➺♥ (i) − (iii)✳ ❑❤✐ ✤â ∂F (x0 ) ⊃ conv( ∂x f (x0 , y)) y∈Y (x0 ) ✸✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ◆❣♦➔✐ r❛ ♥➳✉ ❤➔♠ f (·, y) ❧✐➯♥ tư❝ t↕✐ x0 ✈ỵ✐ ♠å✐ y ∈ Y ✱ t❤➻ ∂F (x0 ) = conv( ∂x f (x0 , y)) ✭✷✳✶✵✮ y∈Y (x0 ) ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû g ∈ ∂xf (x0, y0) ❝❤♦ ♠ët sè y0 ∈ Y (x0)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x t❛ ❝â F (x) = sup f (x, y) ≥ f (x, y0 ) ≥ f (x0 , y0 ) + g, x − xo y∈Y ❉♦ ✤â g ∈ ∂F (x0 ) ữợ ỗ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤ó♥❣ t❛ ✤ó♥❣✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ t➟♣ ❜➯♥ ♣❤↔✐ ❝õ❛ ✭✷✳✶✵✮ ✤â♥❣✳ ❳➨t ♠ët ❞➣② ❤ë✐ tö ❝õ❛ ✈❡❝t♦r sk ∈ ∂x f (x0 , xk )✱ ✈ỵ✐ yk ∈ Y (x0 ) ✈➔ ❝❤♦ s∗ = limk→∞ sk ✳ ❱➻ t➟♣ Y (x0 ) ❧➔ ❝♦♠♣❛❝t✱ t❛ ❣✐↔ sû r➡♥❣ ❝❤✉é✐ yk ❤ë✐ tư✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ♥â✱ y ∗ ✱ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ Y (x0 )✳ ❱ỵ✐ ♠é✐ x tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥❤ä ❝õ❛ x0 ✱ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ f (x, ã) sk ởt ữợrt s r f (x, y ∗ ) ≥ lim sup f (x, yk ) ≥ lim sup[f (x0 , yk ) + sk , x − x0 ] k→∞ k→∞ = f (x0 , y ∗ ) + s∗ , x − x0 ❚r♦♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝✉è✐ ❝ò♥❣✱ t❛ sû ❞ư♥❣ f (x0 , yk ) = F (x0 ) = f (x0 , y ∗ ) s✉② r❛ s∗ ∈ ∂f (x0 , y ∗ )✳ ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ t❤ù ú t s tự tỗ t g ∈ ∂F (x0 ) s❛♦ ❝❤♦ g∈ / conv( ∂x f (x0 , y)) y∈Y (x0 ) ❱➻ t➟♣ ỗ õ tứ r s r tỗ t d = õ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ✈➔ ε > s❛♦ ❝❤♦ g, d ≥ s, d + ε ✭✷✳✶✶✮ ✈ỵ✐ ♠å✐ s ∈ ∂x f (x0 , y) ✈➔ ♠å✐ y ∈ Y (x0 )✳ ❳➨t ❝→❝ ✤✐➸♠ ❝õ❛ ❝æ♥❣ t❤ù❝ x0 + τk d✱ tr♦♥❣ ✤â k t ỗ F t õ F (x0 + τk d) − F (x0 ) ≥ g, d τk ✣à♥❤ ♥❣❤➽❛ ❝→❝ t➟♣ Yk = {y ∈ Y : f (x0 + τk d, y) − F (x0 ) ≥ g, d }, k = 1, 2, τk ❈❤ó♥❣ ✤â♥❣ ✈➻ f (x, ·) ♥û❛ ❧✐➯♥ tư❝ tr➯♥✳ ❈❤ó♥❣ ❦❤ỉ♥❣ ré♥❣✱ ✈➻ Y (x0 + τk d) ⊂ Yk ✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ♠å✐ y ∈ Y ✤➥♥❣ t❤ù❝ f (x0 + τ d, y) − F (x0 ) f (x0 + τ d, y) − f (x0 , y) f (x0 , y) − F (x0 ) = + τ τ τ ✤à♥❤ ♥❣❤➽❛ ♠ët ❤➔♠ t➠♥❣ ❝õ❛ τ ✳ ❚❤➟t ✈➟②✱ ♣❤➙♥ sè ✤➛✉ t✐➯♥ ❧➔ t❤÷ì♥❣ ❝õ❛ ♠ët ❤➔♠ ỗ ổ tự số tự ❝â tû sè ❦❤ỉ♥❣ ❞÷ì♥❣ ❝è ✤à♥❤✳ ✣✐➲✉ ♥➔② s✉② r❛ ✤÷đ❝ Y1 ⊃ Y2 ⊃ Y3 ⊃ ❈→❝ t➟♣ Yk ❝♦♠♣❛❝t ✈➔ ❦❤æ♥❣ ré♥❣✱ ✈➔ ❞♦ õ tỗ t ởt y tở tt t➟♣✳ ❑❤✐ ✤â f (x0 + τk d, y) − F (x0 ) ≥ g, d , k = 1, 2, τk ❇ð✐ ✈➻ f (x0 +τk d, y) → f (x0 , y) ❦❤✐ k → ∞✱ ❝❤ó♥❣ t❛ ❝â f (x0 , y) = F (x0 )✱ ✸✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ♥❣❤➽❛ ❧➔✱ y ∈ Y (x0 )✳ ✣✐ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✈ỵ✐ k sỷ ỵ t ❦➳t ❧✉➟♥ r➡♥❣ f (x0 + τk d, y) − F (x0 ) = s, d ≥ g, d , k→∞ τk lim ✈ỵ✐ ♠ët sè s ∈ ∂x f (x0 , y)✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✶✮✳ ❱➼ ❞ư ✷✳✼✳ ❳➨t ❤➔♠ ❣✐→ trà r✐➯♥❣ ❧ỵ♥ ♥❤➜t λmax (·) ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ Sn ❝õ❛ ❝→❝ tr ố ự õ tữợ n ì n õ ỗ max (A) = max y, Ay , y =1 ❈❤ó♥❣ t❛ ❝â t❤➸ t t ữợ õ ỵ ✷✳✸✱ ❚➟♣ Y (A) = {y ∈ Rn : y, Ay = λmax (A), y = 1}, ❧➔ t➟♣ t➜t tr r A tữỡ ự ợ trà r✐➯♥❣ ❧ỵ♥ ♥❤➜t ✈➔ ❝â ❝❤✐➲✉ ❞➔✐ ✶✳ ❚r♦♥❣ ❬✸✱ ❊①❛♠♣❧❡ ✷✳✸✶❪✱ t❛ sû ❞ö♥❣ t➼❝❤ tr♦♥❣ ❋r♦❜❡♥✐✉s✱ n A, H S n = tr(AH) = aij hij , i=1 i=1 ✈➔ t❛ ✈✐➳t ❧↕✐ ❝→❝ ❤➔♠ ♥❤÷ s❛✉✿ λmax (A) = max yy T , A S y =1 ❱ỵ✐ ♠é✐ y ∈ Rn ❤➔♠ fy (A) + yy T , A ✸✽ S ❧➔ t✉②➳♥ t➼♥❤ ✈➔ ❣r❛❞✐❡♥t ❝õ❛ ♥â ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚rà♥❤ ❚❤à ❚❤❛♥❤ ❍✐➳✉ ❜➡♥❣ ∇fy (A) = yy T ✳ ❉♦ ✤â ∂λmax (A) = conv{yy T : Ay = λmax (A)y, y = 1} ❑❤✐ ✤â✱ t➟♣ ❝→❝ ♠❛ tr➟♥ ❝â ❞↕♥❣ W = yy T ❝â t❤➸ ✤÷đ❝ ✤➦❝ tr÷♥❣ ♥❤÷ s❛✉✿ W ❝â ❤↕♥❣ ✶ ✈➔ W ∈ Sn+ ✳ ❑❤✐ ✤â tr(W ) = y ✈➔ A, W S = λmax (A) ❚❛ ✤÷đ❝ ∂λmax (A) = conv{W ∈ Sn+ : A, W S = λmax , tr(W ) = 1, rank(W ) = 1} õ t ọ q ỗ tr ố ữợ ∂λmax (A) = {W ∈ Sn+ : A, W S = λmax , tr(W ) = 1} ❚❤➟t ✈➟②✱ ♠é✐ ♣❤➛♥ tû ❝õ❛ t➟♣ ♥➔② ❝â ❞↕♥❣ W = n T j=1 àj yj yj ợ tr r trỹ ❣✐❛♦ yj ❝õ❛ W ❝â ❝❤✉➞♥ ✶✱ ✈➔ ✈ỵ✐ ❣✐→ tr r tữợ ự àj tr(W ) = ữ r n j=1 àj n A, W S = 1✳ ❉♦ ✤â✱ n µj yj , Ayj ≤ = j=1 µj λmax (A) = λmax (A) j=1 ❇➜t ✤➥♥❣ t❤ù❝ ①↔② r❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tữỡ ự yj ợ àj ữỡ tr r A tữỡ ự ợ tr r ợ ♥❤➜t ❝õ❛ ♥â✳ ❑❤✐ ✤â W ❧➔ ♠ët tê ❤ñ♣ ỗ tr yj yjT t tứ tr r ú ợ trồ ữủ àj ✸✾ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ♥➔② tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ t ỗ ỗ õ ữợrt ữợ t t ữợ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ ❝ö t❤➸ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ ❈✉è✐ ❝ò♥❣ ❦❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ✈➲ ữợ t ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❱✐➺t ❬✶❪ ❍✉ý♥❤ ❚❤➳ P❤ò♥❣✱ ỡ s t ỗ t ✷✵✶✷✳ ❬❇❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤ ❬✷❪ ❘✳ ❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡②✱ ✶✾✼✵✳ ❬✸❪ ❆✳ ❘✉s③❝③②✁ ♥s❦✐✱ ◆♦♥❧✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥✱ Pr❡ss✱ ✷✵✵✻✳ ✹✶ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t②