Điều kiện cần tối ưu bậc nhất cho bài toán tối ưu đa mục tiêu lipschitz

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣ ✣■➋❯ ❑■➏◆ ❈❺◆❚➮■ ×❯ ❇❾❈ ◆❍❻❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ✣❆ ▼Ư❈ ❚■➊❯ ▲■P❙❈❍■❚❩ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✾ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣ ✣■➋❯ ❑■➏◆ ❈❺◆❚➮■ ×❯ ❇❾❈ ◆❍❻❚ ❈❍❖ ❇⑨■ ❚❖⑩◆ ❚➮■ ×❯ ✣❆ ▼Ư❈ ❚■➊❯ ▲■P❙❈❍■❚❩ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✾ ▲❮■ ❈❷▼ ❒◆ ❊♠ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ỡ tợ t ổ trữớ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ ❣✐â❛ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐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➯♥✱ ♥❣÷í✐ t❤➛② ✤➣ tr✉②➲♥ t❤ư ❦✐➳♥ t❤ù❝✱ t➟♥ t ú ù ữợ tr sốt q tr ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤÷♥❣ s❛✐ sât ✈➔ ❤↕♥ ❝❤➳✳ ❊♠ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ õ ỵ t ổ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣ ▲❮■ ❈❆▼ ữợ sỹ ữợ t❤➛② ❣✐→♦ ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❜➜t ❦➻ ✤➲ t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ ❧➔♠ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ❍♦➔♥❣ ❚❤✐➯♥ ❚r❛♥❣ ▼ö❝ ❧ö❝ ▲í✐ ♠ð ✤➛✉ ✷ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✹ ✶✳✶ ❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ữợ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ◆â♥ t✐➳♣ t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r ✷✳✶ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ✷✳✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ✶✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✶ ✷✸ ✶ ▼ð ✤➛✉ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ sỹ tỗ t ỳ ✤à❛ ♣❤÷ì♥❣ ✈➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ (P ) ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ ởt r t ủ tũ ỵ Min f (x) = (f1 (x), f1 (x), , f1 (x)) ✈ỵ✐ x ∈ C, gj ≤ 0, j = 1, 2, 3, , m, hk (x) = 0, k ∈ {1, 2, , n}, ð ✤â✱ ❝→❝ ❤➔♠ sè✿ J = {1, 2, , m} ❇❛♥❛❝❤ ❤✐➺✉ x¯ fi : U → R, i ∈ I = {1, 2, , l} ✈➔ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ t➟♣ ❝♦♥ ♠ð X ✱ h : X → R, k ∈ K = {1, 2, , n} ✈➔ ✭P✮ gj : U → R, j ∈ U ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ ỳ C U rữợ t ú tổ tr ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❝❤♦ ❜➔✐ t♦→♥ (P ) s õ sỷ ỵ ❧✉➙♥ ♣❤✐➯♥ ✤➸ ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❞↕♥❣ ♥❤➙♥ tû ▲❛❣r❛♥❣❡ ✭❞↕♥❣ ✤è✐ ♥❣➝✉✮✳ ◆❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉ ✈➻ ❝❤ó♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ✤↔♠ ❜↔♦ sü t❤❛♠ ❣✐❛ ❝õ❛ ❤➔♠ ♠ö❝ t✐➯✉ tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❞↕♥❣ ❋r✐t③ ❏♦❤♥ ❝❤♦ ♠ët ✤✐➸♠ ❧➔ ✤✐➸♠ ❝ü❝ trà✳ ❙ü t❤❛♠ ❣✐❛ ❝õ❛ ♠ët ❤➔♠ ♠ư❝ t✐➯✉ ✤÷đ❝ ❝❤➾ r❛ ✷ ❜ð✐ t➼♥❤ ❞÷ì♥❣ ❝õ❛ ❝→❝ ♥❤➙♥ tû r tữỡ ự ợ õ t t ởt tr tỷ r tữỡ ự ợ ♠ư❝ t✐➯✉ ❧➔ ❞÷ì♥❣✱ t❛ ♥â✐ ✤✐➲✉ ❦✐➺♥ ♥➔② ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❞↕♥❣ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r ②➳✉ ✭❲❑❑❚✮✳ ❑❤✐ tt tỷ r tữỡ ự ợ ❤➔♠ ♠ư❝ t✐➯✉ ❧➔ ❞÷ì♥❣✱ t❛ ♥â✐ ✤➙② ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❞↕♥❣ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r ♠↕♥❤ ✭❙❑❑❚✮✳ ❚r♦♥❣ tè✐ ÷✉ ♠ët ♠ö❝ t✐➯✉✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝❤✉➞♥ ❤â❛ r➔♥❣ ❜✉ë❝ ✭❝→❝ ✤✐➲✉ ❦✐➺♥ ✤÷đ❝ ✤↔♠ ❜↔♦ ❜ð✐ ❝→❝ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥✮ ❧➔ ✤õ ✤➸ ✤↔♠ ❜↔♦ ❙❑❑❚✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② t❤÷í♥❣ ❦❤ỉ♥❣ ✤õ ❝❤♦ ❙❑❑❚ ❝õ❛ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ư❝ t✐➯✉✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ✤÷đ❝ sû ❞ư♥❣ ✤➸ ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❞↕♥❣ ❲❑❑❚ ❤❛② ❙❑❑❚ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ❞ü❛ tr➯♥ ❦➳t q✉↔ ❣➛♥ ✤➙② ❝õ❛ ❈♦♥st❛♥t✐♥ ❬✹❪✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❧➔ sü ♠ð rë♥❣ ❝õ❛ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ❝õ❛ ❇✉r❛❝❤✐❦ ✈➔ ❘✐③✈✐ ❬✷❪✱ ❝õ❛ ❈❤❛♥❞r❛ ✈➔ ❝ë♥❣ sü ❬✺❪ ✈➔ ❝õ❛ ●✐♦r❣✐ ✈➔ ❝ë♥❣ sü ❬✸❪✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð t t õ ỗ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ t♦➔♥ ❜ë õ ỹ ỵ tê♥❣ q✉→t✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝❤♦ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ö❝ t✐➯✉✳ ▼ö❝ ✷✳✶ tr➻♥❤ ❜➔② ♠ët ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝❤♦ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ữỡ t P ỵ ▼♦t③❦✐♥ tê♥❣ q✉→t ❣✐ó♣ ❝❤ó♥❣ t❛ ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❞↕♥❣ ❲❑❑❚ ❝❤♦ ❜➔✐ t♦→♥ ♥➔②✳ ▼ư❝ ✷✳✷ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝❤♦ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ữỡ P s õ ỵ ❧✉➙♥ ♣❤✐➯♥ ❚✉❝❦❡r tê♥❣ q✉→t ✤➸ t❤✉ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❞↕♥❣ ❙❑❑❚ ❝❤♦ ✭P✮✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ❈❤♦ Rl ✱ Rl ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ l ❝❤✐➲✉✳ ❱ỵ✐ x = (x1 , , xl ), y = (y1 , , yl ) ∈ t❛ sỷ q ữợ s y, ❦❤✐ xi ≥ yi , i = 1, , l, x ≥ y, ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xi x > y, ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xi > yi , i = 1, , l x yi , x = y, ❳➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✤❛ ♠ö❝ t✐➯✉ ▼✐♥ f (x) ð ✤â✱ ❝→❝ ❤➔♠ sè✿ ❇❛♥❛❝❤ ❑❤✐ fi : U → R, i ∈ I = {1, 2, , l} ✈➔ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ t➟♣ ❝♦♥ ♠ð X ✱ hk : X → R, k ∈ K = {1, 2, , n} C = X✱ ✭P✮ x ∈ C, gj (x) ≤ 0, j = 1, , m, hk (x) = 0, k = 1, , n, ✈ỵ✐ r➔♥❣ ❜✉ë❝ J = {1, 2, , m} = (f1 (x), f1 (x), , f1 (x)) t❤➻ ❜➔✐ t♦→♥ ✭P✮ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✹ gj : U → R, j ∈ U ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ✈➔ (P1 )✳ C ⊆ U✳ D = {x ∈ X : gj (x) ≤ 0, j ∈ J} ❑➼ ❤✐➺✉ ✈➔ S = D ∩ C ∩ Ch , ð ✤â Ch = {x ∈ X : h(x) = 0}✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ (P ) ởt x S xS ổ tỗ t↕✐ ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ fi (x) ≤ fi (¯ x) ✈ỵ✐ ♠å✐ fi (x) < fi (¯ x) ✈ỵ✐ ➼t ♥❤➜t ♠ët ❝❤➾ sè i✳ ❚❛ ❝ô♥❣ ♥â✐ r➡♥❣ ❤ú✉ ❤✐➺✉ ❝õ❛ f ❝õ❛ ❜➔✐ t♦→♥ (P ) ❤✐➺✉ ❝õ❛ f ❈❤♦ ✈➔ lin B ✱ tr➯♥ tr➯♥ B S✳ ✣✐➸♠ x¯ ∈ S x¯ ∈ S i = 1, , l tỗ t ởt ởt V ❝õ❛ x¯ s❛♦ ❝❤♦ x¯ ❧➔ ♥❣❤✐➺♠ ❤ú✉ S∩V✳ ❧➔ t➟♣ ❝♦♥ ❝õ❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ X ✳ ◆❤÷ t❤÷í♥❣ ❧➺✱ t❛ ❦➼ ❤✐➺✉ ❜➡♥❣ cone B ✱ conv B ▼ët ✤✐➸♠ x¯ ∈ S ✤÷đ❝ ❣å✐ ❧➔ ♠ët B✳ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ♥➳✉ õ ỳ tỗ t ợ ♠é✐ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ tữỡ ự õ ỗ t t ❝õ❛ ●❡♦❢❢r✐♦♥ ❝õ❛ ❜➔✐ t♦→♥ M >0 s❛♦ i✱ fi (x) − fi (¯ x) ≤ M, fj (¯ x) − fj (x) ❝❤♦ ♠ët sè ◆➳✉ j t❤ä❛ ♠➣♥ x¯ ∈ S fj (¯ x) < fj (x) ✈ỵ✐ ♠å✐ x∈S ♥❤÷ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥✱ t❤➻ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛ f tr➯♥ S✳ x¯ ✈➔ fi (¯ x) > fi (x)✳ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❧➔ ♠ët ❈❤ó♥❣ t❛ ♥â✐ r➡♥❣ x¯ ∈ S ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ P tỗ t ởt V x ♠➔ x¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛ f tr➯♥ S∩V✳ ❚ò ✤à♥❤ ♥❣❤➽❛✱ t❛ t❤➜② r➡♥❣ ♠å✐ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ▼ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤ỉ♥❣ t❤ü❝ sü ✭✐♠♣r♦♣❡r❧② ❡❢❢✐❝✐❡♥t✮✳ ◆❤÷ ✈➟②✱ ♠ët ✤✐➸♠ ✺ x¯ ∈ X ✭t❤❡♦ ♥❣❤➽❛ ❝õ❛ ●❡♦❢❢r✐♦♥✮ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❦❤ỉ♥❣ t❤ü❝ sü ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✱ ♥➳✉ ✈ỵ✐ ♠å✐ fi (¯ x) > fi (x) M > 0✱ tỗ t xX iI s fi (x) − fi (¯ x) >M fj (¯ x) − fj (x) ✈ỵ✐ t➜t ❝↔ j t❤ä❛ ♠➣♥ fj (¯ x) < fj (x) ữợ r ✶✳✸✳ U ◆➳✉ F ❧➔ ♠ët ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ ♠ët t➟♣ ♠ð ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✭✐✮ X ✈➔ x¯ ∈ U ✱ t❤➻ ✣↕♦ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ❈❧❛r❦❡ ❝õ❛ F t↕✐ x¯ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ F (x + tv) − F (x) , v ∈ X + t (x,t)→(¯ x,0 ) F ◦ ( x, v) = lim sup ữợ ❈❧❛r❦❡ ❝õ❛ F t↕✐ x¯ ❧➔ ∂C F (¯ x) = {ξ ∈ Rp : ξ, v ≤ F ◦ (¯ x, v), ∀v ∈ Rp }, ð ✤â X = Rp ⑩♥❤ ①↕ ✈➔ ξ, v ❜✐➸✉ t❤à t➼❝❤ ổ ữợ v F ( x, v) ξ ∂F ◦ (¯ x, )(0)✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ Y✳ ❑❤✐ ✤â F v tr♦♥❣ Rp t t ữỡ ữợ t ữợ õ t t ỗ t v=0 tỗ t ữủ ❦➼ ∂C F (¯ x) = ∂F ◦ (¯ x, )(0)✳ ❧➔ ♠ët →♥❤ ①↕ tø ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❈❤ó♥❣ t❛ ♥â✐ r➡♥❣ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tø X ✤➳♥ ✻ F Y ❦❤↔ ✈✐ ❝❤➦t t↕✐ x ữủ ỵ X ởt tỗ t F ( x) s t tr ỵ tữỡ ữỡ ợ s ỗ t (u, v, w) Rl ì Rm × Rn l m ui fi (d) + i=1 s❛♦ ❝❤♦ ✈➔ n wk hk (d) ≥ 0, ∀d ∈ Rp vj gj (d) + j=1 u > 0, v k=1 ✶✵ ❈❤÷ì♥❣ ✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r ✷✳✶ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝õ❛ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ q✉↔ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝✱ ✤➥♥❣ t❤ù❝ ởt r t tũ ỵ s ✤➙② ❧➔ ❝➛♥ t❤✐➳t tr♦♥❣ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❦➳t q✉↔ ❝❤➼♥❤✳ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ F : U → R ❧➔ ♠ët ❤➔♠ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ t➟♣ ❝♦♥ ♠ð U ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✳ ●✐↔ sû r➡♥❣✿ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ xn → x¯✱ F (xn ) ≥ F (¯ x)✱ (xn n→+∞ tn u = lim − x¯) ✈ỵ✐ tn → 0, tn > ❑❤✐ ✤â✱ F ◦ (¯ x; u) ≥ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t un = tn (xn − x¯)✱ ❤❛② ❧➔ ✶✶ xn = x¯ + tn un ✳ ❉♦ F ❧➔ ▲✐♣s❝❤✐t③ x¯ ✤à❛ ♣❤÷ì♥❣ t↕✐ ✈➔ lim (¯ x + tn un ) = lim (¯ x + tn u) = x, n tỗ t L n0 ∈ N n→∞ s❛♦ ❝❤♦ |F (¯ x + tn un ) − F (¯ x + tn u)| Ltn uk − u ❢♦r ❛❧❧ n n0 ❱➻ ✈➟②✱ F (¯ x + tn un ) − F (¯ x) = [F (¯ x + tn un ) − F (¯ x + tn u)] + [F (¯ x + tn u) − F (¯ x)] Ltk un − u + F (¯ x + tk u) − F (¯ x) ✈ỵ✐ ♠å✐ n n0 ✳ ✣✐➲✉ ✤â ❦➨♦ t❤❡♦ lim L un − u + lim sup n→∞ n→∞ lim sup x→¯ x F (¯ x + tn u) − F (¯ x) tn F (x + tu) − F (x) t t↓0 ❉♦ ✈➟②✱ t❛ ❝â F ◦ (¯ x, u) ❱ỵ✐ ♠é✐ ✤✐➸♠ 0✳ x¯ ∈ D✱ ❣å✐ J(¯ x) ❧➔ t➟♣ ❤đ♣ ❝→❝ r➔♥❣ ❜✉ë❝ ❤♦↕t t↕✐ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ J(¯ x) = {j ∈ {1, 2, , m} : gj (¯ x) = 0} ❱ỵ✐ ♠é✐ i ∈ I✱ t❛ ✤➦t M i = {x ∈ X : gj (x) ≤ 0, ∀j ∈ J, fi (x) ≤ fj (¯ x)} ✶✷ x ✈➔ ✈➔ M i M= i∈I ◆â♥ t✉②➳♥ t➼♥❤ ❤â❛ ❝õ❛ Mi t↕✐ x¯ ∈ M i ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ L(M i ; x¯) = {d ∈ X : fi◦ (¯ x; d) ≤ 0, i = 1, 2, , l, gj◦ (¯ x; d) ≤ 0, j ∈ J(¯ x)}, ✈➔ ♥â♥ t✉②➳♥ t➼♥❤ ❤â❛ ❝õ❛ ▼ t↕✐ x¯ ∈ M ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ L(M ; x¯) = {d ∈ X : fi◦ (¯ x; d) ≤ 0, i = 1, 2, , l, gj◦ (¯ x; d) ≤ 0, j ∈ J(¯ x)} ỵ s t ởt tố ÷✉ ❦✐➸✉ ❑❛r✉s❤✕❑✉❤♥✕❚✉❝❦❡r ②➳✉ ✭❲❑❑❚✮ ❝❤♦ ♠ët ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ỳ ữỡ t P ỵ ✷✳✶✳ ❳➨t ❜➔✐ t♦→♥ ✭P✮✳ ❈❤♦ x¯ ∈ S = D ∩ C ∩ Ch ✳ ●✐↔ sû fi , i ∈ I ✱ ✈➔ gj , j ∈ J ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U ✈➔ h : X → Rn ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x¯ ✈ỵ✐ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t h (¯ x)✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② (GRC) s❛✉ L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯ x) ⊆ cl conv T (M i ∩ C ∩ Ch , x¯), ✈ỵ✐ ➼t ♥❤➜t ♠ët i, ✭✷✳✶✮ ✤ó♥❣ ✈ỵ✐ i0 ∈ I ✱ ð ✤â Ker h (¯ x) = {d ∈ X : h (¯ x)(d) = 0}✳ ●✐↔ sû r➡♥❣ ❝→❝ ❤➔♠ sè fi , i ∈ I \ {i0 } ❦❤↔ ✈✐ ❝❤➦t t↕✐ x¯✳ ◆➳✉ x¯ ∈ S ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✱ t❤➻ ❤➺ fi◦ (¯ x; d) < 0, i = 1, 2, , l, ✭✷✳✷✮ gj◦ (¯ x; d) ≤ 0, j ∈ J(¯ x), ✭✷✳✸✮ h (¯ x)(d) = 0, ✭✷✳✹✮ ✶✸ d ∈ T (C, x¯), ✭✷✳✺✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ d ∈ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ x¯ ∈ S ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ t P V tỗ t ởt ❝➟♥ ❝õ❛ x¯ x¯ s❛♦ ❝❤♦ f tr➯♥ ❉♦ ✤â✱ d ∈ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ V ∩ S✳ d ∈ X✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ ❤➺ ✭✷✳✷✮✕✭✷✳✺✮ ❝â ♥❣❤✐➺♠ L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯ x) Ch , x¯)✱ ✈➔ tø t õ õ tỗ t ởt d ∈ cl conv T (M i0 ∩ C ∩ {dim0 } ⊆ conv T (M i0 ∩ C ∩ Ch , x¯) s❛♦ ❝❤♦ lim dim0 = d m→+∞ ❱ỵ✐ ♠é✐ dim0 ∈ conv T (M i0 ∩ C ∩ Ch , x¯), m = 1, 2, ỗ tỗ t i0 i0 i0 , dm,k ∈ T (M i0 ∩ C ∩ Ch , x¯), k = 1, 2, , Km Km i Km0 i Km0 i0 i0 k=1 λm,k dm,k , dim0 = ❚ø i0 k=1 λm,k ✈➔ xnm,k,i0 ∈ M i0 ∩ C ∩ Ch ∩ V ✣✐➲✉ ✤â ❦➨♦ t❤❡♦✱ n✳ ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ ❦❤✐ n p fr (xm,k,i ) ≥ fr (¯ x)✱ {dnm,k,i0 }n n → ∞ {tnm,k,i0 }n ✈➔ s❛♦ ❝❤♦ ✈ỵ✐ xnm,k,i0 = x + N0 N õ tỗ t↕✐ s❛♦ ❝❤♦ n ≥ N0 ✳ fi0 (xnm,k,i0 ) − fi0 (¯ x) ≤ 0✱ ❈â t❤➸ ❝❤➾ r❛ r tỗ t s tỗ t tnm,k,i0 > 0, tnm,k,i0 → 0✱ tnm,k,i0 dnm,k,i0 ∈ M i0 ∩ C ∩ Ch s❛♦ ❝❤♦ = 1, λim,k ≥ 0 dim,k ∈ T (M i0 ∩ C ∩ Ch , x¯)✱ dnm,k,i0 → dim,k t❤❡♦ ✤à♥❤ ♥❣❤➽❛ r = i0 ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ ✈➔ ♠ët ❞➣② ❝♦♥ np n ≥ N0 ✳ n p {xm,k,i } ❝õ❛ {xnm,k,i0 } ✤õ ❧ỵ♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✱ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ♣❤↔♥ ❝❤ù♥❣✳ ◆➳✉ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣✱ t❤➻ ✈ỵ✐ ♠å✐ ✣➦t r = i0 tỗ t Nr s ợ ˜ = max{N0 , Nr : r = i0 }✳ N ˜✱ fr (¯ x), ∀n ≥ N ✈➔ tø ✤â t❛ ❝â ✈ỵ✐ ❣✐↔ t❤✐➳t ❧➔ ✤✐➸♠ x¯ n ≥ Nr , fr (xnm,k,i0 ) < fr (¯ x)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ r = i0 ✱ fr (xnm,k,i0 ) < t❛ ❝â ˜ ≥ N0 ✱ fi0 (xnm,k,i0 ) ≤ fi0 (¯ x), ∀n ≥ N ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ❝õ❛ ❉♦ ✤â✱ ♠➺♥❤ ✤➲ ❧➔ ✤ó♥❣ ✈➔ tỗ t r = i0 f tr S ∩V ✭✈ỵ✐ ✈➔ ♠ët ❞➣② ❝♦♥ ♠➙✉ t❤✉➝♥ ˜ x = xN m,k,i0 ✮✳ n p {xm,k,i } ❝õ❛ {xnm,k,i0 } s❛♦ ❝❤♦ n p fr (xm,k,i ) ≥ fr (¯ x)✱ ✈ỵ✐ ♠å✐ np ✤õ ❧ỵ♥✳ ❑❤✐ ✤â✱ tø ❇ê ✤➲ ✷✳✶ ✈➔ t➼♥❤ ❦❤↔ ✈✐ ❝❤➦t ❝õ❛ ❤➔♠ sè 0 ∇fr (¯ x)(dim,k ) = fr◦ (¯ x; dim,k ) ≥ 0✳ fr t↕✐ ▲↕✐ ❞♦ t➼♥❤ ❦❤↔ ✈✐ ❝❤➦t ❝õ❛ ❤➔♠ sè x¯✱ t❛ ❝â fr x¯✱ t❛ ❝â i i Km0 Km0 0 λim,k dim,k ∇fr (¯ x)(dim0 ) = ∇fr (¯ x) k=1 ✈➔ ❞♦ ✤â fr◦ (¯ x; d) = ∇fr (¯ x)(d) ≥ 0, 0 λim,k ∇fr (¯ x)(dim,k )≥0 = k=1 ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✸✮✳ ❉♦ ✤â✱ ❤➺ ✭✷✳✷✮✕✭✷✳✺✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠✳ ❚✐➳♣ t❤❡♦✱ t❛ ①➙② ❞ü♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❦✐➸✉ ❝❤♦ ❜➔✐ t♦→♥ (P1 ) ❦❤✐ ❦❤æ♥❣ ❝â r➔♥❣ ❜✉ë❝ t➟♣✱ tù❝ ❧➔ (W KKT ) ✤è✐ ♥❣➝✉ C = X✳ ỵ t t (P1 ) ợ X = Rp , p ≥ 1✳ ❈❤♦ x¯ ∈ S = D ∩ Ch ✱ ð ✤â ❤➔♠ sè h : Rp → Rn ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x¯ ∈ S ❈❤♦ ❝→❝ ❤➔♠ fi , i ∈ I ✈➔ gj , j ∈ J ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U ✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② (GRC) ✤ó♥❣ ✈ỵ✐ i0 ∈ I ✱ tù❝ ❧➔ L(M ; x¯) ∩ Ker h (¯ x) ⊆ cl conv T (M i0 ∩ Ch , x¯) ●✐↔ sû r➡♥❣ ❝→❝ ❤➔♠ fi , i ∈ I \ {i0 }✱ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x¯ ✈➔ ♥â♥ B ❧➔ ✤â♥❣✱ ð ✤â B = cone conv(∪m x)) + lin{hk (¯ x) : k ∈ K} j=1 ∂C gj (¯ ◆➳✉ x¯ ∈ S ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ữỡ t (P1 ) t tỗ t tỡ u = (u1 , , ul ) ∈ Rl , v = (v1 , , vm ) ∈ Rm ✱ ✈➔ w = (w1, , wn) ∈ Rn s❛♦ ❝❤♦ u ≥ 0, v 0✱ vj gj (¯ x) = 0, j = 1, 2, , m, ✈➔, ✶✺ ✭✷✳✻✮ l 0∈ m ui ∂C fi (¯ x) + i=1 n wk hk (¯ x), vj ∂C gj (¯ x) + j=1 ✭✷✳✼✮ k=1 ❤♦➦❝✱ t÷ì♥❣ ✤÷ì♥❣✱ l m ui fi◦ (¯ x, d) i=1 ❈❤ù♥❣ ♠✐♥❤✳ n vj gj◦ (¯ x, d) + j=1 ❉♦ x¯ wk hk (¯ x)(d) ≥ 0, ∀d ∈ Rp , + k=1 ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t t ỵ ổ õ t tờ qt ỵ tỗ t d X ỵ u = (u1 , , ul ) ∈ Rl , u ≥ 0✱ vj ∈ R, vj ≥ 0, j ∈ J(¯ x), w = (w1 , , wn ) ∈ Rn ❝→❝❤ ❧➜② (P1 )✱ fi◦ (¯ x; d) < 0, i = 1, , l, gj◦ (¯ x; d) ≤ 0, j ∈ J(¯ x), hk (¯ x)(d) = 0, k = 1, 2, , n ✈➔ ✭✷✳✽✮ s❛♦ ❝❤♦ ✭✷✳✼✮ ✤ó♥❣✳ ❇➡♥❣ vj = 0, j ∈ / J(¯ x) ✈➔ ❞♦ gj (¯ x) = ✈ỵ✐ j J( x) t õ ỵ ♠➺♥❤ ✤➲ ✭✷✳✼✮ ✈➔ ✭✷✳✽✮ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✷✳✷ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢✲ ❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ s➩ ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❜➟❝ ♥❤➜t ❝❤♦ ♠ët ✤✐➸♠ ❧➔ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ❜➜t ✤➥♥❣ t❤ù❝✱ ✤➥♥❣ t❤ù❝ ✈➔ r➔♥❣ ❜✉ë❝ t➟♣✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤è✐ ♥❣➝✉ ❑❑❚ ♠↕♥❤ (SKKT ) ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ổ õ r t ỵ t t♦→♥ ✭P✮✳ ❈❤♦ x¯ ∈ S = D ∩ C ∩ Ch ✳ ●✐↔ sû ❝→❝ ❤➔♠ sè fi , i ∈ I ✈➔ gj , j ∈ J ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U ✈➔ ❤➔♠ h ❧➔ ❦❤↔ ✈✐ ✶✻ ❋r➨❝❤❡t t↕✐ x¯✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② (GARC) s❛✉ ✤ó♥❣ t↕✐ x¯✿ l T (M i ∩ C ∩ Ch , x¯), L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯ x) ⊆ i=1 ◆➳✉ x¯ ∈ S ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭P✮✱ ❦❤✐ ✤â ❤➺ fi◦ (¯ x; d) ≤ 0, i = 1, 2, , l, ∃i0 ∈ {1, 2, , l} s❛♦ ❝❤♦ fi◦0 (¯ x; d) < ✭✷✳✾✮ gj◦ (¯ x; d) ≤ 0, j ∈ J(¯ x), ✭✷✳✶✵✮ h (¯ x)(d) = 0, ✭✷✳✶✶✮ d ∈ T (C, x¯), ✭✷✳✶✷✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ d ∈ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ x¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ t P tỗ t ởt f ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛ tr➯♥ V ❝õ❛ x¯ s❛♦ ❝❤♦ x¯ ❧➔ ♠ët ♠ët ♥❣❤✐➺♠ V ∩ S✳ ●✐↔ sû ♣❤↔♥ ❝❤ù♥❣✱ ❤➺ ✭✷✳✾✮✕ ✭✷✳✶✷✮ ❝â ♥❣❤✐➺♠ d ∈ X✳ ❑❤ỉ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû f1◦ (¯ x; d) < 0, fi◦ (¯ x; d) ≤ 0, i = 2, , l ✭✷✳✶✸✮ ❚ø ✭✷✳✶✶✮✕✭✷✳✶✸✮✱ t❛ ♥❤➟♥ ✤÷đ❝ d ∈ L(M ; x¯) ∩ T (C, x¯) ∩ Ker h (¯ x) d ∈ T (M i ∩ C ∩ Ch , x¯) i∈I {dni }n ⊆ M i ∩ C ∩ Ch ✈ỵ✐ ♠å✐ ✈➔ {tni }n ✈➻ s❛♦ ❝❤♦ ✶✼ (GARC) ✈➔ ❦❤✐ ✤â ✤ó♥❣✳ ❚ø ✤â✱ tỗ t tni > 0, tni dni → d ✈ỵ✐ n→∞ ✈➔ xni = x¯ + dni tni ∈ M i ∩ C ∩ Ch , n õ tỗ t no N s❛♦ ❝❤♦ xni ∈ M i ∩ C ∩ Ch ∩ V, n ≥ no ✳ i0 ∈ {2, 3, , l}✳ ❈è ✤à♥❤ Mi0 ✱ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❱ỵ✐ ♠å✐ n ≥ n0 ✱ ✈ỵ✐ i0 ợ i0 tữỡ ự õ t ❝â xni0 ♥❤÷ tr➯♥✳ fi0 (xni0 ) − fi0 (¯ x) ≤ 0✳ t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ An = {k ≥ : fk (xni0 ) > fk (¯ x)} ⊂ {2, 3, , l} ❚❛ ❦❤➥♥❣ ✤à♥❤ An = ∅✱ An = ∅ t❤➻ ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ k≥2 n ≥ n0 ✳ ❚❤➟t ✈➟②✱ ♥➳✉ tỗ t t ợ k = i0 n ≥ n0 s❛♦ ❝❤♦ t❛ ❝â fk (xni0 ) ≤ fk (¯ x) ✭✷✳✶✹✮ ❚ø f1 (¯ x + tni0 dni0 ) − f1 (¯ x) f1 (¯ x + tni0 d) − f1 (¯ x) lim sup ≤ lim sup n→∞ n→∞ tni0 tni0 f1 (¯ x + tni0 dni0 ) − f1 (¯ x + tni0 d) + lim sup n→∞ tni0 ≤ f1◦ (¯ x; d) + lim sup L1 dni0 − d n→∞ = f1◦ (¯ x; d) < 0, tỗ t ni0 n0 s ❝❤♦ f1 (¯ x + tni0 dni0 ) − f1 (¯ x) < 0, ✈ỵ✐ ♠å✐ n ≥ ni0 ✱ ð ✤â L1 > ❧➔ ❤➺ sè ▲✐♣s❝❤✐t③ ❝õ❛ ✭✷✳✶✺✮ f1 ✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ✭✷✳✶✺✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❤ú✉ ❤✐➺✉ ❝õ❛ tr➯♥ V ∩S ✈➔ ❞♦ ✤â An = ∅ ✈ỵ✐ ♠å✐ n ≥ n0 ✶✽ ✳ x¯ ❇➙② ❣✐í t❛ ❝è ✤à♥❤ {xni0r }nr ❝õ❛ {xni0 }n≥n0 k¯ ∈ {2, 3, , l} s❛♦ ❝❤♦ fk¯ (xni0r ) > fk¯ (¯ x) ❝❤♦ t➜t ❝↔ nr ✱ tù❝ ❧➔✱ k¯ ∈ Anr = {k ∈ {2, 3, , l} : fk (xni0r ) > fk (¯ x)} ❚❤❡♦ ❇ê ✤➲ ✷✳✶✱ t❛ ❝â ✳ ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ①➨t ❞➣② ❝♦♥ ✱ ✈ỵ✐ ♠å✐ fk¯◦ (¯ x, d) ≥ 0✳ nr ≥ n0 ✳ ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈➔ ✭✷✳✶✸✮ s✉② r❛ fk¯◦ (¯ x, d) = ❚ø ❝→❝ ❤➔♠ sè fi , i ∈ I ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U✱ fi (¯ x + tni0 dni0 ) − fi (¯ x + tni0 d) = 0, ✈ỵ✐ lim n→∞ tni0 t❛ ❝â ♠å✐ i∈I ✈➻ fi (¯ x + tni0 dni0 ) − fi (¯ x + tni0 d) lim ≤ lim Li dni0 − d = n n→∞ n→∞ ti0 ð ✤â Li > ❧➔ ❤➺ sè ▲✐♣s❝❤✐t③ ❝õ❛ fi ✳ ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ❝â fi (¯ x + tni0 dni0 ) − fi (¯ x) fi (¯ x + tni0 d) − fi (¯ x) lim sup ≤ lim sup n→∞ n→∞ tni0 tni0 fi (¯ x + tni0 dni0 ) − fi (¯ x + tni0 d) + lim sup n→∞ tni0 ≤ fi◦ (¯ x; d), ∀i ∈ I ❚ø f1◦ (¯ x, d) < 0✱ t❛ t❤✉ ✤÷đ❝ f1 (¯ x) − f1 (xni0 ) lim inf > 0, n→∞ tni0 ✈➔ ❞♦ ✤â✱ ✣è✐ ✈ỵ✐ ❝❤➾ sè f1 (¯ x) − f1 (xni0r ) > lim inf nr tni0r k ố trữợ ✤â t❛ ❦✐➸♠ tr❛ ✶✾ k¯ ∈ Anr ✱ ✈ỵ✐ ♠å✐ nr ≥ n0 ✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ✈✐➳t fk¯ (xni0r ) − fk¯ (¯ fk¯ (xni0r ) − fk¯ (¯ x) x) ≤ lim inf ≤ lim sup ≤ fk¯◦ (¯ x; d) = 0, nr nr nr →∞ n →∞ ti0 ti0 r ❞♦ fk¯ (xni0r ) > fk¯ (¯ x) ✈➔ tni0r > 0✳ ❱➻ ✈➟②✱ fk¯ (xni0r ) − fk¯ (¯ x) lim = nr nr →∞ ti0 ✣✐➲✉ ✤â ❦➨♦ t❤❡♦ f1 (xni0r ) f1 (¯ x) − = lim n nr →∞ fk¯ (x r ) − fk¯ (¯ x) nr →∞ i0 lim ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t r➡♥❣ f tr➯♥ V ∩ S✳ f1 (¯ x)−f1 (xni0r ) tni0r x) fk¯ (xni0r )−fk¯ (¯ nr ti0 = +∞, x¯ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ❝õ❛ ❉♦ ✤â✱ ❤➺ ✭✷✳✾✮✕ ✭✷✳✶✷✮ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✳ ❱➼ ❞ư ✷✳✶✳ ❳➨t ❜➔✐ t♦→♥ Min f (x) = (|x1 |, x2 , x1 +x42 )✱ ð ✤â x = (x1 , x2 ) ∈ R2 ✳ ❍➔♠ f = (f1 , f2 , f3 ) ✈✐ ❧✐➯♥ tö❝ t↕✐ ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ x¯ = (0, 0)✱ R2 ✳ ❈→❝ ❤➔♠ ✈➻ ✈➟② ❝❤ó♥❣ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x¯ = (0, 0)✳ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉✳ ❚❛ ❝â M = {(0, x2 ) : x2 ∈ R}, M = {(x1 , x2 ) ∈ R2 : x2 ≤ 0}, M = {(x1 , x2 ) ∈ R2 : x1 + x42 ≤ 0} ✷✵ f2 ✈➔ f3 ✣✐➸♠ ❦❤↔ x¯ ❧➔ ❑❤✐ ✤â T (M , x¯) = {(0, d2 ) : d2 ∈ R}, T (M , x¯) = {(d1 , d2 ) ∈ R2 : d2 ≤ 0}, T (M3 , x¯) = {d = (d1 , d2 ) ∈ R2 : d1 ≤ 0} ✈➔ T (M i , x¯) = {(0, d2 ) : d2 ≤ 0} i=1 ✣✐➲✉ ✤â s✉② r❛ r➡♥❣ ✤✐➲✉ ❦✐➺♥ (GRC) i = ✤ó♥❣ ✈ỵ✐ x; d) ≤ 0}, i = {1, 2, 3} = {(0, d2 ) : d2 ≤ 0}✱ R2 : fi◦ (¯ ✈➻ L(M ; x¯) = {d ∈ ✈➔ ❞♦ ✤â L(M, x¯) ⊆ cl conv T (M , x) t ỵ ✷✳✶ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❈❤ó♥❣ t❛ ❝â t❤➸ t❤➜② r➡♥❣ ❤➺ 0, d1 < 0✱ x; d) < 0, i = 1, 2, 3, fi◦ (¯ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ d ∈ R2 ✳ L(M ; x¯) = ∩3i=1 T (M i , x) ỗ t x d ∈ R2 (d = (0, d2 ), d2 < 0) x; d) = 0✱ f3◦ (¯ |d1 | < 0, d2 < ❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ỵ ữủ tọ t ỳ ỵ (GARC) s ❝❤♦ ✤÷đ❝ t❤ä❛ ♠➣♥✳ x; d) < 0✱ x; d) = 0, f2◦ (¯ f1◦ (¯ ♥❣❤➽❛ ❧➔ ❤➺ ❝õ❛ ỵ õ ởt d R2 ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❜➟❝ ♥❤➜t ✤è✐ ✈ỵ✐ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❦❤ỉ♥❣ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❚â♠ ❧↕✐✱ x¯ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✤❛♥❣ ①➨t✳ ✣✐➲✉ ♥➔② ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ ❦✐➸♠ tr❛ trü❝ t✐➳♣ ❜➡♥❣ ❝→❝❤ ①➨t f2 (x)−f2 (¯ x) f3 (¯ x)−f3 (x) = ✣✐➸♠ −a −a4 x¯ = a3 ✱ t✐➳♥ tỵ✐ +∞ x = (0, −a), a > ✈ỵ✐ ✈➔ ♥❤➟♥ t❤➜② r➡♥❣ a → 0, a > 0✳ ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ ♥❤÷♥❣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ỳ ữỡ ỵ tr♦♥❣ [✸] ❧➔ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✈➻ ❤➺✿ ✷✶ fi◦ (¯ x; d) ≤ 0✱ ✈ỵ✐ ♠å✐ i ∈ I ✈➔ fi◦ (¯ x; d) < 0✱ ✈ỵ✐ ➼t ♥❤➜t ởt i õ d R2 ỵ ❧➔ L(Q; x¯) ⊆ ∩li=1 T (Qi , x¯) ✈➻ ✤✐➲✉ ❦✐➺♥ ❝❤✉➞♥ ❤â❛ r➔♥❣ ❜✉ë❝ ❆❜❛❞✐❡ tê♥❣ q✉→t ✤÷đ❝ ỏ ọ tr tt ỵ ✹✳✶❪ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ tr♦♥❣ ✈➼ ❞ư ♥➔②✳ ❚❤➟t ✈➟②✱ Q1 = M ∩ M = {x = (x1 , x2 ) ∈ R2 : x2 ≤ 0, x1 + x42 ≤ 0}, Q2 = M ∩ M = {¯ x}, Q3 = M ∩ M = {(0, x2 ) ∈ R2 : x2 ≤ 0}, T (Q1 , x¯) = {(d1 , d2 ) ∈ R2 : d1 ≤ 0, d2 ≤ 0}, T (Q2 , x¯) = {¯ x}, T (Q3 , x¯) = {(0, d2 ) ∈ R2 : d2 ≤ 0}, ✈➔ ❞♦ ✤â ✈ỵ✐ L(Q; x¯) = L(M ; x¯)✱ t❛ ♥❤➟♥ ✤÷đ❝ T (Qi , x¯) = {¯ x} ⊇ {(0, d2 ) : d2 ≤ 0} = L(Q; x¯) i=1 ◆❤➟♥ ①➨t ✷✳✶✳ ❚r♦♥❣ ❱➼ ❞ö ✷✳✶✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❇✉r❛❝❤✐❦ ✈➔ ❘✐③✈✐ [✷] ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝ ✈➻ ❦❤ỉ♥❣ ♣❤↔✐ t➜t ❝↔ ❝→❝ ❤➔♠ ♠ư❝ t✐➯✉ ✤➲✉ ❦❤↔ ✈✐ ❧✐➯♥ tư❝✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ❞↕♥❣ (SKKT ) ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ữỡ t (P1 ) ợ r t tự tự ỵ ❳➨t ❜➔✐ t♦→♥ (P1 ) ✈ỵ✐ X = Rp , p ≥ 1✳ ❈❤♦ x¯ ∈ S = D ∩ Ch ✳ ❈❤♦ ❝→❝ ❤➔♠ sè fi , i ∈ I ✈➔ gj , j ∈ J ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ tr➯♥ U ✈➔ ❤➔♠ sè h ❧➔ ❋r➨❝❤❡t ❦❤↔ ✈✐ t↕✐ x¯✳ ●✐↔ sû r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② (GARC) ✤ó♥❣ ✷✷ t↕✐ x¯✱ tù❝ ❧➔✱ L(M ; x¯) ∩ Ker h (¯ x) ⊆ ∩li=1 T (M i ∩ Ch , x¯) ●✐↔ sû t❤➯♠ r➡♥❣ ❝→❝ ♥â♥ Bi , i ∈ I ✱ ❧➔ ✤â♥❣✱ Bi = cone conv(∪j=i ∂C fj (¯ x))+cone conv(∪j∈J\(¯x) ∂C gj (¯ x))+lin{hk (¯ x) : k ∈ K} ◆➳✉ x¯ ∈ S ❧➔ ♠ët ♥❣❤✐➺♠ ❤ú✉ ❤✐➺✉ t❤ü❝ sü ●❡♦❢❢r✐♦♥ ✤à❛ ữỡ t (P1 ) t tỗ t tỡ u = (u1 , , ul ) ∈ Rl , v = (v1 , , vm ) ∈ Rm ✱ ✈➔ w = (w1 , , wn ) ∈ Rn s❛♦ ❝❤♦ u > 0, v l 0∈ 0, vj gj (¯ x) = 0, j ∈ J ✈➔ m ui ∂C fi (¯ x) + i=1 n vj ∂C gj (¯ x) + j=1 wk hk (¯ x), k=1 ❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣✱ l m ui fi◦ (¯ x, d) i=1 ❈❤ù♥❣ ♠✐♥❤✳ n vj gj◦ (¯ x, d) + wk hk (¯ x)(d) ≥ 0, ∀d ∈ Rp + j=1 k=1 ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ ỵ ữủ tứ ỵ ỵ r tờ qt ỵ t❤❛♠ ❦❤↔♦ ❬✶❪ ●❡♦❢❢r✐♦♥✱ ❆✳ ▼✳✿ Pr♦♣❡r ❡❢❢✐❝✐❡♥❝② ❛♥❞ t❤❡ t❤❡♦r② ♦❢ ✈❡❝t♦r ♠❛①✐♠✐③❛✲ t✐♦♥✳ ❏ ▼❛t❤ ❆♥❛❧✳ ✶✾✻✽❀✷✷✿✻✶✽✕✻✸✵✳ ❬✷❪ ❇✉r❛❝❤✐❦ ❘✳ ❙✳✱ ❘✐③✈✐ ▼✳ ▼✳✿ ❖♥ ✇❡❛❦ ❛♥❞ str♦♥❣ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥✲ ❞✐t✐♦♥s ❢♦r s♠♦♦t❤ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✳ ❏ ❖♣t✐♠ ❚❤❡♦r② ❆♣♣❧✳ ✷✵✶✷❀✶✺✺✭✷✮✿✹✼✼✕✹✾✶✳ ❬✸❪ ●✐♦r❣✐ ●✳✱ ❏✐♠❡♥❡③ ❇✳✱ ◆♦✈♦ ❱✳✿ ❙tr♦♥❣ ❑✉❤♥✲❚✉❝❦❡r ❝♦♥❞✐t✐♦♥s ❛♥❞ ❝♦♥str❛✐♥t q✉❛❧✐❢✐❝❛t✐♦♥s ✐♥ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③ ♠✉❧t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✳ ❚♦♣✳ ✷✵✵✾❀✶✼✿✷✽✽✕✸✵✹✳ ❬✹❪ ❈♦♥st❛♥t✐♥ ❊✳✿ ❋✐rst✲♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✐♥ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③ ♠✉❧✲ t✐♦❜❥❡❝t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥✳ ❖♣t✐♠✐③❛t✐♦♥✳ ✷✵✶✽❀✻✼✭✾✮✿✶✹✹✼✕✶✹✻✵✳ ❬✺❪ ❈❤❛♥❞r❛ ❙✳✱ ❉✉tt❛ ❏✳✱ ▲❛❧✐t❤❛ ❈✳ ❙✳✿ ❘❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s ❛♥❞ ♦♣t✐♠❛❧✐t② ✐♥ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥✳ ◆✉♠ ❋✉♥❝t ❆♥❛❧ ❖♣t✐♠✳ ✷✵✵✹❀✷✺✭✺✕✻✮✿✹✼✾✕✺✵✶✳ ✷✹