Dưới vi phân của hàm lồi và một số ứng dụng tối ưu

✶ ➜➵✐ ❤ä❝ t❤➳✐ ♥❣✉②➟♥ ❚r➢ê♥❣ ➤➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ◆➠♥❣ ❚❤Þ ▼❛✐ ❉➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ ❤➭♠ ❧å✐ ✈➭ ♠ét sè ø♥❣ ❞ô♥❣ tr♦♥❣ tè✐ ➢✉ ❈❤✉②➟♥ ♥❣➭♥❤✿ ●✐➯✐ tÝ❝❤ ▼➲ sè✿✻✵✳✹✻✳✵✶ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ●❙ ✲❚❙❑❍ ▲➟ ❉ị♥❣ ▼➢✉ ❚❤➳✐ ♥❣✉②➟♥ ✲ ◆➝♠ ✷✵✵✽ ✷ ▼ô❝ ❧ô❝ ❚r❛♥❣ ❚r❛♥❣ ♣❤ơ ❜×❛ ✶ ▼ơ❝ ❧ơ❝ ✷ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❦ý ❤✐Ư✉✱ ❝➳❝ ❝❤÷ ✈✐Õt t➽t ✸ ▲ê✐ ♥ã✐ ➤➬✉ ✹ ❈❤➢➡♥❣✶✳ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐ ✶✳✶✳ ❚❐♣ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✺ ✶✳✷✳ ❍➭♠ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✶✳ ❍➭♠ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷✳✷✳ ❚Ý♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ❤➭♠ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✸✳ ❈➳❝ ♣❤Ð♣ t♦➳♥ ❜➯♦ t♦➭♥ tÝ♥❤ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✹✳ ❇✃t ➤➻♥❣ t❤ø❝ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✺✳ ❍➭♠ ❧✐➟♥ ❤ỵ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ❈❤➢➡♥❣✷✳ ❉➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ ❤➭♠ ❧å✐ ✶✽ ✷✳✶✳ ➜➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳ ❉➢í✐ ✈✐ ♣❤➞♥ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✶✳ ❉➢í✐ ✈✐ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✷✳ ❚Ý♥❤ ❦❤➯ ✈✐ ❝ñ❛ ❤➭♠ ❧å✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷✳✸✳ ❚Ý♥❤ ➤➡♥ ➤✐Ư✉ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷✳✹✳ ❚Ý♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳✺✳ P❤Ð♣ tÝ♥❤ ✈í✐ ❞➢í✐ ➤➵♦ ❤➭♠ ✷✳✸✳ ❉➢í✐ ✈✐ ♣❤➞♥ ①✃♣ ①Ø ❈❤➢➡♥❣✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ▼ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ tr♦♥❣ tè✐ ➢✉ ❤♦➳ ✸✳✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷✳ ❇➭✐ t♦➳♥ ❧å✐ ❦❤➠♥❣ ❝ã r➺♥❣ ❜✉é❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✸✳ ❇➭✐ t♦➳♥ ❧å✐ ✈í✐ r➺♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✹✳ ❇➭✐ t♦➳♥ ❧å✐ ✈í✐ r➺♥❣ ❜✉é❝ ❜✃t ➤➻♥❣ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❑Õt ❧✉❐♥ ✻✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✻✹ ✸ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❦ý ❤✐Ư✉✱ ❝➳❝ ❝❤÷ ✈✐Õt t➽t ❱í✐ ♥ ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣✱ ❦ý ❤✐Ư✉✿ n R ✿ ❦❤➠♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ♥✲❝❤✐Ị✉ tr➟♥ tr➢ê♥❣ sè t❤ù❝❀ n R+ ✿ ❣ã❝ ❦❤➠♥❣ ➞♠ ❝ñ❛ Rn ✭t❐♣ ❝➳❝ ✈Ð❝✲t➡ ❝ã ♠ä✐ t♦➵ ➤é ➤Ò✉ ❦❤➠♥❣ ➞♠ ✮❀ R✿ trô❝ sè t❤ù❝ (R = R1 )❀ R✿ trô❝ sè t❤ù❝ ♠ë ré♥❣ (R = R ∪ {−∞, +∞})❀ N ✿ t❐♣ ❤ỵ♣ sè ♥❣✉②➟♥ ❞➢➡♥❣❀ n 2R ✿ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ ❝đ❛ Rn ❀ n ❱í✐ ♠ä✐ ✈Ð❝✲t➡ x, y ∈ R ✱ ❦ý ❤✐Ư✉✿ xi ✿ t♦➵ ➤é t❤ø ✐ ❝đ❛ ①❀ xT ✿ ✈Ð❝✲t➡ ❤➭♥❣ ✭❝❤✉②Ĩ♥ ✈Þ ❝đ❛ ①✮❀ x, y = xT y = xy := nj=1 xj yj ✿ tÝ❝❤ ✈➠ ❤➢í♥❣ ❝đ❛ ❤❛✐ ✈Ð❝✲t➡ ① ✈➭ ②❀ ||x|| = n j=1 xj ✿ ❝❤✉➮♥ ❊✉❝❧✐❞❡ ❝ñ❛ ①❀ [x, y]✿ ➤♦➵♥ t❤➻♥❣ ➤ã♥❣ ♥è✐ ① ✈➭ ②❀ (x, y)✿ ➤♦➵♥ t❤➻♥❣ ♠ë ♥è✐ ① ✈➭ ②❀ ❱í✐ t❐♣ A✱ ❦ý ❤✐Ư✉✿ A✿ ❜❛♦ ➤ã♥❣ ❝đ❛ A❀ coA✿ ❜❛♦ ❧å✐ ❝ñ❛ A❀ aff A✿ ❜❛♦ ❛✲♣❤✐♥ ❝ñ❛ A❀ intA✿ t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ A❀ ri A✿ t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ tr♦♥❣ t➢➡♥❣ ➤è✐ ❝đ❛ A❀ ❱í✐ ❤➭♠ f ❝đ❛ ♥ ❜✐Õ♥✱ ❦ý ❤✐Ư✉✿ f ✿ ❤➭♠ ❜❛♦ ➤ã♥❣ ❝đ❛ f ❀ dom f ✿ t❐♣ ❤÷✉ ❞ơ♥❣ ❝đ❛ f ❀ f ∗ ✿ ❤➭♠ ❧✐➟♥ ❤ỵ♣ ❝đ❛ f ❀ epi f ✿ tr➟♥ ➤å t❤Þ ❝đ❛ f ❀ ∂f (x)✿ ❞➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ f t➵✐ ①❀ ∂ f (x)✿ ✲ ❞➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ f t➵✐ ①❀ f (x) ❤♦➷❝ f (x)✿ ➤➵♦ ❤➭♠ ❝ñ❛ f t➵✐ ①❀ f (x, d)✿ ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣ d ❝ñ❛ f t➵✐ ①❀ ✹ ▲ê✐ ♥ã✐ ➤➬✉ ●✐➯✐ tÝ❝❤ ❧å✐ ❧➭ ♠ét ❜é ♠➠♥ q✉❛♥ trä♥❣ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥ ❤✐Ư♥ ➤➵✐✳ ●✐➯✐ tÝ❝❤ ❧å✐ ♥❣❤✐➟♥ ❝ø✉ ♥❤÷♥❣ ❦❤Ý❛ ❝➵♥❤ ❣✐➯✐ tÝ❝❤ ❝ñ❛ t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐✳ ❉➢í✐ ✈✐ ♣❤➞♥ ❧➭ ♠ét ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ❝đ❛ ❣✐➯✐ tÝ❝❤ ❧å✐✳ ➜➞② ❧➭ ♠ë ré♥❣ ❝❤♦ ➤➵♦ ❤➭♠ ❦❤✐ ❤➭♠ ❦❤➠♥❣ ❦❤➯ ✈✐✳ ➜✐Ò✉ ♥➭② ❝❤♦ t❤✃② ✈❛✐ trß ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ❤✐Ư♥ ➤➵✐ ❝ị♥❣ ❝ã t➬♠ q✉❛♥ trä♥❣ ♥❤➢ ✈❛✐ trß ❝đ❛ ➤➵♦ ❤➭♠ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ❝ỉ ➤✐Ĩ♥✳ ❉➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ ❤➭♠ ❧å✐ ❝ã r✃t ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥ ✈➭ ➤➷❝ ❜✐Öt tr♦♥❣ ❝➳❝ ❜é ♠➠♥ t♦➳♥ ø♥❣ ❞ô♥❣✱ ♥❤➢ tè✐ ➢✉ ❤♦➳✱ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✱ ❝➞♥ ❜➺♥❣ ✈✳✳✳✈✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝ã ❤Ư t❤è♥❣✱ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈➭ q✉❛♥ trä♥❣ ♥❤✃t ✈Ị ❞➢í✐ ✈✐ ♣❤➞♥ ❝đ❛ ❤➭♠ ❧å✐ ✈➭ ①Ðt ♠ét sè ø♥❣ ❞ơ♥❣ ➤✐Ĩ♥ ❤×♥❤ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ tr♦♥❣ tè✐ ➢✉ ❤♦➳✳ ▲✉❐♥ ✈➝♥ ❣å♠ ✸ ❝❤➢➡♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ✶ sÏ tr×♥❤ ❜➭② ♥❤÷♥❣ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐✳ ➜➞② ❧➭ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❜ỉ trỵ ❝❤♦ ❝❤➢➡♥❣ ✷ ✈➭ ❞♦ ➤ã sÏ ❦❤➠♥❣ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚r♦♥❣ ❝❤➢➡♥❣ ✷ sÏ ➤Ò ❝❐♣ ✈Ò ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣✱ ❞➢í✐ ✈✐ ♣❤➞♥✱ ❞➢í✐ ✈✐ ♣❤➞♥ ①✃♣ ①Ø ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❝❤ó♥❣✳ ❉ù❛ tr➟♥ ❝➳❝ ❦Õt q✉➯ ➤➲ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ tr➢í❝✱ tr♦♥❣ ❝❤➢➡♥❣ ✸ sÏ tr×♥❤ ❜➭② ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝ù❝ trÞ ❝❤♦ ❝➳❝ ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ ❧å✐ ✈í✐ ❝➳❝ r➺♥❣ ❜✉é❝ ❦❤➳❝ ♥❤❛✉ ✭❦❤➠♥❣ r➺♥❣ ❜✉é❝✱ r➺♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ r➺♥❣ ❜✉é❝ ❜✃t ➤➻♥❣ t❤ø❝✮✳ ❇➯♥ ❧✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❝đ❛ ●❙ ✲❚❙❑❍ ▲➟ ❉ị♥❣ ▼➢✉✳ ◆❤➞♥ ➤➞② ❡♠ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ t❤➬② ➤➲ ❤➢í♥❣ ❞➱♥✱ ➤é♥❣ ✈✐➟♥✱ ❦❤✉②Õ♥ ❦❤Ý❝❤ ❡♠ ❤ä❝ t❐♣✱ ♥❣❤✐➟♥ ❝ø✉ ➤Ó ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❈❤➢➡♥❣ ✶ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t❛ sÏ ❧➭♠ ✈✐Ư❝ ✈í✐ ❦❤➠♥❣ ❣✐❛♥ ❡✉❝❧✐❞✲♥ ❝❤✐Ò✉ tr➟♥ tr➢ê♥❣ sè t❤ù❝ R✳ ❑❤➠♥❣ ❣✐❛♥ ♥➭② ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ Rn ✳ ❈❤➢➡♥❣ ♥➭② ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ♥❤÷♥❣ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ♥❤✃t ❝đ❛ t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐ ❝ï♥❣ ✈í✐ ♥❤÷♥❣ tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ♥ã✳ ❈➳❝ ❦✐Õ♥ t❤ø❝ ë tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ➤✉ỵ❝ ❧✃② ë t➭✐ ❧✐Ư✉ ✿ ✰ ●✐➳♦ tr×♥❤ ✧◆❤❐♣ ♠➠♥ ❣✐➯✐ tÝ❝❤ ❧å✐ ø♥❣ ❞ơ♥❣✧ ❝đ❛ t➳❝ ❣✐➯ ▲➟ ❉ị♥❣ ▼➢✉ ✈➭ ◆❣✉②Ơ♥ ❱➝♥ ❍✐Ị♥✳ ✰ ❈✉è♥ ✧❈♦♥✈❡① ❆♥❛❧②s✐s✧ ❝đ❛ t➳❝ ❣✐➯ ❚✳❘♦❝❦❛❢❡❧❧❛r✳ ❉♦ ❝❤➢➡♥❣ ♥➭② ❝❤Ø ♠❛♥❣ tÝ♥❤ ❝❤✃t ❜ỉ trỵ✱ ♥➟♥ t❛ ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ ♥➟✉ ë ➤➞②✳ ✶✳✶ ❚❐♣ ❧å✐ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳ ➜♦➵♥ t❤➻♥❣ ♥è✐ ❤❛✐ ➤✐Ó♠ ❛ ✈➭ ❜ tr♦♥❣ Rn ❧➭ t❐♣ ❤ỵ♣ ❝➳❝ ✈Ð❝✲t➡ ① ❝ã ❞➵♥❣ {x ∈ Rn | x = αa + βb , α ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳ ▼ét t❐♣ 0, β , α + β = 1} C ⊆ Rn ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét t❐♣ ❧å✐ ♥Õ✉ C ❝❤ø❛ ♠ä✐ ➤♦➵♥ t❤➻♥❣ ➤✐ q✉❛ ❤❛✐ ➤✐Ĩ♠ ❜✃t ❦ú ❝đ❛ ♥ã✳ ❚ø❝ ❧➭ C ❧å✐ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ∀x, y ∈ C, λ ∈ [0, 1] =⇒ λx + (1 − λ)y ∈ C ✺ ✻ ❱Ý ❞ơ ✶✳✶✳ ✭❱Ị t❐♣ ❧å✐✮✳ ❛✮ ❚❐♣ C = R+ ❧➭ t❐♣ ❧å✐✳ ❜✮ ❚❐♣ C = [−2; 3) ❧➭ t❐♣ ❧å✐✳ ❝✮ ❚❐♣ C ≡ oxy tr♦♥❣ R3 ❧➭ t❐♣ ❧å✐✳ ❞✮ ❈➳❝ t❛♠ ❣✐➳❝✱ ❤×♥❤ trß♥ tr♦♥❣ ♠➷t ♣❤➻♥❣ ❧➭ ❝➳❝ t❐♣ ❧å✐✳ ❱Ý ❞ơ ✶✳✷✳ ✭❱Ò t❐♣ ❦❤➠♥❣ ❧å✐✮✳ ❛✮ ❚❐♣ C = (−2; 0) ∪ (0; 3) ❦❤➠♥❣ ❧➭ t❐♣ ❧å✐✳ ❜✮ ❚❐♣ C = {(x, y) ∈ R2 | xy = 0} ❦❤➠♥❣ ❧➭ t❐♣ ❧å✐✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳ ❚❛ ♥ã✐ ① ❧➭ tỉ ❤ỵ♣ ❧å✐ ❝đ❛ ❝➳❝ ➤✐Ĩ♠ ✭✈Ð❝✲t➡✮ k k j λj x , λj x= , ∀j = 1, , k , j=1 ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✹✳ x1 , , xk ♥Õ✉ λj = j=1 ❙✐➟✉ ♣❤➻♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ Rn ❧➭ ♠ét t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ ❝ã ❞➵♥❣ {x ∈ Rn | aT x = α}, tr♦♥❣ ➤ã a ∈ Rn ❧➭ ♠ét ✈Ð❝✲t➡ ❦❤➳❝ ✵ ✈➭ α ∈ R✳ ❱Ð❝✲t➡ ❛ t❤➢ê♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ✈Ð❝✲t➡ ♣❤➳♣ t✉②Õ♥ ❝ñ❛ s✐➟✉ ♣❤➻♥❣✳ ▼ét s✐➟✉ ♣❤➻♥❣ sÏ ❝❤✐❛ ❦❤➠♥❣ ❣✐❛♥ r❛ ❤❛✐ ♥ö❛ ❦❤➠♥❣ ❣✐❛♥✳ ◆ö❛ ❦❤➠♥❣ ❣✐❛♥ ợ ị ĩ s ị ĩ ❣✐❛♥ ❧➭ ♠ét t❐♣ ❤ỵ♣ ❝ã ❞➵♥❣ {x | aT x tr♦♥❣ ➤ã α}, a = ✈➭ α ∈ R✳ ➜➞② ❧➭ ♥ư❛ ❦❤➠♥❣ ❣✐❛♥ ➤ã♥❣✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✻✳ ❈❤♦ C ⊆ Rn ❧➭ ♠ét t❐♣ ❧å✐ ✈➭ x ∈ C ✳ ❚❐♣ NC (x) := {ω | ω, y − x ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ã♥ ♣❤➳♣ t✉②Õ♥ ♥❣♦➭✐ ❝ñ❛ ◆❤❐♥ ①Ðt✳ , ∀y ∈ C}, C t➵✐ ①✳ NC (x) ❧➭ ♠ét ♥ã♥ ❧å✐ ➤ã♥❣✳ ✼ ❱Ý ❞ô ✶✳✸✳ ❚r♦♥❣ R2 ✱ ①Ðt t❐♣ C = R+ ✳ NC (0) = {ω | ω, y − 0 , ∀y ∈ C} = {ω | ωi y i 0} i=1 = {ω | ωi ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✼✳ ▼ét ➤✐Ĩ♠ ♥Õ✉ ♥ã ❧➭ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ 0} a ∈ C ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ tr♦♥❣ t➢➡♥❣ ➤è✐ ❝ñ❛ C C t❤❡♦ t➠✲♣➠ ❝➯♠ s✐♥❤ ❜ë✐ aff C ✳ ❚❛ sÏ ❦ý ❤✐Ư✉ t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ tr♦♥❣ t➢➡♥❣ ➤è✐ ❝ñ❛ C ❧➭ ri C ✳ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥ t❛ ❝ã✿ ri C := {a ∈ C | ∃B : (a + B) ∩ aff C ⊂ C}, tr♦♥❣ ➤ã B ❧➭ ♠ét ❧➞♥ ❝❐♥ ♠ë ❝đ❛ ❣è❝✳ ❍✐Ĩ♥ ♥❤✐➟♥ ri C := {a ∈ aff C | ∃B : (a + B) ∩ aff C ⊂ C} ◆❤➢ t❤➢ê♥❣ ❧Ö✱ t❛ ❦ý ❤✐Ö✉ ❣ä✐ ❧➭ ❜✐➟♥ t➢➡♥❣ ➤è✐ ❝đ❛ ▼Ư♥❤ ➤Ị ✶✳✶✳ y∈C ❈❤♦ C ✱ ❧➭ ❜❛♦ ➤ã♥❣ ❝đ❛ C ✳ ❚❐♣ ❤ỵ♣ C \ ri C ➤➢ỵ❝ C✳ C ⊆ Rn ❧➭ ♠ét t❐♣ ❧å✐✳ ●✐➯ sö x ∈ ri C ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ t✃t ❝➯ ❝➳❝ ➤✐Ĩ♠ tr➟♥ ➤♦➵♥ t❤➻♥❣ ♥è✐ ① ✈➭ ②✱ ❝ã t❤Ĩ trõ ②✱ ➤Ị✉ t❤✉é❝ ri C ✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ✈í✐ ♠ä✐ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✽✳ λ < 1✱ t❤× (1 − λ) ri C + λC ⊂ ri C ✳ ▼ét ➤➢ê♥❣ t❤➻♥❣ ♥è✐ ❤❛✐ ➤✐Ĩ♠ ✭❤❛✐ ✈Ð❝✲t➡✮ ❛✱❜ tr♦♥❣ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ✈Ð❝✲t➡ Rn ❧➭ x ∈ Rn ❝ã ❞➵♥❣ {x ∈ Rn | x = αa + βb , α , β ∈ R , α + β = 1} ị ĩ ột t C ợ ọ t❐♣ ❛✲♣❤✐♥ ♥Õ✉ ♥ã ❝❤ø❛ ♠ä✐ ➤➢ê♥❣ t❤➻♥❣ ➤✐ q✉❛ ❤❛✐ ➤✐Ĩ♠ ❜✃t ❦ú ❝đ❛ ♥ã✱ tø❝ ❧➭ ∀x, y ∈ C , ∀λ ∈ R =⇒ λx + (1 − λ)y ∈ C ❱Ý ❞ơ ✶✳✹✳ ❚❐♣ ✭❱Ị t❐♣ ❛✲♣❤✐♥✮✳ C = R2 ❧➭ t❐♣ ❛✲♣❤✐♥✱ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❧➭ ♠ét t❐♣ ❛❢❢✐♥❡ ✽ ◆❤❐♥ ①Ðt✳ ❚❐♣ ❛✲♣❤✐♥ ❧➭ ột trờ ợ r ủ t ị ĩ ❇❛♦ ❧å✐ ❝ñ❛ ♠ét t❐♣ E ❧➭ ❣✐❛♦ ❝ñ❛ t✃t ❝➯ ❝➳❝ t❐♣ ❧å✐ ❝❤ø❛ E ✳ ❇❛♦ ❧å✐ ❝ñ❛ ♠ét t❐♣ E sÏ ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭ coE ✳ ❇❛♦ ❧å✐ ➤ã♥❣ ❝ñ❛ ♠ét t❐♣ E ❧➭ t❐♣ ❧å✐ ➤ã♥❣ ♥❤á ♥❤✃t ❝❤ø❛ E ✳ ❚❛ sÏ ❦ý ❤✐Ö✉ ❜❛♦ ❧å✐ ➤ã♥❣ ❝ñ❛ ♠ét t❐♣ ❇❛♦ ❛✲♣❤✐♥ ❝ñ❛ ❝ñ❛ ♠ét t❐♣ E ❧➭ coE ✳ E ❧➭ ❣✐❛♦ ❝ñ❛ t✃t ❝➯ ❝➳❝ t❐♣ ❛✲♣❤✐♥ ❝❤ø❛ E ✳ ❇❛♦ ❛✲♣❤✐♥ E ợ ý ệ aff E ị ♥❣❤Ü❛ ✶✳✶✶✳ ❈❤♦ E ⊆ Rn ✳ ➜✐Ĩ♠ ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ ❝đ❛ ❛ s❛♦ ❝❤♦ E ♥Õ✉ tå♥ t➵✐ ♠ét ❧➞♥ ❝❐♥ ♠ë U (a) U (a) ⊂ E ✳ ❑ý ❤✐Ư✉ t❐♣ ❤ỵ♣ ❝➳❝ ➤✐Ĩ♠ tr♦♥❣ ❝ñ❛ t❐♣ E ❧➭ intE ✈➭ B ❧➭ q✉➯ ❝➬✉ ➤➡♥ ✈Þ t➞♠ ë ❣è❝✳ ❑❤✐ ➤ã t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ t❛ ❝ã intE = {x | ∃r > : x + rB ⊂ E} ➜✐Ĩ♠ ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✐➟♥ ❝đ❛ t❤✉é❝ E ♥Õ✉ ♠ä✐ ❧➞♥ ❝❐♥ ❝đ❛ ❛ ➤Ị✉ ❝ã ➤✐Ĩ♠ E ✈➭ ➤✐Ĩ♠ ❦❤➠♥❣ t❤✉é❝ E ✳ ❚❐♣ E ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ♠ë ♥Õ✉ ♠ä✐ ➤✐Ĩ♠ ❝đ❛ E ➤Ị✉ ❧➭ ➤✐Ĩ♠ tr♦♥❣ ❝đ❛ E ✳ ❚❐♣ E ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ➤ã♥❣ ♥Õ✉ E ❝❤ø❛ ♠ä✐ ➤✐Ĩ♠ ❜✐➟♥ ❝đ❛ ♥ã✳ ❚❐♣ E ➤➢ỵ❝ ❣ä✐ ị ế tồ t ột ì ứ E ✳ ❚r♦♥❣ Rn t❐♣ E ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ❝♦♠♣➽❝ ♥Õ✉ E ❧➭ ♠ét t❐♣ ➤ã♥❣ ✈➭ ❜Þ ❝❤➷♥✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✷✳ ▼ét t❐♣ ❈❤♦ ❈ ❧➭ ♠ét t❐♣ ❧å✐✳ F ⊂ C ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞✐Ư♥ ❝đ❛ ♠ét t❐♣ ❧å✐ C ♥Õ✉ F ❧➭ t❐♣ ❧å✐ ✈➭ ∀x, y ∈ C , tx + (1 − t)y ∈ F , < t < =⇒ [x, y] ⊂ F ❱Ý ❞ô ✶✳✺✳ ❈❤♦ C := {(x, y, z) ∈ R3 | x, y, z ∈ [0, 1]}✳ ❚❐♣ F1 := {(x, y, z) ∈ R3 | x, y ∈ [0, 1], z = 0} ❧➭ ♠ét ❞✐Ư♥ ❝đ❛ t❐♣ C ✳ ❚❐♣ F2 := {(x, y, z) ∈ R3 | y ∈ [0, 1], x = 1, z = 0} ❧➭ ♠ét ❞✐Ư♥ ❝đ❛ t❐♣ C✳ ➜✐Ĩ♠ ❝ù❝ ❜✐➟♥ ❧➭ ❞✐Ư♥ ❝ã t❤ø ♥❣✉②➟♥ ✭❝❤✐Ị✉✮ ❜➺♥❣ ✵✳ ✾ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✸✳ ❈❤♦ x0 ∈ C ✳ ❚❛ ♥ã✐ aT x = α ❧➭ s✐➟✉ ♣❤➻♥❣ tù❛ ❝ñ❛ C t➵✐ x0 ✱ ♥Õ✉ aT x0 = α , aT x ◆❤➢ ✈❐② s✐➟✉ ♣❤➻♥❣ tù❛ ❝ñ❛ t❐♣ C t➵✐ x0 ∈ C ❧➭ s✐➟✉ ♣❤➻♥❣ ➤✐ q✉❛ x0 ✈➭ ➤Ĩ C ✈Ị ♠ét ♣❤Ý❛✳ ◆ư❛ ❦❤➠♥❣ ❣✐❛♥ aT x ❧➭ ♥ư❛ ❦❤➠♥❣ ❣✐❛♥ tù❛ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✶✳ α ∀x ∈ C α tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥✱ ➤➢ỵ❝ ❣ä✐ C t➵✐ x0 ✳ ✭❑r❡✐♥✲▼✐❧♠❛♥✮✳ ▼ä✐ t❐♣ ❧å✐ ó rỗ ứ t ề ó ể ❝ù❝ ❜✐➟♥✳ ➜Þ♥❤ ❧ý ✶✳✷✳ ✭❳✃♣ ①Ø t✉②Õ♥ tÝ♥❤ t❐♣ ọ t ó rỗ trù ✈í✐ t♦➭♥ ❜é ❦❤➠♥❣ ❣✐❛♥ ➤Ị✉ ❧➭ ❣✐❛♦ ❝đ❛ t✃t ❝➯ ❝➳❝ ♥ư❛ ❦❤➠♥❣ ❣✐❛♥ tù❛ ❝đ❛ ♥ã✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✹✳ ❈❤♦ ❤❛✐ t❐♣ ❚❛ ♥ã✐ s✐➟✉ ♣❤➻♥❣ aT x aT x ❚❛ ♥ã✐ s✐➟✉ ♣❤➻♥❣ aT x C ✈➭ D rỗ = t C D ế α aT y , ∀x ∈ C , ∀y ∈ D = α t➳❝❤ ❝❤➷t C ✈➭ D ♥Õ✉ aT x < α < aT y , ∀x ∈ C , ∀y ∈ D ❚❛ ♥ã✐ s✐➟✉ ♣❤➻♥❣ aT x = α t➳❝❤ ♠➵♥❤ C ✈➭ D ♥Õ✉ Supx∈C aT x < α < inf y∈D aT y ❱Ý ❞ô ✶✳✻✳ ✭❚➳❝❤ ♥❤➢♥❣ ❦❤➠♥❣ t➳❝❤ ❝❤➷t✮✳ ❈❤♦ t❐♣ C = {(x, y) ∈ R2 | x2 + y 1}, ✈➭ D = {(x, y) ∈ R2 | − ❚❛ ❝ã✿ x 1, y 3} ✶✵ ✰ C D rỗ C, D t ợ ✈× tå♥ t➵✐ s✐➟✉ ♣❤➻♥❣ (0, 1)(x, y) = t❤♦➯ ♠➲♥ (0, 1)(x, y) (0, 1)(x , y ) ∀(x, y) ∈ C, ∀(x , y ) ∈ D y y ❍❛② ✰ ∀(x, y) ∈ C, ∀(x , y ) ∈ D C, D ❦❤➠♥❣ t➳❝❤ t ợ ì tồ t s (a1 , a2 )(x, y) = α ♥➭♦ t❤♦➯ ♠➲♥ (a1 , a2 )(x, y) < α < (a1 , a2 )(x , y ) ∀(x, y) ∈ C, ∀(x , y ) ∈ D ✭❚➳❝❤ ♥❤➢♥❣ ❦❤➠♥❣ t➳❝❤ ♠➵♥❤✮✳ ❱Ý ❞ô ✶✳✼✳ ❈❤♦ t❐♣ C = {(x, y) ∈ R2 | x ✈➭ D = {(x, y) ∈ R2 | y 0, y = 0}, , y > 0, x > 0} x ❚❛ ❝ã✿ ✰ C ✈➭ D ❦❤➳❝ rỗ C, D t ợ ì tồ t s ♣❤➻♥❣ (0, 1)(x, y) = t❤♦➯ ♠➲♥ (0, 1)(x, y) = (0, 1)(x , y ) ∀(x, y) ∈ C, ∀(x , y ) ∈ D ❍❛② y=0 ✰ ∀(x, y) ∈ C, ∀(x , y ) ∈ D y C, D t ợ ì Sup(x,y)C (0, 1)(x, y) = 0, inf (x ,y )∈D (0, 1)(x , y ) = ➜Þ♥❤ ❧ý ✶✳✸✳ ❈❤♦ C ✈➭ ✭➜Þ♥❤ ❧ý t➳❝❤ ✶✮✳ D ❧➭ ❤❛✐ t❐♣ rỗ tr ó ó ột s t C ✈➭ D✳ Rn s❛♦ ❝❤♦ C ∩ D = ∅✳ ❑❤✐ ✺✵ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ ①✃♣ ①Ø✱ t❛ ❝ã✿ x∗ ∈ ∂ (f1 + f2 )(x0 ) ⇔ x∗ , x − x0 + (f1 + f2 )(x0 ) (f1 + f2 )(x) + ⇔ x∗ , x − x0 + f1 (x0 ) + f2 (x0 ) ∀x f1 (x) + f2 (x) + ∀x ⇔f1 (x) + f2 (x) − f1 (x0 ) − f2 (x0 ) − x∗ , x − x0 + ∀x f1 (x) + f2 (y) − f1 (x0 ) − f2 (x0 ) − x∗ , x − x0 + < ⇒❤Ö x=y ❦❤➠♥❣ ❝ã ♥❣❤✐Ö♠ ✭✷✳✶✺✮ ▲✃② D = dom f1 × dom f2 , A(x, y) = x − y, f (x, y) = f1 (x) + f2 (y) − f1 (x0 ) − f2 (x0 ) − x∗ , x − x0 + ❚❤❡♦ ❣✐➯ t❤✐Õt f1 ❧✐➟♥ tơ❝ t➵✐ ♠ét ➤✐Ĩ♠ ♠ét ❧➞♥ ❝❐♥ a ∈ dom f1 ∩ dom f2 ✱ ♥➟♥ tå♥ t➵✐ U ❝ñ❛ ❣è❝ s❛♦ ❝❤♦ U = (a + U ) − a ⊂ dom f1 − dom f2 = A(D) ❱❐② ∈ intA(D) ▲ó❝ ♥➭② ✭✷✳✶✺✮ ❝ã ❞➵♥❣✿ ❤Ö   f (x, y) < A(x, y) =   (x, y) ∈ D ❦❤➠♥❣ ❝ã ♠❣❤✐Ư♠ ❆♣ ❞ơ♥❣ ♠Ư♥❤ ➤Ị ✶✳✹ t❛ ❝ã✿ t, A(x, y) − + f (x, y) ∀(x, y) ∈ D ⇔ t, x − y + f1 (x) + f2 (y) − f1 (x0 ) − f2 (x0 ) + x∗ , x − x0 + ∀x ∈ dom f1 , ∀y ∈ dom f2 ✳ 0, ✺✶ ➜è✐ ✈í✐ x ∈ dom f1 ✈➭ y ∈ dom f2 t❤× ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳ ❱❐② t, x − y + f1 (x) + f2 (y) − f1 (x0 ) − f2 (x0 ) + x∗ , x − x0 + ∀x, y x = x0 t❛ ❝ã ✿ ▲✃② t, x0 − y + f2 (y) − f2 (x0 ) + ⇔ t, y − x0 + f2 (x0 ) f2 (y) + ∀y ∀y ⇔ t ∈ ∂ f2 (x0 ) y = x0 t❛ ❝ã✿ ▲✃② t, x − x0 + f1 (x) − f1 (x0 ) − x∗ , x − x0 + ⇔ x∗ − t, x − x0 + f1 (x0 ) f1 (x) + ∀x ⇔ x∗ − t ∈ ∂ f1 (x0 ) ❉♦ ➤ã x∗ = (x∗ − t) + t ⊆ ∂ f1 (x0 ) + ∂ f2 (x0 ) ❱❐② ∂ (f1 (x0 ) + f2 (x0 )) ⊆ ∂ f1 (x0 ) + ∂ f2 (x0 ) ∀x ❈❤➢➡♥❣ ✸ ▼ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❞➢í✐ ✈✐ ♣❤➞♥ tr♦♥❣ tè✐ ➢✉ ❤♦➳ ❈❤➢➡♥❣ ♥➭② tr➢í❝ ❤Õt ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ❝❤✉♥❣ ✈Ị ❝ù❝ t✐Ĩ✉✱ ✲ ❝ù❝ t✐Ĩ✉ ❝đ❛ ♠ét ❤➭♠ ❧å✐✳ ❚✐Õ♣ t❤❡♦ tr×♥❤ ❜➭② ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ❝đ❛ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝đ❛ ❜➭✐ t♦➳♥ ❧å✐ ✈í✐ ❝➳❝ r➭♥❣ ❜✉é❝ ❦❤➳❝ ♥❤❛✉ ✭❑❤➠♥❣ r➭♥❣ ❜✉é❝✱ r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝✱ r➭♥❣ ❜✉é❝ ❜✃t ➤➻♥❣ t❤ø❝✮✳ ❈✉è✐ ❝❤➢➡♥❣ tr×♥❤ ❜➭② ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ❝đ❛ ♥❣❤✐Ư♠ tè✐ ➢✉ ①✃♣ ①Ø ❝đ❛ ❜➭✐ t♦➳♥ ❧å✐ ✈í✐ ❝➳❝ r➭♥❣ ❜✉é❝ ❦❤➳❝ ♥❤❛✉✳ ✸✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✶✳ ❛✮ ➜✐Ĩ♠ ♠ét ❧➞♥ ❝❐♥ ❈❤♦ C Rn rỗ f : Rn R ∪ {+∞}✳ x∗ ∈ C ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ù❝ t✐Ĩ✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ f tr➟♥ C ♥Õ✉ tå♥ t➵✐ U ❝đ❛ x∗ s❛♦ ❝❤♦ f (x∗ ) ❜✮ ➜✐Ĩ♠ ❝đ❛ f (x), ∀x ∈ U ∩ C x∗ ∈ C ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ù❝ t✐Ĩ✉ t♦➭♥ ❝ơ❝ ✭❤❛② ❝ù❝ t✐Ĩ✉ t✉②Öt ➤è✐ ✮ f tr➟♥ C ♥Õ✉ f (x∗ ) ❝✮ ➜✐Ó♠ f (x), ∀x ∈ C x ∈ C ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳ t♦➭♥ ❝ơ❝ ❝đ❛ ❈❤♦ > 0✳ ▼ét ➤✐Ĩ♠ x ∈ C ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ✲❝ù❝ t✐Ĩ✉ f tr➟♥ C ♥Õ✉ f (x ) f (x) + , ∀x ∈ C ✺✷ ✺✸ ✸✳✷ ❇➭✐ t♦➳♥ ❧å✐ ❦❤➠♥❣ ❝ã r➭♥❣ ❜✉é❝ ❳Ðt ❜➭✐ t♦➳♥ {min h(x) | x ∈ Rn } (P 1) ❚r♦♥❣ ➤ã h ❧➭ ♠ét ❤➭♠ ❧å✐ ❝❤Ý♥❤ t❤➢ê♥❣ tr➟♥ Rn ✳ ▼Ư♥❤ ➤Ị ✸✳✶✳ ❈❤ø♥❣ ♠✐♥❤✳ x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✶✮ ⇔ ∈ ∂h(x∗ )✳ ❚❛ ❝ã✿ x∗ ❧➭ ♥❣❤✐Ö♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✶✮ ⇔ x∗ ❧➭ ➤✐Ĩ♠ ❝ù❝ t✐Ĩ✉ ❝ñ❛ ❤ tr➟♥ Rn ⇔ h(x∗ ) h(x) , ∀x ∈ Rn ⇔ 0, x − x∗ + h(x∗ ) h(x) , ∀x ∈ Rn ⇔ ∈ ∂h(x∗ ) ✸✳✸ ❇➭✐ t♦➳♥ ❧å✐ ✈í✐ r➭♥❣ ❜✉é❝ ➤➻♥❣ t❤ø❝ ❳Ðt ❜➭✐ t♦➳♥ {min f (x) | x ∈ C} (P 2) ❚r♦♥❣ ➤ã C ⊆ Rn ❧➭ ♠ét t❐♣ ❧å✐ rỗ f ột tr C ✳ ▼Ư♥❤ ➤Ị ✸✳✷✳ x∗ ∈ C tr♦♥❣ ➤ã ❝đ❛ C ●✐➯ sö ri(dom f ) ∩ ri C = ∅✳ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✷✮⇔ NC (x∗ ) := {ω | ω, x − x∗ t➵✐ ∈ ∂f (x∗ ) + NC (x∗ )✱ , ∀x ∈ C} ❧➭ ♥ã♥ ♣❤➳♣ t✉②Õ♥ ♥❣♦➭✐ x∗ ✳ ❈❤ø♥❣ ♠✐♥❤✳ ●ä✐ δC (.) ❧➭ ❤➭♠ ❝❤Ø ❝ñ❛ t❐♣ δC (x) := C ✱ tø❝ ❧➭ ♥Õ✉ x ∈ C, +∞ ♥Õ✉ x ∈ C ✺✹ ❑❤✐ ➤ã x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✷✮ ⇔x∗ ❧➭ ➤✐Ĩ♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ ❢ tr➟♥ ❈ ⇔x∗ ❧➭ ➤✐Ĩ♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ h(x) := f (x) + δC (x) tr➟♥ Rn ⇔0 ∈ ∂h(x∗ ) ❉♦ ✭t❤❡♦ ♠Ư♥❤ ➤Ị ✸✳✶✮ ri(dom f ) ∩ ri C = ∅✱ t❤❡♦ ➤Þ♥❤ ❧ý ▼♦r❡❛✉✲❘♦❝❦❛❢❡❧❧❛r t❛ ❝ã✿ ∂h(x∗ ) = ∂[f (x∗ ) + δC (x∗ )] = ∂f (x∗ ) + ∂δC (x∗ ) ❱× x∗ ∈ C ♥➟♥ ∂δC (x∗ ) = NC (x∗ )✳ ❱❐② ∂h(x∗ ) = ∂f (x∗ ) + NC (x∗ )✳ ❙✉② r❛ ✸✳✹ ∈ ∂f (x∗ ) + NC (x∗ )✳ ❇➭✐ t♦➳♥ ❧å✐ ✈í✐ r➭♥❣ ❜✉é❝ ❜✃t ➤➻♥❣ t❤ø❝ ❳Ðt ❜➭✐ t♦➳♥ t×♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ ♠ét ❤➭♠ ❧å✐ tr➟♥ ♠ét t❐♣ ❧å✐ ❝ã ❞➵♥❣ s❛✉✿ {min f (x) | gi (x) (OP ) ❚r♦♥❣ ➤ã (i = 1, m), x ∈ X} X ⊆ Rn ❧➭ ♠ét t❐♣ ❧å✐ ➤ã♥❣ ❦❤➳❝ rỗ f, gi ữ ❤➵♥ tr➟♥ X ✳ ❚❛ sÏ ❧✉➠♥ ❣✐➯ sñ r➺♥❣ X ❝ã ➤✐Ĩ♠ tr♦♥❣✳ ❇➭✐ t♦➳♥ ✭❖P✮ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét q✉② ❤♦➵❝❤ ❧å✐✳ ❍➭♠ f ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ♠ơ❝ t✐➟✉✳ ❈➳❝ ➤✐Ị✉ ❦✐Ư♥ ❚❐♣ x ∈ X, gi (x) D := {x ∈ X | gi (x) (i = 1, m) ➤➢ỵ❝ ❣ä✐ ❧➭ ❝➳❝ r➭♥❣ ❜✉é❝✳ i = 1, m} ➤➢ỵ❝ ❣ä✐ ❧➭ ♠✐Ị♥ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ▼ét ➤✐Ĩ♠ ❉♦ x ∈ D ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮✳ X ❧➭ t❐♣ ❧å✐✱ ❝➳❝ ❤➭♠ gi ✭✐❂✶✱✳✳✱♠✮ ❧å✐ tr➟♥ X ♥➟♥ D ❧➭ ♠ét t❐♣ ❧å✐✳ ➜✐Ó♠ ❝ù❝ t✐Ó✉ ❝đ❛ ✭❖P✮✳ f tr➟♥ D ❝ị♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝ñ❛ ❜➭✐ t♦➳♥ ✺✺ ❚❛ ①➞② ❞ù♥❣ ❤➭♠ s❛✉✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ▲❛❣r❛♥❣❡✱ ❝❤♦ ❜➭✐ t♦➳♥ ✭❖P✮✿ m L(x, λ) := λ0 f (x) + λi gi (x), i=1 ✈í✐ λ = (λ0 , , λm ) ❉ù❛ ✈➭♦ ❤➭♠ ▲❛❣r❛♥❣❡ t❛ ❝ã ❦Õt qñ❛ s❛✉✿ ✭❑❛r✉s❤✲ ❑✉❤♥✲ ❚✉❝❦❡r✮ ➜Þ♥❤ ❧ý ✸✳✶✳ ●✐➯ sư ❛✮ ◆Õ✉ ri(dom f ) ∩ ri(dom gi ) ∩ ri X = ∅ x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮ t❤× tå♥ t➵✐ λ∗i ✭✐❂✵✱✳✳✳✱♠✮ ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵ s❛♦ ❝❤♦✿ ✶✮ L(x∗ , λ∗ ) = minx∈X L(x, λ∗ ) ✭➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Ưt t✐➟✉ ✮ m (⇔ ∈ λ∗0 ∂f (x∗ ) λ∗i ∂gi (x∗ ) + NX (x∗ ) ) + i=1 NX (x∗ ) ❧➭ ♥ã♥ ♣❤➳♣ t✉②Õ♥ ♥❣♦➭✐ ❝ñ❛ ❳ t➵✐ x∗ ❚r♦♥❣ ➤ã ✳ ✷✮ λ∗i gi (x∗ ) = (i = 1, , m) ✭➤✐Ị✉ ❦✐Ư♥ ➤é ❧Ư❝❤ ❜ï✮ ❍➡♥ ♥÷❛ ♥Õ✉ ➤✐Ị✉ ❦✐Ư♥ ❙❧❛t❡r s❛✉ t❤♦➯ ♠➲♥✿ ∃x0 ∈ X : gi (x0 ) < (i = 1, m) t❤× λ∗0 > ❜✮ ◆Õ✉ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Öt t✐➟✉ ✈➭ ➤é ❧Ö❝❤ ❜ï ë tr➟♥ ợ t > tì ể ♥❤❐♥ ➤➢ỵ❝ x∗ ❧➭ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮ ❈❤ø♥❣ ♠✐♥❤✳ ❛✮ ●✐➯ sư x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮✳ ➜➷t C :={(λ0 , λ1 , , λm ) ∈ Rm+1 |∃x ∈ X : f (x) − f (x∗ ) < λ0 , gi (x) ❉♦ λi , i = 1, , m} X = ∅ ❧å✐✱ f, gi ❧å✐ tr➟♥ X ✱ ♥➟♥ C ❧➭ ♠ét t❐♣ ❧å✐ ✳ ✺✻ ❚❛ ❝ã C = ∅✳ ❚❤❐t ✈❐②✿ ✰ ▲✃② m+1 (λ0 , , λm ) ∈ intR+ ✳ ❑❤✐ ➤ã λi > (i = 1, , m)✳ ✰ ❱í✐ x = x∗ ✱ t❛ ❝ã f (x∗ ) − f (x∗ ) = < λ0 gi (x∗ ) < λi (i = 1, , m) ⇒ (λ0 , , λm ) ∈ C m+1 ⇒ intR+ ⊂ C ⇒ C = ∅ tr♦♥❣ Rm+1 ❍➡♥ ♥÷❛ ∈ C ✳ ❚❤❐t ✈❐②✱ ♥Õ✉ ∈ C t❤× ∃x ∈ X : f (x) − f (x∗ ) < gi (x) (i = 1, , m) x∗ ❦❤➠♥❣ ❧➭ ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ✭❖P✮⇒ ♠➞✉ t❤✉➱♥✳ ❱❐② ∈ C ✳ ❉♦ ➤ã ❚❤❡♦ ➤Þ♥❤ ❧ý t➳❝❤ t❤ø ✶✱ ❝ã t❤Ó t➳❝❤ ❝➳❝ t❐♣ ❈ ✈➭ {0}✱ tø❝ ❧➭ ∃λ∗i ✭✐❂✵✱✳✳✳♠✮ ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵ s❛♦ ❝❤♦ m λ∗i λi ✭✸✳✶✮ ∀(λ0 , , λm ) ∈ C i=0 ❉♦ m+1 intR+ ⊂ C ✱ t❛ s✉② r❛ λ∗i ❱í✐ 0✳ > ✈➭ x ∈ X ✱ ❧✃② λ0 = f (x) − f (x∗ ) + λi = gi (x) (i = 1, , m) ❚❤❛② ✈➭♦ ✭✸✳✶✮ t❛ ❝ã m λ∗0 [f (x) ∗ λ∗i gi (x) − f (x ) + ] + ∀x ∈ X i=1 ❈❤♦ → t❛ ➤➢ỵ❝ m λ∗0 f (x) λ∗i gi (x) + i=1 λ∗0 f (x∗ ) ∀x ∈ X ✭✸✳✷✮ ✺✼ ❉♦ x∗ ❧➭ ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ♥➟♥ t❛ ❝ã gi (x∗ ) (i = 1, , m)✳ ❱❐② m λ∗0 f (x∗ ) λ∗0 f (x∗ ) λ∗i gi (x∗ ) + ✭✸✳✸✮ i=1 ❚õ ✭✸✳✷✮ ✈➭ ✭✸✳✸✮ t❛ ❝ã m λ∗0 f (x) m λ∗i gi (x) + λ∗0 f (x∗ ) i=1 ⇔L(x, λ∗ ) ∀x ∈ X ⇔L(x∗ , λ∗ ) = minx∈X L(x, λ∗ ) ❝❤Ø ❦❤✐ ∀x ∈ X i=1 L(x∗ , λ∗ ) ❚❛ ❝❤ó ý r➺♥❣ λ∗i gi (x∗ ) + ✭➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Ưt t✐➟✉✮ x∗ ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ {min L(x, λ∗ ), x ∈ X} ❦❤✐ ✈➭ x∗ ❧➭ ➤✐Ĩ♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ ❤➭♠ L(x, λ∗ ) tr➟♥ ❳ ⇔x∗ ❧➭ ➤✐Ĩ♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ ❤➭♠ L1 (x, λ∗ ) := L(x, λ∗ ) + δX (x) tr➟♥ Rn ⇔0 ∈ ∂L1 (x∗ , λ∗ ) ❉♦ ✭t❤❡♦ ♠Ư♥❤ ➤Ị ✸✳✶✮ ri(dom f ) ∩ ri(dom gi ) ∩ ri X = ∅ ✈➭ f, gi ✭✐✿❂✶✱✳✳✳✱♠✮ ❧➭ ữ tr t ị ❧ý ▼♦r❡❛✉✲❘♦❝❦❛❢❡❧❧❛r t❛ ❝ã ∂L1 (x∗ , λ∗ ) = ∂[L(x∗ , λ∗ ) + δX (x∗ )] m = ∂[λ∗0 f (x∗ ) λ∗i gi (x∗ )] + ∂δX (x∗ ) + i=1 m = λ∗0 ∂f (x∗ ) + λ∗i ∂gi (x∗ ) + NX (x∗ ) i=1 ✭✈× ❱❐② ∂δX (x∗ ) = NX (x∗ ) ✮ m 0∈ λ∗0 ∂f (x∗ ) λ∗i ∂gi (x∗ ) + NX (x∗ ) + i=1 ❉♦ x∗ ❧➭ ➤✐Ó♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ♥➟♥ gi (x∗ ) (i = 1, , m)✳ ◆Õ✉ ∃i ∈ {1, , m} : gi (x∗ ) = ξ < t❤× ∀ > , f (x∗ ) − f (x∗ ) = < gj (x∗ ) < (j = 1, , i − 1, i + 1, , m) ✺✽ ✭ξ ë ✈Þ trÝ t❤ø ✐✮✳ ⇒ ( , , , ξ, , , ) ∈ C ⇒ λ∗i ξ ✭ t❤❛② ✈➭♦ ✭✸✳✶✮ ✈➭ ❝❤♦ → 0✮ ⇒ λ∗i 0✳ ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ❝ã ◆❤➢ ✈❐② ❧➭✱ ♥Õ✉ gi (x∗ ) ❉♦ ➤ã λ∗i 0✳ ❱❐② λ∗i = 0✳ < t❤× λ∗i = 0✳ λ∗i gi (x∗ ) = (i = 1, , m) ✭➤✐Ị✉ ❦✐Ư♥ ➤é ❧Ư❝❤ ❜ï ✮✳ ●✐➯ sư ➤✐Ị✉ ❦✐Ư♥ ❙❧❛t❡r ➤➢ỵ❝ t❤♦➯ ♠➲♥✿ ❑❤✐ ➤ã ♥Õ✉ ∃x0 ∈ X : gi (x0 ) < 0✳ λ∗0 = t❤× ❞♦ ➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Ưt t✐➟✉ ✈➭ ➤é ❧Ö❝❤ ❜ï t❛ ❝ã m 0= λ∗0 f (x∗ ) m λ∗i gi (x∗ ) + λ∗0 f (x) λ∗i gi (x) , ∀x ∈ X + i=1 ❉♦ i=1 λ∗0 = ♥➟♥ ♣❤➯✐ ❝ã Ýt ♥❤✃t ♠ét λ∗i > 0✳ ❚❤❛② x0 ✈➭♦ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ sÏ ➤➢ỵ❝ m m 0= λ∗0 f (x∗ ) λ∗i gi (x∗ ) + λ∗0 f (x0 ) i=1 i=1 ❙✉② r❛ ♠➞✉ t❤✉➱♥✳ ❱❐② λ∗i gi (x0 ) < + λ∗0 = tø❝ ❧➭ λ∗0 > 0✳ ❜✮ ●✐➯ sư x∗ ❧➭ ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ t❤♦➯ ♠➲♥ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Ưt t✐➟✉ ✈➭ ➤é ❧Ư❝❤ ❜ï ë tr➟♥ ✈í✐ ❉♦ λ∗0 > 0, λ∗i (i = 1, , m)✳ λ∗0 > 0✱ ♥➟♥ ❜➺♥❣ ❝➳❝❤ ❝❤✐❛ ❝❤♦ λ∗0 ✱ t❛ ❝ã t❤Ó ❝♦✐ ❤➭♠ ▲❛❣r❛♥❣❡ ❧➭ m L(x, λ) = f (x) + λi gi (x) i=1 ❚õ ➤✐Ị✉ ❦✐Ư♥ ➤➵♦ ❤➭♠ tr✐Ưt t✐➟✉ ✈➭ ➤é ❧Ö❝❤ ❜ï✱ t❛ ❝ã✿ m ∗ m λ∗i gi (x∗ ) f (x ) + λ∗i gi (x) f (x) + i=1 ∀x ∈ X i=1 λ∗i gi (x∗ ) = (i = 1, , m) ❙✉② r❛ m ∗ f (x ) λ∗i gi (x) , ∀x ∈ X f (x) + i=1 ✭✸✳✹✮ ✺✾ ❱í✐ ♠ä✐ ① ❧➭ ➤✐Ĩ♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ tø❝ ❧➭✿ x ∈ X : gi (x) < , i = 1, , m, t❛ ❝ã m λ∗i gi (x) f (x) + ✭✸✳✺✮ f (x) i=1 ❚õ ✭✸✳✹✮ ✈➭ ✭✸✳✺✮ s✉② r❛ f (x∗ ) f (x) , ∀x ∈ X ✳❈❤ø♥❣ tá x∗ ❧➭ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮✳ ❱Ý ❞ơ ✸✳✶✳ ❆♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ❝❤♦ ❜➭✐ t♦➳♥ s❛✉✿ {min f (x) | gi (x) tr♦♥❣ ➤ã (i = 1, 2) , x ∈ X}, (OP ) f (x) = x2 , g1 (x) = x2 − x, g2 (x) = −x, X = [− 12 , 21 ]✳ ●✐➯✐✿ ❚❛ ❝ã ♠✐Ò♥ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ D = {x ∈ X | gi (x) ●✐➯ sö tå♥ t➵✐ ✶✮ λ∗i (i = 1, 2)} = [0, ] (i = 0, , 2) ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵ s❛♦ ❝❤♦✿ L(x∗ , λ∗ ) = minx∈X L(x, λ∗ ) (⇔ ∈ λ∗0 ∂f (x∗ ) + ∗ ∗ i=1 λi ∂gi (x ) ✷✮ λ∗i gi (x∗ ) = 0, i = 1, ✸✮ λ∗0 > ❚õ ➤Þ♥❤ ❧Ý ✸✳✶✱ s✉② r❛ + NX (x∗ ) ) x∗ ❧➭ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮ ⇔f (x∗ ) f (x), ∀x ∈ D ⇔x∗ x2 , ∀x ∈ D ⇔x∗ ⇔x∗ = ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ tå♥ t➵✐ λ∗i x∗ = ❧➭ ♥❣❤✐Ö♠ ủ t P tì từ ị í s r❛ (i = 0, , 2) ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵ s❛♦ ❝❤♦ ✿ ✻✵ ✶✮ L(x∗ , λ∗ ) = minx∈X L(x, λ∗ ) ∗ ∗ i=1 λi ∂gi (x ) (⇔ ∈ λ∗0 ∂f (x∗ ) + ✷✮ + NX (x∗ ) )✳ λ∗i gi (x∗ ) = 0, i = 1, 2✳ ❚❛ ❝ã L(x∗ , λ∗ ) = minx∈X L(x, λ∗ ) ⇔L(x, λ∗ ) L(x∗ , λ∗ ), ∀x ∈ X ⇔λ∗0 f (x) + ⇔λ∗0 x2 i=1 ∗ λ1 (x ❚❛ ❝ã ❉♦ λ∗i ✰ ❈❤ä♥ + λ∗i gi (x) λ∗0 f (x∗ ) λ∗i gi (x∗ ), ∀x ∈ X + i=1 − x) − λ∗2 x 0, ∀x ∈ X λ∗i gi (x∗ ) = 0, i = 1, ⇔ λ∗i = 0, i = 1, ⇔ λ∗i (i = 0, , 2) ❦❤➠♥❣ ➤å♥❣ t❤ê✐ ❜➺♥❣ ✵ ♥➟♥✿ λ∗1 = λ∗2 = 0✳ ❚❛ ❝ã λ∗0 x2 + λ∗1 (x2 − x) − λ∗2 x ⇔λ∗0 x2 0, ∀x ∈ X 0, ∀x ∈ X ⇔λ∗0 > ⇒❈❤ä♥λ∗0 = ✰ ❈❤ä♥ λ∗1 = λ∗2 = 1✳ ❚❛ ❝ã λ∗0 x2 + λ∗1 (x2 − x) − λ∗2 x ⇔(λ∗0 + 1)x2 − 2x 0, ∀x ∈ X 0, ∀x ∈ X ⇒❑❤➠♥❣ tå♥ t➵✐ λ∗0 ✰ ❈❤ä♥ λ∗1 = 0, λ∗2 = 1✳ ❚❛ ❝ã λ∗0 x2 + λ∗1 (x2 − x) − λ∗2 x ⇔λ∗0 x2 − x 0, ∀x ∈ X ⇒❑❤➠♥❣ tå♥ t➵✐ λ∗0 0, ∀x ∈ X 0, i = 1, 2✳ ✻✶ ✰ ❈❤ä♥ λ∗1 = 1, λ∗2 = 0✳ ❚❛ ❝ã λ∗0 x2 + λ∗1 (x2 − x) − λ∗2 x ⇔(λ∗0 + 1)x2 − x 0, ∀x ∈ X 0, ∀x ∈ X ⇒❑❤➠♥❣ tå♥ t➵✐ λ∗0 ❱❐② x∗ = ❧➭ ♥❣❤✐Ư♠ tè✐ ➢✉ ❝đ❛ ❜➭✐ t♦➳♥ ✭❖P✮ ✈➭ λ∗0 = 1, λ∗1 = λ∗2 = ❧➭ ❝➳❝ ♥❤➞♥ tư ▲❛❣r❛♥❣ t➢➡♥❣ ø♥❣✳ ❈❤ó ý ✸✳✶✳ ❚r♦♥❣ ♥❤✐Ị✉ tr➢ê♥❣ ❤ỵ♣ ❜➭✐ t♦➳♥ ✭P✶✮✱ ✭P✷✮ ✈➭ ✭❖P✮ ❝ã t❤Ó ❦❤➠♥❣ ❝ã ❧ê✐ ❣✐➯✐ tè✐ ➢✉ ❝❤Ý♥❤ ①➳❝✳ ❍➡♥ ữ tr tự tế tờ t tí ợ ❧ê✐ ❣✐➯✐ tè✐ ➢✉ ✭❝❤Ý♥❤ ①➳❝✮✱ ♠➭ ❝❤Ø tÝ♥❤ ➤➢ỵ❝ ❧ê✐ ❣✐➯✐ ①✃♣ ①Ø✳ ❑❤✐ ➤ã t❛ ❞ï♥❣ ❦❤➳✐ ♥✐Ö♠ ❧ê✐ ❣✐➯✐ tè✐ ➢✉ ①✃♣ ①Ø ❤❛② ❝ß♥ ❣ä✐ ❧➭ ✲ tè✐ ➢✉✳ ▼Ư♥❤ ➤Ị ✸✳✸✳ x ❧➭ ✲♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✶✮ ⇔ ∈ ∂ h(x ) ❚❛ ❝ã✿ ❈❤ø♥❣ ♠✐♥❤✳ x ❧➭ − ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✶✮ ⇔x ❧➭ ➤✐Ĩ♠ − ❝ù❝ t✐Ĩ✉ ❝đ❛ ❤ tr➟♥ Rn h(x) + , ∀x ∈ Rn ⇔h(x ) ⇔0 ∈ ∂ h(x ) ▼Ư♥❤ ➤Ị ✸✳✹✳ x ∈C tr♦♥❣ ➤ã ❧➭ ●✐➯ sư ✭t❤❡♦ ➤Þ♥❤ ❧ý✷✳✶✮ ri(dom f ) ∩ ri C = ∅✳ ❑❤✐ ➤ã ✲♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✷✮ NC, (x ) := {ω | ω, x − x ♥❣♦➭✐ ❝ñ❛ ❈ t➵✐ ❈❤ø♥❣ ♠✐♥❤✳ x ✭t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛✸✳✷✮ =⇒ ∈ ∂ f (x ) + NC, (x )✱ , ∀x ∈ C} ✳ ●ä✐ δC (.) ❧➭ ❤➭♠ ❝❤Ø ❝ñ❛ t❐♣ ❈ ✱ tø❝ ❧➭ δC (x) := ♥Õ✉ x ∈ C, +∞ ♥Õ✉ x ∈ C ❧➭ ✲♥ã♥ ♣❤➳♣ t✉②Õ♥ ✻✷ ❑❤✐ ➤ã x ❧➭ − ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ✭P✷✮ ⇔x ❧➭ ➤✐Ĩ♠ − ❝ù❝ t✐Ĩ✉ ❝đ❛ ❢ tr➟♥ ❈ ⇔x ❧➭ ➤✐Ĩ♠ − ❝ù❝ t✐Ĩ✉ ❝đ❛ h(x) := f (x) + δC (x) tr➟♥ Rn ⇔0 ∈ ∂ h(x ) ❉♦ ✭t❤❡♦ ♠Ư♥❤ ➤Ị ✸✳✸✮ ri(dom f ) ∩ ri C = ∅✱ t❛ ❝ã✿ ∂ h(x ) = ∂ [f (x ) + δC (x )] ⊆ ∂ f (x ) + ∂ δC (x ) ✭t❤❡♦ ♠Ư♥❤ ➤Ị ✷✳✶✹✮ = ∂ f (x ) + NC, (x ) ✭❱× ❱❐② x ∈ C ♥➟♥ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ✷✳✾ ∂ δC (x ) = NC, (x ) ✮ ∈ ∂ f (x ) + NC, (x )✳ ✻✸ ❑Õt ❧✉❐♥ ◆❤➢ ✈❐②✱ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ t❐♣ ❧å✐ ✈➭ ❤➭♠ ❧å✐✳ ❙❛✉ ➤ã ❧➵✐ ➤Ị ❝❐♣ ✈Ị ➤➵♦ ❤➭♠ t❤❡♦ ♣❤➢➡♥❣✱ ❞➢í✐ ✈✐ ♣❤➞♥✱ ❞➢í✐ ✈✐ ♣❤➞♥ ①✃♣ ①Ø ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❝➳❝❤ ❝ô t❤Ĩ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ❈✉è✐ ❝ï♥❣ ❧✉❐♥ trì ề ệ ự trị ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ ❧å✐ ✈í✐ ❝➳❝ r➺♥❣ ❜✉é❝ ❦❤➳❝ ♥❤❛✉✳ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▲➟ ❉ị♥❣ ▼➢✉ ✈➭ ◆❣✉②Ơ♥ ❱➝♥ ❍✐Ị♥ ✭✷✵✵✸✮✱ ◆❤❐♣ ♠➠♥ ❣✐➯✐ tÝ❝❤ ❧å✐ ø♥❣ ❞ơ♥❣✱ ●✐➳♦ tr×♥❤✳ ❬✷❪ ✳ ❚➵ ◗✉❛♥❣ ❙➡♥ ✭✷✵✵✽✮✱ ❙♦♠❡ ◗✉❛❧✐t❛t✐✈❡ Pr♦❜❧❡♠s ■♥ ❖♣t✐♠✐③❛t✐♦♥✱ ▲✉❐♥ ➳♥ t✐Õ♥ sÜ✳ ❬✸❪ ❚✳ ❘♦❝❦❛❢❡❧❧❛r ✭✶✾✼✵✮✱ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡②✳ ❬✹❪ ❏✳ ❍✐r✐❛rt✲❯rr✉t② ❛♥❞ ❈✳ ▲❡♠❛r❡❝❤❛❧✱ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ▼✐♥✐♠✐③❛t✐♦♥ ❆❧❣♦r✐t❤♠s✳ ✻✹ ... }α∈I ❧➭ ♠ét ❤ä t✉ú ý ❝➳❝ ❤➭♠ sè tr➟♥ Rn ✈➭ E ⊆ Rn ✳ ❍➭♠ ❝❐♥ tr➟♥ ❝ñ❛ ❤ä ❤➭♠ ♥➭② tr➟♥ coE ✱ ý ệ VI f số ợ ị ♥❣❤Ü❛ ♥❤➢ s❛✉✿ (Vα∈I fα )(x) := Supα∈I fα (x) ỗ x coE ệ ề ●✐➯ sö {fα }α∈I ❧➭ ♠ét... ♥❣❤Ü❛ ✶✳✶✻✳ ❈❤♦ f : Rn −→ R ∪ {+∞} ✭❦❤➠♥❣ ♥❤✃t t❤✐Õt ❧å✐✮✱ C ⊆ Rn ❧➭ ♠ét t❐♣ ❧å✐ ❦❤➳❝ rỗ ột số tự ó η ❧➭ ❤Ư sè ❧å✐ ❝đ❛ f tr➟♥ C ✱ ♥Õ✉ ✈í✐ ♠ä✐ λ ∈ (0, 1)✱ ✈í✐ ♠ä✐ x, y ∈ C ✱ t❛ ❝ã✿ f [(1