Phương pháp hiệu chỉnh lặp giải hệ phương trình toán tử đơn điệu

❇é ❣✐➳♦ ❞ô❝ ✈➭ ➤➭♦ t➵♦ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ❚❤đ② P❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣ ❣✐➯✐ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ ➜Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ❝✃♣ ❜é ▼➲ sè✿ ❇✷✵✵✾✲❚◆✵✼✲✵✸ ❚❤➳✐ ◆❣✉②➟♥ ✲ ✷✵✶✵ ✶ ❉❛♥❤ s➳❝❤ ♥❤÷♥❣ ♥❣➢ê✐ t❤❛♠ ❣✐❛ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ❙❚❚ ❍ä ✈➭ t➟♥ ◆é✐ ❞✉♥❣ ➤➲ t❤ù❝ ❤✐Ư♥ ✶ ◆❣✉②Ơ♥ ❚❤❛♥❤ ▼❛✐ ❚ỉ ❝❤ø❝ ❙❡♠✐♥❛r✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ị ✷ ◆❣✉②Ơ♥ ❚✃t ❚❤➽♥❣ ❉Þ❝❤ t➭✐ ❧✐Ư✉✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ị ✸ ❚r➢➡♥❣ ▼✐♥❤ ❚✉②➟♥ ❚ỉ ❝❤ø❝ ❙❡♠✐♥❛r✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ị ✹ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❍➢ê♥❣ ự ề ị ố ợ í ị ộ ố ợ ứ ❱✐Ö♥ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥ ❚❤➯♦ ❧✉❐♥✱ ❙❡♠✐♥❛r ❱✐Õt ❝❤✉♥❣ ❜➭✐ ❜➳♦ ❦❤♦❛ ❤ä❝ ✷ ❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱ ❚r❛♦ ➤æ✐✱ ❚❤➯♦ ❧✉❐♥ ❚r➢ê♥❣ ➜❍❑❍❚◆ ❍➭ ◆é✐ ✸ ❑❤♦❛ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ✷ ❚r❛♦ ➤ỉ✐✱ ❚❤➯♦ ❧✉❐♥ ▼ơ❝ ❧ơ❝ ▼ë ➤➬✉ ✶✶ ❈❤➢➡♥❣ ✶✳ ✶✳✶✳ ✶✳✷✳ ❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤ ✈➭ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ✶✺ ❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✶✳✶✳ ❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✶✳✷✳ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ❍Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✷✳✶✳ P❤➳t ❜✐Ó✉ ❜➭✐ t♦➳♥ ✶✳✷✳✷✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ö♠ ✶✳✷✳✸✳ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❈❤➢➡♥❣ ✷✳ ✷✳✶✳ ❍✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈í✐ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ✈➭ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣ ❜❐❝ ❦❤➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✶✳✶✳ P❤➢➡♥❣ tr×♥❤ ❤✐Ư✉ ❝❤Ø♥❤ ✷✳✶✳✷✳ ❙ù ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ✷✳✶✳✸✳ ❚❤❛♠ sè ❤✐Ư✉ ❝❤Ø♥❤ ✷✳✶✳✹✳ ❚è❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ✷✳✷✳ ✷✽ ✷✳✷✳✶✳ ▼➠ t➯ ♣❤➢➡♥❣ ♣❤➳♣ ✷✳✷✳✷✳ ❙ù ❤é✐ tô ❈❤➢➡♥❣ ✸✳ ❑Õt q✉➯ tÝ♥❤ t♦➳♥ t❤ư ♥❣❤✐Ư♠ ✸ ✹✼ ✸✳✶✳ ❱Ý ❞ô ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷✳ ❱Ý ❞ô ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ❑Õt ❧✉❐♥ ❝❤✉♥❣ ✺✹ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✺✺ ✹ ❚ã♠ t➽t ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ ✶✳ ❚❤➠♥❣ t✐♥ ❝❤✉♥❣ ✲ ❚➟♥ ➤Ị t➭✐✿ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣ ❣✐➯✐ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ ✲ ▼➲ sè✿ ❇✷✵✵✾✲❚◆✵✼✲✵✸ ✲ ❚❤ê✐ ❣✐❛♥ t❤ù❝ ❤✐Ö♥✿ ✷✵✵✾✲✷✵✶✵ ✲ ❈❤đ ♥❤✐Ư♠ ➤Ị t➭✐✿ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ❚❤đ② ➜✐Ư♥ t❤♦➵✐✿ ✵✾✶✷✷✶✶✽✺✽❀ ❊✲♠❛✐❧✿ t❤✉t❤✉②✷✷✵✸✻✾❅❣♠❛✐❧✳❝♦♠ ✲ ❈➡ q✉❛♥ ❝❤đ tr×✿ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❑❤♦❛ ❤ä❝✱ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ✷✳ ▼ơ❝ t✐➟✉ ❝đ❛ ➤Ị t➭✐ ✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ổ ị ệ trì t tử ➤➡♥ ➤✐Ö✉✳ ✲ ◆➞♥❣ ❝❛♦ ♥➝♥❣ ❧ù❝ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ♥❤ã♠ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✳ ✲ P❤ơ❝ ✈ơ ❝❤♦ ❝➠♥❣ t➳❝ ◆❈❑❍✱ ➤➭♦ t➵♦ ➜❍ ✈➭ ❙➜❍ ❝❤✉②➟♥ ♥❣➭♥❤ ❚♦➳♥ ø♥❣ ❞ơ♥❣ ❝đ❛ ➜➵✐ ❤ä❝✳ ✸✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ✲ ◆❣❤✐➟♥ ❝ø✉ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉❀ ✲ ◆❣❤✐➟♥ ❝ø✉ sù ❤é✐ tơ ✈➭ ➤➳♥❤ ❣✐➳ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ tr➟♥ ❝➡ së ❝❤ä♥ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤❀ ✹✳ ❑Õt q✉➯ ❝❤Ý♥❤ ➤➵t ➤➢ỵ❝ ✲ ❳➞② ❞ù♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ ❞ù❛ tr➟♥ ♠ét ➤Ị ①✉✃t ❝đ❛ ◆❣✉②Ơ♥ ❇➢ê♥❣❀ ✲ ➜➢❛ r❛ ❝➳❝❤ ❝❤ä♥ ❣✐➳ trÞ ❝đ❛ t❤❛♠ sè ❤✐Ư✉ ❝❤Ø♥❤ ❤❐✉ ♥❣❤✐Ư♠ t❤❡♦ ♥❣✉②➟♥ ❧Ý ➤é ❧Ö❝❤ s✉② ré♥❣❀ ➜➳♥❤ ❣✐➳ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ø♥❣ ✈í✐ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤ ➤➲ ❝❤ä♥❀ ✺ ✲ ❳➞② ❞ù♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣✱ ❝❤ø♥❣ ♠✐♥❤ sù ❤é✐ tơ ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣❀ ✲ ➜➢❛ r❛ ✈Ý ❞ô sè ♠✐♥❤ ❤ä❛ ❝❤♦ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉✳ ✺✳ ❙➯♥ ♣❤➮♠ ❝đ❛ ➤Ị t➭✐ ✺✳✶✳ ❙➯♥ ♣❤➮♠ ❦❤♦❛ ❤ä❝ • ❈➳❝ ❦Õt q✉➯ ủ ề t ợ ố tr trì ❬✶❪✳ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ▲✳ ❚ ❉✉♦♥❣ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥✲s❡❧❢ str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ✹✾✭✶✮✱ ❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ♣♣✳ ✷✼✲✸✶✳ ❬✷❪✳ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ❚✳ ▼✳ ❚✉②❡♥ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ✹✾✭✶✮✱ ❚➵♣ ♣♣✳ ✸✷✲✸✻✳ ❬✸❪✳ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ t♦ ❛ ❝♦♠♠♦♥ s♦❧✉t✐♦♥ ♦❢ ✐♥✈❡rs❡✲str♦♥❣❧② ♣r♦❜❧❡♠s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ✸✱ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ♣♣✳ ✶✻✺✲✶✼✹✳ ❬✹❪✳ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❈♦♥✈❡r❣❡♥❝❡ r❛t❡s ♦❢ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉✲ ❧❛r✐③❛t✐♦♥ ❢♦r ✐❧❧✲♣♦s❡❞ ♠✐①❡❞ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ✐♥✈❡rs❡✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ♣❡rt✉r❜❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✭➜➲ ♥❤❐♥ ➤➝♥❣ ♥➝♠ ✷✵✶✵✮✳ ❬✺❪✳ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ❚❤đ②✱ ➜➷♥❣ ❚ó ❍å✐ ✭✷✵✶✵✮✱ ✧❑Õt q✉➯ sè ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✧✱ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ✼✵✭✽✮✱ ❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭ ♣♣✳ ✻✶✲✻✹✳ • ❈➳❝ ❦Õt q✉➯ ❝đ❛ ➤Ị t➭✐ ➤➲ ➤➢ỵ❝ ❜➳♦ ❝➳♦ t➵✐✿ ❚❤❡ ✽t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❙♣r✐♥❣ ❙❝❤♦♦❧✴ ❲♦r❦s❤♦♣ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ◆❤❛ ❚r❛♥❣✱ ▼❛r❝❤ ✶✲✸✱ ✷✵✶✵✳ ❚r➢ê♥❣ t♦➳♥ ❈■▼P❆✲❯◆❊❙❈❖✲❱■❊❚◆❆▼✱ ✧❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈➭ ❝➳❝ ✈✃♥ ➤Ò ❝ã ❧✐➟♥ q✉❛♥✧✱ ❍➭ ◆é✐✱ ✶✵✲✷✶✴✵✺✴✷✵✶✵✳ ✻ ❍é✐ t❤➯♦ ❚è✐ ➢✉ ✈➭ ❚Ý♥❤ t♦➳♥ ❦❤♦❛ ❤ä❝ ❧➬♥ t❤ø ✽✱ ❇❛ ✈×✱ ✷✵✲✷✸✴✵✹✴✷✵✶✵✳ ✺✳✷✳ ❙➯♥ ♣❤➮♠ ➤➭♦ t➵♦ ❍➢í♥❣ ❞➱♥ ✵✺ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ✱ tr♦♥❣ ➤ã ✵✷ ❧✉❐♥ ✈➝♥ ➤➲ ❜➯♦ ✈Ö t❤➭♥❤ ❝➠♥❣ ♥➝♠ ✷✵✵✾❀ ✵✸ ❧✉❐♥ ✈➝♥ sÏ ❜➯♦ ✈Ö ✈➭♦ t❤➳♥❣ ✶✶✴✷✵✶✵✳ ❍➢í♥❣ ❞➱♥ ✵✸ s✐♥❤ ✈✐➟♥ ◆❈❑❍✱ ➤➲ ❜➯♦ ✈Ö ✈➭ ➤➵t ❦Õt q✉➯ tèt✱ tr♦♥❣ ➤ã ❝ã ✵✶ s✐♥❤ ✈✐➟♥ ➤➢ỵ❝ ❣✐➯✐ ❦❤✉②Õ♥ ❦❤Ý❝❤ tr♦♥❣ ❝✉é❝ t❤✐ ✧❙✐♥❤ ✈✐➟♥ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ t♦➭♥ q✉è❝ ✷✵✵✾✧✱ ✵✶ s✐♥❤ ✈✐➟♥ ➤➢ỵ❝ ➤Ị ❝ư ❞ù t❤✐ ✧❙✐♥❤ ✈✐➟♥ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ t♦➭♥ q✉è❝ ✷✵✶✵✧✳ ❍➢í♥❣ ❞➱♥ ✵✸ s✐♥❤ ✈✐➟♥ ❧➭♠ ❦❤ã❛ ❧✉❐♥ tèt ♥❣❤✐Ö♣ ➤➲ ❜➯♦ ✈Ư ✈➭ ➤➵t ➤✐Ĩ♠ ①✉✃t s➽❝✳ ➜Ị t➭✐ ❣ã♣ ♣❤➬♥ ♣❤ơ❝ ✈ơ ❝❤♦ ✈✐Ư❝ ❣✐➯♥❣ ❞➵② ❝❤✉②➟♥ ➤Ị ✧❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤✧ ❝❤♦ s✐♥❤ ✈✐➟♥ ✈➭ ❤ä❝ ✈✐➟♥ ❈❛♦ ❤ä❝ ❚♦➳♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❑❤♦❛ ❤ä❝✱ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✳ ✼ s✉♠♠❛r② ✶✳ ●❡♥❡r❛❧ ✐♥❢♦r♠❛t✐♦♥ ✲ Pr♦❥❡❝t t✐t❧❡✿ ■♥t❡r❛t✐✈❡ r❛❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥s ✲ ❈♦❞❡ ♥✉♠❜❡r✿ ❇✷✵✵✾✲❚◆✵✼✲✵✸ ✲ ❉✉r❛t✐♦♥✿ ❋r♦♠ ✷✵✵✾ t♦ ✷✵✶✵ ✲ Pr♦❥❡❝t ♠❛♥❛❣❡r✿ ❉♦❝t♦r ◆❣✉②❡♥ ❚❤✐ ❚❤✉ ❚❤✉② ❚❡❧✿ ✵✾✶✷✷✶✶✽✺✽❀ ❊✲♠❛✐❧✿ t❤✉t❤✉②✷✷✵✸✻✾❅❣♠❛✐❧✳❝♦♠ ✲ ■♠♣❧❡♠❡♥t✐♥❣ ✐♥st✐t✉t✐♦♥✿ ❈♦❧❧❡❣❡ ♦❢ ❙❝✐❡♥❝❡s✱ ❚❤❛✐♥❣✉②❡♥ ❯♥✐✈❡rs✐t② ✷✳ ❖❜❥❡❝t✐✈❡ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣r♦❥❡❝t ✐s t♦ st✉❞② s♦♠❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥s✳ ✸✳ ▼❛✐♥ ❝♦♥t❡♥❞s ✲ ❙t✉❞②✐♥❣ t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛✲ t✐♦♥s❀ ✲ ❙t✉❞②✐♥❣ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s ♦❢ t❤❡ r❡❣✉❧❛r✐③❡❞ s♦❧✉✲ t✐♦♥ ♦♥ t❤❡ ❜❛s❡ ♦❢ ❝❤♦♦s✐♥❣ t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❜② t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡✳ ✹✳ ❘❡s✉❧ts ♦❜t❛✐♥❡❞ ✲ ●✐✈✐♥❣ ❛ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛✲ t✐♦♥s❀ ✲ ❲❡ ❛r❡ s❤♦✇❡❞ t❤❛t t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❝❛♥ ❜❡ ❝❤♦♦s❡♥ ❜② t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❞✐s❝r❡♣❛♥❝② ♣r✐♥❝✐♣❧❡❀ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s ♦❢ t❤❡ r❡❣✉❧❛r✐③❡❞ s♦❧✉t✐♦♥ ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ♦♥ t❤❡ ❜❛s❡ ♦❢ ❝❤♦♦s✐♥❣ t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r❀ ✲ ●✐✈✐♥❣ ❛♥ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s②st❡♠ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥s❀ ✽ ✲ ●✐✈✐♥❣ s♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s✳ • ❙❝✐❡♥t✐❢✐❝ ♣✉❜❧✐❝❛t✐♦♥s✿ ❬✶❪✳ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ▲✳ ❚ ❉✉♦♥❣ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥✲s❡❧❢ str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ✹✾✭✶✮✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t②✱ ♣♣✳ ✷✼✲✸✶✳ ❬✷❪✳ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ❚✳ ▼✳ ❚✉②❡♥ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t②✱ ✹✾✭✶✮✱ ♣♣✳ ✸✷✲✸✻✳ ❬✸❪✳ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ t♦ ❛ ❝♦♠♠♦♥ s♦❧✉t✐♦♥ ♦❢ ✐♥✈❡rs❡✲str♦♥❣❧② ♣r♦❜❧❡♠s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ✸✱ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ♣♣✳ ✶✻✺✲✶✼✹✳ ❬✹❪✳ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❈♦♥✈❡r❣❡♥❝❡ r❛t❡s ♦❢ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉✲ ❧❛r✐③❛t✐♦♥ ❢♦r ✐❧❧✲♣♦s❡❞ ♠✐①❡❞ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ✐♥✈❡rs❡✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ♣❡rt✉r❜❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✭t♦ ❛♣❡❛r✮✳ ❬✺❪✳ ◆❣✳ ❚✳ ❚✳ ❚❤✉②✱ ➜✳ ❚✳ ❍♦✐ ✭✷✵✶✵✮✱ ✧◆✉♠❡r✐❝❛❧ r❡s✉❧ts ✐♥ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ✐❧❧✲♣♦s❡❞ ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥✧✱ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t②✱ ✼✵✭✽✮✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡ ♣♣✳ ✻✶✲✻✹✳ • ❚r❛✐♥✐♥❣ r❡s✉❧ts✿ ✲ ■♥str✉❝t ✵✺ ▼❛st❡r t❤❡s❡s✱ ✵✸ s❝✐❡♥t✐❢✐❝ r❡s❡❛r❝❤ ♣r♦❥❡❝ts ❛♥❞ ✵✸ ❞✐ss❡rt❛✲ t✐♦♥s ♦❢ ✉♥❞❡r❣r❛❞✉❛t❡ st✉❞❡♥ts✳ ✲ ❚❤❡ ♣r♦❥❡❝t ✐s ❤❡❧♣❢✉❧ ❢♦r t❡❛❝❤✐♥❣ ✧■❧❧✲♣♦s❡❞ Pr♦❜❧❡♠✧ ✐♥ ❈♦❧❧❡❣❡ ♦❢ ❙❝✐✲ ❡♥❝❡s✱ ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t②✳ ✾ ▼ét sè ❦ý ❤✐Ư✉ ✈➭ ❝❤÷ ✈✐Õt t➽t H ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ù❝ X ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ù❝ X∗ ❦❤➠♥❣ ❣✐❛♥ ❧✐➟♥ ❤ỵ♣ ❝đ❛ Rn ❦❤➠♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ Rn+ t❐♣ ❝➳❝ ✈Ð❝ t rtt ủ t rỗ X n ề x := y x ợ ị ĩ ❜➺♥❣ y ∀x ✈í✐ ♠ä✐ ∃x tå♥ t➵✐ inf F (x) x∈X x x ✐♥❢✐♠✉♠ ❝ñ❛ t❐♣ {F (x) : x ∈ X} ❛r❣ F (x) t❐♣ ❝➳❝ ➤✐Ó♠ ❝ù❝ t✐Ĩ✉ ❝đ❛ ❤➭♠ I ➳♥❤ ①➵ ➤➡♥ ✈Þ x∈X A∩B ❆ ❣✐❛♦ ✈í✐ ❇ AT ♠❛ tr❐♥ ❝❤✉②Ĩ♥ ✈Þ ❝đ❛ ♠❛ tr❐♥ a∼b a t➢➡♥❣ ➤➢➡♥❣ ✈í✐ b A∗ t♦➳♥ tư ❧✐➟♥ ❤ỵ♣ ❝đ❛ t♦➳♥ tư D(A) ♠✐Ị♥ ①➳❝ ➤Þ♥❤ ❝đ❛ t♦➳♥ tư R(A) ♠✐Ị♥ ❣✐➳ trÞ ❝đ❛ t♦➳♥ tö xk → x xk x F tr➟♥ X ❞➲② A A {xk } ❤é✐ tơ ♠➵♥❤ tí✐ x ❞➲② {xk } ❤é✐ tơ ②Õ✉ tí✐ x ✶✵ A A Rn uk+1 ≤ (1 − ak )uk + bk , ≤ ak ≤ 1✱ ∞ bk ✐✐✮ ak = +∞, lim = 0✳ k→+∞ ak k=1 ❑❤✐ ➤ã✱ lim uk = 0✳ ✐✮ k→+∞ ➜Þ♥❤ ❧ý ✷✳✶✵✳ ●✐➯ sư ❝➳❝ ❞➲② {αn } ✈➭ {βn } tr♦♥❣ ❜➭✐ t♦➳♥ ✭✷✳✶✻✮ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ 0✱ βn → ❦❤✐ n → +∞ ❀ |αn+1 − αn | βn ✭✐✐✮ lim = 0❀ = 0, lim n→+∞ n→+∞ αn βn αn ✭✐✮ ≥ αn ∞ αn βn = +∞✳ ✭✐✐✐✮ n=1 ❑❤✐ ➤ã ❞➲② {zn } s✐♥❤ r❛ tõ ✭✷✳✶✻✮ ❤é✐ tơ tr♦♥❣ H tí✐ ♣❤➬♥ tö x0 ∈ S ❦❤✐ n → +∞✳ zn − x0 ≤ zn − xn + xn − x0 ✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✷✳✾ sè ❤➵♥❣ t❤ø ❤❛✐ tr♦♥❣ ✈Õ ♣❤➯✐ ❝ñ❛ ➤➳♥❤ ❣✐➳ ♥➭② ❞➬♥ ➤Õ♥ ❦❤✐ n → ∞✳ ❉♦ ➤ã t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ zn ①✃♣ ①Ø xn ❦❤✐ n → ∞✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❐t ✈❐②✱ ➤➷t ❚r➢í❝ ❤Õt t❛ ❝ã ∆n = zn − xn ✳ ❘â r➭♥❣✱ ∆n+1 = zn+1 − xn+1 N αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) = zn − xn − βn j=1 ✭✷✳✷✵✮ − (xn+1 − xn ) , N αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) ≤ zn − xn − βn j=1 + xn+1 − xn , ✹✹ ë ➤➞② N αnλj (Aj (zn ) zn − xn − βn − fj ) + αn (zn − x∗ ) = j=1 = zn − xn N + βn2 αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) j=1 N αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) − 2βn zn − xn , j=1 N αnλj (Aj (xn ) − fj ) + αn (xn − x∗ ) − j=1 ≤ (1 − 2βn αn ) zn − xn N + βn2 αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) j=1 ✭✷✳✷✶✮ ❱× Aj ❧➭ t♦➳♥ tư ♥❣➢ỵ❝ ➤➡♥ ➤✐Ư✉ ♠➵♥❤ ♥➟♥ Aj ❧✐➟♥ tơ❝ ▲✐♣s❝❤✐t③ ✈➭ N αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) j=1 N αnλj (Aj (zn ) − fj ) + αn (zn − x∗ ) = j=1 N αnλj (Aj (xn ) − − fj ) − αn (xn − x∗ ) j=1 N αnλj ≤ j=1 zn − xn mAj N αnλj + 2αn j=1 ≤c1 zn − xn , ✹✺ + αn2 zn − xn zn − xn mAj 2 ë ➤➞② c1 ❧➭ ♠ét ❤➺♥❣ sè ❞➢➡♥❣✳ ❑Õt ❤ỵ♣ ✭✷✳✷✵✮✱ ✭✷✳✷✶✮✱ ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ❝ï♥❣ ✈➭ ➜Þ♥❤ ❧ý ✷✳✾ s✉② r❛ 1/2 ∆n+1 ≤ ∆2n (1 − 2βn αn + cβn2 ) +M |αn+1 − αn | αn ❇×♥❤ ♣❤➢➡♥❣ ❤❛✐ ✈Õ ❝đ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② s❛✉ ➤ã ➳♣ ❞ơ♥❣ ➤➳♥❤ ❣✐➳ s➡ ❝✃♣ ✭①❡♠ ❬✷❪✮ (a + b)2 ≤ (1 + αn βn )a2 + (1 + )b2 αn βn t❛ ♥❤❐♥ ➤➢ỵ❝ ∆2n+1 ≤ ∆2n (1 − βn αn + cβn2 − 2αn2 βn2 + cαn βn3 ) 2 |αn+1 − αn | + 1+ M βn αn αn2 ✭✷✳✷✷✮ ❉➲② {∆n } t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❇ỉ ➤Ị ✷✳✷ ✈× ✭✷✳✷✷✮ ✈➭ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ (i) − (iii) ✈í✐ an = αn βn − cβn2 + 2αn2 βn2 − cαn βn3 bn = + |αn+1 − αn |2 M2 βn αn αn2 ✷ ❈❤ó ý✿ ❉➲② βn = (1 + n)−1/2 ✈➭ αn = (1 + n)−p ✱ < 2p < 1/N t❤á❛ ♠➲♥ t✃t ❝➯ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✵✳ ❑Õt ❧✉❐♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ tr×♥❤ ❜➭② ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ tr➟♥ ❝➡ së ❣✐➯✐ ♠ét ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ♣❤ơ t❤✉é❝ t❤❛♠ sè✳ ❈❤ó♥❣ t➠✐ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ sù ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤✱ ➤➢❛ r❛ ❝➳❝❤ ❝❤ä♥ ❣✐➳ trÞ ❝đ❛ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤ ❤❐✉ ♥❣❤✐Ö♠ t❤❡♦ ♥❣✉②➟♥ ❧ý ➤é ❧Ö❝❤ ➤å♥❣ t❤ê✐ ➤➳♥❤ ❣✐➳ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤✳ ❈❤ó♥❣ t➠✐ ❝ị♥❣ ①➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣ ❜❐❝ ❦❤➠♥❣ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ù❝ H ①✃♣ ①Ø ♥❣❤✐Ư♠ ❝❤♦ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✳ ✹✻ ❈❤➢➡♥❣ ✸ ❑Õt q✉➯ tÝ♥❤ t♦➳♥ t❤ư ♥❣❤✐Ư♠ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét ✈➭✐ t❤ư ♥❣❤✐Ư♠ sè tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝ị♥❣ ♥❤➢ ❦❤➠♥❣ ❣✐❛♥ ✈➠ ❤➵♥ ❝❤✐Ị✉ ♠✐♥❤ ❤ä❛ ❝❤♦ ệ ỉ ợ trì tr ❑Õt q✉➯ ➤➵t ➤➢ỵ❝ ❝❤♦ t❤✃② ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ➤➢❛ r❛ ❝ã ❦❤➯ ♥➝♥❣ t❤ù❝ t❤✐ ✈➭ ❦❤➳ ❤✐Ö✉ q✉➯✳ ❈❤ó♥❣ t➠✐ ➤➲ ✈✐Õt ❝❤➢➡♥❣ tr×♥❤ t❤ù❝ ♥❣❤✐Ư♠ ❜➺♥❣ ♥❣➠♥ ♥❣÷ ▼❆❚▲❆❇ ✼✳✵ ✈➭ ➤➲ t❤ư ♥❣❤✐Ư♠ ❝❤➵② tr➟♥ ♠➳② tÝ♥❤ ❆❈❊❘ ✶✳✼✸ ●❍③✳ ❘❛♠ ✺✵✹ ▼❇ ❝❤♦ ❝➳❝ ✈Ý ❞ơ s❛✉ ➤➞②✳ ✸✳✶✳ ❱Ý ❞ơ ✸✳✶ ❳Ðt ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ✭✷✳✶✮ ✈í✐ Aj ❧➭ ♠❛ tr❐♥ M = ợ ị Aj = BjT Bj , j = 1, 2, 3, ë ➤➞②  −1  2 −2  B1 =  3 −3  1 −1  −1  −2  B2 =  −3  0 −1 1 −1 −2 −3 −1 ✹✼ −1 −2 −3 −2 2   2  3   2 −1  −1  −2  −3   1   0  B3 =  1  1  −1 −1  −3 −2 −1  −1 −1   −1 −3 −2 0 fj = 0✱ j = 1, 2, 3✳ ❚❛ ❝ã A1 ✱ A2 ✱ A3 ❧➭ ❝➳❝ ♠❛ tr❐♥ ➤è✐ ①ø♥❣ ①➳❝ ➤Þ♥❤ ❦❤➠♥❣ ➞♠ ✈í✐ r(A1 ) = r(A2 ) = 3✱ r(A3 ) = 4✳ ❉Ô t❤✃② r➺♥❣ x0 = (0, 0, 0, 0, 0)T ∈ R5 ❧➭ ♥❣❤✐Ư♠ ❝ã ❝❤✉➮♥ ♥❤á ♥❤✃t ❝đ❛ ❤Ư ✭✷✳✶✮ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✳ ❇➞② ❣✐ê sư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣ ✭✷✳✶✻✮ ➤Ĩ t×♠ ♥❣❤✐Ư♠ ①✃♣ ①Ø ❝❤♦ ✈Ý ❞ô ♥➭② ❦❤✐ x∗ = θ ♥❤➢ s❛✉✳ ✰ ❱í✐ ①✃♣ ①Ø ❜❛♥ ➤➬✉ z0 tï② ý t❤✉é❝ R5 ✱ ❝❤ä♥ ❤❛✐ ❞➲② sè βm = (1 + m)−1/2 ✈➭ αm = (1 + m)−1/12 t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✵✳ ✰ ❚r♦♥❣ tÝ♥❤ t♦➳♥ t❤ư ♥❣❤✐Ư♠✱ ♥Õ✉ tÝ♥❤ t♦➳♥✱ ✈í✐ (m+1) max zj 1≤j≤5 (m) − zj ≤ err t❤× ❞õ♥❣ err ❧➭ s❛✐ sè ❝❤♦ tr➢í❝✳ ❙❛✉ ➤➞② ❧➭ ❦Õt q✉➯ tÝ♥❤ t♦➳♥✳ ❙è ❧➬♥ ❧➷♣ err m x0 − zm 5.9299 × 10−5 29 0.00077717 9.5917 × 10−6 52 0.00017816 9.5457 × 10−7 97 2.8739 × 10−5 9.7834 × 10−8 168 4.4859 × 10−6 ❇➯♥❣ ✸✳✶✿ ✈í✐ z0 = (1.5, 1.5, 1.5, 1.5, 1.5)T ∈ R5 ✈➭ zm+1 = zm − βm A0 zm + αm A1 zm + αm A2 zm + αm zm ✹✽ ❙è ❧➬♥ ❧➷♣ err m x0 − zm 8.5032 × 10−5 27 0.00091206 9.5351 × 10−6 52 0.00017744 9.5213 × 10−7 97 2.8692 × 10−5 9.7746 × 10−8 168 4.4834 × 10−6 ❇➯♥❣ ✸✳✷✿ ✈í✐ z0 = (1.5, 1.5, 1.5, 1.5, 1.5)T ∈ R5 ✈➭ zm+1 = zm − βm A1 zm + αm A2 zm + αm A0 zm + αm zm err ❙è ❧➬♥ ❧➷♣ m x0 − zm 9.2541 × 10−5 38 0.0013443 9.9215 × 10−6 72 0.00023732 9.7432 × 10−7 130 3.6725 × 10−5 9.893 × 10−8 219 5.5511 × 10−6 ❇➯♥❣ ✸✳✸✿ ✈í✐ z0 = (5, 5, 5, 5, 5)T ∈ R5 ✈➭ zm A2 zm + αm zm+1 = zm − βm A0 zm + αm A1 zm + αm err ❙è ❧➬♥ ❧➷♣ m x0 − zm 9.18 × 10−5 38 0.0013366 9.8821 × 10−6 72 0.0002367 9.7284 × 10−7 130 3.669 × 10−5 9.8882 × 10−8 219 5.5494 × 10−6 ❇➯♥❣ ✸✳✹✿ ✈í✐ z0 = (5, 5, 5, 5, 5)T ∈ R5 ✈➭ zm+1 = zm − βm A1 zm + αm A2 zm + αm A0 zm + αm zm ❉ù❛ tr➟♥ ❦Õt q✉➯ tr♦♥❣ ❇➯♥❣ ✸✳✶✱ ✸✳✷✱ ✸✳✸ ✈➭ ✸✳✹ t❛ rót r❛ ♠ét sè ♥❤❐♥ ①Ðt s❛✉✿ • ❚Ý♥❤ ❤é✐ tơ ❝đ❛ ❞➲② ❧➷♣ ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ➤✐Ó♠ ❝❤ä♥ ❜❛♥ ➤➬✉✱ t✉② ♥❤✐➟♥ ➤✐Ó♠ ❝❤ä♥ ❜❛♥ ➤➬✉ ❝ã ➯♥❤ ❤➢ë♥❣ ➤Õ♥ ❤✐Ư✉ q✉➯ ❝đ❛ ❞➲② ❧➷♣✳ ❚❤ù❝ tÕ✱ ✹✾ ♥Õ✉ ➤✐Ó♠ ①✉✃t ♣❤➳t ❜❛♥ ➤➬✉ ❣➬♥ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ t❤× ❝❤ó♥❣ t❛ ❝➬♥ Ýt ❧➬♥ ❧➷♣ ❤➡♥ s♦ ✈í✐ ✈✐Ư❝ ❝❤ä♥ ➤✐Ĩ♠ ❜❛♥ ➤➬✉ ①❛ ✈í✐ ♥❣❤✐Ư♠ ➤Ĩ ➤➵t ➤➢ỵ❝ ♥❣❤✐Ư♠ ①✃♣ ①Ø ✈í✐ s❛✐ số trớ ã trò ủ A1 A2 ✱ A3 ❧➭ ♥❤➢ ♥❤❛✉ tr♦♥❣ tõ♥❣ ❞➲② ❧➷♣✳ ✸✳✷✳ í ụ ét t ự trị tì ♠ét ♣❤➬♥ tö x0 ∈ H s❛♦ ❝❤♦ ϕj (x0 ) = ϕj (x), j = 1, , N, x∈H ✭✸✳✶✮ ë ➤➞② ϕj ❧➭ ♣❤✐Õ♠ ❤➭♠ ❧å✐ ❝❤Ý♥❤ t❤➢ê♥❣✱ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ②Õ✉ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ù❝ H ✳ ❚r♦♥❣ ✈Ý ❞ô ♥➭② t❛ ①Ðt ❤➭♠ ϕj : L2 [0, 1] → R ∪ {+∞} ①➳❝ ➤Þ♥❤ ❜ë✐ ϕj (x) = f ✈í✐ Bj x, x , j = 1, f : R → R ➤➢ỵ❝ ❝❤ä♥ ♥❤➢ s❛✉   , t ≤ b0 ,    (t − b )2 f (t) = , b0 < t ≤ b0 + ,     t − b0 − , t > b0 + , ë ➤➞② b0 ❧➭ ❤➺♥❣ sè ❞➢➡♥❣✱ > ➤ñ ❜Ð ✈➭ Bj : L2 [0, 1] → L2 [0, 1] t tử ợ ị B1 x(t) = k1 (t, s)x(s)ds, B2 x(t) = k2 (t, s)x(s)ds ë ➤➞② k1 (t, s) = t(1 − s) , t ≤ s, s(1 − t) , s < t, ✺✵ ✈➭ k2 (t, s) =                    (1 − s)2 st2 (1 − s)2 t3 (1 + 2s) − + (t − s)3 + , s2 (1 − s)(1 − t)2 s2 (1 − t3 )(2s − 3) + + (s − t)3 , + ❧➭ ❝➳❝ ❤➭♠ ❤➵❝❤ ➤è✐ ①ø♥❣ ①➳❝ ị tr ì 1} ó j (x) = f ➳♣ t ≤ s, s ≤ t, {0 ≤ t, s ≤ Bj x, x Bj (x) ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣ ✭✷✳✶✻✮ ë ❞➵♥❣ zm , z0 ∈ RM zm+1 = zm − βm A˜1 zm + αm A˜2 zm + αm ➤Ĩ t×♠ ♥❣❤✐Ư♠ ①✃♣ ①Ø ❝❤♦ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ϕj (x) = θ, ∀j = 1, ✈í✐ αm = (1 + m)−p , < p < ✱ βm = (1 + m)−1/2 ✈➭ ˜ ˜j (˜ Bj x˜, x˜ B x), A˜j (x) = f ë ➤➞② ˜j = B k(tı , t ) M , ı,=1 x˜ = (˜ x1 , x˜2 , , x˜M )T , x˜ı ∼ x(tı ), ı = 1, 2, , M, = M ❱✐Ư❝ ❦✐Ĩ♠ tr❛ tÝ♥❤ ➤ó♥❣ ➤➽♥ ❝đ❛ s➡ ➤å ❧➷♣ ❞ù❛ ✈➭♦ trị s số ữ ỉ tế max |z(m−1) − z(m) | ≤ err 0≤≤M ❙❛✉ ➤➞② ❧➭ ❦Õt q✉➯ ✈Ị ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ sè ❧➬♥ ❧➷♣ ✈➭ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❧➷♣ ✭✷✳✶✻✮ ❝❤♦ ❱Ý ❞ơ ✸✳✷✳ ✺✶ ❑Õt q✉➯ tr♦♥❣ ❇➯♥❣ ✸✳✺ ➤➢ỵ❝ tÝ♥❤ ✈í✐ t❤❛♠ sè p = ✱ tr♦♥❣ ❇➯♥❣ ✸✳✻✱ ✳ ❚r♦♥❣ t✃t ❝➯ ❝➳❝ ✈Ý ❞ơ ➤➢ỵ❝ tÝ♥❤ t♦➳♥✱ ➤✐Ĩ♠ ①✃♣ ①Ø ❜❛♥ ➤➬✉ ➤➢ỵ❝ 16 10−3 T M ❝❤ä♥ ❧➭ z0 = (5, 5, 5) ∈ R ✱ b0 = ✱ = 10−2 ✈➭ M = 50✳ p= err x0 − zm 32 0.00091935 0.011739 62 9.8652 × 10−5 0.0021074 114 9.6658 × 10−6 0.00032977 194 9.9717 × 10−7 5.1038 × 10−5 316 9.8618 × 10−8 7.3102 × 10−6 ❙è ❧➬♥ ❧➷♣ m ❇➯♥❣ ✸✳✺✿ p= err x0 − zm 22 0.00084675 0.0051627 37 8.9838 × 10−5 0.00078278 58 9.4758 × 10−6 0.00011169 86 9.9791 × 10−7 1.5268 × 10−5 123 9.921 × 10−8 ❙è ❧➬♥ ❧➷♣ m ❇➯♥❣ ✸✳✻✿ p= 1.9189 × 10−6 16 ◆❣♦➭✐ ❝➳❝ ♥❤❐♥ ①Ðt ♥❤➢ tr♦♥❣ ❱Ý ❞ơ ✸✳✶✱ t❛ t❤✃②✿ • ❚Ý♥❤ ❤✐Ư✉ q✉➯ ❝đ❛ ♣❤➢➡♥❣ ò ụ tộ ệ ọ trị ❝ñ❛ t❤❛♠ sè p tr♦♥❣ ❞➲② αm ✳ ❚r♦♥❣ ✈Ý ❞ơ ✈í✐ M = 50✱ t❛ ❝❤ä♥ p = t❤× 16 ❝➬♥ Ýt ❧➬♥ ❧➷♣ ❤➡♥ s♦ ✈í✐ tr➢ê♥❣ ❤ỵ♣ ❝❤ä♥ p = tÝ♥❤ t♦➳♥ ♥❣❤✐Ư♠ ①✃♣ ①Ø ✈í✐ ❝ï♥❣ s❛✐ sè ❝❤♦ tr➢í❝✳ • ❑❤✐ sè ❧➬♥ ❧➷♣ ❝➭♥❣ ❧í♥ t❤× ♥❣❤✐Ư♠ ①✃♣ ①Ø ❝➭♥❣ ❣➬♥ ✈í✐ ♥❣❤✐Ư♠ ❝❤Ý♥❤ ①➳❝ ❝đ❛ ❜➭✐ t♦➳♥ ❜❛♥ ➤➬✉✳ ✺✷ ❑Õt ❧✉❐♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ ➤➢❛ r❛ ✷ ✈Ý ❞ơ sè tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ✈➠ ❤➵♥ ❝❤✐Ị✉ t❤ư ♥❣❤✐Ư♠ ❝❤♦ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✳ P❤➬♥ ♠Ị♠ ➤➢ỵ❝ sư ❞ơ♥❣ ❧➭ ▼❆❚▲❆❇ ✼✳✵✳ ❑Õt q✉➯ ➤➵t ➤➢ỵ❝ ❝❤♦ t❤✃② ♣❤➢➡♥❣ ♣❤➳♣ ❝ã ❦❤➯ ♥➝♥❣ t❤ù❝ t❤✐ ✈➭ ❦❤➳ ❤✐Ư✉ q✉➯✳ ✺✸ ❑Õt ❧✉❐♥ ❝❤✉♥❣ ➜Ị t➭✐ ➤➲ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ s❛✉ ➤➞②✿ ➜➢❛ r❛ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ tr➟♥ ❝➡ së ❣✐➯✐ ♠ét ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ♣❤ơ t❤✉é❝ t❤❛♠ sè ❞ù❛ tr➟♥ ➤Ị ①✉✃t ❝đ❛ ◆❣✉②Ơ♥ ❇➢ê♥❣ ❬✻❪✳ ❈❤Ø r❛ ❝➳❝❤ ❝❤ä♥ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤ ❤❐✉ ♥❣❤✐Ö♠ t❤❡♦ ♥❣✉②➟♥ ❧ý ➤é ❧Ö❝❤ s✉② ré♥❣✳ ❉ù❛ tr➟♥ ❝➳❝❤ ❝❤ä♥ ♥➭②✱ tè❝ ➤é ❤é✐ tơ ❝đ❛ ♥❣❤✐Ư♠ ❤✐Ư✉ ❝❤Ø♥❤ ➤➢ỵ❝ ➤➳♥❤ ❣✐➳ ❦❤✐ ❝➳❝ t❤❛♠ sè ①✃♣ ①Ø ❞➬♥ ➤Õ♥ 0✳ ❳➞② ❞ù♥❣ ♠ét ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❣✐➯✐ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✳ ❙ù ❤é✐ tơ ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ➤➢ỵ❝ t❤✐Õt ❧❐♣ ❞ù❛ tr➟♥ ❝➡ së ❝❤ä♥ ❞➲② t❤❛♠ sè t❤Ý❝❤ ❤ỵ♣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ➤➡♥ ➤✐Ư✉ ▲✐♣s❝❤✐t③ ❝đ❛ ❝➳❝ t♦➳♥ tư✳ ➜➢❛ r❛ ✈Ý ❞ơ sè ♠✐♥❤ ❤ä❛ tÝ♥❤ ❦❤➯ t❤✐ ✈➭ ❤✐Ư✉ q ủ ế ị ề ữ ứ t✐Õ♣ t❤❡♦ ❈➳❝ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ❜➭✐ t♦➳♥ ♠ë ❝ã t❤Ĩ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ ✈í✐ ♣❤➢➡♥❣ ♣❤➳♣ t✐Õ♣ ❝❐♥ ✈➭ ❦Õt q✉➯ ❝đ❛ ➤Ị t➭✐ ❝❤♦ ❝➳❝ tr➢ê♥❣ ợ s ì ệ N t tứ ế ỗ ợ j = 1, , N ✈í✐ N ≥ 1✳ ▼ë ré♥❣ ❝➳❝ ❦Õt q✉➯ t❤✉ ➤➢ỵ❝ ❧➟♥ ❧í♣ ❜➭✐ t♦➳♥ ré♥❣ ❤➡♥ ➤ã ❧➭ ❤Ư ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣✿ t×♠ ë ➤➞② x∗ ∈ K s❛♦ ❝❤♦ Fj (x∗ , y) ≥ 0, ∀y ∈ K, j = 1, , N, K ❧➭ ♠ét t ó rỗ ủ X ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❨✳ ❆❧❜❡r ❛♥❞ ■✳ ❘②❛③❛♥ts❡✈❛✱ ◆♦♥❧✐♥❡❛r ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ♦❢ ♠♦♥♦t♦♥❡ t②♣❡✱ ❙♣r✐♥❣❡r✱ ✷✵✵✻✳ ❬✷❪ ❆✳ ❇✳ ❇❛❦✉s❤✐♥s❦✐✐ ❛♥❞ ❆✳ ●✳ ●♦♥❝❤❛rs❦✐✐✱ ♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s✿ ❚❤❡✲ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs ❉♦r❞r❡❝❤t✱ ❇♦st♦♥✱ ▲♦♥❞♦♥ ✭✶✾✾✹✮✳ ❬✸❪ ❱✳ ❇❛r❜✉✱ ❙♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❙❡♠✐❣r♦✉♣s ❛♥❞ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ ◆♦♦r❞❤♦❢❢ ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ▲❡②❞❡♥ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✱ ✶✾✼✻✳ ❬✹❪ ❋✳ ❇r♦✇❞❡r ✭✶✾✻✻✮✱ ✧❊①✐st❡♥❝❡ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♦❧✉t✐♦♥s ♦❢ ♥♦♥✲ ❧✐♥❡❛r ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ Pr♦❝✳ ◆❛t✳ ❆❝❛❞✳ ❙❝✐✳ ❯❙❆✱ ✺✻✭✹✮✱ ♣♣✳ ✶✵✽✵✲✶✵✽✻✳ ❬✺❪ ◆❣✳ ❇✉♦♥❣ ✭✷✵✵✺✮✱ ✧❖♥ ♠♦♥♦t♦♥❡ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s✧✱ ♠❛t✐❝❛ ❙✐♥✐❝❛✱ ✷✶✭✺✮✱ ❆❝t❛ ▼❛t❤❡✲ ♣♣✳ ✶✵✵✶✲✶✵✵✹✳ ❬✻❪ ◆❣✳ ❇✉♦♥❣ ✭✷✵✵✻✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ✉♥❝♦♥str❛✐♥❡❞ ✈❡❝t♦r ♦♣t✐♠✐③❛✲ t✐♦♥ ♦❢ ❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ♠❛t✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✹✻✭✸✮✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛t❤❡✲ ♣♣✳ ✸✺✹✲✸✻✵✳ ❬✼❪ ◆❣✳ ❇✉♦♥❣ ✭✷✵✵✼✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❡①tr❛❣r❛❞✐❡♥t ♠❡t❤♦❞ ❢♦r s②st❡♠s ♦❢ ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s✧✱ ♠❛t✐❝s✱ ✼✭✸✮✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡✲ ♣♣✳ ✶✲✾✳ ❬✽❪ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ▲✳ ❚ ❉✉♦♥❣ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥✲s❡❧❢ str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ✺✺ ♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ◆❣✉②➟♥✱ ✹✾✭✶✮✱ ❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ♣♣✳ ✷✼✲✸✶✳ ❬✾❪ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ❛♥❞ ❚✳ ▼✳ ❚✉②❡♥ ✭✷✵✵✾✮✱ ✧❘❡❣✉❧❛r✐③❛t✐♦♥ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ✹✾✭✶✮✱ ❬✶✵❪ ■✳ ❊❦❡❧❛♥❞ ❛♥❞ ❘✳ ❚❡♠❛♠ ✭✶✾✼✵✮✱ Pr♦❜❧❡♠s✱ ♣♣✳ ✸✷✲✸✻✳ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ❱❛r✐❛t✐♦♥❛❧ ◆♦rt❤✲❍♦❧❧❛♥❞ P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥②✱ ❆♠st❡r❞❛♠✱ ❍♦❧❧❛♥❞✳ ❬✶✶❪ ❱✳ ❑✳ ■✈❛♥♦✈✱ ❱✳ ❱✳ ❱❛s✐♥ ❛♥❞ ❱✳ P✳ ❚❛♥❛♥❛✱ Pr♦❜❧❡♠s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❚❤❡♦r② ♦❢ ▲✐♥❡❛r ■❧❧✲P♦s❡❞ ▼♦s❝♦✇ ◆❛✉❦❛ ✭✐♥ ❘✉ss✐❛♥✮✱ ✶✾✼✽✳ ❬✶✷❪ ❏✳ ❙✳ ❏✉♥❣✱ ❨✳ ❏✳ ❈❤♦ ❛♥❞ ❘✳ P✳ ❆❣❛r✇❛❧ ✭✷✵✵✺✮✱ ✧■t❡r❛t✐✈❡ s❝❤❡♠❡s ✇✐t❤ s♦♠❡ ❝♦♥tr♦❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ❢❛♠✐❧② ♦❢ ❢✐♥✐t❡ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✱ ♣♣✳ ✶✷✺✲ ✶✸✺✳ ❬✶✸❪ ▼✳ ▼✳ ▲❛✈r❡♥t✐❡✈ ✭✶✾✻✼✮✱ ♠❛t✐❝❛❧ P❤②s✐❝s✱ ❙♦♠❡ ■♠♣r♦♣❡r❧② P♦s❡❞ Pr♦❜❧❡♠s ✐♥ ▼❛t❤❡✲ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✶✹❪ ❲✳ ❚❛❦❛❤❛s❤✐✱ ❚✳ ❚❛♠✉r❛ ❛♥❞ ▼✳ ❚♦②♦❞❛ ✭✷✵✵✷✮✱ ✧❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ❛ ❢❛♠✐❧② ♦❢ ❢✐♥✐t❡ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❙❝✐✳ ▼❛t❤✳ ❏♣♥✱ ✺✻✱ ♣♣✳ ✹✼✺✲✹✽✵✳ ❬✶✺❪ ❆✳ ◆✳ ❚✐❦❤♦♥♦✈ ✭✶✾✻✸✮✱ ✧❖♥ t❤❡ s♦❧✉t✐♦♥ ♦❢ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ❛♥❞ t❤❡ ♠❡t❤♦❞ ♦❢ r❡❣✉❧❛r✐③❛t✐♦♥✧✱ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ❙❙❙❆✱ ✶✺✶✱ ♣♣✳ ✺✵✶✲✺✵✹ ✭❘✉ss✐❛♥✮✳ ❬✶✻❪ ❆✳ ◆✳ ❚✐❦❤♦♥♦✈ ❛♥❞ ❱✳ ■✳ ❆rs❡♥✐♥✱ ❙♦❧✉t✐♦♥s ♦❢ ■❧❧✲♣♦s❡❞ Pr♦❜❧❡♠s✱ ❲✐❧❡② ◆❡✇ ❨♦r❦✱ ✶✾✼✼✳ ❬✶✼❪ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ t♦ ❛ ❝♦♠♠♦♥ s♦❧✉t✐♦♥ ♦❢ ✐♥✈❡rs❡✲str♦♥❣❧② ♣r♦❜❧❡♠s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ t✐♦♥s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ✸✱ ✺✻ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛✲ ♣♣✳ ✶✻✺✲✶✼✹✳ ❬✶✽❪ ◆❣✳ ❚✳ ❚✳ ❚❤✉② ✭✷✵✶✵✮✱ ✧❈♦♥✈❡r❣❡♥❝❡ r❛t❡s ♦❢ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛✲ t✐♦♥ ❢♦r ✐❧❧✲♣♦s❡❞ ♠✐①❡❞ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ✐♥✈❡rs❡✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ♣❡rt✉r❜❛t✐♦♥s✧✱ t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛✲ ✭➜➲ ♥❤❐♥ ➤➝♥❣ ♥➝♠ ✷✵✶✵✮✳ ❬✶✾❪ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ❚❤đ②✱ ➜➷♥❣ ❚ó ❍å✐ ✭✷✵✶✵✮✱ ✧❑Õt q✉➯ sè ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ư✉ ❝❤Ø♥❤ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✧✱ ✈➭ ❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ✼✵✭✽✮✱ ❬✷✵❪ ▼✳ ▼✳ ❱❛✐♥❜❡r❣✱ ❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ♣♣✳✻✶✲✻✹✳ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞ ❛♥❞ ▼❡t❤♦❞ ♦❢ ▼♦♥♦t♦♥❡ ❖♣❡r✲ ❛t♦rs ✐♥ t❤❡ ❚❤❡♦r② ♦❢ ◆♦♥❧✐♥❡❛r ❊q✉❛t✐♦♥s✱ ◆❡✇ ❨♦r❦✱ ❏♦❤♥ ❲✐❧❡②✱ ✶✾✼✸✳ ❬✷✶❪ ❊✳ ❩❡✐❞❧❡r✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✶✾✽✺✳ ❬✷✷❪ ▲✳ ❈✳ ❩❡♥❣ ❛♥❞ ❏✳ ❈✳ ❨❛♦ ✭✷✵✵✻✮✱ ✧■♠♣❧✐❝✐t ✐t❡r❛t✐♦♥ s❝❤❡♠❡ ✇✐t❤ ♣❡r✲ t✉r❜❡❞ ♠❛♣♣✐♥❣ ❢♦r ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ❛ ❢✐♥✐t❡ ❢❛♠✐❧② ♦❢ ♥♦♥❡①✲ ♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✻✹✱ ♣♣✳ ✷✺✵✼✲✷✺✶✺✳ ❬✷✸❪ ❍✳ ❑✳ ❳✉ ✭✷✵✵✸✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ q✉❛❞r❛t✐❝ ♦♣t✐♠✐③❛t✐♦♥✧✱ ❏♦✉r♥❛❧ ♦❢ ❖♣t✐♠✐③❛t✐♦♥ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✶✻✱ ♣♣✳ ✻✺✾✲✻✼✽✳ ❬✷✹❪ ❙✳ ❳✉✱ ❨✳ ▲✐✉ ❛♥❞ ❘✳ ❈❤❡♥ ✭✷✵✵✼✮✱ ✧❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛♥ ❡①♣❧✐❝✐t ✐t❡r❛t✐♦♥ s❡q✉❡♥❝❡ ❢♦r str✐❝t❧② ❛s②♠♣t♦t✐❝❛❧❧② ♣s❡✉❞♦✲❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s✧✱ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ✺✻✭✶✮✱ ✺✼ ✷✼✾✾✲✷✽✵✹✳ ❈❤ñ ♥❤✐Ư♠ ➤Ị t➭✐ ❳➳❝ ♥❤❐♥ ❝đ❛ ❚❤đ tr➢ë♥❣ ➤➡♥ ✈Þ ◆❣✉②Ơ♥ ❚❤Þ ❚❤✉ ❚❤đ② ✺✽ ... ❧✐➟♥ tơ❝ ✈➭♦ ❞÷ ❦✐Ư♥ ❜❛♥ ➤➬✉✳ ❚✉② ♥❤✐➟♥✱ ❝ị♥❣ ❝ã ♠ét trờ ợ ệt trì t tử ✈í✐ t♦➳♥ tư ❧✐➟♥ tơ❝ ♠➵♥❤✳ ❈❤➻♥❣ ❤➵♥✱ ♥Õ✉ ♠✐Ị♥ ị t tử D(A) ủ A ữ ❝❤✐Ị✉ t❤× ♠ä✐ ❞➲② ❤é✐ tơ ②Õ✉ ➤Ị✉ ❤é✐ tơ ♠➵♥❤✱ ❞♦ ➤ã... ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❬✽❪✱ ❬✾❪✱ ❬✷✹❪✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ➤Ị t➭✐ ♥❤➺♠ ♥❣❤✐➟♥ ❝ø✉ ♠ét sè ♣❤➢➡♥❣ ổ ị ệ trì t tử ❉ù❛ tr➟♥ ✈✐Ư❝ sư ❞ơ♥❣ ♣❤➢➡♥❣ tr×♥❤ ✭✵✳✹✮ ➤Ĩ ❤✐Ư✉ ỉ ỗ trì tr ề t ❝đ❛ ◆❣✉②Ơ♥ ❇➢ê♥❣ ❬✻❪✱ ❝❤ó♥❣... ❤ä❝✱ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ✷✳ ▼ơ❝ t✐➟✉ ❝đ❛ ➤Ị t➭✐ ✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ æ♥ ị ệ trì t tử ệ ❝❛♦ ♥➝♥❣ ❧ù❝ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ♥❤ã♠ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✳ ✲ P❤ô❝ ✈ô ❝❤♦ ❝➠♥❣ t➳❝ ◆❈❑❍✱ ➤➭♦ t➵♦