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

❇é ❣✐➳♦ ❞ô❝ ✈➭ ➤➭♦ t➵♦
➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥

◆❣✉②Ô♥ ❚❤Þ ❚❤✉ ❚❤ñ②

P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣
❣✐➯✐ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥ ➤✐Ö✉

➜Ò t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ❝✃♣ ❜é
▼➲ sè✿ ❇✷✵✵✾✲❚◆✵✼✲✵✸

❚❤➳✐ ◆❣✉②➟♥ ✲ ✷✵✶✵

✶


❉❛♥❤ s➳❝❤ ♥❤÷♥❣ ♥❣➢ê✐
t❤❛♠ ❣✐❛ t❤ù❝ ❤✐Ö♥ ➤Ò t➭✐
❙❚❚

❍ä ✈➭ t➟♥

◆é✐ ❞✉♥❣ ➤➲ t❤ù❝ ❤✐Ö♥

✶

◆❣✉②Ô♥ ❚❤❛♥❤ ▼❛✐

❚æ ❝❤ø❝ ❙❡♠✐♥❛r✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ò

✷

◆❣✉②Ô♥ ❚✃t ❚❤➽♥❣

❉Þ❝❤ t➭✐ ❧✐Ö✉✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ò

✸

❚r➢➡♥❣ ▼✐♥❤ ❚✉②➟♥

❚æ ❝❤ø❝ ❙❡♠✐♥❛r✱ ①➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ò

✹

◆❣✉②Ô♥ ❚❤❛♥❤ ❍➢ê♥❣

❳➞② ❞ù♥❣ ❝❤✉②➟♥ ➤Ò

➜➡♥ ✈Þ ♣❤è✐ ❤î♣ ❝❤Ý♥❤
❙❚❚

❚➟♥ ➤➡♥ ✈Þ

◆é✐ ❞✉♥❣ ♣❤è✐ ❤î♣ ♥❣❤✐➟♥ ❝ø✉

✶

❱✐Ö♥ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥

❚❤➯♦ ❧✉❐♥✱ ❙❡♠✐♥❛r
❱✐Õt ❝❤✉♥❣ ❜➭✐ ❜➳♦ ❦❤♦❛ ❤ä❝

✷

❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱

❚r❛♦ ➤æ✐✱ ❚❤➯♦ ❧✉❐♥

❚r➢ê♥❣ ➜❍❑❍❚◆ ❍➭ ◆é✐
✸

❑❤♦❛ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥
➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥

✷

❚r❛♦ ➤æ✐✱ ❚❤➯♦ ❧✉❐♥


▼ô❝ ❧ô❝

▼ë ➤➬✉

✶✶

❈❤➢➡♥❣ ✶✳

✶✳✶✳

✶✳✷✳

❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤ ✈➭ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö

✶✺

❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✶✳✶✳ ❇➭✐ t♦➳♥ ➤➷t ❦❤➠♥❣ ❝❤Ø♥❤

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✶✳✷✳ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

❍Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✶✳✷✳✶✳ P❤➳t ❜✐Ó✉ ❜➭✐ t♦➳♥
✶✳✷✳✷✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ö♠

✶✳✷✳✸✳ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ tr♦♥❣ tr➢ê♥❣ ❤î♣ ➤➷❝ ❜✐Öt

❈❤➢➡♥❣ ✷✳

✷✳✶✳

❍✐Ö✉ ❝❤Ø♥❤ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈í✐ t♦➳♥ tö ➤➡♥ ➤✐Ö✉

P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ✈➭ tè❝ ➤é ❤é✐ tô ❝ñ❛ ♥❣❤✐Ö♠ ❤✐Ö✉
❝❤Ø♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣ ❜❐❝ ❦❤➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✷✳✶✳✶✳ P❤➢➡♥❣ tr×♥❤ ❤✐Ö✉ ❝❤Ø♥❤

✷✳✶✳✷✳ ❙ù ❤é✐ tô ❝ñ❛ ♥❣❤✐Ö♠ ❤✐Ö✉ ❝❤Ø♥❤
✷✳✶✳✸✳ ❚❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤

✷✳✶✳✹✳ ❚è❝ ➤é ❤é✐ tô ❝ñ❛ ♥❣❤✐Ö♠ ❤✐Ö✉ ❝❤Ø♥❤

✷✳✷✳

✷✽

✷✳✷✳✶✳ ▼➠ t➯ ♣❤➢➡♥❣ ♣❤➳♣
✷✳✷✳✷✳ ❙ù ❤é✐ tô

❈❤➢➡♥❣ ✸✳

❑Õt q✉➯ tÝ♥❤ t♦➳♥ t❤ö ♥❣❤✐Ö♠

✸

✹✼


✸✳✶✳

❱Ý ❞ô ✸✳✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✸✳✷✳

❱Ý ❞ô ✸✳✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵

❑Õt ❧✉❐♥ ❝❤✉♥❣

✺✹

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✺✺

✹


❚ã♠ t➽t ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉
✶✳ ❚❤➠♥❣ t✐♥ ❝❤✉♥❣

✲ ❚➟♥ ➤Ò t➭✐✿ P❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣ ❣✐➯✐ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥
➤✐Ö✉
✲ ▼➲ sè✿ ❇✷✵✵✾✲❚◆✵✼✲✵✸
✲ ❚❤ê✐ ❣✐❛♥ t❤ù❝ ❤✐Ö♥✿ ✷✵✵✾✲✷✵✶✵
✲ ❈❤ñ ♥❤✐Ö♠ ➤Ò t➭✐✿ ❚❙✳ ◆❣✉②Ô♥ ❚❤Þ ❚❤✉ ❚❤ñ②
➜✐Ö♥ t❤♦➵✐✿ ✵✾✶✷✷✶✶✽✺✽❀ ❊✲♠❛✐❧✿ t❤✉t❤✉②✷✷✵✸✻✾❅❣♠❛✐❧✳❝♦♠
✲ ❈➡ q✉❛♥ ❝❤ñ tr×✿ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❑❤♦❛ ❤ä❝✱ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥
✷✳ ▼ô❝ t✐➟✉ ❝ñ❛ ➤Ò t➭✐

✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ æ♥ ➤Þ♥❤ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥
➤✐Ö✉✳
✲ ◆➞♥❣ ❝❛♦ ♥➝♥❣ ❧ù❝ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ♥❤ã♠ t❤ù❝ ❤✐Ö♥ ➤Ò t➭✐✳
✲ P❤ô❝ ✈ô ❝❤♦ ❝➠♥❣ t➳❝ ◆❈❑❍✱ ➤➭♦ t➵♦ ➜❍ ✈➭ ❙➜❍ ❝❤✉②➟♥ ♥❣➭♥❤ ❚♦➳♥
ø♥❣ ❞ô♥❣ ❝ñ❛ ➜➵✐ ❤ä❝✳
✸✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤

✲ ◆❣❤✐➟♥ ❝ø✉ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥ ➤✐Ö✉❀

✲ ◆❣❤✐➟♥ ❝ø✉ sù ❤é✐ tô ✈➭ ➤➳♥❤ ❣✐➳ tè❝ ➤é ❤é✐ tô ❝ñ❛ ♥❣❤✐Ö♠ ❤✐Ö✉ ❝❤Ø♥❤ tr➟♥
❝➡ së ❝❤ä♥ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤❀
✹✳ ❑Õt q✉➯ ❝❤Ý♥❤ ➤➵t ➤➢î❝

✲ ❳➞② ❞ù♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❤Ö ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥ ➤✐Ö✉ ❞ù❛
tr➟♥ ♠ét ➤Ò ①✉✃t ❝ñ❛ ◆❣✉②Ô♥ ❇➢ê♥❣❀
✲ ➜➢❛ r❛ ❝➳❝❤ ❝❤ä♥ ❣✐➳ trÞ ❝ñ❛ t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤ ❤❐✉ ♥❣❤✐Ö♠ t❤❡♦ ♥❣✉②➟♥
❧Ý ➤é ❧Ö❝❤ s✉② ré♥❣❀ ➜➳♥❤ ❣✐➳ tè❝ ➤é ❤é✐ tô ❝ñ❛ ♥❣❤✐Ö♠ ❤✐Ö✉ ❝❤Ø♥❤ ø♥❣ ✈í✐
t❤❛♠ sè ❤✐Ö✉ ❝❤Ø♥❤ ➤➲ ❝❤ä♥❀

✺


✲ ❳➞② ❞ù♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❧➷♣✱ ❝❤ø♥❣ ♠✐♥❤ sù ❤é✐ tô ❝ñ❛ ♣❤➢➡♥❣
♣❤➳♣❀
✲ ➜➢❛ r❛ ✈Ý ❞ô sè ♠✐♥❤ ❤ä❛ ❝❤♦ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉✳
✺✳ ❙➯♥ ♣❤➮♠ ❝ñ❛ ➤Ò t➭✐

✺✳✶✳ ❙➯♥ ♣❤➮♠ ❦❤♦❛ ❤ä❝

• ❈➳❝ ❦Õt q✉➯ ❝ñ❛ ➤Ò t➭✐ ➤➢î❝ ❝➠♥❣ ❜è tr♦♥❣ ❝➳❝ ❝➠♥❣ tr×♥❤
❬✶❪✳ ◆❣✳ ❇✉♦♥❣✱ ◆❣✳ ❚✳ ❚✳ ❚❤✉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✱

✭➜➲ ♥❤❐♥ ➤➝♥❣ ♥➝♠ ✷✵✶✵✮✳
❬✺❪✳ ◆❣✉②Ô♥ ❚❤Þ ❚❤✉ ❚❤ñ②✱ ➜➷♥❣ ❚ó ❍å✐ ✭✷✵✶✵✮✱ ✧❑Õt q✉➯ sè ❝ñ❛ ♣❤➢➡♥❣
♣❤➳♣ ❤✐Ö✉ ❝❤Ø♥❤ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ t♦➳♥ tö ➤➡♥ ➤✐Ö✉✧✱
❈➠♥❣ ♥❣❤Ö ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✱ ✼✵✭✽✮✱

❚➵♣ ❝❤Ý ❑❤♦❛ ❤ä❝ ✈➭

♣♣✳ ✻✶✲✻✹✳

• ❈➳❝ ❦Õt q✉➯ ❝ñ❛ ➤Ò t➭✐ ➤➲ ➤➢î❝ ❜➳♦ ❝➳♦ t➵✐✿
1. ❚❤❡ ✽t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❙♣r✐♥❣ ❙❝❤♦♦❧✴ ❲♦r❦s❤♦♣ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ■ts
❆♣♣❧✐❝❛t✐♦♥s✱ ◆❤❛ ❚r❛♥❣✱ ▼❛r❝❤ ✶✲✸✱ ✷✵✶✵✳

2. ❚r➢ê♥❣ t♦➳♥ ❈■▼P❆✲❯◆❊❙❈❖✲❱■❊❚◆❆▼✱ ✧❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈➭
❝➳❝ ✈✃♥ ➤Ò ❝ã ❧✐➟♥ q✉❛♥✧✱ ❍➭ ◆é✐✱ ✶✵✲✷✶✴✵✺✴✷✵✶✵✳

✻


3. ❍é✐ t❤➯♦ ❚è✐ ➢✉ ✈➭ ❚Ý♥❤ t♦➳♥ ❦❤♦❛ ❤ä❝ ❧➬♥ t❤ø ✽✱ ❇❛ ✈×✱ ✷✵✲✷✸✴✵✹✴✷✵✶✵✳
✺✳✷✳ ❙➯♥ ♣❤➮♠ ➤➭♦ t➵♦

1. ❍➢í♥❣ ❞➱♥ ✵✺ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ✱ tr♦♥❣ ➤ã ✵✷ ❧✉❐♥ ✈➝♥ ➤➲ ❜➯♦ ✈Ö t❤➭♥❤
❝➠♥❣ ♥➝♠ ✷✵✵✾❀ ✵✸ ❧✉❐♥ ✈➝♥ sÏ ❜➯♦ ✈Ö ✈➭♦ t❤➳♥❣ ✶✶✴✷✵✶✵✳

2. ❍➢í♥❣ ❞➱♥ ✵✸ s✐♥❤ ✈✐➟♥ ◆❈❑❍✱ ➤➲ ❜➯♦ ✈Ö ✈➭ ➤➵t ❦Õt q✉➯ tèt✱ tr♦♥❣ ➤ã
❝ã ✵✶ s✐♥❤ ✈✐➟♥ ➤➢î❝ ❣✐➯✐ ❦❤✉②Õ♥ ❦❤Ý❝❤ tr♦♥❣ ❝✉é❝ t❤✐ ✧❙✐♥❤ ✈✐➟♥ ♥❣❤✐➟♥
❝ø✉ ❦❤♦❛ ❤ä❝ t♦➭♥ q✉è❝ ✷✵✵✾✧✱ ✵✶ s✐♥❤ ✈✐➟♥ ➤➢î❝ ➤Ò ❝ö ❞ù t❤✐ ✧❙✐♥❤ ✈✐➟♥
♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ t♦➭♥ q✉è❝ ✷✵✶✵✧✳

3. ❍➢í♥❣ ❞➱♥ ✵✸ s✐♥❤ ✈✐➟♥ ❧➭♠ ❦❤ã❛ ❧✉❐♥ tèt ♥❣❤✐Ö♣ ➤➲ ❜➯♦ ✈Ö ✈➭ ➤➵t ➤✐Ó♠
①✉✃t s➽❝✳

4. ➜Ò t➭✐ ❣ã♣ ♣❤➬♥ ♣❤ô❝ ✈ô ❝❤♦ ✈✐Ö❝ ❣✐➯♥❣ ❞➵② ❝❤✉②➟♥ ➤Ò ✧❇➭✐ 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 ❤✐Ö✉ ✈➭ ❝❤÷ ✈✐Õt t➽t

H

❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ù❝

X

❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ù❝

X∗

❦❤➠♥❣ ❣✐❛♥ ❧✐➟♥ ❤î♣ ❝ñ❛

Rn

❦❤➠♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡

Rn+

t❐♣ ❝➳❝ ✈Ð❝ t➡ ❦❤➠♥❣ ➞♠ ✭♦rt❤❛♥t ❦❤➠♥❣ ➞♠✮ ❝ñ❛

∅

t❐♣ rç♥❣

X

n ❝❤✐Ò✉

x := y

x ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ❜➺♥❣ y

∀x

✈í✐ ♠ä✐

∃x

tå♥ t➵✐

inf F (x)

x∈X

x

x

✐♥❢✐♠✉♠ ❝ñ❛ t❐♣

{F (x) : x ∈ X}

❛r❣ min F (x)

t❐♣ ❝➳❝ ➤✐Ó♠ ❝ù❝ t✐Ó✉ ❝ñ❛ ❤➭♠

I

➳♥❤ ①➵ ➤➡♥ ✈Þ

x∈X

A∩B

❆ ❣✐❛♦ ✈í✐ ❇

AT

♠❛ tr❐♥ ❝❤✉②Ó♥ ✈Þ ❝ñ❛ ♠❛ tr❐♥

a∼b

a t➢➡♥❣ ➤➢➡♥❣ ✈í✐ b

A∗

t♦➳♥ tö ❧✐➟♥ ❤î♣ ❝ñ❛ t♦➳♥ tö

D(A)

♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❝ñ❛ t♦➳♥ tö

R(A)

♠✐Ò♥ ❣✐➳ trÞ ❝ñ❛ t♦➳♥ tö

xk → x
xk

x

F tr➟♥ X

❞➲②

A

A

{xk } ❤é✐ tô ♠➵♥❤ tí✐ x

❞➲②

{xk } ❤é✐ tô ②Õ✉ tí✐ x

✶✵

A

A

Rn


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