❑Õt ❤î♣ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ✈➭ ❤➭♠ ♣❤➵t ❣✐➯✐ ❜➭✐ t♦➳♥ ❜✃t
➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ➤➡♥ ➤✐Ö✉
➜❐✉ ❳✉➞♥ ▲➢➡♥❣
✭❛✮
❚ã♠ t➽t✳
❚r♦♥❣ ❜➭✐ ❜➳♦ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❦Õt ❤î♣ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ♣❤➵t ✭❬✷❪✮ ✈➭ ♣❤➢➡♥❣
♣❤➳♣ ❝❤✐Õ✉ ➤Ó ❣✐➯✐ ♠ét ❧í♣ ❝➳❝ ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✱ ❦Ý ❤✐Ö✉ ❱■P
(D, F )
✱
tr♦♥❣ ➤ã
D
❧➭ ♠ét t❐♣ ❝♦♥ ❧å✐ ➤ã♥❣ ❦❤➳❝ rç♥❣ ❝ñ❛
Rn
✱
F:K→Rn
❧➭ ♠ét ❤➭♠ ➤➡♥ ➤✐Ö✉
✈➭ ❧✐➟♥ tô❝ ▲✐♣s❝❤✐t③ tr➟♥ ♠✐Ò♥
K
❝❤ø❛
D
✳ ❚r➢í❝ t✐➟♥✱ ❜➭✐ t♦➳♥ ❜❛♥ ➤➬✉ ➤➢î❝ ➤➢❛ ✈Ò ♠ét
❞➲② ❝➳❝ ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ tr➟♥ ♠✐Ò♥
K
❝❤ø❛
D
✱ tr♦♥❣ ➤ã
K
t❤á❛ ♠➲♥
tÝ♥❤ ❝❤✃t ❤×♥❤ ❝❤✐Õ✉ ❊✉❝❧✐❞ ❝ñ❛ ♠ét ➤✐Ó♠ ❜✃t ❦ú ❧➟♥
K
❝ã ❝➠♥❣ t❤ø❝ tÝ♥❤ ➤➡♥ ❣✐➯♥✳ ❚✐Õ♣
➤ã✱ sö ❞ô♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ➤Ó ❣✐➯✐ ❞➲② ❝➳❝ ❜➭✐ t♦➳♥ ♥➭②✳ ❑❤✐ ➤ã✱ ♥Õ✉ ♠✐Ò♥
D
t❤á❛
♠➲♥ ♠ét ✈➭✐ ❣✐➯ t❤✐Õt ♥❤✃t ➤Þ♥❤✱ t❤× ➤✐Ó♠ ❣✐í✐ ❤➵♥ ❜✃t ❦ú ❝ñ❛ ❞➲② ♥❣❤✐Ö♠ ❝ñ❛ ❝➳❝ ❜➭✐
t♦➳♥ ♥➭② ❧➭ ♠ét ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❜❛♥ ➤➬✉✳ ❇➺♥❣ ❝➳❝❤ ♥➭②✱ t❛ ❧♦➵✐ ❜á ➤➢î❝ ❦❤ã ❦❤➝♥
❦❤✐ tÝ♥❤ t♦➳♥ ❤×♥❤ ❝❤✐Õ✉ tr♦♥❣ ❝➳❝ t❤✉❐t t♦➳♥ ❝❤✐Õ✉ ❣✐➯✐ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✳ ❈❤ó♥❣
t➠✐ ❝ò♥❣ ➤➢❛ r❛ ♠ét ✈➭✐ ✈Ý ❞ô ➤Ó ♠✐♥❤ ❤ä❛ ♣❤➢➡♥❣ ♣❤➳♣ ♥➭②✳
✶✳
●✐í✐ t❤✐Ö✉
❈❤♦
D⊂Rn
❧➭ ♠ét t❐♣ ❧å✐ ➤ã♥❣ ❦❤➳❝ rç♥❣ ✈➭ ♠ét ➳♥❤ ①➵
F:Rn→Rn
✳ ❳Ðt ❜➭✐
t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ s❛✉✿
❚×♠
x∗∈D,
s❛♦ ❝❤♦
hF(x∗), x −x∗i>0 ; ∀x∈D. (V IP (D, F ))
❚❐♣ ♥❣❤✐Ö♠ ❝ñ❛ ❱■P✭
D, F
✮ ➤➢î❝ ❦Ý ❤✐Ö✉ ❧➭ ❙❖▲✲❱■P✭
D, F
✮✳
◆Õ✉
F
❧➭ ➤➵♦ ❤➭♠ ❝ñ❛ ♠ét ❤➭♠ ❧å✐
f
✱ t❤× ❜➭✐ t♦➳♥ ❱■P
(D, F )
t➢➡♥❣ ➤➢➡♥❣ ✈í✐
❜➭✐ t♦➳♥ t×♠ ❝ù❝ t✐Ó✉ ❝ñ❛
f
tr➟♥
D
✳ ❚✉② ♥❤✐➟♥ ❦❤➠♥❣ ♣❤➯✐ ♠ä✐ ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝
❜✐Õ♥ ♣❤➞♥ ❱■P✭
D, F )
➤Ò✉ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ ❧å✐✳
❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ❝ã ♥❤✐Ò✉ ø♥❣ ❞ô♥❣ ré♥❣ r➲✐ tr♦♥❣ t❤ù❝ tÕ✿ ❝➳❝ ❜➭✐ t♦➳♥
❝➞♥ ❜➺♥❣ ♠➵♥❣ ❣✐❛♦ t❤➠♥❣ ✭❬✺❪✮✱ ❝➳❝ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ❦✐♥❤ tÕ ✭❬✻❪✮✱ ❝➳❝ ❜➭✐ t♦➳♥ ❝➞♥
❜➺♥❣ ❞✐ ❝➢ ✭❬✼❪✮✱ ❝➳❝ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ t➭✐ ❝❤Ý♥❤✱ ♠➵♥❣ ❦✐Õ♥ t❤ø❝ ✭❬✽❪✮✱ ✈✳✈✳ ➤Ò✉ ❝ã t❤Ó
♠➠ t➯ ❞➢í✐ ❞➵♥❣ ♠ét ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥✳
P❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ✭❬✶❪✮ ❧➭ ♠ét ♣❤➢➡♥❣ ♣❤➳♣ ❝➡ ❜➯♥ ✈➭ ❦❤➳ ❤✐Ö✉ q✉➯ ➤Ó ❣✐➯✐ ❝➳❝
❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈í✐ ❣✐➯ t❤✐Õt
F
❣✐➯ ➤➡♥ ➤✐Ö✉ ✈➭ ❧✐➟♥ tô❝✳ ❚rë ♥❣➵✐
❝❤Ý♥❤ tr♦♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ♥➭② ❧➭ ✈✐Ö❝ tÝ♥❤ t♦➳♥ ❤×♥❤ ❝❤✐Õ✉ ❧➟♥ ♠ét t❐♣ ❧å✐ ❜✃t ❦ú ❦❤➠♥❣
❤Ò ➤➡♥ ❣✐➯♥✳ ➜ã ❧➭ ♠ét ❜➭✐ t♦➳♥ q✉② ❤♦➵❝❤ t♦➭♥ ♣❤➢➡♥❣ ✈í✐ ♠✐Ò♥ ①➳❝ ➤Þ♥❤ ❧å✐✳ ◆Õ✉
D
❦❤➠♥❣ ❝ã ❤×♥❤ ❞➵♥❣ ➤➷❝ ❜✐Öt t❤× ✈✐Ö❝ ①➳❝ ➤Þ♥❤ ❤×♥❤ ❝❤✐Õ✉ ❧➟♥
D
sÏ ❣➷♣ ❦❤ã ❦❤➝♥✳
▼ét ❝➳❝❤ t✐Õ♣ ❝❐♥ ➤Ó ❣✐➯✐ q✉②Õt ❦❤ã ❦❤➝♥ ♥➭② ❧➭ ❦ü t❤✉❐t ❤➭♠ ♣❤➵t ❝❤♦ ♣❤Ð♣
➤➢❛ ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ tr➟♥ ♠✐Ò♥ ❧å✐ ➤ã♥❣ ❜✃t ❦ú ✈Ò ❝➳❝ ❜➭✐ t♦➳♥ ❜✃t
➤➻♥❣ t❤ø❝ tr➟♥ ♠ét ♠✐Ò♥
K
❝❤ø❛ ♠✐Ò♥ ❧å✐ ❜❛♥ ➤➬✉✳
ý
t➢ë♥❣ ❝ñ❛ ❝❤ó♥❣ t➠✐ ❧➭ ➤è✐ ✈í✐ ❱■P✭
D, F
✮ tr♦♥❣ ➤ã
D
❧➭ ♠ét ♠✐Ò♥ ❧å✐ ➤ã♥❣ ❜✃t
✶
◆❤❐♥ ❜➭✐ ♥❣➭② ✵✽✴✽✴✷✵✵✽✳ ❙ö❛ ❝❤÷❛ ①♦♥❣ ✸✵✴✾✴✷✵✵✽✳
❦ú✱ tr➢í❝ t✐➟♥ ❞ï♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ♣❤➵t ➤Ó ➤➢❛ ♥ã ✈Ò ♠ét ❞➲② ❝➳❝ ❜➭✐ t♦➳♥ tr➟♥ ♠ét
♠✐Ò♥
K
❝❤ø❛
D
✱ s❛✉ ➤ã ❞ï♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ➤Ó ❣✐➯✐ ♠ç✐ ❜➭✐ t♦➳♥ tr➟♥
K
✳ ▼✐Ò♥
K
➤➢î❝ ①➳❝ ➤Þ♥❤ s❛♦ ❝❤♦ ✈✐Ö❝ tÝ♥❤ t♦➳♥ ❤×♥❤ ❝❤✐Õ✉ ❝ñ❛ ♠ét ➤✐Ó♠ ❧➟♥
K
❧➭ ❞Ô ❞➭♥❣✳ ❚r♦♥❣
tr➢ê♥❣ ❤î♣
K
❧➭ ❤×♥❤ ❤é♣✱ ❤×♥❤ ❝➬✉ ❤❛② ❦❤➠♥❣ ❣✐❛♥ ❝♦♥✱ ✈× ♣❤Ð♣ ❝❤✐Õ✉ ❝ñ❛ ♠ét ➤✐Ó♠
❧➟♥
K
❝ã ❝➠♥❣ t❤ø❝ ❤✐Ó♥ ➤➡♥ ❣✐➯♥ ♥➟♥ trë ♥❣➵✐ ❝❤Ý♥❤ ❝ñ❛ ♣❤➢➡♥❣ ♣❤➳♣ ➤➢î❝ ❦❤➽❝ ♣❤ô❝✳
✷✳
❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ
✷✳✶✳ ❈➳❝ ❦Õt q✉➯ ✈Ò sù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❱■P✭❉✱❋✮
➜Þ♥❤ ❧ý ✶✳
✭❬✶❪✮ ✭✐✮✳
◆Õ✉
D
❧➭ t❐♣ ❧å✐ ❝♦♠♣➝❝ ❦❤➳❝ rç♥❣ ✈➭
F
❧✐➟♥ tô❝ tr➟♥
D
t❤× ❱■P✭
D, F
✮
❝ã Ýt ♥❤✃t ♠ét ♥❣❤✐Ö♠✳
✭✐✐✮✳
◆Õ✉
F
t❤♦➯ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❜ø❝
lim
x∈D;||x||→∞ hF(x)−F(x), x −xi
||x−x|| =∞
✈í✐
x∈D
♥➭♦ ➤ã✱ t❤× ❜➭✐ t♦➳♥ ❱■P✭
D, F
✮ ❧✉➠♥ ❝ã ♥❣❤✐Ö♠✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳
✭❬✶❪✮ ✭✐✮✳ ❈❤♦
F:D→Rn
✱
F
➤➢î❝ ❣ä✐ ❧➭
➤➡♥ ➤✐Ö✉
tr➟♥
D
♥Õ✉
hF(x)−F(y), x −yi>0 ; ∀x, y ∈D.
✭✐✐✮✳
F
➤➢î❝ ❣ä✐ ❧➭
➤➡♥ ➤✐Ö✉ ♥❣➷t
tr➟♥
D
♥Õ✉
hF(x)−F(y), x −yi>0 ; ∀x, y ∈D;x6=y.
✭✐✐✐✮✳
F
➤➢î❝ ❣ä✐ ❧➭
➤➡♥ ➤✐Ö✉ ♠➵♥❤
tr➟♥
D
♥Õ✉ tå♥ t➵✐
α > 0
s❛♦ ❝❤♦
hF(x)−F(y), x −yi>α||x−y||2;∀x, y ∈D.
✭✐✈✮✳
F
➤➢î❝ ❣ä✐ ❧➭
❧✐➟♥ tô❝ ▲✐♣s❝❤✐t③
tr➟♥
D
♥Õ✉ tå♥ t➵✐
L > 0
s❛♦ ❝❤♦
||F(x)−F(y)|| 6L.||x−y|| ;∀x, y ∈D.
✭✈✮✳
F
➤➢î❝ ❣ä✐ ❧➭
❣✐➯ ➤➡♥ ➤✐Ö✉
tr➟♥
D
ø♥❣ ✈í✐ ❙❖▲✲❱■P✭
D, F
✮ ♥Õ✉ ❙❖▲✲❱■P
(D, F )6=∅
✈➭
∀x∗∈SOL −V IP (D, F )
t❤♦➯
hF(x), x −x∗i>0; ∀x∈D.
tr♦♥❣ ➤ã
x∗
❧➭ ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❱■P✭
D, F )
✳
➜Þ♥❤ ❧ý ✷✳
✭❬✶❪✮ ✭✐✮✳
●✐➯ sö
F
➤➡♥ ➤✐Ö✉ ♥❣➷t tr➟♥
D
✳ ❑❤✐ ➤ã ♥❣❤✐Ö♠ ❝ñ❛ ❱■P✭
D, F
✮ ♥Õ✉
❝ã ❧➭ ❞✉② ♥❤✃t✳
✭✐✐✮✳
●✐➯ sö
F
➤➡♥ ➤✐Ö✉ ♠➵♥❤ tr➟♥
D
✳ ❑❤✐ ➤ã ❱■P✭
D, F
✮ ❝ã ➤ó♥❣ ♠ét ♥❣❤✐Ö♠✳
✷✳✷✳ P❤Ð♣ ❝❤✐Õ✉ ✈➭ ♠è✐ q✉❛♥ ❤Ö ✈í✐ ❱■P
➜Þ♥❤ ♥❣❤Ü❛ ✷✳
✭❬✶❪✮ ❈❤♦
D
❧➭ t❐♣ ❝♦♥ ❧å✐ ➤ã♥❣✱ ❦❤➳❝ rç♥❣ ❝ñ❛
Rn
✱ ✈í✐
x∈Rn
t❛ ➤➷t
d(D, x) = inf
y∈D||x−y||.
❍➭♠
d(D, .)
➤➢î❝ ❣ä✐ ❧➭
❤➭♠ ❦❤♦➯♥❣ ❝➳❝❤ t❤❡♦ ❝❤✉➮♥ ❊✉❝❧✐❞
tõ
➤✐Ó♠ tr♦♥❣
Rn
➤Õ♥
D
✳ ◆Õ✉ tå♥ t➵✐
y∈D
s❛♦ ❝❤♦
d(D, x) = ||y−x||
t❤× ② ➤➢î❝ ❣ä✐ ❧➭
❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝
❤❛②
❤×♥❤ ❝❤✐Õ✉ t❤❡♦ ❝❤✉➮♥ ❊✉❝❧✐❞
❝ñ❛ ① tr➟♥
D
✈➭ ➤➢î❝ ❦Ý ❤✐Ö✉
❧➭
PD(x).
◆❤➢ t❛ ➤➲ ❜✐Õt ♥Õ✉
D
❧➭ t❐♣ ❧å✐ ➤ã♥❣✱ ❦❤➳❝ rç♥❣ t❤× ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ ❜❛♦ ❣✐ê
❝ò♥❣ tå♥ t➵✐ ❞✉② ♥❤✃t✳
❚r♦♥❣ tr➢ê♥❣ ❤î♣
K
❧➭ ❤×♥❤ ❤é♣✱ ✈í✐ ❝❤✉➮♥ ❊✉❝❧✐❞✱ ✈✃♥ ➤Ò tÝ♥❤ t♦➳♥ ❤×♥❤ ❝❤✐Õ✉
❝ñ❛ ♠ét ➤✐Ó♠ ❧➟♥
K
trë ❧➟♥ ➤➡♥ ❣✐➯♥✳ ❚❤❐t ✈❐②✱ ❣✐➯ sö
K
❧➭ ❤×♥❤ ❤é♣ s❛✉
K={x= (x1, x2, ..., xn)T∈Rn|ai6xi6bi;i= 1,2, ..., n},
a= (a1, a2, ..., an)T;b= (b1, b2, ..., bn)T∈Rn.
❑❤✐ ➤ã ❤×♥❤ ❝❤✐Õ✉ ❝ñ❛
x
❧➟♥
K
➤➢î❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉
(PK(x))i=
ai, xi< ai
xi, xi∈[ai;bi]
bi, xi> bi.
●✐➯ sö
C
❧➭ ❤×♥❤ ❝➬✉ t➞♠
I(a1, a2, ..., an)∈Rn
✱ ❜➳♥ ❦Ý♥❤
R
✱ ❝ã ♣❤➢➡♥❣ tr×♥❤
C={x∈Rn|
n
X
i=1
(xi−ai)26R2}
✈➭
b= (b1, b2, ..., bn)T∈Rn
✳ ❈➬♥ t×♠ ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝
PC(b)
❝ñ❛ ❜ ❧➟♥
C
✳ ◆Õ✉
b∈C
t❤×
PC(b)≡b
✳ ◆Õ✉ ❜ ♥➺♠ ♥❣♦➭✐
C
✱ ❤×♥❤ ❝❤✐Õ✉ ❝ñ❛ ❜ ❧➟♥
C
❧➭ ❣✐❛♦ ➤✐Ó♠ ❝ñ❛ ➤➢ê♥❣ t❤➻♥❣
♥è✐ ❜ ✈➭ t➞♠
I
❝ñ❛
C
✈í✐ ♠➷t ❝➬✉
S={x∈Rn|
n
P
i=1
(xi−ai)2=R2}
✳
P❤➢➡♥❣ tr×♥❤ t❤❛♠ sè ❝ñ❛ ➤➢ê♥❣ t❤➻♥❣ ♥➭② ♥❤➢ s❛✉✿
∆ = {x∈Rn|xi=ai+t(bi−ai) ; i= 1,2, ..., n ;t∈R}.
❚❤❛②
xi=ai+t(bi−ai)
✈➭♦ ♣❤➢➡♥❣ tr×♥❤
S
✱ t❛ ❝ã
t2n
P
i=1
(bi−ai)2=R2
✳
❉♦ ➤ã
t=R
sn
P
i=1
(bi−ai)2
.
❱❐② ❤×♥❤ ❝❤✐Õ✉
PC(b)
❝ñ❛ ❜ ❧➟♥ ❈ ❝ã t♦➵ ➤é
xi=ai+ (bi−ai)R
sn
P
i=1
(bi−ai)2
.
❑❤✐
L⊂Rn
❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❦ ❝❤✐Ò✉ ✈í✐ ❝➡ së
B={η1, η2, ..., ηk}
✳ ●✐➯ sö
b∈Rn
❧➭ ♠ét
➤✐Ó♠ ❜✃t ❦×✱
a=
k
P
j=1
cjηj∈L;cj
❧➭ ❝➳❝ ❤➺♥❣ sè t❤ù❝✱ s❛♦ ❝❤♦
f=b−a
t❤♦➯
hf, ηji= 0
✱
✈í✐ ♠ä✐
j= 1,2, ..., k
✳ ❑❤✐ ➤ã ❛ ❧➭ ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ ❝ñ❛ ❜ ❧➟♥
L
✳
❚❤❐t ✈❐②✱ ❣✐➯ sö
c∈L
✱ ✈×
f
✈✉➠♥❣ ❣ã❝ ✈í✐ ♠ä✐ ✈Ð❝t➡ tr♦♥❣ ❝➡ së ❝ñ❛
L
✱ ♥➟♥ ♥ã
✈✉➠♥❣ ❣ã❝ ✈í✐ ♠ä✐ ✈Ð❝t➡ t❤✉é❝
L
✳ ❉♦ ➤ã
||b−c||2=hb−a+a−c, b −a+a−ci
=hb−a, b −ai+ha−c, a −ci+ 2hb−a, a −ci
=||b−a||2+||a−c||2
>||b−a||2.
❉♦ ➤ã ❛ ❧➭ ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ ❝ñ❛ ❜ ❧➟♥
L
✳
●✐ê t❛ t×♠ ❛❀ ✈í✐ ♠ä✐
i= 1, . . . , k
✱ t❛ ❝ã✿
hf, ηii= 0,
hb−
k
X
j=1
cjηj, ηii= 0,
k
X
j=1hηi, ηjicj=hb, ηii.
➜➷t
Aij =hηi, ηji;Di=hb, ηii;i= 1,2, ..., k ;j= 1,2, ..., k
✳ ❑❤✐ ➤ã t❛ t❤✉ ➤➢î❝ ♠ét ❤Ö
♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ ❦ ♣❤➢➡♥❣ tr×♥❤✱ ❦ ➮♥
Ac =D ,
tr♦♥❣ ➤ã
A= (Aij ),
c= (c1, c2, ..., ck)T;D= (D1, D2, ..., Dk)T.
❍➡♥ ♥÷❛✱ ❞♦ ❝➳❝❤ ①➳❝ ➤Þ♥❤
A
✱ ➤➞② ❧➭ ♠ét ♠❛ tr❐♥ ①➳❝ ➤Þ♥❤ ❞➢➡♥❣✱ s✉② r❛
det(A)6= 0
✱
♥❣❤Ü❛ ❧➭ ❤Ö tr➟♥ ❧✉➠♥ ❝❤♦ ♥❣❤✐Ö♠ ❞✉② ♥❤✃t✳ ❱×
a=
k
P
j=1
cjηj
✱ s❛✉ ❦❤✐ tÝ♥❤ ➤➢î❝
ci
✱ t❛ tÝ♥❤
➤➢î❝ ❛✳ ◆Õ✉ ❝❤ä♥
B
❧➭ ❝➡ së trù❝ ❝❤✉➮♥ ♥❣❤Ü❛ ❧➭
hηi, ηji= 0
♥Õ✉
i6=j
❀ ❂✶ ♥Õ✉
i=j
✱ t❤×
A
❧➭ ♠❛ tr❐♥ ➤➡♥ ✈Þ✱ ❞♦ ➤ã ❝ã t❤Ó tÝ♥❤ ♥❣❛② ➤➢î❝ ♥❣❤✐Ö♠
ci=Di=hb, ηii
✳
▲✐➟♥ q✉❛♥ ➤Õ♥ ❤×♥❤ ❝❤✐Õ✉ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ t❛ ❝ã ➤Þ♥❤ ❧ý s❛✉✳
▼Ö♥❤ ➤Ò ✶
✳✭❬✶❪✮
●✐➯ sö
D⊂Rn
❧➭ ♠ét t❐♣ ❧å✐ ➤ã♥❣ ✈➭
F:Rn→Rn
❧➭ ♠ét ➳♥❤ ①➵✳ ❑❤✐
➤ã t❛ ❝ã
x∗∈SOL −V IP (D, F )⇔Fnat
D(x∗) = 0,
tr♦♥❣ ➤ã
Fnat
D(x∗)≡x−PD(x−ξF (x))
✈í✐
ξ > 0
♥➭♦ ➤ã✳
✸✳
P❤➢➡♥❣ ♣❤➳♣ ❦Õt ❤î♣ ♣❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ♣❤➵t ✈➭ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉
✸✳✶✳ P❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉
ë
➤➞② t❛ ❝❤Ø ➤Ò ❝❐♣ ➤Õ♥ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ❤❛✐ ❧➬♥ ✭❡①tr❛❣r❛❞✐❡♥t ♠❡t❤♦❞✮ ✈×
♣❤➢➡♥❣ ♣❤➳♣ ♥➭② t✉② ❝➬♥ ❤❛✐ ❧➬♥ ❝❤✐Õ✉ tr♦♥❣ ♠ç✐ ❜➢í❝ ❧➷♣ ♥❤➢♥❣ ❝ã t❤Ó ➳♣ ❞ô♥❣ ❝❤♦
❱■P✭
D, F
✮ ✈í✐
F
❣✐➯ ➤➡♥ ➤✐Ö✉ ✈➭ ❧✐➟♥ tô❝ ▲✐♣s❝❤✐t③✳ ❚r♦♥❣ ❦❤✐ ➤ã ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉
♠ét ❧➬♥ ➤ß✐ ❤á✐
F
➤➡♥ ➤✐Ö✉ ♠➵♥❤✱ ❧✐➟♥ tô❝ ▲✐♣s❝❤✐t③❀ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ❜❛ ❧➬♥ ✭❤②♣❡r✲
♣❧❛♥❡ ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞✮ t✉② ➳♣ ❞ô♥❣ ➤➢î❝ ❝❤♦ ❱■P
(D, F )
✈í✐
F
❣✐➯ ➤➡♥ ➤✐Ö✉ ✈➭ ❧✐➟♥
tô❝ ✭t❤❛② ✈× ❧✐➟♥ tô❝ ▲✐♣s❝❤✐t③ ♥❤➢ ❤❛✐ ♣❤➢➡♥❣ ♣❤➳♣ ➤➬✉✮✱ ♥❤➢♥❣ ❦❤è✐ ❧➢î♥❣ tÝ♥❤ t♦➳♥
tr♦♥❣ ♠ç✐ ❜➢í❝ ❧➷♣ ❧➵✐ t➝♥❣ ❧➟♥ ➤➳♥❣ ❦Ó ❞♦ ♣❤➯✐ sö ❞ô♥❣ t❤➟♠ t❤✉❐t t♦➳♥ t×♠ ❦✐Õ♠✳
❱✐Ö❝ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ❝ã t❤Ó t❤❛♠ ❦❤➯♦ tr♦♥❣ ✭❬✶❪✮✳ ◆Õ✉ t❛ sö ❞ô♥❣
♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ❤❛✐ ❧➬♥ ❝❤♦ ❱■P ✈í✐ r➭♥❣ ❜✉é❝ ❤é♣ ❤♦➷❝ ❤×♥❤ ❝➬✉✱ ❤♦➷❝ ❦❤➠♥❣ ❣✐❛♥
❝♦♥✱ t❤× ❦❤è✐ ❧➢î♥❣ tÝ♥❤ t♦➳♥ tr♦♥❣ ♠ç✐ ✈ß♥❣ ❧➷♣ t➝♥❣ ❧➟♥ ❦❤➠♥❣ ➤➳♥❣ ❦Ó s♦ ✈í✐ ♣❤➢➡♥❣
♣❤➳♣ ❝❤✐Õ✉ ♠ét ❧➬♥✱ ✈× ❤×♥❤ ❝❤✐Õ✉ ❝ñ❛ ♠ét ➤✐Ó♠ ❧➟♥ ❝➳❝ t❐♣ ❦✐Ó✉ ♥➭② ❝ã ❝➠♥❣ t❤ø❝ ❤✐Ó♥
♥❤➢ tr➟♥ ➤➲ t❤✃②✳ ❚r♦♥❣ ❦❤✐ ➤ã✱ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤✐Õ✉ ♠ét ❧➬♥ ❝ß♥ ➤ß✐ ❤á✐ ✈✐Ö❝ tÝ♥❤ t♦➳♥
❝➳❝ ❤Ö sè ➤➡♥ ➤✐Ö✉
α
❝ñ❛ ❤➭♠
F
✳ ➜å♥❣ t❤ê✐✱ ➤Ó t❤✉❐t t♦➳♥ ❤é✐ tô✱ ❝➳❝ ❤Ö sè ❜➢í❝
tk
♣❤➯✐
♥❤á ❤➡♥
α/L2
✱ tr♦♥❣ ➤ã
L
❧➭ ❤➺♥❣ sè ▲✐♣s❝❤✐t③ ❝ñ❛
F
✭①❡♠ ❬✶❪✮✳ ❚r♦♥❣ ♥❤✐Ò✉ tr➢ê♥❣
❤î♣✱
L
trë ❧➟♥ ❦❤➳ ❧í♥ ❦❤✐ sè ❝❤✐Ò✉ ❝ñ❛ ❜➭✐ t♦➳♥ t➝♥❣ ❧➟♥✱ ♠➱✉ sè
L2
❧í♥ sÏ ❧➭♠ ❜➢í❝
tk
trë ♥➟♥ r✃t ♥❤á✳ ➜✐Ò✉ ♥➭② t❤➢ê♥❣ ➯♥❤ ❤➢ë♥❣ tí✐ tè❝ ➤é ❤é✐ tô ✈➭ ➤é ❝❤Ý♥❤ ①➳❝ ❝ñ❛
t❤✉❐t t♦➳♥ ❝❤✐Õ✉✳
❚❤✉❐t t♦➳♥ ✶✳
✭❬✶❪✮ ▲✃②
x0∈K
✈➭ ❝❤ä♥
η > 0
✳
❇➢í❝ ✵✿ ➜➷t ❦ ❂ ✵✳
❇➢í❝ ✶✿ ◆Õ✉
xk=PD(xk−ηF (xk))
✱ t❤×
xk∈SOL −V IP (D, F )
✿ ❞õ♥❣ t❤✉❐t t♦➳♥ ✈í✐
♥❣❤✐Ö♠ t❤✉ ➤➢î❝ ❧➭
xk
✳
❇➢í❝ ✷✿ ◆Õ✉
xk6=PD(xk−ηF (xk))
t❤× tÝ♥❤
xk+1
2=PD(xk−ηF (xk)),
xk+1 =PD(xk−ηF (xk+1
2)).
❈❤♦
k:= k+ 1
✈➭ q✉❛② ❧➵✐ ❜➢í❝ ✶✳
❙ù ❤é✐ tô ❝ñ❛ t❤✉❐t t♦➳♥ ♥➭② ➤➢î❝ ❜➯♦ ➤➯♠ ❜ë✐ ➤Þ♥❤ ❧ý s❛✉✳
➜Þ♥❤ ❧ý ✸
✳✭❬✶❪✮ ✭✐✮✳
●✐➯ sö
D⊂Rn
❧å✐ ➤ã♥❣ ✈➭
F:D→Rn
❧➭ ♠ét ➳♥❤ ①➵ ❣✐➯ ➤➡♥
➤✐Ö✉ tr➟♥
D
t➢➡♥❣ ø♥❣ ✈í✐ ❙❖▲✲❱■P✭
D, F
✮ ✈➭
L
✲▲✐♣s❝❤✐t③ ❧✐➟♥ tô❝ tr➟♥
D
✳ ●✐➯ sö
x∗∈
SOL −V IP (D, F )
✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐
k∈N
||xk+1 −x∗||26||xk−x∗||2−(1 −η2L2)||xk+1
2−xk||2.
✭✐✐✮✳
●✐➯ sö
F:D→Rn
❣✐➯ ➤➡♥ ➤✐Ö✉ tr➟♥
D
t➢➡♥❣ ø♥❣ ✈í✐ ❙❖▲✲❱■P✭
D, F
✮ ✈➭
L
✲
▲✐♣s❝❤✐t③ tr➟♥
D
✳ ◆Õ✉
0< η < 1/L
t❤× ❞➲②
{xk}
s✐♥❤ ❜ë✐ t❤✉❐t t♦➳♥ ✶ ❤é✐ tô ✈Ò ♠ét
♥❣❤✐Ö♠ ❝ñ❛ ❱■P✭
D, F
✮✳
✸✳✷✳ P❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ♣❤➵t
P❤➢➡♥❣ ♣❤➳♣ ❤➭♠ ♣❤➵t tr×♥❤ ❜➭② s❛✉ ➤➞② ➤➢î❝ ➤Ò ①✉✃t tr♦♥❣ ✭❬✷❪✮✳
✸✳✷✳✶✳ ❳➞② ❞ù♥❣ ❤➭♠ ♣❤➵t ❦❤➯ ✈✐
❳Ðt ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ s❛✉✿