V½ dö 2.2.5 X²t b i to¡n (VIP)(SFPP) vîiH1 =R2,H2 =R3,A : R2 −→R3
x¡c ành bði Ax = (x1+ 2x2,3x1−x2, x2), vîi to¡n tû li¶n hñp A∗ : R3 −→R2, x¡c ành bði A∗y = (y1+ 3y2,2y1−y2+y3). Cho C =x∈ R2 |2x1 +x2 ≤ 0 ,
Q = x∈ R3 | 2x1 +x2 −x3+ 1 = 0 t÷ìng ùng l c¡c tªp con lçi, âng, kh¡c réng cõa R2 v R3. nh x¤ gi¡ F = I, vîi I l to¡n tû ìn và trong H1 (thäa m¢n 1-ìn i»u m¤nh v 1-li¶n töc Lipschitz tr¶n C). nh x¤ T : C −→ C v
S : Q −→Q l¦n l÷ñt x¡c ành bði:
T(x) =PC(x), S(x) = PQ(x).
Khi â b i to¡n (VIP)(SFPP) trð th nh:
T¼m x∗ ∈ Ω sao cho hx∗, x−x∗i ≥ 0 ∀x ∈ Ω, (VIP1) trong â Ω l tªp nghi»m cõa b i to¡n iºm b§t ëng t¡ch
t¼m x∗ ∈C sao cho x∗ = PC(x∗), Ax∗ ∈ Q v Ax∗ =PQ(Ax∗). (SFP1) Gi£i b i to¡n c§p d÷îi (SFP1). D¹ th§y x =T(x) vîi måi x ∈ C. Ta câ:
Ax ∈ Q ⇔2(x1 + 2x2) + (3x1−x2)−x2 + 1 = 0. ⇔5x1 + 2x2 + 1 = 0.
çng thíi u =S(u),∀u∈ Q. Do â, tªp nghi»m cõa b i to¡n (SFP1) l :
Ω =C ∩
x∈ R2 | 5x1 + 2x2+ 1 = 0
= x ∈ R2 |5x1 + 2x2 + 1 = 0, x1 ≥ −1 .
D¹ th§y Ω kh¡c réng. Ti¸p theo, ta gi£i b i to¡n c§p tr¶n (VIP1). Ta th§y:
hx∗, x−x∗i ≥ 0 ∀x∈ Ω⇔ hx∗, x∗i ≤ hx∗, xi ∀x ∈Ω
⇔ kx∗k2 ≤ kx∗kkxk ∀x∈ Ω (CauchySchwarz)
⇔ kx∗k ≤ kxk ∀x ∈ Ω,
ngh¾a l b i to¡n (VIP1) trð th nh b i to¡n: t¼mx∗ ∈ Ωsao chox∗ l ph¦n tû câ chu©n nhä nh§t trongΩ. Tùc l nghi»m óng cõa b i to¡n l x∗ =
− 5
29,− 2
29
(ch½nh l h¼nh chi¸u cõa O tr¶n Ω). Thªt vªy: L§y x∗ = (x∗1, x∗2) ∈ C, ta câ kx∗k = q x∗12 +x∗22. V¼ Ax∗ ∈ Q n¶n x∗2 = −1−5x∗1 2 , khi â ta câ: kx∗k = q x∗12 +x∗22 = s x∗12 + −1−5x∗1 2 2 = s 29x∗12 + 10x∗1+ 1 4 = 1 2 s 29 x∗1+ 5 29 2 + 4 29 ≥ √ 29 29 .
D§u “ = ” x£y ra khi x∗1 = −5 29, x
∗
2 = −2 29
Sû döng Thuªt to¡n 2.2.2 vîi iºm xu§t ph¡t x0 = (0,0)>, c¡c tham sè h¬ng
δ = 1
2, µ = 2. i·u ki»n døng l sai sè giúa nghi»m x§p x¿ v nghi»m óng õ nhä, tùc l kxk−x∗k ≤ ε, ð ¥y chån ε = 10−6. Chån c¡c λk v αk kh¡c nhau º xem x²t sü thay êi thíi gian ch¤y ch÷ìng tr¼nh.
Vîi λk = 1
k+ 2, αk = k+ 1
2(k+ 3), ta thu ÷ñc k¸t qu£ trong B£ng 2.3. Vîi λk = 1
1.7k+ 2, αk = k
0.01+ 1
2(k0.01+ 3), ta thu ÷ñc k¸t qu£ trong B£ng 2.4. B÷îc l°p(k) xk 1 xk 2 kxk−xk−1k kxk−x∗k Thíi gian (s) 1000 −0.17227111 −0.06890844 1.538323×10−7 0.000153 0.07814 10000 −0.17239952 −0.06895981 1.536942×10−9 1.536788×10−5 0.59372 100000 −0.17241237 −0.06896495 1.536815×10−11 1.536789×10−6 5.68423 153678 −0.17241286 −0.06896515 6.507027×10−12 9.999993×10−7 8.845065
B£ng 2.3: K¸t qu£ ch¤y ch÷ìng tr¼nh vîiλk = 1
k+ 2,αk = k+ 1 2(k+ 3).
B÷îc l°p(k) xk 1 xk 2 kxk−xk−1k kxk−x∗k Thíi gian (s) 500 −0.1722459 −0.06889836 3.624207×10−7 0.000180 0.03124 1000 −0.17232985 −0.06893194 9.050217×10−8 9.040552×10−5 0.06249 10000 −0.1724054 −0.06896216 9.040966×10−10 9.039997×10−6 0.57810 90399 −0.17241286 −0.06896515 1.106215×10−11 9.999936×10−7 5.25717
B£ng 2.4: K¸t qu£ ch¤y ch÷ìng tr¼nh vîiλk= 1
1.7k+ 2, αk= k
0.01+ 1 2(k0.01+ 3).
Nghi»m x§p x¿ hëi tö d¦n v· nghi»m óng:
x∗ = − 5 29,− 2 29 ≈ (−0.17241379,−0.06896551).
K¸t luªn
Luªn v«n ¢ ¤t ÷ñc möc ti¶u · ra
"Nghi¶n cùu mët ph÷ìng ph¡p l°p gi£i mët lîp b i to¡n b§t ¯ng thùc bi¸n ph¥n hai c§p trong khæng gian Hilbert thüc; ÷a ra v t½nh to¡n v½ dö minh håa".
K¸t qu£ cõa luªn v«n
Luªn v«n ¢ tr¼nh b y mët sè ph÷ìng ph¡p l°p gi£i mët lîp b§t ¯ng thùc bi¸n ph¥n hai c§p. Cö thº:
1. Giîi thi»u v· b i to¡n iºm b§t ëng, b i to¡n b§t ¯ng thùc bi¸n ph¥n, b i to¡n b§t ¯ng thùc bi¸n ph¥n hai c§p, n¶u v½ dö v mèi li¶n h» giúa b i to¡n b§t ¯ng thùc bi¸n ph¥n v b i to¡n iºm b§t ëng.
2. Mæ t£ ph÷ìng ph¡p l°p x§p x¿ nghi»m b i to¡n b§t ¯ng thùc bi¸n ph¥n hai c§p trong hai tr÷íng hñp: b i to¡n c§p d÷îi l b§t ¯ng thùc bi¸n ph¥n gi£ ìn i»u v b i to¡n iºm b§t ëng t¡ch, chùng minh sü hëi tö m¤nh cõa c¡c ph÷ìng ph¡p.
3. ÷a ra hai v½ dö sè minh håa cho sü hëi tö m¤nh cõa c¡c ph÷ìng ph¡p, ch÷ìng tr¼nh thüc nghi»m ÷ñc thüc hi»n b¬ng ngæn ngú MATLAB.
H÷îng ph¡t triºn cõa luªn v«n trong t÷ìng lai
1. Nghi¶n cùu mët sè b i to¡n thüc t¸ ÷ñc mæ t£ d÷îi d¤ng b i to¡n b§t ¯ng thùc bi¸n ph¥n hai c§p.
2. Nghi¶n cùu c£i ti¸n ph÷ìng ph¡p l°p hi»n gi£i b i to¡n b§t ¯ng thùc bi¸n ph¥n hai c§p v mët sè b i to¡n li¶n quan.
T i li»u tham kh£o Ti¸ng Vi»t
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Ti¸ng Anh
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