Nội dung luận án gồm 4 chương: chương 1 - Bài toán quy hoạch lồi, toàn phương và hàm lồi thô; chương 2 - Điểm infimum toàn cục của Bài toán P; chương 3 - tính T-lồi ngoài của hàm mục tiêu và điểm infimum toàn cục của bài toán P; chương 4 - điểm supremum của bài toán Q. Để hiểu rõ hơn về đề tài, mời các bạn cùng tham khảo nội dung chi tiết luận án!
Tˆoi xin cam d¯oan nh˜u.ng kˆe´t qua˙’ d¯u.o.
. c su.
. c tr`ınh b`ay trong luˆa. n ´an l`a m´o.i, d¯˜a d¯u.o. . c cˆong bˆo´ trˆen c´ac ta. p ch´ı To´an ho. c quˆo´c tˆe´. C´ac kˆe´t qua˙’ viˆe´t chung v´o.i GS. TSKH. Ho`ang Xuˆan Ph´u v`a PGS. TS. Phan Th`anh An d¯˜a . d¯ˆo`ng ´y cu˙’a c´ac d¯ˆo`ng t´ac gia˙’ khi d¯u.a v`ao luˆa. n ´an. C´ac kˆe´t qua˙’ d¯u.o. . c v`a chu.a t`u.ng d¯u.o. nˆeu trong luˆa. n ´an l`a trung thu. . c ai cˆong bˆo´ trong bˆa´t k`y cˆong tr`ınh n`ao kh´ac tru.´o.c d¯´o.
Nghiˆen c´u.u sinh
. . I CA˙’ M O L `O N
Luˆa. n ´an d¯u.o. . c ho`an th`anh du.´o.i su.
. hu.´o.ng dˆa˜n, chı˙’ ba˙’o cu˙’a GS. TSKH. Ho`ang Xuˆan Ph´u v`a PGS. TS. Phan Thanh An. T´ac gia˙’ chˆan th`anh ca˙’m o.n su. . gi´up d¯˜o. mo. i mˇa. t m`a c´ac Thˆa` y d¯˜a d`anh cho. T´ac gia˙’ b`ay to˙’ l`ong biˆe´t o.n sˆau sˇa´c v`a chˆan th`anh t´o.i GS. TSKH. Ho`ang Xuˆan Ph´u, Thˆa` y d¯˜a quan tˆam, hu.´o.ng dˆa˜n tˆa. n t`ınh, nghiˆem khˇa´c v`a ta. o mo. i d¯iˆe` u kiˆe.n d¯ˆe˙’ t´ac gia˙’ c´o thˆe˙’ ho`an th`anh nh˜u.ng mu. c tiˆeu d¯ˇa. t ra cho luˆa. n ´an. T´ac gia˙’ xin b`ay to˙’ l`ong biˆe´t o.n d¯ˆe´n GS. TSKH. Nguyˆe˜n D- ˆong Yˆen, PGS. TS. Ta. Duy Phu.o. . ng, PGS. TS. Nguyˆe˜n Nˇang Tˆam v`a c´ac d¯ˆo`ng nghiˆe.p thuˆo. c Ph`ong Gia˙’i t´ıch sˆo´ v`a T´ınh to´an Khoa ho. c Viˆe.n To´an ho. c v`ı d¯˜a c´o nh˜u.ng ´y kiˆe´n qu´y b´au cho t´ac gia˙’ trong qu´a tr`ınh nghiˆen c´u.u.
T´ac gia˙’ xin d¯u.o.
. d¯˜a ta. o mo. i d¯iˆe` u kiˆe.n thuˆa. n lo.
. c b`ay to˙’ l`ong ca˙’m o.n d¯ˆe´n Ban chu˙’ nhiˆe.m Khoa Cˆong Nghˆe. thˆong tin, Ph`ong Sau d¯a. i ho. c v`a Ban Gi´am d¯ˆo´c Ho. c viˆe.n K˜y thuˆa. t . i d¯ˆe˙’ t´ac gia˙’ c´o nhiˆe` u th`o.i gian thu. Quˆan su. . c hiˆe.n luˆa. n ´an.
T´ac gia˙’ c˜ung b`ay to˙’ l`ong biˆe´t o.n d¯ˆe´n PGS. TS. D- `ao Thanh T˜ınh, PGS. TS. Nguyˆe˜n D- ´u.c Hiˆe´u, PGS. TS. Nguyˆe˜n Thiˆe.n Luˆa. n, PGS. TS. Tˆo Vˇan Ban, TS. Nguyˆe˜n Nam Hˆo`ng, TS. Nguyˆe˜n H˜u.u Mˆo. ng, TS. V˜u Thanh H`a, TS. Nguyˆe˜n Ma. nh H`ung, TS. Nguyˆe˜n Tro. ng To`an, TS. Ngˆo H˜u.u Ph´uc, TS. Tˆo´ng Minh D- ´u.c, TS. Lˆe D- `ınh So.n, TS. Trˆa` n Nguyˆen Ngo. c v`a tˆa´t ca˙’ c´ac d¯ˆo`ng nghiˆe.p trong Khoa Cˆong Nghˆe. thˆong tin, HVKTQS, d¯˜a d¯ˆo. ng viˆen, kh´ıch lˆe. v`a c´o nh˜u.ng trao d¯ˆo˙’i h˜u.u ´ıch trong suˆo´t th`o.i gian nghiˆen c´u.u v`a cˆong t´ac.
T´ac gia˙’ ca˙’m o.n sˆau sˇa´c GS. TSKH. Pha. m Thˆe´ Long, Gi´am d¯ˆo´c Ho. c Viˆe.n KTQS, ngu.`o.i d¯˜a ta. o mo. i d¯iˆe` u kiˆe.n vˆe` mˇa. t thu˙’ tu. c c˜ung nhu. chuyˆen mˆon d¯ˆe˙’ t´ac gia˙’ c´o thˆe˙’ ho`an th`anh luˆa. n ´an n`ay. Cuˆo´i c`ung t´ac gia˙’ gu.˙’ i l`o.i c´am o.n t´o.i vo.
. v`a c´ac con, nh˜u.ng ngu.`o.i d¯˜a d¯ˆo. ng viˆen, chˇam s´oc v`a ta. o mo. i d¯iˆe` u kiˆe.n cho t´ac gia˙’ trong qu´a tr`ınh l`am luˆa. n ´an.
Mu. c lu. c
L`o.i cam d¯oan 1
2 L`o.i ca˙’m o.n
5 Danh mu. c c´ac k´y hiˆe. u thu.`o.ng d`ung
Mo.˙’ d¯ˆa` u 1
1 B`ai to´an quy hoa. ch lˆo`i, quy hoa. ch to`an phu.o.ng v`a h`am lˆo`i
8
9
12
thˆo 1.1. B`ai to´an quy hoa. ch lˆo`i, quy hoa. ch to`an phu.o.ng . . . . . . 1.2. H`am lˆo`i suy rˆo. ng thˆo . . . . . . . . . . . . . . . . . . . . . 1.3. H`am γ-lˆo`i ngo`ai . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4. H`am Γ-lˆo`i ngo`ai . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5. H`am γ-lˆo`i trong . . . . . . . . . . . . . . . . . . . . . . . . 17
20 2 D- iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P )
20
27
28
33 2.1. T´ınh γ-lˆo`i ngo`ai cu˙’a h`am bi. nhiˆe˜u . . . . . . . . . . . . . . .c tiˆe˙’u to`an cu. c v`a d¯iˆe˙’m infimum to`an cu. c . . . . . 2.2. D- iˆe˙’m cu. 2.3. C´ac t´ınh chˆa´t cu˙’a d¯iˆe˙’m infimum to`an cu. c . . . . . . . . . .a v`a d¯iˆe` u kiˆe.n tˆo´i u.u . . . . . . . . . . . . . . 2.4. T´ınh chˆa´t tu.
3 T´ınh Γ-lˆo`i ngo`ai cu˙’a h`am bi. nhiˆe˜u v`a d¯iˆe˙’m infimum to`an
43
43
52
55
58 cu. c cu˙’a B`ai to´an ( ˜P ) 3.1. T´ınh Γ-lˆo`i ngo`ai cu˙’a h`am bi. nhiˆe˜u . . . . . . . . . . . . . . 3.2. D- iˆe˙’m infimum to`an cu. c cu˙’a b`ai to´an nhiˆe˜u . . . . . . . . . 3.3. T´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c . . . . . 3.4. Du.´o.i vi phˆan suy rˆo. ng thˆo v`a d¯iˆe` u kiˆe.n tˆo´i u.u . . . . . . .
4 D- iˆe˙’m supremum cu˙’a B`ai to´an ( ˜Q) 64
64
66
73
86 4.1. T´ınh γ-lˆo`i trong cu˙’a h`am bi. nhiˆe˜u . . . . . . . . . . . . . . 4.2. D- iˆe˙’m supremum to`an cu. c cu˙’a h`am bi. nhiˆe˜u . . . . . . . . 4.3. T´ınh chˆa´t cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c . . . . . . 4.4. T´ınh chˆa´t cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng . . . .
94 Kˆe´t luˆa. n chung
96 Danh mu. c cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa. n ´an
97 T`ai liˆe. u tham kha˙’o
. .`O NG D `UNG DANH MU. C C ´AC K ´Y HIˆE. U THU
• IRn : Khˆong gian Euclide n chiˆe` u
• (cid:107) · (cid:107) : Chuˆa˙’n Euclide trong IRn
• (cid:104)x, y(cid:105) : T´ıch vˆo hu.´o.ng cu˙’a v´ec to. x, y
• B(x, r) := {y | (cid:107)y − x(cid:107) < r} : H`ınh cˆa` u mo.˙’ b´an k´ınh r tˆam x
• ¯B(x, r) := {y | (cid:107)y − x(cid:107) ≤ r} : H`ınh cˆa` u d¯´ong b´an k´ınh r tˆam x
• A ∈ IRn×n, A (cid:31) 0 : Ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng
• AT : Ma trˆa. n chuyˆe˙’n vi. cu˙’a ma trˆa. n A • λmin, (λmax) : Gi´a tri. riˆeng nho˙’ nhˆa´t (l´o.n nhˆa´t) cu˙’a ma trˆa. n A
• λ(A) : Tˆa. p c´ac gi´a tri. riˆeng cu˙’a ma trˆa. n A
√ • (cid:107)A(cid:107) = { max λ | λ ∈ λ(AT A)} : Chuˆa˙’n cu˙’a ma trˆa. n A trong IRn×n
• f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) : H`am to`an phu.o.ng lˆo`i ngˇa. t • p(x), supx∈D |p(x)| ≤ s v´o.i s ∈ [0, +∞[ : H`am nhiˆe˜u gi´o.i nˆo. i • ˜f = f + p : H`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i
• f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) → inf, x ∈ D : B`ai to´an quy hoa. ch to`an
phu.o.ng (P )
• f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) → sup, x ∈ D : B`ai to´an quy hoa. ch to`an
phu.o.ng (Q)
• f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + p(x) → inf, x ∈ D : B`ai to´an quy hoa. ch
to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u ( ˜P )
• f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + p(x) → sup, x ∈ D : B`ai to´an quy hoa. ch
to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u ( ˜Q) • ∂g(x∗) : Du.´o.i vi phˆan cu˙’a g ta. i d¯iˆe˙’m x∗
i=0 µigi(x) : H`am Lagrange
• L(x, µ0, . . . , µm) := (cid:80)m
• T´ınh chˆa´t (Mγ) : Mˆo˜i d¯iˆe˙’m γ-cu. . c tiˆe˙’u x∗ cu˙’a f l`a d¯iˆe˙’m cu. . c tiˆe˙’u
to`an cu. c
• T´ınh chˆa´t (Iγ) : Mˆo˜i d¯iˆe˙’m γ-infimum x∗ cu˙’a f l`a d¯iˆe˙’m infimum
to`an cu. c
• Lα( ˜f ) := {x | x ∈ D, ˜f (x) ≤ α}, α ∈ IR : Tˆa. p m´u.c du.´o.i cu˙’a h`am
˜f = f + p
• h1(γ) := inf x0, x1∈D, (cid:107)x0−x1(cid:107)=γ (cid:16) 1 2(f (x0) + f (x1)) − f ( 1 (cid:17) 2(x0 + x1))
(cid:16) (cid:17) f (x0)−2f (x1)+f (−x0+2x1) • h2(γ) := inf x0, x1∈D, (cid:107)x0−x1(cid:107)=γ,−x0+2x1∈D
. c biˆen cu˙’a tˆa. p lˆo`i d¯a diˆe.n D • aff D : Bao aphin cu˙’a tˆa. p D • ext D : Tˆa. p c´ac d¯iˆe˙’m cu.
• JD(x∗) := ext D \ {x∗}, x∗ ∈ ext D
• d(x, D) := inf y∈D (cid:107)x − y(cid:107) : Khoa˙’ng c´ach t`u. x d¯ˆe´n D
• conv D : Bao lˆo`i cu˙’a tˆa. p D • dD := minx∗∈ext D{d(cid:0)x∗, conv JD(x∗)(cid:1)}
• D(x∗, β) := {x ∈ D | x = (1 − α)x∗ + αy, y ∈ D, 0 ≤ α ≤ 1 − β},
x∗ ∈ ext D, β ∈ [0, 1]
• C 0(D) := {p : D → IR | (cid:107)p(cid:107)C 0 := supx∈D |p(x)| < +∞}
• ¯BC 0(0, r) : H`ınh cˆa` u d¯´ong b´an k´ınh r tˆam 0 trong C 0(D)
.˙’ D- ˆA` U MO
B`ai to´an quy hoa. ch to`an phu.o.ng truyˆe` n thˆo´ng c´o da. ng
f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) → inf, x ∈ D
trong d¯´o A ∈ IRn×n l`a ma trˆa. n vuˆong, b ∈ IRn l`a v´ec to. v`a D ⊂ IRn l`a tˆa. p lˆo`i.
C`ung v´o.i b`ai to´an quy hoa. ch lˆo`i, b`ai to´an quy hoa. ch to`an phu.o.ng . c nhiˆe` u nh`a to´an ho. c Viˆe.t nam v`a quˆo´c tˆe´ nghiˆen c´u.u, v´ı du. nhu. H. d¯u.o. W. Kuhn v`a A. W. Tucker [22], B. Bank v`a R. Hasel [5], E. Blum v`a W.
Oettli [7], B. C. Eaves [12], M. Frank v`a P. Wolfe [13], O. L. Magasarian
[26], G. M. Lee, N. N. Tam v`a N. D. Yen [31], H. X. Phu [45], H. X. Phu
v`a N. D. Yen [53], M. Schweighofer [57], H. Tuy [63], [64], [72], H. H. Vui
v`a P. T. Son [66]. . .
C´ac kˆe´t qua˙’ quan tro. ng d¯˜a thu d¯u.o.
. c khi nghiˆen c´u.u c´ac b`ai to´an quy hoa. ch to`an phu.o.ng cu˙’a c´ac nh`a to´an ho. c l`a vˆe` su. . tˆo`n ta. i nghiˆe.m tˆo´i u.u, d¯iˆe` u kiˆe.n cˆa` n tˆo´i u.u, d¯iˆe` u kiˆe.n d¯u˙’ tˆo´i u.u, thuˆa. t to´an t`ım nghiˆe.m tˆo´i u.u, t´ınh ˆo˙’n d¯i.nh cu˙’a nghiˆe.m tˆo´i u.u khi c´ac b`ai to´an trˆen bi. t´ac d¯ˆo. ng bo.˙’ i . c ´u.ng du. ng d¯ˆe˙’ nhiˆe˜u. Nhiˆe` u kˆe´t qua˙’ nghiˆen c´u.u vˆe` b`ai to´an trˆen d¯˜a d¯u.o. gia˙’i c´ac b`ai to´an trong kinh tˆe´ v`a k˜y thuˆa. t, nhu. b`ai to´an lu. . a cho. n d¯ˆa` u tu. (portfolio selection) ([27], [28]), b`ai to´an ph´at d¯iˆe.n tˆo´i u.u (economic power dispatch) ([6], [11], [69]), b`ai to´an kinh tˆe´ d¯ˆo´i s´anh (matching economic), ([17]), b`ai to´an m´ay hˆo˜ tro. . v´ec to. (support vector machine) ([29]). . .
Khi A l`a nu.˙’ a x´ac d¯i.nh du.o.ng hoˇa. c nu.˙’ a x´ac d¯i.nh ˆam th`ı b`ai to´an trˆen
c´o thˆe˙’ phˆan r˜a th`anh hai b`ai to´an kh´ac nhau sau:
f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) → inf, x ∈ D (P )
v`a
f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) → sup, x ∈ D. (Q)
Luˆa. n ´an n`ay nghiˆen c´u.u c´ac b`ai to´an quy hoa. ch to`an phu.o.ng lˆo`i ngˇa. t
v´o.i nhiˆe˜u gi´o.i nˆo. i sau:
˜f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + p(x) → inf, x ∈ D ( ˜P )
v`a
x ∈ D, ˜f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + p(x) → sup,
( ˜Q) trong d¯´o p : D → IR tho˙’a m˜an d¯iˆe` u kiˆe.n supx∈D |p(x)| ≤ s v´o.i gi´a tri. s ∈ [0, +∞[ v`a A trong c´ac b`ai to´an (P ), (Q), ( ˜P ) v`a ( ˜Q) d¯u.o. . c gia˙’ thiˆe´t l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng. V`ı sao c´ac b`ai to´an trˆen d¯u.o.
. c cho. n d¯ˆe˙’ nghiˆen c´u.u? R˜o r`ang, khi s = 0 th`ı c´ac b`ai to´an ( ˜P ) v`a ( ˜Q) ch´ınh l`a c´ac b`ai to´an (P ) v`a (Q), hay n´oi c´ach kh´ac c´ac b`ai to´an (P ) v`a (Q) l`a c´ac tru.`o.ng ho. . p riˆeng cu˙’a c´ac b`ai to´an ( ˜P ) v`a ( ˜Q). D- ˆay l`a l´y do d¯ˆe˙’ tiˆe´n h`anh nghiˆen c´u.u c´ac b`ai to´an trˆen, tˆo´i thiˆe˙’u . c tˆe´ kh´ac du.´o.i t`u. quan d¯iˆe˙’m l´y thuyˆe´t. Tuy nhiˆen, c`on mˆo. t sˆo´ l´y do thu. . c su. d¯ˆay, cho thˆa´y viˆe.c nghiˆen c´u.u c´ac b`ai to´an ( ˜P ), ( ˜Q) l`a thu. . cˆa` n.
. c tho˙’a m˜an trong nhiˆe` u b`ai to´an thu.
L´y do th´u. nhˆa´t: f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) l`a h`am mu. c tiˆeu ban d¯ˆa` u v`a p l`a h`am nhiˆe˜u n`ao d¯´o. H`am nhiˆe˜u p c´o thˆe˙’ bao gˆo`m c´ac t´ac d¯ˆo. ng bˆo˙’ sung (tˆa´t d¯i.nh hoˇa. c ngˆa˜u nhiˆen) lˆen h`am mu. c tiˆeu v`a c´ac lˆo˜i gˆay ra trong qu´a tr`ınh mˆo h`ınh h´oa, d¯o d¯a. c, t´ınh to´an. . . D- iˆe˙’m d¯ˇa. c biˆe.t l`a o.˙’ chˆo˜, ch´ung ta ha. n chˆe´ chı˙’ x´et nhiˆe˜u gi´o.i nˆo. i. Ha. n chˆe´ n`ay l`a khˆong qu´a ngˇa. t, c´o thˆe˙’ . c tˆe´, chˇa˙’ ng ha. n nhu. trong hai v´ı d¯u.o. du. minh ho. a sau d¯ˆay.
Mˆo. t trong nh˜u.ng ´u.ng du. ng nˆo˙’i bˆa. t cu˙’a quy hoa. ch to`an phu.o.ng l`a b`ai to´an lu. . a cho. n d¯ˆa` u tu. (H. M. Markowitz [27], [28]). B`ai to´an ph´at biˆe˙’u nhu. sau: Phˆan phˆo´i vˆo´n qua n ch´u.ng kho´an (asset) c´o sˇa˜n d¯ˆe˙’ . i nhuˆa. n, t´u.c l`a t`ım v´ec to. tı˙’ lˆe. c´o thˆe˙’ gia˙’m thiˆe˙’u ru˙’i ro v`a tˆo´i d¯a lo. x ∈ D, D := {x = (x1, x2, . . . , xn) | (cid:80)n j=1 xj = 1} d¯ˆe˙’ f (x) = ωxT Σx − ρT x d¯a. t gi´a tri. nho˙’ nhˆa´t, trong d¯´o xj, j = 1, . . . , n, l`a ty˙’ lˆe. ch´u.ng kho´an th´u. j trong danh mu. c d¯ˆa` u tu., ω l`a tham sˆo´ ru˙’i ro, Σ ∈ IRn×n l`a ma trˆa. n . i nhuˆa. n k`y vo. ng. V`ı Σ v`a ρ thu.`o.ng hiˆe.p phu.o.ng sai, ρ ∈ IRn l`a v´ec to. lo.
. c x´ac d¯i.nh ch´ınh x´ac m`a chı˙’ xˆa´p xı˙’ bo.˙’ i ˜Σ v`a ˜ρ, do d¯´o ch´ung khˆong d¯u.o. ta pha˙’i cu. . c tiˆe˙’u h´oa h`am ˜f (x) = ωxT ˜Σx − ˜ρT x = f (x) + p(x), trong d¯´o p(x) = ωxT ( ˜Σ − Σ)x − (˜ρ − ρ)T x. Khi quy d¯i.nh, khˆong d¯u.o. . c b´an khˆo´ng, . c D l`a gi´o.i nˆo. i. V`ı vˆa. y t´u.c l`a xj ≥ 0, j = 1, . . . , n, th`ı tˆa. p chˆa´p nhˆa. n d¯u.o. nhiˆe˜u p c˜ung gi´o.i nˆo. i trˆen D. N´oi mˆo. t c´ach tˆo˙’ng qu´at, t´ınh gi´o.i nˆo. i cu˙’a nhiˆe˜u luˆon d¯u.o. . c d¯a˙’m ba˙’o khi D gi´o.i nˆo. i v`a p liˆen tu. c trˆen D. Gia˙’ thiˆe´t . p v´o.i nhiˆe` u b`ai to´an thu. n`ay c˜ung ph`u ho. . c tˆe´.
. ng d¯u.o.
Mˆo. t v´ı du. n˜u.a cho thˆa´y l`a nhiˆe˜u gi´o.i nˆo. i luˆon xuˆa´t hiˆe.n khi gia˙’i mˆo. t b`ai to´an tˆo´i u.u (P ) hoˇa. c (Q) n`ao d¯´o bˇa`ng m´ay t´ınh. Do phˆa` n l´o.n c´ac sˆo´ thu. . c khˆong thˆe˙’ biˆe˙’u diˆe˜n ch´ınh x´ac bˇa`ng m´ay t´ınh, nˆen d¯ˆo´i v´o.i hˆa` u hˆe´t x ∈ D ta khˆong thˆe˙’ t´ınh ch´ınh x´ac d¯a. i lu.o. . ng f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) m`a chı˙’ c´o thˆe˙’ xˆa´p xı˙’ f (x) bo.˙’ i mˆo. t sˆo´ dˆa´u chˆa´m d¯ˆo. ng ˜f (x) n`ao d¯´o. H`am ˜f khˆong lˆo`i, khˆong to`an phu.o.ng v`a thˆa. m ch´ı l`a khˆong liˆen tu. c trˆen D. Khi d¯´o h`am p := ˜f − f mˆo ta˙’ c´ac lˆo˜i t´ınh to´an. C´ac lˆo˜i d¯´o bi. chˇa. n bo.˙’ i mˆo. t cˆa. n . c, t´u.c l`a supx∈D |p(x)| ≤ s. trˆen s ∈ [0, +∞[ n`ao d¯´o c´o thˆe˙’ u.´o.c lu.o. Ngo`ai ra, bˇa`ng c´ach su.˙’ du. ng c´ac sˆo´ dˆa´u chˆa´m d¯ˆo. ng d`ai ho.n v`a/hoˇa. c c´ac thuˆa. t to´an tˆo´t ho.n, ta c´o thˆe˙’ gia˙’m cˆa. n trˆen s.
L´y do th´u. hai: ˜f l`a h`am mu. c tiˆeu d¯´ıch thu. . c l´y tu.o.˙’ ng h´oa hoˇa. c l`a h`am mu. c tiˆeu thay thˆe´. Trong thu.
. c tiˆe˜n d¯u.o.
. c v`a f l`a h`am mu. c tiˆeu d¯u.o. . c tˆe´, nhiˆe` u h`am thˆe˙’ hiˆe.n mˆo. t sˆo´ mu. c tiˆeu thu. . c gia˙’ d¯i.nh l`a lˆo`i, hoˇa. c to`an . c nghiˆen c´u.u k˜y, hoˇa. c phu.o.ng, hoˇa. c c´o mˆo. t sˆo´ t´ınh chˆa´t thuˆa. n tiˆe.n d¯˜a d¯u.o. . c ra th`ı khˆong pha˙’i l`a nhu. vˆa. y. D- iˆe` u n`ay d¯˜a d¯u.o. dˆe˜ nghiˆen c´u.u, nhu.ng thu. . c H. X. Phu, H. G. Bock v`a S. Pickenhain d¯ˆe` cˆa. p d¯ˆe´n trong [48]. Trong bˆo´i ca˙’nh d¯´o, p = ˜f − f l`a h`am hiˆe.u chı˙’nh. C´o thˆe˙’ gia˙’ thiˆe´t p l`a gi´o.i nˆo. i (tˆo´i thiˆe˙’u trˆen tˆa. p chˆa´p nhˆa. n d¯u.o. . c) bo.˙’ i mˆo. t sˆo´ du.o.ng kh´a b´e s, v`ı nˆe´u |p(x)| qu´a l´o.n th`ı su. . thay thˆe´ khˆong c`on ph`u ho. . p n˜u.a.
D- ˆe˙’ gia˙’i th´ıch d¯iˆe` u n`ay, ta d¯ˆe` cˆa. p d¯ˆe´n vˆa´n d¯ˆe` thu.`o.ng d¯u.o.
cu˙’a ph´at d¯iˆe.n tˆo´i u.u, t´u.c l`a b`ai to´an phˆan bˆo´ lu.o. tˆo˙’ m´ay ph´at nhiˆe.t d¯iˆe.n sao cho tˆo˙’ng chi ph´ı (gi´a th`anh) l`a cu. th`o.i vˆa˜n d¯´ap ´u.ng d¯u.o. . c nghiˆen c´u.u . ng d¯iˆe.n nˇang cho t`u.ng . c tiˆe˙’u, d¯ˆo`ng . ng d¯iˆe.n nˇang v`a thoa˙’ m˜an r`ang buˆo. c . c nhu cˆa` u lu.o.
n (cid:88)
vˆe` cˆong suˆa´t ph´at ra cu˙’a mˆo˜i tˆo˙’ m´ay. Ngu.`o.i ta thu.`o.ng gia˙’ thiˆe´t (xem [6], [11], [69],. . . ) h`am chi ph´ı tˆo˙’ng cˆo. ng (bao gˆo`m c´ac chi ph´ı nhiˆen liˆe.u (fuel cost), chi ph´ı ta˙’i sau (load-following cost), chi ph´ı du. . ph`ong quay (sprinning-reserve cost), chi ph´ı du. . ph`ong bˆo˙’ sung (supplemental-reserve cost), chi ph´ı tˆo˙’n thˆa´t ph´at v`a truyˆe` n dˆa˜n d¯iˆe.n nˇang) l`a h`am to`an phu.o.ng, lˆo`i ngˇa. t v`a c´o da. ng
i
i=1 trong d¯´o n l`a sˆo´ tˆo˙’ m´ay ph´at, P := (P1, P2, . . . , Pn), Pi ∈ [Pi min, Pi max] l`a lu.o. . ng d¯iˆe.n nˇang ph´at ra cu˙’a tˆo˙’ m´ay th´u. i, Pi min, Pi max l`a cˆong suˆa´t ph´at nho˙’ nhˆa´t v`a l´o.n nhˆa´t cu˙’a tˆo˙’ m´ay ph´at th´u. i, Fi(Pi) = ai + biPi + ciP 2 l`a h`am chi ph´ı cu˙’a tˆo˙’ m´ay ph´at th´u. i v`a ai, bi, ci l`a c´ac hˆe. sˆo´ gi´a cu˙’a tˆo˙’ m´ay ph´at th´u. i ∈ {1, 2, . . . , n}.
F (P ) = Fi(Pi),
. c x´et d¯ˆe´n th`ı h`am chi ph´ı to`an phu.o.ng pha˙’i d¯u.o.
n (cid:88)
D˜ı nhiˆen, gia˙’ thiˆe´t to`an phu.o.ng, lˆo`i ngˇa. t cu˙’a h`am mu. c tiˆeu l`a qu´a l´y . c tˆe´ c´o thˆe˙’ khˆong l`a h`am to`an phu.o.ng v`a c˜ung khˆong tu.o.˙’ ng. Chi ph´ı thu. l`a h`am lˆo`i ngˇa. t. Nhu. vˆa. y, d¯ˆe˙’ gia˙’ thiˆe´t vˆe` t´ınh to`an phu.o.ng v`a lˆo`i ngˇa. t cu˙’a h`am mu. c tiˆeu d¯u.o. . c tho˙’a m˜an, cˆa` n h`am gi´o.i nˆo. i p hiˆe.u chı˙’nh h`am chi . c tˆe´. D- ˇa. c biˆe.t (xem [62], [6], [11], [69],. . . ), nˆe´u hiˆe.u ´u.ng d¯iˆe˙’m-van ph´ı thu. d¯u.o. . c hiˆe.u chı˙’nh bo.˙’ i tˆo˙’ng h˜u.u ha. n c´ac h`am da. ng sin, t´u.c l`a
i=1
F (P ) = (cid:0)Fi(Pi) + |ei sin(fi(Pi min − Pi))|(cid:1),
i=1 |ei sin(fi(Pi min − Pi))| l`a gi´o.i nˆo. i.
trong d¯´o ei, fi l`a c´ac hˆe. sˆo´ hiˆe.u ´u.ng d¯iˆe˙’m-van. R˜o r`ang h`am hiˆe.u chı˙’nh p := (cid:80)n
D- ˆe˙’ ngˇa´n go. n, ta thu.`o.ng go. i p l`a h`am nhiˆe˜u (mˇa. c d`u n´o khˆong chı˙’ d¯´ong vai tr`o d¯´o nhu. d¯˜a gia˙’i th´ıch o.˙’ trˆen), ˜f l`a h`am bi. nhiˆe˜u v`a ( ˜P ) v`a ( ˜Q) l`a c´ac b`ai to´an nhiˆe˜u. Thˆa. t ra, ch´ung chı˙’ l`a c´ac thuˆa. t ng˜u. vay mu.o. . n, khˆong pha˙’i l´uc n`ao c˜ung ch´ınh x´ac nhu. thu.`o.ng lˆe..
Nh˜u.ng vˆa´n d¯ˆe` g`ı l`a m´o.i cu˙’a c´ac b`ai to´an ( ˜P ) v`a ( ˜Q) cˆa` n d¯u.o. . c nghiˆen c´u.u? Cˆau ho˙’i n`ay l`a cˆa` n thiˆe´t, v`ı d¯˜a c´o nh˜u.ng kˆe´t qua˙’ nghiˆen c´u.u d¯ˇa. c
sˇa´c theo c´ac kh´ıa ca. nh kh´ac nhau vˆe` t´ınh ˆo˙’n d¯i.nh cu˙’a c´ac b`ai to´an nhiˆe˜u lˆo`i v`a/hoˇa. c nhiˆe˜u to`an phu.o.ng. D- iˆe˙’m chung cu˙’a phˆa` n l´o.n c´ac cˆong tr`ınh nghiˆen c´u.u t`u. tru.´o.c d¯ˆe´n nay l`a nhiˆe˜u khˆong l`am thay d¯ˆo˙’i nh˜u.ng thuˆo. c t´ınh tiˆeu biˆe˙’u cu˙’a b`ai to´an ban d¯ˆa` u. V´ı du. b`ai to´an lˆo`i bi. nhiˆe˜u vˆa˜n gi˜u. nguyˆen t´ınh lˆo`i (nhu. trong c´ac nghiˆen c´u.u cu˙’a M. J Canovas [8], D. Klatte [21], B. Kumer [23]. . . ) v`a c´ac b`ai to´an to`an phu.o.ng gi˜u. d¯u.o. . c t´ınh to`an phu.o.ng (nhu. trong c´ac nghiˆen c´u.u cu˙’a J. V. Daniel [10], G. M. Lee, N. N. Tam v`a N. D. Yen [31], K. Mirnia v`a A. Ghaffari-Hadigheh [30], H. X. Phu [45], H. X. Phu v`a N. D. Yen [53]. . . ). D- iˆe` u kh´ac biˆe.t l`a, h`am mu. c tiˆeu ˜f cu˙’a c´ac b`ai to´an nhiˆe˜u trong luˆa. n ´an n`ay khˆong lˆo`i, khˆong to`an phu.o.ng mˇa. c d`u h`am f l`a lˆo`i ngˇa. t v`a to`an phu.o.ng. Ho.n n˜u.a, v`ı nhiˆe˜u p chı˙’ gia˙’ thiˆe´t l`a gi´o.i nˆo. i, nˆen h`am bi. nhiˆe˜u ˜f c´o thˆe˙’ khˆong liˆen tu. c ta. i bˆa´t c´u. d¯iˆe˙’m n`ao. V´o.i nh˜u.ng h`am mu. c tiˆeu nhu. vˆa. y, du.`o.ng nhu. s˜e khˆong thˆe˙’ thu d¯u.o. . c kˆe´t qua˙’ g`ı d¯ˇa. c biˆe.t. Mu. c tiˆeu cu˙’a luˆa. n ´an l`a chı˙’ ra d¯iˆe` u ngu.o. . c la. i.
Luˆa. n ´an gˆo`m 4 chu.o.ng. Chu.o.ng 1 v´o.i tiˆeu d¯ˆe` “B`ai to´an quy hoa. ch lˆo`i, to`an phu.o.ng v`a h`am lˆo`i thˆo” tr`ınh b`ay D- i.nh l´y Kuhn-Tucker cu˙’a b`ai to´an quy hoa. ch lˆo`i, D- i.nh l´y vˆe` d¯iˆe` u kiˆe.n cu. . c tri. cu˙’a b`ai to´an quy hoa. ch to`an phu.o.ng v`a mˆo. t sˆo´ loa. i h`am lˆo`i thˆo nhu. γ-lˆo`i ngo`ai, Γ-lˆo`i ngo`ai, γ-lˆo`i trong c`ung mˆo. t sˆo´ t´ınh chˆa´t tˆo´i u.u cu˙’a ch´ung.
C´ac kh´ai niˆe.m, c´ac t´ınh chˆa´t, c´ac d¯i.nh l´y d¯u.o.
. c dˆa˜n ra trong chu.o.ng . c su.˙’ du. ng d¯ˆe˙’ nghiˆen c´u.u c´ac vˆa´n d¯ˆe` d¯ˇa. t ra trong c´ac chu.o.ng
n`ay s˜e d¯u.o. sau.
Chu.o.ng 2 v´o.i tiˆeu d¯ˆe` “D- iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P )” nghiˆen c´u.u t´ınh γ-lˆo`i ngo`ai cu˙’a h`am to`an phu.o.ng v´o.i nhiˆe˜u gi´o.i nˆo. i, d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c, d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), kha˙’o s´at t´ınh ˆo˙’n d¯i.nh nghiˆe.m v`a mo.˙’ rˆo. ng D- i.nh l´y Kuhn-Tucker cho b`ai to´an n`ay.
Chu.o.ng 3 v´o.i tiˆeu d¯ˆe` “T´ınh Γ-lˆo`i ngo`ai cu˙’a h`am mu. c tiˆeu v`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P )” nghiˆen c´u.u t´ınh Γ-lˆo`i ngo`ai cu˙’a h`am
. c mˆo. t sˆo´ kˆe´t qua˙’ . c tiˆe˙’u to`an cu. c, d¯iˆe˙’m
. c chı˙’ ra trong Chu.o.ng 2. mu. c tiˆeu ˜f (theo c´ach tiˆe´p cˆa. n tˆo pˆo), qua d¯´o nhˆa. n d¯u.o. ma. nh ho.n nh˜u.ng kˆe´t qua˙’ nghiˆen c´u.u vˆe` d¯iˆe˙’m cu. infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) d¯u.o.
Chu.o.ng 4 cu˙’a luˆa. n ´an c´o tiˆeu d¯ˆe` “D- iˆe˙’m supremum cu˙’a B`ai to´an ( ˜Q)” nghiˆen c´u.u t´ınh chˆa´t v`a t´ınh ˆo˙’n d¯i.nh cu˙’a c´ac d¯iˆe˙’m supremum to`an cu. c v`a d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q).
C´ac kˆe´t qua˙’ d¯a. t d¯u.o. . c trong luˆa. n ´an bao gˆo`m:
• Chı˙’ ra c´ac d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai, Γ-lˆo`i
ngo`ai v`a γ-lˆo`i trong.
• Ch´u.ng minh d¯u.o. . c d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c
cu˙’a B`ai to´an ( ˜P ) khˆong vu.o. . t qu´a γ∗ = 2(cid:112)2s/λmin.
• Chı˙’ ra t´ınh ˆo˙’n d¯i.nh nghiˆe.m cu˙’a B`ai to´an ( ˜P ) theo cˆa. n trˆen s cu˙’a h`am
nhiˆe˜u.
• Mo.˙’ rˆo. ng D- i.nh l´y Kuhn-Tucker cho B`ai to´an ( ˜P ). • Chı˙’ ra c´ac t´ınh chˆa´t (ma. nh ho.n c´ac t´ınh chˆa´t d¯˜a c´o) cu˙’a c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c v`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) khi su.˙’ du. ng t´ınh Γ-lˆo`i ngo`ai cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p.
. c su. . tˆo`n ta. i v`a vi. tr´ı cu˙’a c´ac d¯iˆe˙’m supremum to`an
• Ch´u.ng minh d¯u.o. cu. c trˆen miˆe` n D.
• Khˇa˙’ ng d¯i.nh t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c khi D l`a d¯a diˆe.n lˆo`i v`a tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng khi D l`a tˆa. p lˆo`i d¯a diˆe.n cu˙’a B`ai to´an ( ˜Q) theo nhiˆe˜u p.
C´ac kˆe´t qua˙’ ch´ınh cu˙’a luˆa. n ´an d¯˜a d¯u.o.
. c tr`ınh b`ay ta. i c´ac xemina “Tˆo´i u.u h´oa v`a T´ınh to´an hiˆe.n d¯a. i” cu˙’a Khoa Cˆong nghˆe. thˆong tin (Ho. c viˆe.n KTQS), “Tˆo´i u.u v`a T´ınh to´an khoa ho. c” cu˙’a Ph`ong Gia˙’i t´ıch sˆo´
v`a T´ınh to´an khoa ho. c (Viˆe.n To´an ho. c), Hˆo. i tha˙’o “Tˆo´i u.u v`a T´ınh to´an Khoa ho. c” (Ba V`ı, H`a Nˆo. i, th´ang 4 nˇam 2010). C´ac kˆe´t qua˙’ n`ay c˜ung d¯˜a d¯u.o. . c cˆong bˆo´ trˆen c´ac ta. p ch´ı Optimization, Mathematical Methods of Operations Research v`a Journal of Optimization Theory and Applications. Ch´ung tˆoi d¯ang tiˆe´p tu. c nghiˆen c´u.u mˆo. t sˆo´ vˆa´n d¯ˆe` vˆe` l´y thuyˆe´t v`a . c tˆe´ cu˙’a c´ac b`ai to´an ( ˜P ) v`a ( ˜Q), hy vo. ng
t´ınh to´an ´u.ng du. ng trong thu. rˇa`ng trong th`o.i gian t´o.i s˜e c´o thˆem mˆo. t sˆo´ kˆe´t qua˙’ m´o.i.
. . CHU NG 1 O
B `AI TO ´AN QUY HOA. CH L ˆO` I, . . NG V `A H `AM L ˆO` I TH ˆO QUY HOA. CH TO `AN PHU O
Trong chu.o.ng n`ay, ch´ung tˆoi nhˇa´c la. i D- i.nh l´y Kuhn-Tucker cho b`ai to´an quy hoa. ch lˆo`i, D- i.nh l´y vˆe` d¯iˆe` u kiˆe.n cˆa` n cu. . c tri. cho b`ai to´an quy hoa. ch to`an phu.o.ng. D- ˆo`ng th`o.i ch´ung tˆoi c˜ung tr`ınh b`ay la. i mˆo. t sˆo´ kh´ai niˆe.m, t´ınh chˆa´t cu˙’a h`am lˆo`i thˆo nhu. γ-lˆo`i ngo`ai, Γ-lˆo`i ngo`ai v`a γ-lˆo`i trong. C´ac kh´ai niˆe.m, c´ac kˆe´t qua˙’ dˆa˜n ra o.˙’ trong chu.o.ng n`ay, s˜e d¯u.o. . c su.˙’
du. ng nhiˆe` u lˆa` n trong c´ac chu.o.ng sau.
Trong suˆo´t luˆa. n ´an n`ay, IRn l`a khˆong gian Euclide n-chiˆe` u, D ⊆ IRn . c gia˙’ thiˆe´t l`a tˆa. p lˆo`i d¯a . p D d¯u.o.
l`a c´ac tˆa. p lˆo`i, v`a trong nhiˆe` u tru.`o.ng ho. diˆe.n. V´o.i x0, x1 ∈ IRn, λ ∈ IR, ta k´y hiˆe.u
xλ := (1 − λ)x0 + λx1, := {xλ | 0 ≤ λ ≤ 1}, := [x0, x1] \ {x0}. [x0, x1] ]x0, x1]
C´ac tˆa. p ho. . c d¯i.nh ngh˜ıa tu.o.ng tu. . .
V´o.i r l`a sˆo´ thu. . p [x0, x1[ v`a ]x0, x1[ c˜ung d¯u.o. . c du.o.ng, c´ac tˆa. p ho. . p
B(x, r) := {y | (cid:107)y − x(cid:107) < r}, ¯B(x, r) := {y | (cid:107)y − x(cid:107) ≤ r}, S(x, r) := {y | (cid:107)y − x(cid:107) = r},
. t d¯u.o. . c go. i l`a c´ac h`ınh cˆa` u mo.˙’ , h`ınh cˆa` u d¯´ong v`a mˇa. t cˆa` u tˆam x
lˆa` n lu.o. b´an k´ınh r. Ngo`ai ra, trong luˆa. n ´an n`ay ch´ung tˆoi luˆon k´y hiˆe.u:
• f l`a h`am to`an phu.o.ng lˆo`i ngˇa. t c´o da. ng
f (x) := (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105), x ∈ D (1.0.1)
2(A + AT )).
trong d¯´o A ∈ IRn×n l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng (nˆe´u A khˆong d¯ˆo´i x´u.ng ta c´o thˆe˙’ thay A bo.˙’ i 1
• p(x) l`a h`am nhiˆe˜u gi´o.i nˆo. i, t´u.c l`a
|p(x)| ≤ s < +∞. (1.0.2) sup x∈D
• ˜f (x) := f (x) + p(x) d¯u.o. . c go. i l`a h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u
. t l`a c´ac gi´a tri. riˆeng nho˙’ gi´o.i nˆo. i trˆen D, go. i tˇa´t l`a h`am bi. nhiˆe˜u. • Ta c˜ung k´y hiˆe.u λmin, λmax v`a λ(A) lˆa` n lu.o.
nhˆa´t, l´o.n nhˆa´t v`a tˆa. p c´ac gi´a tri. riˆeng cu˙’a ma trˆa. n A.
Trong mu. c n`ay, ch´ung tˆoi tr`ınh b`ay D- i.nh l´y Kuhn-Tucker cho b`ai to´an
quy hoa. ch lˆo`i sau:
x ∈ D (L1) g0(x) → inf, D = {x ∈ S | gi(x) ≤ 0, i = 1, . . . , m},
trong d¯´o gi : IRn → IR, i = 0, . . . , m, l`a c´ac h`am h`am lˆo`i, S ⊂ IRn l`a tˆa. p lˆo`i.
B`ai to´an trˆen d¯˜a d¯u.o.
m (cid:88)
. c nghiˆen c´u.u t`u. rˆa´t s´o.m, mˆo. t trong nh˜u.ng kˆe´t qua˙’ quan tro. ng l`a d¯i.nh l´y Kuhn-Tucker do W. H. Kuhn v`a A. W. Tucker d¯u.a ra v`ao nˇam 1951 trong [22] cˆong tr`ınh khai ph´a cu˙’a Quy hoa. ch lˆo`i. Trong B`ai to´an (L1) h`am Lagrange d¯u.o. . c d¯i.nh ngh˜ıa nhu. sau:
i=0
(1.1.3) L(x, µ0, . . . , µm) := µigi(x),
m (cid:88)
trong d¯´o µi, i = 0, 1, . . . , m, nhˆa. n c´ac gi´a tri. thu. . c, x ∈ D. Nˆe´u tˆa. p D cu˙’a B`ai to´an (P ) tr`ung v´o.i tˆa. p D cu˙’a B`ai to´an (L1) th`ı h`am Lagrange cu˙’a B`ai to´an (P ) c´o da. ng
i=1
(1.1.4) L(x, µ0, . . . , µm) := f (x) + µigi(x),
D- i.nh l´y 1.1.1. (D- i.nh l´y Kuhn-Tucker, xem [74]). X´et B`ai to´an (L).
(a) Nˆe´u x∗ l`a nghiˆe. m cu.
. c tiˆe˙’u cu˙’a b`ai to´an th`ı tˆo`n ta. i c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe. t tiˆeu, tho˙’a m˜an d¯iˆe` u kiˆe. n Kuhn-Tucker
(1.1.5) L(x, µ0, . . . , µm) L(x∗, µ0, . . . , µm) = min x∈S
v`a d¯iˆe` u kiˆe. n b`u
(1.1.6) µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m.
Nˆe´u thˆem d¯iˆe` u kiˆe. n Slater
(1.1.7) ∃z ∈ S : gi(z) < 0 v´o.i mo. i i = 1, . . . , m,
tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) Nˆe´u tˆo`n ta. i x∗ tho˙’a m˜an (1.1.5), (1.1.6) v´o.i µ0 = 1 th`ı x∗ l`a nghiˆe. m
cu. . c tiˆe˙’u cu˙’a B`ai to´an (L1). Da. ng du.´o.i vi phˆan cu˙’a D- i.nh l´y Kuhn-Tucker d¯u.o.
. c cu˙’a B`ai to´an (L1).
. c ph´at biˆe˙’u nhu. sau: D- i.nh l´y 1.1.2. (xem [74]) Gia˙’ thiˆe´t rˇa`ng gi : IRn → IR, i = 1, . . . , m, l`a c´ac h`am lˆo`i, c`ung liˆen tu. c ´ıt nhˆa´t ta. i mˆo. t d¯iˆe˙’m cu˙’a tˆa. p lˆo`i S ⊂ IRn. Cho x∗ l`a mˆo. t nghiˆe. m chˆa´p nhˆa. n d¯u.o. (a) Nˆe´u x∗ l`a nghiˆe. m cu.
m (cid:88)
. c tiˆe˙’u cu˙’a b`ai to´an th`ı tˆo`n ta. i c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe. t tiˆeu, tho˙’a m˜an phu.o.ng tr`ınh
i=0
0 ∈ (1.1.8) µi∂gi(x∗) + N (x∗|S)
v`a
(1.1.9) µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m,
trong d¯´o tˆa. p
∂gi(x∗) := {ξ | gi(x) − gi(x∗) ≥ (cid:104)ξ, x − x∗(cid:105) ∀x ∈ IRn}
l`a du.´o.i vi phˆan cu˙’a gi ta. i x∗ v`a tˆa. p
N (x∗|S) := {ξ | (cid:104)ξ, x − x∗(cid:105) ≤ 0 ∀x ∈ S}
l`a n´on ph´ap tuyˆe´n cu˙’a S ta. i x∗. Nˆe´u d¯iˆe` u kiˆe. n Slater (1.1.7) tho˙’a m˜an, th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1. (b) Nˆe´u tˆo`n ta. i x∗ tho˙’a m˜an (1.1.8), (1.1.9) v´o.i µ0 = 1 th`ı x∗ l`a nghiˆe. m
cu. . c tiˆe˙’u cu˙’a B`ai to´an (L1).
m (cid:88)
Nhˆa. n x´et 1.1.1. Nˆe´u S = IRn th`ı khi d¯´o N (x∗|S) = {0}, nˆen biˆe˙’u th´u.c (1.1.8) d¯u.o. . c thay bo.˙’ i
i=0
0 ∈ (1.1.10) µi∂gi(x∗).
D- ˆo´i v´o.i b`ai to´an quy hoa. ch to`an phu.o.ng ta c´o d¯i.nh l´y sau: D- i.nh l´y 1.1.3. (Xem [31]). X´et b`ai to´an quy hoa. ch to`an phu.o.ng
x ∈ D (L2)
m (cid:88)
(cid:104)M x, x(cid:105) + (cid:104)b, x(cid:105) → inf, D = {x ∈ IRn | (cid:104)ci, x(cid:105) ≤ di, i = 1, . . . , m}, trong d¯´o M ∈ IRn×n l`a ma trˆa. n d¯ˆo´i x´u.ng, ci ∈ IRn, i = 1, . . . , m. Khi d¯´o, nˆe´u x∗ l`a nghiˆe. m cu. . c tiˆe˙’u d¯i.a phu.o.ng th`ı tˆo`n ta. i c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 1, . . . , m, sao cho ch´ung tho˙’a m˜an c´ac d¯iˆe` u kiˆe. n
i=1
(2M x∗ + b) + (1.1.6) µici = 0,
v`a
i = 1, . . . , m. (1.1.7) µi((cid:104)ci, x∗(cid:105) − di) = 0 v´o.i mo. i
D- i.nh l´y 1.1.4. (xem [31], trang 79). Cho D l`a tˆa. p lˆo`i d¯a diˆe. n, khi d¯´o
(a) Nˆe´u M l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng v`a D (cid:54)= ∅ th`ı B`ai to´an
(L2) c´o d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t.
. c tiˆe˙’u d¯i.a phu.o.ng
(b) Nˆe´u M l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh ˆam th`ı d¯iˆe˙’m cu. cu˙’a B`ai to´an (L2) (nˆe´u tˆo`n ta. i) l`a mˆo. t d¯iˆe˙’m cu. . c biˆen cu˙’a D.
Nhˆa. n x´et 1.1.2. Kˆe´t luˆa. n (b) cu˙’a d¯i.nh l´y trˆen tu.o.ng d¯u.o.ng v´o.i ph´at . c d¯a. i d¯i.a phu.o.ng biˆe˙’u sau “Nˆe´u M d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng th`ı d¯iˆe˙’m cu. cu˙’a B`ai to´an (Q) l`a d¯iˆe˙’m cu. . c biˆen cu˙’a D”.
H`am g : D ⊂ IRn → IR d¯u.o. . c go. i l`a lˆo`i, nˆe´u x0, x1 ∈ D, th`ı bˆa´t d¯ˇa˙’ ng
th´u.c
. a lˆo`i [71], tu.
g(xλ) ≤ (1 − λ)g(x0) + λg(x1), (1.2.8) tho˙’a m˜an v´o.i mo. i d¯iˆe˙’m xλ ∈ [x0, x1]. H`am lˆo`i c´o nhiˆe` u t´ınh chˆa´t th´u vi. khˆong nh˜u.ng vˆe` phu.o.ng diˆe.n gia˙’i t´ıch m`a c`on vˆe` phu.o.ng diˆe.n tˆo´i u.u h´oa nhu.: tˆa. p m´u.c du.´o.i cu˙’a h`am lˆo`i d¯ang x´et l`a lˆo`i; mˆo˜i d¯iˆe˙’m cu. . c tiˆe˙’u d¯i.a . c tiˆe˙’u to`an cu. c; mˆo˜i d¯iˆe˙’m d`u.ng cu˙’a phu.o.ng cu˙’a h`am d¯ang x´et l`a d¯iˆe˙’m cu. h`am d¯ang x´et l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c; nˆe´u h`am d¯ang x´et d¯a. t gi´a tri. . c d¯a. i trˆen miˆe` n lˆo`i compact th`ı c˜ung d¯a. t gi´a tri. cu. cu. . c d¯a. i ta. i ´ıt nhˆa´t mˆo. t d¯iˆe˙’m cu. . c biˆen. Tuy nhiˆen trong nhiˆe` u b`ai to´an thu. . c tˆe´, h`am cˆa` n x´et c´o mˆo. t sˆo´ t´ınh chˆa´t trˆen nhu.ng khˆong pha˙’i l`a h`am lˆo`i. Do d¯´o, d¯˜a xuˆa´t hiˆe.n . c d¯ˇa. c tru.ng bo.˙’ i mˆo. t trong c´ac t´ınh chˆa´t nhiˆe` u loa. i h`am lˆo`i suy rˆo. ng d¯u.o. cu˙’a h`am lˆo`i nhu.: h`am tu. . a lˆo`i hiˆe.n [18], [26], gia˙’ lˆo`i [25], [72], lˆo`i bˆa´t biˆe´n [14] . . .
T`u. nˇam 1989 xuˆa´t hiˆe.n mˆo. t hu.´o.ng m´o.i mo.˙’ rˆo. ng kh´ai niˆe.m h`am lˆo`i go. i l`a h`am lˆo`i thˆo. Mˆo. t h`am P -lˆo`i d¯u.o. . c H. X. Phu go. i l`a lˆo`i thˆo nˆe´u nhu. t´ınh chˆa´t P tho˙’a m˜an v´o.i mo. i x0, x1 ∈ D m`a (cid:107)x0 − x1(cid:107) ≥ γ, trong d¯´o γ
. a lˆo`i, δ-lˆo`i gi˜u.a l`a mˆo. t sˆo´ du.o.ng cˆo´ d¯i.nh cho tru.´o.c. H`am lˆo`i thˆo δ-lˆo`i, δ-tu. . c T. C. Hu, V. Klee v`a D. Larman [16] d¯u.a ra v`ao nˇam 1989. Tiˆe´p d¯´o d¯u.o. . c nghiˆen c´u.u bo.˙’ i H. nˇam 1991 R. Kl¨otzler d¯ˆe` xuˆa´t kh´ai niˆe.m ρ-lˆo`i v`a d¯u.o. . a lˆo`i, γ-lˆo`i d¯ˆo´i x´u.ng, Hartwig [15] v`a B. S¨ollner [73]. C´ac h`am γ-lˆo`i, γ-tu. γ-lˆo`i nhe., γ-lˆo`i gi˜u.a d¯u.o. . c d¯ˆe` xuˆa´t v`a nghiˆen c´u.u bo.˙’ i H. X. Phu [34]–[37], H. X. Phu v`a N. N. Hai [49]. Trong luˆa. n ´an n`ay ch´ung tˆoi quan tˆam v`a su.˙’ du. ng nhiˆe` u lˆa` n c´ac t´ınh chˆa´t tˆo´i u.u cu˙’a c´ac h`am γ-lˆo`i ngo`ai [47], Γ-lˆo`i ngo`ai [44] v`a γ-lˆo`i trong [41]–[43]. C´ac l´o.p h`am n`ay d¯ˆe` u do H. X. Phu d¯ˆe` xuˆa´t v`a nghiˆen c´u.u.
. c biˆen, mˆo. t kh´ai niˆe.m d¯u.o.
Tru.´o.c khi tr`ınh b`ay mu. c tiˆe´p theo, ch´ung tˆoi nhˇa´c la. i d¯i.nh ngh˜ıa vˆe` . c H. X. Phu gi´o.i thiˆe.u lˆa` n d¯ˆa` u tiˆen . c su.˙’ du. ng
d¯iˆe˙’m γ-cu. v`ao nˇam 1994 v`a nghiˆen c´u.u trong [35]. Kh´ai niˆe.m n`ay s˜e d¯u.o. trong Chu.o.ng 4 cu˙’a luˆa. n ´an.
D- i.nh ngh˜ıa 1.2.1. ([35]) Cho γ > 0 v`a D ⊂ X l`a tˆa. p lˆo`i trong khˆong gian . c biˆen (tu.o.ng ´u.ng tuyˆe´n t´ınh d¯i.nh chuˆa˙’n X. D- iˆe˙’m x ∈ D go. i l`a d¯iˆe˙’m γ-cu. γ-cu. . c biˆen ngˇa. t) cu˙’a D nˆe´u x(cid:48), x(cid:48)(cid:48) ∈ D tho˙’a m˜an x = 0.5(x(cid:48) + x(cid:48)(cid:48)) th`ı suy ra (cid:107)x(cid:48) − x(cid:48)(cid:48)(cid:107) ≤ 2γ (tu.o.ng ´u.ng (cid:107)x(cid:48) − x(cid:48)(cid:48)(cid:107) < 2γ).
Trong mu. c n`ay ch´ung tˆoi tr`ınh b`ay vˆe` h`am γ-lˆo`i ngo`ai ([46]). C´ac t´ınh chˆa´t tˆo´i u.u cu˙’a l´o.p h`am n`ay ch´ung tˆoi s˜e khai th´ac su.˙’ du. ng trong Chu.o.ng 2.
D- i.nh ngh˜ıa 1.3.2. ([46]) Cho γ > 0. H`am g : D ⊂ IRn → IR d¯u.o. . c go. i l`a γ-lˆo`i ngo`ai (hoˇa. c γ-lˆo`i ngo`ai ngˇa. t) v´o.i d¯ˆo. thˆo γ, nˆe´u v´o.i mo. i x0, x1 ∈ D tˆo`n ta. i k ∈ IN v`a
khi i = 0, 1, . . . , k − 1, λi ∈ [0, 1], i = 0, 1, . . . , k, λ0 = 0, λk = 1, γ 0 ≤ λi+1 − λi ≤ (cid:107)x0 − x1(cid:107)
sao cho v´o.i xλi = (1 − λi)x0 + λix1, i = 0, 1, . . . , k, th`ı
i = 0, 1, . . . , k, g(xλi) ≤ (1 − λi)g(x0) + λig(x1) v´o.i
(hoˇa. c
g(xλi) < (1 − λi)g(x0) + λig(x1) v´o.i i = 1, . . . , k − 1).
D- i.nh l´y 1.3.5. ([46]) Nˆe´u g : D ⊂ IRn →]−∞, +∞] l`a γ-lˆo`i ngo`ai th`ı lsc g c˜ung l`a γ-lˆo`i ngo`ai, trong d¯´o lsc g(x) := lim inf y→x g(y) v´o.i mo. i x ∈ D. D- i.nh ngh˜ıa 1.3.3. ([46]) Cho γ > 0, M ⊂ IRn, M (cid:54)= ∅, M d¯u.o. . c go. i l`a γ-lˆo`i ngo`ai v´o.i d¯ˆo. thˆo γ nˆe´u x0, x1 ∈ M v`a (cid:107)x0 − x1(cid:107) > γ suy ra tˆo`n ta. i z0 := x0, z1, . . . , zk := x1 ∈ [x0, x1] ∩ M sao cho
(cid:107)zi+1 − zi(cid:107) ≤ γ v´o.i i=0, 1,. . . , k-1.
. c go. i l`a D- i.nh l´y 1.3.6. ([46]) K´y hiˆe.u L(g, α) := {x ∈ D : g(x) ≤ α}, v´o.i α ∈ IR v`a go. i l`a tˆa. p m´u.c du.´o.i cu˙’a h`am g. Khi d¯´o, nˆe´u g l`a h`am γ-lˆo`i ngo`ai th`ı L(g, α) l`a tˆa. p γ-lˆo`i ngo`ai. D- i.nh ngh˜ıa 1.3.4. (xem [1], [38]) x∗ ∈ D d¯u.o.
1) d¯iˆe˙’m γ-cu. . c tiˆe˙’u cu˙’a g nˆe´u tˆo`n ta. i (cid:15) > 0 sao cho g(x∗) ≤ g(x) v´o.i
mo. i x ∈ B(x∗, γ + (cid:15)) ∩ D;
2) d¯iˆe˙’m γ-infimum cu˙’a g nˆe´u tˆo`n ta. i (cid:15) > 0 sao cho
g(x); g(x) = lim inf x→x∗ inf x∈B(x∗,γ+(cid:15))∩D
3) d¯iˆe˙’m inf imum to`an cu. c cu˙’a g nˆe´u
g(x). lim inf x→x∗ g(x) = inf x∈D
Mˆe. nh d¯ˆe` 1.3.1. ([1], [38]) x∗ l`a d¯iˆe˙’m γ-infimum cu˙’a g khi v`a chı˙’ khi d¯iˆe˙’m n`ay l`a d¯iˆe˙’m γ-cu. . c tiˆe˙’u cu˙’a lsc g.
T´ınh chˆa´t tˆo´i u.u cu˙’a h`am γ-lˆo`i ngo`ai d¯u.o. . c chı˙’ ra bo.˙’ i d¯i.nh l´y sau:
D- i.nh l´y 1.3.7. ([1], [38]) Nˆe´u g l`a γ-lˆo`i ngo`ai th`ı c´o c´ac t´ınh chˆa´t
(Mγ) Mˆo˜i d¯iˆe˙’m γ-cu. . c tiˆe˙’u x∗ cu˙’a g l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c.
(Iγ) Mˆo˜i d¯iˆe˙’m γ-infimum x∗ cu˙’a g l`a d¯iˆe˙’m infimum to`an cu. c.
. c tiˆe˙’u to`an Mˆe.nh d¯ˆe` sau nˆeu cho ta d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m cu.
cu. c cu˙’a h`am γ-lˆo`i ngˇa. t.
Mˆe. nh d¯ˆe` 1.3.2. ([42]) Nˆe´u g : D ⊂ IRn → IR l`a h`am γ-lˆo`i ngo`ai ngˇa. t, th`ı d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c khˆong vu.o. . t qu´a γ.
D- ˆo´i v´o.i h`am lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ta c´o mˆe.nh d¯ˆe` vˆe` t´ınh γ-lˆo`i
ngo`ai sau d¯ˆay.
Mˆe. nh d¯ˆe` 1.3.3. ([42]) Cho γ > 0, g : D ⊂ IRn → IR l`a h`am lˆo`i v`a
(cid:17) > 0. (g(x0) + g(x1)) − g( (x0 + x1)) h1(γ) := (cid:16)1 2 1 2 inf x0,x1∈D, (cid:107)x0−x1(cid:107)=γ
Khi d¯´o, nˆe´u h`am nhiˆe˜u p tho˙’a m˜an
|p(x)| ≤ h1(γ)/2 v´o.i mo. i x ∈ D
th`ı h`am bi. nhiˆe˜u ˜g = g + p l`a γ-lˆo`i ngo`ai v`a nˆe´u
|p(x)| < h1(γ)/2 v´o.i mo. i x ∈ D
th`ı ˜g = g + p l`a γ-lˆo`i ngo`ai ngˇa. t.
Kh´ai niˆe.m h`am Γ-lˆo`i ngo`ai do H. X. Phu d¯ˆe` xuˆa´t v`a nghiˆen c´u.u trong [44]. Trong mu. c n`ay ch´ung tˆoi tr`ınh b`ay la. i mˆo. t sˆo´ t´ınh chˆa´t cu˙’a l´o.p h`am Γ-lˆo`i ngo`ai m`a H. X. Phu d¯˜a chı˙’ ra. Mˆo. t sˆo´ t´ınh chˆa´t tˆo´i u.u cu˙’a l´o.p h`am n`ay s˜e l`a co. so.˙’ cho viˆe.c nghiˆen c´u.u B`ai to´an ( ˜P ) trong Chu.o.ng 3.
D- i.nh ngh˜ıa 1.4.5. ([44]) Cho X l`a khˆong gian v´ec to. trˆen tru.`o.ng sˆo´ thu. . c, Γ l`a tˆa. p cˆan trong X t´u.c l`a λΓ ⊂ Γ v´o.i mo. i |λ| ≤ 1, v`a D l`a tˆa. p lˆo`i trong X. H`am g : D → IR d¯u.o. . c go. i l`a Γ-lˆo`i ngo`ai nˆe´u v´o.i mo. i x0, x1 ∈ D tˆo`n ta. i tˆa. p d¯´ong Λ ⊂ [0, 1] v`a ch´u.a {0, 1} sao cho
(1.4.9) [x0, x1] ⊂ {xλ | λ ∈ Λ} + 0.5Γ
v`a
(1.4.10) ∀λ ∈ Λ : g(xλ) ≤ (1 − λ)g(x0) + λg(x1).
V´o.i d¯i.nh ngh˜ıa h`am Γ-lˆo`i ngo`ai nhu. trˆen th`ı hˆa` u hˆe´t c´ac h`am lˆo`i thˆo . c d¯i.nh ngh˜ıa o.˙’ c´ac mu. c trˆen nhu. δ-lˆo`i, ρ-lˆo`i, γ-lˆo`i, γ-lˆo`i d¯ˆo´i x´u.ng . . .
. c go. i l`a Γ-lˆo`i ngo`ai nˆe´u v´o.i mo. i
d¯u.o. . p riˆeng cu˙’a l´o.p h`am n`ay. l`a c´ac tru.`o.ng ho. D- i.nh ngh˜ıa 1.4.6. ([44]) Tˆa. p S ⊂ X d¯u.o. x0, x1 ∈ S
[x0, x1] ⊂ ([x0, x1] ∩ S) + 0.5Γ,
t´u.c l`a tˆo`n ta. i Λ ⊂ [0, 1] sao cho
(1.4.11) {xλ | λ ∈ Λ} ⊂ S, [x0, x1] ⊂ {xλ | λ ∈ Λ} + 0.5Γ.
V´ı du. 1.4.1. ([44]) Gia˙’ su.˙’ zi ∈ IR, Z l`a tˆa. p c´ac sˆo´ nguyˆen, i ∈ Z tho˙’a m˜an 0 < zi+1 − zi ≤ γ, i ∈ Z v`a g : IR → IR sao cho
g(x) ≥ g(zi) ∀x ∈ IR v`a i ∈ Z.
Khi d¯´o g(x) l`a Γ-lˆo`i ngo`ai v´o.i Γ = ¯B(0, γ). Mˆe. nh d¯ˆe` 1.4.4. ([44]) Tˆa. p m´u.c du.´o.i cu˙’a h`am Γ-lˆo`i ngo`ai l`a Γ-lˆo`i ngo`ai. D- i.nh l´y 1.4.8. ([44]) Cho B l`a tˆa. p cˆan trong khˆong gian v´ec to. X. Khi d¯´o g : D ⊂ X → IR l`a h`am Γ-lˆo`i ngo`ai v´o.i Γ = B khi v`a chı˙’ khi epi g l`a tˆa. p Γ-lˆo`i ngo`ai v´o.i Γ = B × IR. D- i.nh ngh˜ıa 1.4.7. ([44]) Cho g : D → IR. D- iˆe˙’m x∗ ∈ D go. i l`a d¯iˆe˙’m Γ-cu. . c tiˆe˙’u cu˙’a g nˆe´u
g(x∗) = g(x) inf x∈(x∗+Γ)∩D
v`a go. i l`a Γ-infimum cu˙’a g nˆe´u
g(x) = g(x). lim inf x∈X, x→x∗ inf x∈(x∗+Γ)∩D
T´ınh chˆa´t tˆo´i u.u quan tro. ng cu˙’a h`am Γ-lˆo`i ngo`ai d¯u.o. . c chı˙’ ra bo.˙’ i d¯i.nh
l´y sau: D- i.nh l´y 1.4.9. ([44]) Gia˙’ su.˙’ 0 l`a d¯iˆe˙’m trong cu˙’a tˆa. p Γ v`a g : D → IR l`a h`am Γ-lˆo`i ngo`ai. Khi d¯´o
g(x), (1.4.12) g(x∗) = g(x) =⇒ g(x∗) = inf x∈D inf x∈D∩({x∗}+Γ)
t´u.c l`a nˆe´u x∗ l`a d¯iˆe˙’m Γ-cu. . c tiˆe˙’u th`ı x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c.
Kh´ai niˆe.m h`am γ-lˆo`i trong d¯u.o.
. c H. X. Phu d¯u.a ra nhˇa`m nghiˆen c´u.u c´ac d¯iˆe˙’m cu. . c d¯a. i to`an cu. c v`a d¯iˆe˙’m supremum to`an cu. c. Trong mu. c n`ay ch´ung tˆoi d¯iˆe˙’m qua mˆo. t sˆo´ kˆe´t qua˙’ nghiˆen c´u.u cu˙’a H. X. Phu trong c´ac b`ai b´ao [41], [42] v`a [43]. Ch´ung tˆoi s˜e su.˙’ du. ng c´ac kˆe´t qua˙’ d¯´o d¯ˆe˙’ nghiˆen c´u.u d¯iˆe˙’m cu. . c d¯a. i to`an cu. c, supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q) trong Chu.o.ng 4 cu˙’a luˆa. n ´an n`ay. D- i.nh ngh˜ıa 1.5.8. ([42]) H`am g : D ⊂ IRn → IR go. i l`a h`am γ-lˆo`i trong (hoˇa. c γ-lˆo`i trong ngˇa. t) trˆen D v´o.i d¯ˆo. thˆo γ > 0, nˆe´u tˆo`n ta. i d¯ˆo. tinh cˆo´ d¯i.nh ν ∈]0, 1] sao cho
v´o.i mo. i x0, x1 ∈ D tho˙’a m˜an (cid:107)x0 − x1(cid:107) = νγ v`a x1+1/ν = −(1/ν)x0 + (1 + 1/ν)x1 ∈ D,
th`ı
(cid:16) (cid:17) ≥ 0, g((1 − λ)x0 + λx1) − (1 − λ)g(x0) − λg(x1) sup λ∈[2,1+1/ν]
(hoˇa. c
(cid:16) (cid:17) > 0). g((1 − λ)x0 + λx1) − (1 − λ)g(x0) − λg(x1) sup λ∈[2,1+1/ν]
V´ı du. 1.5.2. ([41]) Cho
(cid:40)
g(x) = a nˆe´u x l`a h˜u.u ty˙’ nˆe´u x l`a vˆo ty˙’, b
nˆe´u νγ l`a h˜u.u ty˙’ th`ı g l`a h`am γ-lˆo`i trong v´o.i γ > 0.
Nhˆa. n x´et 1.5.3. Khi ν = 1 th`ı h`am g l`a γ-lˆo`i trong (hoˇa. c γ-lˆo`i trong ngˇa. t) nˆe´u v´o.i x0, x1 ∈ D tho˙’a m˜an (cid:107)x0 − x1(cid:107) = γ v`a −x0 + 2x1 ∈ D k´eo theo
−g(x0) + 2g(x1) ≤ g(−x0 + 2x1),
(hoˇa. c
−g(x0) + 2g(x1) < g(−x0 + 2x1)).
Mˆe. nh d¯ˆe` 1.5.5. ([41]) Gia˙’ su.˙’ g : D → IR l`a γ-lˆo`i trong v´o.i d¯ˆo. tinh ν. Nˆe´u x1 ∈ D l`a d¯iˆe˙’m cu. . c d¯a. i cu˙’a g th`ı mo. i d¯iˆe˙’m x0 tho˙’a m˜an
(cid:107)x0 − x1(cid:107) = νγ, x1+1/ν = −(1/ν)x0 + (1 + 1/ν)x1 ∈ D
c˜ung l`a d¯iˆe˙’m cu. . c d¯a. i cu˙’a g trˆen D.
. c d¯a. i th`ı c´o ´ıt nhˆa´t mˆo. t d¯iˆe˙’m cu.
D- i.nh l´y 1.5.10. ([41]) Cho D ⊂ IRn l`a tˆa. p lˆo`i, gi´o.i nˆo. i v`a g : D → IR l`a h`am γ-lˆo`i trong. Nˆe´u g c´o d¯iˆe˙’m cu. . c d¯a. i l`a d¯iˆe˙’m γ-cu. . c biˆen ngˇa. t cu˙’a D.
. c d¯a. i l`a d¯iˆe˙’m γ-cu. . c d¯a. i trˆen D th`ı d¯iˆe˙’m cu. . c biˆen ngˇa. t cu˙’a D.
D- i.nh l´y 1.5.11. ([41]) Cho g : D → IR l`a h`am γ-lˆo`i trong ngˇa. t. Nˆe´u g d¯a. t cu. Mˆe. nh d¯ˆe` 1.5.6. ([41]) Cho g : D → IR, D l`a tˆa. p mo.˙’ tu.o.ng d¯ˆo´i theo bao aphin cu˙’a D (k´y hiˆe. u l`a aff D) l`a h`am bi. chˇa. n trˆen v`a γ-lˆo`i trong v´o.i d¯ˆo. tinh ν ∈ [0, 1]. Nˆe´u x1 l`a d¯iˆe˙’m supremum cu˙’a g, th`ı v´o.i mo. i x0 ∈ D tho˙’a m˜an
(cid:107)x0 − x1(cid:107) = νγ, x1+1/ν = −(1/ν)x0 + (1 + 1/ν)x1 ∈ D
c˜ung l`a d¯iˆe˙’m supremum cu˙’a g.
. c biˆen ngˇa. t cu˙’a D. D- i.nh l´y 1.5.12. ([41]) Cho D ⊂ IRn l`a tˆa. p mo.˙’ tu.o.ng d¯ˆo´i theo aff D, g : D → IR bi. chˇa. n trˆen v`a γ-lˆo`i trong. Nˆe´u g d¯a. t supremum trˆen D, th`ı c´o ´ıt nhˆa´t mˆo. t d¯iˆe˙’m supremum l`a d¯iˆe˙’m γ-cu.
Hˆe. qua˙’ 1.5.1. Cho D ⊂ IRn l`a tˆa. p compact, g : D → IR bi. chˇa. n trˆen v`a lˆo`i trong. Khi d¯´o g c´o tˆo´i thiˆe˙’u mˆo. t d¯iˆe˙’m supremum l`a d¯iˆe˙’m biˆen tu.o.ng d¯ˆo´i cu˙’a D theo aff D hoˇa. c l`a d¯iˆe˙’m γ-cu. . c biˆen ngˇa. t cu˙’a D.
D- ˆo´i v´o.i h`am lˆo`i ngˇa. t bi. nhiˆe˜u, ta c´o mˆe.nh d¯ˆe` quan tro. ng vˆe` t´ınh γ-lˆo`i
trong sau d¯ˆay.
g : D ⊂ IRn → IR l`a h`am lˆo`i v`a Mˆe. nh d¯ˆe` 1.5.7. ([42]) Cho λ > 0,
h2(γ) := (cid:0)g(x0) − 2g(x1) + g(−x0 + 2x1)(cid:1) > 0. inf x0,x1∈D, (cid:107)x0−x1(cid:107)=γ,−x0+2x1∈D
Khi d¯´o, nˆe´u h`am nhiˆe˜u p tho˙’a m˜an
|p(x)| ≤ h2(γ)/4 v´o.i mo. i x ∈ D
th`ı h`am bi. nhiˆe˜u ˜g = g + p l`a γ-lˆo`i trong v`a nˆe´u
|p(x)| < h2(γ)/4 v´o.i mo. i x ∈ D
th`ı h`am bi. nhiˆe˜u ˜g = g + p l`a γ-lˆo`i trong ngˇa. t.
Kˆe´t luˆa. n: Trong chu.o.ng n`ay ch´ung tˆoi d¯˜a tr`ınh b`ay D- i.nh l´y Kuhn- Tucker cho b`ai to´an lˆo`i, d¯i.nh l´y vˆe` d¯iˆe` u kiˆe.n cˆa` n cu. . c tri. cho b`ai to´an to`an phu.o.ng, tˆo˙’ng quan vˆe` c´ac loa. i h`am lˆo`i thˆo v`a mˆo. t sˆo´ t´ınh chˆa´t tˆo´i u.u cu˙’a . c su.˙’ du. ng nhiˆe` u lˆa` n trong . c tr´ıch dˆa˜n s˜e d¯u.o. ch´ung. Nh˜u.ng kˆe´t qua˙’ d¯u.o. c´ac chu.o.ng sau. Vˆe` su. . tˆo`n ta. i nghiˆe.m v`a t´ınh ˆo˙’n d¯i.nh nghiˆe.m cu˙’a b`ai to´an to`an phu.o.ng c´o thˆe˙’ t`ım thˆa´y trong [5], [7], [10], [13], [31]. . . v`a vˆe` c´ac loa. i h`am lˆo`i thˆo c`ung c´ac t´ınh chˆa´t cu˙’a ch´ung c´o thˆe˙’ t`ım thˆa´y trong [1], [3], [38], [41], [42], [44], [46] v`a [49],. . .
. . CHU O NG 2 D- IˆE˙’M INFIMUM TO `AN CU. C CU˙’ A B `AI TO ´AN ( ˜P )
Chu.o.ng n`ay chu˙’ yˆe´u nghiˆen c´u.u t´ınh γ-lˆo`i ngo`ai cu˙’a h`am bi. nhiˆe˜u ˜f = f + p; c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ); d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ); t´ınh ˆo˙’n d¯i.nh nghiˆe.m tˆo´i u.u cu˙’a B`ai to´an ( ˜P ); t´ınh chˆa´t tu. . a v`a D- i.nh l´y Kuhn-Tucker suy rˆo. ng cho B`ai to´an ( ˜P ).
Ch´ung tˆoi nhˇa´c la. i, ˜f = f + p l`a h`am bi. nhiˆe˜u, trong d¯´o f d¯u.o. . c cho bo.˙’ i cˆong th´u.c (1.0.1), t´u.c l`a f (x) = (cid:104)A, x(cid:105) + (cid:104)b, x(cid:105), A ∈ IRn×n l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng v`a h`am nhiˆe˜u p tho˙’a m˜an (1.0.2), ngh˜ıa l`a
|p(x)| ≤ s < +∞. sup x∈D
Ngo`ai ra, trong suˆo´t chu.o.ng n`ay ta k´y hiˆe.u γ∗ := 2(cid:112)2s/λmin trong d¯´o λmin l`a gi´a tri. riˆeng nho˙’ nhˆa´t cu˙’a A.
Phˆa` n l´o.n c´ac t´ınh chˆa´t d¯ˇa. c tru.ng cu˙’a c´ac h`am lˆo`i suy rˆo. ng khˆong c`on d¯´ung khi bi. nhiˆe˜u, trong khi nhiˆe` u ´u.ng du. ng thu. . c tˆe´ thu.`o.ng bi. a˙’nh hu.o.˙’ ng bo.˙’ i nhiˆe˜u tuyˆe´n t´ınh hoˇa. c nhiˆe˜u phi tuyˆe´n. C´ac t´ınh chˆa´t cu˙’a h`am γ-lˆo`i ngo`ai v`a t´ınh ˆo˙’n d¯i.nh cu˙’a l´o.p h`am n`ay theo t´ınh chˆa´t lˆo`i d¯ˇa. c tru.ng cu˙’a . c nghiˆen c´u.u trong [47]. Trong mu. c n`ay, n´o khi bi. nhiˆe˜u tuyˆe´n t´ınh d¯˜a d¯u.o. ch´ung tˆoi nghiˆen c´u.u c´ac d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai.
Mˆe.nh d¯ˆe` sau cho ta gi´a tri. cu. thˆe˙’ cu˙’a h`am h1(γ), h`am n`ay d¯u.o. . c d¯i.nh
ngh˜ıa trong Mˆe.nh d¯ˆe` 1.3.3 Chu.o.ng 1. Mˆe. nh d¯ˆe` 2.1.8. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1) v`a γ > 0. Khi d¯´o
(cid:17) ) h1(γ) := f (x0) + f (x1) − f ( ≥ λminγ2/4, (cid:16)1 2 1 2 x0 + x1 2 inf x0, x1∈D, (cid:107)x0−x1(cid:107)=γ
(2.1.1)
nˆe´u D = IRn th`ı
h1(γ) = λminγ2/4.
Lˆa´y cˇa. p x0, x1 bˆa´t k`y trong D tho˙’a m˜an d¯iˆe` u kiˆe.n
Ch´u.ng minh. (cid:107)x0 − x1(cid:107) = γ. Ta c´o
f (x0) + x0 + x1 2
1 2 = (cid:104)b, x1(cid:105) 1 2
(cid:104)b, x0 + x1(cid:105)
= 1 2 (cid:104)Ax0, x0(cid:105) + 1 4 (cid:104)Ax0, x0(cid:105) + (cid:104)Ax1, x1(cid:105) −
= (cid:104)Ax1, x1(cid:105) − (cid:104)Ax0, x0(cid:105) + (cid:104)Ax0, x1(cid:105), f (x1) − f ( 1 2 (cid:10)A(x0 + x1), x0 + x1 1 2 1 4 ) 1 (cid:104)Ax1, x1(cid:105) + (cid:104)b, x0(cid:105) + 2 1 (cid:11) − 2 1 (cid:104)A(x0 + x1), x0 + x1(cid:105) 4 1 2 1 2 − 1 2 1 4
do d¯´o
) = (2.1.2) f (x0) + f (x1) − f ( (cid:104)A(x0 − x1), x0 − x1(cid:105). 1 2 x0 + x1 2 1 4 1 2
n (cid:88)
n (cid:88)
Go. i λi, i = 1, . . . , n, l`a c´ac gi´a tri. riˆeng cu˙’a ma trˆa. n x´ac d¯i.nh du.o.ng A (c´o thˆe˙’ c´o mˆo. t sˆo´ gi´a tri. tr`ung nhau), {ei | i = 1, 2, . . . , n} l`a co. so.˙’ tru. . c chuˆa˙’n trong IRn v`a ei l`a v´ec to. riˆeng ´u.ng v´o.i gi´a tri. riˆeng λi, i = 1, 2, . . . , n (xem [77]). Khi d¯´o, ta c´o thˆe˙’ viˆe´t
i=1
i=1
x0 = ζ i 0ei, x1 = ζ i 1ei
n (cid:88)
n (cid:88)
v`a
0 − ζ i
1)ei,
0 − ζ j
1)ej(cid:105)
i=1
j=1
(ζ j (cid:104)A(x0 − x1), x0 − x1(cid:105) = (cid:104) λi(ζ i
n (cid:88)
n (cid:88)
0 − ζ i
1)(ζ j
0 − ζ j
1)(cid:104)ei, ej(cid:105)
j=1
i=1 n (cid:88)
= λi(ζ i
0 − ζ i
1)2
i=1
n (cid:88)
= λi(ζ i
0 − ζ i
1)2
(ζ i ≥ λmin
i=1 = λmin(cid:107)x0 − x1(cid:107)2 ≥ λminγ2.
(2.1.3)
Mˇa. t kh´ac, theo (2.1.1) th`ı
(cid:17) ) f (x1) − f ( f (x0) + h1(γ) = x0 + x1 2 inf x0, x1∈D, (cid:107)x0−x1(cid:107)=γ
= (cid:11), (cid:16)1 1 2 2 (cid:10)A(x0 − x1), x0 − x1 1 4 inf x0, x1∈D, (cid:107)x0−x1(cid:107)=γ
nˆen
h1(γ) ≥ λminγ2/4. (2.1.4) Nˆe´u D = IRn th`ı ta c´o thˆe˙’ cho. n cˇa. p x0, x1 ∈ IRn tho˙’a m˜an (cid:107)x0 − x1(cid:107) = γ v`a x0 − x1 d¯ˆo`ng phu.o.ng v´o.i ei0, trong d¯´o ei0 l`a v´ec to. riˆeng trong co. so.˙’ tru. . c chuˆa˙’n ´u.ng v´o.i v´ec to. riˆeng λmin cu˙’a ma trˆa. n A. Do d¯´o
(2.1.5) (cid:10)A(x0 − x1), x0 − x1 (cid:11) = λmin(cid:107)x0 − x1(cid:107)2 = λminγ2.
Kˆe´t ho. . p (2.1.1)–(2.1.5) ta suy ra d¯iˆe` u cˆa` n ch´u.ng minh.
V´ı du. 2.1.3. Cho ma trˆa. n 15 −5 −5 3
−5 15 3 −5 A = . −5 3 11 −9 3 −5 −9 11
Khi d¯´o
15 − λ −5 −5 3
−5 15 − λ 3 −5 |A − λI| = . −5 3 11 − λ −9
3 −5 −9 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 11 − λ (cid:12)
Phu.o.ng tr`ınh d¯ˇa. c tru.ng |A − λI| = 0 tu.o.ng d¯u.o.ng v´o.i
λ4 − 52λ3 + 832λ2 − 4672λ + 5376 = 0.
Gia˙’i ra ta d¯u.o. . c c´ac gi´a tri. riˆeng sau:
√ √ 5 λ1 = 12; λ2 = 28; λ3 = 6 + 2 5; λ4 = 6 − 2
v`a c´ac v´ec to. riˆeng tru.
(−1, 1, −1, 1), e1 =
(1, −1, −1, 1), e2 = . c chuˆa˙’n tu.o.ng ´u.ng l`a 1 2 1 2 √ √ 1 (−2 − 5, −2 − 5, 1, 1), e3 = √ (cid:112) 5 √ √ 20 + 8 1 5, −2 + 5, −1, 1). (−2 + e4 = √ (cid:112) 5
√ 5 nˆen 20 − 8 V`ı ma trˆa. n A d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng v`a λmin = 6 − 2
√ 5)γ2/2. h1(γ) = (3 −
Hai mˆe.nh d¯ˆe` sau d¯ˆay chı˙’ ra c´ac d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am to`an phu.o.ng lˆo`i
ngˇa. t v´o.i nhiˆe˜u l`a γ-lˆo`i ngo`ai v`a γ-lˆo`i ngo`ai ngˇa. t.
Mˆe. nh d¯ˆe` 2.1.9. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1), p : D ⊂ IRn → IR l`a h`am nhiˆe˜u v`a γ > 0. Khi d¯´o h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai nˆe´u h`am nhiˆe˜u p tho˙’a m˜an d¯iˆe` u kiˆe. n
(2.1.6) |p(x)| ≤ λminγ2/8 v´o.i mo. i x ∈ D.
Ch´u.ng minh. Theo (2.1.1) th`ı λminγ2/8 ≤ h1(γ)/2, nˆen t`u. gia˙’ thiˆe´t ta c´o
|p(x)| ≤ h1(γ)/2 v´o.i mo. i x ∈ D,
t´u.c l`a h`am nhiˆe˜u p tho˙’a m˜an d¯iˆe` u kiˆe.n cu˙’a Mˆe.nh d¯ˆe` 1.3.3. ´Ap du. ng mˆe.nh d¯ˆe` n`ay ta suy ra ˜f = f + p l`a γ-lˆo`i ngo`ai.
Mˆe. nh d¯ˆe` 2.1.10. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1), p : D ⊂ IRn → IR l`a h`am nhiˆe˜u v`a γ > 0. Khi d¯´o h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai ngˇa. t nˆe´u h`am nhiˆe˜u p tho˙’a m˜an d¯iˆe` u kiˆe. n
(2.1.7) |p(x)| < λminγ2/8 v´o.i mo. i x ∈ D.
. p v´o.i gia˙’
Ch´u.ng minh. T`u.(2.1.1) suy ra λminγ2/8 ≤ h1(γ)/2 nˆen kˆe´t ho. thiˆe´t ta nhˆa. n d¯u.o. . c
|p(x)| < h1(γ)/2 v´o.i mo. i x ∈ D.
Do d¯´o h`am nhiˆe˜u p tho˙’a m˜an Mˆe.nh d¯ˆe` 1.3.3, v`ı thˆe´ ˜f = f + p l`a γ-lˆo`i ngo`ai ngˇa. t.
V`ı γ∗ = 2(cid:112)2s/λmin v`a supx∈D |p(x)| ≤ s < +∞, nˆen
|p(x)| ≤ s = . λminγ∗2 8 sup x∈D
Do d¯´o
|p(x)| ≤ v´o.i mo. i x ∈ D. λminγ∗2 8
Mˇa. t kh´ac, nˆe´u γ > γ∗ th`ı
|p(x)| = < , λminγ∗2 8 λminγ2 8 sup x∈D
suy ra
|p(x)| < v´o.i mo. i x ∈ D. λminγ2 8
T`u. hai kˆe´t qua˙’ trˆen, suy ra h`am nhiˆe˜u p tho˙’a m˜an (2.1.6) cu˙’a Mˆe.nh d¯ˆe` 2.1.9 v`a bˆa´t d¯ˇa˙’ ng th´u.c (2.1.7) cu˙’a Mˆe.nh d¯ˆe` 2.1.10, do d¯´o ta nhˆa. n d¯u.o. . c mˆe.nh d¯ˆe` quan tro. ng sau: Mˆe. nh d¯ˆe` 2.1.11. X´et h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Khi d¯´o ˜f = f + p l`a γ-lˆo`i ngo`ai v´o.i γ ≥ γ∗ v`a γ-lˆo`i ngo`ai ngˇa. t v´o.i γ > γ∗.
V´ı du. 2.1.4 du.´o.i d¯ˆay chı˙’ ra rˇa`ng γ∗ l`a gi´a tri. nho˙’ nhˆa´t d¯ˆe˙’ mo. i h`am
to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u l`a γ-lˆo`i ngo`ai.
√ √ 2. X´et c´ac h`am 2, cho. n γ1 sao cho γ < γ1 < 2 V´ı du. 2.1.4. Lˆa´y γ < 2
f (x) = x2, (cid:40) 1 p(x) = nˆe´u x (cid:54)= γ1i, i = 0, ±1, ±2, . . . −1 nˆe´u x = γ1i, i = 0, ±1, ±2, . . .
√
2. Theo Mˆe.nh d¯ˆe` 2.1.11 . c ˜f (x0) = −1 khi x0 = 0 v`a
2 − 1 khi x1 = γ1. Mˇa. t kh´ac, v´o.i mo. i λ ∈ ] 0, 1[ th`ı
2
2 − 1 − λγ1
2 − 1) = γ1
V`ı supx∈IR |p(x)| = 1, λmin = 1 nˆen γ∗ = 2 th`ı ˜f = f + p l`a γ∗-lˆo`i ngo`ai. Ta c˜ung t´ınh d¯u.o. ˜f (x1) = γ1
λ ˜f (x0) + (1 − λ) ˜f (x1) = −λ + (1 − λ)(γ1
v`a
2 + 1
2 − 2γ1
˜f (xλ) = ˜f (λx0 + (1 − λ)x1)
= ˜f ((1 − λ)γ1) = (1 − λ)2γ1 2λ2 + 1. 2λ + γ1 = γ1
D- ˆe˙’ ch´u.ng minh ˜f = f + p khˆong l`a h`am γ-lˆo`i ngo`ai, ta cˆa` n chı˙’ ra
˜f (xλ) > λ ˜f (x0) + (1 − λ) ˜f (x1) v´o.i mo. i λ ∈ ] 0, 1[,
2
2 − 2γ1
2λ + γ1
2λ2 + 1 > γ1
2 − 1 − λγ1
t´u.c l`a
2λ + 2 > 0.
2 − γ1
γ1 tu.o.ng d¯u.o.ng v´o.i
2(γ1
2 = γ1
4 − 8γ1
2λ1 Bˆa´t d¯ˇa˙’ ng th´u.c cuˆo´i c`ung l`a hiˆe˙’n nhiˆen, v`ı v´o.i γ1 < 2 ∆ = γ1
2 − 8) nhˆa. n gi´a tri. ˆam.
γ1 √ 2, biˆe.t th´u.c
V´ı du. sau cho thˆa´y h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i c´o thˆe˙’
khˆong γ-lˆo`i ngo`ai ngˇa. t khi γ = γ∗.
V´ı du. 2.1.5. Cho c´ac h`am
√ f (x) = x2, (cid:40) 2i, i = 1, 2, . . . 1 p(x) = √ nˆe´u x (cid:54)= ± −1 nˆe´u x = ± 2i, i = 1, 2, . . .
√ V`ı supx∈IR |p(x)| = 1, λmin = 1 nˆen γ∗ = 2
√ √ 2, x1 =
2. Theo Mˆe.nh d¯ˆe` 2.1.11 th`ı ˜f = f + p l`a γ-lˆo`i ngo`ai v´o.i γ = γ∗. Ta dˆe˜ d`ang t´ınh d¯u.o. . c ˜f (x0) = 1, ˜f (x1) = 1 v´o.i x0 = − 2. Mˇa. t kh´ac, nˆe´u ˜f = f + p l`a γ-lˆo`i ngo`ai ngˇa. t v´o.i γ = γ∗ th`ı tˆo`n ta. i ´ıt nhˆa´t mˆo. t gi´a tri. λ ∈ ] 0, 1[ sao cho
˜f (xλ) < λ ˜f (x0) + (1 − λ) ˜f (x1).
Tuy nhiˆen
λ + 1 ≥ 1 = λ ˜f (x0) + (1 − λ) ˜f (x1)
˜f (xλ) = x2
nˆen ˜f = f + p khˆong l`a h`am γ-lˆo`i ngo`ai ngˇa. t v´o.i γ = γ∗.
Mˆo. t trong nh˜u.ng t´ınh chˆa´t cu˙’a h`am lˆo`i l`a tˆa. p m´u.c du.´o.i cu˙’a h`am lˆo`i l`a lˆo`i. H. X. Phu d¯˜a d¯u.a ra kh´ai niˆe.m tˆa. p γ-lˆo`i ngo`ai [47] (D- i.nh ngh˜ıa 1.3.4, . : Tˆa. p m´u.c du.´o.i cu˙’a h`am γ-lˆo`i Chu.o.ng 1) v`a chı˙’ ra t´ınh chˆa´t tu.o.ng tu. ngo`ai l`a tˆa. p γ-lˆo`i ngo`ai. C´ac t´ınh chˆa´t cu˙’a tˆa. p γ-lˆo`i ngo`ai d¯u.o. . c nghiˆen c´u.u v`a tr`ınh b`ay k˜y trong [1] v`a [47]. D- ˆo´i v´o.i l´o.p h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i, ta c´o thˆem t´ınh chˆa´t sau d¯ˆay cu˙’a tˆa. p m´u.c du.´o.i
Mˆe. nh d¯ˆe` 2.1.12. Cho γ > 0, k´y hiˆe. u
Lα( ˜f ) := {x | x ∈ D, ˜f (x) ≤ α}
l`a tˆa. p m´u.c du.´o.i cu˙’a h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Khi d¯´o, nˆe´u h`am nhiˆe˜u p tho˙’a m˜an d¯iˆe` u kiˆe. n
|p(x)| ≤ λminγ2/8 v´o.i mo. i x ∈ D,
th`ı tˆa. p Lα( ˜f ) l`a γ-lˆo`i ngo`ai.
Ch´u.ng minh. Gia˙’ thiˆe´t trˆen tho˙’a m˜an Mˆe.nh d¯ˆe` 2.1.9, nˆen suy ra ˜f = f + p l`a γ-lˆo`i ngo`ai. Theo D- i.nh l´y 1.3.6 (hoˇa. c Mˆe.nh d¯ˆe` 2.2 [47]) th`ı tˆa. p m´u.c du.´o.i cu˙’a h`am γ-lˆo`i ngo`ai l`a γ-lˆo`i ngo`ai, do d¯´o Lα( ˜f ) l`a tˆa. p γ-lˆo`i ngo`ai. Mˆe. nh d¯ˆe` 2.1.13. Tˆa. p m´u.c du.´o.i Lα( ˜f ) cu˙’a h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p l`a tˆa. p γ-lˆo`i ngo`ai v´o.i γ ≥ γ∗.
Ch´u.ng minh. Theo Mˆe.nh d¯ˆe` 2.1.11 th`ı ˜f = f + p l`a h`am γ-lˆo`i ngo`ai v´o.i γ ≥ γ∗ nˆen theo Mˆe.nh d¯ˆe` 2.1.12 ta suy ra Lα( ˜f ) l`a tˆa. p γ-lˆo`i ngo`ai v´o.i γ ≥ γ∗.
Mˆe.nh d¯ˆe` du.´o.i d¯ˆay (d¯u.o.
. c suy ra t`u. c´ac d¯i.nh l´y 1.3.7 Chu.o.ng 1 v`a Mˆe.nh . c tiˆe˙’u to`an cu. c v`a d¯iˆe˙’m infimum . c tiˆe˙’u v`a γ-infimum, cu˙’a
d¯ˆe` 2.1.11), cho ph´ep ta x´ac d¯i.nh d¯iˆe˙’m cu. to`an cu. c thˆong qua viˆe.c t`ım kiˆe´m c´ac d¯iˆe˙’m γ-cu. h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i. Mˆe. nh d¯ˆe` 2.2.14. X´et h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Khi d¯´o (a) Nˆe´u x∗ ∈ D l`a d¯iˆe˙’m γ- cu. . c tiˆe˙’u cu˙’a ˜f = f + p v´o.i γ ≥ γ∗, th`ı x∗ ∈ D
l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a ˜f = f + p.
(b) Nˆe´u x∗ l`a d¯iˆe˙’m γ-infimum cu˙’a ˜f = f + p v´o.i γ ≥ γ∗ th`ı x∗ l`a d¯iˆe˙’m
infimum to`an cu. c cu˙’a ˜f = f + p. D- ˆo´i v´o.i h`am γ-lˆo`i ngo`ai, n´oi chung tˆa. p c´ac d¯iˆe˙’m cu.
. c tiˆe˙’u to`an cu. c
. c tiˆe˙’u to`an cu. c c´o thˆe˙’ khˆong gi´o.i nˆo. i, d¯iˆe` u n`ay c´o thˆe˙’ thˆa´y r˜o qua h`am g(x) := x−[x], x ∈ IR, l`a h`am γ-lˆo`i ngo`ai v´o.i γ = 1 v`a tˆa. p c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c l`a {i | i ∈ IN }. Tuy nhiˆen, d¯ˆo´i v´o.i h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i ˜f = f + p ta c´o mˆe.nh d¯ˆe` sau: Mˆe. nh d¯ˆe` 2.2.15. K´y hiˆe. u arg min ˜f l`a tˆa. p c´ac d¯iˆe˙’m cu. cu˙’a B`ai to´an ( ˜P ). Khi d¯´o
(cid:107)˜x1 − ˜x2(cid:107) ≤ γ∗ v´o.i mo. i ˜x1, ˜x2 ∈ arg min ˜f ,
t´u.c l`a
diam(arg min ˜f ) ≤ γ∗ .
Ch´u.ng minh. Theo Mˆe.nh d¯ˆe` 2.1.11 th`ı ˜f = f + p l`a γ-lˆo`i ngo`ai ngˇa. t khi γ > γ∗ = 2(cid:112)2s/λmin ,
nˆen ˜f = f + p tho˙’a m˜an Mˆe.nh d¯ˆe` 1.3.2, v`ı vˆa. y
diam(arg min ˜f ) ≤ γ v´o.i mo. i γ > γ∗.
Do d¯´o suy ra
diam(arg min ˜f ) ≤ γ∗.
.˙’ mu. c tru.´o.c, trong Mˆe.nh d¯ˆe` 2.2.15 ch´ung tˆoi d¯˜a nghiˆen c´u.u d¯u.`o.ng O k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an quy hoa. ch to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i ( ˜P ). Trong mu. c n`ay, ch´ung tˆoi nghiˆen c´u.u d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c v`a t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) theo cˆa. n trˆen s cu˙’a h`am nhiˆe˜u p.
˜f (y) v`a c´o bˆo˙’ d¯ˆe` sau:
Nghiˆen c´u.u c´ac d¯iˆe˙’m infimum to`an cu. c, trong mu. c n`ay ta su.˙’ du. ng h`am bao d¯´ong nu.˙’a liˆen tu. c du.´o.i (xem [54], trang 68-90) lsc ˜f (x) := lim infy→x, y∈D Bˆo˙’ d¯ˆe` 2.3.1. X´et h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Khi d¯´o
(a) V´o.i mo. i x ∈ D th`ı lsc ˜f (x) = f (x) + lsc p(x),
(b) supx∈D |lsc p(x)| ≤ supx∈D |p(x)| ≤ s.
Ch´u.ng minh. (a) Ta c´o
y→x, y∈D
lsc ˜f (x) = lim inf ˜f (y)
˜f (yn)}. = inf{η : ∃yn → x, yn ∈ D, η = lim n→∞
V`ı f (x) liˆen tu. c nˆen
y→x, y∈D
lsc ˜f (x) = lim inf ˜f (y)
p(yn)}. = f (x) + inf{η(cid:48) : ∃yn → x, yn ∈ D, η(cid:48) = lim n→∞
Do d¯´o, d¯ˆo´i v´o.i h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i th`ı
lsc ˜f (x) = f (x) + lsc p(x).
(b) Ta c´o
p(x)| |lsc p(x)| = | lim inf y→x, x∈D
|p(x)|
|p(x)|, ≤ lim inf y→x, x∈D ≤ lim inf y→x, x∈D sup x∈D
nˆen
|p(x)| ≤ s < +∞. sup x∈D |lsc p(x)| ≤ sup x∈D
Bˆo˙’ d¯ˆe` d¯u.o. . c ch´u.ng minh.
V`ı tˆa. p c´ac d¯iˆe˙’m cu.
1, ˜x∗
2 l`a hai d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a
to`an cu. c, nˆen mˆe.nh d¯ˆe` sau l`a tru.`o.ng ho. . c tiˆe˙’u to`an cu. c l`a tˆa. p con cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum . p tˆo˙’ng qu´at cu˙’a Mˆe.nh d¯ˆe` 2.2.15.
Mˆe. nh d¯ˆe` 2.3.16. Nˆe´u ˜x∗ B`ai to´an ( ˜P ) th`ı
1 − ˜x∗
2(cid:107) ≤ γ∗.
1, ˜x∗
2 l`a hai d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a B`ai to´an 2 l`a c´ac
1, ˜x∗
(cid:107)˜x∗
Ch´u.ng minh. V`ı ˜x∗ ( ˜P ), nˆen theo Mˆe.nh d¯ˆe` 1.3.1 v`a (a) cu˙’a Bˆo˙’ d¯ˆe` 2.3.1 ta suy ra ˜x∗ d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a lsc ˜f = f + lsc p.
(cid:114) Mˇa. t kh´ac, t`u. (b) cu˙’a Bˆo˙’ d¯ˆe` ta c´o (cid:114) 2 |lsc p(x)|/λmin ≤ 2 |p(x)|/λmin ≤ 2(cid:112)2s/λmin = γ∗, 2 sup x∈D 2 sup x∈D
nˆen theo Mˆe.nh d¯ˆe` 2.1.11 ta suy ra lsc ˜f = f + lsc p l`a γ-lˆo`i ngo`ai ngˇa. t khi γ > γ∗. ´Ap du. ng Mˆe.nh d¯ˆe` 2.2.15 cho h`am lsc ˜f = f + lsc p ta nhˆa. n d¯u.o. . c
1 − ˜x∗
2(cid:107) ≤ γ∗.
(cid:107)˜x∗
Mˆe.nh d¯ˆe` d¯˜a d¯u.o. . c ch´u.ng minh.
V´ı du. 2.3.6. X´et c´ac h`am
f (x) = x2, (cid:40)
p(x) = −0.5 nˆe´u |x| ≥ 1 nˆe´u |x| < 1. 0.5
Khi d¯´o
(cid:40)
˜f (x) = f (x) + p(x) = x2 − 0.5 nˆe´u |x| ≥ 1 x2 + 0.5 nˆe´u |x| < 1,
√
c´o ba d¯iˆe˙’m infimum to`an cu. c l`a x1 = −1, x2 = 0, x3 = 1, s = supx∈IR |p(x)| = 0.5, λmin = 1 v`a γ∗ = 2(cid:112)2s/λmin = 2 2 × 0.5 = 2. Theo mˆe.nh d¯ˆe` trˆen th`ı
2 = max{(cid:107)xi − xj(cid:107) | i, j = 1, 2, 3} ≤ 2 = 2(cid:112)2s/λmin ,
suy ra
max{(cid:107)xi − xj(cid:107) | i, j = 1, 2, 3} = γ∗. Biˆe˙’u th´u.c cuˆo´i cho ph´ep kˆe´t luˆa. n d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u n´oi chung khˆong nho˙’ ho.n γ∗.
Khi x´et l´o.p h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i, cˆau ho˙’i d¯u.o. . c d¯ˇa. t ra l`a: C´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a h`am n`ay thay d¯ˆo˙’i nhu. thˆe´ n`ao . c tiˆe˙’u to`an cu. c duy nhˆa´t x∗ cu˙’a h`am to`an phu.o.ng (nˆe´u tˆo`n so v´o.i d¯iˆe˙’m cu. ta. i)? c´o thˆe˙’ d¯´anh gi´a d¯u.o. . c khoa˙’ng c´ach gi˜u.a ch´ung hay khˆong? D- i.nh l´y sau s˜e tra˙’ l`o.i cho ta cˆau ho˙’i n`ay.
. c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an
D- i.nh l´y 2.3.13. Nˆe´u x∗ ∈ D l`a d¯iˆe˙’m cu. (P ), ˜x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a B`ai to´an ( ˜P ), th`ı
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2.
Ch´u.ng minh. Ta x´et c´ac tru.`o.ng ho. . p sau:
i) ˜x∗ l`a cu. . c tiˆe˙’u to`an cu. c cu˙’a ˜f = f + p. D- ˇa. t
ϕ(t) : = f (x∗ + t(˜x∗ − x∗)) − f (x∗)
= (cid:104)Ax∗, x∗(cid:105) + (cid:104)b, x∗(cid:105) + (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105)t
+(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105)t2 − (cid:104)Ax∗, x∗(cid:105) + (cid:104)b, x∗(cid:105)
= (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105)t + (cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105)t2.
Do d¯´o
ϕ(cid:48)(t) = (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105) + 2(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105)t (2.3.8)
v`a
ϕ(cid:48)(cid:48)(t) = 2(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105). (2.3.9) . c tiˆe˙’u to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t f trˆen
≥ 0. ϕ(cid:48)(0) = lim t(cid:38)0 = lim t(cid:38)0 Mˇa. t kh´ac, v`ı x∗ l`a cu. D, nˆen ϕ(t) ≥ 0 v´o.i mo. i t ∈ [0, 1]. Ta c´o ϕ(t) − ϕ(0) t ϕ(t) t
Kˆe´t ho. . p biˆe˙’u th´u.c (2.3.8) khi t = 0 v´o.i bˆa´t d¯ˇa˙’ ng th´u.c trˆen ta suy ra
(cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105) ≥ 0. (2.3.10)
Theo cˆong th´u.c Tay lo th`ı
ϕ(t) = ϕ(0) + ϕ(cid:48)(0)t + t2 ϕ(cid:48)(cid:48)(0) 2
2 ta suy ra
nˆen, v´o.i t = 1
+ ϕ(cid:48)(cid:48)(0) ) = ϕ(cid:48)(0) . ϕ( 1 2 1 8
f ( (cid:104)A(˜x∗ −x∗), ˜x∗ −x∗(cid:105). (2.3.11) (cid:104)2Ax∗ +b, ˜x∗ −x∗(cid:105)+ ) = f (x∗)+ 1 2 Thay c´ac gi´a tri. cu˙’a ϕ(cid:48)(0), ϕ(cid:48)(cid:48)(0) theo c´ac cˆong th´u.c (2.3.8) v`a (2.3.9) ta nhˆa. n d¯u.o. . c ˜x∗ + x∗ 2 1 4 1 2
(cid:68) = Theo biˆe˙’u th´u.c (2.1.2) th`ı A(˜x∗ − x∗), ˜x∗ − x∗(cid:69) f (˜x∗) + f (x∗) − f ( (2.3.12) ), 1 4 1 2 1 2 ˜x∗ + x∗ 2
2
nˆen thay f ( ˜x∗+x∗ ) o.˙’ (2.3.11) v`ao (2.3.12), chuyˆe˙’n vˆe´ v`a r´ut go. n ta d¯u.o. . c
(cid:68) A(˜x∗ − x∗), ˜x∗ − x∗(cid:69) = f (˜x∗) − f (x∗) − (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105).
Kˆe´t ho. . p v´o.i (2.3.10) suy ra
(cid:10)A(˜x∗ − x∗), ˜x∗ − x∗(cid:11) ≤ f (˜x∗) − f (x∗). (2.3.13)
D- ˇa. t η := (cid:107)˜x∗ − x∗(cid:107), t`u. (2.1.3) v`a (2.3.13) suy ra
(2.3.14) λminη2 ≤ f (˜x∗) − f (x∗).
Mˇa. t kh´ac, v`ı ˜x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a ˜f = f + p nˆen
f (˜x∗) + p(˜x∗) ≤ f (x∗) + p(x∗).
Kˆe´t ho. . p v´o.i supx∈D |p(x)| ≤ s, ta suy ra
0 ≤ f (˜x∗) − f (x∗) ≤ 2s. (2.3.15)
Thay (2.3.15) v`ao (2.3.14) ta nhˆa. n d¯u.o. . c
λminη2 ≤ 2s
tu.o.ng d¯u.o.ng v´o.i
η ≤ (cid:112)2s/λmin .
ii) ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c (khˆong l`a d¯iˆe˙’m cu.
. c tiˆe˙’u to`an cu. c) cu˙’a . c tiˆe˙’u to`an cu. c cu˙’a lsc ˜f = f + lsc p. Su.˙’
˜f = f + p. Khi d¯´o ˜x∗ l`a d¯iˆe˙’m cu. du. ng i) cho h`am lsc ˜f ta c˜ung nhˆa. n d¯u.o. . c
f (˜x∗) − f (x∗) ≤ lsc p(˜x∗) − lsc p(x∗).
D- ˇa. t η := (cid:107)˜x∗ − x∗(cid:107). Theo Bˆo˙’ d¯ˆe` 2.3.1 th`ı
lsc p(˜x∗) − lsc p(x∗) ≤ 2s,
nˆen
f (˜x∗) − f (x∗) ≤ 2s. (2.3.16)
Kˆe´t ho. . p (2.3.16) v´o.i (2.3.13), ta suy ra
λminη2 ≤ 2s,
v`ı vˆa. y
η ≤ (cid:112)2s/λmin,
t´u.c l`a
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2.
. c ch´u.ng minh. D- i.nh l´y d¯˜a d¯u.o.
D- i.nh l´y trˆen d¯˜a d¯u.o. . c H. X. Phu ch´u.ng minh rˆa´t go. n trong [51].
Trong mu. c n`ay, ta nghiˆen c´u.u t´ınh chˆa´t tu.
. a cu˙’a h`am ˜f = f + p v`a su. . tˆo`n ta. i c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), cu. thˆe˙’ l`a D- i.nh l´y Kuhn-Tucker suy rˆo. ng cho B`ai to´an ( ˜P ) khi D l`a tˆa. p lˆo`i tho˙’a m˜an mˆo. t trong hai tru.`o.ng ho. . p sau:
(2.4.17) D = {x ∈ S | gi(x) ≤ 0, i = 1, . . . , m},
trong d¯´o gi : IRn → IR, i = 1, . . . , m, l`a c´ac h`am lˆo`i v`a S ⊂ IRn l`a tˆa. p lˆo`i d¯´ong, hoˇa. c
(2.4.18) D = {x ∈ IRn | (cid:104)ci, x(cid:105) ≤ di, i = 1, . . . , m}.
Mˆo. t t´ınh chˆa´t d¯ˇa. c biˆe.t cu˙’a h`am lˆo`i bˆa´t k`y g : IRn → IR l`a v´o.i x∗ ∈ IRn
n`ao d¯´o, tˆo`n ta. i ξ ∈ IRn go. i l`a du.´o.i vi phˆan sao cho
g(x) ≥ g(x∗) + (cid:104)ξ, x − x∗(cid:105), v´o.i mo. i x ∈ IRn,
t´u.c l`a ta. i mo. i d¯iˆe˙’m x∗ ∈ IRn h`am lˆo`i g tu. . a trˆen mˆo. t h`am tuyˆe´n t´ınh . a cu˙’a g. Trong tru.`o.ng ho. g(x∗) + (cid:104)ξ, x − x∗(cid:105) v`a ta go. i l`a t´ınh chˆa´t tu. . p g = f th`ı ξ = 2Ax∗ + b. V`ı h`am nhiˆe˜u p chı˙’ gia˙’ thiˆe´t gi´o.i nˆo. i nˆen khˆong . a nhu. trˆen. Tuy nhiˆen, ta hy vo. ng h`am bi. nhiˆe˜u ˜f = f + p c´o t´ınh chˆa´t tu.
. a thˆo. Muˆo´n vˆa. y, ta viˆe´t la. i
s˜e chı˙’ ra h`am lˆo`i thˆo ˜f s˜e c˜ung c´o t´ınh chˆa´t tu. t´ınh chˆa´t tu. . a nhu. sau:
(2.4.19) g(x∗) + (cid:104)ξ, x∗(cid:105) ≤ g(x) + (cid:104)ξ, x(cid:105), v´o.i mo. i x ∈ IRn.
Thay thˆe´ vˆe´ tr´ai cu˙’a (2.4.19) bo.˙’ i
x(cid:48)∈ ¯B(x∗,r)
(cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) (cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) hoˇa. c min inf x(cid:48)∈B(x∗,r)
. p l´y n`ao d¯´o v`a vˆe´ pha˙’i cu˙’a (2.4.19) bo.˙’ i ˜f (x) − (cid:104)ξ, x(cid:105) ta
v´o.i mˆo. t r > 0 ho. d¯u.o. . c
(cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) ≤ ˜f (x) − (cid:104)ξ, x(cid:105) v´o.i mo. i x ∈ IRn, inf x(cid:48)∈B(x∗,r)
hoˇa. c
(cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) ≤ ˜f (x) − (cid:104)ξ, x(cid:105) v´o.i mo. i x ∈ IRn. min x(cid:48)∈ ¯B(x∗,r)
Nh˜u.ng cˆong th´u.c trˆen mˆo ta˙’ t´ınh chˆa´t tu. . a thˆo cu˙’a h`am ˜f . T´ınh chˆa´t n`ay d¯˜a d¯u.o. . c H. X. Phu chı˙’ ra khi nghiˆen c´u.u c´ac h`am γ-lˆo`i ngo`ai tˆo˙’ng qu´at [44]. Bˇa`ng c´ach su.˙’ du. ng D- i.nh l´y 2.3.13 cho h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i ˜f = f + p, mˆe.nh d¯ˆe` du.´o.i d¯ˆay cho ta kˆe´t qua˙’ tˆo´t ho.n, t´u.c l`a chı˙’ ra r = γ∗/2 v`a ξ = 2Ax∗ + b.
Mˆe. nh d¯ˆe` 2.4.17. ([51]) Cho D = IRn. Khi d¯´o v´o.i x∗ ∈ IRn v`a (cid:15) > 0 th`ı
(cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) ≤ ˜f (x) − (cid:104)ξ, x(cid:105) v´o.i mo. i x ∈ IRn, inf x(cid:48)∈B(x∗,γ∗/2+(cid:15))
D- ˇa. c biˆe. t, nˆe´u p l`a nu.˙’a liˆen tu. c du.´o.i th`ı
(cid:0) ˜f (x(cid:48)) − (cid:104)ξ, x(cid:48)(cid:105)(cid:1) ≤ ˜f (x) − (cid:104)ξ, x(cid:105) v´o.i mo. i x ∈ IRn. min x(cid:48)∈ ¯B(x∗,γ∗)
Mˆe.nh d¯ˆe` trˆen d¯u.o. . c H. X. Phu chı˙’ ra. Ch´u.ng minh chi tiˆe´t c´o thˆe˙’ xem
trong [51]
Trong qu´a tr`ınh kha˙’o s´at d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ),
ch´ung tˆoi su.˙’ du. ng bˆo˙’ d¯ˆe` sau:
Bˆo˙’ d¯ˆe` 2.4.2. Cho h`am g : D ⊂ IRn → IR l`a nu.˙’a liˆen tu. c du.´o.i, bi. chˇa. n du.´o.i, D l`a tˆa. p d¯´ong v`a lim(cid:107)x(cid:107)→+∞, x∈D g(x) = +∞. Khi d¯´o tˆo`n ta. i x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a g trˆen D.
Ch´u.ng minh. Bˆo˙’ d¯ˆe` n`ay c´o thˆe˙’ ch´u.ng minh nhu. l`a hˆe. qua˙’ cu˙’a D- i.nh l´y . c tiˆe´p nhu. 8.2 (xem [76], trang 119-121). Tuy nhiˆen c´o thˆe˙’ ch´u.ng minh tru. sau:
Cˆo´ d¯i.nh d¯iˆe˙’m x0 ∈ D. V`ı lim(cid:107)x(cid:107)→∞, x∈D g(x) = +∞ nˆen
∃r ∈ [(cid:107)x0(cid:107), +∞[ : x ∈ D, (cid:107)x(cid:107) > r ⇒ g(x) > g(x0).
. c tiˆe˙’u to`an cu. c nˆe´u c´o th`ı s˜e chı˙’ nˇa`m trong miˆe` n B(0, r)∩D.
Do d¯´o, d¯iˆe˙’m cu. V`ı g bi. chˇa. n du.´o.i nˆen
g(x) > −∞. inf x∈B(0,r)∩D
Mˇa. t kh´ac, tˆo`n ta. i d˜ay (xi) ⊂ B(0, r) ∩ D sao cho
g(x). g(xi) = lim i→∞ inf x∈B(0,r)∩D
Tˆa. p B(0, r) ∩ D l`a d¯´ong, gi´o.i nˆo. i trong IRn nˆen l`a tˆa. p compact, v`ı vˆa. y t`u. d˜ay (xi) ⊂ B(0, r) ∩ D c´o thˆe˙’ tr´ıch d˜ay con hˆo. i tu. . Khˆong gia˙’m tˆo˙’ng qu´at, ta coi ch´ınh d˜ay d¯´o hˆo. i tu. , t´u.c l`a limi→∞ xi = x∗ v`a x∗ ∈ B(0, r) ∩ D. V`ı g l`a nu.˙’a liˆen tu. c du.´o.i nˆen
g(x). g(xi) = g(x∗) ≤ lim i→∞ inf x∈B(0,r)∩D
. c tiˆe˙’u trˆen B(0, r) ∩ D. V`ı d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c, nˆen x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u cu˙’a g trˆen . c tiˆe˙’u to`an cu. c
Do d¯´o x∗ l`a d¯iˆe˙’m cu. B(0, r) ∩ D l`a d¯iˆe˙’m cu. cu˙’a g trˆen D.
Tru.´o.c khi ph´at biˆe˙’u v`a ch´u.ng minh D- i.nh l´y Kuhn-Tucker suy rˆo. ng . c d¯i.nh
m (cid:88)
cho B`ai to´an ( ˜P ), ta nhˇa´c la. i h`am Lagrange cho B`ai to´an ( ˜P ) d¯u.o. ngh˜ıa theo cˆong th´u.c (1.1.4), t´u.c l`a
i=1
L(x, µ0, . . . , µm) = µ0f (x) + µigi(x).
. c cho bo.˙’ i cˆong th´u.c (2.4.17). D- i.nh l´y 2.4.14. Gia˙’ su.˙’ D d¯u.o.
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), th`ı tˆo`n ta. i duy
nhˆa´t d¯iˆe˙’m x∗ ∈ D sao cho
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, khˆong c`ung triˆe. t tiˆeu, tho˙’a m˜an d¯iˆe` u kiˆe. n Kuhn-Tucker
(2.4.20) L(x, µ0, . . . , µm) L(x∗, µ0, . . . , µm) = min x∈S
v`a d¯iˆe` u kiˆe. n b`u
i = 1, . . . , m. (2.4.21) µigi(x∗) = 0 v´o.i mo. i
Nˆe´u d¯iˆe` u kiˆe. n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1. (b) Nˆe´u tˆo`n ta. i x∗ ∈ D tho˙’a m˜an (2.4.20), (2.4.21) v´o.i µ0 = 1 th`ı tˆo`n ta. i ˜x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) trˆen D, tho˙’a m˜an
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
v`a khˆong c´o d¯iˆe˙’m infimum to`an cu. c n`ao cu˙’a B`ai to´an ( ˜P ) nˇa`m ngo`ai h`ınh cˆa` u B(x∗, γ∗/2).
Ch´u.ng minh. (a) X´et tˆa. p D, v`ı S l`a lˆo`i d¯´ong, gi(x), i = 1, . . . , m, l`a c´ac h`am lˆo`i nˆen D = {x ∈ S | gi(x) ≤ 0, i = 1, . . . m} c˜ung l`a lˆo`i d¯´ong. Khi d¯´o i) Nˆe´u D gi´o.i nˆo. i, th`ı v`ı f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) l`a lˆo`i ngˇa. t, liˆen tu. c . c tiˆe˙’u to`an cu. c x∗ ∈ D. nˆen tˆo`n ta. i duy nhˆa´t d¯iˆe˙’m cu.
ii) Nˆe´u D khˆong gi´o.i nˆo. i, th`ı
f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105)
≥ λmin(cid:107)x(cid:107)2 − (cid:107)b(cid:107)(cid:107)x(cid:107),
nˆen
f (x) = +∞. lim (cid:107)x(cid:107)→+∞, x∈D
H`am to`an phu.o.ng lˆo`i ngˇa. t f trˆen tˆa. p lˆo`i D tho˙’a m˜an c´ac d¯iˆe` u kiˆe.n cu˙’a Bˆo˙’ d¯ˆe` 2.4.2, do d¯´o tˆo`n ta. i d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c v`a duy nhˆa´t x∗ trˆen D. V`ı ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f = f + p, nˆen theo D- i.nh l´y 2.3.13, ta nhˆa. n d¯u.o. . c
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2.
Mˇa. t kh´ac, x∗ l`a cu.
. c tiˆe˙’u to`an cu. c cu˙’a f trˆen D, t´u.c l`a x∗ l`a nghiˆe.m cu˙’a B`ai to´an (P ), do d¯´o theo (a) cu˙’a D- i.nh l´y Kuhn-Tucker 1.1.1 suy ra tˆo`n ta. i µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an
L(x, µ0, . . . , µm) L(x∗, µ0, . . . , µm) = min x∈S
v`a
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m.
Nˆe´u d¯iˆe` u kiˆe.n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0, nˆen c´o thˆe˙’ cho. n
µ0 = 1.
(b) V`ı x∗ ∈ D tho˙’a m˜an d¯iˆe` u kiˆe.n (2.4.20), (2.4.21) v´o.i µ0 = 1 nˆen x∗ tho˙’a m˜an (b) cu˙’a D- i.nh l´y Kuhn-Tucker 1.1.1 cho B`ai to´an (P ). Do d¯´o x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a f trˆen D v`a v`ı f l`a lˆo`i ngˇa. t nˆen x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a f trˆen D.
Ta ch´u.ng minh tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). Thˆa. t
vˆa. y, x´et h`am lsc ˜f = f + lsc p trˆen D, xa˙’y ra c´ac tru.`o.ng ho. . p sau:
i) D gi´o.i nˆo. i, khi d¯´o tˆo`n ta. i M > 0 sao cho (cid:107)x(cid:107) ≤ M v´o.i mo. i x ∈ D.
V`ı vˆa. y
|p(x)| lsc ˜f (x) ≥ (cid:104)Ax, x(cid:105) − (cid:107)b(cid:107)(cid:107)x(cid:107) − sup x∈D |p(x)|, ≥ −(cid:107)b(cid:107)M − sup x∈D
suy ra h`am bi. chˇa. n du.´o.i trˆen D. V`ı h`am lsc ˜f = f + lsc p nu.˙’a liˆen tu. c du.´o.i, bi. chˇa. n du.´o.i trˆen tˆa. p compact D (d¯´ong, gi´o.i nˆo. i trong IRn) nˆen tˆo`n ta. i d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c ˜x∗ cu˙’a lsc ˜f = f + lsc p trˆen D.
|p(x)|. ii) D khˆong gi´o.i nˆo. i. Ta c´o lsc ˜f (x) = f (x) + lsc p(x) ≥ (cid:104)Ax, x(cid:105) − (cid:107)b(cid:107)(cid:107)x(cid:107) − |p(x)| ≥ λmin(cid:107)x(cid:107)2 − (cid:107)b(cid:107)(cid:107)x(cid:107) − sup x∈D
Do d¯´o
lsc ˜f (x) = +∞. lim (cid:107)x(cid:107)→+∞, x∈D
Mˇa. t kh´ac, v`ı h`am lsc ˜f = f + lsc p bi. chˇa. n du.´o.i v`a nu.˙’ a liˆen tu. c du.´o.i nˆen lsc ˜f = f + lsc p tho˙’a m˜an Bˆo˙’ d¯ˆe` 2.4.2, v`ı vˆa. y tˆo`n ta. i d¯iˆe˙’m ˜x∗ l`a cu. . c tiˆe˙’u to`an cu. c cu˙’a lsc ˜f = f + lsc p trˆen D.
Kˆe´t ho. . p ca˙’ hai tru.`o.ng ho. . p i), ii) suy ra tˆo`n ta. i ˜x∗ l`a d¯iˆe˙’m infimum
to`an cu. c cu˙’a B`ai to´an ( ˜P ). Ngo`ai ra ´ap du. ng D- i.nh l´y 2.3.13 ta c´o
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
v`a c˜ung theo D- i.nh l´y 2.3.13 th`ı khˆong thˆe˙’ tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c kh´ac cu˙’a B`ai to´an ( ˜P ) nˇa`m ngo`ai h`ınh cˆa` u B(x∗, γ∗/2).
. c cho bo.˙’ i (2.4.17) v`a gi : IRn → IR, i =
D- i.nh l´y sau l`a mˆo. t mo.˙’ rˆo. ng D- i.nh l´y 1.1.2, n´o chı˙’ ra d¯iˆe` u kiˆe.n cˆa` n v`a d¯u˙’ d¯ˆe˙’ tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), d¯i.nh l´y ph´at biˆe˙’u nhu. sau: D- i.nh l´y 2.4.15. Gia˙’ su.˙’ D d¯u.o. 1, . . . , m, l`a c´ac h`am lˆo`i, c`ung liˆen tu. c ´ıt nhˆa´t ta. i mˆo. t d¯iˆe˙’m cu˙’a tˆa. p S.
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), th`ı tˆo`n ta. i x∗ v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, khˆong c`ung triˆe. t tiˆeu, sao cho
m (cid:88)
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2,
i=1
(2.4.22) 0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗) + N (x∗|S)
v`a
i = 1, . . . , m, (2.4.23) µigi(x∗) = 0 v´o.i mo. i
trong d¯´o N (x∗|S) l`a n´on ph´ap tuyˆe´n cu˙’a S ta. i x∗. Nˆe´u d¯iˆe` u kiˆe. n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) Nˆe´u tˆo`n ta. i x∗ ∈ D tho˙’a m˜an (2.4.22), (2.4.23) v´o.i µ0 = 1 th`ı tˆo`n ta. i ˜x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) tho˙’a m˜an
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
v`a khˆong c´o d¯iˆe˙’m infimum to`an cu. c kh´ac cu˙’a ˜f = f + p trˆen D nˇa`m ngo`ai h`ınh cˆa` u B(x∗, γ∗/2).
Ch´u.ng minh. (a) V`ı f = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) nˆen ∂f (x∗) = 2Ax∗ + b. Su.˙’ du. ng Bˆo˙’ d¯ˆe` 2.4.2, ch´u.ng minh tu.o.ng tu. . nhu. o.˙’ D- i.nh l´y 2.4.14, ta suy ra tˆo`n ta. i duy nhˆa´t x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ).
m (cid:88)
. c su. ´Ap du. ng D- i.nh l´y Kuhn-Tuker 1.1.2 cho B`ai to´an (P ) ta nhˆa. n d¯u.o. . tˆo`n ta. i c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an c´ac d¯iˆe` u kiˆe.n sau:
i=1
0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗) + N (x∗|S)
v`a
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m.
Nˆe´u thˆem d¯iˆe` u kiˆe.n Slater (1.1.7) th`ı µ0 (cid:54)= 0, nˆen c´o thˆe˙’ coi µ0 = 1.
. c tiˆe˙’u to`an cu. c cu˙’a ˜f = f + p pha˙’i Theo D- i.nh l´y 2.3.13, mo. i d¯iˆe˙’m cu.
tho˙’a m˜an:
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2.
. c d¯ˆo` ch´u.ng minh tu.o.ng tu.
(b) Nˆe´u µ0 = 1 v`a x∗ tho˙’a m˜an (2.4.22), (2.4.23), th`ı theo D- i.nh l´y Kuhn-Tucker 1.1.2 cho B`ai to´an (P ) suy ra x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a f trˆen D. ´Ap du. ng lu.o. . phˆa` n (b) cu˙’a D- i.nh l´y 2.4.14, suy ra tˆo`n ta. i ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f = f + p trˆen D. Ngo`ai ra, theo D- i.nh l´y 2.3.13 th`ı mo. i d¯iˆe˙’m infimum to`an cu. c ˜x∗ d¯ˆe` u pha˙’i tho˙’a m˜an:
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2, t´u.c l`a khˆong c´o d¯iˆe˙’m infimum to`an cu. c n`ao cu˙’a ˜f = f + p nˇa`m ngo`ai h`ınh cˆa` u B(x∗, γ∗/2).
m (cid:88)
Nhˆa. n x´et 2.4.4. Nˆe´u S = IRn th`ı N (x∗|S) = {0}, nˆen biˆe˙’u th´u.c (2.4.22) d¯u.o. . c thay bo.˙’ i
i=1
(2.4.24) 0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗).
. tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c
. c x´ac d¯i.nh theo cˆong th´u.c (2.4.18). Tiˆe´p theo, ch´ung tˆoi ch´u.ng minh su. cu˙’a B`ai to´an ( ˜P ) khi D l`a tˆa. p lˆo`i d¯a diˆe.n. D- i.nh l´y 2.4.16. Gia˙’ su.˙’ D d¯u.o.
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), th`ı tˆo`n ta. i duy nhˆa´t x∗ ∈ D v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 1, . . . , m, sao cho
m (cid:88)
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2,
i=1
(2.4.25) (2Ax∗ + b) + µici = 0,
v`a
i = 1, . . . , m. (2.4.26) µi((cid:104)ci, x∗(cid:105) − di) = 0 v´o.i mo. i
(b) Nˆe´u c´o x∗ ∈ D tho˙’a m˜an (2.4.25), (2.4.26) th`ı tˆo`n ta. i ˜x∗ l`a d¯iˆe˙’m
infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) tho˙’a m˜an
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
m (cid:88)
v`a
i=1
(cid:107)2A˜x∗ + b + µici(cid:107) ≤ λmaxγ∗.
. c tiˆe˙’u to`an cu. c l`a D (cid:54)= ∅, nˆen tˆo`n ta. i x∗ l`a cu.
Ch´u.ng minh. (a) T`u. gia˙’ thiˆe´t cu˙’a mˆe.nh d¯ˆe` suy ra D (cid:54)= ∅. Do f l`a lˆo`i ngˇa. t nˆen nˆe´u tˆo`n ta. i d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c trˆen D th`ı d¯iˆe˙’m d¯´o l`a duy nhˆa´t. Theo Hˆe. qua˙’ 2.3 (xem [31], trang 41) th`ı d¯iˆe` u kiˆe.n cˆa` n v`a d¯u˙’ d¯ˆe˙’ B`ai to´an (P ) c´o nghiˆe.m cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a f trˆen D. V`ı x∗ l`a d¯iˆe˙’m cu.
. c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ) nˆen theo D- i.nh l´y 1.1.3 suy ra tˆo`n ta. i c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 1, . . . , m, sao cho
m (cid:88)
ch´ung tho˙’a m˜an c´ac d¯iˆe` u kiˆe.n
i=1
(2Ax∗ + b) + µici = 0
v`a
µi((cid:104)ci, x∗(cid:105) − di) = 0 v´o.i mo. i i = 1, . . . , m.
Mˇa. t kh´ac, theo D- i.nh l´y 2.3.13 ta suy ra
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2
v´o.i mo. i d¯iˆe˙’m infimum to`an cu. c ˜x∗ cu˙’a B`ai to´an ( ˜P ).
(b) X´et B`ai to´an (P ) v´o.i gi(x) := (cid:104)ci, x(cid:105) − di, i = 1, . . . , m, khi d¯´o ∂gi(x∗) = ci, i = 1, 2, . . . , m. Theo D- i.nh l´y 2.4.15 th`ı c´ac d¯iˆe` u kiˆe.n (2.4.25) . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ). Ch´u.ng v`a (2.4.26) l`a d¯u˙’ d¯ˆe˙’ x∗ l`a d¯iˆe˙’m cu. minh tu.o.ng tu. . nhu. phˆa` n (b) cu˙’a D- i.nh l´y 2.4.14, ta suy ra tˆo`n ta. i ˜x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f = f + p trˆen D. Theo D- i.nh l´y 2.3.13 th`ı mo. i d¯iˆe˙’m infimum to`an cu. c ˜x cu˙’a ˜f = f + p pha˙’i tho˙’a m˜an
(cid:107)˜x∗ − x∗(cid:107) ≤ γ∗/2.
√ λ | λ ∈ λ(AT A)} = λmax
m (cid:88)
m (cid:88)
Ho.n thˆe´, t`u. biˆe˙’u th´u.c (2.4.25) v`a (cid:107)A(cid:107) = max{ (xem [2]) cho ph´ep ta biˆe´n d¯ˆo˙’i
i=1
i=1
(cid:107)2A˜x∗ + b + µici(cid:107) = (cid:107)2Ax∗ + b + µici + 2A˜x∗ − 2Ax∗(cid:107)
= 2(cid:107)A˜x∗ − Ax∗(cid:107)
≤ 2(cid:107)A(cid:107)(cid:107)˜x∗ − x∗(cid:107) ≤ 2λmaxγ∗/2 = λmaxγ∗.
Do d¯´o d¯i.nh l´y d¯u.o. . c ch´u.ng minh.
Trong [51], ngo`ai c´ac kˆe´t luˆa. n o.˙’ (b), H. X. Phu c`on chı˙’ ra: V´o.i 1 ≤ i ≤ m, nˆe´u (cid:104)ci, ˜x∗(cid:105) < di − (2s/λmin)1/2(cid:107)ci(cid:107) th`ı
µi = 0.
Kˆe´t luˆa. n: Trong chu.o.ng n`ay ch´ung tˆoi d¯˜a chı˙’ ra: c´ac d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai (c´ac Mˆe.nh d¯ˆe` 2.1.9–2.1.11); d¯iˆe` u kiˆe.n tˆo`n ta. i d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c v`a infimum to`an cu. c (Mˆe.nh d¯ˆe` 2.2.14); thiˆe´t lˆa. p cˆa. n trˆen d¯´ung cu˙’a d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c, infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ); (c´ac mˆe.nh d¯ˆe` 2.2.15, 2.3.16); t´ınh ˆo˙’n d¯i.nh nghiˆe.m cu˙’a B`ai to´an quy hoa. ch to`an phu.o.ng bi. nhiˆe˜u gi´o.i nˆo. i ( ˜P ) (D- i.nh l´y 2.3.13); D- i.nh l´y Kuhn-Tucker suy rˆo. ng cho B`ai to´an ( ˜P ) (c´ac d¯i.nh l´y 2.4.14–2.4.16).
. . CHU NG 3 O
T´INH Γ-L ˆO` I NGO `AI CU˙’ A H `AM BI. NHIˆE˜ U V `A D- IˆE˙’M INFIMUM TO `AN CU. C CU˙’ A B `AI TO ´AN ( ˜P )
Trong chu.o.ng n`ay ch´ung tˆoi su.˙’ du. ng phu.o.ng ph´ap tiˆe´p cˆa. n tˆo pˆo d¯ˆe˙’ nghiˆen c´u.u: c´ac t´ınh chˆa´t cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i ˜f = f + p; quan hˆe. gi˜u.a c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ); t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ); t´ınh chˆa´t tu. . a v`a d¯iˆe` u kiˆe.n tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ).
Trong mu. c n`ay ta nghiˆen c´u.u t´ınh Γ-lˆo`i ngo`ai cu˙’a h`am to`an phu.o.ng . c cho bo.˙’ i
lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f (x) = f (x) + p(x), trong d¯´o f, p d¯u.o. c´ac cˆong th´u.c (1.0.1) v`a (1.0.2), tu.o.ng ´u.ng.
D- i.nh ngh˜ıa 3.1.9. Cho f tho˙’a m˜an cˆong th´u.c (1.0.1). H`am h1(., z) theo hu.´o.ng z d¯u.o.
f (x + µz) − f (x + (3.1.1) µz)(cid:1), f (x) + h1(µ, z) := inf x∈IRn 1 2 . c d¯i.nh ngh˜ıa nhu. sau: 1 (cid:0)1 2 2
trong d¯´o µ ∈ IR v`a z ∈ IRn.
Ta k´y hiˆe.u
(3.1.2) m(γ, z) := inf{µ | h1(µ, z) > γ}
v`a
M (γ) := {tz | z ∈ IRn, |t| ≤ m(γ, z)}. (3.1.3)
Bˆo˙’ d¯ˆe` sau cho ta c´ac gi´a tri. cu˙’a h1(µ, z), m(γ, z) v`a c´ac t´ınh chˆa´t cu˙’a tˆa. p M (γ).
Bˆo˙’ d¯ˆe` 3.1.3. V´o.i mo. i z ∈ IRn v`a z (cid:54)= 0, ta c´o
4 (cid:104)Az, z(cid:105). (cid:113) γ
(a) h1(µ, z) = µ2
(cid:104)Az,z(cid:105).
(b) m(γ, z) = 2
(c) M (γ) = {x | x ∈ IRn, (cid:104)Ax, x(cid:105) ≤ 4γ}.
(d) M (γ) l`a tˆa. p lˆo`i, d¯´ong v`a cˆan.
(e) 0 ∈ M (γ) l`a d¯iˆe˙’m trong cu˙’a tˆa. p M (γ).
Ch´u.ng minh. (a) Ta thˆa´y rˇa`ng
f (x) + f (x + µz) − f (x + µz) 1 2 1 2 1 2
(cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + (cid:104)A(x + µz), (x + µz)(cid:105) + (cid:104)b, (x + µz)(cid:105) = 1 2 1 2 1 2
−(cid:104)A(x + µz), (x + µz)(cid:105) − (cid:104)b, (x + µz)(cid:105) 1 2 1 2 1 2 1 2
= (cid:104)Ax, x(cid:105) + (cid:104)Ax, x(cid:105) + µ(cid:104)Ax, z(cid:105) + (cid:104)Az, z(cid:105) − (cid:104)Ax, x(cid:105) µ2 2 1 2 1 2
−µ(cid:104)Ax, z(cid:105) − (cid:104)Az, z(cid:105) µ2 4
= (cid:104)Az, z(cid:105). (3.1.4) µ2 4
Do d¯´o, theo (3.1.1) suy ra
(cid:104)Az, z(cid:105). h1(µ, z) = µ2 4
(b) Theo (3.1.2) th`ı
m(γ, z) = inf{µ | h1(µ, z) > γ},
tu.o.ng d¯u.o.ng v´o.i
m(γ, z) = inf{µ | (cid:104)Az, z(cid:105) > γ}, µ2 4
nˆen (cid:114) γ . m(γ, z) = 2 (cid:104)Az, z(cid:105)
(c) Gia˙’ su.˙’ x ∈ M (γ), khi d¯´o tˆo`n ta. i z ∈ IRn v`a t ∈ IR tho˙’a m˜an x = tz, |t| ≤ 2(cid:112)γ/(cid:104)Az, z(cid:105).
Ta c´o
(cid:104)Ax, x(cid:105) = t2(cid:104)Az, z(cid:105) ≤ ((cid:104)Az, z(cid:105))4γ/(cid:104)Az, z(cid:105) = 4γ.
. c la. i, gia˙’ su.˙’ x ∈ IRn tho˙’a m˜an (cid:104)Ax, x(cid:105) ≤ 4γ. Do d¯´o (cid:104)Ax, x(cid:105) ≤ 4γ. Ngu.o. Nˆe´u x = 0, theo c´ach xˆay du. . ng m(z, γ) v`a M (γ) o.˙’ c´ac cˆong th´u.c (3.1.2) v`a (3.1.3) th`ı x ∈ M (γ). Nˆe´u x (cid:54)= 0 th`ı (cid:104)Ax, x(cid:105) > 0. D- ˇa. t z = lx, khi d¯´o (cid:104)Az, z(cid:105) = l2(cid:104)Ax, x(cid:105), v`ı vˆa. y c´o thˆe˙’ cho. n l sao cho
(cid:104)Az, z(cid:105) = l2(cid:104)Ax, x(cid:105) > γ. h1(µ, z) = µ2 4 µ2 4
Mˇa. t kh´ac, theo (3.1.2) th`ı
m(γ, z) = inf{µ | h1(µ, z) > γ}
= inf{µ | (cid:104)Az, z(cid:105) > γ}
l2(cid:104)Ax, x(cid:105) > γ}. = inf{µ | µ2 4 µ2 4
Do d¯´o (cid:114) γ m(γ, z) = ≥ . 2 |l| (cid:104)Ax, x(cid:105)
l z th`ı | 1
1 |l| l | ≤ m(γ, z). Theo d¯i.nh ngh˜ıa M (γ) trong cˆong
Nhu. vˆa. y, v´o.i x = 1 th´u.c (3.1.3) ta suy ra x ∈ M (γ).
(d) M (γ) d¯ˆo´i x´u.ng l`a hiˆe˙’n nhiˆen. Ta ch´u.ng minh M (γ) l`a d¯´ong. Thˆa. t vˆa. y, gia˙’ su.˙’ xn ∈ M (γ) v`a limn→∞ xn = x, khi d¯´o theo (c) Bˆo˙’ d¯ˆe` 3.1.3 ta c´o
(cid:104)Axn, xn(cid:105) ≤ 4γ v´o.i mo. i n ∈ IN,
cho n → ∞, ta suy ra x ∈ M (γ). V`ı h`am (cid:104)Ax, x(cid:105) l`a lˆo`i ngˇa. t, nˆen tˆa. p m´u.c du.´o.i
M (γ) = {x ∈ IRn | (cid:104)Ax, x(cid:105) ≤ 4γ}
l`a tˆa. p lˆo`i.
(e) Cho. n δ := 2(cid:112)γ/λmax. V´o.i mo. i x = (ξ1, ξ2, . . . , ξn) ∈ B(0, δ), biˆe˙’u
n (cid:88)
n (cid:88)
diˆe˜n x theo co. so.˙’ tru. . c chuˆa˙’n l`a c´ac v´ec to. riˆeng cu˙’a A ta d¯u.o. . c
j=1
i=1 n (cid:88)
2
(cid:104)Ax, x(cid:105) = (cid:104) λiξiei, ξjej(cid:105)
i=1 ≤ λmax(cid:107)x(cid:107)2
= λiξi
≤ 4γ.
Suy ra B(0, δ) ⊆ M (γ). V`ı thˆe´ 0 l`a d¯iˆe˙’m trong cu˙’a M (γ).
Mˆe. nh d¯ˆe` 3.1.18. Cho f tho˙’a m˜an cˆong th´u.c (1.0.1), γ > 0 v`a Γ = M (γ). Khi d¯´o h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u ˜f = f + p l`a Γ-lˆo`i ngo`ai nˆe´u
|p(x)| ≤ γ/2 v´o.i mo. i x ∈ D.
Ch´u.ng minh. Lˆa´y x0, x1 bˆa´t k`y trong D, khi d¯´o xa˙’y ra c´ac tru.`o.ng ho. . p sau:
i) x0 − x1 ∈ M (γ). D- ˇa. t Λ = {0, 1}, ta c´o
[x0, x1] ⊂ {xλ | λ ∈ Λ} + 0.5M (γ)
v`a hiˆe˙’n nhiˆen v´o.i λ ∈ {0, 1} th`ı
˜f (xλ) = λ ˜f (x0) + (1 − λ) ˜f (x1).
ii) x0 −x1 /∈ M (γ). D- ˇa. t z := x0 −x1, α := ((cid:104)Az, z(cid:105))/4 v`a l := m(γ, z).
´Ap du. ng lˆa` n lu.o.
. t (c) v`a (b) cu˙’a Bˆo˙’ d¯ˆe` 3.1.3 ta d¯u.o. . c (cid:114) γ α = (cid:104)Az, z(cid:105) > 4γ v`a l = m(γ, z) = 2 . = (cid:114) γ α (cid:104)Az, z(cid:105)
Do d¯´o 0 < l < 1.
Mˇa. t kh´ac, v`ı
(cid:10)A(l(x0 − x1)), l(x0 − x1)(cid:11) = l2(cid:104)A(x0 − x1), x0 − x1(cid:105)
= 4l2α
= 4γ, (3.1.5)
nˆen
(3.1.6) l(x0 − x1) ∈ M (γ)
v`a
tl(x0 − x1) /∈ M (γ) v´o.i mo. i |t| > 1. V`ı 0 < l < 1 nˆen Λ := [l/2, 1 − l/2] ∪ {0, 1} l`a tˆa. p d¯´ong kh´ac rˆo˜ng nˇa`m trong d¯oa. n [0, 1] v`a
(3.1.7) {xλ | λ ∈ Λ} = [xl/2, x1−l/2] ∪ {x0, x1}.
Ta cˆa` n ch´u.ng minh
(3.1.8) [x0, x1] ⊂ {xλ | λ ∈ Λ} + 0.5M (γ).
V`ı
[xl/2, x1−l/2] = (cid:8)xλ | λ ∈ [l/2, 1 − l/2](cid:9)
(cid:3) = (cid:2)(1 − )x0 + x1, x0 + (1 − )x1 l 2 l 2
= (cid:2)x0 + (x1 − x0), x0 + (1 − l 2 )(x1 − x0)(cid:3) l 2 l 2 l 2
nˆen
[x0, x1] = (cid:2)x0, x0 + (x1 − x0)(cid:3) ∪ [xl/2, x1−l/2]
(cid:3). (3.1.9) l 2 ∪(cid:2)x0 + (1 − )(x1 − x0), x1 l 2
Lˆa´y x ∈ [x0, x1], t`u. biˆe˙’u th´u.c (3.1.9) ta x´et c´ac tru.`o.ng ho. . p sau:
i) x ∈ [xl/2, x1−l/2]. V`ı 0 ∈ M (γ) v`a [xl/2, x1−l/2] ⊂ {xλ | λ ∈ Λ} theo
(3.1.7), nˆen
x ∈ {xλ | λ ∈ Λ} + 0.5 M (γ).
ii) x ∈ (cid:2)x0, x0 + l
x = (1 − t)x0 + t(x0 + (x1 − x0))
2(x1 − x0)(cid:3). Khi d¯´o tˆo`n ta. i t ∈ [0, 1] sao cho l 2 l 2
= (1 − t)x0 + tx0 + t (x1 − x0)
(3.1.10) = x0 + t (x1 − x0). l 2
Theo (3.1.6) th`ı l(x1 − x0) ∈ M (γ) v`a v`ı t ∈ [0, 1] nˆen tl(x1 − x0) ∈ M (γ). Do d¯´o t l 2(x1 − x0) ∈ 0.5M (γ). Mˇa. t kh´ac, x0 ∈ {xλ | λ ∈ Λ} theo d¯i.nh ngh˜ıa tˆa. p Λ. V`ı vˆa. y, t`u. cˆong th´u.c (3.1.10) ta suy ra
x ∈ {xλ | λ ∈ Λ} + 0.5 M (γ).
2)(x1 − x0), x1 x = t(cid:0)x0 + (1 −
iii) x ∈ (cid:2)x0 + (1 − l
)(x1 − x0)(cid:1) + (1 − t)x1
= tx0 + t(1 −
(3.1.11) (cid:3). Khi d¯´o tˆo`n ta. i t ∈ [0, 1] sao cho l 2 l )(x1 − x0) + x1 − tx1 2 (x0 − x1).
l = x1 + t 2 . nhu. trong tru.`o.ng ho. . p ii), v`ı x1 ∈ {xλ | λ ∈ Λ} v`a
2(x0 − x1) ∈ M (γ), nˆen t`u. cˆong th´u.c (3.1.11) suy ra
L´y luˆa. n tu.o.ng tu. 2(x1 − x0), t l t l
x ∈ {xλ | λ ∈ Λ} + 0.5 M (γ).
Kˆe´t ho. . p ca˙’ ba tru.`o.ng ho. . p i), ii) v`a iii) ta suy ra cˆong th´u.c (3.1.8).
(cid:48)
(cid:48)(cid:48)
X´et bˆa´t k`y λ ∈ Λ\{0, 1}, k´y hiˆe.u
λ := λ − ; λ := λ + , l 2 l 2
(cid:48)(cid:48)
(cid:48)
(cid:48)
(cid:48)(cid:48)
xλ = (1 − )x0 + x1
= ta dˆe˜ d`ang nhˆa. n thˆa´y λ(cid:48), λ(cid:48)(cid:48) ∈ [0, 1], λ = (λ(cid:48) + λ(cid:48)(cid:48))/2. V`ı λ(cid:48) + λ(cid:48)(cid:48) 2 (cid:0)(1 − λ (cid:1), λ(cid:48) + λ(cid:48)(cid:48) 2 x1 + (1 − λ )x0 + λ )x0 + λ x1 1 2
nˆen
. (3.1.12) xλ = xλ(cid:48) + xλ(cid:48)(cid:48) 2
Ta c˜ung c´o
)x0 + (1 − λ + )x1 − (λ + )x0 − (1 − λ − )x1 xλ(cid:48) − xλ(cid:48)(cid:48) = (λ − l 2 l 2 l 2 l 2 = l(x1 − x0).
Theo (3.1.6) th`ı l(x1 − x0) ∈ M (γ), vˆa. y
xλ(cid:48) − xλ(cid:48)(cid:48) ∈ M (γ).
Mˇa. t kh´ac, v`ı
(cid:10)A(xλ(cid:48) − xλ(cid:48)(cid:48) ), xλ(cid:48) − xλ(cid:48)(cid:48) (cid:11) = (cid:10)A(l(x1 − x0)), l(x1 − x0)(cid:11)
nˆen theo (3.1.5) th`ı
(3.1.13) (cid:10)A(xλ(cid:48) − xλ(cid:48)(cid:48) ), xλ(cid:48) − xλ(cid:48)(cid:48) (cid:11) = 4γ.
(1 − λ)f (x0) + λf (x1) = (1 − )f (x0) + ( )f (x1), X´et h`am to`an phu.o.ng lˆo`i ngˇa. t f (x) = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105), v`ı λ(cid:48) + λ(cid:48)(cid:48) 2 λ(cid:48) + λ(cid:48)(cid:48) 2
v`a
(cid:48)
(cid:48)
(cid:48)(cid:48)
(cid:48)(cid:48)
(1 − )f (x0) + (
λ(cid:48) + λ(cid:48)(cid:48) 2 (cid:0)(1 − λ = (cid:0)(1 − λ )f (x0) + λ λ(cid:48) + λ(cid:48)(cid:48) 2 f (x1)(cid:1) + )f (x0) + λ f (x1)(cid:1) 1 2 )f (x1) 1 2
nˆen
(3.1.14) (1 − λ)f (x0) + λf (x1) ≥ (cid:0)f (xλ(cid:48) ) + f (xλ(cid:48)(cid:48) )(cid:1). 1 2
T`u. c´ac biˆe˙’u th´u.c (3.1.12), (3.1.14) v`a (2.1.2) suy ra
≥ ) xλ(cid:48) + xλ(cid:48)(cid:48) 2
= (1 − λ)f (x0) + λf (x1) − f (xλ) (cid:0)f (xλ(cid:48) ) + f (xλ(cid:48)(cid:48) )(cid:1) − f ( (cid:10)A(xλ(cid:48)(cid:48) − xλ(cid:48) ), xλ(cid:48)(cid:48) − xλ(cid:48) (cid:11).
Do d¯´o kˆe´t ho. 1 2 1 4 . p v´o.i (2.1.6) ta d¯u.o. . c
(3.1.15) (1 − λ)f (x0) + λf (x1) − f (xλ) ≥ γ
X´et h`am bi. nhiˆe˜u ˜f = f + p ta c´o
(1 − λ) ˜f (x0) + λ ˜f (x1) − ˜f (xλ) = (1 − λ)(cid:0)f (x0) + p(x0)) + λ(f (x1) + p(x1)(cid:1) − (cid:0)f (xλ) + p(xλ)(cid:1) ≥ (1 − λ)(cid:0)f (x0) − γ/2(cid:1) + λ(cid:0)f (x1) − γ/2(cid:1) − (cid:0)f (xλ) + γ/2(cid:1) = (1 − λ)f (x0) + λf (x1) − f (xλ) − γ.
´Ap du. ng (3.1.15) ta suy ra
(1 − λ) ˜f (x0) + λ ˜f (x1) − ˜f (xλ) ≥ 0.
Tru.`o.ng ho. . p λ = 0, λ = 1 th`ı biˆe˙’u th´u.c trˆen luˆon d¯´ung.
T´om la. i, v´o.i x0, x1 ∈ D, tˆo`n ta. i tˆa. p d¯´ong Λ ⊂ [0, 1] ch´u.a {0, 1} sao
cho tho˙’a m˜an (2.3.12) v`a
∀λ ∈ Λ : ˜f (xλ) ≤ (1 − λ) ˜f (x0) + λ ˜f (x1).
Theo d¯i.nh ngh˜ıa suy ra ˜f = f + p l`a Γ-lˆo`i ngo`ai, v´o.i Γ = M (γ). Nhˆa. n x´et 3.1.5. Trong IRn v´o.i chuˆa˙’n Euclide, ta nhˆa. n thˆa´y
(3.1.16) ¯B(0, 2(cid:112)γ/λmax) ⊆ M (γ) ⊆ ¯B(0, 2(cid:112)γ/λmin).
i = 1, 2, . . . , n} l`a co. so.˙’ tru. Thˆa. t vˆa. y, go. i {ei |
i=1 ζiei v`a n (cid:88)
n (cid:88)
. c chuˆa˙’n trong IRn, trong d¯´o ei l`a v´ec to. riˆeng d¯o.n vi. ´u.ng v´o.i gi´a tri. riˆeng λi, i = 1, 2 . . . , n, cu˙’a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng A (c´o thˆe˙’ c´o mˆo. t sˆo´ gi´a tri. tr`ung nhau). Khi d¯´o v´o.i mo. i x ∈ IRn th`ı x = (cid:80)n
j=1
i=1 n (cid:88)
n (cid:88)
(cid:11) (cid:104)Ax, x(cid:105) = (cid:10) ζjej λiζiei,
j=1
i=1 n (cid:88)
= λiζiζj(cid:104)ei, ej(cid:105)
i=1
= λiζ 2 i .
nˆen
(3.1.17) λmin(cid:107)x(cid:107)2 ≤ (cid:104)Ax, x(cid:105) ≤ λmax(cid:107)x(cid:107)2.
Do d¯´o, nˆe´u x ∈ ¯B(0, 2(cid:112)γ/λmax) th`ı
(cid:104)Ax, x(cid:105) ≤ λmax(cid:107)x(cid:107)2 ≤ 4γ,
nˆen suy ra
B(0, 2(cid:112)γ/λmax) ⊆ M (γ).
(cid:104)Ax, x(cid:105) ≤ 4γ, do d¯´o kˆe´t ho. (3.1.18) . p v´o.i
Nˆe´u x ∈ M (γ), theo (c) Bˆo˙’ d¯ˆe` 3.1.3 th`ı vˆe´ tr´ai cu˙’a (3.1.17) ta d¯u.o. . c
λmin(cid:107)x(cid:107)2 ≤ (cid:104)Ax, x(cid:105) ≤ 4γ,
nˆen
(cid:107)x(cid:107) ≤ 2(cid:112)γ/λmin .
Biˆe˙’u th´u.c cuˆo´i cho ta
(3.1.19) M (γ) ⊂ B(0, 2(cid:112)γ/λmin).
Kˆe´t ho. . p c´ac biˆe˙’u th´u.c (3.1.18) v`a (3.1.19) ta nhˆa. n d¯u.o. . c (3.1.16).
l`a v´ec to. riˆeng ´u.ng v´o.i gi´a tri. riˆeng b´e nhˆa´t λmin. D- ˇa. t Go. i ei0
±y0 = ±2(cid:112)γ/λmin ei0, ta c´o (cid:104)Ay0, y0(cid:105) = 4γ, t´u.c l`a ±y0 ∈ M (γ). V`ı
(cid:107) ± y0(cid:107)2 = 4(γ/λmin)(cid:107)eio(cid:107)2 = 4γ/λmin,
nˆen ±y0 ∈ ¯B(0, 2(cid:112)γ/λmin). Do d¯´o suy ra
(3.1.20) ± y0 ∈ M (γ) v`a ± y0 ∈ S(0, 2(cid:112)γ/λmin).
Kˆe´t ho. . p (3.1.16) v`a (3.1.20) ta suy ra M (γ) nˇa`m trong h`ınh cˆa` u B(0, 2(cid:112)γ/λmin), tiˆe´p x´uc v´o.i mˇa. t cˆa` u n`ay ta. i 2 d¯iˆe˙’m riˆeng biˆe.t d¯ˆo´i x´u.ng v´o.i nhau qua tˆam cu˙’a n´o.
Mˇa. t kh´ac, nˆe´u λmax > λmin, go. i ej0 l`a v´ec to. riˆeng ´u.ng v´o.i gi´a tri. riˆeng l´o.n nhˆa´t λmax cu˙’a ma trˆa. n A. D- ˇa. t y := tej0, 2(cid:112)γ/λmax < t ≤ 2(cid:112)γ/λmin, khi d¯´o
(cid:104)Ay, y(cid:105) = t2λmax(cid:104)ej0, ej0(cid:105) = t2λmax > λmax4γ/λmax = 4γ,
. c su.
t´u.c l`a y /∈ M (γ). Mˇa. t kh´ac, v`ı (cid:107)y(cid:107) = (cid:107)tej0(cid:107) ≤ 2(cid:112)γ/λmax ≤ 2(cid:112)γ/λmin nˆen y ∈ B(0, 2(cid:112)γ/λmin). Do vˆa. y, nˆe´u λmax > λmin th`ı tˆa. p M (γ) thu. . nˇa`m trong h`ınh cˆa` u B(0, 2(cid:112)γ/λmin).
Mˆe. nh d¯ˆe` 3.1.19. H`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i ˜f = f +p l`a Γ-lˆo`i ngo`ai v´o.i Γ = M (2s).
Ch´u.ng minh. Hiˆe˙’n nhiˆen ta c´o
|p(x)| v´o.i mo. i x ∈ D, |p(x)| ≤ sup x∈D
t´u.c l`a
|p(x)| ≤ s = 2s/2 v´o.i mo. i x ∈ D.
Theo Mˆe.nh d¯ˆe` 3.1.18, h`am ˜f = f + p l`a Γ-lˆo`i ngo`ai v´o.i Γ = M (2s).
Trong [44] khi nghiˆen c´u.u tˆa. p Γ-lˆo`i ngo`ai (d¯i.nh ngh˜ıa 1.4.6), H. X. Phu d¯˜a chı˙’ ra mˆo. t sˆo´ t´ınh chˆa´t co. ba˙’n cu˙’a tˆa. p n`ay. Mˆe.nh d¯ˆe` du.´o.i d¯ˆay chı˙’ nˆeu thˆem t´ınh chˆa´t Γ-lˆo`i ngo`ai cu˙’a tˆa. p m´u.c du.´o.i cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i, v´o.i tˆa. p Γ cu. thˆe˙’ phu. thuˆo. c v`ao cˆa. n trˆen s cu˙’a |p|.
Mˆe. nh d¯ˆe` 3.1.20. Tˆa. p m´u.c du.´o.i Lα( ˜f ) cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p l`a Γ-lˆo`i ngo`ai, v´o.i Γ = M (2s).
Ch´u.ng minh. Theo Mˆe.nh d¯ˆe` 3.1.19 th`ı h`am ˜f = f + p l`a Γ-lˆo`i ngo`ai v´o.i Γ = M (2s). Mˇa. t kh´ac, Mˆe.nh d¯ˆe` 1.4.4 Chu.o.ng 1, khˇa˙’ ng d¯i.nh tˆa. p m´u.c du.´o.i cu˙’a h`am Γ-lˆo`i ngo`ai l`a Γ-lˆo`i ngo`ai, nˆen suy ra Lα( ˜f ) l`a Γ-lˆo`i ngo`ai, v´o.i Γ = M (2s).
Mˆo. t t´ınh chˆa´t quan tro. ng cu˙’a h`am lˆo`i l`a cu. . c tiˆe˙’u d¯i.a phu.o.ng l`a cu. . c
tiˆe˙’u to`an cu. c. D- ˆo´i v´o.i h`am Γ-lˆo`i ngo`ai ta c´o t´ınh chˆa´t gˆa` n giˆo´ng sau:
Mˆe. nh d¯ˆe` 3.2.21. Cho Γ = M (2s) nˆe´u x∗ ∈ D l`a d¯iˆe˙’m Γ-cu. h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p, th`ı x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u cu˙’a . c tiˆe˙’u to`an cu. c cu˙’a ˜f .
Ch´u.ng minh. D- i.nh l´y 1.4.9 Chu.o.ng 1 khˇa˙’ ng d¯i.nh, nˆe´u Γ c´o 0 l`a d¯iˆe˙’m trong, g : D ⊂ IRn → IR l`a Γ-lˆo`i ngo`ai v`a x∗ l`a d¯iˆe˙’m Γ-cu. . c tiˆe˙’u th`ı x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a g trˆen D. Theo Mˆe.nh d¯ˆe` 3.1.18 th`ı ˜f = f + p l`a h`am Γ-lˆo`i ngo`ai v´o.i Γ = M (2s) v`a theo (e) cu˙’a Bˆo˙’ d¯ˆe` 3.1.3 th`ı 0 l`a d¯iˆe˙’m trong cu˙’a M (2s), nˆen c´ac d¯iˆe` u kiˆe.n cu˙’a D- i.nh l´y 1.4.9 tho˙’a m˜an, do d¯´o x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a ˜f = f + p trˆen D.
Trong [50] H. X. Phu chı˙’ ra d¯i.nh l´y sau:
D- i.nh l´y 3.2.17. Cho Γ = M (2s) v`a x∗ ∈ D l`a d¯iˆe˙’m Γ-infimum cu˙’a h`am bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Khi d¯´o x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f = f + p.
Ch´u.ng minh chi tiˆe´t d¯u.o. . c tr`ınh b`ay trong [50]. Nghiˆen c´u.u hiˆe.u cu˙’a c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) ta
1, ˜x∗
2 l`a hai d¯iˆe˙’m infimum to`an cu. c bˆa´t k`y cu˙’a
c´o mˆe.nh d¯ˆe` sau:
Mˆe. nh d¯ˆe` 3.2.22. Nˆe´u ˜x∗ B`ai to´an ( ˜P ) th`ı
2 ∈ M (2s).
˜x∗ 1 − ˜x∗
Ch´u.ng minh. Theo (2.1.2) th`ı
1) +
2) − f (
1 − ˜x∗
2), ˜x∗
1 − ˜x∗ 2
) = f (˜x∗ f (˜x∗ (cid:10)A(˜x∗ (cid:11). (3.2.21) 1 4 1 2
1) +
) ˜x∗ 1 + ˜x∗ 2 2
1) +
)
2) − lsc ˜f ( 1 + ˜x∗ ˜x∗ 2 2 2) − lsc p(
1 + ˜x∗ ˜x∗ 2 2
lsc ˜f (˜x∗ 1 2 + lsc p(˜x∗ ). ˜x∗ 1 + ˜x∗ 1 2 2 2 Ta la. i c´o lsc ˜f = f + lsc p, nˆen 1 1 lsc ˜f (˜x∗ 2 2 1 = 2 lsc p(˜x∗ 1) + f (˜x∗ 1 2 f (˜x∗ 2) − f ( 1 2
Kˆe´t ho. . p v´o.i (3.2.21) ta suy ra
1 + ˜x∗ ˜x∗ 2 2 lsc p(˜x∗
lsc ˜f (˜x∗ )
1 − ˜x∗
1) +
2) − lsc p(
2) − lsc ˜f ( 1 2)(cid:11) + 2
lsc p(˜x∗ ). 1 2 = 1 2 2), (˜x∗ 1 − ˜x∗ lsc ˜f (˜x1) + 1 (cid:10)A(˜x∗ 4 1 2 ˜x∗ 1 + ˜x2 2
2), ˜x1 − ˜x∗ 2
2) − ˜f (
1) +
(cid:11) − 2s. lsc ˜f (˜x∗ lsc ˜f (˜x∗ ) ≥ (cid:10)A(˜x∗ V`ı supx∈D |p(x)| ≤ s < +∞, nˆen theo (b) Bˆo˙’ d¯ˆe` 2.3.1 suy ra supx∈D |lsc p(x)| ≤ s. Thay v`ao biˆe˙’u th´u.c trˆen ta d¯u.o. . c 1 + ˜x∗ ˜x∗ 2 2 1 2 1 2
. c tiˆe˙’u to`an cu. c
Mˇa. t kh´ac, theo Mˆe.nh d¯ˆe` 3.4.3 [1] th`ı ˜x∗ cu˙’a lsc ˜f (x), t´u.c l`a lsc ˜f (˜x∗ 1 1 − ˜x∗ 4 2 l`a c´ac d¯iˆe˙’m cu. 1, ˜x∗ 2), nˆen t`u. biˆe˙’u th´u.c trˆen suy ra
1) − lsc ˜f (
1 − ˜x∗
2), ˜x∗
1 − ˜x∗ 2
1) = lsc ˜f (˜x∗ 1 4
1 + ˜x∗ ˜x∗ 2 2
lsc ˜f (˜x∗ ) ≥ (cid:10)A(˜x∗ (cid:11) − 2s. (3.2.22)
1) − lsc ˜f (
Thay
1 − ˜x∗
1 − ˜x∗
2), (˜x∗
˜x1 + ˜x∗ 2 2
1 − ˜x∗ ˜x∗
2 ∈ M (2s).
lsc ˜f (˜x∗ ) ≤ 0 v`ao vˆe´ tr´ai cu˙’a (3.2.22) v`a chuyˆe˙’n vˆe´ ta nhˆa. n d¯u.o. . c 2)(cid:11) ≤ 8s. (cid:10)A(˜x∗ Do d¯´o t`u. (c) Bˆo˙’ d¯ˆe` 3.1.3 v`a biˆe˙’u th´u.c trˆen ta suy ra
Mˆe.nh d¯ˆe` d¯u.o. . c ch´u.ng minh.
T`u. Nhˆa. n x´et 3.1.5, khi λmax > λmin ta suy ra
2 + ξ2
2, x = (ξ1, ξ2)
Nhˆa. n x´et 3.2.6. Mˆe. nh d¯ˆe` 3.2.22 ma. nh ho.n Mˆe. nh d¯ˆe` 2.3.16 Chu.o.ng 2. V´ı du. 3.2.7. X´et h`am to`an phu.o.ng lˆo`i ngˇa. t
f (x) = 0.25 ξ1
v`a h`am nhiˆe˜u
(cid:40) −0.5 nˆe´u 2 ≤ (cid:107)x(cid:107) ≤ 100 p(x) = 0.5 nˆe´u (cid:107)x(cid:107) < 2.
Khi d¯´o
(cid:107)x(cid:107)≤100
s := sup |p(x)| = 0.5
v`a
(cid:40)
2 − 0.5 nˆe´u 2 ≤ (cid:107)x(cid:107) ≤ 100 2 + 0.5 nˆe´u (cid:107)x(cid:107) < 2.
2 + ξ2 2 + ξ2
˜f (x) = f (x) + p(x) = 0.25 ξ1 0.25 ξ1
2 = 1}
Ta thˆa´y rˇa`ng
2 + ξ2
{(0, 0)} ∪ {x = (ξ1, ξ2) | 0.25 ξ1
l`a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f (x).
Lˆa´y z ∈ IRn, z (cid:54)= 0 bˆa´t k`y, tia {tz | t ∈ IR} cˇa´t tˆa. p c´ac d¯iˆe˙’m infimum
2
2
to`an cu. c ta. i ˜x0 = (0, 0) v`a ta. i ˜x1 = ( ˜ξ1, ˜ξ2), ˜x2 = (− ˜ξ1, − ˜ξ2) tho˙’a m˜an
= 1. 0.25 ˜ξ1 + ˜ξ2
Ta thˆa´y rˇa`ng
˜x1 − ˜x2 = (2 ˜ξ1, 2 ˜ξ2), ˜x2 − ˜x1 = (−2 ˜ξ1, −2 ˜ξ2)
2 = 4} ⊂ M (1) = M (2s),
2 + ξ2
nˆen ˜x1 − ˜x2, ˜x2 − ˜x1 d¯ˆo´i x´u.ng v´o.i nhau qua ˜x0 = (0, 0) v`a
˜x1 − ˜x2, ˜x2 − ˜x1 ∈ {x = (ξ1, ξ2) | 0.25 ξ1
t´u.c l`a v´o.i mo. i phu.o.ng z (cid:54)= 0 luˆon tˆo`n ta. i 2 d¯iˆe˙’m infimum to`an cu. c ˜x1, ˜x2 cu˙’a ˜f d¯ˆe˙’ ˜x1 − ˜x2, ˜x2 − ˜x1 d¯ˆo´i x´u.ng qua (0, 0) v`a nˇa`m trˆen biˆen cu˙’a M (1). Do d¯´o suy ra tˆa. p M (1) l`a tˆa. p nho˙’ nhˆa´t ch´u.a hiˆe.u cu˙’a hai d¯iˆe˙’m infimum bˆa´t k`y cu˙’a v´ı du. trˆen, v`ı thˆe´ d¯´anh gi´a o.˙’ Mˆe.nh d¯ˆe` 3.2.22 l`a khˆong thˆe˙’ tˆo´t ho.n.
T´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) . c x´et d¯ˆe´n trong Chu.o.ng 2. Hai d¯i.nh l´y du.´o.i d¯ˆay l`a nh˜u.ng kˆe´t qua˙’
d¯˜a d¯u.o. co. ba˙’n cu˙’a mu. c n`ay.
. c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ), ˜x∗
D- i.nh l´y 3.3.18. Nˆe´u x∗ l`a d¯iˆe˙’m cu. l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). Khi d¯´o
˜x∗ ∈ x∗ + 0.5M (2s). (3.3.23)
Ch´u.ng minh. V`ı f (˜x∗) ≥ f (x∗) nˆen
f (˜x∗) + f (x∗) − f (x∗) f (˜x∗) − f (x∗) ≥
= f (x∗) f (˜x∗) − 1 2 1 2
(cid:0)(cid:104)A˜x∗, ˜x∗(cid:105) + (cid:104)b, ˜x∗(cid:105) − (cid:104)Ax∗, x∗(cid:105) − (cid:104)b, x∗(cid:105)(cid:1) =
=
=
1 2 1 2 1 2 1 (cid:0)(cid:104)A(x∗ + ˜x∗ − x∗), x∗ + ˜x∗ − x∗(cid:105) + (cid:104)b, x∗ + ˜x∗ − x∗(cid:105) 2 −(cid:104)Ax∗, x∗(cid:105) − (cid:104)b, x∗(cid:105)(cid:1) 1 (cid:0)(cid:104)Ax∗, x∗(cid:105) + 2(cid:104)Ax∗, ˜x∗ − x∗(cid:105) + (cid:104)A˜x∗ − x∗, ˜x∗ − x∗(cid:105) 2 +(cid:104)b, ˜x∗ − x∗(cid:105) + (cid:104)b, x∗(cid:105) − (cid:104)Ax∗, x∗(cid:105) − (cid:104)b, x∗(cid:105)(cid:1).
R´ut go. n biˆe˙’u th´u.c trˆen ta d¯u.o. . c
(cid:0)(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105) + (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105)(cid:1) ≤ f (˜x∗) − f (x∗). 1 2
Theo (2.3.10) Chu.o.ng 2 th`ı (cid:104)2Ax∗ + b, ˜x∗ − x∗(cid:105) ≥ 0 nˆen t`u. biˆe˙’u th´u.c trˆen ta suy ra
(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105) ≤ f (˜x∗) − f (x∗). (3.3.24)
1 2 Nˆe´u ˜x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a ˜f th`ı ˜f (˜x∗) − ˜f (x∗) ≤ 0 v`a c´o thˆe˙’ biˆe´n d¯ˆo˙’i
f (˜x∗) − f (x∗) = ˜f (˜x∗) − ˜f (x∗) − p(˜x∗) + p(x∗)
≤ ˜f (˜x∗) − ˜f (x∗) + 2s
≤ 2s.
Do d¯´o kˆe´t ho. . p v´o.i (3.3.24) ta d¯u.o. . c
(cid:104)A(˜x∗ − x∗), ˜x∗ − x∗(cid:105) ≤ 4s. (3.3.25)
. c tiˆe˙’u . p v´o.i
Nˆe´u ˜x∗ chı˙’ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a ˜f th`ı n´o l`a d¯iˆe˙’m cu. to`an cu. c cu˙’a lsc ˜f = f + lsc p nˆen lsc ˜f (˜x∗) − lsc ˜f (x∗) ≤ 0. Kˆe´t ho. supx∈D |lsc p(x)| ≤ s ta suy ra
f (˜x∗) − f (x∗) = lsc ˜f (˜x∗) − lsc ˜f (x∗) − lsc p(˜x∗) + lsc p(x∗)
≤ lsc ˜f (˜x∗) − lsc ˜f (x∗)
≤ 2s.
. c (3.3.25).
Thay bˆa´t d¯ˇa˙’ ng th´u.c trˆen v`ao (3.3.24) ta la. i nhˆa. n d¯u.o. T`u. (3.3.25) v`a d¯i.nh ngh˜ıa tˆa. p M (γ) ta suy ra
˜x∗ − x∗ ∈ 0.5M (2s),
t´u.c l`a ˜x∗ ∈ x∗ + 0.5M (2s).
Nhˆa. n x´et 3.3.7. D- i.nh l´y 3.3.18 ma. nh ho.n D- i.nh l´y 2.3.13 o.˙’ Mu. c 2.3 Chu.o.ng 2.
V´ı du. sau d¯ˆay cho ta thˆa´y tˆa. p 0.5M (2s) l`a nho˙’ nhˆa´t trong c´ac tˆa. p
ch´u.a ˜x∗ − x∗.
2
V´ı du. 3.3.8. X´et h`am to`an phu.o.ng lˆo`i ngˇa. t
2 + 2ξ2
f (x) = ξ1
trˆen IR2 v`a h`am nhiˆe˜u
(cid:40) s − f (x) nˆe´u x = (ξ1, ξ2) ∈ 0.5M (2s) p(x) = 0 nˆe´u x = (ξ1, ξ2) /∈ 0.5M (2s) .
Khi d¯´o
(cid:40)
˜f (x) = nˆe´u x = (ξ1, ξ2) ∈ 0.5M (2s) s f (x) nˆe´u x = (ξ1, ξ2) ∈ IR2 \ 0.5M (2s).
. c tiˆe˙’u duy nhˆa´t cu˙’a f trˆen IR2 v`a ˜f d¯a. t cu. Ta thˆa´y, x∗ = 0 l`a d¯iˆe˙’m cu. . c tiˆe˙’u ta. i mo. i ˜x∗ ∈ 0.5M (s). Do d¯´o ta suy ra 0.5M (2s) l`a tˆa. p nho˙’ nhˆa´t ch´u.a ˜x∗ − x∗.
Go. i S0 l`a tˆa. p c´ac d¯iˆe˙’m cu.
. c tiˆe˙’u cu˙’a B`ai to´an (P ) v`a Ss l`a tˆa. p c´ac d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). Khoa˙’ng c´ach Hausdorff l`a d¯a. i lu.o. . ng:
(cid:107)x − y(cid:107)}. inf y∈Ss inf x∈S0 dH(S0, Ss) = max{sup x∈S0 (cid:107)x − y(cid:107), sup y∈Ss
. c tiˆe˙’u x∗ v`a Trong [50] H. X. Phu d¯˜a ph´at biˆe˙’u v`a ch´u.ng minh d¯i.nh l´y sau: D- i.nh l´y 3.3.19. Gia˙’ su.˙’ B`ai to´an (P ) c´o d¯iˆe˙’m cu.
(x∗ + ¯B(0, r)) ∩ D l`a d¯´ong v´o.i gi´a tri. r > 0 n`ao d¯´o .
Nˆe´u
|p(x)| ≤ s ≤ r2λmin, 1 2 sup x∈D
th`ı tˆa. p Ss l`a kh´ac rˆo˜ng v`a
dH({x∗}, Ss) ≤ (cid:112)2s/λmin .
Trong mu. c 2.4 Chu.o.ng 2, ch´ung tˆoi d¯˜a tr`ınh b`ay t´ınh chˆa´t tu.
. a v`a .˙’ mu. c n`ay ta x´et la. i c´ac t´ınh chˆa´t trˆen
d¯iˆe` u kiˆe.n tˆo´i u.u cu˙’a B`ai to´an ( ˜P ). O v`a nhˆa. n d¯u.o. . c c´ac kˆe´t qua˙’ tˆo˙’ng qu´at ho.n c´ac kˆe´t qua˙’ tru.´o.c d¯´o. D- i.nh ngh˜ıa 3.4.10. ([50]) Cho tˆa. p cˆan Γ ta n´oi ξ l`a du.´o.i vi phˆan suy rˆo. ng thˆo cu˙’a h`am g : D → IR ta. i d¯iˆe˙’m x∗ ∈ D nˆe´u
(cid:0)g(x(cid:48)) + (cid:104)ξ, x(cid:48)(cid:105)(cid:1) ≤ g(x) + (cid:104)ξ, x(cid:105) v´o.i mo. i x ∈ D. inf x(cid:48)∈(x∗+Γ)∩D
Khi g = ˜f = f + p ta c´o d¯i.nh l´y sau:
D- i.nh l´y 3.4.20. Gia˙’ su.˙’ 0 < supx∈D |p(x)| ≤ s < +∞, f (x) = (cid:104)Ax, x(cid:105) − (cid:104)b, x(cid:105). Khi d¯´o, v´o.i x∗ ∈ D n`ao d¯´o th`ı
(cid:0) ˜f (x(cid:48))−(cid:104)2Ax∗+b, x(cid:48)(cid:105)(cid:1) ≤ ˜f (x)−(cid:104)2Ax∗+b, x(cid:105) v´o.i mo. i x ∈ D. inf x(cid:48)∈(x∗+0.5M (2s))∩D
. p d¯ˇa. c biˆe. t, nˆe´u D d¯´ong v`a p l`a nu.˙’a liˆen tu. c du.´o.i, th`ı v´o.i
Trong tru.`o.ng ho. mˆo˜i x∗ ∈ D tˆo`n ta. i
˜x∗ ∈ (cid:0)x∗ + 0.5M (2s)(cid:1) ∩ D
sao cho
(cid:0) ˜f (x(cid:48)) − (cid:104)2Ax∗ + b, x(cid:48)(cid:105)(cid:1) ˜f (˜x∗) − (cid:104)2Ax∗ + b, ˜x∗(cid:105) = min x(cid:48)∈(x∗+0.5M (2s))∩D
v`a
˜f (˜x∗) − (cid:104)2Ax∗ + b, ˜x∗(cid:105) ≤ ˜f (x) − (cid:104)2Ax∗ + b, x(cid:105) v´o.i mo. i x ∈ D,
hoˇa. c tu.o.ng d¯u.o.ng l`a
˜f (x) ≥ ˜f (˜x∗) + (cid:104)2Ax∗ + b, x − ˜x∗(cid:105) v´o.i mo. i x ∈ D.
D- i.nh l´y n`ay do H. X. Phu ph´at biˆe˙’u v`a ch´u.ng minh trong [50]. Nghiˆen c´u.u su. . tˆo`n ta. i d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), ta c´o
. c cho bo.˙’ i (2.4.17).
D- i.nh l´y Kuhn-Tucker suy rˆo. ng nhu. sau: D- i.nh l´y 3.4.21. X´et B`ai to´an ( ˜P ) v´o.i miˆe` n D d¯u.o. Khi d¯´o
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c, th`ı tˆo`n ta. i duy nhˆa´t x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D v`a c´ac nhˆan tu.˙’ Lagrange µ0 ≥ 0, µ1 ≥ 0, . . . , µm ≥ 0, sao cho ch´ung khˆong c`ung triˆe. t tiˆeu, tho˙’a m˜an d¯iˆe` u kiˆe. n Kuhn-Tucker
(3.4.26) L(x, µ0, . . . , µm), L(x∗, µ0, . . . , µm) = min x∈S
d¯iˆe` u kiˆe. n b`u
i = 1, . . . , m. (3.4.27) µigi(x∗) = 0 v´o.i mo. i
Nˆe´u d¯iˆe` u kiˆe. n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1. (b) Nˆe´u tˆo`n ta. i x∗ tho˙’a m˜an (3.4.26) v`a (3.4.27) v´o.i µ0 = 1 th`ı tˆo`n ta. i ˜x∗ ∈ (x∗ + 0.5M (2s)) ∩ D l`a infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ).
Ch´u.ng minh. (a) Gia˙’ thiˆe´t (a) cu˙’a d¯i.nh l´y c˜ung l`a gia˙’ thiˆe´t (a) cu˙’a D- i.nh l´y 2.4.14 Chu.o.ng 2, nˆen suy ra tˆo`n ta. i duy nhˆa´t d¯iˆe˙’m x∗ ∈ D v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho ch´ung khˆong c`ung triˆe.t tiˆeu, tho˙’a m˜an d¯iˆe` u kiˆe.n Kuhn-Tucker
L(x, µ0, . . . , µm) L(x∗, µ0, . . . , µm) = min x∈S
v`a
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m. Nˆe´u d¯iˆe` u kiˆe.n Slater (1.1.7) tho˙’a m˜an, th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1. Ngo`ai ra, x∗ l`a cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ) trˆen D, nˆen theo D- i.nh l´y 3.3.18 ta suy ra x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D.
(b) V`ı c´ac gia˙’ thiˆe´t cu˙’a d¯i.nh l´y c˜ung l`a c´ac gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.14 Chu.o.ng 2, nˆen tˆo`n ta. i ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). Mˇa. t kh´ac, theo D- i.nh l´y 1.1.5 th`ı x∗ l`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ) trˆen D. Do d¯´o, ´ap du. ng D- i.nh l´y 3.3.18 ta suy ra
˜x∗ ∈ (x∗+ ∈ 0.5M (2s)) ∩ D.
. c ch´u.ng minh. D- i.nh l´y d¯u.o.
Mo.˙’ rˆo. ng D- i.nh l´y 2.4.15 l`a d¯i.nh l´y sau:
D- i.nh l´y 3.4.22. Gia˙’ su.˙’ D d¯u.o. . c cho bo.˙’ i cˆong th´u.c (2.4.17), gi : IRn → IR, i = 1, . . . , m, l`a c´ac h`am lˆo`i, c`ung liˆen tu. c ´ıt nhˆa´t ta. i mˆo. t d¯iˆe˙’m cu˙’a tˆa. p lˆo`i, d¯´ong S ⊂ IRn. Khi d¯´o
m (cid:88)
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) th`ı tˆo`n ta. i duy nhˆa´t x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D v`a c´ac nhˆan tu.˙’ Lagrange µ0 ≥ 0, µ1 ≥ 0, . . . , µm ≥ 0, ch´ung khˆong c`ung triˆe. t tiˆeu tho˙’a m˜an
i=1
(3.4.28) 0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗) + N (x∗|S)
v`a
i = 1, . . . , m, (3.4.29) µigi(x∗) = 0 v´o.i mo. i
trong d¯´o N (x∗|S) l`a n´on ph´ap tuyˆe´n cu˙’a S ta. i x∗. Nˆe´u d¯iˆe` u kiˆe. n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1. (b) Nˆe´u c´o x∗ ∈ D tho˙’a m˜an (3.4.28) v`a (3.4.29) v´o.i µ0 = 1 th`ı tˆo`n ta. i ˜x∗ ∈ (x∗ + 0.5M (2s)) ∩ D l`a d¯iˆe˙’m infimum to`an cu. c duy nhˆa´t cu˙’a B`ai to´an ( ˜P ).
m (cid:88)
Ch´u.ng minh. (a) V`ı gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.15 c˜ung l`a gia˙’ thiˆe´t cu˙’a d¯i.nh l´y n`ay, nˆen tˆo`n ta. i x∗ ∈ D v`a c´ac nhˆan tu.˙’ Lagrange µi, i = 0, . . . , m, tho˙’a m˜an c´ac d¯iˆe` u kiˆe.n
i=1
0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗) + N (x∗|S)
v`a
µigi(x∗) = 0 v´o.i mo. i i = 1, . . . , m, trong d¯´o N (x∗|S) l`a n´on ph´ap tuyˆe´n cu˙’a S ta. i x∗. D- ˆo`ng th`o.i, theo D- i.nh l´y Kuhn-Tucker cho B`ai to´an (P ) ta c˜ung suy ra x∗ l`a cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a B`ai to´an (P ). ´Ap du. ng D- i.nh l´y 3.3.18 cho c´ac d¯iˆe˙’m ˜x∗ v`a x∗ ta nhˆa. n d¯u.o. . c
x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D.
Ta c˜ung suy ra, nˆe´u d¯iˆe` u kiˆe.n Slater (1.1.7) tho˙’a m˜an th`ı µ0 (cid:54)= 0 v`a c´o thˆe˙’ coi µ0 = 1.
(b) C´ac d¯iˆe` u kiˆe.n (3.4.28) v`a (3.4.29) tho˙’a m˜an (b) cu˙’a D- i.nh l´y 2.4.15, nˆen tˆo`n ta. i ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). Mˇa. t kh´ac c´ac d¯iˆe` u kiˆe.n (3.4.28) v`a (3.4.29) tho˙’a m˜an D- i.nh l´y Kuhn-Tucker cho B`ai to´an (P ) nˆen x∗ l`a cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t f. ´Ap du. ng D- i.nh l´y 3.3.18 cho c´ac d¯iˆe˙’m ˜x∗ v`a x∗ ta suy ra
˜x∗ ∈ (x∗ + 0.5M (2s)) ∩ D.
. c ch´u.ng minh. D- i.nh l´y d¯u.o.
Nhˆa. n x´et 3.4.8. (a) Nˆe´u S = IRn th`ı khi d¯´o N (x∗|S) = {0}, nˆen biˆe˙’u
m (cid:88)
th´u.c (3.4.28) d¯u.o. . c thay bo.˙’ i
i=1
0 ∈ µ0(2Ax∗ + b) + µi∂gi(x∗).
m (cid:88)
(b) Nˆe´u gi, i = 1, . . . , m lˆo`i, kha˙’ vi liˆen tu. c th`ı biˆe˙’u th´u.c trˆen c´o da. ng
i=1
0 = µ0(2Ax∗ + b) + µi(cid:53)gi(x∗).
Khi D l`a tˆa. p lˆo`i d¯a diˆe.n ta c´o d¯i.nh l´y sau:
. c cho bo.˙’ i (2.4.18). Khi d¯´o D- i.nh l´y 3.4.23. Gia˙’ su.˙’ D d¯u.o.
m (cid:88)
(a) Nˆe´u ˜x∗ l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ) th`ı tˆo`n ta. i duy nhˆa´t x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D v`a c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 1, . . . , m, sao cho
i=1
(2Ax∗ + b) + (3.4.30) µici = 0,
i = 1, . . . , m. (3.4.31) µi((cid:104)ci, x∗(cid:105) − di) = 0 v´o.i mo. i
m (cid:88)
Ho.n thˆe´ n˜u.a ta c´o
i=1
2A˜x∗ + b + µici ∈ AM (2s).
(b) Nˆe´u c´o x∗ ∈ D tho˙’a m˜an (3.4.30), (3.4.31) th`ı tˆo`n ta. i ˜x∗ ∈ (x∗ +
0.5M (2s)) ∩ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ).
i=1
Ch´u.ng minh. (a) Gia˙’ thiˆe´t cu˙’a d¯i.nh l´y c˜ung l`a gia˙’ thiˆe´t cu˙’a D- i.nh l´y 2.4.16, do d¯´o tˆo`n ta. i duy nhˆa´t x∗ c`ung c´ac nhˆan tu.˙’ Lagrange µi ≥ 0, i = 0, . . . , m, sao cho (cid:88) (2Ax∗ + b) + µici = 0
v`a
µi((cid:104)ci, x∗(cid:105) − di) = 0 v´o.i mo. i i = 1, . . . , m.
. c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a B`ai to´an (P ) nˆen
Mˇa. t kh´ac x∗ l`a d¯iˆe˙’m cu. theo D- i.nh l´y 3.3.18 suy ra
x∗ ∈ (˜x∗ + 0.5M (2s)) ∩ D.
m (cid:88)
m (cid:88)
Ta la. i c´o
i=1
i=1
2A˜x∗ + b + µici = 2Ax∗ + b + µici + 2A˜x∗ − 2Ax∗
= 2A(˜x∗ − x∗) ∈ AM (2s). (3.4.32)
(b) C´ac d¯iˆe` u kiˆe.n (3.4.30) v`a (3.4.31) tho˙’a m˜an (b) cu˙’a D- i.nh l´y 2.4.16, do d¯´o tˆo`n ta. i ˜x∗ ∈ D l`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ). V`ı x∗ l`a cu. . c tiˆe˙’u to`an cu. c duy nhˆa´t cu˙’a B`ai to´an (P ) nˆen theo D- i.nh l´y 3.3.18 suy ra mo. i d¯iˆe˙’m infimum to`an cu. c ˜x∗ ∈ (x∗ + 0.5M (2s)) ∩ D. D- i.nh l´y d¯u.o. . c ch´u.ng minh.
. c tiˆe˙’u (Γ-infimum) l`a d¯iˆe˙’m cu.
. c nh˜u.ng vˆa´n d¯ˆe` Kˆe´t luˆa. n: Trong chu.o.ng n`ay ch´ung tˆoi d¯˜a gia˙’i quyˆe´t d¯u.o. . c d¯ˇa. t ra o.˙’ d¯ˆa` u chu.o.ng l`a: chı˙’ ra c´ac d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am bi. co. ba˙’n d¯u.o. nhiˆe˜u gi´o.i nˆo. i ˜f = f +p l`a Γ-lˆo`i ngo`ai (c´ac Mˆe.nh d¯ˆe` 3.1.18 – 3.1.19); ch´u.ng minh d¯iˆe˙’m Γ-cu. . c tiˆe˙’u to`an cu. c (infimum to`an cu. c) khi Γ = M (2s) (c´ac Mˆe.nh d¯ˆe` 3.2.21 – 3.2.22); x´ac lˆa. p d¯u.o. . c quan hˆe. gi˜u.a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an (P ) v`a d¯iˆe˙’m infimum to`an cu. c cu˙’a B`ai to´an ( ˜P ), d¯ˆo`ng th`o.i ch´u.ng minh d¯u.o. . c t´ınh ˆo˙’n d¯i.nh nghiˆe.m theo khoa˙’ng c´ach Hausdorff (c´ac d¯i.nh l´y 3.3.18, 3.4.20); tr`ınh b`ay du.´o.i vi phˆan suy rˆo. ng thˆo cu˙’a h`am ˜f = f + p v`a chı˙’ ra c´ac d¯iˆe` u kiˆe.n tˆo´i u.u cu˙’a B`ai to´an ( ˜P ) (c´ac d¯i.nh l´y 3.4.21 – 3.4.23).
. . CHU NG 4 O
D- IˆE˙’M SUPREMUM CU˙’ A B `AI TO ´AN ( ˜Q)
B`ai to´an d¯u.o. . c x´et trong chu.o.ng n`ay l`a
˜f (x) := f (x) + p(x) → sup, x ∈ D, ( ˜Q)
trong d¯´o D l`a tˆa. p lˆo`i, f tho˙’a m˜an cˆong th´u.c (1.0.1), t´u.c l`a f = (cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105), A ∈ IRn×n l`a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng, p l`a nhiˆe˜u gi´o.i nˆo. i, t´u.c l`a
|p(x)| ≤ s < +∞. sup x∈D
Trong chu.o.ng n`ay ch´ung tˆoi nghiˆen c´u.u t´ınh γ-lˆo`i trong cu˙’a h`am bi. nhiˆe˜u ˜f = f + p; mˆo. t sˆo´ t´ınh chˆa´t cu˙’a d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q); t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c v`a t´ınh ˆo˙’n d¯i.nh cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q) theo nhiˆe˜u p.
Trong mu. c n`ay, ta tr`ınh b`ay mˆo. t sˆo´ d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p l`a γ-lˆo`i trong. D- ´o l`a c´ac mˆe.nh d¯ˆe` sau: Mˆe. nh d¯ˆe` 4.1.23. Cho γ > 0, f, p x´ac d¯i.nh theo cˆong th´u.c (1.0.1) v`a (1.0.2), tu.o.ng ´u.ng. Khi d¯´o
(a) Nˆe´u supx∈D |p(x)| ≤ λminγ2/2, th`ı ˜f = f + p l`a γ-lˆo`i trong.
(a) Nˆe´u supx∈D |p(x)| < λminγ2/2, th`ı ˜f = f + p l`a γ-lˆo`i trong ngˇa. t.
Ch´u.ng minh. X´et bˆa´t k`y x0, x1 ∈ D tho˙’a m˜an (cid:107)x0 − x1(cid:107) = γ, −x0 + 2x1 ∈ D, ta c´o
f (x0) − 2f (x1) + f (−x0 + 2x1) = (cid:104)Ax0, x0(cid:105) + (cid:104)b, x0(cid:105) − 2(cid:104)Ax1, x1(cid:105) − 2(cid:104)b, x1(cid:105)
+(cid:104)A(−x0 + 2x1), (−x0 + 2x1)(cid:105) + (cid:104)b, −x0 + 2x1(cid:105)
= 2(cid:104)Ax0, x0(cid:105) − 4(cid:104)Ax0, x1(cid:105) + 2(cid:104)Ax1, x1(cid:105) = 2(cid:104)A(x0 − x1), x0 − x1(cid:105).
n (cid:88)
n (cid:88)
Khˆong gia˙’m t´ınh tˆo˙’ng qu´at ta go. i λi, i = 1, . . . , n, l`a c´ac gi´a tri. riˆeng cu˙’a ma trˆa. n d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng A (c´o thˆe˙’ c´o mˆo. t sˆo´ gi´a tri. tr`ung nhau), . c chuˆa˙’n trong IRn v`a d¯ˆo`ng th`o.i ei l`a v´ec {ei | i = 1, 2, . . . , n} l`a co. so.˙’ tru. to. riˆeng ´u.ng v´o.i gi´a tri. riˆeng λi, i = 1, 2 . . . , n. Khi d¯´o
i=1
i=1
x0 = ζ i 0ei, x1 = ζ i 1ei
n (cid:88)
n (cid:88)
v`a
0 − ζ i
1)ei,
0 − ζ j
1)ej(cid:105)
j=1
i=1 n (cid:88)
n (cid:88)
(ζ j 2(cid:104)A(x0 − x1), x0 − x1(cid:105) = 2(cid:104) λi(ζ i
0 − ζ i
1)(ζ j
0 − ζ j
1)(cid:104)ei, ej(cid:105)
j=1
i=1 n (cid:88)
= 2 λi(ζ i
0 − ζ i
1)2
i=1
= 2 λi(ζ i
≥ 2λmin(cid:107)x0 − x1(cid:107)2.
T`u. biˆe˙’u th´u.c cuˆo´i v`a d¯i.nh ngh˜ıa h`am h2(γ) trong Mˆe.nh d¯ˆe` 1.5.7 ta
nhˆa. n d¯u.o. . c
h2(γ) := inf x0,x1∈D, (cid:107)x0−x1(cid:107)=γ,−x0+2x1∈D
(cid:0)f (x0)−2f (x1)+f (−x0+2x1)(cid:1) ≥ 2λminγ2. (4.1.1)
Theo gia˙’ thiˆe´t (a) v`a biˆe˙’u th´u.c (4.1.1) th`ı
|p(x)| ≤ h2(γ)/4 v´o.i mo. i x ∈ D, nˆen ´ap du. ng Mˆe.nh d¯ˆe` 1.5.7 ta suy ra ˜f l`a γ-lˆo`i trong.
Tu.o.ng tu. . theo gia˙’ thiˆe´t (b) v`a biˆe˙’u th´u.c (4.1.1) ta c˜ung c´o
|p(x)| < h2(γ)/4 v´o.i mo. i x ∈ D.
Vˆa. y (a), (b) d¯u.o. ´Ap du. ng Mˆe.nh d¯ˆe` 1.5.7 ta suy ra ˜f l`a γ-lˆo`i trong ngˇa. t. . c ch´u.ng minh.
Mˆe. nh d¯ˆe` 4.1.24. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1), p tho˙’a m˜an (1.0.2). Khi d¯´o h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i trong v´o.i γ ≥ (cid:112)2s/λmin v`a γ-lˆo`i trong ngˇa. t v´o.i γ > (cid:112)2s/λmin . Ch´u.ng minh. Ta thˆa´y v´o.i (cid:112)2s/λmin ≤ γ (hoˇa. c (cid:112)2s/λmin < γ) tu.o.ng d¯u.o.ng v´o.i
(cid:114) (cid:114) |p(x)|/λmin < γ), |p(x)|/λmin ≤ γ (hoˇa. c sup x∈D sup x∈D
t´u.c l`a
|p(x)| < λminγ2/2). sup x∈D
´Ap du. ng Mˆe.nh d¯ˆe` 4.1.23 ta suy ra d¯u.o. |p(x)| ≤ λminγ2/2 (hoˇa. c sup x∈D . c mˆe.nh d¯ˆe` trˆen.
Trong mu. c n`ay ch´ung tˆoi nghiˆen c´u.u mˆo. t sˆo´ t´ınh chˆa´t cu˙’a d¯iˆe˙’m cu. . c d¯a. i to`an cu. c v`a supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p. Cu. thˆe˙’ l`a chı˙’ ra vi. tr´ı cu˙’a c´ac d¯iˆe˙’m cu. . c d¯a. i to`an cu. c, supremum to`an cu. c v`a mˆo. t sˆo´ d¯iˆe` u kiˆe.n tˆo`n ta. i d¯iˆe˙’m cu. . c d¯a. i to`an cu. c, d¯iˆe˙’m supremum to`an cu. c cu˙’a l´o.p h`am n`ay trˆen tˆa. p lˆo`i D.
. ng du. ng c´ac d¯i.nh l´y 1.5.10 v`a 1.5.11 cho l´o.p h`am to`an phu.o.ng bi. ´U
. c c´ac mˆe.nh d¯ˆe` sau:
. c d¯a. i, th`ı n´o d¯a. t cu.
nhiˆe˜u ta nhˆa. n d¯u.o. Mˆe. nh d¯ˆe` 4.2.25. Cho f : D ⊂ IRn → IR x´ac d¯i.nh theo cˆong th´u.c (1.0.1). Nˆe´u h`am bi. nhiˆe˜u ˜f = f + p d¯a. t gi´a tri. cu. . c d¯a. i to`an . c biˆen ngˇa. t n`ao d¯´o cu˙’a D, v´o.i γ = (cid:112)2s/λmin . cu. c ta. i mˆo. t sˆo´ d¯iˆe˙’m γ-cu.
Ch´u.ng minh. Tru.´o.c tiˆen, v´o.i c´ac gia˙’ thiˆe´t trˆen ta ch´u.ng minh D gi´o.i nˆo. i. Gia˙’ su.˙’ D khˆong gi´o.i nˆo. i, t´u.c l`a tˆo`n ta. i d˜ay (xk) ⊂ D sao cho limk→+∞ (cid:107)xk(cid:107) = +∞. Ta c´o
|p(x)|, | ˜f (x)| = |(cid:104)Ax, x(cid:105) + (cid:104)b, x(cid:105) + p(x)| ≥ (cid:104)Ax, x(cid:105) − (cid:107)b(cid:107)(cid:107)x(cid:107) − sup x∈D
v`a (cid:104)Ax, x(cid:105) ≥ λmin(cid:107)x(cid:107)2 do A l`a d¯ˆo´i x´u.ng x´ac d¯i.nh du.o.ng, nˆen
| ˜f (x)| ≥ λmin(cid:107)x(cid:107)2 − (cid:107)b(cid:107)(cid:107)x(cid:107) − s.
Biˆe˙’u th´u.c cuˆo´i cho ta limk→+∞ | ˜f (xk)| = +∞, d¯iˆe` u n`ay tr´ai v´o.i gia˙’ thiˆe´t tˆo`n ta. i gi´a tri. cu. . c d¯a. i to`an cu. c cu˙’a h`am ˜f . Mˇa. t kh´ac, theo Mˆe.nh d¯ˆe` 4.1.24 th`ı ˜f = f + p l`a γ-lˆo`i trong v´o.i γ = (cid:112)2s/λmin . ´Ap du. ng D- i.nh l´y 1.5.10, ta suy ra d¯iˆe` u cˆa` n ch´u.ng minh. Mˆe. nh d¯ˆe` 4.2.26. Cho f : D ⊂ IRn → IR x´ac d¯i.nh theo cˆong th´u.c (1.0.1). Khi d¯´o nˆe´u h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p d¯a. t gi´a tri. cu. . c d¯a. i to`an cu. c ta. i nh˜u.ng d¯iˆe˙’m γ-cu. . c d¯a. i to`an cu. c trˆen D, th`ı chı˙’ c´o thˆe˙’ d¯a. t cu. . c biˆen cu˙’a D, v´o.i γ = (cid:112)2s/λmin .
Ch´u.ng minh. Ta ch´u.ng minh bˇa`ng pha˙’n ch´u.ng. Gia˙’ su.˙’ , tˆo`n ta. i y ∈ D l`a . c d¯a. i to`an cu. c cu˙’a ˜f = f + p, nhu.ng y khˆong pha˙’i l`a d¯iˆe˙’m γ-cu. d¯iˆe˙’m cu. . c biˆen v´o.i γ = (cid:112)2s/λmin , t´u.c l`a
. c ch´u.ng minh. ∃y(cid:48), y(cid:48)(cid:48) ∈ D : y = 0.5(y(cid:48) + y(cid:48)(cid:48)), (cid:107)y(cid:48) − y(cid:48)(cid:48)(cid:107) > 2(cid:112)2s/λmin. Cho. n (cid:15) > 0 sao cho 2γ1 := (cid:107)y(cid:48) − y(cid:48)(cid:48)(cid:107) − 2(cid:15) > 2(cid:112)2s/λmin . Theo Mˆe.nh d¯ˆe` 3.1.19 th`ı h`am to`an phu.o.ng bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i trong ngˇa. t v´o.i γ = γ1, nˆen ˜f = f + p tho˙’a m˜an D- i.nh l´y 1.5.11. V`ı vˆa. y, d¯iˆe˙’m cu. . c d¯a. i to`an . c biˆen ngˇa. t v´o.i γ = γ1, t´u.c l`a cu. c y cu˙’a ˜f = f + p chı˙’ c´o thˆe˙’ l`a d¯iˆe˙’m γ-cu. t`u. y = 0.5(y(cid:48) + y(cid:48)(cid:48)) suy ra (cid:107)y(cid:48) − y(cid:48)(cid:48)(cid:107) < 2γ1. D- iˆe` u nhˆa. n d¯u.o. . c l`a mˆau thuˆa˜n v´o.i gia˙’ thiˆe´t (cid:107)y(cid:48) − y(cid:48)(cid:48)(cid:107) > 2γ1. Do d¯´o mˆe.nh d¯ˆe` d¯˜a d¯u.o.
l´o.p h`am n`ay, su. .˙’ mu. c tru.´o.c ta d¯˜a x´et c´ac d¯iˆe˙’m cu. O . tˆo`n ta. i d¯iˆe˙’m cu. . c d¯a. i cu˙’a h`am γ-lˆo`i trong. D- ˆo´i v´o.i . c d¯a. i to`an cu. c n´oi chung khˆong ba˙’o d¯a˙’m,
. c d¯a. i to`an cu. c d¯u.o. . c mo.˙’ rˆo. ng ra nhu. sau:
. c go. i l`a d¯iˆe˙’m supremum to`an cu. c
thˆa. m ch´ı khi h`am l`a γ-lˆo`i trong x´ac d¯i.nh trˆen tˆa. p compact. V`ı vˆa. y kh´ai niˆe.m d¯iˆe˙’m cu. D- i.nh ngh˜ıa 4.2.11. ([43]) x∗ ∈ D d¯u.o. cu˙’a g : D ⊂ IRn → IR nˆe´u
g(y) ≥ g(x) v´o.i mo. i x ∈ D. lim sup y→x∗, y∈D
. c go. i l`a d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a
D- i.nh ngh˜ıa 4.2.12. x∗ ∈ D d¯u.o. g : D ⊂ IRn → IR nˆe´u
∃δ > 0 : g(y) ≥ g(x) v´o.i mo. i x ∈ ¯B(x∗, δ) ∩ D. lim sup y→x∗, y∈D
V´o.i d¯i.nh ngh˜ıa trˆen ta thˆa´y mˆo˜i d¯iˆe˙’m cu.
. c d¯a. i to`an cu. c l`a d¯iˆe˙’m supremum to`an cu. c, nhu.ng ngu.o. . c la. i th`ı khˆong, d¯iˆe` u d¯´o c´o thˆe˙’ thˆa´y qua v´ı du. d¯o.n gia˙’n g(x) = x − [x] x´ac d¯i.nh trˆen IR, [x] l`a phˆa` n nguyˆen cu˙’a x. D- ˆe˙’ nghiˆen c´u.u c´ac d¯iˆe˙’m supremum to`an cu. c ta s˜e su.˙’ du. ng mˆo. t sˆo´ t´ınh . c d¯i.nh ngh˜ıa nhu. chˆa´t cu˙’a h`am bao d¯´ong nu.˙’a liˆen tu. c trˆen, h`am d¯´o d¯u.o. sau:
D- i.nh ngh˜ıa 4.2.13. (xem [43]) Cho g : D ⊂ IRn → IR, h`am usc g x´ac d¯i.nh nhu. sau
g(y), usc g(x) := lim sup y→x, y∈D
. c go. i l`a h`am bao d¯´ong nu.˙’a liˆen tu. c trˆen cu˙’a g trˆen D.
d¯u.o. Bˆo˙’ d¯ˆe` 4.2.4. Ta c´o c´ac t´ınh chˆa´t sau cu˙’a h`am bao d¯´ong nu.˙’a liˆen tu. c trˆen:
(a) x∗ ∈ D l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a g trˆen D khi v`a chı˙’ khi x∗
l`a d¯iˆe˙’m cu. . c d¯a. i to`an cu. c cu˙’a h`am usc g trˆen D.
x∗ l`a d¯iˆe˙’m cu.
(b) x∗ ∈ D l`a d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a g trˆen D khi v`a chı˙’ khi . c d¯a. i d¯i.a phu.o.ng cu˙’a h`am usc g trˆen D. (c) Nˆe´u ˜f = f + p th`ı usc ˜f = f + usc p v`a usc ˜f l`a γ-lˆo`i trong v´o.i
γ ≥ (cid:112)2s/λmin v`a γ-lˆo`i trong ngˇa. t v´o.i γ > (cid:112)2s/λmin .
Ch´u.ng minh. (a) Gia˙’ su.˙’ x∗ l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a g trˆen D. Khi d¯´o
g(y) ≥ g(x) v´o.i mo. i x ∈ D. usc g(x∗) = lim sup y→x∗, y∈D
Mˇa. t kh´ac, v´o.i x ∈ D th`ı
g(y), usc g(x) = lim sup y→x, y∈D
nˆen
usc g(x) ≤ usc g(x∗) v´o.i mo. i x ∈ D.
Do d¯´o suy ra x∗ l`a d¯iˆe˙’m d¯a. t cu.
Ngu.o. . c la. i, nˆe´u x∗ l`a d¯iˆe˙’m d¯a. t cu. . c d¯a. i to`an cu. c cu˙’a usc g(x) trˆen D. . c d¯a. i cu˙’a usc g, t´u.c l`a
usc g(x∗) ≥ usc g(x) v´o.i mo. i x ∈ D,
th`ı
g(x) ≥ usc g(x) ≥ g(x) v´o.i mo. i x ∈ D. usc g(x∗) = lim sup x→x∗, x∈D
Ta suy ra x∗ l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a g trˆen D.
(b) Ch´u.ng minh tu.o.ng tu. . (a).
(c) Lˆa´y x ∈ D, ta c´o
usc ˜f (x) = lim supy→x, y∈D
˜f (yn)}
˜f (y) = sup{η : ∃yn → x, yn ∈ D, η = limn→+∞ = f (x) + sup{η(cid:48) : ∃yn → x, yn ∈ D, η(cid:48) = limn→+∞ p(yn)},
nˆen
usc ˜f (x) = f (x) + usc p(x).
Mˇa. t kh´ac, v´o.i mo. i x ∈ D th`ı
p(y)| |usc p(x)| = | lim sup y→x, y∈D
|p(y)| p(y)| = | lim δ(cid:38)0 ≤ lim δ(cid:38)0 sup y∈B(x,δ) sup y∈B(x,δ)
|p(y)|
sup y∈D s ≤ lim δ(cid:38)0 ≤ lim δ(cid:38)0 = s.
Do d¯´o, ´ap du. ng Mˆe.nh d¯ˆe` 4.1.24 cho h`am usc ˜f = f + usc p ta suy ra d¯iˆe` u pha˙’i ch´u.ng minh.
Nhˆa. n x´et 4.2.9. N´oi chung t`u. mˆo. t h`am l`a γ-lˆo`i trong khˆong suy ra h`am bao d¯´ong nu.˙’a liˆen tu. c trˆen cu˙’a n´o l`a γ-lˆo`i trong (xem V´ı du. 2.4 [43]). Hˆe. qua˙’ 4.2.2. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1). Nˆe´u ˜f = f + p c´o d¯iˆe˙’m supremum to`an cu. c trˆen D, th`ı tˆo`n ta. i d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p l`a d¯iˆe˙’m γ-cu. . c biˆen ngˇa. t, v´o.i γ = (cid:112)2s/λmin .
. c d¯a. i to`an cu. c l`a d¯iˆe˙’m γ-cu.
Ch´u.ng minh. Nˆe´u h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u ˜f = f + p c´o d¯iˆe˙’m supremum to`an cu. c trˆen D, th`ı theo Bˆo˙’ d¯ˆe` 4.2.4, d¯iˆe˙’m d¯´o l`a d¯iˆe˙’m cu. . c d¯a. i to`an cu. c cu˙’a h`am usc ˜f = f + usc p. V`ı vˆa. y, t`u. Mˆe.nh d¯ˆe` 1.3.1 suy ra h`am usc ˜f = f + usc p c´o d¯iˆe˙’m cu. . c biˆen ngˇa. t v´o.i γ = (cid:112)2s/λmin . ´Ap du. ng la. i (c) cu˙’a Bˆo˙’ d¯ˆe` 4.2.4 ta suy ra . c biˆen ngˇa. t v´o.i ˜f = f + p c´o mˆo. t sˆo´ d¯iˆe˙’m supremum to`an cu. c l`a d¯iˆe˙’m γ-cu. γ = (cid:112)2s/λmin .
Hˆe. qua˙’ 4.2.3. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1). Khi d¯´o h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p chı˙’ d¯a. t supremum to`an cu. c . c biˆen cu˙’a D, v´o.i γ = (cid:112)2s/λmin . ta. i nh˜u.ng d¯iˆe˙’m γ-cu.
Ch´u.ng minh. Theo Bˆo˙’ d¯ˆe` 4.2.4, nˆe´u ˜f = f + p c´o d¯iˆe˙’m supremum to`an cu. c th`ı d¯iˆe˙’m d¯´o l`a d¯iˆe˙’m cu. . c d¯a. i cu˙’a h`am usc ˜f = f + usc p v`a h`am usc ˜f = f + usc p l`a γ-lˆo`i trong v´o.i γ = (cid:112)2s/λmin . Do d¯´o ´ap du. ng Mˆe.nh d¯ˆe` 2.1.10 ta suy ra, tˆa. p c´ac d¯iˆe˙’m cu. . c d¯a. i cu˙’a usc ˜f = f + usc p, chı˙’ c´o thˆe˙’ l`a c´ac d¯iˆe˙’m γ-cu. . c biˆen cu˙’a D. ´Ap du. ng la. i (c) Bˆo˙’ d¯ˆe` 4.2.4 ta suy ra c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p chı˙’ c´o thˆe˙’ l`a c´ac d¯iˆe˙’m γ-cu. . c biˆen v´o.i γ = (cid:112)2s/λmin .
Hˆe. qua˙’ 4.2.4. Cho f x´ac d¯i.nh theo cˆong th´u.c (1.0.1), D l`a tˆa. p compact. Nˆe´u h`am nhiˆe˜u ˜f = f + p c´o d¯iˆe˙’m supremum to`an cu. c trˆen D th`ı c´o ´ıt nhˆa´t mˆo. t d¯iˆe˙’m l`a d¯iˆe˙’m biˆen tu.o.ng d¯ˆo´i D theo aff D trong d¯´o aff D l`a bao . c biˆen cu˙’a D v´o.i γ = (cid:112)2s/λmin , tuyˆe´n t´ınh cu˙’a tˆa. p D, hoˇa. c l`a d¯iˆe˙’m γ-cu. l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p trˆen D.
Ch´u.ng minh. Theo Mˆe.nh d¯ˆe` 4.1.24 th`ı ˜f = f + p l`a γ-lˆo`i trong v´o.i γ = (cid:112)2s/λmin . Nˆe´u D l`a compact, th`ı suy ra h`am ˜f = f + p bi. chˇa. n trˆen trˆen D. Theo Hˆe. qua˙’ 1.5.1 suy ra tˆo`n ta. i d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p trˆen D. T`u. D- i.nh l´y 2.4 [43] suy ra, c´o ´ıt nhˆa´t mˆo. t d¯iˆe˙’m supremum l`a d¯iˆe˙’m biˆen tu.o.ng d¯ˆo´i D theo aff D hoˇa. c l`a d¯iˆe˙’m γ-cu. . c biˆen cu˙’a D v´o.i γ = (cid:112)2s/λmin .
V´ı du. 4.2.9. Cho
f (x) = x2, x ∈ [−1, 1], (cid:40) −0.5 nˆe´u x ∈ [−1, −0.5] ∪ [0.5, 1] p(x) = 0.5 − 2x2 nˆe´u x ∈ ] − 0.5, 0.5[ .
Khi d¯´o
x∈[−1,1]
s = sup |p(x)| = 0.5, λmin = 1, γ = (cid:112)2s/λmin = 1,
v`a
x0 = −1, x1 = 0, x2 = −x0 + 2x1 = 1, l`a c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p, trong d¯´o x0, x1 l`a d¯iˆe˙’m . c biˆen v´o.i γ = 1. V´ı du. trˆen cho thˆa´y, . c biˆen, c`on x1 l`a d¯iˆe˙’m γ-cu. cu. c´o thˆe˙’ tˆo`n ta. i d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t . c biˆen ngˇa. t v´o.i bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p khˆong pha˙’i l`a d¯iˆe˙’m γ-cu. γ = (cid:112)2s/λmin. V´ı du. c˜ung cho thˆa´y, nˆe´u x0, x1 ∈ D, (cid:107)x0 − x1(cid:107) = γ th`ı c´o thˆe˙’ tˆa. p {−x0 + 2x1 | −x0 + 2x1 ∈ D} (cid:54)= ∅.
Mˆe. nh d¯ˆe` 4.2.27. Cho γ > 0, x1 ∈ D l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p v`a supx∈D |p(x)| ≤
λminγ2/2. Khi d¯´o nˆe´u x0 ∈ D, (cid:107)x0 − x1(cid:107) = γ, −x0 + 2x1 ∈ D th`ı x0 v`a −x0 + 2x1 c˜ung l`a c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p.
Ch´u.ng minh. V`ı supx∈D |p(x)| ≤ λminγ2/2, nˆen ´ap du. ng Mˆe.nh d¯ˆe` 2.1.9 ta suy ra ˜f = f + p l`a γ-lˆo`i trong. V`ı (cid:107)x0 − x1(cid:107) = γ v`a −x0 + 2x1 ∈ D, suy ra
f (x0) − 2f (x1) − f (−x0 + 2x1) = 2(cid:104)A(x0 − x1), (x0 − x1)(cid:105)
(4.2.2) ≥ 2λminγ2.
Do d¯´o
(cid:16) (cid:17) (cid:16) (cid:17) f (x0) − 2f (x1) + f (−x0 + 2x1) = − 2 f (x1) + usc p(x1)
f (x0) + usc p(x0) (cid:16) (cid:17) + f (−x0 + 2x1) + usc p(−x0 + 2x1)
−usc p(x0) + 2usc p(x1) − usc p(−x0 + 2x1) = usc ˜f (x0) − 2usc ˜f (x1) + usc ˜f (−x0 + 2x1)
−usc p(x0) + 2usc p(x1) − usc p(−x0 + 2x1).
Chuyˆe˙’n vˆe´ biˆe˙’u th´u.c cuˆo´i v`a kˆe´t ho. . p v´o.i (4.2.2) ta suy ra
|usc p(x)| usc ˜f (x0) − 2usc ˜f (x1) + usc ˜f (−x0 + 2x1) ≥ 2λminγ2 − 4 sup x∈D
≥ 2λminγ2 − 2λminγ2 = 0.
Do d¯´o
usc ˜f (x0) − 2usc ˜f (x1) + usc ˜f (−x0 + 2x1) ≥ 0,
t´u.c l`a
usc ˜f (x0) + usc ˜f (−x0 + 2x1) ≥ 2usc ˜f (x1).
. c d¯a. i to`an cu. c cu˙’a h`am usc ˜f nˆen t`u. bˆa´t d¯ˇa˙’ ng th´u.c . c d¯a. i to`an cu. c cu˙’a usc ˜f v`a v`ı vˆa. y x0, −x0 +2x1
V`ı x1 l`a c´ac gi´a tri. cu. trˆen suy ra x0, −x0 +2x1 cu. l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p trˆen D.
Mˆe. nh d¯ˆe` 4.2.28. Cho x1 l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p, trong d¯´o supx∈D |p(x)| ≤ s < +∞. Nˆe´u x0 ∈ D v`a −x0 + 2x1 tho˙’a m˜an
(cid:107)x0 − x1(cid:107) = (cid:112)2s/λmin , −x0 + 2x1 ∈ D
th`ı x0 v`a −x0 + 2x1 c˜ung l`a c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p.
Ch´u.ng minh. Theo Mˆe.nh d¯ˆe` 4.1.24 th`ı ˜f = f + p l`a γ-lˆo`i trong v´o.i γ = (cid:112)2s/λmin , nˆen tho˙’a m˜an Mˆe.nh d¯ˆe` 4.2.27, do d¯´o ta suy ra d¯iˆe` u pha˙’i ch´u.ng minh.
Trong mu. c n`ay, ch´ung tˆoi nghiˆen c´u.u quan hˆe. gi˜u.a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q) v´o.i tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an (Q), khi D l`a tˆa. p lˆo`i d¯a diˆe.n, t´u.c l`a
D := {x ∈ IRn | (cid:104)ci, x(cid:105) ≤ di, ci ∈ IRn, i = 1, . . . , m}.
Bˇa´t d¯ˆa` u t`u. phˆa` n n`ay cu˙’a chu.o.ng ta su.˙’ du. ng mˆo. t sˆo´ k´y hiˆe.u sau:
. c biˆen cu˙’a tˆa. p lˆo`i d¯a diˆe.n D}.
(cid:107)x − y(cid:107). ext D := {x∗ | x∗ l`a d¯iˆe˙’m cu. JD(x∗) := ext D \ {x∗}, x∗ ∈ ext D. d(x, D) := inf y∈D
x∗∈ext D
{d(cid:0)x∗, conv JD(x∗)(cid:1)}. dD := min
D(x∗, β) := {x ∈ D | x = (1 − α)x∗ + αy, y ∈ D,
0 ≤ α ≤ 1 − β}, x∗ ∈ ext D, β ∈ [0, 1].
v`a |ext D| l`a sˆo´ d¯iˆe˙’m cu. . c biˆen cu˙’a D.
Ta c´o bˆo˙’ d¯ˆe` sau:
Bˆo˙’ d¯ˆe` 4.3.5. Cho D ⊂ IRn l`a d¯a diˆe. n lˆo`i, khi d¯´o
x∗∈ext D
(cid:91) (a) Tˆo`n ta. i β0 > 0 d¯ˆe˙’ v´o.i mo. i β ∈ [ 0, β0] ta c´o D(x∗, β). D =
(b) Tˆo`n ta. i γ0 > 0 sao cho v´o.i mo. i γ ∈ [0, γ0] tˆa. p c´ac d¯iˆe˙’m γ-cu. . c biˆen
cu˙’a miˆe` n D nˇa`m trong tˆa. p
x∗∈ext D
(cid:91) ¯B(x∗, γ) ∩ D.
(c) Tˆo`n ta. i s0 > 0 sao cho v´o.i mo. i s ∈ [0, s0] tˆa. p c´ac d¯iˆe˙’m γ-cu. . c biˆen
cu˙’a D v´o.i γ = (cid:112)2s/λmin nˇa`m trong tˆa. p
x∗∈ext D Ch´u.ng minh. Tru.`o.ng ho. . p |ext D| = 1. Theo gia˙’ thiˆe´t D l`a d¯a diˆe.n lˆo`i nˆen D = ext D = {x∗}. Do d¯´o c´ac kˆe´t luˆa. n (a), (b), (c) l`a hiˆe˙’n nhiˆen. Ta x´et tru.`o.ng ho.
(cid:91) ¯B(x∗, (cid:112)2s/λmin ) ∩ D.
. p |ext D| ≥ 2.
(a) Nˆe´u β = 0 th`ı D = D(x∗, 0) v´o.i mo. i x∗ ∈ ext D nˆen
x∗∈ext D
(cid:91) D = D(x∗, 0). (4.3.3)
khˇa˙’ ng d¯i.nh (a) d¯´ung khi β = 0. X´et tru.`o.ng ho. D(x∗, β) ⊆ D v´o.i mo. i x∗ ∈ ext D v`a β > 0 nˆen (cid:83) . p β > 0. Ta thˆa´y rˇa`ng x∗∈ext D D(x∗, β) ⊆ D.
Ngu.o. . c la. i, ta ch´u.ng minh bˇa`ng pha˙’n ch´u.ng. Gia˙’ su.˙’ v´o.i mo. i β0 > 0,
x∗∈ext D
tˆo`n ta. i β ∈]0, β0] v`a x ∈ D sao cho (cid:91) x /∈ D(x∗, β),
khi d¯´o
y∗∈ext D α(y∗)y∗ = 1, sao cho
x /∈ D(x∗, β) v´o.i mo. i x∗ ∈ ext D. Cˆo´ d¯i.nh x∗ ∈ ext D, v`ı D l`a d¯a diˆe.n lˆo`i v`a x ∈ D nˆen x c´o thˆe˙’ biˆe˙’u diˆe˜n thˆong qua c´ac d¯iˆe˙’m cu. . c biˆen, t´u.c l`a tˆo`n ta. i α(y∗), y∗ ∈ ext D tho˙’a m˜an α(y∗) ≥ 0, (cid:80)
y∗∈ext D
y∗(cid:54)=x∗
(cid:88) (cid:88) x = α(y∗)y∗ = α(x∗)x∗ + α(y∗)y∗. (4.3.4)
Nˆe´u α(x∗) = 1 th`ı x ≡ x∗, suy ra x ∈ D(x∗, β). D- iˆe` u n`ay tr´ai v´o.i gia˙’ thiˆe´t, do d¯´o α(x∗) < 1. Biˆe˙’u th´u.c (4.3.4) c´o thˆe˙’ viˆe´t la. i nhu. sau
y∗∈ext D
(cid:88) x = α(y∗)y∗
y∗(cid:54)=x∗
= α(x∗)x∗ + (cid:0)1 − α(x∗)(cid:1) (cid:88) y∗ α(y∗) 1 − α(x∗)
= α(x∗)x∗ + (cid:0)1 − α(x∗)(cid:1)x(cid:48) = (cid:0)1 − α(x∗)(cid:1)x∗ + α(x∗)x(cid:48), (4.3.5)
trong d¯´o
y∗(cid:54)=x∗
(cid:88) α(x∗) := 1 − α(x∗), x(cid:48) := y∗ ∈ conv JD(x∗) ⊂ D. α(y∗) 1 − α(x∗)
V`ı x /∈ D(x∗, β) v´o.i mo. i x∗ ∈ ext D, nˆen t`u. d¯i.nh ngh˜ıa D(x∗, β) v`a (4.3.5) suy ra
1 − β < α(x∗) ≤ 1 v´o.i mo. i x∗ ∈ ext D.
Nˆe´u lˆa´y β0 < 1/|ext D| th`ı v´o.i mo. i β ∈ ] 0, β0] ta c´o
x∗∈ext D
(cid:88) α(x∗) ≥ |ext D| min x∗ ∈ ext D{α(x∗)} ≥ |ext D| − |ext D|β
> |ext D| − 1
≥ 1.
x∗∈ext D α(x∗) = 1. Do d¯´o kˆe´t ho.
. p v´o.i (4.3.3)
D- iˆe` u n`ay tr´ai v´o.i gia˙’ thiˆe´t (cid:80) ta suy ra (a).
(b) Lˆa´y β ∈ ]0, β0], khi d¯´o theo (a) th`ı
x∗∈ext D
(cid:91) D = D(x∗, β).
D- ˇa. t γ1 := dD − dDβ. Ta ch´u.ng minh, v´o.i γ ∈ ]0, γ1] v`a v´o.i mo. i x∗ ∈ ext D th`ı
¯B(x∗, γ) ∩ D ⊆ D(x∗, β).
Thˆa. t vˆa. y, lˆa´y x∗ ∈ ext D, gia˙’ su.˙’ x ∈ ¯B(x∗, γ) ∩ D. Khi d¯´o (cid:88) α(y∗)y∗ x =
y∗∈ext D = α(x∗)x∗ +
y∗(cid:54)=x∗, y∗∈ext D
(cid:88) α(y∗)y∗.
Nˆe´u α(x∗) = 1 th`ı x ≡ x∗, do d¯´o suy ra x ∈ D(x∗, β). Nˆe´u α(x∗) < 1, viˆe´t la. i biˆe˙’u th´u.c trˆen giˆo´ng nhu. o.˙’ (4.3.5), ta d¯u.o. . c
x = (cid:0)1 − α(x∗)(cid:1)x∗ + α(x∗)x(cid:48),
trong d¯´o x(cid:48) ∈ conv JD(x∗). Biˆe˙’u th´u.c trˆen c´o thˆe˙’ biˆe´n d¯ˆo˙’i th`anh
x∗ − x = α(x∗)(x∗ − x(cid:48)),
nˆen kˆe´t ho. . p v´o.i x ∈ ¯B(x∗, γ) ta suy ra
α(x∗)(cid:107)x∗ − x(cid:48)(cid:107) ≤ γ.
V`ı (cid:107)x∗ − x(cid:48)(cid:107) ≥ d nˆen α(x∗)d ≤ γ v`a do d¯´o
α(x∗) ≤ = ≤ 1 − β. γ d γ1 d
T`u. d¯i.nh ngh˜ıa D(x∗, β) v`a biˆe˙’u th´u.c n`ay ta suy ra x ∈ D(x∗, β), v`ı vˆa. y
(4.3.6) ¯B(x∗, γ) ∩ D ⊆ D(x∗, β), v´o.i mo. i x∗ ∈ ext D.
. c γ2 < γ1
x∗∈ext D
V`ı D = ∪x∗∈ext DD(x∗, β) nˆen t`u. (4.3.6) suy ra c´o thˆe˙’ cho. n d¯u.o. sao cho v´o.i mo. i γ ∈ [ 0, γ2] d¯ˆe˙’ (cid:91) (cid:0) ¯B(x∗, γ) ∩ D(cid:1) (cid:54)= ∅. D \ (4.3.7)
Thˆa. t vˆa. y, lˆa´y x ∈ D, x /∈ ext D, d¯ˇa. t t := minx∗∈ext D (cid:107)x − x∗(cid:107) > 0, khi d¯´o v´o.i mo. i γ < t v`a x∗ ∈ ext D th`ı x /∈ ¯B(x∗, γ). Do d¯´o ta c´o (4.3.7).
x∗∈ext D
(cid:0) ¯B(x∗, γ) ∩ D(cid:1) khˆong pha˙’i l`a d¯iˆe˙’m γ-cu. D- ˇa. t γ0 := min{γ1, γ2, βdD/2}. Ta ch´u.ng minh rˇa`ng nˆe´u γ ∈ [0, γ0] th`ı . c biˆen cu˙’a
mo. i x ∈ D \ (cid:83) miˆe` n D. Ta x´et c´ac tru.`o.ng ho. . p sau:
i) γ ∈]0, γ0]. Theo (a) x ∈ D\∪x∗∈ext D
(cid:0) ¯B(x∗, γ)∩D(cid:1) suy ra tˆo`n ta. i y∗ ∈ ext D sao cho x ∈ D(y∗, β). V`ı D, D(y∗, β) l`a c´ac tˆa. p compact nˆen go. i t0 := max {t | y∗ + t(x − y∗) ∈ D} v`a t1 := max {t | y∗ + t(x − y∗) ∈ D(y∗, β)}. Ta k´y hiˆe.u
x(cid:48)(cid:48) := y∗ + t0(x − y∗) v`a x(cid:48) := y∗ + t1(x − y∗),
khi d¯´o x(cid:48)(cid:48) ∈ D, x(cid:48) ∈ D(y∗, β) v`a
[y∗, x] ⊂ [y∗, x(cid:48)] ⊂ [y∗, x(cid:48)(cid:48)[. (4.3.8)
Ta khˇa˙’ ng d¯i.nh x(cid:48)(cid:48) ∈ convJD(y∗). Thˆa. t vˆa. y, nˆe´u d¯iˆe` u n`ay khˆong xa˙’y ra th`ı x(cid:48)(cid:48) ∈ D \ convJD(y∗) v`a c´o thˆe˙’ biˆe˙’u diˆe˜n nhu. (4.3.5), t´u.c l`a
x(cid:48)(cid:48) = (cid:0)1 − α(y∗)(cid:1)y∗ + α(y∗)y, trong d¯´o α(y∗) < 1 v`a y ∈ conv JD(y∗).
Do d¯´o x(cid:48)(cid:48) ∈ ] y∗, y [ , d¯iˆe` u n`ay tr´ai v´o.i c´ach cho. n x(cid:48)(cid:48), nˆen x(cid:48)(cid:48) ∈ conv JD(y∗).
Do x(cid:48) ∈ D(y∗, β) nˆen tˆo`n ta. i α1 ∈ [0, 1 − β] v`a y ∈ D sao cho
(4.3.9) x(cid:48) = (1 − α1)y∗ + α1y.
T`u. (4.3.8) ta suy ra y ∈]y∗, x(cid:48)(cid:48)] do d¯´o c´o thˆe˙’ viˆe´t
y = α2y∗ + (1 − α2)x(cid:48)(cid:48), α2 ∈]0, 1].
Thay y t`u. cˆong th´u.c trˆen v`ao (4.3.9) ta d¯u.o. . c
x(cid:48) = (1 − α1)y∗ + α1y
(4.3.10) = (1 − α1)y∗ + α1(α2y∗ + (1 − α2)x(cid:48)(cid:48)) = (1 − α1 + α1α2)y∗ + (α1 − α1α2)x(cid:48)(cid:48) = (1 − α0)y∗ + α0x(cid:48)(cid:48)
trong d¯´o α0 := α1 − α1α2 v`a dˆe˜ nhˆa. n thˆa´y α0 ≤ 1 − β.
Mˇa. t kh´ac, x(cid:48)(cid:48) ∈ conv JD(y∗) v`a biˆe˙’u th´u.c (4.3.10) c´o da. ng tu.o.ng
d¯u.o.ng x(cid:48)(cid:48) − x(cid:48) = (1 − α0)(x(cid:48)(cid:48) − y∗) nˆen
(4.3.11) (cid:107)x(cid:48)(cid:48) − x(cid:48)(cid:107) = (1 − α0)(cid:107)x(cid:48)(cid:48) − y∗(cid:107) ≥ βd ≥ γ0 > γ.
x∗∈ext D
Kˆe´t ho. . p biˆe˙’u th´u.c (4.3.11) v´o.i x /∈ (cid:83) ¯B(x∗, γ) ta d¯u.o. . c
(cid:107)x − y∗(cid:107) > γ v`a (cid:107)x − x(cid:48)(cid:48)(cid:107) > γ.
Nhu. vˆa. y khoa˙’ng c´ach t`u. d¯iˆe˙’m x ∈ [y∗, x(cid:48)(cid:48)] d¯ˆe´n hai d¯iˆe˙’m y∗ v`a x(cid:48)(cid:48) l´o.n ho.n γ nˆen suy ra tˆo`n ta. i y(cid:48), y(cid:48)(cid:48) ∈ [y∗, x(cid:48)(cid:48)] ⊂ D sao cho x = 0.5(y(cid:48) + y(cid:48)(cid:48)) v`a (cid:107)y(cid:48) − y(cid:48)(cid:48)(cid:107) > 2γ. Do d¯´o x khˆong pha˙’i l`a d¯iˆe˙’m γ-cu. . c biˆen cu˙’a miˆe` n D.
x∗∈ext D
¯B(x∗, γ) ∩ D.
ii) γ = 0, khi d¯´o ext D = (cid:83) . p ca˙’ hai tru.`o.ng ho. Kˆe´t ho. . p i), ii) ta suy ra (b).
0/2, v´o.i γ0 tho˙’a m˜an (b) cu˙’a bˆo˙’ d¯ˆe` . Ta thˆa´y s ∈ [0, s0] khi v`a chı˙’ khi (cid:112)2s/λmin ∈ [0, γ0], nˆen ´ap du. ng (b) ta suy ra tˆa. p . c biˆen cu˙’a miˆe` n D v´o.i γ = (cid:112)2s/λmin nˇa`m trong c´ac d¯iˆe˙’m γ-cu.
(c) D- ˇa. t s0 = λminγ2
x∗∈ext D
(cid:91) ¯B(x∗, (cid:112)2s/λmin) ∩ D.
Bˆo˙’ d¯ˆe` d¯u.o. . c ch´u.ng minh.
V´ı du. sau d¯ˆay cho thˆa´y tˆa. p c´ac d¯iˆe˙’m γ-cu. . c biˆen c´o thˆe˙’ nho˙’ ho.n thu. . c
x∗∈ext D
¯B(x∗, γ) ∩ D. su. . tˆa. p (cid:83)
V´ı du. 4.3.10. Cho D ⊂ IR2 l`a tam gi´ac c´o c´ac d¯ı˙’nh √ √ 3, 1). 3, 1), x2 = (− x0 = (0, 0), x1 = (
Cho γ = 0.25, khi d¯´o tˆa. p c´ac d¯iˆe˙’m γ-cu.
2 = (
1 = (−
2 − x(cid:48)
√ √ 3/5, 0.2), khi d¯´o x(cid:48)
. c biˆen cu˙’a D v´o.i γ = 0.25 0 = (0, 0.2) ∈ 1 + x(cid:48) 0 = 0.5(x(cid:48) 2) 0 ∈ ¯B(x0, 0.25) nhu.ng
nˇa`m trong (cid:0) ¯B(x0, 0.25)∪ ¯B(x1, 0.25)∪ ¯B(x2, 0.25)(cid:1)∩D. Cho x(cid:48) ¯B(x0, 0.25), x(cid:48) 1 − x(cid:48) v`a (cid:107)x(cid:48) 0(cid:107) = (cid:107)x(cid:48) khˆong pha˙’i l`a d¯iˆe˙’m γ-cu. 3/5, 0.2), x(cid:48) √ 3/5 > 0.25. Ta suy ra x(cid:48) 0(cid:107) = . c biˆen cu˙’a D v´o.i γ = 0.25.
Ta k´y hiˆe.u
|p(x)| < +∞}. C 0(D) := {p : D → IR | sup x∈D
. c trang bi. c´ac ph´ep to´an (p1 + p2)(x) := :=
Bˆo˙’ d¯ˆe` 4.3.6. Nˆe´u C 0(D) d¯u.o. p1(x) + p2(x), (αp)(x) := αp(x), x ∈ D, α ∈ IR v`a chuˆa˙’n (cid:107)p(cid:107)C 0 supx∈D |p(x)| < +∞ th`ı (C 0(D), (cid:107).(cid:107)C 0) l`a khˆong gian Banach.
. c suy tru. Ch´u.ng minh. C 0(D) l`a khˆong gian tuyˆe´n t´ınh d¯i.nh chuˆa˙’n d¯u.o. . c tiˆe´p t`u. d¯i.nh ngh˜ıa c´ac ph´ep to´an v`a chuˆa˙’n trˆen C 0(D). Ta ch´u.ng minh C 0(D) l`a khˆong gian Banach. X´et d˜ay Cosi {pi} trong C 0(D), khi d¯´o theo d¯i.nh ngh˜ıa
∀(cid:15) > 0 ∃ N : i, j ≥ N ⇒ (cid:107)pi − pj(cid:107)C 0 ≤ (cid:15).
T`u. biˆe˙’u th´u.c trˆen suy ra v´o.i mo. i x ∈ D
(4.3.12) ∀(cid:15) > 0 ∃ N : i, j ≥ N ⇒ |pi(x) − pj(x)| ≤ (cid:15)
nˆen theo d¯i.nh l´y Cosi tˆo`n ta. i p(x) sao cho limi→+∞ pi(x) = p(x).
T`u. biˆe˙’u th´u.c (4.3.12) khi j → +∞ th`ı
∀(cid:15) > 0 ∃ N : i ≥ N ⇒ |pi(x) − p(x)| ≤ (cid:15)
v´o.i mo. i x ∈ D. Suy ra
∀(cid:15) > 0 ∃ N : i ≥ N ⇒ (cid:107)pi − p(cid:107)C 0 ≤ (cid:15),
t´u.c l`a pi hˆo. i tu. d¯ˆe´n p theo chuˆa˙’n cu˙’a C 0(D).
Mˇa. t kh´ac, cˆo´ d¯i.nh i ≥ N, v`ı
(cid:107)p(cid:107)C 0 ≤ (cid:107)pi(cid:107)C 0 + (cid:107)pi − p(cid:107)C 0 ≤ (cid:107)pi(cid:107)C 0 + (cid:15) ≤ (cid:107)pN (cid:107) + 2(cid:15)
nˆen p gi´o.i nˆo. i trˆen D. Do d¯´o, d˜ay Cosi {pi} hˆo. i tu. vˆe` p ∈ C 0(D), vˆa. y C 0(D) l`a khˆong gian Banach.
K´y hiˆe.u ¯BC 0(0, s) l`a h`ınh cˆa` u d¯´ong tˆam 0 b´an k´ınh s cu˙’a khˆong gian
C 0(D).
D- i.nh ngh˜ıa 4.3.14. (xem [4]) Cho X, Y l`a c´ac khˆong gian tuyˆe´n t´ınh d¯i.nh . c go. i l`a nu.˙’a liˆen tu. c trˆen ta. i x0 nˆe´u chuˆa˙’n. H`am d¯a tri. F : X → 2Y , d¯u.o. v´o.i mo. i tˆa. p mo.˙’ V ⊂ Y tho˙’a m˜an F (x0) ⊂ V tˆo`n ta. i lˆan cˆa. n U (x0) sao cho
x ∈ U (x0) =⇒ F (x) ⊂ V
. c go. i l`a nu.˙’a liˆen tu. c du.´o.i ta. i x0 nˆe´u v´o.i mo. i tˆa. p mo.˙’ V ⊂ Y tho˙’a
v`a d¯u.o. m˜an F (x0) ∩ V (cid:54)= ∅ tˆo`n ta. i lˆan cˆa. n U (x0) sao cho
x ∈ U (x0) =⇒ F (x) ∩ V (cid:54)= ∅.
Go. i Sglobal(p) l`a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q), khi d¯´o Sglobal : C 0(D) → 2IRn v`a dˆe˜ thˆa´y Sglobal(0) l`a tˆa. p c´ac d¯iˆe˙’m cu. . c d¯a. i to`an cu. c cu˙’a B`ai to´an (P ). V`ı | ˜f (x)| = |f (x)+p(x)| ≥ λmin(cid:107)x(cid:107)2−(cid:107)b(cid:107)(cid:107)x(cid:107)−(cid:107)p(cid:107)C 0, |f (x)| ≥ λmin(cid:107)x(cid:107)2 − (cid:107)b(cid:107)(cid:107)x(cid:107) v`a D l`a tˆa. p lˆo`i d¯a diˆe.n nˆen Sglobal(p), Sglobal(0) kh´ac ∅ khi v`a chı˙’ khi D l`a d¯a diˆe.n lˆo`i trong IRn. T´ınh ˆo˙’n d¯i.nh cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i d¯u.o. . c thˆe˙’ hiˆe.n qua mˆe.nh d¯ˆe` sau:
D- i.nh l´y 4.3.24. X´et B`ai to´an ( ˜Q). Khi d¯´o
∃s0 > 0 ∀p ∈ ¯B(0, s0) : Sglobal(p) ⊆ Sglobal(0) + (cid:112)2(cid:107)p(cid:107)C 0/λmin ¯B(0, 1).
(4.3.13)
Ch´u.ng minh. Tru.´o.c tiˆen ta nhˆa. n x´et rˇa`ng, nˆe´u D khˆong gi´o.i nˆo. i hoˇa. c D = ext D = {x∗} th`ı Sglobal(p) = Sglobal(0) = ∅ hoˇa. c Sglobal(p) = Sglobal(0) = {x∗} v´o.i mo. i p ∈ C 0(D), nˆen kˆe´t luˆa. n cu˙’a d¯i.nh l´y l`a d¯´ung. Do d¯´o ta chı˙’ cˆa` n x´et tru.`o.ng ho. . p |ext D| ≥ 2 v`a D l`a d¯a diˆe.n lˆo`i. Ngo`ai ra khi p ≡ 0 th`ı kˆe´t luˆa. n cu˙’a mˆe.nh d¯ˆe` l`a hiˆe˙’n nhiˆen.
V`ı h`am to`an phu.o.ng f lˆo`i ngˇa. t trˆen D, ext D l`a tˆa. p c´ac d¯ı˙’nh cu˙’a D, . c d¯a. i cu˙’a f nˇa`m trong ext D, ngh˜ıa l`a Sglobal(0) ⊆ ext D.
nˆen c´ac d¯iˆe˙’m cu. Ta x´et c´ac tru.`o.ng ho. . p sau:
i) Tru.`o.ng ho.
s0 > 0 sao cho v´o.i mo. i s ∈ [0, s0], tˆa. p c´ac d¯iˆe˙’m γ-cu. . p ext D = Sglobal(0). Theo (c) cu˙’a Bˆo˙’ d¯ˆe` 4.3.5, tˆo`n ta. i . c biˆen cu˙’a miˆe` n D v´o.i
γ = (cid:112)2s/λmin nˇa`m trong tˆa. p
x∗∈ext D
(cid:91) (4.3.14) ¯B(x∗, (cid:112)2s/λmin) ∩ D.
Lˆa´y p ∈ C 0(D) v`ı (cid:107)p(cid:107)C 0 = supx∈D |p(x)| nˆen theo Hˆe. qua˙’ 4.2.3 d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i . c biˆen cu˙’a D v´o.i γ = (cid:112)2(cid:107)p(cid:107)C 0/λmin . Kˆe´t ho. ˜f = f + p l`a d¯iˆe˙’m γ cu. . p v´o.i (4.3.14) ta suy ra v´o.i mo. i p ∈ ¯BC 0(0, s0), t´u.c l`a (cid:107)p(cid:107)C 0 ≤ s0, th`ı
x∗∈ext D (cid:91)
(cid:91) (cid:1) ∩ D Sglobal(p) ⊆ ¯B(cid:0)x∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin
x∗∈Sglobal(0)
(cid:1) ⊆ ¯B(cid:0)x∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin
= Sglobal(0) + ¯B(0, (cid:112)2(cid:107)p(cid:107)C 0/λmin ).
Do d¯´o
∃s0 ∀p ∈ ¯BC 0(0, s0) : Sglobal(p) ⊆ Sglobal(0) + (cid:112)2(cid:107)p(cid:107)C 0/λmin ¯B(0, 1).
Vˆa. y tru.`o.ng ho. . p ext D = Sglobal(0) d¯˜a d¯u.o. . c ch´u.ng minh.
. p Sglobal(0) ⊂ ext D. Gia˙’ su.˙’ γ0 tho˙’a m˜an (b) cu˙’a Bˆo˙’ 0/2. Theo (c) cu˙’a Bˆo˙’ d¯ˆe` 4.3.5 ta suy ra, v´o.i mo. i . c biˆen cu˙’a miˆe` n D v´o.i γ = (cid:112)2s/λmin nˇa`m
x∗∈ext D
ii) Tru.`o.ng ho. d¯ˆe` 4.3.5, d¯ˇa. t s1 := λminγ2 s ∈ [0, s1] tˆa. p c´ac d¯iˆe˙’m γ-cu. trong tˆa. p (cid:91) (cid:1) ∩ D. ¯B(cid:0)x∗, (cid:112)2s/λmin
x∗∈ext D
Mˇa. t kh´ac, nˆe´u p ∈ C 0(D) th`ı d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f + p l`a d¯iˆe˙’m γ-cu. . c biˆen cu˙’a miˆe` n D v´o.i γ = (cid:112)2(cid:107)p(cid:107)C 0/λmin . Do d¯´o (cid:91) (cid:1) ∩ D. (4.3.15) ¯B(cid:0)x∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin ∀p ∈ ¯BC 0(0, s1) : Sglobal(p) ⊆
D- ˇa. t
f (x), k := f (y∗), K := max x∈D max y∗∈ext D\Sglobal(0)
khi d¯´o v´o.i mo. i x∗ ∈ Sglobal(0) th`ı f (x∗) = K v`a k < K.
4 ]. V`ı h`am to`an phu.o.ng lˆo`i ngˇa. t f liˆen tu. c ta. i mo. i d¯iˆe˙’m
Lˆa´y (cid:15) ∈ ]0, K−k
trˆen D nˆen v´o.i mˆo˜i y∗ ∈ ext D \ Sglobal(0) ta c´o
∃δ = δ(y∗, (cid:15)) > 0, ∀x ∈ D : (cid:107)x − y∗(cid:107) ≤ δ =⇒ f (x) ≤ f (y∗) + (cid:15).
D- ˇa. t δ := miny∗∈ext D\Sglobal(0) δ(y∗, (cid:15)). V`ı tˆa. p ext D \ Sglobal(0) l`a h˜u.u ha. n nˆen δ > 0. Do d¯´o
∀y∗ ∈ ext D \ Sglobal(0), ∀x ∈ D : (cid:107)x − y∗(cid:107) ≤ δ =⇒ f (x) ≤ f (y∗) + (cid:15).
Thay f (y∗) ≤ k < K − 4(cid:15) v`ao biˆe˙’u th´u.c trˆen ta d¯u.o. . c
2
∀y∗ ∈ ext D \ Sglobal(0), ∀x ∈ D : (cid:107)x − y∗(cid:107) ≤ δ =⇒ f (x) ≤ K − 3(cid:15).
:= min{λminδ /2, (cid:15)}. Khi d¯´o, nˆe´u p ∈ ¯B(0, s2) th`ı v´o.i mo. i
D- ˇa. t s2 y∗ ∈ ext D \ Sglobal(0) v`a x ∈ D, tho˙’a m˜an (cid:107)x − y∗(cid:107) ≤ δ, suy ra
˜f (x) = f (x) + p(x) ≤ f (x) + (cid:107)p(cid:107)C 0 ≤ K − 3(cid:15) + s2 ≤ f (x∗) − 2(cid:15) ≤ ˜f (x∗) − (cid:15),
trong d¯´o x∗ ∈ Sglobal(0).
Do vˆa. y, v´o.i mo. i p ∈ ¯BC 0(0, s2) th`ı
˜f (x) − (cid:15). ∀y∗ ∈ ext D \ Sglobal(0), ∀x ∈ D : (cid:107)x − y∗(cid:107) ≤ δ =⇒ ˜f (x) ≤ sup x∈D
T`u. biˆe˙’u th´u.c trˆen suy ra, nˆe´u p ∈ ¯BC 0(0, s2) th`ı v´o.i mo. i y∗ ∈ ext D \ Sglobal(0), tˆa. p ¯B(y∗, δ) khˆong ch´u.a d¯iˆe˙’m supremum to`an cu. c cu˙’a h`am ˜f = f + p, t´u.c l`a
∀p ∈ ¯BC 0(0, s2) : Sglobal(p) ∩ ¯B(y∗, δ) = ∅
v´o.i mo. i y∗ ∈ ext D \ Sglobal(0).
Mˇa. t kh´ac, (cid:112)2(cid:107)p(cid:107)C 0/λmin ≤ δ nˆen ¯B(y∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin ) ⊆ ¯B(y∗, δ).
Do d¯´o
y∗∈ext D\Sglobal(0)
(cid:91) (cid:1) = ∅. ¯B(cid:0)y∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin ∀p ∈ ¯BC 0(0, s2) : Sglobal(p) ∩
(4.3.16) D- ˇa. t s0 := min{s1, s2}, su.˙’ du. ng (4.3.15) v`a (4.3.16) ta suy ra v´o.i mo. i p ∈ ¯BC 0(0, s0), tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c Sglobal(p) tho˙’a m˜an
y∗∈ext D
(cid:16) (cid:91) (cid:1) ∩ D(cid:1) Sglobal(p) ⊆ ¯B(cid:0)y∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin
y∗∈ext D\Sglobal(0)
(cid:17) (cid:91) \ ¯B(cid:0)y∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin
x∗∈Sglobal(0)
(cid:91) (cid:1) ⊆ ¯B(cid:0)x∗, (cid:112)2(cid:107)p(cid:107)C 0/λmin
= Sglobal(0) + (cid:112)2(cid:107)p(cid:107)C 0/λmin ¯B(0, 1).
. c ch´u.ng minh.
T´om la. i ta nhˆa. n d¯u.o. . c ∃s0 > 0 ∀p ∈ ¯BC 0(0, s0) : Sglobal(p) ⊆ Sglobal(0) + (cid:112)2(cid:107)p(cid:107)C 0/λmin ¯B(0, 1). D- i.nh l´y d¯˜a d¯u.o. Nhˆa. n x´et 4.3.10. Biˆe˙’u th´u.c (4.3.13) tu.o.ng d¯u.o.ng v´o.i
(cid:0)∃s0 > 0 ∀p ∈ ¯BC 0(0, s0)(cid:1) =⇒
(cid:1).(4.3.17) (cid:0)∀˜x∗ ∈ Sglobal(p), ∃x∗ ∈ Sglobal(0) : (cid:107)˜x∗ − x∗(cid:107) ≤ (cid:112)2(cid:107)p(cid:107)C 0/λmin
. ng (cid:112)2(cid:107)p(cid:107)C 0/λmin o.˙’ d¯´anh gi´a trˆen l`a tˆo´t nhˆa´t. Kˆe´t Ngo`ai ra d¯a. i lu.o.
. a trˆen V´ı du. 4.3.12. luˆa. n n`ay du.
Ta c´o thˆe˙’ t´ınh s0 thˆong qua v´ı du. sau:
V´ı du. 4.3.11. Cho
f (x) = x2, x ∈ [−0.5, 1]
1 = −0.5, x∗
Ta c´o D = [−0.5, 1] nˆen c´ac d¯iˆe˙’m cu.
v`a λmin = 1, Sglobal(0) = {x∗ . c biˆen l`a x∗ 2 = 1 2} = {1}. Theo mˆe.nh d¯ˆe` trˆen th`ı c´o thˆe˙’ cho. n
0/2 ≈ 0.28125. Ta c˜ung c´o K = 1, k = 0.5 nˆen 0.375]
√
2
√ 0.375 ≈ 1.1123.
/2, (cid:15)} = 0.125. nˆen s0 = min{s1, s2} = 0.125. Do d¯´o, trong
γ0 = 0.75 v`a do d¯´o s1 = λminγ2 c´o thˆe˙’ cho. n (cid:15) = (1 − 0.5)/4 = 0.125. Ngo`ai ra v´o.i mo. i x ∈ [−0.5, th`ı f (x) ≤ f (−0.5) + (cid:15) = 0.25 + 0.125 nˆen cho. n δ = 0.5 + V`ı s2 := min{δ v´ı du. trˆen v´o.i s0 = 0.125 biˆe˙’u th´u.c (4.3.17) tho˙’a m˜an. V´ı du. 4.3.12. Cho
1 = −1, x∗
√ −s nˆe´u x ∈ [−1, − s ] √ √ p(x) = s [ s, s − x2 nˆe´u x ∈ ] − √ −s nˆe´u x ∈ [ s, 1 ]. f (x) = x2, x ∈ [−1, 1],
V`ı D = [−1, 1] nˆen c´ac d¯iˆe˙’m cu. 1, x∗ . c biˆen l`a x∗ 2 = 1 v`a 2} = {−1, 1}. Dˆe˜ thˆa´y khi γ0 = 1
λmin = 1, (cid:107)p(cid:107)C 0 = s, Sglobal(0) = {x∗ th`ı v´o.i mo. i γ ≤ γ0, tˆa. p
[−1, −1 + γ] ∪ [1 − γ, 1]
. c biˆen cu˙’a [−1, 1]. Theo (c) cu˙’a Bˆo˙’ d¯ˆe` 4.3.5 v`a v`ı
0/2 = 0.5, tˆa. p
l`a tˆa. p c´ac d¯iˆe˙’m γ-cu. λmin = 1 nˆen v´o.i mo. i 0 ≤ s ≤ s0 = γ2 √ √
1| = |˜x∗
3 − x∗
√ [−1, −1 + 2s ] ∪ [1 − 2s, 1] . c biˆen cu˙’a [−1, 1] v´o.i γ = √ 0.5,
ch´u.a tˆa. p c´ac d¯iˆe˙’m γ-cu. 2s. Mˇa. t kh´ac, khi √ 0.5[. Nˆen s = s0 = 0.5 th`ı (cid:107)p(cid:107)C 0 = 0.5, Sglobal(p) = {−1, 1} ∪ ] − 3 := 0 c˜ung l`a d¯iˆe˙’m supremum to`an cu. c cu˙’a ˜f = f + p, t´u.c l`a ˜x∗ suy ra ˜x∗ 3 ∈ 2| = |0 − 1| = 1 = (cid:112)2(cid:107)p(cid:107)C 0/λmin. Sglobal(p). Ta c˜ung c´o |˜x∗ 3 − x∗ Do d¯´o, bˆa´t d¯ˇa˙’ ng th´u.c (4.3.17) tro.˙’ th`anh d¯ˇa˙’ ng th´u.c khi s = s0 = 0.5. Vˆa. y ta kˆe´t luˆa. n d¯´anh gi´a o.˙’ D- i.nh l´y 4.3.24 l`a tˆo´t nhˆa´t. Mˆe. nh d¯ˆe` 4.3.29. X´et B`ai to´an ( ˜Q). Khi d¯´o Sglobal(p) l`a h`am nu.˙’a liˆen tu. c trˆen ta. i 0.
Ch´u.ng minh. Gia˙’ su.˙’ V ⊂ IRn l`a tˆa. p mo.˙’ bˆa´t k`y tho˙’a m˜an Sglobal(0) ⊂ V. Khi d¯´o
∀x∗ ∈ Sglobal(0), ∃δ(x∗) > 0 : ¯B(cid:0)x∗, δ(x∗)(cid:1) ⊂ V.
D- ˇa. t δ := minx∗∈Sglobal(0) δ(x∗), v`ı tˆa. p Sglobal(0) l`a h˜u.u ha. n nˆen δ > 0 v`a
∀x∗ ∈ Sglobal(0) : x∗ + ¯B(cid:0)0, δ(cid:1) ⊂ V,
t´u.c l`a
2
(4.3.18)
Sglobal(0) + δ ¯B(0, 1) ⊂ V. . c x´ac d¯i.nh o.˙’ D- i.nh l´y 4.3.24, d¯ˇa. t s1 := min{λminδ V´o.i s0 d¯u.o. /2, s0}, khi d¯´o (cid:112)2s1/λmin ≤ δ v`a s1 ≤ s0. V`ı vˆa. y, v´o.i mo. i p ∈ C 0(D) sao cho 0 < (cid:107)p(cid:107)C 0 ≤ s1 c´ac biˆe˙’u th´u.c (4.3.15), (4.3.18) d¯ˆo`ng th`o.i tho˙’a m˜an. Kˆe´t . p la. i ta d¯u.o. ho. . c
∀p ∈ ¯BC 0(0, s1) : Sglobal(p) ⊆ Sglobal(0) + (cid:112)2(cid:107)p(cid:107)C 0/λmin ¯B(0, 1)
⊆ Sglobal(0) + δ ¯B(0, 1) ⊂ V.
Vˆa. y ta suy ra
∀p ∈ ¯BC 0(0, s1) : Sglobal(p) ⊂ V,
t´u.c l`a Sglobal(p) nu.˙’a liˆen tu. c trˆen ta. i 0.
T`u. d¯i.nh ngh˜ıa suy ra h`am ˜f = f + p khˆong nu.˙’a liˆen tu. c du.´o.i ta. i 0 nˆe´u tˆo`n ta. i lˆan cˆa. n V tho˙’a m˜an Sglobal(0) ∩ V (cid:54)= ∅ v`a d˜ay (pi), i = 1, 2 . . . hˆo. i tu. vˆe` 0 trong C 0(D) sao cho Sglobal(pi) ∩ V = ∅ v´o.i mo. i i = 1, 2, . . . . V´ı du. sau d¯ˆay chı˙’ ra rˇa`ng h`am Sglobal(p) khˆong nu.˙’a liˆen tu. c du.´o.i ta. i 0.
V´ı du. 4.3.13. Cho
f (x) = x2, x ∈ [−4, 4] −1/i nˆe´u x ∈ [−4, −3.8] nˆe´u x ∈ ] − 3.8, 3.8[ pi(x) =
0 1/i nˆe´u x ∈ [3.8, 4],
i = 1, 2 . . . .
T`u. v´ı du. ta c´o Sglobal(0) = {−4, 4}, D = [−4, 4], pi ∈ C 0(D), (cid:107)pi(cid:107)C 0(D) = 1/i, Sglobal(pi) = {4} v´o.i mo. i sˆo´ nguyˆen du.o.ng i. Lˆa´y tˆa. p
mo.˙’ V = ]-4.1,0.5[ khi d¯´o Sglobal(0) ∩ V = {−4} (cid:54)= ∅. X´et d˜ay (pi) ⊂ C 0(D) ta c´o limi→∞ pi = 0 v`a Sglobal(pi) = {4} nˆen Sglobal(pi) ∩ V = ∅. Khi d¯´o theo d¯i.nh ngh˜ıa suy ra Sglobal(p) khˆong nu.˙’a liˆen tu. c du.´o.i ta. i 0.
Ta biˆe´t rˇa`ng l´o.p h`am lˆo`i c´o t´ınh chˆa´t: d¯iˆe˙’m cu.
. c tiˆe˙’u d¯i.a phu.o.ng l`a . c tiˆe˙’u to`an cu. c. D- ˆo´i v´o.i mˆo. t sˆo´ l´o.p h`am lˆo`i suy rˆo. ng thˆo nhu. d¯iˆe˙’m cu. . c, t´u.c l`a nˆe´u x∗ ∈ D γ-lˆo`i ngo`ai th`ı t´ınh chˆa´t gˆa` n nhu. trˆen vˆa˜n gi˜u. d¯u.o. l`a d¯iˆe˙’m γ-cu. . c tiˆe˙’u, γ-infimum, th`ı x∗ l`a cu. . c tiˆe˙’u to`an cu. c, infimum to`an cu. c tu.o.ng ´u.ng [47], su. . liˆen hˆe. d¯´o cho ph´ep ta tˆa. p trung v`ao viˆe.c nghiˆen . c tiˆe˙’u to`an cu. c, infimum to`an cu. c. Tuy nhiˆen khˆong c´o su. c´u.u c´ac d¯iˆe˙’m cu. . . c d¯a. i to`an cu. c, supremum to`an cu. c v´o.i liˆen hˆe. nhu. thˆe´ gi˜u.a c´ac d¯iˆe˙’m cu. c´ac d¯iˆe˙’m cu. . c d¯a. i d¯i.a phu.o.ng, supremum d¯i.a phu.o.ng trong c´ac l´o.p h`am lˆo`i suy rˆo. ng. Do d¯´o trong phˆa` n n`ay, ch´ung tˆoi tˆa. p trung nghiˆen c´u.u tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q).
Go. i Slocal(p) l`a tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q) khi d¯´o Slocal : C 0(D) → 2IRn v`a dˆe˜ thˆa´y Slocal(0) l`a tˆa. p c´ac d¯iˆe˙’m cu. . c d¯a. i d¯i.a phu.o.ng cu˙’a B`ai to´an (Q). Do f l`a h`am to`an phu.o.ng lˆo`i ngˇa. t trˆen tˆa. p lˆo`i d¯a diˆe.n D nˆen Slocal(0) chı˙’ c´o thˆe˙’ l`a c´ac d¯iˆe˙’m cu. . c biˆen cu˙’a D, t´u.c l`a mˆo. t sˆo´ d¯ı˙’nh cu˙’a D.
Tru.´o.c khi nghiˆen c´u.u d¯iˆe˙’m cu. . c d¯a. i, supremum d¯i.a phu.o.ng ch´ung ta
c´o nhˆa. n x´et sau: Nhˆa. n x´et 4.4.11. Slocal(0) c´o thˆe˙’ bˇa`ng ∅ khi D khˆong gi´o.i nˆo. i v`a Slocal(0) = ∅ khi D ch´u.a d¯u.`o.ng thˇa˙’ ng
Thˆa. t vˆa. y, nˆe´u D khˆong gi´o.i nˆo. i th`ı Slocal(0) = ∅, Slocal(0) (cid:54)= ∅ c´o thˆe˙’ thˆa´y r˜o qua viˆe.c x´et h`am f (x) = x2 trˆen c´ac miˆe` n D = [0, +∞[ v`a D = [−1, +∞[, tu.o.ng ´u.ng.
Tru.`o.ng ho. . p D ch´u.a d¯u.`o.ng thˇa˙’ ng. Theo lu.o.
. c d¯ˆo` ch´u.ng minh trong D- i.nh l´y 19.1 [54] th`ı D c´o thˆe˙’ biˆe˙’u diˆe˜n du.´o.i da. ng D = D0 + L, trong d¯´o D0 = D ∩ L⊥ l`a tˆa. p lˆo`i d¯´ong khˆong ch´u.a d¯u.`o.ng thˇa˙’ ng, L l`a khˆong gian con tuyˆe´n t´ınh, L⊥ l`a khˆong gian con b`u vuˆong g´oc v´o.i L. Gia˙’ su.˙’ x ∈ D, nˆe´u x /∈ D0 th`ı tˆo`n ta. i x(cid:48) ∈ D0 ⊆ D, y (cid:54)= 0, y ∈ L sao cho x = x(cid:48) + y, d¯ˇa. t x(cid:48)(cid:48) := x(cid:48) + 2y, ta nhˆa. n d¯u.o. . c x(cid:48), x(cid:48)(cid:48) ∈ D v`a (cid:107)x(cid:48) − x(cid:107) = (cid:107)x(cid:48)(cid:48) − x(cid:107) = (cid:107)y(cid:107). Nˆe´u x ∈ D0 th`ı x(cid:48) := x − y ∈ D v`a x(cid:48)(cid:48) := x + y ∈ D v´o.i mo. i y ∈ L khi d¯´o (cid:107)x(cid:48) − x(cid:107) = (cid:107)x(cid:48)(cid:48) − x(cid:107) = (cid:107)y(cid:107) nˆen suy ra D khˆong c´o d¯iˆe˙’m cu. . c biˆen. V`ı vˆa. y Slocal(0) = ∅.
V`ı nh˜u.ng l´y do trˆen nˆen khi nghiˆen c´u.u c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng ta luˆon gia˙’ thiˆe´t tˆo`n ta. i d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a h`am to`an phu.o.ng lˆo`i ngˇa. t f, t´u.c l`a Slocal(0) (cid:54)= ∅.
K´y hiˆe.u
x∈D, x(cid:54)=x∗
(cid:68) (cid:69) η(x∗) := sup 2Ax∗ + b, . x − x∗ (cid:107)x − x∗(cid:107)
Mˆo. t sˆo´ bˆo˙’ d¯ˆe` sau s˜e d¯u.o. . c d`ung khi nghiˆen c´u.u h`am Slocal(p).
. c biˆen cu˙’a tˆa. p . p l´y khi ta viˆe´t maxx∗∈Slocal(0) η(x∗)
Bˆo˙’ d¯ˆe` 4.4.7. D- ˇa. t η0 := maxx∗∈Slocal(0) η(x∗). Khi d¯´o η0 < 0. Ch´u.ng minh. Hˆe. qua˙’ 19.1.1 [54] khˇa˙’ ng d¯i.nh, tˆa. p c´ac d¯iˆe˙’m cu. lˆo`i d¯a diˆe.n l`a h˜u.u ha. n, nˆen ho`an to`an ho. thay cho supx∗∈Slocal(0) η(x∗). Ta k´y hiˆe.u
Pi
:= {x ∈ IRn | (cid:104)ci, x(cid:105) = di, i = 1, . . . , m} := {x ∈ IRn | (cid:104)ci, x(cid:105) ≤ di, i = 1, . . . , m}.
. c biˆen cu˙’a D nˆen tˆo`n ta. i i1, i2, . . . , ik ∈
Fi V`ı x∗ ∈ ext D l`a d¯iˆe˙’m cu. {1, 2, . . . , m} sao cho
j=1Pij.
j=1Fij ∩ ¯B(x∗, 1)
{x∗} = ∩k
V`ı h`am (cid:104)2Ax∗+b, u(cid:105) liˆen tu. c theo biˆe´n u trˆen tˆa. p compact ∩k j=1Fij ∩ ¯B(x∗, 1), sao cho nˆen tˆo`n ta. i u0 ∈ ∩k
u∈∩k
j=1Fij ∩ ¯B(x∗,1)
max (cid:104)2Ax∗ + b, u(cid:105). (cid:104)2Ax∗ + b, u0(cid:105) =
Mˇa. t kh´ac, v´o.i mo. i x ∈ D, x (cid:54)= x∗ th`ı
− (cid:10)cij, x∗ + (cid:11) = (cid:104)cij, x∗(cid:105) + x − x∗ (cid:107)x − x∗(cid:107) (cid:104)cij, x(cid:105) (cid:107)x − x∗(cid:107) (cid:104)cij, x∗(cid:105) (cid:107)x − x∗(cid:107)
≤ dij + dij − dij (cid:107)x − x∗(cid:107)
≤ dij
v´o.i mo. i j = 1, . . . , k nˆen
j=1Fij ∩ ¯B(x∗, 1).
x∗ + ∈ ∩k x − x∗ (cid:107)x − x∗(cid:107)
Do d¯´o
(cid:10)2Ax∗ + b, x∗ + (cid:11) ≤ (cid:104)2Ax∗ + b, u0(cid:105). x − x∗ (cid:107)x − x∗(cid:107) sup x∈D,x(cid:54)=x∗
Biˆe˙’u th´u.c trˆen k´eo theo
(cid:10)2Ax∗ + b, (4.4.19) (cid:11) ≤ (cid:104)2Ax∗ + b, u0 − x∗(cid:105). x − x∗ (cid:107)x − x∗(cid:107) sup x∈D,x(cid:54)=x∗
X´et tia x∗ + t(u0 − x∗). Ta khˇa˙’ ng d¯i.nh tˆo`n ta. i δ1 > 0 sao cho v´o.i mo. i t ∈ [0, δ1] th`ı x∗ + tu0 ∈ D.
Thˆa. t vˆa. y, nˆe´u u0 ∈ D th`ı
(4.4.20) x∗ + t(u0 − x∗) ∈ D v´o.i mo. i t ∈ [0, 1].
Nˆe´u u0 /∈ D ta x´et c´ac tru.`o.ng ho. . p sau:
i) i ∈ {i1, . . . , ik}. V`ı (cid:104)ci, x∗(cid:105) = di v`a (cid:104)ci, u0(cid:105) ≤ di nˆen v´o.i mo. i t ≥ 0
th`ı
(4.4.21) (cid:104)ci, x∗ + t(u0 − x∗)(cid:105) = (cid:104)ci, x∗(cid:105) + t(cid:104)ci, u0 − x∗(cid:105) ≤ di
ii) i /∈ {i1, . . . , ik}. Khi d¯´o (cid:104)ci, x∗(cid:105) < di. D- ˇa. t l := maxi /∈{i1,...,ik}(cid:104)ci, u0 −
x∗(cid:105), ta c´o
(cid:104)ci, x∗ + t(u0 − x∗)(cid:105) = (cid:104)ci, x∗(cid:105) + t(cid:104)ci, u0 − x∗(cid:105)
(4.4.22) ≤ (cid:104)ci, x∗(cid:105) + tl.
Nˆe´u l ≤ 0, th`ı hiˆe˙’n nhiˆen
(4.4.23)
(cid:104)ci, x∗ + t(u0 − x∗)(cid:105) ≤ di ∀t ≥ 0. Nˆe´u l > 0 th`ı d¯ˇa. t δ1 := min{mini /∈{i1,...,ik} (di − (cid:104)ci, x∗(cid:105)), 1} > 0. Do d¯´o, t`u. (4.4.22) ta suy ra
(4.4.24)
v´o.i mo. i t ∈ [0, δ1]. Kˆe´t ho. (cid:104)ci, x∗ + t(u0 − x∗)(cid:105) ≤ di . p (4.4.20)–(4.4.24) ta d¯u.o. . c khˇa˙’ ng d¯i.nh trˆen.
X´et h`am φ(t) := f (x∗+t(u0−x∗))−f (x∗) trˆen d¯oa. n [0, δ1]. V`ı φ(0) = 0 . c d¯a. i d¯i.a phu.o.ng . c d¯a. i d¯i.a phu.o.ng ta. i x∗ nˆen φ(t) c˜ung d¯a. t cu.
v`a f (x) d¯a. t cu. ta. i 0, t´u.c l`a
∃δ ∈]0, δ1] : φ(t) ≤ 0 ∀t ∈ [0, δ].
Nhu. vˆa. y, v´o.i mo. i t ∈ [0, δ] th`ı
φ(t) = (cid:10)A(cid:0)x∗ + t(u0 − x∗)(cid:1), x∗ + t(u0 − x∗)(cid:11) − (cid:10)Ax∗, x∗(cid:11) − (cid:104)b, x∗(cid:105)
= (cid:10)2Ax∗ + b, u0 − x∗(cid:11)t + (cid:10)A(u0 − x∗), u0 − x∗(cid:11)t2 ≤ 0.
V`ı u0 − x∗ (cid:54)= 0 nˆen (cid:104)A(u0 − x∗), u0 − x∗(cid:105) > 0. Do d¯´o t`u. bˆa´t d¯ˇa˙’ ng th´u.c trˆen cho ta (cid:10)2Ax∗ + b, u0 − x∗(cid:11) < 0. Kˆe´t ho. . p bˆa´t d¯ˇa˙’ ng th´u.c n`ay v´o.i biˆe˙’u th´u.c (4.4.19) v`a Slocal(0) ⊆ ext D ta suy ra
x∗∈Slocal(0)
η(x∗) < 0. η0 = max
Bˆo˙’ d¯ˆe` d¯˜a d¯u.o. . c ch´u.ng minh.
D- ˆe˙’ tiˆe.n theo d˜oi t`u. d¯ˆay ta luˆon k´y hiˆe.u
(4.4.25) s0
0/(12λmax), (cid:113) 0 − 12λmaxs (cid:1)/(2λmax), η2
(4.4.26) := η2 ξ(s) := (cid:0) − η0 −
v`a nhˆa. n thˆa´y rˇa`ng nˆe´u s ∈]0, s0] th`ı ξ(s) > 0. Ta c´o bˆo˙’ d¯ˆe` sau: Bˆo˙’ d¯ˆe` 4.4.8. V´o.i mˆo˜i s ∈ [0, s0] th`ı
∀x∗ ∈ Slocal(0), ∀x ∈ D : (cid:107)x − x∗(cid:107) = ξ(s) =⇒ f (x) ≤ f (x∗) − 3s.
Ch´u.ng minh. Nˆe´u s = 0 th`ı ξ(s) = 0 nˆen kˆe´t luˆa. n cu˙’a bˆo˙’ d¯ˆe` l`a hiˆe˙’n nhiˆen. Nˆe´u s > 0, lˆa´y bˆa´t k`y x∗ ∈ Slocal(0). V´o.i mo. i x ∈ D, x (cid:54)= x∗ ta c´o
(cid:105)(cid:107)x − x∗(cid:107) + (cid:104)A(x − x∗), x − x∗(cid:105), f (x) = f (x∗) + (cid:104)2Ax∗ + b, x − x∗ (cid:107)x − x∗(cid:107)
nˆen, v´o.i mo. i x ∈ D, x (cid:54)= x∗ th`ı
(cid:105)(cid:107)x − x∗(cid:107) + (cid:104)A(x − x∗), x − x∗(cid:105) + 3s + f (x∗) − 3s. f (x) = (cid:104)2Ax∗ + b, x − x∗ (cid:107)x − x∗(cid:107)
(4.4.27)
X´et biˆe˙’u th´u.c
(cid:104)2Ax∗ + b, (cid:105)(cid:107)x − x∗(cid:107) + (cid:104)A(x − x∗), x − x∗(cid:105) + 3s. x − x∗ (cid:107)x − x∗(cid:107)
V`ı (cid:104)A(x − x∗), x − x∗(cid:105) ≤ λmax(cid:107)x − x∗(cid:107)2 v`a ξ(s) > 0, nˆen v´o.i mo. i x ∈ D tho˙’a m˜an (cid:107)x − x∗(cid:107) = ξ(s), ta d¯u.o. . c
(cid:104)2Ax∗ + b, (cid:105)(cid:107)x − x∗(cid:107) + (cid:104)A(x − x∗), x − x∗(cid:105) + 3s x − x∗ (cid:107)x − x∗(cid:107)
≤ λmaxξ(s)2 + η0ξ(s) + 3s.
V`ı s ≤ s0 v`a ξ(s) ≤ ξ(s0) nˆen
λmaxξ(s)2 + η0ξ(s) + 3s ≤ λmaxξ(s0)2 + η0ξ(s0) + 3s0.
Thay s0, ξ(s) d¯u.o. . c x´ac d¯i.nh theo (4.4.25) v`a (4.4.26) ta suy ra
λmaxξ(s0)2 + η0ξ(s0) + 3s0 = 0.
Do d¯´o
(cid:105)(cid:107)x − x∗(cid:107) + (cid:104)A(x − x∗), x − x∗(cid:105) + 3s ≤ 0. (cid:104)2Ax∗ + b, x − x∗ (cid:107)x − x∗(cid:107)
Kˆe´t ho. . p biˆe˙’u th´u.c n`ay v´o.i (4.4.27) ta suy ra kˆe´t luˆa. n cu˙’a bˆo˙’ d¯ˆe` .
D- i.nh l´y 4.4.25. X´et B`ai to´an ( ˜Q). Khi d¯´o
x∗∈Slocal(0)
∀p ∈ ¯BC 0(0, s0) : max d(cid:0)x∗, Slocal(p)(cid:1) ≤ ξ((cid:107)p(cid:107)C 0).
Ch´u.ng minh. Nˆe´u p ≡ 0 th`ı d¯iˆe` u khˇa˙’ ng d¯i.nh l`a hiˆe˙’n nhiˆen.
˜f (x). Lˆa´y bˆa´t k`y p ∈ C 0(D) sao cho 0 < (cid:107)p(cid:107)C 0 ≤ s0. D- ˇa. t s := (cid:107)p(cid:107)C 0. Do D l`a tˆa. p lˆo`i d¯a diˆe.n nˆen ¯B(cid:0)x∗, ξ(s)(cid:1)∩D l`a tˆa. p compact. V`ı h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i ˜f = f +p bi. chˇa. n (trˆen) trˆen tˆa. p ¯B(cid:0)x∗, ξ(s)(cid:1)∩D, nˆen ˜f (x) < +∞. Mˇa. t kh´ac, tˆo`n ta. i (˜xi) trong ¯B(cid:0)x∗, ξ(s)(cid:1) ∩ D supx∈ ¯B(x∗, ξ(s))∩D ˜f (˜xi) = supx∈ ¯B(x∗, ξ(s))∩D sao cho limi→∞ ˜xi = ˜x∗ v`a limi→∞
Ta khˇa˙’ ng d¯i.nh ˜x∗ ∈ B(x∗, ξ(s)) ∩ D. Thˆa. t vˆa. y gia˙’ su.˙’ kˆe´t luˆa. n trˆen l`a
sai, khi d¯´o ˜x∗ ∈ (cid:16) ¯B(cid:0)x∗, ξ(s)(cid:1) \ B(x∗, ξ(s)(cid:1)(cid:17) ∩ D. Ta c´o
˜f (˜xi) ˜f (x) = lim i→∞ sup x∈ ¯B(x∗, ξ(s))∩D
(4.4.28) (cid:0)f (˜xi) + p(˜xi)(cid:1). = lim i→∞
˜f (˜xi) = limi→∞
(cid:0)f (˜xi) + p(˜xi)(cid:1), m`a h`am to`an phu.o.ng lˆo`i V`ı tˆo`n ta. i limi→∞ ngˇa. t f liˆen tu. c ta. i mo. i d¯iˆe˙’m thuˆo. c D nˆen tˆo`n ta. i limi→∞ p(˜xi). Do d¯´o thay (cid:0) ˜f (˜xi) + p(˜xi)(cid:1) = limi→∞ f (˜xi) + limi→∞ p(˜xi) v`ao (4.4.28) v`a ch´u limi→∞ ´y rˇa`ng f (˜x∗) ≤ f (x∗) − 3s theo Bˆo˙’ d¯ˆe` 4.4.8, ta d¯u.o. . c
i→∞
p(˜xi)(cid:1). (cid:0)f (˜xi) + lim sup x∈ ¯B(x∗, ξ(s))∩D
˜f (x) = lim i→∞ ≤ f (˜x∗) + s ≤ f (x∗) − 3s + s ≤ f (x∗) + p(x∗) − s = ˜f (x∗) − s.
Suy ra
˜f (x) ≤ ˜f (x) − s. sup x∈ ¯B(x∗, ξ(s))∩D sup x∈ ¯B(x∗, ξ(s))∩D
Biˆe˙’u th´u.c nhˆa. n d¯u.o. . c l`a vˆo l´y, nˆen ˜x∗ ∈ B(x∗, ξ(s)) ∩ D, do d¯´o suy ra ˜x∗ ∈ D l`a d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a ˜f = f + p trˆen D. Ngo`ai ra v`ı
˜x∗ ∈ ¯B(x∗, ξ(s)) nˆen
(cid:107)˜y∗ − x∗(cid:107) max x∗∈Slocal(0)
inf ˜y∗∈Slocal(p) (cid:107)˜x∗ − x∗(cid:107) d(cid:0)x∗, Slocal(p)(cid:1) = ≤
ξ(s) ≤
= ξ((cid:107)p(cid:107)C 0). max x∗∈Slocal(0) max x∗∈Slocal(0) max x∗∈Slocal(0) max x∗∈Slocal(0)
T´om la. i ta nhˆa. n d¯u.o. . c
x∗∈Slocal(0)
d(cid:0)x∗, Slocal(p)(cid:1) ≤ ξ((cid:107)p(cid:107)C 0). ∀p ∈ ¯BC 0(0, s0) : max
. c ch´u.ng minh.
D- i.nh l´y d¯˜a d¯u.o. Mˆe. nh d¯ˆe` 4.4.30. H`am d¯a tri. Slocal(p) l`a nu.˙’a liˆen tu. c du.´o.i ta. i d¯iˆe˙’m 0.
Ch´u.ng minh. Theo d¯i.nh ngh˜ıa h`am nu.˙’a liˆen tu. c du.´o.i, nˆe´u Slocal(0) = ∅ th`ı Slocal(p) nu.˙’a liˆen tu. c du.´o.i ta. i 0 l`a hiˆe˙’n nhiˆen.
. p Slocal(0) (cid:54)= ∅. Lˆa´y tˆa. p mo.˙’ bˆa´t k`y V ⊂ IRn tho˙’a
Ta x´et tru.`o.ng ho. m˜an Slocal(0) ∩ V (cid:54)= ∅. Khi d¯´o
∃ x∗ ∈ Slocal(0) : x∗ ∈ V.
V`ı V l`a tˆa. p mo.˙’ , ta suy ra tˆo`n ta. i δ > 0 sao cho ¯B(x∗, δ) ⊂ V. Mˇa. t kh´ac, do ξ(s) d¯u.o. . c x´ac d¯i.nh bo.˙’ i biˆe˙’u th´u.c (4.4.26) nˆen
(cid:113) (cid:16) (cid:17) − η0 − /(2λmax) lim s→0 ξ(s) = lim s→0
η2 0 − 12λmaxs = (cid:0) − η0 − (−η0)(cid:1)/(2λmax) = 0.
Biˆe˙’u th´u.c n`ay cho ph´ep ta cho. n sˆo´ du.o.ng s1 ≤ s0 sao cho v´o.i mo. i s ∈ ]0, s1] th`ı ξ(s) ≤ δ, v`ı vˆa. y
¯B(cid:0)x∗, ξ(s)(cid:1) ∩ D ⊂ ¯B(x∗, δ) ∩ D ⊂ V. (4.4.29)
Mˇa. t kh´ac, theo D- i.nh l´y 4.4.25, v´o.i mo. i p ∈ C 0(D) tho˙’a m˜an (cid:107)p(cid:107)C 0 ≤ s0 . c x´ac d¯i.nh theo (4.4.25)) tˆo`n ta. i ˜x∗ ∈ ¯B(cid:0)x∗, ξ((cid:107)p(cid:107)C 0)(cid:1) ∩ D l`a d¯iˆe˙’m (d¯u.o.
supremum d¯i.a phu.o.ng cu˙’a ˜f = f + p, t´u.c l`a ˜x∗ ∈ Slocal(p). V`ı s1 ≤ s0 nˆen kˆe´t ho. . p v´o.i biˆe˙’u th´u.c (4.4.29) ta nhˆa. n d¯u.o. . c
∀p ∈ ¯BC 0(0, s1) : V ∩ Slocal(p) (cid:54)= ∅,
Do d¯´o Slocal(p) l`a nu.˙’a liˆen tu. c du.´o.i ta. i d¯iˆe˙’m 0.
T`u. d¯i.nh ngh˜ıa 4.3.14 ta suy ra h`am Slocal(p) khˆong nu.˙’a liˆen tu. c trˆen ta. i 0 nˆe´u tˆo`n ta. i lˆan cˆa. n V tho˙’a m˜an Slocal(0) ⊆ V v`a d˜ay (pi), i = 1, 2 . . . hˆo. i tu. vˆe` 0 trong C 0(D) sao cho Sglobal(pi) \ V (cid:54)= ∅ v´o.i mo. i i = 1, 2, . . . . V´ı du. sau d¯ˆay chı˙’ ra Slocal(p) khˆong nu.˙’a liˆen tu. c trˆen ta. i 0.
V´ı du. 4.4.14. Cho
f (x) = x2, x ∈ [ 0, 2 ] (cid:40) 1/i − 2x2 nˆe´u x ∈ [ 0, (cid:112)1/i ] pi(x) = 0 nˆe´u x ∈ [ 0, 2 ] \ [ 0, (cid:112)1/i ]
i = 1, 2 . . .
Ta t´ınh d¯u.o. . c (cid:107)pi(cid:107)0
C(D) = 1/i, Slocal(0)={2} v`a Slocal(pi) = {0, 2}, i = 1, 2, . . . . Lˆa´y tˆa. p mo.˙’ V = ]1.5, 2.1[ ta c´o {2} = Slocal(0) ⊂ V = ]1.5, 2.1[ . Trong khi d¯´o v´o.i mo. i i th`ı 0 ∈ Slocal(pi) nhu.ng 0 /∈ V, nˆen suy ra Slocal(pi) \ V (cid:54)= ∅. Do d¯´o Slocal(p) khˆong nu.˙’a liˆen tu. c trˆen ta. i 0. Kˆe´t luˆa. n: C´ac kˆe´t qua˙’ d¯a. t d¯u.o. . c tr`ınh b`ay trong c´ac Mu. c 4.1–4.4, ch´ung bao gˆo`m: mˆo. t sˆo´ d¯iˆe` u kiˆe.n d¯u˙’ d¯ˆe˙’ h`am to`an phu.o.ng lˆo`i ngˇa. t bi. nhiˆe˜u gi´o.i nˆo. i l`a γ-lˆo`i trong (Mˆe.nh d¯ˆe` 4.1.24); c´ac t´ınh chˆa´t cu˙’a c´ac d¯iˆe˙’m cu. . c d¯a. i v`a supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q) (c´ac mˆe.nh d¯ˆe` 4.2.25, 4.2.26, 4.2.27, 4.2.28); t´ınh ˆo˙’n d¯i.nh, nu.˙’a liˆen tu. c trˆen cu˙’a h`am tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q) (D- i.nh l´y 4.3.24, Mˆe.nh d¯ˆe` 4.3.29); t´ınh ˆo˙’n d¯i.nh, nu.˙’a liˆen tu. c du.´o.i cu˙’a h`am tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜Q) (D- i.nh l´y 4.4.25, Mˆe.nh d¯ˆe` 4.4.30).
. c, co. ba˙’n d¯u.o.
KˆE´T LU ˆA. N CHUNG
. c c´ac vˆa´n d¯ˆe` : 1. Luˆa. n ´an d¯˜a gia˙’i quyˆe´t d¯u.o.
. c tiˆe˙’u cu˙’a ˜f l`a d¯iˆe˙’m cu.
. c tr`ınh b`ay. C´ac kˆe´t qua˙’ trˆen d¯˜a d¯u.o.
• Chı˙’ ra h`am bi. nhiˆe˜u ˜f = f + p l`a γ-lˆo`i ngo`ai v´o.i mo. i γ ≥ γ∗, trong d¯´o γ∗ = 2(cid:112)2s/λmin; d¯iˆe˙’m γ∗-cu. . c tiˆe˙’u to`an cu. c; d¯u.`o.ng k´ınh cu˙’a tˆa. p c´ac d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an ( ˜P ) nho˙’ ho.n hoˇa. c bˇa`ng γ∗; khoa˙’ng c´ach gi˜u.a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a B`ai to´an ( ˜P ) v`a d¯iˆe˙’m cu. . c tiˆe˙’u to`an cu. c cu˙’a h`am f nho˙’ ho.n hoˇa. c bˇa`ng γ∗. Ngo`ai ra t´ınh chˆa´t tu. . a thˆo v`a mˆo. t sˆo´ d¯iˆe` u kiˆe.n tˆo´i u.u suy rˆo. ng cu˙’a h`am ˜f c˜ung d¯u.o. . c cˆong bˆo´ trong b`ai b´ao “Global infimum of strictly convex quadratic functions with bounded perturbation” (xem Danh mu. c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa. n ´an).
2Γ nˆe´u x∗ l`a nghiˆe.m cu.
• Ch´u.ng minh d¯u.o.
. c, h`am ˜f l`a Γ-lˆo`i ngo`ai v´o.i tˆa. p cˆan d¯ˇa. c biˆe.t Γ ⊂ IRn; d¯iˆe˙’m Γ-tˆo´i u.u d¯i.a phu.o.ng cu˙’a B`ai to´an ( ˜P ) l`a d¯iˆe˙’m tˆo´i u.u to`an cu. c; hiˆe.u cu˙’a hai nghiˆe.m tˆo´i u.u bˆa´t k`y cu˙’a B`ai to´an ( ˜P ) n`am trong tˆa. p . c tiˆe˙’u to`an cu. c cu˙’a f trˆen D v`a Γ; x∗ − ˜x∗ ∈ 1 ˜x∗ l`a nghiˆe.m tˆo´i u.u to`an cu. c bˆa´t k`y cu˙’a B`ai to´an ( ˜P ); tˆa. p nghiˆe.m tˆo´i u.u Ss cu˙’a ( ˜P ) l`a ˆo˙’n d¯i.nh theo khoa˙’ng c´ach Hausdorff dH(.,.). D- i.nh l´y Kuhn-Tucker suy rˆo. ng cho B`ai to´an ( ˜P ) c˜ung d¯u.o. . c ch´u.ng minh. C´ac kˆe´t qua˙’ trˆen d¯˜a d¯u.o. . c d¯ˇang ta˙’i trong b`ai b´ao “ Some properties of boundedly disturbed strictly convex quadratic functions” (xem Danh mu. c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa. n ´an).
2 v`a γ-lˆo`i trong ngˇa. t v´o.i γ > (2/λmin) 1 2 , mo. i d¯iˆe˙’m supremum to`an cu. c cu˙’a B`ai to´an ( ˜Q) chı˙’ c´o thˆe˙’ l`a d¯iˆe˙’m γ- cu. . c biˆen cu˙’a D v`a c´o ´ıt nhˆa´t mˆo. t d¯iˆe˙’m l`a γ-cu. . c biˆen ngˇa. t. Mˆo. t sˆo´ t´ınh chˆa´t quan tro. ng cu˙’a tˆa. p c´ac d¯iˆe˙’m supremum to`an cu. c Sglobal(p) v`a tˆa. p c´ac d¯iˆe˙’m supremum d¯i.a phu.o.ng Slocal(p) cu˙’a B`ai to´an ˜Q nhu. t´ınh ˆo˙’n d¯i.nh v`a t´ınh nu.˙’ a
• Chı˙’ ra h`am ˜f l`a γ-lˆo`i trong v´o.i γ ≥ (2/λmin) 1 2 ; khi D bi. chˇa. n v`a γ = (2/λmin) 1
. c chı˙’ ra. Phˆa` n l´o.n c´ac kˆe´t qua˙’ d¯u.o.
. c liˆe.t kˆe o.˙’ trˆen liˆen tu. c c˜ung d¯u.o. d¯˜a d¯u.o. . c cˆong bˆo´ trong b`ai b´ao “Maximizing strictly convex quadratic functions with bounded perturbation” (xem Danh mu. c c´ac cˆong tr`ınh cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa. n ´an).
2. Nh˜u.ng vˆa´n cˆa` n tiˆe´p tu. c nghiˆen c´u.u:
Luˆa. n ´an chı˙’ m´o.i d¯ˆe` cˆa. p d¯ˆe´n mˆo. t sˆo´ vˆa´n d¯ˆe` vˆe` l´y thuyˆe´t cu˙’a B`ai to´an quy hoa. ch to`an phu.o.ng lˆo`i ngˇa. t v´o.i nhiˆe˜u gi´o.i nˆo. i. Do d¯´o ch´ung tˆoi c`on tiˆe´p tu. c nghiˆen c´u.u nh˜u.ng vˆa´n d¯ˆe` sau d¯ˆay.
• Xˆay du. . ng thuˆa. t to´an t´ınh to´an t`ım l`o.i gia˙’i tˆo´i u.u cu˙’a c´ac b`ai to´an
( ˜P ) v`a ( ˜Q).
• ´Ap du. ng thuˆa. t to´an t´ınh to´an t`ım l`o.i gia˙’i tˆo´i u.u cu˙’a c´ac b`ai to´an ( ˜P ) . c tˆe´ nhu. b`ai to´an ph´at d¯iˆe.n tˆo´i u.u, kinh
v`a ( ˜Q) v`ao c´ac b`ai to´an thu. tˆe´ d¯ˆo´i s´anh,. . .
. c c´ac . c gia˙’i
Tuy nhiˆen, v`ı th`o.i gian ha. n he.p nˆen ch´ung tˆoi c˜ung chu.a tra˙’ l`o.i d¯u.o. vˆa´n d¯ˆe` trˆen. Ch´ung tˆoi hy vo. ng rˇa`ng c´ac vˆa´n d¯ˆe` n`ay s˜e s´o.m d¯u.o. quyˆe´t.
DANH MU. C C ˆONG TR`INH CU˙’ A T ´AC GIA˙’ LIˆEN QUAN D- ˆE´N LU ˆA. N ´AN
C´ac b`ai b´ao cu˙’a t´ac gia˙’ liˆen quan d¯ˆe´n luˆa. n ´an l`a:
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functions with bounded perturbation, Mathematical Methods of Oper-
ations Research, 72(2), 2010, 327–345.
2. H. X. Phu and V. M. Pho, Some properties of boundedly disturbed
strictly convex quadratic functions, Optimization, DOI 10.1080/02331-
93100746114, Published online: 07 May 2010.
3. H. X. Phu, V. M. Pho and P. T. An, Maximizing strictly convex
quadratic functions with bounded perturbation, Journal of Optimiza-
tion Theory and Applications, 149(1) 2011, 1–25.
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