tóm tắt bài toán quy hoạch toàn phương lồi ngặt với nhiễu giới nội

MO
.
˙’
D
-
ˆ
A
`
U
B`ai to´an quy hoa
.
ch to`an phu
.
o
.
ng truyˆe
`
n thˆo
´
ng c´o da
.
ng
f(x) := Ax, x + b, x → inf, x ∈ D
trong d¯´o A ∈ IR
n×n
l`a ma trˆa
.
n vuˆong, b ∈ IR
n
l`a v´ec to
.

v`a D ⊂ IR
n
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 d¯u
.
o
.
.
c
nhiˆe
`
u nh`a to´an ho
.

c trong v`a ngo`ai nu
.
´o
.
c nghiˆen c´u
.
u, v´ı du
.
nhu
.
H. W. Kuhn
v`a A. W. Tucker (1951), B. Bank v`a R. Hasel (1984), E. Blum v`a W. Oettli
(1973), B. C. Eaves (1971), M. Frank v`a P. Wolfe (1956), O. L. Magasarian
(1980), G. M. Lee, N. N. Tam v`a N. D. Yen (2005), H. X. Phu (2007), H.
X. Phu v`a N. D. Yen (2001), M. Schweighofer (2006), H. Tuy (1964, 1983,
2007), H. H. Vui v`a P. T. Son (2008). . .
Khi A l`a ma trˆa
.
n 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 phˆan r˜a th`anh c´ac b`ai to´an kh´ac nhau sau:
f(x) := Ax, x + b, x → inf, x ∈ D (P )
v`a
f(x) := Ax, x + b, x → 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) := Ax, x + b, x + p(x) → inf, x ∈ D (
˜
P )
v`a
˜
f(x) := Ax, x + b, x + p(x) → sup, x ∈ D, (
˜
Q)
trong d¯´o p : D → IR l`a tho
˙’
a m˜an d¯iˆe
`
u kiˆe
.
n sup
x∈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, 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
˙’
ch´ung tˆoi tiˆe
´
n h`anh nghiˆen c´u
.
u c´ac b`ai to´an trˆen, tˆo
´
i thiˆe
˙’
u t`u

.
1
quan d¯iˆe
˙’
m l´y thuyˆe
´
t. Tuy nhiˆen, c`on mˆo
.
t sˆo
´
l´y do thu
.
.
c tˆe
´
kh´ac du
.
´o
.
i d¯ˆay,
cho thˆa
´
y viˆe
.
c nghiˆen c´u
.
u c´ac b`ai to´an (
˜
P ), (
˜

Q) l`a thu
.
.
c su
.
.
cˆa
`
n.
L´y do th´u
.
nhˆa
´
t: f(x) = Ax, x + b, x 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
˙’
d¯u
.
o
.
.
c
tho
˙’
a m˜an trong nhiˆe
`

u b`ai to´an thu
.
.
c tˆe
´
chˇa
˙’
ng ha
.
n nhu
.
hai v´ı 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 (1952, 1959)). 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
˙’
c´o thˆe
˙’
gia
˙’
m thiˆe
˙’
u ru
˙’
i ro v`a tˆo
´
i d¯a lo
.
.
i nhuˆa
.
n, t´u
.
c l`a t`ım v´ec to
.
tı
˙’
lˆe
.
x ∈ D, D := {x = (x
1
, x
2

, . . . , x
n
) |

n
j=1
x
j
= 1} d¯ˆe
˙’
f(x) = ωx
T
Ax − ρ
T
x
d¯a
.
t gi´a tri
.
nho
˙’
nhˆa
´
t, trong d¯´o x
j
, 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, A ∈ IR
n×n
l`a ma trˆa
.
n hiˆe
.
p
phu
.
o
.
ng sai, ρ ∈ IR
n
l`a v´ec to
.
lo

.
.
i nhuˆa
.
n k`y vo
.
ng. V`ı A v`a ρ thu
.
`o
.
ng
khˆong d¯u
.
o
.
.
c x´ac d¯i
.
nh ch´ınh x´ac m`a chı
˙’
xˆa
´
p xı
˙’
bo
.
˙’
i
˜
A v`a ˜ρ, do d¯´o ch´ung

ta pha
˙’
i cu
.
.
c tiˆe
˙’
u h´oa h`am
˜
f(x) = ωx
T
˜
Ax − ˜ρ
T
x = f(x) + p(x), trong d¯´o
p(x) = ωx
T
(
˜
A − A)x − (˜ρ − ρ)
T
x. Khi quy d¯i
.
nh khˆong d¯u
.
o
.
.
c b´an khˆo
´

ng, t´u
.
c
l`a x
j
≥ 0, j = 1, . . . , n, th`ı tˆa
.
p chˆa
´
p nhˆa
.
n d¯u
.
o
.
.
c D l`a gi´o
.
i nˆo
.
i. V`ı vˆa
.
y 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¯a
˙’
m ba
˙’
o khi D gi´o
.
i nˆo
.
i v`a p liˆen tu
.
c trˆen D. Gia
˙’
thiˆe
´
t n`ay l`a ph`u ho
.
.
p v´o

.
i
nhiˆe
`
u b`ai to´an thu
.
.
c tˆe
´
.
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) = Ax, x+ b, x 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 trˆen s ∈ [0, +∞[ n`ao
d¯´o c´o thˆe
˙’
u
.
´o
.
c lu
.
o
.
.
ng d¯u
.
o
.

.
c, t´u
.
c l`a sup
x∈D
|p(x)| ≤ s. 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.
2
L´y do th´u
.
hai:
˜
f l`a h`am mu
.
c tiˆeu d¯´ıch thu
.
.
c v`a f l`a h`am mu
.
c tiˆeu d¯u
.
o
.
.
c
l´y tu
.
o
.

˙’
ng h´oa hoˇa
.
c l`a h`am mu
.
c tiˆeu thay thˆe
´
. Trong thu
.
.
c tiˆe
˜
n, nhiˆe
`
u h`am
thˆe
˙’
hiˆe
.
n mˆo
.
t sˆo
´
mu
.
c tiˆeu thu
.
.
c tˆe
´

d¯u
.
o
.
.
c gia
˙’
thiˆe
´
t l`a lˆo
`
i, hoˇa
.
c to`an phu
.
o
.
ng,
hoˇa
.
c c´o mˆo
.
t sˆo
´
t´ınh chˆa
´
t thuˆa
.
n tiˆe
.

n d¯˜a d¯u
.
o
.
.
c nghiˆen c´u
.
u k˜y, hoˇa
.
c dˆe
˜
nghiˆen
c´u
.
u, nhu
.
ng thu
.
.
c tˆe
´
khˆong pha
˙’
i l`a nhu
.
vˆa
.
y. D
-
iˆe

`
u n`ay d¯˜a d¯u
.
o
.
.
c H. X. Phu,
H. G. Bock v`a S. Pickenhain (2000) d¯ˆe
`
cˆa
.
p d¯ˆe
´
n. 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
.
.
c nghiˆen c´u
.
u
cu
˙’
a ph´at d¯iˆe
.
n tˆo
´
i u
.
u, t´u
.
c l`a vˆa
´
n d¯ˆe

`
phˆan bˆo
´
lu
.
o
.
.
ng d¯iˆe
.
n nˇang cho t`u
.
ng tˆo
˙’
m´ay ph´at nhiˆe
.
t d¯iˆe
.
n sao cho tˆo
˙’
ng chi ph´ı (gi´a th`anh) l`a cu
.
.
c tiˆe
˙’
u, d¯ˆo
`
ng th`o
.
i

vˆa
˜
n d¯´ap ´u
.
ng d¯u
.
o
.
.
c nhu cˆa
`
u lu
.
o
.
.
ng d¯iˆe
.
n nˇang v`a thoa
˙’
m˜an r`ang buˆo
.
c vˆe
`
cˆong
suˆa
´
t ph´at ra cu
˙’
a mˆo

˜
i tˆo
˙’
m´ay. Ngu
.
`o
.
i ta thu
.
`o
.
ng gia
˙’
thiˆe
´
t (P. P. J. Van den
Bosch v`a F. A. Lootsma (1987), R. M. S. Danaraj v`a F. Gajendran (2005)),
h`am chi ph´ı tˆo
˙’
ng cˆo
.
ng (bao gˆo
`
m 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
F (P ) =

n

i=1
F
i
(P
i
),
trong d¯´o n l`a sˆo
´
tˆo
˙’
m´ay ph´at, P := (P
1
, P
2
, . . . , P
n
), P
i
∈ [P
i min
, P
i max
] l`a
lu
.
o
.
.

ng d¯iˆe
.
n nˇang ph´at ra cu
˙’
a tˆo
˙’
m´ay th´u
.
i, P
i min
, P
i 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 v`a F

i
(P
i
) = a
i
+ b
i
P
i
+ c
i
P
2
i
l`a
h`am chi ph´ı cu
˙’
a tˆo
˙’
m´ay ph´at th´u
.
i ∈ {1, 2, . . . , n}.
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
tu
.
o
.
˙’
ng. Chi ph´ı thu
.
.
c tˆe
´
c´o thˆe
˙’
khˆong l`a h`am to`an phu
.
o
.
ng v`a c˜ung khˆong 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 ph´ı thu
.
.
c
tˆe
´
. D
-
ˇa
.
c biˆe
.
t, nˆe
´
u hiˆe
.
u ´u
.

ng d¯iˆe
˙’
m-van d¯u
.
o
.
.
c x´et d¯ˆe
´
n (P. P. J. van den Bosch
v`a F. A. Lootsma (1987), R. M. S. Danaraj v`a F. Gajendran (2005),. . . ) th`ı
h`am chi ph´ı to`an phu
.
o
.
ng pha
˙’
i d¯u
.
o
.
.
c hiˆe
.
u chı
˙’
nh bo
.
˙’
i tˆo

˙’
ng h˜u
.
u ha
.
n c´ac h`am
3
da
.
ng sin, t´u
.
c l`a
F (P ) =
n

i=1

F
i
(P
i
) + |e
i
sin(f
i
(P
i min
− P
i
))|


,
trong d¯´o e
i
, f
i
l`a c´ac hˆe
.
sˆo
´
cu
˙’
a hiˆe
.
u ´u
.
ng d¯iˆe
˙’
m-van. R˜o r`ang h`am hiˆe
.
u chı
˙’
nh
p :=

n
i=1
|e
i
sin(f

i
(P
i min
− P
i
))| l`a gi´o
.
i nˆo
.
i.
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 co
.
ba
˙’
n khi nghiˆen c´u
.

u c´ac b`ai to´an (
˜
P ) v`a (
˜
Q)?
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 (2008), D. Klatte (1997), B. Kumer
(1984), . . . ) 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 (1973), G. M. Lee, N. N. Tam v`a N.
D. Yen (2005), K. Mirnia v`a A. Ghaffari-Hadigheh (2007), H. X. Phu (2007),
H. X. Phu v`a N. D. Yen (2001). . . ). 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 tr`ınh b`ay b`ai to´an quy hoa
.
ch lˆo
`
i, b`ai to´an quy hoa
.
ch to`an
phu
.
o
.

ng, 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.

Chu
.
o
.
ng 2 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, c´ac t´ınh chˆa
´
t cu
˙’
a d¯iˆe
˙’
m cu

.
.
c tiˆe
˙’
u to`an cu
.
c, d¯iˆe
˙’
m infimum to`an cu
.
c cu
˙’
a
4
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 nghiˆen c´u
.
u t´ınh Γ-lˆo
`
i ngo`ai cu
˙’
a h`am mu
.
c tiˆeu
˜
f (theo c´ach
tiˆe
´
p cˆa
.
n tˆo pˆo), qua d¯´o nhˆa
.
n d¯u
.
o
.

.
c mˆo
.
t sˆo
´
kˆe
´
t qua
˙’
ma
.
nh ho
.
n nh˜u
.
ng kˆe
´
t qu
˙’
a
nghiˆen c´u
.
u vˆe
`
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 ) d¯u
.
o
.
.
c chı
˙’
ra trong Chu
.
o
.
ng 2.
Chu
.
o
.
ng 4 nghiˆen c´u
.

u t´ınh γ-lˆo
`
i trong cu
˙’
a h`am mu
.
c tiˆeu
˜
f, 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).
Luˆa
.
n ´an d¯u
.
o
.
.
c ho`an th`anh du
.
´o
.
i su
.
.
hu
.
´o

.
ng dˆa
˜
n 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 t´ac gia
˙’
trong qu´a tr`ınh nghiˆen c´u
.
u. T´ac gia
˙’
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
.
.
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
Quˆan su
.
.
d¯˜a ta
.
o mo
.
i d¯iˆe
`
u kiˆe
.
n thuˆa
.
n lo
.

.
i d¯ˆe
˙’
t´ac gia
˙’
c´o nhiˆe
`
u th`o
.
i gian thu
.
.
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
˙’
xin d¯u
.
o
.
.
c gu
.
˙’

i l`o
.
i ca
˙’
m o
.
n sˆau sˇa
´
c t´o
.
i 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
`
chuyˆen
mˆon c˜ung nhu
.
thu
˙’
tu
.
c h`anh ch´ınh d¯ˆe
˙’
t´ac gia
˙’
c´o thˆe
˙’
ho`an th`anh luˆa
.
n ´an n`ay.
5
CHU
.
O
.
NG 1
B
`
AI TO
´

AN QUY HOA
.
CH L
ˆ
O
`
I,
QUY HOA
.
CH TO
`
AN PHU
.
O
.
NG V
`
A H
`
AM L
ˆ
O
`
I TH
ˆ
O
Trong suˆo
´
t luˆa
.

n ´an n`ay, ta luˆon k´y hiˆe
.
u IR
n
l`a khˆong gian Euclide n
chiˆe
`
u, A ∈ IR
n×n
l`a ma trˆa
.
n d¯ˆo
´
i x´u
.
ng x´ac d¯i
.
nh du
.
o
.
ng, λ
min
, λ
max
tu
.
o
.
ng ´u

.
ng,
l`a c´ac gi´a tri
.
riˆeng nho
˙’
nhˆa
´
t, l´o
.
n nhˆa
´
t cu
˙’
a A, b ∈ IR
n
v`a
• f l`a h`am to`an phu
.
o
.
ng lˆo
`
i ngˇa
.
t c´o da
.
ng
f(x) := Ax, x + b, x, x ∈ D (1.0.1)
• p : D → IR l`a h`am nhiˆe

˜
u gi´o
.
i nˆo
.
i, ngh˜ıa l`a p tho
˙’
a m˜an
sup
x∈D
|p(x)| ≤ s < +∞. (1.0.2)
•
˜
f := f + p d¯u
.
o
.
.
c go
.
i l`a h`am to`an phu
.
o
.
ng lˆo
`
i ngˇa
.
t v´o
.

i nhiˆe
˜
u gi´o
.
i nˆo
.
i, go
.
i
tˇa
´
t l`a h`am bi
.
nhiˆe
˜
u gi´o
.
i nˆo
.
i.
1.1. B`ai to´an quy hoa
.
ch lˆo
`
i, quy hoa
.
ch to`an phu
.
o
.

ng
Trong mu
.
c n`ay, ch´ung tˆoi ph´at biˆe
˙’
u
• D
-
i
.
nh l´y Kuhn-Tucker cho b`ai to´an quy hoa
.
ch lˆo
`
i
g
0
(x) → inf, x ∈ D
D = {x ∈ S | g
i
(x) ≤ 0, i = 1, . . . , m},
(L)
trong d¯´o g
i
: IR
n
→ IR, i = 0, . . . , m, l`a c´ac h`am h`am lˆo
`
i, S ⊂ IR
n

l`a
tˆa
.
p lˆo
`
i.
• D
-
i
.
nh l´y vˆe
`
d¯iˆe
`
u kiˆe
.
n tˆo
`
n ta
.
i nghiˆe
.
m tˆo
´
i u
.
u cho b`ai to´an quy hoa
.
ch to`an
phu

.
o
.
ng
Mx, x + b, x → inf, x ∈ D
D = {x ∈ IR
n
| c
i
, x ≤ d
i
, i = 1, . . . , m},
trong d¯´o M ∈ IR
n×n
l`a ma trˆa
.
n d¯ˆo
´
i x´u
.
ng, c
i
∈ IR
n
, i = 1, . . . , m.
C´ac d¯i
.
nh l´y n`ay s˜e d¯u
.
o

.
.
c mo
.
˙’
rˆo
.
ng trong c´ac chu
.
o
.
ng 2 v`a 3.
6
1.2. H`am lˆo
`
i suy rˆo
.
ng thˆo
Trong mu
.
c n`ay ch´ung tˆoi tr`ınh b`ay tˆo
˙’
ng quan vˆe
`
kh´ai niˆe
.
m h`am lˆo
`
i thˆo
v`a mˆo

.
t sˆo
´
t´ınh chˆa
´
t quan tro
.
ng cu
˙’
a c´ac l´o
.
p h`am n`ay.
1.3. H`am γ-lˆo
`
i ngo`ai
Trong mu
.
c n`ay ch´ung tˆoi tr`ınh b`ay vˆe
`
h`am γ-lˆo
`
i ngo`ai v`a mˆo
.
t sˆo
´
t´ınh
chˆa
´
t tˆo
´

i u
.
u cu
˙’
a l´o
.
p h`am n`ay.
D
-
i
.
nh ngh˜ıa 1.3.3. (H. X. Phu) Cho γ > 0. H`am g : D ⊂ IR
n
→ 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 x
0
, x
1
∈ D
tˆo
`
n ta
.
i k ∈ IN v`a
λ
i
∈ [0, 1], i = 0, 1, . . . , k, λ
0
= 0, λ
k
= 1,
0 ≤ λ
i+1
− λ
i

≤
γ
x
0
− x
1

khi i = 0, 1, . . . , k − 1,
sao cho v´o
.
i x
λ
i
= (1 − λ
i
)x
0
+ λ
i
x
1
, i = 0, 1, . . . , k, th`ı
g(x
λ
i
) ≤ (1 − λ
i
)g(x
0
) + λ

i
g(x
1
) v´o
.
i i = 0, 1, . . . , k,
(hoˇa
.
c
g(x
λ
i
) < (1 − λ
i
)g(x
0
) + λ
i
g(x
1
) v´o
.
i i = 1, . . . , k − 1).
D
-
i
.
nh ngh˜ıa 1.3.4. (H. X. Phu) Cho γ > 0, M ⊂ IR
n
, M = ∅, M d¯u

.
o
.
.
c go
.
i
l`a γ-lˆo
`
i ngo`ai v´o
.
i d¯ˆo
.
thˆo γ nˆe
´
u x
0
, x
1
∈ M v`a x
0
− x
1
 > γ suy ra tˆo
`
n ta
.
i
z
0

:= x
0
, z
1
, . . . , z
k
:= x
1
∈ [x
0
, x
1
] ∩ M sao cho
z
i+1
− z
i
 ≤ γ v´o
.
i i=0, 1,. . . , k-1.
D
-
i
.
nh ngh˜ıa 1.3.5. (H. X. Phu) D
-
iˆe
˙’
m x
∗

∈ D d¯u
.
o
.
.
c go
.
i l`a
1) d¯iˆe
˙’
m γ-cu
.
.
c tiˆe
˙’
u cu
˙’
a g nˆe
´
u tˆo
`
n ta
.
i  > 0 sao cho g(x
∗
) ≤ g(x) v´o
.
i mo
.
i

x ∈ B(x
∗
, γ + ) ∩ D;
2) d¯iˆe
˙’
m γ-infimum cu
˙’
a g nˆe
´
u tˆo
`
n ta
.
i  > 0 sao cho
lim inf
x→x
∗
g(x) = inf
x∈B(x
∗
,γ+)∩D
g(x);
7
3) d¯iˆe
˙’
m infimum to`an cu
.
c cu
˙’
a g nˆe

´
u
lim inf
x→x
∗
g(x) = inf
x∈D
g(x).
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. (H. X. Phu) Nˆe
´
u g l`a γ-lˆo
`
i ngo`ai th`ı c´o c´ac t´ınh chˆa
´
t
(M
γ
) 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.
(I
γ
) Mˆo
˜
i d¯iˆe
˙’
m γ-infimum x
∗
cu
˙’
a g l`a d¯iˆe
˙’
m infimum to`an cu
.
c.
D
-
ˆo
´
i v´o
.
i h`am lˆo
`

i ngˇa
.
t v´o
.
i nhiˆe
˜
u gi´o
.
i nˆo
.
i ta c´o mˆe
.
nh d¯ˆe
`
sau vˆe
`
t´ınh γ-lˆo
`
i
ngo`ai v`a lˆo
`
i ngo`ai ngˇa
.
t.
Mˆe
.
nh d¯ˆe
`
1.3.1. (H. X. Phu) Cho γ > 0, g : IR
n

→ IR l`a h`am lˆo
`
i v`a
h
1
(γ) := inf
x
0
,x
1
∈D, x
0
−x
1
=γ

1
2
(g(x
0
) + g(x
1
)) − g

1
2
(x
0
+ x
1

)


> 0.
Khi d¯´o, nˆe
´
u h`am nhiˆe
˜
u p tho
˙’
a m˜an
|p(x)| ≤ h
1
(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)| < h
1
(γ)/2 v´o

.
i mo
.
i x ∈ D
th`ı ˜g = g + p l`a γ-lˆo
`
i ngo`ai ngˇa
.
t.
1.4. H`am Γ-lˆo
`
i ngo`ai
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. Ch´ung s˜e l`a co

.
so
.
˙’
d¯ˆe
˙’
nghiˆen c´u
.
u B`ai to´an (
˜
P ) trong Chu
.
o
.
ng 3.
D
-
i
.
nh ngh˜ıa 1.4.6. (H. X. Phu) 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 x
0

, x
1
∈ D
tˆo
`
n ta
.
i tˆa
.
p d¯´ong Λ ⊂ [0, 1] v`a ch´u
.
a {0, 1} sao cho
[x
0
, x
1
] ⊂ {x
λ
| λ ∈ Λ} + 0.5Γ (1.4.4)
v`a
∀λ ∈ Λ : g(x
λ
) ≤ (1 − λ)g(x
0
) + λg(x
1
). (1.4.5)
8
D
-

i
.
nh ngh˜ıa 1.4.7. (H. X. Phu) Tˆa
.
p S ⊂ X d¯u
.
o
.
.
c go
.
i l`a Γ-lˆo
`
i ngo`ai nˆe
´
u v´o
.
i
mo
.
i x
0
, x
1
∈ S
[x
0
, x
1
] ⊂ ([x

0
, x
1
] ∩ S) + 0.5Γ,
t´u
.
c l`a tˆo
`
n ta
.
i Λ ⊂ [0, 1] sao cho
{x
λ
| λ ∈ Λ} ⊂ S, [x
0
, x
1
] ⊂ {x
λ
| λ ∈ Λ} + 0.5Γ. (1.4.6)
Mˆe
.
nh d¯ˆe
`
1.4.2. (H. X. Phu) 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. (H. X. Phu) 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 1.4.8. (H. X. Phu) 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
∗

) = inf
x∈(x
∗
+Γ)∩D
g(x)
v`a go
.
i l`a Γ-infimum cu
˙’
a g nˆe
´
u
lim inf
x∈X, x→x
∗
g(x) = inf
x∈(x
∗
Γ
)∩D
g(x).
T´ınh chˆa
´
t tˆo
´
i u
.
u quan tro
.
ng cu

˙’
a h`am Γ-lˆo
`
i ngo`ai l`a d¯i
.
nh l´y sau:
D
-
i
.
nh l´y 1.4.9. (H. X. Phu) 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
∗
) = inf
x∈D∩({x
∗

}+Γ)
g(x) =⇒ g(x
∗
) = inf
x∈D
g(x), (1.4.7)
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.
1.5. H`am γ-lˆo
`
i trong
Trong mu
.
c n`ay ch´ung tˆoi tr`ınh b`ay kh´ai niˆe
.
m v`a mˆo
.
t sˆo
´
kˆe
´
t qua
˙’
vˆe
`
h`am γ-lˆo
`
i trong, ch´ung s˜e d¯u
.
o
.
.
c su
.
˙’
du
.

ng d¯ˆe
˙’
nghiˆen c´u
.
u B`ai to´an (
˜
Q) trong
Chu
.
o
.
ng 4.
9
D
-
i
.
nh ngh˜ıa 1.5.9. (H. X. Phu) H`am g : D ⊂ IR
n
→ 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 x
0
, x
1
∈ D tho
˙’
a m˜an x
0

− x
1
 = νγ
v`a x
1+1/ν
= −(1/ν)x
0
+ (1 + 1/ν)x
1
∈ D,
th`ı
sup
λ∈[2,1+1/ν]

g((1 − λ)x
0
+ λx
1
) − (1 − λ)g(x
0
) − λg(x
1
)

≥ 0,
(hoˇa
.
c
sup
λ∈[2,1+1/ν]


g((1 − λ)x
0
+ λx
1
) − (1 − λ)g(x
0
) − λg(x
1
)

> 0,
tu
.
o
.
ng ´u
.
ng).
Mˆe
.
nh d¯ˆe
`
1.5.6. (H. X. Phu) Gia
˙’
su
.
˙’
g : D → IR l`a γ-lˆo
`

i trong v´o
.
i d¯ˆo
.
tinh
ν. Nˆe
´
u x
1
∈ D l`a d¯iˆe
˙’
m cu
.
.
c d¯a
.
i cu
˙’
a g th`ı mo
.
i d¯iˆe
˙’
m x
0
tho
˙’
a m˜an
x
0
− x

1
 = νγ, x
1+1/ν
= −(1/ν)x
0
+ (1 + 1/ν)x
1
∈ D
c˜ung l`a d¯iˆe
˙’
m cu
.
.
c d¯a
.
i cu
˙’
a g trˆen D.
D
-
i
.
nh l´y 1.5.10. (H. X. Phu) Cho D ⊂ IR
n
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 th`ı c´o ´ıt nhˆa
´
t mˆo
.
t d¯iˆe
˙’
m cu
.
.
c d¯a
.
i
l`a d¯iˆe
˙’
m γ-cu
.

.
c biˆen ngˇa
.
t cu
˙’
a D.
D
-
i
.
nh l´y 1.5.11. (H. X. Phu) Cho g : D → IR l`a h`am γ-lˆo
`
i trong ngˇa
.
t. Nˆe
´
u
g d¯a
.
t cu
.
.
c d¯a
.
i trˆen D th`ı d¯iˆe
˙’
m cu
.
.
c d¯a

.
i l`a d¯iˆe
˙’
m γ-cu
.
.
c biˆen ngˇa
.
t cu
˙’
a D.
Mˆe
.
nh d¯ˆe
`
sau d¯ˆay chı
˙’
ra t´ınh γ-lˆo
`
i trong cu
˙’
a h`am lˆo
`
i ngˇa
.
t bi
.
nhiˆe
˜
u gi´o

.
i
nˆo
.
i.
Mˆe
.
nh d¯ˆe
`
1.5.7. (H. X. Phu) Cho g : IR
n
→ IR l`a h`am lˆo
`
i v`a
h
2
(γ) := inf
x
0
,x
1
∈D, x
0
−x
1
=γ,−x
0
+2x
1
∈D


g(x
0
) − 2g(x
1
) + g(−x
0
+ 2x
1
)

> 0
v`a γ > 0. Khi d¯´o, nˆe
´
u h`am nhiˆe
˜
u p tho
˙’
a m˜an
|p(x)| ≤ h
2
(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)| < h
2
(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.
10
CHU
.
O
.
NG 2
D
-
I

ˆ
E
˙’
M INFIMUM TO
`
AN CU
.
C CU
˙’
A B
`
AI TO
´
AN (
˜
P )
Chu
.
o
.
ng n`ay d`anh cho viˆe
.
c nghiˆen c´u
.
u t´ınh γ-lˆo
`
i ngo`ai cu
˙’
a h`am bi
.

nhiˆe
˜
u
˜
f = f +p; d¯iˆe
˙’
m infimum to`an cu
.
c; cˆa
.
n trˆen cu
˙’
a 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 cu
˙’
a B`ai to´an (
˜
P )
theo s; t´ınh chˆa
´
t tu
.
.
a v`a D
-
i
.
nh l´y Kuhn-Tucker suy rˆo
.
ng cho B`ai to´an (
˜
P ).
Trong suˆo
´
t chu
.
o
.

ng n`ay ch´ung tˆoi k´y hiˆe
.
u γ
∗
:= 2

2s/λ
min
v`a λ
min
l`a
gi´a tri
.
riˆeng nho
˙’
nhˆa
´
t cu
˙’
a ma trˆa
.
n A.
2.1. T´ınh γ-lˆo
`
i ngo`ai cu
˙’
a h`am bi
.
nhiˆe
˜

u gi´o
.
i nˆo
.
i
Mˆe
.
nh d¯ˆe
`
quan tro
.
ng vˆe
`
t´ınh γ-lˆo
`
i ngo`ai cu
˙’
a h`am bi
.
nhiˆe
˜
u ph´at biˆe
˙’
u nhu
.
sau:
Mˆe
.
nh d¯ˆe
`

2.1.11. X´et h`am to`an phu
.
o
.
ng lˆo
`
i ngˇa
.
t v´o
.
i 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
∗
.
´
Ap du
.
ng mˆe
.
nh d¯ˆe
`
trˆen v`a d¯i
.
nh l´y vˆe
`
tˆa
.
p m´u
.
c du
.
´o
.
i cu
˙’
a h`am γ-lˆo
`
i ngo`ai

ta d¯u
.
o
.
.
c:
Mˆe
.
nh d¯ˆe
`
2.1.13. Tˆa
.
p m´u
.
c du
.
´o
.
i L
α
(
˜
f) 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 l`a tˆa
.
p γ-lˆo
`
i ngo`ai v´o
.
i γ ≥ γ
∗
.
2.2. 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. 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.
11
Mˆe
.
nh d¯ˆe
`
2.2.15. K´y hiˆe
.

u arg min
˜
f l`a tˆa
.
p c´ac d¯iˆe
˙’
m cu
.
.
c tiˆe
˙’
u 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. Khi d¯´o
˜x
1
− ˜x
2
 ≤ γ
∗
v´o
.
i mo
.
i ˜x
1
, ˜x
2
∈ arg min
˜
f,
t´u
.
c l`a
diam(arg min
˜
f) ≤ γ
∗
.

2.3. C´ac t´ınh chˆa
´
t cu
˙’
a d¯iˆe
˙’
m infimum to`an cu
.
c
Trong mu
.
c n`ay, ch´ung tˆoi nghiˆen c´u
.
u t´ınh chˆa
´
t 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 s.
Mˆe
.
nh d¯ˆe
`
2.3.16. Nˆe
´
u ˜x
∗
1
, ˜x
∗
2
l`a hai d¯iˆe
˙’
m infimum to`an cu
.
c bˆa
´

t k`y cu
˙’
a B`ai
to´an (
˜
P ) th`ı
˜x
∗
1
− ˜x
∗
2
 ≤ γ
∗
.
D
-
i
.
nh l´y 2.3.13. Nˆe
´
u ˜x
∗
∈ D l`a d¯iˆe
˙’
m infimum to`an cu
.
c bˆa
´
t k`y cu

˙’
a B`ai to´an
(
˜
P ) v`a x
∗
l`a d¯iˆe
˙’
m cu
.
.
c tiˆe
˙’
u to`an cu
˙’
a B`ai to´an (P), th`ı
˜x
∗
− x
∗
 ≤ γ
∗
/2.
2.4. T´ınh chˆa
´
t tu
.
.
a v`a d¯iˆe
`

u kiˆe
.
n tˆo
´
i u
.
u cu
˙’
a B`ai to´an (
˜
P )
Trong mu
.
c n`ay, ch´ung tˆoi nghiˆen c´u
.
u t´ınh chˆa
´
t tu
.
.
a suy rˆo
.
ng cu
˙’
a h`am
˜
f = f + p, D
-
i
.

nh l´y Kuhn-Tucker suy rˆo
.
ng cho B`ai to´an (
˜
P ).
D
-
i
.
nh l´y vˆe
`
t´ınh chˆa
´
t tu
.
.
a suy rˆo
.
ng 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, d¯u
.
o
.
.
c H. X. Phu chı
˙’
ra nhu
.
sau:
Mˆe
.
nh d¯ˆe
`
2.4.17. Cho D = IR
n
. Khi d¯´o v´o
.
i x
∗
∈ IR

n
v`a  > 0 th`ı
inf
x

∈B(x
∗
,γ
∗
/2+)

˜
f(x

) − 2Ax
∗
+ b, x



≤
˜
f(x) − 2Ax
∗
+ b, x v´o
.
i mo
.
i x ∈ IR
n

,
D
-
ˇa
.
c biˆe
.
t, nˆe
´
u p l`a nu
.
˙’
a liˆen tu
.
c du
.
´o
.
i th`ı
min
x

∈
¯
B(x
∗
,γ
∗
)


˜
f(x

) − 2Ax
∗
+ b, x



≤
˜
f(x) − 2Ax
∗
+ b, x v´o
.
i mo
.
i x ∈ IR
n
.
12
Cho
D = {x ∈ S | g
i
(x) ≤ 0, i = 1, . . . , m}, (2.4.8)
trong d¯´o g
i
: IR
n
→ IR, i = 1, . . . , m, l`a c´ac h`am lˆo

`
i v`a S ⊂ IR
n
l`a tˆa
.
p lˆo
`
i
d¯´ong. H`am Lagrange cu
˙’
a B`ai to´an (
˜
P ) c´o da
.
ng sau:
L(x, µ
0
, . . . , µ
m
) := λ
0
˜
f(x) +
m

i=1
λ
i
g
i

(x).
Ta c´o d¯i
.
nh l´y sau:
D
-
i
.
nh l´y 2.4.14. (D
-
i
.
nh l´y Kuhn-Tucker suy rˆo
.
ng) Gia
˙’
su
.
˙’
D d¯u
.
o
.
.
c cho bo
.
˙’
i
cˆong th´u
.

c (2.4.8).
(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
˜x
∗
− x
∗
 ≤ γ

∗
/2
v`a c´ac nhˆan tu
.
˙’
Lagrange µ
i
≥ 0, i = 0, . . . , m, khˆong c`ung triˆe
.
t tiˆeu,
tho
˙’
a m˜an d¯iˆe
`
u kiˆe
.
n Kuhn-Tucker
L(x
∗
, µ
0
, . . . , µ
m
) = min
x∈S
L(x, µ
0
, . . . , µ
m
) (2.4.9)

v`a d¯iˆe
`
u kiˆe
.
n b`u
µ
i
g
i
(x
∗
) = 0 v´o
.
i mo
.
i i = 1, . . . , m. (2.4.10)
Nˆe
´
u d¯iˆe
`
u kiˆe
.
n Slater
∃z ∈ S : g
i
(z) < 0 v´o
.
i mo
.
i i = 1, . . . , m, (2.4.11)

tho
˙’
a m˜an th`ı µ
0
= 0 v`a c´o thˆe
˙’
coi µ
0
= 1.
(b) Nˆe
´
u tˆo
`
n ta
.
i x
∗
∈ D tho
˙’
a m˜an (2.4.9), (2.4.10) v´o
.
i µ
0
= 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
˜x
∗
− x
∗
 ≤ γ
∗
/2
v`a mo
.
i d¯iˆe
˙’
m infimum to`an cu
.
c n`ao cu
˙’
a B`ai to´an (
˜
P ) nˇa
`

m trong h`ınh
cˆa
`
u B(x
∗
, γ
∗
/2).
Da
.
ng kh´ac cu
˙’
a D
-
i
.
nh l´y Kuhn-Tucker cho B`ai to´an (
˜
P ) ph´at biˆe
˙’
u nhu
.
sau:
13
D
-
i
.
nh l´y 2.4.15. Gia
˙’

su
.
˙’
D d¯u
.
o
.
.
c cho bo
.
˙’
i (2.4.8) v`a g
i
: IR
n
→ R, 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 ⊂ IR
n
.
(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 µ
i
≥ 0, i = 0, . . . , m, khˆong c`ung triˆe
.
t tiˆeu, sao cho
˜x
∗
− x
∗
 ≤ γ
∗
/2,
0 ∈ µ
0
(2Ax
∗
+ b) +
m

i=1
µ
i
∂g
i
(x
∗
) + N(x
∗
|S) (2.4.12)
v`a

µ
i
g
i
(x
∗
) = 0 v´o
.
i mo
.
i i = 1, . . . , m, (2.4.13)
trong d¯´o ∂g
i
(x
∗
) := {ξ ∈ IR
n
| g
i
(x) − g
i
(x
∗
) ≥ ξ, x − x
∗
} l`a du
.
´o
.
i vi

phˆan cu
˙’
a g
i
ta
.
i x
∗
v`a N(x
∗
|S) := {ξ ∈ IR
n
| ξ, x − x
∗
 ≤ 0} l`a n´on ph´ap
tuyˆe
´
n cu
˙’
a S ta
.
i x
∗
.
Nˆe
´
u d¯iˆe
`
u kiˆe
.

n Slater tho
˙’
a m˜an th`ı µ
0
= 0 v`a c´o thˆe
˙’
coi µ
0
= 1.
(b) Nˆe
´
u tˆo
`
n ta
.
i x
∗
∈ D tho
˙’
a m˜an (2.4.12), (2.4.13) v´o
.
i µ
0
= 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
˜x
∗
− x
∗
 ≤ γ
∗
/2
v`a mo
.
i d¯iˆe
˙’
m infimum to`an cu
.
c kh´ac cu
˙’
a
˜
f = f + p trˆen D nˇa
`

m trong
h`ınh cˆa
`
u
B(x
∗
, γ
∗
/2).
Cho
D = {x ∈ IR
n
| c
i
, x ≤ d
i
, i = 1, . . . , m}. (2.4.14)
Ta c´o d¯i
.
nh l´y sau:
D
-
i
.
nh l´y 2.4.16. Gia
˙’
su
.
˙’
D d¯u

.
o
.
.
c x´ac d¯i
.
nh theo cˆong th´u
.
c (2.4.14).
(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 µ
i
≥ 0, i = 1, . . . , m, sao cho
˜x
∗
− x
∗
 ≤ γ
∗
/2,
(2Ax
∗
+ b) +
m

i=1
µ
i
c
i
= 0, (2.4.15)
v`a
µ
i
(c
i
, x

∗
 − d
i
) = 0 v´o
.
i mo
.
i i = 1, . . . , m. (2.4.16)
14
(b) Nˆe
´
u c´o x
∗
∈ D tho
˙’
a m˜an (2.4.15), (2.4.16) 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
˜x
∗
− x
∗
 ≤ γ
∗
/2
v`a
2A˜x
∗
+ b +
m

i=1
µ
i
c
i
 ≤ λ
max
γ
∗
.
D
-
i
.

nh l´y n`ay c`on d¯u
.
o
.
.
c H. X. Phu chı
˙’
ra thˆem:
V´o
.
i 1 ≤ i ≤ m, nˆe
´
u c
i
, ˜x
∗
 < d
i
− (2s/λ
min
)
1/2
c
i
 th`ı
µ
i
= 0.
Kˆe
´

t luˆa
.
n: Trong chu
.
o
.
ng n`ay ch´ung tˆoi d¯˜a chı
˙’
ra: t´ınh γ-lˆo
`
i ngo`ai cu
˙’
a h`am
bi
.
nhiˆe
˜
u
˜
f = f +p (Mˆe
.
nh d¯ˆe
`
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); x´ac lˆa
.
p d¯u
.
`o
.
ng k´ınh cu
˙’
a tˆa
.
p c´ac 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 ) khˆong vu
.
o
.
.
t qu´a
γ
∗
(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 (
˜
P ) (D
-
i
.
nh
l´y 2.3.13); D
-
i
.
nh l´y Kuhn-Tucker cho B`ai to´an (
˜
P ) (c´ac d¯i
.
nh l´y 2.4.14–2.4.16).
CHU
.
O
.
NG 3
T
´
INH Γ-L
ˆ

O
`
I NGO
`
AI CU
˙’
A H
`
AM BI
.
NHI
ˆ
E
˜
U
V
`
A D
-
I
ˆ
E
˙’
M INFIMUM TO
`
AN CU
.
C CU
˙’
A B

`
AI TO
´
AN (
˜
P )
Trong chu
.
o
.
ng n`ay, bˇa
`
ng tiˆe
´
p cˆa
.
n m´o
.
i thˆong qua h`am Γ-lˆo
`
i ngo`ai, ch´ung
tˆoi nghiˆen c´u
.
u c´ac t´ınh chˆa
´
t cu
˙’
a h`am bi
.
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; d¯iˆe
`
u kiˆe
.
n tˆo
`
n ta
.
i nghiˆe
.
m cu
˙’
a B`ai to´an (
˜
P ).
3.1. T´ınh Γ-lˆo
`
i ngo`ai cu
˙’
a h`am bi
.
nhiˆe
˜
u gi´o
.
i nˆo
.
i

D
-
i
.
nh ngh˜ıa 3.1.9. Cho f tho
˙’
a m˜an cˆong th´u
.
c (1.0.1). H`am h
1
(., z) theo
hu
.
´o
.
ng z d¯u
.
o
.
.
c d¯i
.
nh ngh˜ıa nhu
.
sau:
h
1
(µ, z) := inf
x∈IR
n


1
2
f(x) +
1
2
f(x + µz) − f(x +
1
2
µz)

, (3.1.17)
trong d¯´o µ ∈ IR v`a z ∈ IR
n
.
15
Ta k´y hiˆe
.
u
m(γ, z) := inf{µ | h
1
(µ, z) > γ} (3.1.18)
v`a
M(γ) := {tz | z ∈ IR
n
, |t| ≤ m(γ, z)}. (3.1.19)
Bˆo
˙’
d¯ˆe
`

sau cho ta c´ac gi´a tri
.
cu
˙’
a h
1
(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 ∈ IR
n
v`a z = 0, ta c´o
(a) h
1
(µ, z) =
µ
2

4
Az, z.
(b) m(γ, z) = 2

γ
Az,z
.
(c) M(γ) = {x | x ∈ IR
n
, x
T
Ax ≤ 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).
Nghiˆen c´u
.
u t´ınh Γ-lˆo
`
i ngo`ai cu
˙’

a h`am bi
.
nhiˆe
˜
u
˜
f = f + p ta c´o c´ac mˆe
.
nh
d¯ˆe
`
sau:
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 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.
Khi p tho
˙’
a m˜an (1.0.2), ta c´o mˆe
.
nh d¯ˆe
`
quan tro
.
ng sau:
Mˆe
.
nh d¯ˆe
`
3.1.19. H`am bi
.
nhiˆe
˜
u gi´o
.
i nˆo
.
i
˜

f = f + p l`a Γ-lˆo
`
i ngo`ai v´o
.
i
Γ = M(2s).
´
Ap du
.
ng mˆe
.
nh d¯ˆe
`
trˆen v`a di
.
nh l´y vˆe
`
tˆa
.
p m´u
.
c du
.
´o
.
i cu
˙’
a h`am Γ-lˆo
`
i ngo`ai,

ta nhˆa
.
n d¯u
.
o
.
.
c mˆe
.
nh d¯ˆe
`
sau:
Mˆe
.
nh d¯ˆe
`
3.1.20. Tˆa
.
p m´u
.
c du
.
´o
.
i cu
˙’
a h`am bi
.
nhiˆe
˜

u gi´o
.
i nˆo
.
i
˜
f = f + p l`a
Γ-lˆo
`
i ngo`ai, v´o
.
i Γ = M(2s).
3.2. D
-
iˆe
˙’
m infimum to`an cu
.
c cu
˙’
a b`ai to´an nhiˆe
˜
u
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:
16
Mˆe
.
nh d¯ˆe
`
3.2.21. Cho Γ = M(2s) nˆe
´
u x
∗
∈ D l`a d¯iˆe
˙’
m Γ-cu
.
.
c tiˆe
˙’
u cu
˙’
a 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 to`an cu
.
c cu
˙’
a
˜
f = f + p.
Ngo`ai ra, H. X. Phu c`on 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.
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 c´o
mˆe
.
nh d¯ˆe
`
sau:
Mˆe
.
nh d¯ˆe
`
3.2.22. Nˆe
´
u ˜x
∗
1
, ˜x
∗
2

l`a hai d¯iˆe
˙’
m infimum to`an cu
.
c bˆa
´
t k`y cu
˙’
a B`ai
to´an (
˜
P ) th`ı
˜x
∗
1
− ˜x
∗
2
∈ M(2s).
3.3. T´ınh ˆo
˙’
n d¯i
.
nh cu
˙’
a tˆa
.
p c´ac d¯iˆe
˙’
m infimum to`an cu

.
c
Trong mu
.
c n`ay, ch´ung tˆoi kha
˙’
o s´at 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´ac kˆe
´
t qua
˙’
thu d¯u
.
o

.
.
c trong mu
.
c n`ay l`a
c´ac d¯i
.
nh l´y sau:
D
-
i
.
nh l´y 3.3.18. Nˆe
´
u x
∗
l`a d¯iˆe
˙’
m cu
.
.
c tiˆe
˙’
u to`an cu
.
c cu
˙’
a B`ai to´an (P ), ˜x
∗
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). (3.3.20)
Kˆe
´
t qua
˙’
nhˆa
.
n d¯u
.
o
.
.
c o
.
˙’
c´ac mu
.

c trˆen l`a ma
.
nh ho
.
n c´ac kˆe
´
t qua
˙’
tu
.
o
.
ng tu
.
.
o
.
˙’
Chu
.
o
.
ng 2.
Go
.
i S
0
l`a tˆa
.
p c´ac d¯iˆe

˙’
m cu
.
.
c tiˆe
˙’
u cu
˙’
a B`ai to´an (P ) v`a S
s
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 sau:
d
H
(S
0
, S
s
) = max{sup
x∈S
0
inf
y∈S
s
x − y, sup
y∈S
s
inf
x∈S
0
x − y}.
H. X. Phu d¯˜a chı
˙’
ra 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
.
.
c tiˆe
˙’
u x
∗
v`a
(x
∗
+
¯
B(0, r)) ∩ D l`a d¯´ong v´o
.
i gi´a tri
.
r > 0 n`ao d¯´o .
17
Nˆe
´
u
sup
x∈D
|p(x)| ≤ s ≤

1
2
r
2
λ
min
,
th`ı tˆa
.
p S
s
l`a kh´ac rˆo
˜
ng v`a
d
H
({x
∗
}, S
s
) ≤

2s/λ
min
.
3.4. Du
.
´o
.
i vi phˆan suy rˆo

.
ng thˆo v`a d¯iˆe
`
u kiˆe
.
n tˆo
´
i u
.
u
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 d¯iˆe
`
u
kiˆe
.
n tˆo
´
i u
.

u cu
˙’
a B`ai to´an (
˜
P ). O
.
˙’
mu
.
c n`ay ta x´et la
.
i c´ac t´ınh chˆa
´
t trˆen v`a
nhˆa
.
n d¯u
.
o
.
.
c c´ac kˆe
´
t qua
˙’
ma
.
nh ho
.
n c´ac kˆe

´
t qua
˙’
tru
.
´o
.
c d¯´o.
D
-
i
.
nh ngh˜ıa 3.4.10. (H. X. Phu) 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
inf
x

∈(x
∗
+Γ)∩D

g(x

) + ξ, x



≤ g(x) + ξ, x v´o
.
i mo
.
i x ∈ 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 < sup
x∈D
|p(x)| ≤ s < +∞, f(x) = Ax, x −
b, x. Khi d¯´o, v´o
.
i x
∗
∈ D n`ao d¯´o th`ı
inf
x

∈(x
∗
+0.5M(2s))∩D

˜
f(x

)−2Ax
∗
+b, x



≤
˜

f(x)−2Ax
∗
+b, x v´o
.
i mo
.
i x ∈ D.
Trong tru
.
`o
.
ng ho
.
.
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
mˆo
˜
i x
∗
∈ D tˆo
`
n ta
.
i
˜x
∗
∈

x
∗
+ 0.5M(2s)

∩ D
sao cho
˜
f(˜x
∗
) − 2Ax
∗
+ b, ˜x
∗
 = min
x


∈(x
∗
+0.5M(2s))∩D

˜
f(x

) − 2Ax
∗
+ b, x



v`a
˜
f(˜x
∗
) − 2Ax
∗
+ b, ˜x
∗
 ≤
˜
f(x) − 2Ax
∗
+ b, x v´o
.
i mo
.

i x ∈ D,
hoˇa
.
c tu
.
o
.
ng d¯u
.
o
.
ng l`a
˜
f(x) ≥
˜
f(˜x
∗
) + 2Ax
∗
+ b, x − ˜x
∗
 v´o
.
i mo
.
i x ∈ D.
18
H. X. Phu d¯˜a chı
˙’
ra d¯i

.
nh l´y n`ay.
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
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
.
.
c cho bo
.
˙’
i (2.4.8). 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 µ
0
≥ 0, µ
1
≥ 0, . . . µ
m
≥ 0, sao
cho ch´ung khˆong c`ung triˆe
.
t tiˆeu, tho
˙’
a m˜an d¯iˆe
`
u kiˆe
.
n Kuhn-Tucker
L(x
∗
, µ

0
, . . . , µ
m
) = min
x∈S
L(x, µ
0
, . . . , µ
m
), (3.4.21)
d¯iˆe
`
u kiˆe
.
n b`u
µ
i
g
i
(x
∗
) = 0 v´o
.
i mo
.
i i = 1, . . . , m. (3.4.22)
Nˆe
´
u d¯iˆe
`

u kiˆe
.
n Slater (2.4.11) tho
˙’
a m˜an th`ı µ
0
= 0 v`a c´o thˆe
˙’
coi µ
0
= 1.
(b) Nˆe
´
u tˆo
`
n ta
.
i x
∗
tho
˙’
a m˜an (3.4.21) v`a (3.4.22) v´o
.
i µ
0
= 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 ).
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 (2.4.8), g
i
: IR
n
→
R, 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 ⊂ IR
n
. Khi d¯´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
∗

∈ (˜x
∗
+ 0.5M(2s)) ∩ D v`a c´ac nhˆan tu
.
˙’
Lagrange µ
0
≥ 0, µ
1
≥
0, . . . , µ
m
≥ 0, ch´ung khˆong c`ung triˆe
.
t tiˆeu tho
˙’
a m˜an
0 ∈ µ
0
(2Ax
∗
+ b) +
m

i=1
µ
i
∂g
i
(x

∗
) + N(x
∗
|S) (3.4.23)
v`a
µ
i
g
i
(x
∗
) = 0 v´o
.
i mo
.
i i = 1, . . . , m, (3.4.24)
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 (2.4.11) tho
˙’
a m˜an th`ı µ
0
= 0 v`a c´o thˆe
˙’
coi µ
0
= 1.
19
(b) Nˆe
´
u c´o x
∗
∈ D tho
˙’
a m˜an (3.4.23) v`a (3.4.24) v´o
.
i µ
0
= 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 ).
Khi D l`a tˆa
.
p lˆo
`
i d¯a diˆe
.
n ta c´o d¯i
.
nh l´y sau:
D
-
i
.
nh l´y 3.4.23. Gia
˙’
su

.
˙’
D d¯u
.
o
.
.
c cho bo
.
˙’
i (2.4.14). Khi d¯´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
∗
∈ (˜x
∗
+0.5M(2s))∩D v`a c´ac nhˆan tu
.
˙’
Lagrange µ
i
≥ 0, i = 1, . . . , m,
sao cho
(2Ax
∗
+ b) +
m

i=1
µ
i
c
i
= 0, (3.4.25)
µ
i
(c
i
, x
∗
 − d
i

) = 0 v´o
.
i mo
.
i i = 1, . . . , m. (3.4.26)
Ho
.
n thˆe
´
n˜u
.
a ta c´o
2A˜x
∗
+ b +
m

i=1
µ
i
c
i
∈ AM(2s).
(b) Nˆe
´
u c´o x
∗
∈ D tho
˙’
a m˜an (3.4.25), (3.4.26) 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 ).
Kˆe
´
t luˆa
.
n: Bˇa
`
ng c´ach tiˆe
´
p cˆa
.
n tˆo pˆo ch´ung tˆoi d¯˜a chı
˙’
ra: t´ınh Γ-lˆo

`
i
ngo`ai cu
˙’
a h`am bi
.
nhiˆe
˜
u
˜
f = f + p (c´ac Mˆe
.
nh d¯ˆe
`
3.1.18–3.2.21); tˆa
.
p M(2s)
ch´u
.
a hiˆe
.
u c´ac d¯iˆe
˙’
m infimum to`an cu
.
c cu
˙’
a B`ai to´an (
˜
P ) (Mˆe

.
nh d¯ˆe
`
3.2.22);
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 ); 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); 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
.
O
.
NG 4

D
-
I
ˆ
E
˙’
M SUPREMUM CU
˙’
A B
`
AI TO
´
AN (
˜
Q)
Trong chu
.
o
.
ng n`ay ch´ung tˆoi nghiˆen c´u
.
u: t´ınh γ-lˆo
`
i trong cu
˙’
a h`am
˜
f = f + p; t´ınh ˆo
˙’
n d¯i

.
nh, t´ınh liˆen tu
.
c cu
˙’
a h`am tˆa
.
p c´ac d¯iˆe
˙’
m supremum
to`an cu
.
c; t´ınh ˆo
˙’
n d¯i
.
nh, t´ınh liˆen tu
.
c 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) theo nhiˆe
˜
u p.
20
4.1. T´ınh γ-lˆo
`
i trong cu
˙’
a h`am bi
.
nhiˆe
˜
u
Nghiˆen c´u
.
u h`am bi
.
nhiˆe
˜
u
˜
f = f + p ta c´o c´ac mˆe
.
nh d¯ˆe
`

sau:
Mˆe
.
nh d¯ˆe
`
4.1.23. Cho γ > 0, f x´ac d¯i
.
nh theo cˆong th´u
.
c (1.0.1). Khi d¯´o
(a) Nˆe
´
u sup
x∈D
|p(x)| ≤ λ
min
γ
2
/2, th`ı
˜
f = f + p l`a γ-lˆo
`
i trong.
(a) Nˆe
´
u sup
x∈D
|p(x)| < λ
min
γ

2
/2, th`ı
˜
f = f + p l`a γ-lˆo
`
i trong ngˇa
.
t.
´
Ap du
.
ng Mˆe
.
nh d¯ˆe
`
4.1.23 v`a t´ınh chˆa
´
t cu
˙’
a h`am p ta c´o d¯u
.
o
.
.
c mˆe
.
nh d¯ˆe
`
quan tro
.

ng sau:
Mˆe
.
nh d¯ˆe
`
4.1.24. Cho f x´ac d¯i
.
nh theo cˆong th´u
.
c (1.0.1) v`a 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 ≥

2s/λ
min
v`a
γ-lˆo
`
i trong ngˇa

.
t v´o
.
i γ >

2s/λ
min
.
4.2. D
-
iˆe
˙’
m supremum to`an cu
.
c cu
˙’
a h`am bi
.
nhiˆe
˜
u
Mˆe
.
nh d¯ˆe
`
4.2.25. Nˆe
´
u h`am bi
.
nhiˆe

˜
u gi´o
.
i nˆo
.
i
˜
f = f + p d¯a
.
t gi´a tri
.
cu
.
.
c d¯a
.
i,
th`ı n´o chı
˙’
d¯a
.
t cu
.
.
c d¯a
.
i to`an cu
.
c ta
.

i mˆo
.
t sˆo
´
d¯iˆe
˙’
m γ-cu
.
.
c biˆen n`ao d¯´o cu
˙’
a D,
v´o
.
i γ =

2s/λ
min
.
D
-
i
.
nh ngh˜ıa 4.2.11. (H. X. Phu) x
∗
∈ D d¯u
.
o
.
.

c go
.
i l`a d¯iˆe
˙’
m supremum to`an
cu
.
c cu
˙’
a g : D ⊂ IR
n
→ IR nˆe
´
u
lim sup
y→x
∗
, y∈D
g(y) ≥ g(x) v´o
.
i mo
.
i x ∈ D.
D
-
i
.
nh ngh˜ıa 4.2.12. x
∗
∈ D d¯u

.
o
.
.
c go
.
i l`a d¯iˆe
˙’
m supremum d¯i
.
a phu
.
o
.
ng cu
˙’
a
g : D ⊂ IR
n
→ IR nˆe
´
u
∃δ > 0 : lim sup
y→x
∗
, y∈D
g(y) ≥ g(x) v´o
.
i mo
.

i x ∈
¯
B(x
∗
, δ) ∩ D.
Mˆe
.
nh d¯ˆe
`
4.2.28. Cho x
1
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 sup
x∈D
|p(x)| ≤ s < +∞. Nˆe
´
u
x
0
∈ D v`a −x
0
+ 2x
1
tho
˙’
a m˜an
x
0
− x
1
 =

2s/λ
min
, −x
0

+ 2x
1
∈ D
th`ı x
0
v`a −x
0
+ 2x
1
c˜ung l`a c´ac d¯iˆe
˙’
m supremum to`an cu
.
c cu
˙’
a
˜
f = f + p.
21
4.3. T´ınh chˆa
´
t cu
˙’
a tˆa
.
p c´ac d¯iˆe
˙’
m supremum to`an cu
.
c

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 := {x ∈ IR
n
| c
i
, x ≤ d
i
, c
i
∈ IR
n
, i = 1, . . . , m}.
Go
.
i
ext D := {x
∗
| x
∗
l`a d¯iˆe
˙’
m cu
.
.
c biˆen cu
˙’
a tˆa
.
p lˆo
`

i d¯a diˆe
.
n D}.
D(x
∗
, β) := {x ∈ D | x = (1 − α)x
∗
+ αy, y ∈ D,
0 ≤ α ≤ 1 − β}, x
∗
∈ ext D, β ∈ [0, 1].
Ta c´o bˆo
˙’
d¯ˆe
`
sau:
Bˆo
˙’
d¯ˆe
`
4.3.5. Cho D ⊂ IR
n
l`a d¯a diˆe
.
n lˆo
`
i, khi d¯´o
(a) Tˆo
`
n ta

.
i β
0
> 0 d¯ˆe
˙’
v´o
.
i mo
.
i β ∈ [ 0, β
0
] ta c´o
D =

x
∗
∈ext D
D(x
∗
, β).
(b) Tˆo
`
n ta
.
i γ
0
> 0 sao cho v´o
.
i mo
.

i γ ∈ [0, γ
0
] tˆa
.
p c´ac d¯iˆe
˙’
m γ-cu
.
.
c biˆen cu
˙’
a
miˆe
`
n D nˇa
`
m trong tˆa
.
p

x
∗
∈ext D
¯
B(x
∗
, γ) ∩ D.
(c) Tˆo
`
n ta

.
i s
0
> 0 sao cho v´o
.
i mo
.
i s ∈ [0, s
0
] tˆa
.
p c´ac d¯iˆe
˙’
m γ-cu
.
.
c biˆen cu
˙’
a
D v´o
.
i γ =

2s/λ
min
nˇa
`
m trong tˆa
.
p


x
∗
∈ext D
¯
B(x
∗
,

2s/λ
min
) ∩ D.
K´y hiˆe
.
u
C
0
(D) := {p : D → IR | sup
x∈D
|p(x)| < +∞}.
Nˆe
´
u trang bi
.
chuˆa
˙’
n p
C
0
(D)

:= sup
x∈D
|p(x)| th`ı C
0
(D) l`a khˆong gian tuyˆe
´
n
t´ınh d¯i
.
nh chuˆa
˙’
n v´o
.
i c´ac ph´ep to´an cˆo
.
ng h`am sˆo
´
v`a nhˆan h`am sˆo
´
v´o
.
i sˆo
´
thu
.
.
c.
Go
.
i S

global
(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
S
global
: C
0
(D) ⇒ IR
n
v`a dˆe
˜
thˆa
´
y S
global
(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 ).
22
D
-
i
.
nh l´y 4.3.24. X´et B`ai to´an (
˜
Q). Khi d¯´o
∃s
0
> 0 ∀p ∈
¯
B(0, s
0
) : S
global
(p) ⊆ S
global
(0) +

2p

C
0
/λ
min
¯
B(0, 1).
(4.3.27)
Mˆe
.
nh d¯ˆe
`
4.3.29. X´et B`ai to´an (
˜
Q). Khi d¯´o S
global
(p) l`a h`am nu
.
˙’
a liˆen tu
.
c
trˆen ta
.
i 0.
4.4. T´ınh chˆa
´
t cu
˙’
a tˆa
.

p c´ac d¯iˆe
˙’
m supremum d¯i
.
a phu
.
o
.
ng
Go
.
i S
local
(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 S
local

: C
0
(D) ⇒ IR
n
.
K´y hiˆe
.
u
η(x
∗
) := sup
x∈D, x=x
∗

2Ax
∗
+ b,
x − x
∗
x − x
∗


.
Bˆo
˙’
d¯ˆe
`
4.4.7. D
-

ˇa
.
t η
0
:= max
x
∗
∈S
local
(0)
η(x
∗
). Khi d¯´o η
0
< 0.
Bˆo
˙’
d¯ˆe
`
4.4.8. V´o
.
i mˆo
˜
i s ∈ [0, s
0
] th`ı
∀x
∗
∈ S
local

(0), ∀x ∈ D : x − x
∗
 = ξ(s) =⇒ f(x) ≤ f(x
∗
) − 3s.
D
-
i
.
nh l´y 4.4.25. X´et B`ai to´an (
˜
Q). Khi d¯´o
∀p ∈
¯
B
C
0
(0, s
0
) : max
x
∗
∈S
local
(0)
d

x
∗
, S

local
(p)

≤ ξ(p
C
0
).
Mˆe
.
nh d¯ˆe
`
4.4.30. H`am d¯a tri
.
S
local
(p) l`a nu
.
˙’
a liˆen tu
.
c du
.
´o
.
i ta
.
i d¯iˆe
˙’
m 0.
Kˆe

´
t luˆa
.
n: Chu
.
o
.
ng n`ay ch´ung tˆoi d¯˜a chı
˙’
ra: t´ınh γ-lˆo
`
i trong cu
˙’
a h`am
bi
.
nhiˆe
˜
u
˜
f = f + p (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, 4.2.28); 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 v`a 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 v`a Mˆe
.
nh d¯ˆe
`
4.4.30).
K
ˆ
E

´
T LU
ˆ
A
.
N CHUNG
1. Luˆa
.
n ´an d¯˜a gia
˙’
i quyˆe
´
t d¯u
.
o
.
.
c c´ac vˆa
´
n d¯ˆe
`
:
23
• 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

2s/λ
min
; d¯iˆe
˙’
m γ
∗
-cu
.
.
c tiˆe
˙’
u cu
˙’
a
˜
f l`a d¯iˆe

˙’
m cu
.
.
c tiˆe
˙’
u to`an cu
.
c; x´ac
lˆa
.
p 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 tr`ınh b`ay. C´ac kˆe
´
t qua
˙’
trˆen d¯˜a d¯u
.
o
.

.
c cˆong bˆo
´
trong
b`ai b´ao [1].
• 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 ⊂ IR
n
;
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
Γ; x
∗
− ˜x
∗
∈
1
2
Γ nˆe
´
u x
∗
l`a nghiˆe
.
m cu
.
.
c tiˆe
˙’
u to`an cu

.
c cu
˙’
a f trˆen D v`a
˜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 S

s
cu
˙’
a (
˜
P ) l`a ˆo
˙’
n d¯i
.
nh theo khoa
˙’
ng c´ach Hausdorff d
H
(.,.). 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 [2].
• Chı
˙’
ra h`am
˜
f l`a γ-lˆo
`
i trong v´o
.
i γ ≥ (2/λ
min
)
1
2
v`a γ-lˆo
`
i trong ngˇa
.
t v´o

.
i
γ > (2/λ
min
)
1
2
; khi D bi
.
chˇa
.
n v`a γ = (2/λ
min
)
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 S
global
(p) v`a tˆa
.
p c´ac d¯iˆe
˙’
m supremum
d¯i
.
a phu
.
o
.
ng S
local
(p) cu
˙’
a B`ai to´an (
˜
Q) nhu
.
t´ınh ˆo
˙’

n d¯i
.
nh v`a t´ınh nu
.
˙’
a liˆen
tu
.
c c˜ung d¯u
.
o
.
.
c chı
˙’
ra. Phˆa
`
n l´o
.
n c´ac kˆe
´
t qua
˙’
d¯u
.
o
.
.
c liˆe
.

t kˆe o
.
˙’
trˆen d¯˜a d¯u
.
o
.
.
c
cˆong bˆo
´
trong b`ai b´ao [3].
2. Nh˜u
.
ng vˆa
´
n d¯ˆe
`
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 t`ım l`o
.
i gia
˙’
i tˆo
´
i u

.
u cu
˙’
a c´ac b`ai to´an (
˜
P ) v`a (
˜
Q) v`ao c´ac
b`ai to´an thu
.
.
c tˆe
´
nhu
.
b`ai to´an ph´at d¯iˆe
.
n tˆo
´
i u
.
u, kinh tˆe
´
d¯ˆo
´
i s´anh,. . .
24
C
ˆ
ONG TR

`
INH CU
˙’
A T
´
AC GIA
˙’
LI
ˆ
EN QUAN D
-
ˆ
E
´
N LU
ˆ
A
.
N
´
AN
1. H. X. Phu and V. M. Pho, Global infimum of strictly convex quadratic
functions with bounded perturbation, Mathematical Methods of Opera-
tions 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 10.1080/023319-
3100746114, 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.

M
ˆ
O
.
T S
ˆ
O
´
K
ˆ
E
´
T QUA
˙’
CU
˙’
A LU
ˆ
A
.
N
´
AN D
-
U
.
O
.
.
C B

´
AO C
´
AO TA
.
I
1. Xe mi na “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,
2. Xe mi na “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,
3. Hˆo
.
i tha
˙’
o “Tˆo
´
i u
.
u v`a T´ınh to´an Khoa ho
.
c” ta
.
i Ba v`ı, H`a nˆo
.
i, th´ang 4
nˇam 2010,
4. Xe mi na “T´ınh to´an hiˆe

.
n d¯a
.
i” cu
˙’
a Bˆo
.
mˆon To´an, Khoa Cˆong nghˆe
.
thˆong tin Ho
.
c viˆe
.
n KTQS,
5. Hˆo
.
i tha
˙’
o “Tˆo
´
i u
.
u v`a T´ınh to´an Khoa ho
.
c” ta
.
i Ba v`ı, H`a nˆo
.
i, th´ang 4
nˇam 2011.