Báo cáo nghiên cứu khoa học: "Một số kết quả về sự hiệu chỉnh đầy đủ và hiệu chỉnh nửa đầy đủ trong các phương pháp xấp xỉ giải bài toán quy hoạch ngẫu nhiên." docx

1
1
2
2.1 m × n
m × 1 = [w
1
w
2
. . . w
n
]
T
n × 1
w
j
≥ 0, ∀j = 1, , n
m × n
= [r
1
r
2
. . . r
n
]
T
n × 1 r
j
> 0, ∀j = 1, , n
W
1
, W

2
, . . . , W
n
{W
1
, W
2
, . . . , W
n
}
rank(W) ≤ min{m, n − 1}.
2.2 R
k
{
−→
v
1
,
−→
v
2
, . . . ,
−→
v
s
}
−−→
v
s+1
rank{

−→
v
1
,
−→
v
2
, . . . ,
−−→
v
s+1
} = rank{
−→
v
1
,
−→
v
2
, . . . ,
−→
v
s
} + 1,
−−→
v
s+1

−→
v

1
,
−→
v
2
, . . . ,
−→
v
s

{
−→
v
1
,
−→
v
2
, . . . ,
−→
v
s
}
2.3
∗
= [r
1
r
2
. . . r

n
]
T
n × 1 r
j
> 0, ∀j = 1, , n
m × 1
∗
W
1
=

0 0
I I

; W
2
=

−1 1
1 −1

.
3
3.1 m × n m < n
W
1
, W
2
, . . . , W

n
R
m
m < n
m > n rank{W
1
, W
2
, . . . , W
n
} < m
t ∈ R
m
rank{W
1
, W
2
, . . . , W
n
, t} = rank{W
1
, W
2
, . . . , W
n
} + 1,
W
1
, W
2

, . . . , W
n
 {W
1
, W
2
, . . . , W
n
}
m = n rank(W) < m
m = n rank(W) = m
{W
1
, W
2
, . . . , W
n
} R
m
= [r
1
r
2
. . . r
n
]
T
n × 1 r
j
> 0, ∀j = 1, , n

r
1
W
1
+ r
2
W
2
+ · · · + r
n
W
n
= 0.
{W
1
, W
2
, . . . , W
n
}
m rank{W
1
, W
2
, . . . , W
n
} < m
t ∈ R
m
rank{W

1
, W
2
, . . . , W
n
, t} = rank{W
1
, W
2
, . . . , W
n
} + 1,
W
1
, W
2
, . . . , W
n
 {W
1
, W
2
, . . . , W
n
}
3.2
m × n n > 1
k ≤ min{m, n − 1} m × n
rank(W) = k
∗ n = 1

R
m
∗ n > 1 W
1
, W
2
, . . . , W
n−1
∈ R
m
k r
j
> 0, ∀j = 1, , n
W
n
= −
r
1
r
n
W
1
−
r
2
r
n
W
2
− · · · −

r
n−1
r
n
W
n−1
∈ R
m
(1)
⇒ r
1
W
1
+ r
2
W
2
+ · · · + r
n
W
n
= 0 (2)
W
1
, W
2
, . . . , W
n
Wr = 0
m × n W

1
, W
2
, . . . , W
n−1
∈ R
m
k W
n
W
1
, W
2
, . . . , W
n
−
1
rank(W) = k
m × n k
3.3 m × n (m < n)
= [r
1
r
2
. . . r
n
]
T
n × 1 r
j

≥ 0, ∀j = 1, , n
0 r
j
> 0
R
m
W
1
, W
2
, . . . , W
n
W
1
, W
2
, . . . , W
m
r
1
> 0, r
2
>
0, . . . , r
m
> 0
r
1
W
1

+ r
2
W
2
+ · · · + r
n
W
n
= 0 (3)
⇔ r
1
W
1
+ r
2
W
2
+ · · · + r
m
W
m
= −r
m+1
W
m+1
− r
m+2
W
m+2
− · · · − r

n
W
n
. (4)
m × 1 {W
1
, W
2
, . . . , W
m
}
t = t
1
W
1
+ t
2
W
2
+ · · · + t
m
W
m
. (5)
∗ t
j
≥ 0, j = 1, , m

w
j

= t
j
, j = 1, , m
w
j
= 0, j = m + 1, , n
(6)
⇒ t = w
1
W
1
+ w
2
W
2
+ · · · + w
n
W
n
,
(w
j
≥ 0, j = 1, , n)
∗ ∃t
j
< 0 r
j
> 0, j = 1, , m min{
t
j

r
j
| t
j
< 0, j =
1, m}
min{
t
j
r
j
| t
j
< 0, j = 1, , m =
t
m
r
m
. (7)
t
m
< 0, r
m
> 0
t
m
r
m
< 0.
w

j
= t
j
−
t
m
r
m
.r
j
, (j = 1, , m)
(8)
⇔ w
j
= r
j

t
j
r
j
−
t
m
r
m

, (j = 1, , m) .
(9)
t

j
< 0
t
j
r
j
≥
t
m
r
m
⇔
t
j
r
j
−
t
m
r
m
≥ 0 ⇒ w
j
≥ 0.
t
j
> 0
t
m
r

m
< 0
w
j
= t
j
−
t
m
r
m
.r
j
≥ 0.
w
j
≥ 0, j = 1, , m. (10)
t
j
= w
j
+
t
m
r
m
.r
j
, j = 1, , m. (11)
t

j
t =
m

j=1
t
j
.W
j
=
m

j=1

w
j
+
t
m
r
m
.r
j

W
j
=
m−1

j=1


w
j
+
t
m
r
m
.r
j

W
j
+ t
m
.W
m
=
m−1

j=1
w
j
.W
j
+
m−1

j=1


t
m
r
m
.r
j

W
j
+
t
m
r
m
r
m
.W
m
=
m−1

j=1
w
j
.W
j
+
m

j=1

t
m
r
m
r
j
.W
j
=
m−1

j=1
w
j
.W
j
+
t
m
r
m

m

j=1
r
j
.W
j


=
m−1

j=1
w
j
.W
j
+
t
m
r
m

−
n

j=m+1
r
j
.W
j

=
m−1

j=1
w
j
.W

j
+
n

j=m+1

−
t
m
r
m
r
j

.W
j
.
w
j
= −
t
m
r
m
.r
j
, j = 1, , n (12)
t
m
< 0, r

m
> 0, r
j
≥ 0
w
j
≥ 0, j = m + 1, , n. (13)
w
j
?? w
m
= 0
t =
n

j=1
w
j
.W
j
(w
j
≥ 0, j = 1, , n) ,
3.4 m × n (m < n)
∗
∗
n × 2n
n
= [1 1 . . . 1]
T

3.5 m × n
m < n
∗
∗
Nhn xt.
k = m
3.6
min{f(x)}

Ax = b
x ≥ 0,
(14)
[x
1
x
2
. . . x
n
]
T
n × 1
[b
1
b
2
. . . b
m
]
T
m × 1

[a
ij
] m × n
f
3.7 m × n m ≥ n m < n
{x | Ax = b, x ≥ 0}
(14)
{x | Ax = b, x ≥ 0}
m × n m < n m
3.8 m × n
α m × 1 [w
1
w
2
. . . w
n
]
T
α w
j
≥ α, j = 1, , n
α
m × 1 t
∗
= t − Wα
∗
α
∗
= [α α . . . α]
T

n × 1
[y
1
y
2
. . . y
n
]
T
y
j
≥ 0, j = 1 , , n t
∗
Wα
∗
W(y + α
∗
) = t α
∗
= [w
1
w
2
. . . w
n
]
T
w
j
= y

j
+ α ≥ α α
α = 0
3.9 m × n
α m × 1 [w
1
w
2
. . . w
n
]
T
α w
j
≤ α, j = 1, , n
α
m × 1 t
∗
= −t + Wα
∗
α
∗
= [α α . . . α]
T
n × 1
[y
1
y
2
. . . y

n
]
T
y
j
≥ 0, j = 1, , n t
∗
Wα
∗
W(−y + α
∗
) = t α
∗
= [w
1
w
2
. . . w
n
]
T
w
j
= −y
j
+ α ≤ α α
α = 0 m × 1 t
∗
= −t
[x

1
x
2
. . . x
n
]
T
0
x
j
≤ 0, j = 1, , n t
∗
[w
1
w
2
. . . w
n
]
T
w
j
= −x
j
≥ 0, j = 1, , n
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