Một lớp bài toán quy hoạch ngẫu nhiên nhiều giai đoạn với ràng buộc kỳ vọng

σ
Ω = ∅ P(Ω) Ω F ⊂ P(Ω)
σ
Ω ∈ F
A ∈ F ⇒ A
c
= Ω \A ∈ F
A
n
∈ F ∀n = 1, 2, ⇒
∞

n=1
A
n
∈ F
Ω F σ
Ω (Ω, F)
P : F → R F
P(A) ≥ 0 ∀A ∈ F
P(Ω) = 1
A
n
∈ F, (n = 1, 2, ), A
i

A
j

= A
i
A
j
= ∅, (i = j)
P

∞

n=1
A
n

=
∞

n=1
P(A
n
)
(Ω, F, P)
σ F σ
A ∈ F
Ω ∈ F
∅ ∈ F
A = Ω\A
A
A ∩B = AB = ∅ A, B
(Ω, F, P)
(Ω, F, P) G σ σ F

B(R) σ R X : Ω → R
G B ∈ B(R)
X
−1
(B) = {ω : X(ω) ∈ B} ∈ G.
X F
X
X : (Ω, F, P) → (R, B(R))
X P
X EX
EX =

Ω
XdP.
X ≥ 0 EX ≥ 0
X = C EX = C
EX C ∈ R E(CX) = C(EX)
EX EY E(X ± Y ) = EX ±EY
EX =













i
x
i
p
i
X x
1
, x
2
,
P(X = x
i
) = p
i
,

+∞
−∞
xp(x)dx, X p(x).
X : Ω → R DX = E(X−EX)
2
X
DX = EX
2
− (EX)
2
DX ≥ 0
DX = 0 X = EX
D(CX) = C

2
DX
D(X ± Y ) = DX + DY.
(Ω, F, P) X : Ω → R
G σ F Y
X σ G
Y G
A ∈ G

A
Y dP =

A
XdP
Y = E(X|G)
(Ω, F, P) (X
n
, n ∈ N)
(F
n
, n ∈ N) σ (X
n
, F
n
)
n∈N
(X
n
, F
n

)
n∈N
E|X
n
| < ∞, ∀n ∈ N
m ≤ n, m, n ∈ N E(X
n
|F
m
) = X
n
(Ω, F, P)
G σ F
E|X| < ∞ Y = E(X|G).
X = c E(X|G) = E(c|G) = c
X ≥ Y E(X|G) ≥ E(Y |G)
E(X|{∅, Ω}) = EX .
E(X|F) = X .
a, b
E(aX + bY |G) = aE(X|G) + bE(Y |G).
X G E(X|G) = EX

E(X|G)

= EX.
G
1
⊂ G
2
E(X|G

1
) = E

E(X|G
1
)|G
2

.
X G E(X|G) = X
X
n
↑ C n
0
∈ N
EX
−
n
0
< ∞ E(X
n
|G) ↑ E(X|G)
X
n
↓ C n
0
∈ N EX
+
n
0

< ∞
E(X
n
|G) ↓ E(X|G)
Y
X
n
≤ Y n ≥ 1
E

limX
n
|G

≤ limE(X
n
|G .
X
n
≥ Y
limE

X
n
|G

≤ E

limX
n

|G

.
Y |X
n
| < Y
X
n
→ X
lim
n→∞
E

X
n
|G

= E

lim
n→∞
X
n
|G

= E(X|G) .
E|XY | < ∞, E|X| < ∞, X ∈ G
E(XY |G) = XE(Y |G) .
φ X
φ(X)

E

φ(X)|G

≥ φ

E(X|G)

.
(Ω, F, P) X : Ω → R
P
X
:B(R) → R
B → P
X
(B) = P(X
−1
(B))
X
n
p 0 ≤ p ≤ 1. X
n X
S = {0, 1, , n}
P|X = k| = C
k
n
p
k
(1 − p)
n−k

, k ∈ S.
X n, p
X B(n, p) X ∼ B(n, p))
λ > 0(X ∼ P (λ)) X S = N =
{0, 1, }
P(X = k) =
λ
k
e
−λ
k!
, k = 0, 1,
X
µ ∈ R, σ > 0 X ∼ N(µ, σ
2
) X
p(x) =
1
σ
√
2π
.e
−
(x−µ)
2
2σ
2
µ = 0, σ = 1 X ∼ N(0, 1)
p(x) =
1

√
2π
e
−
x
2
2
.
X
λ > 0 X ∼ ε(λ) X
p(x) =



0 x ≤ 0
λ.e
−λx
x > 0.
X
p(x) =



1
b−a
a ≤ x ≤ b
0 x ∈ [a, b].
(X
n
)

X ε > 0
lim
n→∞
P(|X
n
− X| > ε) = 0.
X
n
P
−→ X.
(X
n
)
X
P( lim
n→∞
|X
n
− X| = 0) = 1.
X
n
h.c.c
−−→ X
(X
n
)
p (p > 0) X
lim
n→∞
E|X

n
− X|
p
= 0.
X
n
L
p
−→ X
(X
n
)
X ε > 0
∞

n=1
P(|X
n
− X| > ε) < ∞.
X
n
c
−→ X.
(X
n
)
X
lim
n→∞
F

n
(x) = F (x) ∀x ∈ C(F ),
F
n
(x) F (x)
X
n
C(F ) F (x)
X
n
D
−→ X.
A
n
, n ≥ 1
∞

n=1
P(A
n
) < ∞
P(lim sup A
n
) = 0
∞

n=1
P(A
n
) = ∞

(A
n
, n ≥ 1) P(lim sup A
n
) = 1
lim sup A
n
=
∞

n=1
∞

k=n
A
k
.
∞

n=1
P(A
n
) < ∞.
P(lim sup A
n
) = P

∞

n=1

∞

k=n
A
k

= lim
n→∞
P

∞

k=n
A
k

≤ lim
n→∞
∞

k=n
P(A
k
) = 0.
0 ≤ x ≤ 1 1 − x ≤ e
−x
.
∞

n=1

P(A
n
) = ∞.
(A
n
) (A
n
)
n = 1, 2, m > n
1−P

m

k=n
A
k

= P

m

k=n
A
k

= P

m

k=n

A
k

=
m

k=n

1−P(A
k
)

≤ e
−
m

k=n
P(A
k
)
.
0 ≤ 1 − P

∞

k=n
A
k

= lim

m→∞

1 − P(
m

k=n
A
k
)

≤ lim
m→∞
e
−
m

k=n
P(A
k
)
= 0.
P(

∞
k=n
A
k
) = 1 n = 1, 2,
P


lim sup A
n

= lim
n→∞
P

∞

k=n
A
k

= 1.

ε > 0
P




n(A)
n
− p



≤ ε

∼ Φ


ε

n
pq

− Φ

− ε

n
pq

,
N(0, 1)
Φ(x) =
1
2π

x
−∞
e
−t
2
2
dt.
pq ≤
1
4
Φ


ε

n
pq

− Φ

− ε

n
pq

≥ 2Φ

2ε
√
n

− 1.
n
P




n(A)
n
− p




≥ ε

≤ 2e
−2nε
2
.
lim P




n(A)
n
− p



≤ ε

= 1,
n(A)
n
p
k x
(1)
, x
(2)
, , x

(k)
∈ R
n
x =
k

i=1
λ
i
x
(i)
,
k

i=1
λ
i
= 1, λ
i
≥ 0, i = 1, 2, , k
x
(1)
, x
(2)
∈ R
n
x
(1)
x
(2)

= {x ∈ R
n
: x = λx
(1)
+ (1 − λx
(2)
, 0 ≤ λ ≤ 1}
x
(1)
x
(2)
= {x ∈ R
n
: x = λx
(1)
+ (1 − λx
(2)
, λ ≥ 0}
x
(2)
x
(1)
x
(2)
= {x ∈ R
n
: x = λx
(1)
+ (1 − λx
(2)

, λ ∈ R}
M ⊂ R
n
M M x
(1)
, x
(2)
∈ M, x = λx
(1)
+
(1 − λ)x
(2)
, 0 ≤ λ ≤ 1 x ∈ M.
M ⊂ R
n
x ∈ M M
x
(1)
, x
(2)
∈ M x = λx
1
+ (1 −λ)x
2
, 0 < λ < 1 x
M
c ∈ R
n
α
H = {x ∈ R

n
: c, x = α}
R
n
. S = {x ∈ R
n
:
c, x ≤ α} H
M = {x ∈ R
n
: A
i
, x ≤ b
i
, i = 1, 2, , m, A
i
= (a
i
j) ∈ R
n
}.
M
x ∈ M
M x n
M.
f(x) M
x, y ∈ M λ ∈ [0, 1]
f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f (y).
min


f(x) : x ∈ M ⊂ R
n

.
f(x) M
M f(x)
f

k

i=1
λ
i
x
(i)

≤
k

i=1
λ
i
f(x
(i)
), ∀x
(i)
∈ M
k

i=1

λ
i
= 1, λ
i
≥ 0, i = 1, 2, , k.
φ : R → R X φ(X)
Eφ(X) ≥ φ(X).
min{f(x) = c
T
x} (LP )



Ax = b,
x ≥ 0,
x = (x
1
, x
2
, , x
n
) c
T
= (c
1
c
2
c
n
)

T
c b = (b
1
b
2
b
m
) A = (a
ij
)
m × n
A, b, c
(LP )
min

c
T
x + E

Q(x, z)

(1.1)



Ax ≤ b,
x ≥ 0,
Q(x, z) = min

q

T
y




A(z)x + Dy = b(z),
y ≥ 0,
Q(x, z) z ∈ R
n
E(Q(x, z)) Q(x, z) x y
D
m×m y = (y
1
, y
2
, , y
m
); Dy
Ax b q = (q
1
, q
2
, , q
m
)
z.
x
y
x z

min

c
T
x + E(q
T
y(z))

























Ax ≤ b,
T (z)x + Dy(z) ≤ h(z)
x ≥ 0
y(z) ≥ 0
y(.) ∈ Y − .
(LP 1) Q
1
= min
x
(1)

c
T
1
x
(1)
+ E
ω∈Ω

Q(x
(1)
, ω)




A
1
x

(1)
= B
1
,
x
(1)
≥ 0,
t, (t = 1, 2, , T )
E[Q
t
(x
(t−1)
)] =
q
t

i=1
p
ti
E
ω
ti
∈Ω
t
[Q
t
(x
(t−1)
), ω
ti

)],
p
ti
w
ti
∈ Ω
t
, i = 1, 2, , q
t
, [Q
t
(x
(t−1)
), ω
ti
)]
(LP t) [Q
t
(x
(t−1)
), ω
t
)] = min
x
(t)
{c
T
t
x
(t)

+ E
ω∈Ω
[Q
t−1
(x
(t)
, ω)]},
A
t
x
(t)
= w
t
− B
t−1
x
(t−1)
,
x
(t)
≥ 0.
α(x, b, A)
α(x, b, A) x b A
c
T
x + E(α(x, b, A)) x (1.1)
b
0
, A
0

b A x
1
, x
2
y
1
, y
2
(x
1
, b
0
, A
0
) (x
2
, b
0
, A
0
)
α(x, b, A) λ ∈ [0, 1]
α(λx
1
+ (1 − λ)x
2
, b
0
, A
0

) ≤ λα(x
1
, b
0
, A
0
) + (1 − λ)α(x
2
, b
0
, A
0
).
y
1
, y
2
≥ 0 0 ≤ λ ≤ 1 y
0
= λy
1
+ (1 − λ)y
2
.
Dy
0
= λDy
1
+ (1 − λ)Dy
2

= λ(A
0
x
1
− b
0
) + (1 − λ)(A
0
x
2
− b
0
)
= A
0
(λx
1
+ (1 − λ)x
2
) − b
0
.
y
0
λx
1
+ (1 −
λ)x
2
), b

0
, A
0
λ(λx
1
+ (1 − λ)x
2
), b
0
, A
0
)
≤ q
T
y
0
= λα(x
1
, b
0
, A
0
) + (1 − λ)α(x
2
, b
0
, A
0
).
α(x, b, A)

c
T
x+Eα(x, b, A)

t = 1, 2, , T
(Ω, F, P)
Π
T
t=1
Z
t
T
Z
t
η ξ
Θ Ξ
Z := Θ × Ξ
F
t
t
σ
1 → t F
t
:= σ(ζ
t
)
F := {F
t
}
T

t=1
R
n
t
, R
m
t
n
t
X X ⊂ Π
T
t=1
R
n
t
X(F) :=

x ∈ Π
T
t=1
L
∞
(Ω, F
t
, P; R
n
t
)



x X

X
0
(F) X(F)
Y (F) :=

y ∈ Π
T
t=1
L
∞
(Ω, F
t
, P; R
m
t
)


y Y

Y Π
T
t=1
R
m
t
). Y (F)
y x

Y
0
(F) Y (F)
U
0
(F) := Π
T
t=1
L
∞
(Ω, F
t
, P; R
m
t
).
c : X ×Θ → R f
t
: X ×Θ →
R
m
t
t = 1, , T
t
∈ L
∞
(Ω, F
t
, P; R
n

t
)
ζ
Z ζ = (ζ
1
, ζ
2
, ζ
3
, , ζ
T
) Z = X
T
t=1
Z
t
ζ
t
(Ω, F)
(Z
t
, B(Z
t
)) Z
t
ζ
t
:= (ζ
1
, ζ

2
, , ζ
t
) Z
t
:= Π
T
τ=1
Z
τ
t = 1, 2, , T.
min
x∈X(F)
E

c(x, η)

, (P)
E(f
t
(x, ξ)|F
t
) ≤ 0 , t = 1, 2, , T.
η P ξ
ζ := (η, ξ) ∈ Z := Θ ×Ξ
P x
X ⊂ X
T
t=1
R

n
t
X(F)
(C
1
) c x η X × Θ
(C
2
) f
t
X × Ξ t = 1, 2, , T
(C
3
X
t
R
n
t
t = 1, 2, , T
(C
4
) P
ε > 0 x
s
∈ X(F)
E(f
t
(x
s
, ξ)|F

t
) ≤ −ε1
t
t = 1, 2, , T.
P
X
P
min
x∈X
0
(F)
F (x, u) (P(u))
F : X
0
(F) × U
0
(F) → [−∞, +∞]
F (x, u) =









E(c(x, η)), x ∈ X(F) t = 1, 2, , T
E(f
t

(x, ξ)|F
t
) ≤ u
t
+∞, .
(C
1
) (C
2
) F P(0)
P y ∈ Y (F)
F Y
., . U
0
(F) × Y
0
(F)
y, u := E(y.u)
U
0
(F) Y
0
(F)
u → y, u
U
0
(F) y ∈ Y
0
(F)
y → y, u

Y
0
(F)
u ∈ U
0
(F)
L : X
0
(F) × Y
0
(F) → [−∞, +∞].
L(x, y) := inf
u∈U
0
(F)
F (x, u) + y, u
L(x, y) =









E

L(x, y; η, ξ)


, x ∈ X(F), y ∈ Y (F)
−∞, x ∈ X(F), y ∈ Y (F)
+∞, x ∈ X(F),
L : X ×Y ×Θ ×Ξ → R
L(x, y; η, ξ) := c(x, η) +
T

t=1
y
t
f
t
(x, ξ).
L (x, ξ) (y, η)
F (x, 0) = sup
y∈Y (F)
L(x, y)
P
min sup
y∈Y (F)
L(x, y) x ∈ X(F). (P)
max inf
x∈X(F)
L(x, y) y ∈ Y (F). (D)
P
P, D
arg max D
ϕ : U
0
(F) → [−∞, +∞] ϕ(u) := inf P(u)

ϕ
(x, y) ∈ X
0
(F) × Y
0
(F)
L(x, y) ≥ L(x, y) ≥ L(x, y), x ∈ X
0
(F), y ∈ Y
0
(F)
x P y D inf P = sup D
ζ
ζ = (η, ξ)
Z = Θ × Ξ Z
ζ = (η, ξ) = (H
o
η, H
c

ξ), (2.2)
ζ := (η,

ξ)

Z :=

Θ ×

Ξ

H
o

Θ → Θ H
c

Ξ → Ξ
H
o
η
t
η
s
s > t
3
(C
1
) −(C
4
)
η
u
ξ
u
Θ Ξ η
u
η ξ
u
ξ
ζ

u
= (η
u
, ξ
u
) Z ζ
u
ζ
F
u
ζ
u
F
u
:= {F
u,t
}
T
t=1
F
u,t
:= σ(ζ
u,t
) F
u
:= F
u,T
ζ
u
E(x|F) ∈ X(F), x ∈ X(F

u
), (2.3)
E(y|F
u
) ∈ Y (F
u
), y ∈ Y (F
u
), (2.4)
E(ξ
u
|F) = ξ, (2.5)
E(η|F
u
) = η
u
. (2.6)
ε > 0
ζ
u
ζ − ζ
u

∞
≤ ε ζ
u
E(y|F) ∈ Y (F) y ∈ Y (F
u
), (2.7)
E(x|F

u
) ∈ X(F
u
) x ∈ X(F), (2.8)
ζ
l
= (η
l
, ξ
l
)
η
l
ξ
l
Θ Ξ F
l
F
l
:= {F
l,t
}
T
t=1
F
l,t
:= σ(ζ
l,t
)
F

l
:= F
l,T
ζ
u
E(x|F
l
) ∈ X(F
l
) x ∈ X(F), (2.9)
E(y|F) ∈ Y (F) y ∈ Y (F
l
), (2.10)
E(ξ|F
l
) = ξ
l
, (2.11)
E(η
l
|F) = η. (2.12)
ε > 0
ζ
l
ζ −ζ
l

∞
≤ ε ζ
l

E(y|F
l
) ∈ Y (F
l
) y ∈ Y (F) (2.13)
E(x|F) ∈ X(F) x ∈ X(F
l
), (2.14)
ζ
l
ζ
u
ζ
u
ζ
l
ζ
ζ ζ
u
F
F
u
P P
u
P
l
ζ
l
ζ F
l

F