Bài toán sản xuất leontief đa mục tiêu và đối ngẫu

R
n
+
x = (x
1
, x
2
, , x
n
) ∈ R
n
+
x
i
≥ 0 i = 1, 2, , n,
i p
i
(p
i
≥ 0) i
p = (p
1
, p
2
, , p
n

)
x γ
γ p
t
x
n

i=1
p
i
x
i
.
m j (j = 1, 2, , m)
b
j
(b
j
≥ 0) j
i a
ij
(a
ij
≥ 0) b
j
j m
x ∈ R
n
+
n


i=1
a
ij
x
i
≤ b
j
, j = 1, 2, , m.
a
j
= (a
1j
, a
2j
, , a
nj
)
t
∈ R
n
+
, j = 1, 2, , m,
m
a
j
t
x ≤ b
j
, j = 1, 2, , m.

max p
t
x
a
j
t
x ≤ b
j
, j = 1, 2, , m,
x ≥ 0.
b
j
> 0,
n

i=1
a
ij
> 0 ∀j = 1, 2, , m
¯a
j
=
1
b
j
a
j
, j = 1, 2, , m,
m


j=1
¯a
j
> 0.
max p
t
x
a
j
t
x ≤ 1, j = 1, 2, , m,
x ≥ 0.
m n
u = (u
1
, u
2
, , u
m
)
t
∈ R
m
+
u
j
(j = 1, 2, , m)
j
max


p
t
x : a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0

= sup
x≥0
min
u≥0

p
t
x +
m

j=1
u
j
(1−a
j
t
x)

= min
u≥0
sup
x≥0


p
t
x +
m

j=1
u
j
(1−a
j
t
x)

= min
u≥0

m

j=1
u
j
+ sup
x≥0

p −
m

j=1
u
j

a
j

t
x

.
m

j=1
u
j
¯a
j
≥ p sup
x≥0

p −
m

j=1
u
j
¯a
j

t
x = +∞.
max


p
t
x : a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0

= min

m

j=1
u
j
:
m

j=1
u
j
a
j
≥ p, u ≥ 0

.
min
m

j=1
u

j
m

j=1
u
j
a
j
≥ p,
u ≥ 0.
(1.5) (1.7)
a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0
m

j=1
u
j
¯a
j
≥ p, u ≥ 0
m

j=1
(1 − a
j
t
x)u

j
= 0,

m

j=1
u
j
a
j
− p

t
x = 0
m

j=1
u
j
=

m

j=1
u
j
¯a
j

t

x = p
t
x.
m

j=1
u
j
= p
t
x.
m

j=1
(1 − a
j
t
x)u
j
≥ 0,

m

j=1
u
j
¯a
j
− p


t
x ≥ 0.
m

j=1
u
j
≥
m

j=1
u
j
a
j
t
x (do (1.12))
≥ p
t
x. (do (1.13))
m

j=1
u
j
=
m

j=1
u

j
a
j
t
x = p
t
x
x u
x
u x u
x u x u
p
t
x
m

j=1
u
j
x u
p
t
x
f(x) R
n
+
f(x

) ≥ f(x) ∀x ∈ R
n

+
, ∀x

∈ R
n
+
: x

≥ x,
f(µx) = µf(x) ∀x ∈ R
n
+
, ∀µ ≥ 0.
f(x)
max f(x) a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0.
L(x, u) = f(x) +
m

j=1
u
j
(1 − a
j
t
x) ∀x ∈ R
n
+

, ∀u ∈ R
m
+
.
max
x≥0
inf
u≥0
L(x, u).
x ≥ 0
inf
u≥0
L(x, u) = inf
u≥0

f(x) +
m

j=1
u
j
(1 − a
j
t
x)

= f(x) + inf
u≥0
m


j=1
u
j
(1 − a
j
t
x)
=

f(x) a
j
t
x ≤ 1, j = 1, 2, , m,
+∞ .
max
x≥0
inf
u≥0
L(x, u) = max {f(x) : a
j
t
x ≤ 1, j = 1, 2, , m}.
inf
u≥0
max
x≥0
L(x, u).
L(x, u)
(¯x, ¯u) ∈ R
n

+
× R
m
+
min
u≥0
max
x≥0
L(x, u) = L(¯x, ¯u) = max
x≥0
min
u≥0
L(x, u).
¯u
1
≥ 0, ¯u
2
≥ 0, , ¯u
m
≥ 0
max {f(x) : a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0}
= max

f(x) :
m

j=1

u
j
(1 − a
j
t
x)

≥ inf
u≥0
max
x≥0
L(x, u).
¯x
m

j=1
u
j
(1 − a
j
t
¯x) = 0.
max
x≥0
inf
u≥0
L(x, u) = max
x≥0
{f(x) : a
j

t
x ≤ 1, j = 1, 2, , m}
= f(x) ( ¯x (1.14))
= f(x) +
m

j=1
u
j
(1 − a
j
t
x) ( (1.19))
= L(x, u)
≥ inf
u≥0
max
x≥0
L(x, u) ( (1.18))
≥ max
x≥0
inf
u≥0
L(x, u).
F = {x ≥ 0 : f(x) ≥ 1}.
F f R
n
+
f(x) F R
n

+
x + R
n
+
⊆ F ∀x ∈ F f(x) f(x)
F
f(x) = sup{θ ≥ 0 : x ∈ θF}.
0R
n
+
= R
n
+
F
∗
F
F
∗
= {v ≥ 0 : v
t
x ≥ 1 ∀x ∈ F }.
F F
∗
F = {x ≥ 0 : v
t
x ≥ 1 ∀v ∈ F
∗
}.
A = {x ≥ 0 : v
t

x ≥ 1 ∀v ∈ F
∗
}.
F
∗
x ∈ F v
t
x ≥ 1 ∀v ∈ F
∗
⇒ x ∈ A
x ≥ 0 x /∈ F v

∈ R
n
α ∈ R
v
t
z ≥ α ∀z ∈ F,
v
t
x < α.
v

≥ 0 α > 0 v =
1
α
v

v
t

z ≥ 1 ∀z ∈ F,
v
t
x < 1
v ∈ F
∗
x ≥ 0 x /∈ F v ∈ F
∗
v
t
x < 1 x /∈ A
F = {x ≥ 0 : v
t
x ≥ 1 ∀v ∈ F
∗
}.
f(x)
f(x) = inf

v
t
x : v ∈ F
∗

.
max
x≥0
L(x, u) = max

f(x) +

m

j=1
u
j
(1 − a
j
t
x)

=
m

j=1
u
j
+ max
x≥0

f(x) −

m

j=1
u
j
a
j

t

x

=
m

j=1
u
j
+ max
x≥0

inf {v
t
x : v ∈ F
∗
} −

m

j=1
u
j
a
j

t
x

=
m


j=1
u
j
+ max
x≥0
inf
v∈F
∗

v −
m

j=1
u
j
a
j

t
x.
v ∈ F
∗
v ≤
m

j=1
u
j
a

j
max
x≥0

v −
m

j=1
u
j
¯a
j

t
x = +∞.
v ≤
m

j=1
u
j
¯a
j
max
x≥0

v −
m

j=1

u
j
¯a
j

t
x = 0.
max
x≥0
L(x, u) =





m

j=1
u
j
v ∈ F v ≤
m

j=1
u
j
¯a
j
,
+∞ .

min
m

j=1
u
j
v ≤
m

j=1
u
j
¯a
j
, v ∈ F
∗
, u ≥ 0,
min
m

j=1
u
j
m

j=1
u
j
¯a
j

∈ F
∗
, u ≥ 0.
f(x) X X
R
n
+
x ∈ X ⇒ y ∈ X ∀y : 0 ≤ y ≤ x
max f(x) x ∈ X.
X f(x)
f
∗
(.) f(x)
f
∗
(v) = inf{v
t
x − f(x) : x ≥ 0} v ∈ R
n
+
.
f
∗
(0) ≤ v
t
0 − f(0) = 0 ∀v ∈ R
n
+
.
f

∗
(v) = inf{v
t
x − inf{v
t
x : v

∈ F
∗
} : x ≥ 0}
= inf
x≥0
sup
v

∈F
∗
(v − v

)
t
x
= sup
v

∈F
∗
inf
x≥0
(v − v


)
t
x.
v ∈ F
∗
f
∗
(v) = sup
v

∈F
∗
inf
x≥0
(v − v

)
t
x ≥ inf
x≥0
(v − v)
t
x = 0.
f
∗
(v) = 0.
v /∈ F
∗
¯x ≥ 0

v
t
¯x > α > v
t
¯x ∀v

∈ F
∗
.
sup
v

∈F
∗
(v − v

)
t
¯x < 0.
f
∗
(v) = inf
x≥0
sup
v

∈F
∗
(v − v


)
t
x ≤ inf
θ≥0
sup
v

∈F
∗
θ(v − v

)
t
¯x ≤ +∞.
f
∗
(v) = −δ(v|F
∗
),
δ(v|F
∗
) F
∗
δ(v|F
∗
) =

0 v ∈ F
∗
,

∞ v /∈ F
∗
.
f(x)
f
∗
(v)
f(x) = inf{v
t
x − f
∗
(v) : v ≥ 0}.
inf{v
t
x − f
∗
(x) : x ≥ 0} = inf{v
t
x + δ(v|F
∗
) : v ≥ 0}
= inf{v
t
x : v ∈ F
∗
} = f(x).
g(x) = δ(x|X) δ(x|X) X g(x)
R
n
+

sup(f(x) − g(x)) x ∈ R
n
+
.
inf(g
∗
(v) − f
∗
(v)) v ∈ R
n
+
,
g
∗
(v) g(x)
g
∗
(v) = sup{v
t
x − g(x) : x ≥ 0} = sup{v
t
x : x ∈ X}.
sup{f(x) − g(x) : x ≥ 0} = inf{g
∗
(v) − f
∗
(v) : v ≥ 0}.
inf{g
∗
(v) − f

∗
(v) : v ≥ 0} = inf {g
∗
(v) : v ∈ F
∗
} ( (1.25))
= inf
v∈F
∗
sup
x∈X
v
t
x
= sup
x∈X
inf
v∈F
∗
v
t
x
= sup{f(x) : x ∈ X}
= sup{f(x) − g(x) : x ≥ 0}.
inf g
∗
(v) v ∈ F
∗
.
X

X =

x ≥ 0 : ¯a
jt
x ≤ 1, j = 1, 2, , m

,
g
∗
(v) = sup{v
t
x : x ∈ X}
= max{v
t
x : a
j
t
x ≤ 1, j = 1, 2, , m, x ≥ 0}
= min

m

j=1
u
j
: v ≤
m

j=1
u

j
, u ≥ 0

.
min
x≥0
min
u≥0:v≤
m

j=1
u
j
¯a
j
m

j=1
u
j
,
min
m

j=1
u
j
m

j=1

u
j
¯a
j
∈ F
∗
, u ≥ 0.
f(x)
c
1
≥ 0, c
2
≥ 0, , c
n
≥ 0
f(x) = min

x
i
c
i
: i = 1, 2, , n

.
X
X =

x ≥ 0 : x ≤
m


i=1
η
i
b
i
,
m

i=1
η
i
≤ 1, η
i
≥ 0, i = 1, 2, , m

,
b
i
= 0, b
i
> 0 (i = 1, 2, , m)
m

i=1
b
i
> 0.
f
∗
(v) = −δ(v|F

∗
)
F
∗
= {v ≥ 0 : c
t
v ≥ 1}
g
∗
(v) = sup{v
t
x : x ∈ X} = max{v
t
b
i
, i = 1, 2, , m}.
f(x) X
min
v≥0
max{v
t
b
j
: j = 1, 2, , m}.
f(x)
R
n
+
f(x)
˜

f(v) f(x)
R
n
+
˜
f(v) =
1
sup{f(x) : v
t
x ≤ 1, x ≥ 0}
∀v ∈ R
n
+
.
˜
f(.) R
n
+
f(.)
˜
f(.)
f(x) =
1
sup{
˜
f(v) : v
t
x ≤ 1, v ≥ 0}
∀x ∈ R
n

+
.

1
+∞
= 0

f(x) f(x) = min

q
i
t
x : i = 1, 2, , s

.
f(x)
f(x)
F = {x ≥ 0 : f(x) ≥ 1} ,
F
∗
F
F
∗
=

v ≥ 0 : v
t
x ≥ 1 ∀x ∈ F

,

F F
∗
F =

x ≥ 0 : v
t
x ≥ 1 ∀v ∈ F
∗

.
g(v) F
∗
g(v) = max {γ ≥ 0 : v ∈ γF
∗
} ∀v ∈ R
n
+
.
{v ≥ 0 : g(v) ≥ γ} = γF
∗
∀γ ≥ 0.
γ = 0 v ∈ R
n
+
1
sup {f(x) : v
t
x ≤ 1, x ≥ 0}
≥ γ ⇔ v ∈ γF
∗

.
γ > 0 v ∈ R
n
+
1
sup {f(x) : v
t
x ≤ 1, x ≥ 0}
≥ γ
⇔ 1 ≥ γsup

f(x) : v
t
x ≤ 1, x ≥ 0

⇔ 1 ≥ sup

f(γx) : v
t
x ≤ 1, x ≥ 0

⇔ 1 ≥ sup

f(x

) : v
t
1
γ
x


≤ 1, x

≥ 0

⇔

x

≥ 0,
1
γ
v
t
x

≤ 1 ⇒ 1 ≥ f(x

)

⇔

x

≥ 0,
1
γ
v
t
x


< 1 ⇒ 1 > f(x

)

⇔

x

≥ 0, 1 ≤ f(x

) ⇒
1
γ
v
t
x

≥ 1

⇔

x

∈ F ⇒
1
γ
v
t
x


≥ 1

⇔
1
γ
v ∈ F
∗
.
1
sup {f(x) : v
t
x ≤ 1, x ≥ 0}
≥ γ ⇔ v ∈ γF
∗
∀γ ≥ 0.
g(v) =
1
sup {f(x) : v
t
x ≤ 1, x ≥ 0}
=
˜
f(v) ∀v ∈ R
n
+
.
˜
f(v) F
∗

f(x) F F F
∗
f(x)
˜
f(v)
f(x) =
1
sup

˜
f(v) : v
t
x ≤ 1, v ≥ 0

∀x ∈ R
n
+
.
X ⊆ R
n
+
X
x ∈ X ⇒ y ∈ X ∀y : 0 ≤ y ≤ x.
V X
V = {v ≥ 0 : v
t
x ≤ 1 ∀x ∈ X}.
V
R
n

+
X X
V
X = {x ≥ 0 : v
t
x ≤ 1 ∀v ∈ V }.
max f(x) x ∈ X.
max
˜
f(v) v ∈ V.
f(x)
˜
f(v) X V
x v
x ∈ X, v ∈ V f(x)
˜
f(v) = 1.
x ∈ X v ∈ V v
t
x ≤ 1. f(x) = 0
f(x)
˜
f(v) = 0 ≤ 1 f(x) > 0 v
t
x ≤ 1
f(x)
˜
f(v) = f(x)
1
sup {f(x) : v

t
x ≤ 1, x ≥ 0}
≤ f(x)
1
f(x)
= 1.
f(x)
˜
f(v) ≤ 1 ∀x ∈ X ∀v ∈ V.
¯x ¯v
f(¯x)
˜
f(¯v) = max

f(x)
˜
f(v) : x ∈ X, v ∈ V

= max
x∈X
f(x) max
v∈V
˜
f(v).
f(¯x) = max
x∈X
f(x)
˜
f(¯v) = max
v∈V

˜
f(v). ¯x ¯v
¯x ¯v ¯x f(x)
X f(¯x)F
v

∈ R
n
+
v
t
x ≤ 1 ∀x ∈ X,
v
t
x ≥ 1 ∀x ∈ f(¯x)F.
v

∈ V
f(¯x)v
t
x ≥ 1 ∀x ∈ F ⇒ f(¯x)v

∈ F
∗
⇔
˜
f(f(¯x)v

) ≥ 1
⇔ f(¯x)

˜
f(v

) ≥ 1.
f(¯x)
˜
f(v

) = 1 v

v

˜
f(v

) =
˜
f(¯v) ⇒ 1 = f(¯x)
˜
f(v

) = f(¯x)
˜
f(¯v).
¯x ¯v ¯v
t
¯x = 1.
¯x ¯v f(¯v) > 0
˜
f(¯v) =

1
sup{f(x) : ¯v
t
x ≤ 1, x ≥ 0}
=
1
f(¯x)
.
f(¯x) = sup{f(x) : ¯v
t
x ≤ 1, x ≥ 0}
f(x) f(x) ≥ 0 ¯v
t
¯x = 1.
f(x) X
˜
f(v) = c
t
v ∀v ∈ R
n
+
,
V = {v ≥ 0 : b
i
t
v ≤ 1, i = 1, 2, , m}.
˜
f(v) V
c
i

i (i = 1, 2, , n)
c = (c
1
, c
2
, , c
n
)
x = (x
1
, x
2
, , x
n
) x
i
i (x
i
≥
0, i = 1, 2, , n) f(x)
f(x) = min

x
i
c
i
: i = 1, 2, , n

.
f(x)

X x
f(x) x ∈ X.
m j (j = 1, 2, , m)
a
j
= (a
1j
, a
2j
, , a
nj
)
t
a
ij
i j
u
j
j j
u
j
a
j
m

j=1
u
j
a
j

X
X =

x ∈ R
n
+
: x ≤ A
t
u,
m

j=1
u
j
≤ 1, u
j
≥ 0, j = 1, 2, , m

,
a
j
, j = 1, 2, , m.
X
X ⊂ R
n
+
, R
n
+
= {x = (x

1
, x
2
, , x
n
) : x
i
≥ 0, i = 1, 2, , n} ;
X
x ∈ X y
0 ≤ y ≤ x y ∈ X
∃x ∈ X : x > 0 (x
i
> 0, i = 1, 2, , n)
X f(x) x
f
∗
f(x) X
X f
∗
> 0