Cực tiểu của hàm lồi – lõm với ràng buộc lồi

N
R
∅
x ∈ M x M
y /∈ M y M
∀x x
∃x x
M ∩ N M N
M ∪ N M N
M ⊂ N M N
M \ N M N
|x| x
||x|| x
sup
x∈K
f(x) {f(x) | x ∈ K}
inf
x∈K
f(x) {f(x) | x ∈ K}
A = (a
ij
) A a
ij
A
T
A
., .
M M
int M M

aff M M
co M M
dim M M
ri M M
con M M
cone M M
co f f
epi f f
dom f f
•
•
•
•
min{f(x, y) : (x, y) ∈ C, x ∈ X, y ∈ Y }
X ⊂ R
n
1
Y ⊂ R
n
2
C X ×Y
f(x, y) : X ×Y → R f(x, y) X x
y Y y x
y
x
A ⊆ R
n
λx + (1 − λ)y ∈ A, ∀x, y ∈ A, λ ∈ R.
A = ∅ A = M + a a ∈ A M
R

n
M
A
A dim A
dim ∅ = −1
x ∈ R
n
x = λ
1
a
1
+ λ
2
a
2
+ + λ
k
a
k
a
i
∈ R
n
λ
1
+ λ
2
+ + λ
k
= 1 a

1
, a
2
, , a
k
A R
n
A A aff A
A
A ⊆ R
n
∀x, y ∈ A, ∀λ ∈ (0, 1) ⇒ λx + (1 − λ)y ∈ A.
∅
A B R
n
A ∩ B αA + βB α, β ∈ R
A dim A
aff A A R
n
dim A = n
x ∈ R
n
x = λ
1
a
1
+ λ
2
a
2

+ + λ
k
a
k
a
i
∈ R
n
λ
i
≥ 0 λ
1
+ λ
2
+ + λ
k
= 1 a
1
, a
2
, , a
k
A ⊂ R
n
x
1
, , x
m
∈ A A
x

1
, , x
m
m = 1
m = 2 λ
1
, λ
2
≥ 0 λ
1
+ λ
2
= 1, x
1
, x
2
∈ A
λ
1
x
1
+ (1 − λ
2
)x
2
∈ A.
m = k − 1 ≥ 2
m = k x
1
, , x

k
∈ A λ
i
≥ 0 (i = 1, , k)
k

i=1
λ
i
= 1
x = λ
1
x
1
+ + λ
k
x
k
∈ A.
λ
k
= 1 λ
k
= 1 λ
1
= = λ
k−1
= 0 x ∈ A
λ =
k−1


i=1
λ
i
= 1 −λ
k
= 0.
k−1

i=1
λ
i
1 − λ
k
= 1
λ
i
1 − λ
k
≥ 0, ∀i = 1, , k −1
y =
λ
1
1 − λ
k
x
1
+ +
λ
k−1

1 − λ
k
x
k−1
∈ A.
y ∈ A x
k
∈ A 1 − λ
k
> 0, (1 − λ
k
) + λ
k
= 1
x = (1 − λ
k
)y + λ
k
x
k
∈ A.

A R
n
A A co A A
a A
x ∈ A λ > 0 a + λ(x −a) ∈ A
A ri A
R
n

{x ∈ R
n
: a
T
x = α},
a ∈ R
n
0 α ∈ R
a ∈ R
n
0 α ∈ R
{x ∈ R
n
: a
T
x ≥ α},
{x ∈ R
n
: a
T
x > α},
a
i1
x
1
+ a
i2
x
2
+ + a

in
x
n
≤ b
i
, i = 1, 2, , m,
x Ax ≤ b A = (a
ij
) ∈ R
m×n
, b = (b
1
, , b
m
)
T
.
D
λ > 0, ∀x ∈ D ⇒ λx ∈ D.
D ⊂ X con D cone D
D
D ⊆ R
n
x
0
∈ D
N
D
(x
0

) := {ω ∈ R
n
: ω, x − x
0
 ≤ 0, ∀x ∈ D}
D x
0
−N
D
(x
0
)
D x
0
.
0 ∈ N
D
(x
0
) N
D
(x
0
)
C D
H := {x : v, x = λ}
C D
v, a ≤ λ ≤ v, b, ∀a ∈ C, b ∈ D.
C D
v, a < λ < v, b, ∀a ∈ C, b ∈ D.

C D
sup
x∈C
v, x < λ < inf
y∈D
v, x.
C D R
n
C ∩ D = ∅ C D
C D
R
n
C ∩ D = ∅ C D
a ∈ R
n
A m × n
a, x ≥ 0 x Ax ≥ 0 y ≥ 0 R
m
a = A
T
y
a, x = 0
Ax ≥ 0 a
A
D = ∅ y
d
D
(y) := inf
x∈D
||x − y||.

d
D
(y) y D π ∈ D d
D
(y) = ||y−π||
π y D π = P
D
(y)
P
D
(y) y D
min{
1
2
||x − y||
2
: x ∈ D}.
y D
||x − y||
2
D D = ∅ d
D
(y)
0 ≤ d
D
(y) ≤ ||x − y||, ∀x ∈ D.
D
y ∈ R
n
, π ∈ D,

π = P
D
(y);
y −π ∈ N
D
(π).
y ∈ R
n
P
D
(y) y D
||P
D
(x) − P
D
(y)|| ≤ ||x − y||, ∀x, y ∈ R
n
||P
D
(x) − P
D
(y)||
2
≤ P
D
(x) − P
D
(y), x − y, ∀x, y ∈ R
n
C ⊂ R

n
, f : C → R ∪{+∞}
epi f := {(x, α) ∈ C ×R : f(x) ≤ α},
dom f := {(x ∈ C : f(x) < +∞},
f
f dom f = ∅ f(x) > −∞, ∀x ∈ C
f x ∈ C {x
k
} ⊂ C
x
k
→ x lim inff(x
k
) ≥ f(x) f C f
x ∈ C
f C epi f R
n
×R
f C −f C
C R
n
C
δ(x, C) :=

0 x ∈ C,
+∞ x ∈ C.
f : C → R ∪{+∞} f
f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y), ∀x, y ∈ C, λ ∈ (0, 1).
f epi f
∀(x, r) ∈ epi f ∀(y, s) ∈ epi f ∀λ ∈ (0, 1)

λ(x, r) + (1 −λ)(y, s) = (λx + (1 − λ)y, λr + (1 −λ)s) ∈ epif
⇔ f(λx + (1 − λ)y) ≤ λr + (1 −λ)s
⇔ f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y), ( r = f (x), s = f(y))
x y dom f f(x) = +∞ f(y) = +∞
(x, r) ∈ epi f, (y, s) ∈
epi f λ ∈ (0, 1)
λ(x, r) + (1 −λ)(y, s) ∈ epi f.
(x, r) ∈ epi f, (y, s) ∈ epi f f(x) ≤ r, f(y) ≤ s
f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y)
≤ λr + (1 −λ)s.
(λx + (1 − λ)y, λr + (1 −λ)s) ∈ epi f
λ(x, r) + (1 −λ)(y, s) ∈ epi f.

f : C → R ∪{+∞} f
λ
i
≥ 0 (i = 1, , m)
m

i=1
λ
i
= 1, ∀x
1
, , x
m
∈ C,
f(λ
1
x

1
+ + λ
m
x
m
) ≤ λ
1
f(x
1
) + + λ
m
f(x
m
).
x
i
∈ dom f, (i = 1, , m) f(x
i
) = +∞, λ
i
f(x
i
) = +∞
x
i
∈ dom f, (i = 1, , m) epi f (x
i
, f(x
i
)) ∈ epif, (i = 1, , m)

(λ
1
x
1
+ + λ
m
x
m
, λ
1
f(x
1
) + + λ
m
f(x
m
)) ∈ epi f.
f(λ
1
x
1
+ + λ
m
x
m
) ≤ λ
1
f(x
1
) + + λ

m
f(x
m
).

f
1
, f
2
f
1
+ f
2
f
1
, f
2
, , f
m
λ
i
λ
1
f
1
+ λ
2
f
2
+ + λ

m
f
m
ω ∈ R
n
f
x
0
∈ R
n
ω, x − x
0
 ≤ f(x) − f(x
0
), ∀x ∈ R
n
.
f x
0
f x
0
∂f (x
0
) := {ω ∈ R
n
: ω, x − x
0
 ≤ f(x) − f(x
0
), ∀x ∈ R

n
}.
f x
0
∂f (x
0
) = ∅.
f
1
, f
2
, , f
m
R
n
m

i=1
∂f
i
(x) ⊆ ∂(
m

i=1
f
i
(x)), ∀x.
∩ri (dom f
i
) = ∅

m

i=1
∂f
i
(x) = ∂(
m

i=1
f
i
(x)), ∀x.
d f
x
f

(x, d) := lim
λ→0
+
f(x + λd) − f(x)
λ
f C x ∈ C d
x + d ∈ C d f x
f

(x, d) ≤ f(x + d) − f(x).
x f

(x, .) {d : x + d ∈ C}
f

f

(x, d) = ∇f(x), d, ∀d.
∂f (x
∗
) f x
∗
D ⊆ R
n
f : R
n
→ R
min{f(x) : x ∈ D}. P
x
∗
∈ D f(x
∗
) ≤ f(x)
x ∈ D x
∗
∈ D (P )
D f
(P ) D
D := {x ∈ X : g
j
(x) ≤ 0, h
i
(x) = 0, j = 1, , m, i = 1, , p},
∅ = X ⊆ R
n

g
j
, h
i
: R
n
→ R (j = 1, , m, i = 1, , p)
(P ) D
(P )
x x
j
j f(x) x (P )
D
x
∗
∈ D
(P ) x
∗
f(x
∗
) ≤ f(x), ∀x ∈ U ∩ D
x
∗
(P )
f(x
∗
) ≤ f(x), ∀x ∈ D.
(P ) D
f
(P )

x
∗
∈ D
(P ) U x
∗
f(x
∗
) ≤ f(x) x ∈ U ∩ D
x ∈ D λ > 0 x
λ
= λx + (1 − λ)x
∗
∈ U ∩ D
f(x
∗
) ≤ f(x
λ
) ≤ λf(x) − (1 − λ)f(x
∗
),
f(x
∗
) ≤ f(x). x
∗
(P )
x
∗
1
, x
∗

2
(P ) λ ∈ (0, 1)
f(λx
∗
1
+ (1 − λ)x
∗
2
) ≤ λf(x
∗
1
) + (1 − λ)f(x
∗
2
) ≤ λf(x) + (1 − λ)f(x), ∀x ∈ D.
f(λx
∗
1
+ (1 − λ)x
∗
2
) ≤ f(x), ∀x ∈ D λx
∗
1
+ (1 − λ)x
∗
2
(P ) 
(P )
• D = ∅

• f D inf
x∈D
f(x) = −∞
• inf
x∈D
f(x) < ∞ D
• x
∗
∈ D f(x
∗
) = min
x∈D
f(x)
(P )
(P )
F
+
:= {t ∈ R : f(x) ≤ t, x ∈ D},
x
∗
F
+
(D) = [f(x
∗
), +∞]
F
+
(D) t
∗
= inf F

+
(D) t > −∞ F
+
(D)
t
∗
∈ F
+
(D) x
∗
∈ D f(x
∗
) = t
∗
x
∗
f D 
D f D
(P )
α := inf
x∈D
f(x)

x
k

∈ D
lim
k→+∞
f(x

k
) = α D x
0
∈ D
x
k
→ x
0
f α > −∞
x
0
∈ D α f(x
0
) ≥ α f(x
0
) = α 
f D
f(x) → +∞ x ∈ D, ||x|| → +∞
f D
D(a) := {x ∈ D : f(x) ≤ f(a)} a ∈ D D(a)
f D(a)
f D 
D (int D = ∅) f
D x
∗
(P )
0 ∈ ∂f(x
∗
) + N
D

(x
∗
)
N
D
(x
∗
) D x
∗
p
∗
p
∗
∈ ∂f (x
∗
) ∩ (−N
D
(x
∗
)).
p
∗
∈ ∂f (x
∗
)
p
∗
, x − x
∗
 ≤ f(x) − f(x

∗
), ∀x
p
∗
∈ N
D
(x
∗
)
p
∗
, x − x
∗
 ≥ 0, ∀x ∈ D.
f(x) − f(x
∗
) ≥ 0, ∀x ∈ D.
x
∗
D D D
int D = ∅
E := {(t, x) ∈ R × R
n
: t > f(x) − f(x
∗
), x ∈ D},
G := {0} × D.
E G D f G ∩ E = ∅
(u
0

, u) = 0 ∈ R × R
n
u
0
t + u
T
x ≤ u
0
0 + u
T
y, ∀(t, x) ∈ E, ∀y ∈ D.
t → +∞ u
0
≤ 0 u
0
= 0
u, x − y ≤ 0, ∀x, y ∈ D.
0 ∈ D 0 ∈ int D
u = 0 u
0
= 0 u
0
< 0
−u
0
> 0
−t + u
T
x ≤ u
T

y, ∀x, y ∈ D.
t → f(x) − f(x
∗
)
−[f(x) − f(x
∗
)] + u
T
x ≤ u
T
y, ∀x, y ∈ D.
y = x
∗
−[f(x) − f(x
∗
)] + u
T
x ≤ u
T
x
∗
, ∀x ∈ D.
f(x
∗
) − f(x) + u
T
(x − x
∗
) ≤ 0, ∀x ∈ D.
x ∈ D f(x) = ∞

f(x
∗
) − f(x) + u
T
(x − x
∗
) ≤ 0, ∀x ∈ X.
u ∈ ∂f (x
∗
)
x = x
∗
u
T
(y −x
∗
) ≥ 0, ∀y ∈ D.
−u ∈ N
D
(x
∗
) u ∈ ∂f(x
∗
)
0 ∈ ∂f(x
∗
) + N
D
(x
∗

).

x
∗
∈ int D
(P ) 0 ∈ ∂f(x
∗
) f D = R
n
0 = f(x
∗
)
(P )
min f(x)
x ∈ D := {x ∈ X : g
i
(x) ≤ 0, h
j
(x) = 0, i = 1, , m, j = 1, , k}
∅ = X ⊆ R
n
f, g
i
, h
j
: R
n
→ R, ∀i, j (P )
X f, g
i

h
j
(P )
L(x, λ, µ) := λ
0
f(x) +
m

i=1
λ
i
g
i
(x) +
k

j=1
µ
j
h
j
(x).
(P )
x
∗
(P ) λ
∗
i
≥ 0 (i = 0, 1, , m) µ
∗

j
(j =
1, , k) 0
L(x
∗
, λ
∗
, µ
∗
) = min
x∈X
L(x, λ
∗
, µ
∗
)
λ
∗
i
g
i
(x
∗
) = 0 (i = 1, , m)
int X = ∅
∃x
0
∈ D : g
i
(x

0
) < 0 (i = 1, , m)
h
j
(j = 1, , k) X λ
∗
0
> 0
x
∗
(P )
x
∗
(P )
C := {(λ
0
, λ
1
, , λ
m
, µ
1
, , µ
k
) : (∃x ∈ X) :
f(x) − f(x
∗
) < λ
0
, g

i
(x) ≤ λ
i
, i = 1, , m, h
j
(x) = µ
j
, j = 1, , k}.
X = ∅ f, g
i
h
j
X C
R
m+k+1
0 ∈ C 0 ∈ C
x f(x) < f(x
∗
) x
∗
(P ) λ
∗
i
(i = 0, 1, , m), µ
∗
j
(j = 1, , k)
0
m


i=0
λ
∗
i
λ
i
+
k

j=1
µ
∗
j
µ
j
≥ 0, ∀(λ
0
, , λ
m
, µ
1
, , µ
k
) ∈ C.
λ
0
, , λ
m
> 0 x = x
∗

(λ
0
, , λ
m
, 0, 0) ∈ C.
λ
∗
0
, λ
∗
1
, , λ
∗
m
≥ 0 ε > 0 x ∈ X
λ
0
= f(x) − f(x
∗
) + ε, λ
i
= g
i
(x) (i = 1, , m), µ
j
= h
j
(x) (j = 1, , k)
ε → 0
λ

∗
0
f(x) +
m

i=1
λ
∗
i
g
i
(x) +
k

j=1
µ
∗
j
h
j
(x) ≥ λ
∗
0
f(x
∗
) +
m

i=1
λ

∗
i
g
i
(x
∗
) +
k

j=1
µ
∗
j
h
j
(x
∗
), ∀x ∈ X.
L(x
∗
, λ
∗
, µ
∗
) ≤ L(x, λ
∗
, µ
∗
), ∀x ∈ X.
x

∗
g
i
(x
∗
) ≤ 0 i i
g
i
(x
∗
) = ξ < 0 ε > 0
(ε, , ξ, ε, , ε, 0, , 0) ∈ C (ξ i + 1).
ε → 0 λ
∗
i
ξ ≥ 0 ξ < 0 λ
∗
i
≤ 0
λ
∗
i
= 0
λ
∗
0
> 0 λ
∗
0
= 0

0 =
m

i=1
λ
∗
i
g
i
(x
∗
) +
k

j=1
µ
∗
j
h
j
(x
∗
) ≤
m

i=1
λ
∗
i
g

i
(x) +
k

j=1
µ
∗
j
h
j
(x), ∀x ∈ X.
λ
∗
0
= 0
• i λ
∗
i
> 0 x = x
0
0 =
m

i=1
λ
∗
i
g
i
(x

∗
) +
k

j=1
µ
∗
j
h
j
(x
∗
) ≤
m

i=1
λ
∗
i
g
i
(x
0
) +
k

j=1
µ
∗
j

h
j
(x
0
) < 0 .
• λ
∗
i
= 0 i j µ
∗
j
> 0
0 =
k

j=0
µ
∗
j
h
j
(x
∗
) ≤
k

j=0
µ
∗
j

h
j
(x), ∀x ∈ X.
int X = ∅ h
j
j
k

j=0
µ
∗
j
h
j
(x) = 0, ∀x ∈ X.
h
j
X µ
∗
j
= 0 j
λ
∗
i
µ
∗
j
0 λ
∗
0

> 0
λ
∗
0
> 0 (P )
L(x, λ, µ) = f(x) +
m

i=1
λ
i
g
i
(x) +
k

j=1
µ
j
h
j
(x)
x
f(x
∗
) = f
0
(x
∗
) +

m

i=1
λ
∗
i
g
i
(x
∗
) +
k

j=1
µ
∗
j
h
j
(x
∗
)
≤ f(x) +
m

i=1
λ
∗
i
g

i
(x) +
k

j=1
µ
∗
j
h
j
(x) ≤ f(x)
x
∗
(P ) 
X X
0 ∈ ∂f(x
∗
) +
m

i=1
λ
∗
i
∂g
i
(x
∗
) +
k


j=1
µ
∗
j
∇h
j
(x
∗
).
f g
i
(i = 1, , m)
0 = λ
∗
0
∇f
0
(x
∗
) +
m

i=1
λ
∗
i
∇g
i
(x

∗
) +
k

j=1
µ
∗
j
∇h
j
(x
∗
).
d = 0
D x
∗
∈ D
x
∗
+ λd ∈ D, ∀λ > 0 .
D(x
∗
) D x
∗
D(x
∗
)
f D x
∗
f D

d
T
∇f(x
∗
) ≥ 0, ∀d ∈ D(x
∗
).
f x
∗
f(x
∗
+ λd) = f(x
∗
) + λ∇f(x
∗
), d + o(λ||d||).
x
∗
(P )
f(x
∗
+ λd) − f(x
∗
) ≥ 0, ∀λ > 0 .
d
T
∇f(x
∗
) +
o(λ||d||)

λ
≥ 0, ∀λ > 0 .

x
∗
∈ D f
D
min f(x) = x
3
D = [−1, 2].
x
∗
= 0 f(x) f(x) D
x = −1
(P )
min f(x)
x ∈ D := {x ∈ X, g
j
(x) ≤ 0, h
i
(x) = 0, j = 1, , m, i = 1, , k}.
x
0
∈ D
A(x
0
) := {j : g
j
(x
0

) = 0}
S(x
0
)

h
i
(x
0
), d = 0, i = 1, , k,
g
j
(x
0
), d ≤ 0, j ∈ A(x
0
).
x
0
∈ D D(x
0
) ⊆ S(x
0
)
d ∈ D(x
0
) d
T
∇g
j

(x
0
) > 0 (j ∈ A(x
0
))
g
j
(x
0
+ d) − g
j
(x
0
) > 0 g
j
(x
0
+ d) > g
j
(x
0
) = 0 j ∈ A(x
0
)
d
g
j
(x
0
), d ≤ 0, j ∈ A(x

0
).
h
i
(x
0
), d = 0, i = 1, , k.
d ∈ S(x
0
) D(x
0
) ⊆ S(x
0
) S(x
0
)
D(x
0
) ⊆ S(x
0
)
x
0
S(x
0
) = D(x
0
).
f, g
j

(j = 1, , m), h
i
(i = 1, , k)
x
∗
(P )
λ
∗
= (λ
∗
1
, , λ
∗
m
) ≥ 0, µ
∗
= (µ
∗
1
, , µ
∗
k
)
∇f(x
∗
) +
m

j=1
λ

∗
j
∇g
j
(x
∗
) +
k

i=1
µ
∗
i
∇h
i
(x
∗
) = 0,
λ
∗
j
g(x
∗
) = 0, ∀j = 1, , m ( ).
f, g
j
j h
i
i
x

∗
∈ D (1.14), (1.15), (1.16) x
∗
(P )
f(x
∗
+ λd) = f(x
∗
) + ∇f(x
∗
), λd + o(λd),
f(x
∗
), d ≥ 0, ∀d ∈ D(x
∗
).
D(x
∗
) = S(x
∗
) ∇f(x
∗
), d ≥ 0 d ∈ S(x
∗
)
−∇g
j
(x
∗
), j ∈ A(x

∗
), ∇h
i
(x
∗
), −∇h
i
(x
∗
), 1 = 1, , k.
λ
∗
j
≥ 0, j ∈ A(x
∗
) α
∗
i
≥ 0, β
∗
i
≥ 0, i = 1, , k
∇f(x
∗
) +

j∈A(x
∗
)
λ

∗
j
∇g
j
(x) +
k

i=1
(α
∗
i
) − β
∗
i
)∇h
i
(x
∗
) = 0.
λ
∗
j
= 0 j ∈ A(x
∗
) µ
∗
i
= α
∗
i

− β
∗
i
i (1.15)
(1.16)
g
j
h
i
j, i
(1.14), (1.15) (1.16) x
∗
∈ D
(P ) x
∗
x ∈ D
f(x) < f(x
∗
) d := x − x
∗
∈ 0
∇f(x
∗
), d = lim
t0
f(x
∗
+ td) − f(x
∗
)

t
< 0.
λ
∗
j
g
j
(x
∗
) = 0 j λ
∗
j
= 0 j ∈ A(x
∗
) x ∈ D
∇g
j
(x
∗
), x − x
∗
 ≤ g
j
(x) − g
j
(x
∗
) ≤ 0, ∀j ∈ A(x
∗
).

λ
∗
j
∇g
j
(x
∗
), d ≤ 0, ∀j ∈ A(x
∗
).
h
i
i = 1, , k
∇h
i
(x
∗
), d = 0.
µ
∗
i
∇h
i
(x
∗
), d = 0, i = 1, , k.
(1.17), (1.18) (1.19)
∇f(x
∗
), d +

m

j=1
λ
∗
j
∇g
j
(x
∗
), d +
k

i=1
µ
∗
i
∇h
i
(x
∗
), d < 0.
(1.15) x
∗
(P ) 
L(x, λ, µ) := f(x) +
m

j=1
λ

j
g
j
(x) +
k

i=1
µ
i
h
i
(x)
(1.15)
∇
x
L(x
∗
, λ
∗
, µ
∗
) ≡ ∇f(x
∗
) +
m

j=1
λ
∗
j

∇g
j
(x
∗
) +
k

i=1
µ
∗
i
∇h
i
(x
∗
) = 0.
min{f(x) = x
1
+ x
2
},
x ∈ D := {x
2
1
+ x
2
2
− 2 ≤ 0}
f(x) = x
1

+ x
2
g(x) := x
2
1
+ x
2
2
− 2
L(x, λ) = f(x) + λg(x) = x
1
+ x
2
+ λ(x
2
1
+ x
2
2
− 2).
∇f(x) =

1
1

∇g(x) =

2x
1
2x

2








∇f(x) + λ∇g(x) = 0,
λg(x) = 0, λ ≥ 0,
x ∈ D.
⇔













1 + 2λx
1
= 0, (1.20a)
1 + 2λx

2
= 0, (1.20b)
λ(x
2
1
+ x
2
2
− 2) = 0, λ ≥ 0 (1.20c)
x
2
1
+ x
2
2
− 2 ≤ 0. (1.20d)
λ = 0 (1.20a)−(1.20b) λ > 0 (1.20c) x
2
1
+x
2
2
−2 =
0 (1.20a) − (1.20b) x
1
= x
2
= −1 λ =
1
2

x
∗
= (−1, −1) (1.20)
min{f(x) : x ∈ X, g
j
(x) ≤ 0, j = 1, , m} P
X ⊆ R
n
max{d(y) : y ∈ Y } D
Y ⊆ R
m
(D) (P )
x (P ) y (D)
f(x) ≥ d(y).
(D) (P ) (D)
(P ) x
∗
∈ (P ), y
∗
∈ (D)
f(x
∗
) ≤ d(y
∗
).
(D)
(P ) f(x
∗
) = d(y
∗

) (P ) (D)
(P )
L(x, y) := f(x) +
m

j=1
y
j
g
j
(x).
d(y) := inf
x∈X
L(x, y) LD
(LD) R
m
+
sup
y≥0
d(y) := sup
y≥0
inf
x∈X
L(x, y).
(LD) (P )
d(y) = inf
x∈X
L(x, y) ≤ f(x) +
m


j=1
y
j
g
j
(x) ≤ f(x), ∀x ∈ X, ∀y ∈ Y.
(LD) (P ) 
min{f(x) = −x
2
, x ∈ X = [0, 2], x −1 ≤ 0}.
f(1) = min f = −1.
L(x, y) = −x
2
+ y(x − 1), y ≥ 0, x ∈ X = [0, 2].
max
y≤0
d(y) = 0.
(P ) (LD)
(P )
f g
j
, (j = 1, , m) X
x
0
g
j
(x
0
) < 0 j
(P ) (LD)

g(x) := (g
1
(x), g
2
(x), , g
m
(x))
T
.
A := {(t, z) ∈ R × R
n
: t > f(x), z ≥ g(x), ∀x ∈ X}.