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An in-depth analysis of sufficiency conditions for equality constrained nonlinear programming problems. It covers the second order sufficiency conditions, convexification using augmented lagrangians, and proof of the sufficiency conditions. Additionally, the document discusses sensitivity analysis to understand how the minimum changes when constraints are modified.
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i
(x) = 0, i = 1,... , m.
n
i
n
i
∗ ∈
n
∗ ∈
m
∗ ∗ ∗ ∗
∇ x
L(x , λ ) = 0, ∇ λ
L(x , λ ) = 0,
y
′
∇
2
xx
L(x
∗
, λ
∗
)y > 0 , ∀ y = 0 with ∇h(x
∗
)
′
y = 0.
∗
1
x 2
x 3
x 3
∗ ∗
x 1
x 2
x 3
∗ = x 2
= x 3
1
λ
∗
∗ ∗
∇
2
xx
L(x , λ ) = − 1 0 − 1.
∗ )
′
1
y 3
∗ ∗
y
′
∇
2
xx
L(x , λ )y = −y 1
(y 2
) − y 2
(y 1
) − y 3
(y 1
2 2 2
= y 1
y 2
y 3
∗
c
Lc(x, λ) = f (x) + λ
′
h(x) + ‖h(x)‖
2
,
∇xLc(x, λ) = ∇xL(x, λ),
xx
Lc(x, λ) = ∇
2 ˜ ∇
2
xx
L(x, λ) + c∇h(x)∇h(x)
′
∗ , λ
∗
∗
, λ
∗
) = 0, ∇
2
xx
Lc(x
∗
, λ
∗
∇xLc(x ) > 0 ,
∗
) +
γ
‖x − x
∗
) ≥ L c
(x
∗
, λ
∗ ∗
‖
2
c
(x, λ
c
(x, λ
∗
∗
) +
γ
‖x − x
∗ ∗
‖
2
∇f(x
)
x
x
∆x
a a'x = b + ∆b
a'x = b
Sensitivity theorem for the problem min a
� x=b
f (x). If b is
∗
changed to b+∆b, the minimum x will change to x
∗ +∆x.
∗
Since b + ∆b = a
′ (x
∗
′ x + a
′ ∆x = b + a
′ ∆x, we
have a
′ ∆x = ∆b. Using the condition ∇f (x
∗ ) = −λ
∗ a,
∗ ∗ ∗ ∆cost = f (x + ∆x) − f (x ) = ∇f (x )
′ ∆x + o(‖∆x‖)
∗
= −λ a
′
∆x + o(‖∆x‖)
Thus ∆cost = −λ
∗ ∆b + o(‖∆x‖), so up to first order
∆cost ∗
λ = −.
∆b
For multiple constraints a
′
i
x = b i
, i = 1,... , n, we have
m
∗
∆cost = − λ i
∆bi + o(‖∆x‖).
i=
p(u)
-1 0
u slope ∇p(0) = - λ
= -
Illustration of the primal function p(u) = f x(u)
for the two-dimensional problem
2 2
minimize f (x) =
1 x 1
− x 2
− x 2 2
subject to h(x) = x 2
Here,
p(u) = min f (x) = −
1
2
u
2
− u
h(x)=u
and λ
∗ = −∇p(0) = 1, consistently with the sensitivity
theorem.
∗
h(x) = x
2
2
∇f (x) + ∇h(x)λ = 0, h(x) = u.
∗ , λ
∗
m
i=
λ i
∗ ∇
2 h i
(x
∗ ) ∇h(x
∗ ∇ )
2 f (x
∗ ) +
∇h(x
∗ )
′ 0
∗
∗
∇f x(u) + ∇h x(u) λ(u) = 0, h x(u) = u.
h(x)=u
∇p(u) = ∇ f x(u) = ∇x(u)∇f x(u). u
