UC Berkeley
Department of Electrical Engineering and Computer Science
EECS 227A
Nonlinear and Convex Optimization
Review 1
Fall 2009
Problem 1.1
Consider the function
f
:
R
2
→
R
given by:
f
(
x, y
) =
(
x
2
−
y
)
2
−
10
x
2
.
(a) Is
f
convex or not?
(b) Show that
f
has exactly one stationary point over
R
2
, and characterize whether it is a
local minimum, a local maximum, or neither.
(c) Now consider the constrained optimization of
f
over the set
X
=
{
(
x, y
)
∈
R
2

0
≤
y
≤
1
}
.
Show that a global minimum exists, and find all points at which it is attained.
Problem 1.2
Consider the inequalityconstrained problem
min
x
∈
R
n
f
(
x
)
such that
x
∈
C
C
=
{
x
∈
R
n

g
j
(
x
)
≤
0
, j
= 1
, . . . , m
}
,
where
f
and
g
j
, j
= 1
, . . . m
are convex and differentiable functions on
R
n
. Suppose that
x
*
is
feasible, and the pair (
x
*
, μ
*
) satisfy the KarushKuhnTucker necessary conditions, including
the complementary slackness condition.
Using the convexity and KKT conditions, show that [
∇
f
(
x
*
)]
T
[
x
−
x
*
]
≥
0 for all
x
∈
C
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Staff
 Electrical Engineering, Optimization, optimization problem, Πk, Convex Optimization Review, projection ΠK

Click to edit the document details