HW
1
EE
588:
Optimization
for
the
information
and
data
sciences
University
of
Southern
California
Assigned
on:
August
23,
2023
Due
date:
beginning
of
class
on
September
11,
2023
2.5
What
is
the
distance
between
two
parallel
hyperplanes
{z
€
R"

a2
=
b;}
and
{z
€
R"

a"z
=
by}?
2.7
Voronoi
description
of
halfspace.
Let
a
and
b
be
distinct
points
in
R™.
Show
that
the
set
of
all
points
that
are
closer
(in
Euclidean
norm)
to
a
than
b,
i.e.,
{z

z
—al2
<
z—b2},
is
a
halfspace.
Describe
it
explicitly
as
an
inequality
of
the
form
¢'z
<
d.
Draw
a
picture.
2.12
Which
of
the
following
sets
are
convex?
(a)
A
slab,
i.e.,
a
set
of
the
form
{z
€
R"

a
<
aTz
<
B}.
(b)
A
rectangle,
i.e.,
a
set
of
the
form
{z
€
R"

a;
<z
<
Bi,
i
=1,...,n}.
A
rectangle
is
sometimes
called
a
hyperrectangle
when
n
>
2.
(c)
A
wedge,
i.e.,
{x
€
R"

alz
<
by,
alz
<
by}.
(d)
The
set
of
points
closer
to
a
given
point
than
a
given
set,
i.e.,
{o

llz

zoll2
<
llo
—
ylla
for
all
y
€
5}
where
S
C
R".
(e)
The
set
of
points
closer
to
one
set
than
another,
i.e.,
{z

dist(z,
S)
<
dist(z,T)},
where
S,
T
C
R",
and
dist(z,S)
=
inf{lz
—
z2

z
€
S}.
(f)
[HUL93,
volume
1,
page
93]
The
set
{z

z
+
Sz
C
S1},
where
S1,S2
C
R"
with
S;
convex.
(g)
The
set
of
points
whose
distance
to
a
does
not
exceed
a
fixed
fraction
6
of
the
distance
to
b,
i.e.,
the
set
{z

z
—
al]2
<
6z
—
b2}.
You
can
assume
a
#
b
and
0<6<1.
2.28
Positive
semidefinite
cone
for
n
=
1,
2,
3.
Give
an
explicit
description
of
the
positive
semidefinite
cone
S,
in
terms
of
the
matrix
coefficients
and
ordinary
inequalities,
for
n
=1,
2,
3.
To
describe
a
general
element
of
S™,
for
n
=
1,
2,
3,
use
the
notation
1
xr2
rs3
1
xro
I,
N
r2
T4
s
.
xr2
xrs3
r3
Ts
Te