6.253:
Convex
Analysis
and
Optimization
Homework
Prof.
Dimitri
P.
Bertsekas
Spring
2010,
M.I.T.
Problem
1
(a)
Show
that
a
nonpolyhedral
closed
convex
cone
need
not
be
retractive,
by
using
as
an
example
the
cone
C
=
{
(
u,v,w
)
(
u,v
)
≤
w
}
,
the
recession
direction
d
=
(1
,
0
,
1),
and
the
corresponding
asymptotic
sequence
{
(
k,

√
k,
√
k
2
+
k
)
}
.
(This
is
the,
socalled,
second
order
cone,
which
plays
an
important
role
in
conic
programming;
see
Chapter
5.)
(b)
Verify
that
the
cone
C
of
part
(a)
can
be
written
as
the
intersection
of
an
inFnite
number
of
closed
halfspaces,
thereby
showing
that
a
nested
set
sequence
obtained
by
intersection
of
an
inFnite
number
of
retractive
nested
set
sequences
need
not
be
retractive.
Problem
2
Let
C
be
a
nonempty
convex
set
in
R
n
,
and
let
M
be
a
nonempty
aﬃne
set
in
R
n
.
Show
that
M
∩
rin
(
C
) =
∅
is
a
necessary
and
suﬃcient
condition
for
the
existence
of
a
hyperplane
H
containing
M
,
and
such
that
rin
(
C
)
is
contained
in
one
of
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 Spring '10
 Rejaei
 Electromagnet, Mathematical analysis, Convex set, Empty set, Convex function, Convex Analysis and Optimization, closed convex cone

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