Digital Signal Processing Reference
In-Depth Information
huge amount of work in the statistics community. Formulation (7.4) is equivalent
1
to the constrained form
min
α
∈R
α
1
s
.
t
.
y
−
F
α
≤
σ,
(7.5)
T
in which case,
F
2
is the indicator function of
the closed convex set
{
α
∈
T
R
. This is a challenging optimization problem that has received
much less interest than equation (7.4), for example, second-order cone program-
ming (Candes et al. 2006c), probing the Pareto curve (Van Den Berg and Fried-
lander 2008), smoothing the dual problem, and using the Nesterov scheme (Becker
et al. 2009). The Dantzig selector (Candes and Tao 2007) is also another challeng-
ing problem that is a special instance of (
P
) when
F
2
is the indicator function of
{
α
∈ R
y
−
F
α
≤
σ
}
T
F
∗
(
y
)
∞
≤
τ
}
, where
F
∗
is the adjoint of
F
.
−
F
α
7.3 MONOTONE OPERATOR SPLITTING FRAMEWORK
7.3.1 Elements of Convex Analysis
We only provide key concepts from convex analysis that will be necessary in this
chapter. A comprehensive account can be found in the work of Rockafellar (1970)
and Lemar echal and Hiriart-Urruty (1996).
Let
H
be a finite-dimensional Hilbert space (typically, the real vector spaces
N
or
T
) equipped with the inner product
R
R
.,.
and associated norm
.
.Let
I
be
H
H
the identity operator on
. The operator spectral norm of
A
:
→H
2
is denoted
1
sup
x
∈H
1
A
x
|||
A
||| =
.Let
.
p
,
p
≥
1bethe
p
norm with the usual adaptation for
x
. Denote
B
p
as the closed
the case
p
=+∞
p
ball of radius
ρ>
0.
A real-valued
function
F
:
H→
(
−∞
,
+∞
]
is
(weakly)
coercive
if
lim
x
→+∞
F
(
x
)
=+∞
. The domain of
F
is defined by dom
F
={
x
∈H
:
F
(
x
)
. We say that a real-valued function
F
is lower semicontinuous (lsc) if lim inf
x
→
x
0
F
(
x
)
<
+∞}
, and
F
is proper if dom
F
=∅
F
(
x
0
). Lower semicontinuity is
weaker than continuity and plays an important role for existence of solutions in
minimization problems.
≥
0
(
H
) is the class of all proper lsc convex functions from
H
to (
−∞
,
+∞
].
Let
C
be a nonempty convex subset of
H
. The indicator function
ı
C
of
C
is
0
,
if
x
∈ C
ı
C
(
x
)
=
otherwise
.
(7.6)
+∞
,
) is the closed convex function
F
∗
de-
The conjugate of a function
F
∈
0
(
H
fined by
F
∗
(
u
)
=
sup
dom
F
u
,
x
−
F
(
x
)
,
(7.7)
x
∈
and we have
F
∗
∈
) and the biconjugate
F
∗∗
=
0
(
H
F
. For instance, a result that
we will use intensively is that the conjugate of the
p
norm is the indicator function
of the unit ball
B
q
, where 1
/
p
+
1
/
q
=
1.
1
In the sense that there exists a bijection
λ
↔
σ
such that the two problems share the same solution.
