|
|
Previous | Next |
[F] = ckbs_kuhn_tucker_L1(
mu, s, y, r, b, d, Bdia, Hdia, Hlow,
pPlus, pMinus)
\mu
-relaxed affine L1 robust smoother problem.
\mu \in \B{R}_+
,
s \in \B{R}^{m \times N}
,
y \in \B{R}^{n \times N}
,
r \in \B{R}^{m \times N}
,
b \in \B{R}^{m \times N}
,
d \in \B{R}^{n \times N}
,
B \in \B{R}^{m \times n \times N}
,
H \in \B{R}^{n \times n \times N}
,
p^+ \in \B{R}^{m \times N}
,
p^- \in \B{R}^{m \times N}
,
the
\mu
-relaxed affine L1 robust Kalman-Bucy smoother problem is:
\[
\begin{array}{ll}
{\rm minimize}
& \frac{1}{2} y^\R{T} H(x) y + d(x)^\R{T} y
+ \sqrt{\B{2}}^\R{T} (p^+ + p^-) -\mu \sum_{i =
1}^{mN} \log(p_i^+) - \sum_{i=1}^{mN} \mu \log(p_i^-)
\;{\rm w.r.t} \; y \in \B{R}^{nN}\; , \; p^+ \in \B{R}_+^{M} \; , \; p^- \in \B{R}_+^{M}
\\
{\rm subject \; to} & b(x) + B(x) y - p^+ + p^- = 0
\end{array}
\]
In addition,
H
is symmetric block tri-diagonal with each block of
size
n \times n
and
B
is block diagonal with each block of size
m \times n
r, \; s \in \B{R}^{m \times N}
to denote
\mu /p^+\;,\; \mu/p^-
, respectively, and
we denote by
q
the lagrange multiplier associated to the
equality constraint. We also use
\B{1}
to denote the vector of length
mN
with all its components
equal to one, and
\B{\sqrt{2}}
to denote the vector of
length
mN
with all its components equal to
\sqrt{2}
.
The corresponding Lagrangian is
\[
L(p^+, p^-, y, q) =
\frac{1}{2} y^\R{T} H y + d^\R{T} y + \B{\sqrt{2}}^T(p^+ + p^-)
- \mu \sum_{i=1}^{mN} \log(p_i^+) - \mu\sum_{i=1}^{mN}\log(p_i^-)
+ q^\R{T} (b + B y - p^+ + p^-)
\]
The partial gradients of the Lagrangian are given by
\[
\begin{array}{rcl}
\nabla_p^+ L(p^+, p^-, y, q ) & = & \B{\sqrt{2}} - q - r \\
\nabla_p^- L(p^+, p^-, y, q) & = & \B{\sqrt{2}} + q - s \\
\nabla_y L(p^+, p^-, y, q ) & = & H y + c + B^\R{T} q \\
\nabla_q L(p^+, p^-, y, q ) & = & b + B y - p^+ + p^- \\
\end{array}
\]
From the first two of the above equations,
we have
q = (r - s)/2
.
D(s)
to denote the diagonal matrix with
s
along its diagonal.
The Kuhn-Tucker Residual function
F : \B{R}^{4mN + nN} \rightarrow \B{R}^{4mN + nN}
is defined by
\[
F(p^+, p^-, r, s, y)
=
\left(
\begin{array}{c}
p^+ - p^- - b - B y \\
D(p^-) D(s) \B{1} - \tau \B{1} \\
r + s - 2 \B{\sqrt{2}} \\
D(p^+) D(r ) \B{1} - \tau \B{1} \\
H y + d + B^\R{T} (r - s)/2
\end{array}
\right)
\]
The Kuhn-Tucker conditions for a solution of the
\mu
-relaxed constrained affine Kalman-Bucy smoother problem is
F(p^+, p^-, r, s, y) = 0
; see Equation (13) in
Aravkin et al 2009
mu
is a positive scalar specifying the
relaxation parameter
\mu
.
s
is an array of size
m \times
N
.
All the elements of
s
are greater than zero.
y
is an array of size
n \times
N
r
is an array of size
m \times
N
. All the elements of
r
are greater than zero.
b
is an array of size
m \times N
.
d
is an array of size
n \times
N
.
Bdia
is an
m \times n \times N
array.
For
k = 1 , \ldots , N
we define
B_k \in \B{R}^{m \times n}
by
\[
B_k = Bdia(:, :, k)
\]
B
is defined by
\[
B
=
\left( \begin{array}{cccc}
B_1 & 0 & 0 & \\
0 & B_2 & 0 & 0 \\
0 & 0 & \ddots & 0 \\
& 0 & 0 & B_N
\end{array} \right)
\]
Hdia
is an
n \times n \times N
array.
For
k = 1 , \ldots , N
we define
H_k \in \B{R}^{n \times n}
by
\[
H_k = Hdia(:, :, k)
\]
Hlow
is an
n \times n \times N
array.
For
k = 1 , \ldots , N
we define
L_k \in \B{R}^{n \times n}
by
\[
L_k = Hlow(:, :, k)
\]
H
is defined by
\[
H
=
\left( \begin{array}{cccc}
H_1 & L_2^\R{T} & 0 & \\
L_2 & H_2 & L_3^\R{T} & 0 \\
0 & \ddots & \ddots & \ddots \\
& 0 & L_N & H_N
\end{array} \right)
\]
F
is a column vector of length
4mN + nN
containing the value of the
ckbs_kuhn_tucker_L1
; Kuhn-Tucker L1 residual, i.e.,
F(p^+, p^-, s^+, s^-, y)
.
ckbs_kuhn_tucker_L1.
It returns true if ckbs_kuhn_tucker_L1 passes the test
and false otherwise.