Previous | Next |
[
F] = ckbs_kuhn_tucker(
mu,
s,
y,
u,
b,
d,
Bdia,
Hdia,
Hlow)
\mu
-relaxed affine constrained Kalman-Bucy smoother problem.
\mu \in \R_+
,
H \in \R^{p \times p}
,
d \in \R^p
,
b \in \R^r
, and
B \in \R^{r \times p}
,
the
\mu
-relaxed affine constrained Kalman-Bucy smoother problem is:
\[
\begin{array}{rl}
{\rm minimize} & \frac{1}{2} y^\T H y + d^\T y
- \mu \sum_{i=1}^r \log(s_i)
\; {\rm w.r.t} \; y \in \R^p \; , \; s \in \R_+^r
\\
{\rm subject \; to} & s + b + B y = 0
\end{array}
\]
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
(there is an integer
N
such that
p = n * N
and
r = m * N
).
u \in \R^r
to denote the Lagrange multipliers corresponding to the constraint equation.
The corresponding Lagrangian is
\[
L(y, s, u) =
\frac{1}{2} y^\T H y + d^\T y
- \mu \sum_{i=1}^r \log(s_i)
+ u^\T (s + b + B y)
\]
The partial gradients of the Lagrangian are given by
\[
\begin{array}{rcl}
\nabla_y L(y, s, u ) & = & H y + B^\T u + d \\
\nabla_s L(y, s, u ) & = & u - \mu / s \\
\nabla_u L(y, s, u ) & = & s + b + B y \\
\end{array}
\]
where
\mu / s
is the component by component division of
\mu
by the components of the
s
.
Note, from the second equation, that we only need consider
u \geq 0
because
s \geq 0
.
D(s)
to denote the diagonal matrix with
s
along its diagonal and
1_r
to denote the vector, of length
r
with all its components
equal to one.
The Kuhn-Tucker Residual function
F : \R^{r + p + r} \rightarrow \R^{r + p + r}
is defined by
\[
F(s, y, u)
=
\left(
\begin{array}{c}
s + b + B y \\
H y + B^\T u + d \\
D(s) D(u) 1_r - \mu 1_r
\end{array}
\right)
\]
The Kuhn-Tucker conditions for a solution of the
\mu
-relaxed constrained affine Kalman-Bucy smoother problem is
F(s, y, u) = 0
.
\mu
.
r
.
All the elements of s are greater than zero.
p
r
.
All the elements of s are greater than zero.
r
.
p
m \times n \times N
array.
For
k = 1 , \ldots , N
we define
B_k \in \R^{m \times n}
by
\[
B_k = Bdia(:, :, k)
\]
B
is defined by
\[
B
=
\left( \begin{array}{cccc}
B_1 & 0 & 0 & \\
0 & B_2 & 0 & 0 \\
0 & 0 & \ddots & 0 \\
& 0 & 0 & B_N
\end{array} \right)
\]
n \times n \times N
array.
For
k = 1 , \ldots , N
we define
H_k \in \R^{n \times n}
by
\[
H_k = Hdia(:, :, k)
\]
n \times n \times N
array.
For
k = 1 , \ldots , N
we define
L_k \in \R^{n \times n}
by
\[
L_k = Hlow(:, :, k)
\]
H
is defined by
\[
H
=
\left( \begin{array}{cccc}
H_1 & L_2^\T & 0 & \\
L_2 & H_2 & L_3^\T & 0 \\
0 & \ddots & \ddots & \ddots \\
& 0 & L_N & H_N
\end{array} \right)
\]
r + p + r
containing the value of the
Kuhn-Tucker residual
; i.e.,
F(s, y, u)
.
ckbs_kuhn_tucker
.
It returns true if ckbs_kuhn_tucker
passes the test
and false otherwise.