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Compute Residual in Kuhn-Tucker Conditions

Syntax
[F] = ckbs_kuhn_tucker(       mu, s, y, u, b, d, Bdia, Hdia, Hlow)

Purpose
This routine computes the residual in the Kuhn-Tucker conditions for the   \mu -relaxed affine constrained Kalman-Bucy smoother problem.

Problem
Given   \mu \in \B{R}_+ ,   H \in \B{R}^{p \times p} ,   d \in \B{R}^p ,   b \in \B{R}^r , and   B \in \B{R}^{r \times p} , the   \mu -relaxed affine constrained Kalman-Bucy smoother problem is:   $\begin{array}{rl} {\rm minimize} & \frac{1}{2} y^\R{T} H y + d^\R{T} y - \mu \sum_{i=1}^r \log(s_i) \; {\rm w.r.t} \; y \in \B{R}^p \; , \; s \in \B{R}_+^r \\ {\rm subject \; to} & s + b + B y = 0 \end{array}$  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 ).

Lagrangian
We use   u \in \B{R}^r to denote the Lagrange multipliers corresponding to the constraint equation. The corresponding Lagrangian is   $L(y, s, u) = \frac{1}{2} y^\R{T} H y + d^\R{T} y - \mu \sum_{i=1}^r \log(s_i) + u^\R{T} (s + b + B y)$  The partial gradients of the Lagrangian are given by   $\begin{array}{rcl} \nabla_y L(y, s, u ) & = & H y + B^\R{T} u + d \\ \nabla_s L(y, s, u ) & = & u - \mu / s \\ \nabla_u L(y, s, u ) & = & s + b + B y \\ \end{array}$  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 .

Kuhn-Tucker Residual
We use   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 : \B{R}^{r + p + r} \rightarrow \B{R}^{r + p + r} is defined by   $F(s, y, u) = \left( \begin{array}{c} s + b + B y \\ H y + B^\R{T} u + d \\ D(s) D(u) 1_r - \mu 1_r \end{array} \right)$  The Kuhn-Tucker conditions for a solution of the   \mu -relaxed constrained affine Kalman-Bucy smoother problem is   F(s, y, u) = 0  .

mu
The argument mu is a positive scalar specifying the relaxation parameter   \mu .

s
The argument s is a column vector of length   r . All the elements of s are greater than zero.

y
The argument y is a column vector of length   p

u
The argument u is a column vector of length   r . All the elements of s are greater than zero.

b
The argument b is a column vector of length   r .

d
The argument d is a column vector of length   p

Bdia
The argument 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
The matrix   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
The argument 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
The argument 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
The matrix   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
The result F is a column vector of length   r + p + r containing the value of the Kuhn-Tucker residual ; i.e.,   F(s, y, u) .

Example
The file kuhn_tucker_ok.m contains an example and test of ckbs_kuhn_tucker. It returns true if ckbs_kuhn_tucker passes the test and false otherwise.
Input File: src/ckbs_kuhn_tucker.m