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$\newcommand{\R}[1]{{\rm #1}}$$\newcommand{\B}[1]{{\bf #1}}$
Newton Step for Combined L1-L2 Fitting

Syntax
dup , dum , dvp , dvm , dw , dx ] = l1_newton_l2( AaBbmuupumvpvmwx )

Notation
We use $I_\ell \in \B{R}^{\ell \times \ell}$ to denote the identity matrix and $1_\ell \in \B{R}^m$ to denote the vector will all elements equal to one. Given a vector $u \in \B{R}^\ell$ we use $D(u) \in \B{R}^{\ell \times \ell}$, to denote the corresponding diagonal matrix. If $v$ is also in $\B{R}^\ell$, $u / v \in \B{R}^\ell$ denotes element-wise division. The notation $u^{-1}$ is defined by $u^{-1} = 1_\ell / u$.

Purpose
We are given a matrix $A \in \B{R}^{\ell \times n}$, a vector $a \in \B{R}^\ell$, a matrix $B \in \B{R}^{m \times n}$, a vector $b \in \B{R}^m$, and scalar $\mu \in \B{R}_+$. The function $F : \B{R}^{5 \ell +n} \rightarrow \B{R}^{5 \ell +n}$ is defined by $$F \left( \begin{array}{c} u_+ , u_- , v_+ , v_- , w , x \end{array} \right) = \left( \begin{array}{c} D ( u_+ ) v_+ - \mu 1_\ell \\ D ( u_- ) v_- - \mu 1_\ell \\ 1_\ell - u_+ - w \\ 1_\ell - u_- + w \\ A x - v_+ + v_- - a \\ A^\R{T} w + B^\R{T} B x - B^\R{T} b \end{array} \right)$$ Given $( u_+ , u_- , v_+ , v_- , w , x ) \in \B{R}_+^{5m+n}$ this routine solves for $\Delta u_+$, $\Delta u_-$, $\Delta v_+$, $\Delta v_-$, $\Delta w$, $\Delta x$ such that $$F ( u_+ , u_- , v_+ , v_- , w , x ) + F^{(1)} ( u_+ , u_- , v_+ , v_- , w , x ) \left( \begin{array}{c} \Delta u_+ \\ \Delta u_- \\ \Delta v_+ \\ \Delta v_- \\ \Delta w \\ \Delta x \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right)$$

Restriction
The matrix $A^\R{T} D( v_+ / u_+ + v_- / u_- ) A + B^\R{T} B$ must have rank $n$

A
The argument A is an $\ell \times n$ matrix where $\ell$ can not be zero.

a
The argument a is real column vector of length $\ell$.

B
The argument B is an $m \times n$ matrix where $m$ can not be zero (see l1_linear for the case where $m$ is zero).

b
The argument b is real column vector of length $m$.

mu
The argument mu is a real scalar. It corresponds to $\mu$ in the discussion above.

up, um, vp, vm
The arguments up, um, vp, and vm are real column vectors of length $\ell$. They specify the value of $u_+$, $u_-$, $v_+$, and $v_-$ respectively. All the elements of these vectors must be greater than zero.

w
The argument w is a column vector of length $m$.

x
The argument x is a column vector of length $n$.

dup, dum, dvp, dvm, dw
The return values dup, dum, dvp, dvm, and dw are column vectors of length $m$. They correspond respectively to $\Delta u_+$, $\Delta u_-$, $\Delta v_+$, $\Delta v_-$, $\Delta w$, in the discussion above.

dx
The return value dx is a column vectors of length $n$. It corresponds to $\Delta x$, in the discussion above.

Example
The file newton_l2_ok.m contains an example and test of l1_newton_l2. It returns true if the test passes and false otherwise.
Input File: l1_newton_l2.m