\section{Introduction}

Conventional approaches to distributed network optimization are based on iterative descent in either the primal or dual domain. The reason for this is that for many types of network optimization problems there exist descent directions that can be computed in a distributed fashion. Subgradient descent algorithms, for example, implement iterations through distributed updates based on local information exchanges with neighboring nodes; see e.g., \cite{mung, kelly, low, Srikant}. However, practical applicability of the resulting algorithms is limited by exceedingly slow convergence rates typical of gradient descent algorithms. Furthermore, since traditional line search methods require global information, fixed stepsizes are used, exacerbating the already slow convergence rate, \cite{averagepaper, CISS-rate}.

Faster distributed descent algorithms have been recently developed by constructing approximations to the Newton direction using iterative local information exchanges, \cite{lowdiag,cdc09,acc11}. These results build on earlier work in \cite{BeG83} and \cite{cgNewt} which present Newton-type algorithms for network flow problems that, different from the more recent versions in \cite{cdc09} and \cite{acc11}, require access to all network variables. To achieve global convergence and recover quadratic rates of centralized Newton's algorithm \cite{cdc09} and \cite{acc11} use distributed backtracking line searches that use average consensus to verify global exit conditions. Since each backtracking line search step requires running a consensus iteration with consequently asymptotic convergence \cite{fagnani, spielman}, the exit conditions of the backtracking line search can only be achieved up to some error. Besides introducing inaccuracies, computing stepsizes with a consensus iteration is not a suitable solution because the consensus iteration itself is slow. Thus, the quadratic convergence rate of the algorithms in \cite{cdc09} and \cite{acc11} is to some extent hampered by the linear convergence rate of the line search. This paper presents a distributed line search algorithm based on local information so that each node in the network can solve its own backtracking line search using only locally available information.

Work on line search methods for descent algorithms can be found in \cite{NMLS,zhang,cgLS}. The focus in \cite{NMLS} and \cite{zhang} is on nonmonotone line searches which improve convergent rates for Newton and Newton-like descent algorithms. The objective in \cite{cgLS} is to avoid local optimal solutions in nonconvex problems. While these works provide insights for developing line searches they do not tackle the problem of dependence on information that is distributed through nodes of a graph.

To simplify discussion we restrict attention to the network flow problem. Network connectivity is modeled as a directed graph and the goal of the network is to support a single information flow specified by incoming rates at an arbitrary number of sources and outgoing rates at an arbitrary number of sinks. Each edge of the network is associated with a concave function that determines the cost of traversing that edge as a function of flow units transmitted across the link. Our objective is to find the optimal flows over all links. Optimal flows can be found by solving a concave optimization problem with linear equality constraints (Section II).  Evaluating a line search algorithm requires us to choose a descent direction. We choose to work with the family of Accelerated Dual Descent (ADD) methods introduced in \cite{acc11}. Algorithms in this family are parameterized by the information dependence between nodes. The $N$th member of the family, shorthanded as ADD-N, relies on information exchanges with nodes not more than $N$ hops away. Similarly, we propose a group of line searches that can be implemented through information exchanges with nodes in this $N$ hop neighborhood.

Our work is based on the Armijo rule which is the workhorse condition used in backtracking line searches, \cite[Section 7.5]{LaNLP}. We construct a local version of the Armijo rule at each node by taking only the terms computable at that node, using information from no more than $N$ hops away(Section III).  Thus the line search always has the same information requirements as the descent direction computed via the ADD-N algorithm.  Our proofs(Section IV) leverage the information dependence properties of the algorithm to show that key properties of the backtracking line search are preserved: (i) We guarantee the selection of unit stepsize within a neighborhood of the optimal value (Section IV-A). (ii) Away from this neighborhood, we guarantee a strict decrease in the optimization objective (Section IV-B). These properties make our algorithm a practical distributed alternative to standard backtracking line search techniques. Simulations further demonstrate that our line search is functionally equivalent to its centralized counterpart (Section V).

\section{Network Optimization}\label{mincost-Newton}

Consider a network represented by a directed graph ${\cal G}=({\cal N},{\cal E})$ with node set ${\cal N}=\{1,\ldots,n\}$, and edge set ${\cal E} = \{1,\ldots,E\}$.  The $i$th component of vector $x$ is denoted as $x^{i}$. The notation $x\ge 0$ means that all components $x^i\ge 0$. The network is deployed to support a single information flow specified by incoming rates $b^{i}>0$ at source nodes and outgoing rates $b^{i}<0$ at sink nodes. Rate requirements are collected in a vector $b$, which to ensure problem feasibility has to satisfy $\sum_{i=1}^{n}b^{i}=1$. Our goal is to determine a flow vector $x=[x^e]_{e\in {\cal E}}$, with $x^e$ denoting the amount of flow on edge $e=(i,j)$.

Flow conservation implies that it must be $Ax=b$, with $A$ the $n\times E$ node-edge incidence matrix defined as
\[ [A]_{ij} = \left\{
\begin{array}{ll}
 1 & \hbox{if edge $j$ leaves node $i$} , \\
-1 & \hbox{if edge $j$ enters node $i$},  \\
 0 & \hbox{otherwise,}
\end{array}\right.\]
where $[A]_{ij}$ denotes the element in the $i$th row and $j$th column of the matrix $A$. We define the reward as the negative of scalar cost function $\phi_e(x^{e})$ denoting the cost of $x^e$ units of flow traversing edge $e$. We assume that the cost functions $\phi_e$ are strictly convex and twice continuously differentiable. The maximum reward network optimization problem is then defined as
\begin{equation}
	\begin{array}{ll}\hbox{maximize }& -f(x)=\sum_{e=1}^E -\phi_e(x^e)\\ \hbox{subject to: }& Ax=b.\label{optnet}\end{array}
\end{equation}
Our goal is to investigate a distributed line search technique for use with Accelerated Dual Descent (ADD) methods for solving the optimization problem in (\ref{optnet}). We begin by discussing the Lagrange dual problem of the formulation in (\ref{optnet}) in Section \ref{subsection:dualform}) and reviewing the ADD method in Section \ref{subsection:ADD}.

\subsection{Dual Formulation} \label{subsection:dualform}
Dual descent algorithms solve (\ref{optnet}) by descending on the Lagrange dual function $q(\lambda)$. To construct the dual function consider the Lagrangian ${\cal L} (x,\lambda) = -\sum_{e=1}^E \phi_e(x^e) +\lambda'(Ax-b)$ and define
\begin{eqnarray} \label{eqn_separable_dual}
	q(\lambda)  \!\!&=&\!\! \sup_{x\in \mathbb{R}^E} {\cal L} (x,\lambda) \nonumber \\
                &=&\sup_{x\in \mathbb{R}^E} \left(-\sum_{e=1}^E \phi_e(x^e) +\lambda'Ax\right) - \lambda'b \nonumber\\
		       &=&\!\! \sum_{e=1}^E \sup_{x^e \in \mathbb{R}} \Big((\lambda'A)^e x^e-\phi_e(x^e)\Big) - \lambda'b,
\end{eqnarray}
where in the last equality we wrote $\lambda'Ax = \sum_{e=1}^{E}(\lambda'A)^e x^e$ and exchanged the order of the sum and supremum operators.

It can be seen from (\ref{eqn_separable_dual}) that the evaluation of the dual function $q(\lambda)$ decomposes into the $E$ one-dimensional optimization problems that appear in the sum. We assume that each of these problems has an optimal solution, which is unique because of the strict convexity of the functions $\phi_e$. Denote this unique solution as $x^e(\lambda)$ and use the first order optimality conditions for these problems in order to write
\begin{equation}
	x^e(\lambda) = (\phi_e')^{-1} (\lambda^i-\lambda^j),\label{primal-sol}
\end{equation}
where $i\in{\cal N}$ and $j\in{\cal N}$ respectively denote the source and destination nodes of edge $e=(i,j)$. As per (\ref{primal-sol}) the evaluation of $x^e(\lambda)$ for each node $e$ is based on local information
about the edge cost function $\phi^e$ and the dual variables of the incident nodes $i$ and $j$.

The dual problem of (\ref{optnet}) is defined as $\min_{\lambda\in \mathbb{R}^n} q(\lambda)$. The dual function is convex, because all dual functions of minimization problems are, and differentiable, because the $\phi_e$ functions are strictly convex. Therefore, the dual problem can be solved using any descent algorithm of the form
\begin{equation}
	\lambda_{k+1} = \lambda_k + \alpha_k d_k\qquad \hbox{for all }k\ge 0, \label{sgit}
\end{equation}
where the descent direction $d_k$ satisfies $g_k'd_k<0$ for all times $k$ with $g_k = g(\lambda_{k})=\nabla q(\lambda_{k})$ denoting the gradient of the dual function $q(\lambda)$ at $\lambda=\lambda_k$.  An important observation here is that we can compute the elements of $g_k$ as
 \begin{equation}\label{eqn_dual_update_distributed}
	g_k^{i} = \sum_{e =(i,j)} x^e(\lambda_{k}) -  \sum_{e =(j,i)} x^e(\lambda_{k})  - b_{i}.
\end{equation}
with the vector $x(\lambda_k)$ having components $x^e(\lambda_k)$ as determined by (\ref{primal-sol}) with $\lambda=\lambda_k$, \cite[Section 6.4]{nlp}.  An important fact that follows from \eqref{eqn_dual_update_distributed} is that the $i$th element $g_k^{i}$ of the gradient $g_k$ can be computed using information that is either locally available $x^{(i,j)}$ or available at neighbors $x^{(j,i)}$.  Thus, the simplest distributed dual descent algorithm, known as subgradient descent takes $d_k=-g_k$. Subgradient descent suffers from slow convergence so we work with an approximate Newton direction.

\subsection{Accelerated Dual Descent} \label{subsection:ADD}
The Accelerated Dual Descent (ADD) method is a parameterized family of dual descent algorithms developed in \cite{acc11}. An algorithm in the ADD family is called ADD-N and each node uses information from $N$-hop neighbors to compute its portion of an approximate Newton direction. Two nodes are $N$-hop neighbors if the shortest undirected path between those nodes is less than or equal to $N$.

The exact Newton direction $d_k$ is defined as the solution of the linear equation $ H_k d_k = -g_k$ where $H_k = H(\lambda_{k})=\nabla^2 q(\lambda_{k})$ denotes the Hessian of the dual function.  We approximate $d_k$ using the ADD-N direction defined as
\begin{equation}
d_k^{(N)} = -\bar H_k^{(N)} g_k \label{ADDd}
\end{equation}
where the approximate Hessian inverse, $\bar H_k^{(N)}$ is defined
\begin{equation}
\bar H_k^{(N)} =\sum_{r=0}^N D_k^{-\frac{1}{2}}\!\left(D_k^{-\frac{1}{2}} B_k D_k^{-\frac{1}{2}}\right)^r\!D_k^{-\frac{1}{2}}\!
\label{ADD_h}
\end{equation}
using a Hessian splitting: $H_k= D_k-B_k$ where $D_k$ is the diagonal matrix $[D_k]_{ii} = [H_k]_{ii}$.  The resulting accelerated dual descent algorithm
\begin{equation}
	\lambda_{k+1} = \lambda_k + \alpha_k d^{(N)}_k\qquad \hbox{for all }k\ge 0, \label{update}
\end{equation}
can be computed using information from $N$-hop neighbors because the dependence structure of $g_k$ shown in equation (\ref{eqn_dual_update_distributed}) causes the Hessian to have a local structure as well: $[H_k]_{ij} \not = 0$ if and only if $(i,j)\in \mathcal{E}$.  since $H_k$ has the sparsity pattern of the network, $B_k$ and thus $D_k^{-\frac{1}{2}} B_k D_k^{-\frac{1}{2}}$ must also have the sparsity pattern of the graph.  Each term $D_k^{-\frac{1}{2}}\!\left(D_k^{-\frac{1}{2}} B_k D_k^{-\frac{1}{2}}\right)^r\!D_k^{-\frac{1}{2}}\!$ is a matrix which is non-zero only for $r$-hop neighbors so the sum is non-zero only for $N$-hop neighbors.

Analysis of the ADD-N algorithm fundamentally depends on a network connectivity coefficient $\bar \rho$, which is defined in \cite{acc11} as the bound
\begin{equation}
 \rho \left(B_k D_k^{-1}\right) \le \bar\rho \in (0,1) \label{rhobar}
 \end{equation}
 where $\rho(\cdot)$ denotes the second largest eigenvalue modulus.  When $\bar\rho$ is small, information in the network spreads efficiently and $d_k^{(N)}$ is a more exact approximation of $d_k$.  See \cite{acc11} for details.

\section{Distributed Backtracking Line Search}

Algorithms ADD-$N$ for different $N$ differ in their information dependence. Our goal is to develop a family of distributed backtracking line searches parameterized by the same $N$ and having the same information dependence. The idea is that the $N^{th}$ member of the family of line searches is used in conjunction with the $N^{th}$ member of the ADD family to determine the step and descent direction in (\ref{update}). As with the ADD-$N$ algorithm, implementing the distributed backtracking line search requires each node to get information from its $N$-hop neighbors.

Centralized backtracking line searches are typically intended as method to find a stepsize  $\alpha$ that satisfies Armijo's rule. This rule requires the stepsize $\alpha$ to satisfy the inequality
\begin{equation}q(\lambda+\alpha d) \le q(\lambda) + \sigma \alpha d'g, \label{armijo}\end{equation}
for given descent direction $d$ and search parameter $\sigma\in (0,1/2)$. The backtracking line search algorithm is then defined as follows:
\begin{algorithm}\label{BLS}
Consider the objective function $q(\cdot)$ and given variable value $\lambda$ and corresponding descent direction $d$ and dual gradient $g$. The backtracking line search algorithm is:
\begin{itemize}
\item[\footnotesize\bf] Initialize $\alpha=1$
\item[\footnotesize\bf] {\bf while} $q(\lambda+\alpha d) > q(\lambda) + \sigma \alpha d'g$
\item[\footnotesize\bf] $\qquad\alpha = \alpha \beta$
\item[\footnotesize\bf] \bf{end}
\end{itemize}
The scalars $\beta\in (0,1)$ and $\sigma \in (0,1/2)$ are given parameters.
\end{algorithm}

This line search algorithm is commonly used with Newton's method because it guarantees a strict decrease in the objective and once in an error neighborhood it always selects $\alpha=1$ allowing for quadratic convergence, \cite[Section 9.5]{boydbook}.

In order to create a distributed version of the backtracking line search we need a local version of the Armijo Rule.  We start by decomposing the dual objective $q(\lambda) = \sum_{i=1}^n q_i(\lambda)$ where the local objectives takes the form \begin{equation}q_i(\lambda)= \sum_{e=(j,i)} \phi_e(x^e)-\lambda_i(a_i'x-b_i).\label{ldep}\end{equation} The vector $a_i'$ is the $i^{th}$ row of the incidence matrix $A$. Thus the local objective $q_i(\lambda)$ depends only on the flows adjacent to node $i$ and $\lambda^i$.

An $N$-parameterized local Armijo rule is therefore given by
\begin{equation}
q_i(\lambda+\alpha_i d)\le q_i(\lambda)+\sigma\alpha_i\sum_{j\in \mathcal{N}_i^{(N)}} {d^j g^j},
\label{armijoN}
\end{equation}
where $\mathcal{N}_j^{(N)}$ is the set of $N$-hop neighbors of node $j$.  The scalar $\sigma\in (0,1/2)$ is the same as in (\ref{armijo}), $g=\nabla q(\lambda)$ and $d$ is a descent direction.  Each node is able to compute a stepsize $\alpha_i$
satisfying (\ref{armijoN}) using $N$-hop information.  The stepsize used for the dual descent update (\ref{sgit}) is
\begin{equation}
\alpha = \min_{i\in \mathcal{N}} \alpha_i.
\end{equation}
Therefore, we define the distributed backtracking line search according to the following algorithm.

\begin{algorithm}
Given local objectives $q_i(\cdot)$, descent direction $d$ and dual gradient $g$.
\begin{itemize}
\item[\footnotesize\bf] \bf{for} $i=1:n$
\item[\footnotesize\bf] $\qquad$Initialize $\alpha_i=1$
\item[\footnotesize\bf] $\qquad${\bf while} $q_i(\lambda+\alpha_i d)> q_i(\lambda)+\sigma\alpha_i\sum_{j\in \mathcal{N}_i^{(N)}} {d^j g^j}$
\item[\footnotesize\bf] $\qquad\qquad\alpha_i = \alpha_i \beta$
\item[\footnotesize\bf] $\qquad$\bf{end}
\item[\footnotesize\bf] \bf{end}
\item[\footnotesize\bf] $\alpha = \min_i \alpha_i$
\end{itemize}
The scalars $\beta\in (0,1)$, $\sigma \in (0,1/2-\bar\rho^{N+1}/2)$ and $N\in \field{Z}^+$ are parameters.
\label{DBLS}
\end{algorithm}

The distributed backtracking line search described in Algorithm \ref{DBLS} works by allowing each node to execute its own modified version of Algorithm \ref{BLS} using only information from $N$-hop neighbors. Minimum consensus of $\alpha_i$ requires at most diameter of $\mathcal{G}$ iterations. If each node shares its current $\alpha_i$ along with $g_k^i$ with its $N$-hop neighbors the maximum number of iterations drops to $\lceil \hbox{diam}(\mathcal{G})/N \rceil$.

The parameter $\sigma$ is restricted by the network connectivity coefficient $\bar\rho$ and the choice of $N$ because these are scalars which encode information availability.  Smaller $\bar\rho^{N+1}$ indicates more accessible information and thus allows for greater $\sigma$ and thus a more aggressive search.  As $\bar\rho^{N+1}$ approaches zero, we recover the condition $\sigma\in(0,1)$ from Algorithm \ref{BLS}.

\section{Analysis}

In this section we show that when implemented with the Accelerated Dual Descent update in (\ref{update}) the distributed backtracking line search defined in Algorithm \ref{DBLS} recovers the key properties of Algorithm \ref{BLS}: strict decrease of the dual objective and selection of $\alpha=1$ within an error neighborhood.

We proceed by outlining our assumptions.  The standard Lipshitz and strict convexity assumptions regarding the dual Hessian are defined here.
\begin{assumption} \label{standard}
The Hessian $H(\lambda)$ of the dual function $q(\lambda)$ satisfies the following conditions
\begin{list}{}{\setlength{\itemsep  }{2pt} \setlength{\parsep    }{2pt}
                                           \setlength{\parskip  }{0in} \setlength{\topsep    }{2pt}
                                           \setlength{\partopsep}{0in} \setlength{\leftmargin}{10pt}
                                           \setlength{\labelsep }{10pt} \setlength{\labelwidth}{-0pt}}
\item[({\it Lipschitz dual Hessian})] There exists some constant $L>0$ such that
\[\|H(\lambda)-H(\bar{\lambda})\| \le L\|\lambda-\bar{\lambda}\| \, \forall \lambda,\bar{\lambda}\in \mathbb{R}^n.\]
\item [({\it Strictly convex dual function})] There exists some constant $M>0$ such that $\|H(\lambda)^{-1}\|\le M \qquad \, \forall \lambda\in \mathbb{R}^n.$
\end{list}\end{assumption}
In addition to assumptions about the dual Hessian we assume that key properties of the inverse Hessian carry forward to our approximation.
\begin{assumption}
The approximate inverse Hessian remains well conditioned,
\[ m \le \|\bar H^{(N)}\| \le M.\]
within the subspace $\mathbf{1}^\perp$.
\label{cond}
\end{assumption}
These assumptions make sense because $\bar H^{(N)}$ is a truncated sum whose limit as $N$ approaches infinity is $H^{-1}$, a matrix we already assume to be well conditioned on $\mathbf{1}^\perp$ even when solving this problem in the centralized case.  Furthermore the first term in the sum is $D^{-1}$ which is well conditioned by construction.

We begin our analysis by characterizing the stepsize $\alpha$ chosen by Algorithm \ref{DBLS} when the descent direction $d$ is chosen according the the ADD-N method.
\begin{lemma}
For any $\alpha_i$ satisfying the distributed Armijo rule in equation (\ref{armijoN})
with descent direction  $d= -\bar H^{(N)}g$ we have
\[q_i(\lambda+\alpha_i d)- q_i(\lambda)\le 0.\]
\label{neg}\vspace{-5mm}
\end{lemma}
\begin{proof}
Recall that $\bar H ^{(N)}$ is non-zero only for elements corresponding to $N$-hop neighbors by construction. Therefore, by defining the local gradient vector $\tilde g^{(i)}$ as a sparse vector with nonzero elements $[\tilde g^{(i)}]_j=g^j$ for $j\in \mathcal{N}_i^{(N)}$ we can write
\begin{equation}\sum_{j\in \mathcal{N}_i^{(N)}} {d^j g^j} = -\left(\tilde g^{(i)}\right)' \bar H ^{(N)}\tilde g^{(i)}\label{quad}\end{equation}
Because $\bar H^{(N)}$ is positive definite the right hand side of (\ref{quad}) is nonpositive from where it follows that $\sum_{j\in \mathcal{N}_i^{(N)}} {d^j g^j}\leq0 $. The desired result follows by noting that $\alpha_i$ and $\sigma$ are positive scalars.\end{proof}

Lemma \ref{neg} tells us that when using the distributed backtracking line search with the ADD-N algorithm, we achieve improvement in each element of the decomposed objective $q_i(\lambda)$.  From the quadratic form in equation (\ref{quad}) it also follows that if equation (\ref{armijoN}) is satisfied by a stepsize $\alpha_i$, then it is also satisfied by any $\alpha\le \alpha_i$ and in particular $\alpha= \min_i \alpha_i$ satisfies equation (\ref{armijoN}) for all $i$.

\subsection{Unit Stepsize Phase}

A fundamental property of the backtracking line search using Armijo's rule summarized in Algorithm \ref{BLS}is that it always selects $\alpha=1$ when iterates $\lambda$ are within a neighborhood of the optimal argument. This property is necessary to ensure quadratic convergence of Newton's method and is therefore a desirable property for the distributed line search summarized in Algorithm \ref{DBLS}. We prove here that this is true as stated in the following theorem.

\begin{theorem}
Consider the distributed line search in Algorithm \ref{DBLS} with parameter $N$, starting point $\lambda=\lambda_k$, and descent direction $d = d_k^{(N)} = -\bar H_k^{(N)} g_k$ computed by the ADD-$N$ algortihm [cf. \eqref{ADDd} and \eqref{ADD_h}. If the search parameter $\sigma$ is chosen such that
\[ \sigma\in \left(0, \frac{1-\bar\rho^{N+1}}{2}\right) \]
and the norm of the dual gradient satisfies
\[\|g_k\|\le\frac{3m}{LM^3}\left({1-\bar\rho^{N+1}}-2\sigma\right), \]
then Algorithm \ref{DBLS} selects stepsize $\alpha=1$.
\label{a1}
\end{theorem}
\begin{proof}
Recall the definition of the local gradient $\tilde g^{(i)}_k$ as the sparse vector with nonzero elements $[\tilde g^{(i)}_k]_j=g_k^j$ for $j\in \mathcal{N}_i^{(N)}$. Further define the local update vector $\tilde d_k^{(i)} := \bar H^{(N)}_k \tilde g^{(i)}_k$ whose sparsity pattern is the same as that of $\tilde g^{(i)}_k$. Due to this and to the fact that the local objective $q_i(\lambda)$ in \eqref{ldep} depends only on values in $\mathcal{N}_i^{(N)}$, we have
\begin{align}\label{eqn_theo_unit_step_size_pf_10}
    q_i(\lambda_k+\alpha d_k) = q_i(\lambda_k+\alpha\tilde d_k^{(i)}).
\end{align}
Applying the Lipschitz dual Hessian assumption to the local update vector $\tilde d_k^{(i)}$ we get
\begin{align}\label{eqn_theo_unit_step_size_pf_20}
    \Vert H(\lambda_k+\alpha \tilde d_k^{(i)})- H(\lambda_k)\Vert \le
        \alpha L \Vert \tilde d_k^{(i)}\Vert.
\end{align}
We further define a reduced Hessian $\nabla^2 q_i(\lambda) = \tilde H^{(i)}$ by setting to zero the rows and columns corresponding to nodes outside of the neighborhood $\mathcal{N}_i^{(N)}$, i.e.,
\begin{equation}
    \left[\tilde H^{(i)}\right]_{ij} :=
        \left\{ \begin{array}{ll} H_{ij} & i,j\in \mathcal{N}_i^{(N)}\\
                                  0 & else\end{array}\right. \label{RH}
\end{equation}
Since the elements of $H$ already satisfy $H_{ij}= 0$ for all ${i,j}\not\in \mathcal{E}$ the resulting $\tilde H^{(i)}$ has the structure of a principal submatrix of $H$ with the deleted rows left as zeros. Since the norm $\Vert H(\lambda_k+\alpha \tilde d_k^{(i)})- H(\lambda_k)\Vert$ in \eqref{eqn_theo_unit_step_size_pf_20} is the maximum eigenvalue modulus of the matrix $H(\lambda_k+\alpha \tilde d_k^{(i)})- H(\lambda_k)$, it is larger than the norm $\Vert \tilde H^{(i)}(\lambda_k+\alpha \tilde d_k^{(i)})- \tilde H^{(i)}(\lambda_k)\Vert$ because the latter is the maximum over a subset of the eigenvalues of the former. Combining this observation with \eqref{eqn_theo_unit_step_size_pf_20} yields
\begin{equation}\Vert \tilde H^{(i)}(\lambda_k+\alpha \tilde d_k^{(i)})- \tilde H^{(i)}(\lambda_k)\Vert\le \alpha L\Vert \tilde d_k^{(i)}\Vert.\label{LL}\end{equation}
Interpret now the update in \eqref{eqn_theo_unit_step_size_pf_10} as a function of $\tilde q_i(\alpha)$ defined as
\begin{equation}\label{eqn_theo_unit_step_size_pf_50}
    \tilde q_i(\alpha) := q_i(\lambda_k+\alpha\tilde d_k^{(i)}).
\end{equation}
Differentiating with respect to $\alpha$ and using the definition of the local gradient $\tilde g^{(i)}_k$ we get the derivative of $\tilde q_i(\alpha)$ as
\begin{equation}\label{eqn_theo_unit_step_size_pf_60}
    \tilde q_i'(\alpha)
        = \nabla q_i (\lambda_k+\alpha\tilde d_k^{(i)}) \tilde d_k^{(i)}
        = \tilde g^{(i)}(\lambda_k+\alpha\tilde d_k^{(i)}) \tilde d_k^{(i)}.
\end{equation}
Differentiating with respect to $\alpha$ a second time and using the definition of $\tilde H^{(i)}$ in \eqref{RH} yields
\begin{align}\label{eqn_theo_unit_step_size_pf_70}
    \tilde q_i''(\alpha)
       &=\ \tilde d_k^{(i)}\phantom{}' \nabla^2 q_i(\lambda_k+\alpha\tilde d_k^{(i)})\tilde d_k^{(i)}\nonumber\\
       &=\ \tilde d_k^{(i)}\phantom{}' \tilde H^{(i)}(\lambda_k+\alpha\tilde d_k^{(i)})\tilde d_k^{(i)}.
\end{align}
Return now to (\ref{LL}) and replace the matrix norm on the right hand side with left and right multiplication by the unit vector $\tilde d_k^{(i)}/\|\tilde d_k^{(i)}\|$. This yields
\begin{align}\label{eqn_theo_unit_step_size_pf_80}
    \tilde d_k^{(i)}\phantom{}'
        \left[\tilde H^{(i)}(\lambda_k+\alpha \tilde d_k^{(i)})\!- \tilde H^{(i)}(\lambda_k)\right]
            \tilde d_k^{(i)}
    \!\le \alpha L\Vert \tilde d_k^{(i)}\Vert^3.
\end{align}
Comparing the expressions for the derivatives $\tilde q_i''(\alpha)$ in \eqref{eqn_theo_unit_step_size_pf_70} with the left hand side of \eqref{eqn_theo_unit_step_size_pf_80} we can simplify the latter to
\[\tilde q_i''(\alpha)-\tilde q_i''(0)\le \alpha L\Vert \tilde d_k^{(i)}\Vert^3.\]
Integrating the above expression with respect to $\alpha$ results in
\[\tilde q_i'(\alpha)-\tilde q_i'(0)\le \frac{\alpha^2}{2} L\Vert \tilde d_k^{(i)}\Vert^3+\alpha\tilde q_i''(0),\]
which upon a second integration with respect to $\alpha$ yields
\[\tilde q_i(\alpha)-\tilde q_i(0)\le \frac{\alpha^3}{6} L\Vert \tilde d_k^{(i)}\Vert^3+\frac{\alpha^2}{2}\tilde q_i''(0)+\alpha\tilde q_i'(0).\]
Since we are interested in unit stepsize substitute $\alpha=1$ and the definitions of the derivatives $\tilde q_i'(0)$ and $\tilde q_i''(0)$ given in \eqref{eqn_theo_unit_step_size_pf_60} and \eqref{eqn_theo_unit_step_size_pf_70} to get
\begin{equation*}
    \tilde q_i(1)-\tilde q_i(0)\le
         \frac{L}{6} \Vert \tilde d_k^{(i)}\Vert^3 \!
       + \frac{1}{2} \tilde d_k^{(i)}\phantom{}'\tilde H^{(i)}(\lambda_k)\tilde d_k^{(i)}
       + \tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}.
\end{equation*}
Since according to (\ref{RH}) the reduced Hessian $\tilde H^{(i)}$ has the structure of a principal submatrix of the Hessian $H$ and $H\succeq 0$ it follows that $0\preceq\tilde H^{(i)}\preceq H$ and that as a consequence \[\tilde d_k^{(i)}\phantom{}' \tilde H^{(i)}(\lambda_k)\tilde d_k^{(i)} \le \tilde d_k^{(i)}\phantom{}' H_k \tilde d_k^{(i)}.\]
Incorporating this latter relation and the definition of the local update $\tilde d_k^{(i)} = \bar H^{(N)}_k \tilde g^{(i)}_k$ in the previous equation we obtain
\begin{align}\label{eqn_theo_unit_step_size_pf_140}
   \tilde q_i(1)-\tilde q_i(0)\le & \
        \frac{L}{6}\Vert \bar H^{(N)}_k \tilde g^{(i)}_k\Vert^3 \\&\hspace{-4mm}\nonumber
      + \frac{1}{2}\left( \bar H^{(N)}_k \tilde g^{(i)}_k\right)' H_k\bar H^{(N)}_k \tilde g^{(i)}_k
      - \tilde g_k^{(i)}\phantom{}' \bar H^{(N)}_k \tilde g^{(i)}_k.
\end{align}
Consider now the last term in the right hand side and recall the sparsity pattern of the local gradient $\tilde g_k^{(i)}$ to write
\begin{align}\label{eqn_theo_unit_step_size_pf_150}
    -\tilde g_k^{(i)} \bar H^{(N)}_k \tilde g^{(i)}_k = \sum_{j\in \mathcal{N}_i^{(N)}} g_k^j d_k^j,
\end{align}
and further split the right hand side of \eqref{eqn_theo_unit_step_size_pf_150} to generate suitable structure
\begin{align}\label{eqn_theo_unit_step_size_pf_160}
    \sum_{j\in \mathcal{N}_i^{(N)}} g_k^j d_k^j
        = \sum_{j\in \mathcal{N}_i^{(N)}} \sigma{g_k^j d_k^j} + (1-\sigma){g_k^j d_k^j}.
\end{align}
Substitute now \eqref{eqn_theo_unit_step_size_pf_160} into \eqref{eqn_theo_unit_step_size_pf_150} and the result into \eqref{eqn_theo_unit_step_size_pf_140} to write
\begin{align*}
    \tilde q_i(1)-\tilde q_i(0)\le \
         \frac{L}{6}&\Vert \bar H^{(N)}_k \tilde g^{(i)}_k\Vert^3
       + \frac{1}{2}\tilde d_k^{(i)}\phantom{}' H\tilde d_k^{(i)}\\&
       + \sigma \sum_{j\in \mathcal{N}_i^{(N)}} {g_k^j d_k^j}
       + (1-\sigma)\sum_{j\in \mathcal{N}_i^{(N)}}{g_k^j d_k^j}.
\end{align*}
Using the expression for the quadratic form in (\ref{eqn_theo_unit_step_size_pf_150}) to substitute the last term in the previous equation yields
\begin{align}\label{terms}
    \tilde q_i(1)-\tilde q_i(0)\le \
          \frac{L}{6}&\Vert\bar H_k ^{(N)}\tilde g_k^{(i)}\Vert^3
         +\frac{1}{2}\tilde d_k^{(i)}\phantom{}' H\tilde d_k^{(i)}\\&
         +\sigma \sum_{j\in \mathcal{N}_i^{(N)}} {g_k^j d_k^j}
         -(1-\sigma)\tilde g^{(i)}_k\phantom{}' \bar H_k ^{(N)}\tilde g_k^{(i)}\nonumber
\end{align}
Further note that from the definition of $\tilde d^{(i)}$ it follows that
\[\tilde d_k^{(i)}\phantom{}' H_k\tilde d_k^{(i)} = \tilde g_k^{(i)}\phantom{}'\bar H_k^{(N)} H_k\bar H_k^{(N)}\tilde g_k^{(i)}.\]
The right hand side of this latter equality can be bounded using Cauchy-Schwarz's inequality and the submultiplicity of matrix norms as
\[\tilde g_k^{(i)}\phantom{}'\bar H_k^{(N)} H\bar H_k^{(N)}\tilde g_k^{(i)}\le \|\tilde g_k^{(i)}\|\, \| \bar H_k^{(N)}\|\, \|H_k \bar H_k^{(N)}\|\, \|\tilde g_k^{(i)}\|.\]
The norm $H_k \bar H_k^{(N)}$ can be further bounded using the result $\|H_k\bar H^{(N)}_k\|\le \bar \rho^{N+1}+1$ from \cite{acc11}. The norm $\|\bar H^{(N)}_k\|$ can be bounded as $\|\bar H^{(N)}_k\|\le M$ according to Assumption \ref{cond}. These two observations substituted in the last displayed equation yield
\begin{equation}\tilde d_k^{(i)}\phantom{}' H_k\tilde d_k^{(i)} \le M(\rho^{N+1}+1)\|\tilde g_k^{(i)}\|^2. \label{term2}\end{equation}
Applying the bound $\|\bar H^{(N)}_k\|\le M$ from Assumption 2 to the norm $\Vert\bar H_k ^{(N)}\tilde g_k^{(i)}\Vert^3$ we get $\Vert\bar H_k ^{(N)}\tilde g_k^{(i)}\Vert^3 \le M^3 \|\tilde g_k^{(i)}\|^3$.  Since Assumption 2 also guarantees that $\|\bar H^{(N)}_k\|\ge m$, we have \[\frac{\tilde g_k^{(i)}\phantom{}'\bar H_k^{(N)} \tilde g_k^{(i)}}{\|\tilde g_k^{(i)}\|^2} \ge m.\]  Therefore, we can write
\begin{equation}
\Vert\bar H_k ^{(N)}\tilde g_k^{(i)}\Vert^3 \le \frac{M^3}{m}\|\tilde g_k^{(i)}\|\ \tilde g_k^{(i)}\phantom{}'\bar H_k^{(N)} \tilde g_k^{(i)}.
\label{term1}
\end{equation}
Substituting the relations (\ref{term2}) and (\ref{term1}) in relation (\ref{terms}) and factoring we get
\begin{align*}
   & \tilde q_i(1)-\tilde q_i(0)\le \
          \sigma\sum_{j\in \mathcal{N}_i^{(N)}} {g_k^j d_k^j}\\&\hspace{3mm}
        + \tilde g^{(i)}_k\phantom{}' \bar H_k ^{(N)}\tilde g_k^{(i)}
              \left[-(1-\sigma)+\frac{LM^3}{6m}\Vert \tilde g_k^{(i)}\Vert+ \frac{\bar\rho^{N+1}}{2} + 1\right].
\end{align*}
Use $\Vert \tilde g_k^{(i)}\Vert\le \|g_k\|\le {6m}/({LM^3})(({1-\bar\rho^{N+1}})/{2}-\sigma)$ to write
\begin{align*}
  & \tilde q_i(1)-\tilde q_i(0)\le
        \sigma\sum_{j\in \mathcal{N}_i^{(N)}} {g_k^j d_k^j} \\&
      + M \Vert \tilde g_k^{(i)}\Vert^2
         \left[-(1-\sigma)
            +\left(\frac{1-\bar\rho^{N+1}}{2}-\sigma\right)
            +\frac{\bar\rho^{N+1}+1}{2} \right].
\end{align*}
Algebraic simplification of the bracketed portion yields
\begin{equation}\left[-(1-\sigma)+\left(\frac{1-\bar\rho^{N+1}}{2}-\sigma\right)+\frac{\bar\rho^{N+1}+1}{2} \right]=0\label{cancel}.\end{equation}
Thus we have \[\tilde q_i(1)-\tilde q_i(0)\le \sigma\sum_{j\in \mathcal{N}_i^{(N)}} {g_k^j d_k^j}.\]
Substituting the definition of $\tilde q_i(\lambda)$ in \eqref{eqn_theo_unit_step_size_pf_50} into this equation we arrive at
\[q_i\left(\lambda_{k}+d_k^{(i)}\right)\le q_i(\lambda_k)+\sigma\sum_{j\in \mathcal{N}_i^{(N)}} {d_k^j g_k^j}\]
which means that the exit condition \eqref{armijoN} in Algorithm \ref{DBLS} is met with $\alpha=1$.
\end{proof}
Theorem \ref{a1} guarantees that for an appropriately chosen line search parameter $\sigma$ the local backtracking line search will always choose a step size of $\alpha=1$ once the norm of the dual gradient becomes small.  Furthermore, the condition on the line search parameter tells us that $\bar\rho$ and our choice of $N$ fully capture the impact of distributing the line search. The distributed Armijo rule requires $\left(1-\bar\rho^{N+1}-2\sigma\right)>0$ while the standard Armijo rule requires $(1-2\sigma)>0$.  It is clear that in the limit $N\rightarrow \infty$ these conditions become the same with a rate controlled by $\bar\rho$.

\subsection{Strict Decrease Phase}
A second fundamental property of the backtracking line search with the Armijo rule is that there is a strict decrease in the objective when iterates are outside of an arbitrary noninfinitesimal neighborhood of the optimal solution. This property is necessary to ensure global convergence of Newton's algorithm as it ensures the quadratic convergence phase is eventually reached. Our goal here is to prove that this strict decrease can be also achieved using the distributed backtracking line search specified by Algorithm \ref{DBLS}.

Traditional analysis of the centralized backtracking line search of Algorithm \ref{BLS} leverages a lower bound on the stepsize $\alpha$ to prove strict decrease. We take the same approach here and begin by finding a global lower bound on the stepsize $\hat\alpha\le\alpha_i$ that holds for all nodes $i$. We do this in the following lemma.
\begin{lemma}
Consider the distributed line search in Algorithm \ref{DBLS} with parameter $N$, starting point $\lambda=\lambda_k$, and descent direction $d = d_k^{(N)} = -\bar H_k^{(N)} g_k$ computed by the ADD-$N$ algortihm [cf. \eqref{ADDd} and \eqref{ADD_h}. The stepsize
\[\hat \alpha = {2(1-\sigma)}\frac{m^2}{M^2}\]
satisfies the local Armijo rule in \eqref{armijoN}, i.e.,
\[q_i(\lambda_{k+1})\le q_i(\lambda_k)+\sigma  \hat{\alpha}\sum_{j\in n_i^{(N)}} {d_k^j g_k^j}\]
for all network nodes $i$ and all $k$.
\label{ahat}
\end{lemma}
\begin{proof}
From the mean value theorem centered at $\lambda_k$ we can write the dual function's value as
\[q_i(\lambda_k+\alpha \tilde d_k^{(i)}) = q_i(\lambda_k)+\alpha\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}+\frac{\alpha^2}{2} \tilde d_k^{(i)}\phantom{}' \tilde H^{(i)} (z) \tilde d_k^{(i)}\]
where the vector $z= \lambda_k + t\alpha\tilde d_k^{(i)}$ for some $t\in (0,1)$; see e.g.,\cite[Section 9.1]{boydbook}.  We use the relation $0\preceq\tilde H^{(i)}\preceq H$ and the bound $\|H^{-1}\|>m$ from Assumption \ref{cond} to transform this equality into the bound
\[q_i(\lambda_k+\alpha \tilde d_k^{(i)}) \le q_i(\lambda_k)+\alpha\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}+\frac{\alpha^2}{2 m} \Vert\tilde d_k^{(i)}\Vert^2.\]
Introduce now a splitting of the term $\alpha\tilde g_k^{(i)} \tilde d_k^{(i)}$ to generate convenient structure
\begin{align*}
    q_i(\lambda_k+\alpha \tilde d_k^{(i)}) \le\ &
          q_i(\lambda_k)\\& \hspace{-8mm}
        +\alpha\sigma\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}
        + \alpha(1-\sigma)\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}
        +\frac{\alpha^2}{2 m}\Vert\tilde d_k^{(i)}\Vert^2.
\end{align*}
Further apply the definition of  the local update vector $\tilde d_k^{(i)} := \bar H^{(N)}_k \tilde g^{(i)}_k$ and use the well-conditioning of the approximate inverse Hessian $\bar H_k^{(N)}$ as per Assumption \ref{cond} to claim that $m\le \|\bar H_k^{(N)}\|\le M$ and obtain
\begin{align*}
   q_i(\lambda_k+\alpha \tilde d_k^{(i)}) & \le \
          q_i(\lambda_k) \\& \hspace{-12mm}
        +\alpha\sigma\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}
        - \alpha(1-\sigma)m\|\tilde g_k^{(i)}\|^2
        + \frac{\alpha^2M^2}{2 m}  \Vert\tilde g_k^{(i)}\Vert^2.
\end{align*}
Factoring common terms in this latter equation yields
\begin{align*}
    q_i(\lambda_k+\alpha \tilde d_k^{(i)}) & \le \
        q_i(\lambda_k)\\&\hspace{-8mm}
      + \alpha\sigma\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}
      +\alpha m\|\tilde g_k^{(i)}\|^2 \left[-(1-\sigma)+\frac{\alpha M^2}{(2m^2)}\right].
\end{align*}
Substitute $\hat \alpha$ for $\alpha$ in this inequality. Observe that by doing so we have $[-(1-\sigma)+{\hat\alpha M^2}/({2m^2})]=0$ implying that the second term vanishes from this expression. Therefore
\[q_i(\lambda_k+\hat\alpha \tilde d_k^{(i)}) \le q_i(\lambda_k)+\hat\alpha\sigma\tilde g_k^{(i)}\phantom{}' \tilde d_k^{(i)}.\]
From the definitions of the local gradient $\tilde g^{(i)}_k$ as the sparse vector with nonzero elements $[\tilde g^{(i)}_k]_j=g_k^j$ for $j\in n_i^{(N)}$ and the local update vector $\tilde d_k^{(i)} := \bar H^{(N)}_k \tilde g^{(i)}_k$ the desired result follows. \end{proof}
We proceed with the second main result using Lemma \ref{ahat}, in the same manner that strict decrease is proven for the Newton method with the standard backtracking line search in \cite{boydbook}[Section 9.5].
\begin{theorem}
Consider the distributed line search in Algorithm \ref{DBLS} with parameter $N$, starting point $\lambda=\lambda_k$, and descent direction $d = d_k^{(N)} = -\bar H_k^{(N)} g_k$ computed by the ADD-$N$ algortihm [cf. \eqref{ADDd} and \eqref{ADD_h}]. If the norm of the dual gradient is bounded away from zero as $\|g_k\|\ge \eta$, the function  value at $\lambda_{k+1} = \lambda_k + \alpha_k d^{(N)}_k$ satisfies
\[q(\lambda_{k+1})-q(\lambda_k) \le -\beta \hat \alpha \sigma m N \eta^2\]
\label{strict}
I.e., the dual function decreases by at least $\alpha \sigma m N \eta^2$
\end{theorem}
\begin{proof}
According to Lemma \ref{ahat} we have
\[q_i(\lambda_{k+1})-q_i(\lambda_k) \le \hat \alpha \sigma \tilde g^{(i)}_k\phantom{}' \tilde d^{(i)}\] because $\hat \alpha$ is a lower bound on $\alpha_i$. Therefore, Algorithm \ref{DBLS} exits with $\alpha\in (\beta \hat \alpha, \hat\alpha)$ and any $\alpha\le \hat\alpha$ satisfies the exit condition in (\ref{armijoN}) therefore
\[q_i(\lambda_{k+1})-q_i(\lambda_k) \le \beta \hat \alpha \sigma \tilde g^{(i)}_k\phantom{}' \tilde d^{(i)}.\]
Applying Assumption \ref{cond} with the definition of $\tilde d^{(i)}$ we get
\[q_i(\lambda_{k+1})-q_i(\lambda_k) \le -\beta\hat \alpha \sigma m\|\tilde g^{(i)}_k\|^2.\]
Summing over all $i$,
\[q(\lambda_{k+1})-q(\lambda_k) \le -\beta\hat \alpha \sigma m \sum_{i=1}^n \|\tilde g^{(i)}_k\|^2.\]
Using the definition of the 2-norm we can write $\sum_{i=1}^n \|\tilde g^{(i)}\|^2 = \sum_{i=1} \sum_{j\in n_i^{(N)}} (g_k^j)^2$. Counting the appearance of each $(g_k^j)^2$ term in this  sum we have that $\sum_{i=1} \sum_{j\in n_i^{(N)}} (g_k^j)^2 = \sum_{i=1}|n_i^{(N)}| (g_k^i)^2$. Notice however that since the network is connected it must be $|n_i^{(N)}|\ge N$, from where it follows $\sum_{i=1} \sum_{j\in n_i^{(N)}} (g_k^j)^2 \leq N \sum_{i=1}^n (g^i_k)^2$. Substituting this expression into the above equation yields
\[q(\lambda_{k+1})-q(\lambda_k) \le -\beta\hat \alpha \sigma m N\sum_{i=1}^n (g^i_k)^2\]
Observe now that $\sum_{i=1}^n (g^i)^2=  \|g_k\|^2$ and substitute the lower bound $\eta\le \|g_k\|$ to obtain the desired relation.
\end{proof}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{example.pdf}
\centering
\caption{\label{ex} The distributed line search results in solution trajectories nearly equivalent to those of the centralized line search. Top: the Primal Objective follows a similar trajectory in both cases.  Middle: Primal Feasibility is achieved asymptotically. Bottom: unit stepsize is achieved in roughly the same number of steps.}
\end{figure}

Theorem \ref{strict} guarantees global convergence into any error neighborhood $\|g_k\|\le \eta$ around the optimal value because the dual objective is strictly decreasing by, at least, the noninfinitesimal quantity $\beta\hat \alpha \sigma m N \eta^2$ while we remain outside of this neighborhood. In particular, we are guaranteed to reach a point inside the neighborhood $\|g_k\| \le \eta = 3m/(LM^3)\left({1-\bar\rho^{N+1}}-2\sigma\right)$ at which point  Theorem \ref{a1} will be true and the ADD-N algorithm with the local line search becomes simply
\[ \lambda_{k+1} = \lambda_k - \bar H^{(N)}_k g_k.\]
This iteration is shown to have quadratic convergence properties in \cite{acc11}.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{data.pdf}
\centering
\caption{
\label{data}The distributed line search reaches unit stepsize in 2 to 3 iterations.  Fifty simulations were done for each algorithm with N=1, N=2 and N=3 and for Networks with 25 nodes and 100 edges (small), 50 nodes and 200 edges (medium) and 100 nodes and 400 edges (large).}
\end{figure}

\section{Numerical results}
Numerical experiments demonstrate that the distributed version of the backtracking line search is functionally equivalent to the centralized backtracking line search when the descent direction is chosen by the ADD method.  The simulations use networks generated by selecting edges are added uniformly at random but are restricted to connected networks.   The primal objective function is given by $\phi^e(x) = e^{c x^e}+e^{-c x^e}$ where $c$ captures the notion of edge capacity.  For simplicity we let $c=1$ for all edges.

Figure \ref{ex} shows an example of a network optimization problem with 25 nodes and 100 edges being solved using ADD-1 with the centralized and distributed backtracking line searches.  The top plot shows that the trajectory of primal objective is not significantly affected by the choice line search.  The middle plot shows that primal feasibility is approached asymptotically at the same rate for both algorithms. The bottom plot shows that a unit stepsize is achieved in roughly the same number of steps.

In Figure \ref{data} we look closer at the number of steps required to reach a unit stepsize.  We compare the distributed backtracking line search to its centralized counterpart on networks with 25 nodes and 100 edges, 50 nodes and 200 edges and 100 nodes and 400 edges.  For each network optimization problem generated we implemented distributed optimization using ADD-1, ADD-2, and ADD-3.  Most trials required only 2 or 3 iterations to reach $\alpha=1$ for both the centralized and distributed line searches.  The variation came from the few trials which required significantly more iterations.  As might be expected, increasing $N$ causes the distributed and centralized algorithms to behave closer to each other. When we increase the size of the network most trials still only require 2 to 3 iterations to reach $\alpha=1$ but for the cases which take more than 2 iterations we jump from around 10 iterations in the 25 nodes networks to around 40 iterations in 100 node networks.

\section{Conclusion}
We presented an alternative version of the backtracking line search using a local version of the Armijo rule which allows the stepsize for the dual update in the single commodity network flow problem to be computed using only local information.  When this distributed backtracking line search technique is paired with the ADD method for selecting the dual descent direction we recover the key properties of the standard centralized backtracking line search: a strict decrease in the dual objective and unit stepsize in a region around the optimal.  We use simulations to demonstrate that the distributed backtracking line search is functionally equivalent to its centralized counterpart.

This work focuses on line searches when the ADD-N method is used to select the descent direction, however the proof method relies primarily on the sparsity structure of the inverse hessian approximation.  This implies that our line search method could be applied with other descent directions provided they have are themselves depend only on local information.

\vfill

\bibliographystyle{amsplain}


\section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics. Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}. Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}. However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature. That usually leads to phase separation and the formation of secondary phases. It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}. For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection. Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors.

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology. In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0.94}$Mn$_{0.06}$ films by Park \textit{et al.} \cite{Park02}. The maximum critical temperature was 116 K. Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}. Cho \textit{et al.} reported a Curie temperature as high as 285 K \cite{Cho02}.
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required. Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films. Local chemical inhomogeneities have been recently reported by Kang \textit{et al.} \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes. Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors. However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}. Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}. This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy. The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}. Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors.
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers. Sugahara \textit{et al.} \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix. Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al.} \cite{Ahle06} which exhibit a Curie temperatures below 200 K. Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices.\\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%). By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases. We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix. We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature. We also discuss the magnetic anisotropy of nanocolumns and
Ge$_3$Mn$_5$ clusters.

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells.  Deposition rate was low ($\approx$ 0.2 \AA.s$^{-1}$). Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega.cm$ respectively. After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig. 1a). Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C). Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon. Mn concentrations range from 1 \% to 11\% Mn.

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears. After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig. 1b) indicate a predominantly two-dimensional growth. For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig. 1c). These spots may correspond to the formation of very small secondary phase crystallites. The nature of these crystallites will be discussed below.

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV. Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter . Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82}.

\begin{figure}[htb]
    \center
    \includegraphics[width=.29\linewidth]{./fig1a.eps}
    \includegraphics[width=.29\linewidth]{./fig1b.eps}
    \includegraphics[width=.29\linewidth]{./fig1c.eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer. (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C). (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C. The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film.}
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width=.49\linewidth]{./fig2a.eps}
	\includegraphics[width=.49\linewidth]{./fig2b.eps}
	 \includegraphics[width=.49\linewidth]{./fig2c.eps}
	 \includegraphics[width=.49\linewidth]{./fig2d.eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese. (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis. (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer. The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium. No defect can be seen which could be caused by the presence of nanocolumns. (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns. (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM). The background was carefully substracted from pre-edge images. Bright areas correspond to Mn-rich regions.}
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i.e.} nanocolumns as shown in Fig. 2a. Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness. From the high resolution TEM image shown in Fig. 2b, we deduce their average diameter around 3 nm. Moreover in Fig. 2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film. The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate. In Fig. 2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film. From this image, we can deduce the size and density of nanocolumns. The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements. In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM. In Fig. 2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film. This map shows that the formation of nanocolumns is a consequence of Mn segregation. Nanocolumns are Mn rich and the surrounding matrix is Mn poor. However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section. Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns. In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}. These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument). Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns. The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \%.\\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix. In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1.3 \%, 2.3 \%, 4 \%, 7 \% and 11.3 \%. Their structural properties have been investigated by plane view TEM observations.

\begin{figure}[htb]
    \center
    \includegraphics[width=.98\linewidth]{./fig3a.eps}
	\includegraphics[width=.45\linewidth]{./fig3b.eps}
		\includegraphics[width=.45\linewidth]{./fig3c.eps}
    \caption{Nanocolumns size and density as a function of growth conditions. Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively. (a) Mn concentration dependence of the size distribution. (b) columns density as a function of Mn concentration. (c) Volume fraction of the nanocolumns as a function of Mn concentration.}
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1.8$\pm$0.2 nm. The evolution of columns density as a fonction of Mn concentration is reported in figure 3b. By increasing the Mn concentration from 1.3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$. For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1.8 nm at 4 \% to 2.8 nm at 11.3 \%. By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%. The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix. For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig. 4a).

Increasing the Mn content in the samples grown at 150$^\circ$C from 1.3 \% to 11.3 \% leads to a decrease of the columns density (fig 3b). Moreover, their average diameter increases significantly and size distributions become very broad (see Fig. 3a). For the highest Mn concentration (11.3 \%) we observe the coexistence of very small columns with a diameter of 2.5 nm and very large columns with a diameter of 9 nm. In samples grown at 150$^\circ$C containing 11.3 \% of Mn, the crystalline structure of nanocolumns is also highly modified. In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures. We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature. Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11.3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig. 4b). Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension. On the same samples (T$_g$=150$^{\circ}$C, 11.3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig. 4c).

\begin{figure}[htb]
    \center
   \includegraphics[width=.31\linewidth]{./fig4a.eps}
	\includegraphics[width=.31\linewidth]{./fig4b.eps}
	\includegraphics[width=.31\linewidth]{./fig4c.eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C. The crystal structure is exactly the same as germanium . (b) Partially relaxed nanocolumn. One can see dislocations at the interface between the columns and the matrix leading to stress relaxation. (c) Amorphous nanocolumn. These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents.}
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature. In particular Mn incorporation is highly inhomogeneous. For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns. Their density mostly depends on Mn concentration and their mean diameter is about 2 nm. These results can be compared with the theoretical predictions of Fukushima \textit{et al.} \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns. This model may also properly describe the formation of Mn rich nanocolumns in our samples. Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth. For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth. Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig. 5). Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode. Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition.

\begin{figure}[htb]
    \center
	\includegraphics[width=.7\linewidth]{./fig5.eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure. This sample has been grown at 130 $^{\circ}$C with 6\% Mn. Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick. We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces. Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films.}
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix. A typical high resolution TEM image is shown in figure 6.
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C. To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode. Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF). Three samples grown at different temperatures and/or annealed at high temperature were investigated. The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively. The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C. By analysing x-ray diffraction spectra, we can evidence two different crystalline structures. For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$. For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical. In addition to the (004) peak of germanium, we observe three additional weak peaks. The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase. The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5.053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate.

\begin{figure}[htb]
    \center
	\includegraphics[width=.7\linewidth]{./fig6.eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C. We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters.}
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films. This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions. We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically. By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn. Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C.

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations. In this section, we focus on Mn concentrations between 2 \% and 11 \%. We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K). The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration. For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures. Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C. The larger columns become ferromagnetic \textit{i.e.} $T_{B}>T_{C}$. Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form. Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears.

\begin{figure}[htb]
\center
   \includegraphics[width=.6\linewidth]{./fig7a.eps}
   \includegraphics[width=.3\linewidth]{./fig7b.eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0.93}$Mn$_{0.07}$ samples for different growth temperatures. The magnetic field is applied in the film plane. The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes. After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters. (b) ZFC-FC measurements performed on Ge$_{0.93}$Mn$_{0.07}$ samples for different growth temperatures. The in-plane applied field is 0.015 T. The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns. This peak widens and shifts towards high blocking temperatures when increasing growth temperature. The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters. The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters. The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes.}
\label{fig7}
\end{figure}

In Fig. 7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C. The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes. Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0.015 Tesla. In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0.015 Tesla while increasing the temperature up to 400 K (ZFC curve). Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve). Three different regimes can be clearly distinguished. \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature. The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix. The Mn concentration dependence of the saturation magnetization is displayed in figure 8. For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11.3 \%). Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns. In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11.3 \% of Mn. This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1.8 nm to 2.8 nm) as discussed in section \ref{structural}.

\begin{figure}[htb]
\center
   \includegraphics[width=.7\linewidth]{./fig8.eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4.1 \%; 7 \%; 8.9 \% and 11.3 \%.}
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic. The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature. In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures. As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured. For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig. 7b). Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM. To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$.

\begin{figure}[htb]
\center
   \includegraphics[width=.6\linewidth]{./fig9.eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures. If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution. In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum. At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic.}
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig. 9, the FC curve deviates from the ZFC curve. Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility. Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures. The results are given in Fig. 10a. ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected. Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00}.

\begin{figure}[htb]
\center
    \includegraphics[width=.35\linewidth]{./fig10a.eps}
    \includegraphics[width=.63\linewidth]{./fig10b.eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0.887}$Mn$_{0.113}$ sample grown at 115$^{\circ}$C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0.015 T).}
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp. out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp. $\chi_{\perp}^{-1}$). The corresponding experimental ZFC-FC curves are reported in Fig. 10b. Since susceptibility measurements are performed at low field (0.015 T), the matrix magnetic signal remains negligible. In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T. However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function. From Fig. 10b, we can conclude that nanocolumns are isotropic. Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$. $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume. The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes. Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$. Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$.
The only fitting parameters are the maximum magnetic moment (\textit{i.e.} at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i.e.} at low temperature): $\mu_{0}H_{eff}$.

\begin{figure}[htb]
\center
   \includegraphics[width=.7\linewidth]{./fig11.eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0.887}$Mn$_{0.113}$ sample grown at 115$^{\circ}$C. Fits were performed assuming isotropic nanostructures or cubic anisotropy. Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility.}
\label{fig11}
\end{figure}

In Fig. 11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility. It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT. Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K. However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column. Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures. The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling. This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime. As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy. First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy. We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes). Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely. We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy. From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ.m$^{-3}$ which is of the same order of magnitude as shape anisotropy.

\begin{figure}[htb]
\center
    \includegraphics[width=.35\linewidth]{./fig12a.eps}
    \includegraphics[width=.63\linewidth]{./fig12b.eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0.93}$Mn$_{0.07}$ sample grown at 122$^{\circ}$C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0.015 T).}
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig. 7a). As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}. We also observe a sharp transition in the ZFC curve (see Fig. 7b, Fig. 12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K). This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn. Therefore the magnetic volume increases sharply as well as blocking temperatures. At the same time, the size distribution widens as observed in TEM. In Fig. 12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures. The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i.e.} $T_{B}\geq T_{C}$).
In Fig. 12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy. Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic. This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref.\cite{Bihl06}. The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane.

\begin{figure}[ht]
\center
   \includegraphics[width=.35\linewidth]{./fig13a.eps}
   \includegraphics[width=.63\linewidth]{./fig13b.eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0.887}$Mn$_{0.113}$ sample grown at 145$^{\circ}$C. The in-plane applied field is 0.015 T. Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K. Curves are shifted up for more clarity. (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0.015 T).}
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig. 13b). Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig. 13a). The first ZFC peak at low temperature \textit{i.e.} $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K). This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K. These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i.e.} higher blocking temperatures) and wider size distributions. ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns. The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic. We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig. 13b). In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties.

\begin{figure}[ht]
\center
    \includegraphics[width=.49\linewidth]{./fig14a.eps}
    \includegraphics[width=.49\linewidth]{./fig14b.eps}
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2.3 \%; 4 \%; 7 \%; 9 \%; 11.3 \%. (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C. The in-plane applied field is 0.025 T for 2.3 \% and 4 \% and 0.015 T for 8 \% and 11.3 \%. }
\label{fig14}
\end{figure}

In Fig. 14a, for low Mn concentrations (2.3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures. Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters. We can derive more information from ZFC-FC measurements (Fig. 14b). Indeed, for 2.3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range. Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations. For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve. We can also derive the Curie temperature of nanocolumns: $\approx$45 K. The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters. Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig. 3a). Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters.\\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist. The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig. 7a).\\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref.\cite{Jame06}.\\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures. For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures. Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations. In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$. Results on magnetic anisotropy seems counter-intuitive. Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic. This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy.

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy. A wide range of growth temperatures and Mn concentrations have been explored. All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition. However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration. For low growth temperatures, nanocolumns are very small (their diameter ranges between 1.8 nm for 1.3 \% of Mn and 2.8 nm for 11.3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures. Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged. For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens. Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn. Some columns are ferromagnetic even if Curie temperatures remain quite low. In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same. In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane. In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling. Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K. Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films.

\section{Aknowledgements}
The authors would like to thank Dr. F. Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility.