text:
\section{INTRODUCTION} \label{sec:Intro}
The broadcast nature of the wireless medium exhibits different challenges in ensuring secure communications in the presence of adversarial users \cite{FTA13,MFHS14}. In particular, it is difficult to protect the transmitted signals from unintended recipients, who may improperly extract information from an ongoing transmission without being detected \cite{LPS08,Baetal13}. Physical layer security \cite{P12,PS17}  has been proposed as a solution to provide security in wireless networks and researchers with a goal being to optimize the secure throughput of a wireless network in the presence of eavesdroppers, which is the difference between the desired user throughput and eavesdroppers' throughput \cite{MFHS14}. Beyond secure throughput, significant interest has recently been put on optimizing the secure energy efficiency (SEE), which is the ratio of the secure throughput to the total network power consumption, measured in terms of bits per Joule per Hertz \cite{WBCH16,NTDP17}.

There has been considerable recent research on physical layer security in wireless communication systems. For example,
assuming the availability of full channel state information (CSI), secrecy optimization has been studied for cooperative relaying networks in \cite{DHPP10,LPW11,Hoetal15}. Energy efficiency (EE) of wireless networks has also drawn attention.  For examples, resource allocation algorithms for the optimization of spectral efficiency as well as EE have been established in \cite{LSYW13}. Keeping EE maximization as an objective, the authors in \cite{RRL17} proposed a precoder design for multi-input-multi-output (MIMO) two-way relay networks. EE maximization for cooperative spectrum sensing in cognitive sensor networks is studied in \cite{ZCLYW17}.

The critical topic of SEE has also been explored very recently \cite{WBCH15,WBCH16,FBSR15,CWMJ16,XYCJ16,OLZZM17,VKT16,NTDP17}.
Specifically, power control algorithms for SEE maximization in decode-and-forward
(DF) and amplify-and-forward (AF) relaying networks have been considered in \cite{WBCH15} and \cite{WBCH16}, respectively. In \cite{FBSR15}, the authors developed a distributed power control algorithm for SEE maximization in DF relaying. The same resource allocation problem for SEE maximization assuming full-duplex relaying is considered in \cite{CWMJ16}.  Recently, the authors in \cite{XYCJ16} and \cite{OLZZM17} also derived the trade-off between SEE and secure spectral efficiency in cognitive radio networks. All these works have assumed the perfect CSI knowledge at the transmitter end, which is not always possible.

It is commonly known that time or frequency resources are generally limited in wireless networks and thus have to be shared among multiple users. This can result in interference among users in the network and thus one has to opt for careful resource allocation or interference alignment schemes \cite{ZKMT14}. Considering a multiuser MIMO interference network,
\cite{VKT16} used the costly interference alignment technique to cancel both information leakage and interference and
then Dinkelbach's method of fractional programming is adopted to optimize EE. As shown in \cite{NTDP17}, both zero-forcing
and interference alignment are not efficient in optimizing the network SEE.

In this paper, we propose novel and efficient resource allocation algorithms for both worst-case secure throughput and worst secure energy efficiency maximization of a highly interference-limited multi-user wireless network. Unlike
many previous works, we do not assume perfect CSI knowledge at the transmitters. In fact, our transmitters only carry channel distribution knowledge for the eavesdropper and imperfect CSI for the users. Particularly, we consider three optimization scenarios to gradually build our algorithms. We start with the ``perfect CSI" scenario. Next, we consider a
``partial CSI" setup where the channel between the transmitters and the eavesdropper is exponentially distributed and only that channel distribution knowledge is available at the transmitters. Finally, we solve for the hardest ``robust optimization" scenario, where in addition to the assumption of only channel distribution knowledge about eavesdroppers, we also assume uncertain channels between the transmitters
and the users with exponentially distributed errors.
We develop various path-following  procedures of low complexity and rapid convergence for the optimal power allocation.  Our extensive simulation results  illustrate their effectiveness and viability.

The rest of the paper is organized as follows. Section II, section III and section IV are devoted to
optimizing the links' worst secrecy throughput and the network secure energy efficiency under the perfect
CSI, partial CSI and imperfectly known CSI, respectively. The simulation is provided in Section V to
show the efficiency of the theoretical developments in the previous section. Appendices provide fundamental rate outage
inequalities and approximations, which are the mathematical base of the theoretical sections II-IV.

\section{Interference networks under perfect CSI}
\begin{figure}
\centering
\includegraphics[width=3.5in]{Secrecy_model.eps}
\caption{System model}
\label{fig:Secrecy_model}
\end{figure}
We consider a cooperative network consisting of $M$ single-antenna transmitters and $M$ single-antenna users
as depicted in Figure \ref{fig:Secrecy_model}, where each transmitter $i$ intends
to send  the information  $s_i$ to user $i$. The information $s_i$
is normalized, i.e. $\mathbb{E}(x_i^2)=1$. Let $\mathbf{p}_i$ be the transmit power allocated to transmitter $i$ and
$\mathbf{p}=(\mathbf{p}_1,\dots,\mathbf{p}_M)^T$. Furthermore, denote by
$h_{ji}$ the channel gain from transmitter $j$ to user $i$. The received signal at user $i$ is
\[
y_i=h_{ii}\mathbf{p}_i s_i+\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j s_j+ n_i,
\]
where $n_i\in {\cal CN}(0,\sigma_i^2)$ is additive noise.

Suppose that there is an eavesdropper (EV), which is also equipped with a single antenna. Denoting by
$h_{ie}$ the channel gain from transmitter $i$ to the EV, the received signal at the EV is
\[
y_e=\sum_{i=1}^Mh_{ie}\mathbf{p}_i s_i+ n_e,
\]
where $n_e\in {\cal CN}(0,\sigma_e^2)$ is additive noise.

Under the perfect CSI at the transmitters, the information throughput
at user $i$ is
\begin{equation}\label{sec1}
f_i(\mathbf{p})\triangleq \ln\left(1+\frac{h_{ii}\mathbf{p}_i}{\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j+\sigma_i^2}\right).
\end{equation}
With the EV considered as  part of the legitimate network, the channel gain
$h_{ie}$ can also be assumed known \cite{Cetal13}. The wiretapped throughput for
user $i$ at the EV is
\begin{equation}\label{dwire1}
g_i(\mathbf{p})\triangleq \ln\left(1+\frac{h_{ie}\mathbf{p}_i}{\sum_{j\neq i}
h_{je}\mathbf{p}_j+\sigma_e^2}\right).
\end{equation}
The secrecy throughput in transmitting information $s_i$ to user $i$ while keeping it confidential from the eavesdropper
is defined as
\begin{equation}\label{sec3}
\max \{f_i(\mathbf{p})-g_i(\mathbf{p}),0\}.
\end{equation}
We consider the following fundamental optimization problems in a such network: the maximin secrecy throughput optimization
\begin{subequations}\label{sec4}
\begin{eqnarray}
\max_{\pmb{p}}\ \Phi_{\sf sp}(\mathbf{p})\triangleq \min_{i=1,...,M} [ f_i(\mathbf{p})-g_i(\mathbf{p}) ]\quad\mbox{s.t.}\label{sec4a}\\
{\color{black}0<}\mathbf{p}_i\leq P_i, i=1,\cdots,M,\label{sec4b}
\end{eqnarray}
\end{subequations}
and the network secure energy efficiency (SEE) maximization under users' secrecy throughput quality-of-service (QoS) requirements
\begin{subequations}\label{esec4}
\begin{eqnarray}
\max_{\pmb{p}}\ \Phi_{\sf ee}(\mathbf{p})\triangleq\frac{\displaystyle\sum_{i=1}^M[f_i(\mathbf{p})-g_i(\mathbf{p})]}{\zeta\displaystyle\sum_{i=1}^M\pmb{p}_i+P_c} \quad\mbox{s.t.}\quad (\ref{sec4b}),\label{esec4a}\\
f_i(\mathbf{p})-g_i(\mathbf{p})\geq c_i,\ i=1,..,M,
\label{esec4b}
\end{eqnarray}
\end{subequations}
or the maximin transmitter EE optimization under users' secrecy throughput QoS requirements
\begin{equation}\label{mesec5}
\max_{\pmb{p}}\min_{i=1,...,M} \frac{f_i(\mathbf{p})-g_i(\mathbf{p})}{{\color{black}\zeta \pmb{p}_i}+P_c^i}\quad\mbox{s.t.}\quad (\ref{sec4b}), (\ref{esec4b}).
\end{equation}
Here $\zeta$ is the reciprocal of the drain efficiency of the power amplifier, $P_c^i$
is the circuit power at transmitter $i$ and $P_c=\sum_{i=1}^MP_c^i$. As the numerator
in the objective function in (\ref{esec4}) is the sum secrecy throughput while  the denominator  is the network power consumption, the objective function in (\ref{esec4}) expresses the network SEE in terms of nats/s/Joule. Similarly, each subfunction in  (\ref{mesec5}) expresses the SEE in
for transmitting the information $s_i$. Moreover,
the constraint (\ref{esec4b}) for given thresholds $c_i$ sets the QoS for the users in terms of
the secrecy throughput. This constraint  is nonconvex, which is in contrast to the throughput constraint
\[
f_i(\mathbf{p})\geq \tilde{c}_i, i=1,\dots, M,
\]
which is equivalent to the linear constraint
\[
h_{ii}\mathbf{p}_i\geq (e^{\tilde{c}_i}-1)(\sum_{j\neq i}h_{ji}\mathbf{p}_j+\sigma_i^2), i=1,\dots, M.
\]
A popular now approach \cite{KTN11} is to treat $f_i-g_i$ in (\ref{sec4})
as a d.c (\underline{d}ifference of two \underline{c}oncave functions) function \cite{Tuybook}:
$f_i(\mathbf{p})-g_i(\mathbf{p})=\tilde{f}_i(\mathbf{p})-\tilde{g}_i(\mathbf{p})$ with
$
\tilde{f}_i(\mathbf{p})=\ln(\sum_{j=1}^Mh_{ji}\mathbf{p}_j+\sigma_i^2)+\ln(\sum_{j\neq i}^Mh_{je}\mathbf{p}_j+\sigma_e^2)
$
and
$\tilde{g}_i(\mathbf{p})=\ln(\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j+\sigma_i^2)+\ln(\sum_{j=1}^Mh_{je}\mathbf{p}_j+\sigma_e^2)$
which are concave. Then at each iteration, $\tilde{f}_i$ is linearized while $\tilde{g}_i$ is innerly approximated
by a concave quadratic function for a lower approximation of $\tilde{f}_i-\tilde{g}_i$ \cite{Naetal17,Taetal17}. As a result,
each iteration invokes solution of a simple convex quadratic optimization problem with the logarithmic function optimization
of high computational complexity avoided.

Our next subsections are devoted to efficient computational approach to solving each of (\ref{sec4}), (\ref{esec4}) and
(\ref{esec5}) without d.c. representation.
\vspace*{-0.3cm}
\subsection{Max-min secrecy throughput optimization}
At every $p^{(\kappa)}\in R^M_+$, applying  inequality (\ref{sec5.2}) in the Appendix II
for $x=1/h_{ii}\mathbf{p}_i, y=\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j+\sigma_i^2$
and $
\bar{x}=1/h_{ii}p_i^{(\kappa)}, \bar{y}=\sum_{j\neq i}^Mh_{ji}p_j^{(\kappa)}+\sigma_i^2$
yields
\begin{eqnarray}
f_i(\mathbf{p})\geq f_i^{(\kappa)}(\pmb{p})\label{dc1}
\end{eqnarray}
for
\begin{eqnarray}
f_i^{(\kappa)}(\pmb{p})&\triangleq&\displaystyle\ln(1+x_i^{(\kappa)})+\displaystyle\frac{x_i^{(\kappa)}}{1+x_i^{(\kappa)}}\left(2-
\frac{p_i^{(\kappa)}}{\mathbf{p}_i}-
\frac{\sum_{j\neq i}h_{ji}\mathbf{p}_j+\sigma_i^2}{\sum_{j\neq i}^Mh_{ji}p_j^{(\kappa)}+\sigma_i^2}\right).\label{dc3}
\end{eqnarray}
On the other hand, applying inequality (\ref{conv1}) in the Appendix II for
$
x=h_{ie}\mathbf{p}_i, y=\sum_{j\neq i}^Mh_{je}\mathbf{p}_j
$
and
$
\bar{x}=h_{ie}p_i^{(\kappa)}, \bar{y}=\sum_{j\neq i}^Mh_{je}p_j^{(\kappa)}
$
yields
\begin{equation}\label{dc2}
g_i(\mathbf{p})\leq g_i^{(\kappa)}(\pmb{p}),
\end{equation}
for
\begin{eqnarray}
g_i^{(\kappa)}(\pmb{p})&=&\ln(1+x_{e,i}^{(\kappa)})+\displaystyle\frac{1}{1+x_{e,i}^{(\kappa)}}\left(\frac{0.5h_{ie}(\mathbf{p}_i^2/p_i^{(\kappa)}+p_i^{(\kappa)})}{\sum_{j\neq i}^M
h_{je}\mathbf{p}_j+\sigma_e^2}  -x_{e,i}^{(\kappa)}\right).
\label{dc4}
\end{eqnarray}
Initialized by a feasible $p^{(0)}$ for the convex constraint (\ref{sec4b}), at the $\kappa$-th iteration we solve the convex optimization problem
\begin{equation}\label{dc5}
\max_{\pmb{p}}\ \Phi_{\sf sp}^{(\kappa)}(\mathbf{p})\triangleq \min_{i=1,...,M} [f_i^{(\kappa)}(\pmb{p})-g_i^{(\kappa)}(\pmb{p})]\quad\mbox{s.t.}\quad (\ref{sec4b})
\end{equation}
to generate the next iterative point $p^{(\kappa+1)}$.\\
One can see that $\Phi_{\sf sp}(\mathbf{p})\geq F_{\sf sp}^{(\kappa)}(\mathbf{p})\ \forall\ \mathbf{p}$ and
$\Phi_{\sf sp}(p^{(\kappa)})=F_{\sf sp}^{(\kappa)}(p^{(\kappa)})$. Futhermore,
$\Phi_{\sf sp}^{(\kappa)}(p^{(\kappa+1)})>\Phi_{\sf sp}^{(\kappa)}(p^{(\kappa)})$ if
$p^{(\kappa+1)}\neq p^{(\kappa)}$ because the former is the optimal solution of
(\ref{dc5}) while the latter is its feasible point. Therefore,
\begin{equation}\label{pathf}
\Phi_{\sf sp}(p^{(\kappa+1)})\geq  \Phi^{(\kappa)}_{\sf sp}(p^{(\kappa+1)})> \Phi^{(\kappa)}_{\sf sp}(p^{(\kappa)})
=\Phi_{\sf sp}(p^{(\kappa)}),
\end{equation}
i.e. $p^{(\kappa+1)}$ is better than $p^{(\kappa)}$; as such $\{p^{(\kappa)}\}$ is a sequence of improved points
that converges at least to a locally optimal solution of (\ref{sec4}) satisfying the first order necessary optimality condition \cite[Prop. 1]{MW78}. In summary,  we propose in Algorithm \ref{alg1}
a path-following computational procedure for the maximin secrecy throughput optimization problem (\ref{sec4}).
\begin{algorithm}
\caption{Path-following algorithm for maximin secrecy throughput optimization} \label{alg1}
\begin{algorithmic}
\STATE \textbf{Initialization}: Set $\kappa=0$. Choose an initial feasible point
$p^{(0)}$ for the convex constraints (\ref{sec4b}).
Calculate $R_{\min}^{(0)}$ as the value of the objective in (\ref{sec4}) at $p^{(0)}$.
\REPEAT \STATE $\bullet$ Set
$\kappa=\kappa+1$.
\STATE $\bullet$ Solve the
convex optimization problem \eqref{dc5} to obtain the
solution $p^{(\kappa)}$.
\STATE $\bullet$ Calculate $R_{\min}^{(\kappa)}$ as the value of the objective in (\ref{sec4}) at $p^{(\kappa)}$.
 \UNTIL{$\frac{R_{\min}^{(\kappa)}-R_{\min}^{(\kappa-1)})}{R_{\min}^{(\kappa-1)}} \leq
 \epsilon_{\rm tol}$}.
\end{algorithmic}
\end{algorithm}
\vspace*{-0.7cm}
\subsection{Secure energy efficient maximization}
Define
\[
\pi(\mathbf{p})=\zeta\displaystyle\sum_{i=1}^M\mathbf{p}_i+P_c.
\]
Applying the inequality (\ref{esec5.2}) in Appendix II for
$
x=1/h_{ii}\mathbf{p}_i$, $y=\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j+\sigma_i^2, t=\pi(\mathbf{p})$,
and $\bar{x}=1/h_{ii}p_i^{(\kappa)}$, $\bar{y}=\sum_{j\neq i}^Mh_{ji}p_j^{(\kappa)}+\sigma_i^2, \bar{t}=\pi(p^{(\kappa)})$
yields
\begin{eqnarray}\label{fedc3}
\displaystyle\frac{f_i(\mathbf{p})}{\pi(\mathbf{p})}&\geq&F_i^{(\kappa)}(\mathbf{p})
\end{eqnarray}
for
\begin{eqnarray}
F_i^{(\kappa)}(\mathbf{p})&\triangleq&  \displaystyle\frac{2\ln(1+x_i^{(\kappa)})}{\pi(p^{(\kappa)})}
+\displaystyle\frac{x_i^{(\kappa)}}{\pi(p^{(\kappa)})(1+x_i^{(\kappa)})}\left(2-
\frac{p_i^{(\kappa)}}{\mathbf{p}_i}-
\frac{\sum_{j\neq i}h_{ji}\mathbf{p}_j+\sigma_i^2}{\sum_{j\neq i}^Mh_{ji}p_j^{(\kappa)}+\sigma_i^2}\right)\nonumber\\
&&-\displaystyle\frac{\ln(1+x_i^{(\kappa)})}{\pi^2(p^{(\kappa)})}\pi(\mathbf{p}).\label{edc3}
\end{eqnarray}
On the other hand, applying inequality (\ref{conv1}) in Appendix II for
$
\alpha=1+\ln(2)$, $x=h_{ie}\mathbf{p}_i/(\sum_{j\neq i}h_{je}\mathbf{p}_j+\sigma_e^2), t=\pi(\mathbf{p})
$
and
$
\bar{x}\triangleq h_{ie}p_i^{(\kappa)}/(\sum_{j\neq i}
h_{je}p_j^{(\kappa)}+\sigma_e^2)$, $\bar{t}=\pi(p^{(\kappa)})$
yields
\begin{eqnarray}
\displaystyle\frac{-g_i(\mathbf{p})}{\pi(\mathbf{p})}&\geq&\displaystyle\ 2\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi(p^{(\kappa)})}+\frac{x_{e,i}^{(\kappa)}}{(1+x_{e,i}^{(\kappa)})\pi(p^{(\kappa)})}
-\displaystyle\frac{1}{(1+x_{e,i}^{(\kappa)})\pi(p^{(\kappa)})}\frac{h_{ie}\mathbf{p}_i}{\sum_{j\neq i}
h_{je}\mathbf{p}_j+\sigma_e^2}\nonumber\\
&&-\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi^2(p^{(\kappa)})}\pi(\mathbf{p})-\frac{\alpha}{\pi(\mathbf{p})}
\end{eqnarray}
which together with (\ref{conv3}) in Appendix II yield
\begin{equation}\label{nin2}
\displaystyle\frac{f_i(\mathbf{p})-g_i(\mathbf{p})}{\pi(\mathbf{p})}\geq G_i^{(\kappa)}(\mathbf{p})
\end{equation}
for the concave function
\begin{eqnarray}
G_i^{(\kappa)}(\mathbf{p})&\triangleq&\displaystyle\ 2\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi(p^{(\kappa)})}+\frac{x_{e,i}^{(\kappa)}}{(1+x_{e,i}^{(\kappa)})\pi(p^{(\kappa)})}
-\displaystyle\frac{1}{(1+x_{e,i}^{(\kappa)})\pi(p^{(\kappa)})}\frac{0.5h_{ie}(\mathbf{p}_i^2/p_i^{(\kappa)}+p_i^{(\kappa)})}{\sum_{j\neq i}
h_{je}\mathbf{p}_j+\sigma_e^2}\nonumber\\
&&-\displaystyle\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi^2(p^{(\kappa)})}\pi(\mathbf{p})-\frac{\alpha}{\pi(\mathbf{p})}.\label{nin3}
\end{eqnarray}
Initialized by a feasible point $p^{(0)}$ for (\ref{esec4}),
we solve the following convex optimization problem at the $\kappa$-th iteration to generate
the next iterative point $p^{(\kappa+1)}$:
\begin{subequations}\label{nin5}
\begin{eqnarray}
\max_{\pmb{p}}\ \Phi_{\sf ee}^{(\kappa)}(\mathbf{p})\triangleq  \sum_{i=1}^M[F_i^{(\kappa)}(\pmb{p})+G_i^{(\kappa)}(\pmb{p})]\quad\mbox{s.t.}\quad
(\ref{sec4b}), \label{nin5a}\\
f_i^{(\kappa)}(\pmb{p})-g_i^{(\kappa)}(\pmb{p})\geq c_i, i=1,\dots,M.\label{nin5b}
\end{eqnarray}
\end{subequations}
Due to (\ref{dc1}) and (\ref{dc2}), the nonconvex constraint (\ref{esec4b}) in (\ref{esec4}) is implied by
the convex constraint (\ref{nin5b}) in (\ref{nin5}). Similarly to (\ref{pathf}), we can show that $\Phi_{\sf ee}(p^{(\kappa+1)})>\Phi_{\sf ee}(p^{(\kappa)})$ whenever
$p^{(\kappa+1)}\neq p^{(\kappa)}$; as such the computational procedure that invokes the convex program
(\ref{nin5}) to generate the next iterative point, is path-following for (\ref{esec4}), which at least converges to its locally optimal solution satisfying the Karush-Kuh-Tucker (KKT) conditions of optimality.

\noindent Recalling the definition (\ref{dc2}) and (\ref{dc4}) of functions $f_i^{(\kappa)}$ and $g_i^{(\kappa)}$, initialized by any feasible point $\tilde{p}^{0)}$ for
the convex constraint (\ref{sec4b}), we generate $\tilde{p}^{(\kappa+1)}$, $\kappa=0, \dots$, as the optimal solution of the convex optimization problem
\begin{equation}\label{feap}
\max_{\mathbf{p}}\ \min_{i=1,\dots,M} \frac{f_i^{({\kappa})}(\mathbf{p})-g_i^{(\kappa)}(\mathbf{p})}{c_i}\quad
\mbox{s.t.}\quad (\ref{sec4b})
\end{equation}
until $\tilde{p}^{(\kappa+1})$ such that
$\min_{i=1,\dots,M} (f_i(p^{(\kappa+1)})-g_i(p^{(\kappa+1))}/c_i
\geq 1$ is found and thus $p^{(0)}=\tilde{p}^{(\kappa+1})$ is feasible for (\ref{esec4}) that is needed for
the initial step.

Analogously, to address the maximin secure energy efficient optimization problem (\ref{esec5}) define
\[
\pi_i(\mathbf{p}_i)=\zeta \mathbf{p}_i +P_c^i.
\]
Similarly to (\ref{fedc3}) and (\ref{nin2}) the following inequalities can be obtained:
\begin{eqnarray}
\displaystyle\frac{f_i(\mathbf{p})}{\pi_i(\pmb{p}_i)}&\geq&\tilde{F}_i^{(\kappa)}(\mathbf{p}_i)
\label{ext1}\\
\displaystyle\frac{-g_i(\mathbf{p})}{\pi_i(\mathbf{p}_i)}&\geq& \tilde{G}_i^{(\kappa)}(\mathbf{p}_i)\label{ext2}
\end{eqnarray}
for
\begin{eqnarray}
\tilde{F}_i^{(\kappa)}(\mathbf{p}_i)&\triangleq&  \displaystyle\frac{2\ln(1+x_i^{(\kappa)})}{\pi_i(p_i^{(\kappa)})}+\displaystyle\frac{x_i^{(\kappa)}}{\pi_i(p_i^{(\kappa)})(1+x_i^{(\kappa)})}\left(2-
\frac{p_i^{(\kappa)}}{\mathbf{p}_i}-
\frac{\sum_{j\neq i}h_{ji}\mathbf{p}_j+\sigma_i^2}{\sum_{j\neq i}^Mh_{ji}p_j^{(\kappa)}+\sigma_i^2}\right)\nonumber\\
&&-\displaystyle\frac{\ln(1+x_i^{(\kappa)})}{\pi_i^2(p_i^{(\kappa)})}\pi_i(\mathbf{p}_i)\label{ext3}\\
\tilde{G}_i^{(\kappa)}(\mathbf{p}_i)&\triangleq& \displaystyle\ 2\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi_i(p_i^{(\kappa)})}+\frac{x_{e,i}^{(\kappa)}}{(1+x_{e,i}^{(\kappa)})\pi_i(p_i^{(\kappa)})}
-\displaystyle\frac{1}{(1+x_{e,i}^{(\kappa)})\pi_i(p_i^{(\kappa)})}\frac{0.5h_{ie}(\mathbf{p}_i^2/p_i^{(\kappa)}+p_i^{(\kappa)})}{\sum_{j\neq i}
h_{je}\mathbf{p}_j+\sigma_e^2}\nonumber\\
&&-\displaystyle\frac{\alpha-\ln(1+x_{e,i}^{(\kappa)})}{\pi_i^2(p_i^{(\kappa)})}\pi_i(\mathbf{p}_i)-\frac{\alpha}{\pi_i(\mathbf{p}_i)}.\label{ext4}
\end{eqnarray}
Initialized  a feasible point $p^{(0)}$ for (\ref{esec5}), which is found by using the generation (\ref{feap}),
the following convex optimization problem at the $\kappa$-th iteration is proposed to generate
the next iterative point $p^{(\kappa+1)}$:
\begin{equation}\label{mmee}
\max_{\pmb{p}}\min_{i=1,...,M}  [\tilde{F}_i^{(\kappa)}(\pmb{p})+\tilde{G}_i^{(\kappa)}(\pmb{p})]\quad\mbox{s.t.}\quad
(\ref{sec4b}), f_i^{(\kappa)}(\pmb{p})-g_i^{(\kappa)}(\pmb{p})\geq c_i, i=1,\dots,M.
\end{equation}
The computational procedure that invokes the convex program
(\ref{mmee}) to generate the next iterative point, is path-following for (\ref{esec5}), which at least converges to its locally optimal solution satisfying the first order necessary optimality condition.
\section{Interference networks under partial wiretap CSI}
When the EV is not part of the legitimate network, it is almost impossible to estimate
channels $h_{ie}$ from the transmitters to it. It is common to  assume
that $h_{ie}\sim \bar{h}_{ie}\chi_{ie}$, where $\chi_{ie}$ is an exponential distribution with the unit mean
and $\bar{h}_{ie}$ is a known deterministic quantity. Accordingly, instead of (\ref{dwire1}),
the wiretapped throughput for user $i$ at the
EV is defined via the following throughput outage  \cite{KB02,GRZ10,LCLC13,LCC15a,LCC15b}:
\begin{equation}\label{wire1}
g_{i,o}(\mathbf{p})\triangleq \max\ \{ \ln(1+\mathbf{r}_i)\ :\ \mbox{Prob}\left(\frac{h_{ie}\mathbf{p}_i}{\sum_{j\neq i}
h_{je}\mathbf{p}_j+\sigma_e^2}<\mathbf{r}_i\right)\leq \epsilon_{EV} \}
\end{equation}
for  $\epsilon_{EV}>0$. Using (\ref{root1}) in Appendix I, it follows that
\[
g_{i,o}(\mathbf{p})=\ln(1+\mathbf{r}_i)
\]
where
\begin{equation}\label{esec5b}
\mathbf{p}_i\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle\mathbf{r}_{i}\sigma_e^2
\mathbf{p}_i\bar{h}_{ie}\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i}\right)= 0,
\ i=1,...,M.
\end{equation}
Therefore, the problem of maximin secrecy throughput optimization can be formulated as
\begin{subequations}\label{esec5}
\begin{eqnarray}
\max_{\pmb{p},\mathbf{r}}\min_{i=1,...,M} [f_i(\mathbf{p})-\ln(1+\mathbf{r}_i)]\quad\mbox{s.t}\quad (\ref{sec4b}), (\ref{esec5b}),\label{esec5a}\\
{\color{black}\mathbf{r}_i>0, i=1,...,M.}\label{sec5c}
\end{eqnarray}
\end{subequations}
The following result unravels the computationally intractable nonlinear equality constraints in (\ref{esec5b}).
\begin{mypro}\label{outpro} The problem (\ref{esec5}) is equivalent to the following problem
\begin{subequations}\label{sec5}
\begin{eqnarray}
\max_{\pmb{p},\mathbf{r}}\min_{i=1,...,M} [f_i(\mathbf{p})-\ln(1+\mathbf{r}_i)]\quad\mbox{s.t}\quad (\ref{sec4b}), (\ref{sec5c}),\label{sec5a}\\
\mathbf{p}_i\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle\mathbf{r}_{i}\sigma_e^2+
\mathbf{p}_i\bar{h}_{ie}\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i}\right){\color{black}\geq} 0,
\ i=1,...,M.
\label{sec5b}
\end{eqnarray}
\end{subequations}
\end{mypro}
{\it Proof: \ } Since the equality constraint (\ref{esec5b}) implies the inequality constraint (\ref{sec5b}), it is true that
\[
\max\ (\ref{esec5}) \leq \max\ (\ref{sec5}).
\]
We now show that there is an optimal solution of (\ref{sec5}) satisfies the equality constraint (\ref{esec5b}) and thus
\[
\max\ (\ref{sec5}) \leq \max\ (\ref{esec5}),
\]
showing the equivalence between (\ref{sec5}) and (\ref{esec5}). Indeed, suppose that at the optimality,
\[
\mathbf{p}_i\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle\mathbf{r}_{i}\sigma_e^2+
\mathbf{p}_i\bar{h}_{ie}\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i}\right)> 0
\]
for some $i=1, \dots, M$. Then there is $0<\gamma_i<1$ such that
\[
\mathbf{p}_i\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle(\gamma\mathbf{r}_{i})\sigma_e^2+
\mathbf{p}_i\bar{h}_{ie}\sum_{j\neq i}^M\ln\left(1+\frac{\gamma\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i}\right)=0,
\]
that yields
\[
f_i(\mathbf{p})-\ln(1+\gamma\mathbf{r}_i)> f_i(\mathbf{p})-\ln(1+\mathbf{r}_i),
\]
so $\gamma_i\mathbf{r}_i$ is also the optimal solution of (\ref{sec5}), which satisfies the equality constraint
(\ref{esec5b}).\qed

To address problem (\ref{sec5}), note that a lower bounding function for the first term in (\ref{sec5a}) is $f_i^{(\kappa)}(\pmb{p})$ defined by (\ref{dc3}), while an upper bounding function
for the second term in (\ref{sec5a}) is the following linear function
\begin{equation}\label{dc6}
a_i^{(\kappa)}(\mathbf{r}_i)=\ln(1+r_i^{(\kappa)})-\displaystyle\frac{r_i^{(\kappa)}}{r_i^{(\kappa)}+1}+\frac{\mathbf{r}_i}{r_i^{(\kappa)}+1}.
\end{equation}
The main difficulty now is to develop a lower bounding approximation for the function in the left hand side (LHS)
of constraint (\ref{sec5b}). Applying  inequality (\ref{sec5.2}) in Appendix II
for
$x=1/\mathbf{r}_i\bar{h}_{je}\mathbf{p}_j$, $y=\bar{h}_{ie}\mathbf{p}_i$
and
$\bar{x}=1/r_i^{(\kappa)}\bar{h}_{je}p_j^{(\kappa)}$, $\bar{y}=\bar{h}_{ie}p_i^{(\kappa)}$
yields
\begin{eqnarray}
\displaystyle\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i}\right)&\geq&\lambda_{ij}^{(\kappa)}(\mathbf{r}_i,\pmb{p}_j,\pmb{p}_i)
\label{sec5.3}
\end{eqnarray}
for
\begin{eqnarray}\label{dsec5.3}
\lambda_{ij}^{(\kappa)}(\mathbf{r}_i,\pmb{p}_j,\pmb{p}_i)
&\triangleq&\displaystyle\ln(1+x_{ij}^{(\kappa)})+y_{ij}^{(\kappa)}\left(2-\frac{\rikp_j^{(\kappa)}}{\mathbf{r}_i\mathbf{p}_j}-\frac{\mathbf{p}_i}{p_i^{(\kappa)}}\right)
\end{eqnarray}
with $x_{ij}^{(\kappa)}\triangleq
r_i^{(\kappa)}\bar{h}_{je}p_j^{(\kappa)}/\bar{h}_{ie}p_i^{(\kappa)}$ and $y_{ij}^{(\kappa)}\triangleq x_{ij}^{(\kappa)}/(x_{ij}^{(\kappa)}+1)$.
Therefore, over the trust region
\begin{equation}\label{tr1}
\begin{array}{c}
\lambda_{ij}^{(\kappa)}(\mathbf{r}_i,\pmb{p}_j,\pmb{p}_i)   \geq 0,\\
2.5-\displaystyle\frac{\mathbf{r}_i}{r_i^{(\kappa)}}-\frac{\mathbf{p}_j}{p_j^{(\kappa)}}\geq 0
\end{array}
\end{equation}
it is true that
\begin{eqnarray}
\mathbf{p}_i\displaystyle\ln(1+\frac{\mathbf{r}_{i}\bar{h}_{je}\mathbf{p}_j}{\bar{h}_{ie}\mathbf{p}_i})&\geq&
\mathbf{p}_i\displaystyle\ln(1+x_{ij}^{(\kappa)})+y_{ij}^{(\kappa)}\left(2\mathbf{p}_i-\frac{\rikp_j^{(\kappa)}\mathbf{p}_i}{\mathbf{r}_i\mathbf{p}_j}-\frac{\mathbf{p}_i^2}{p_i^{(\kappa)}}\right)
\nonumber\\
&=&\displaystyle\left(\ln(1+x_{ij}^{(\kappa)})+2y_{ij}^{(\kappa)}\right)\mathbf{p}_i-0.5y_{ij}^{(\kappa)}\left[2\frac{\mathbf{p}_i^2}{p_i^{(\kappa)}}
\right.\nonumber\\
&&\left.\displaystyle+(\frac{\sqrt{2}\mathbf{p}_i}{\sqrt{p_i^{(\kappa)}}}+\frac{\sqrt{p_i^{(\kappa)}}\rikp_j^{(\kappa)}}{\sqrt{2}\mathbf{r}_i\mathbf{p}_j})^2-
\frac{2\mathbf{p}_i^2}{p_i^{(\kappa)}}-\frac{p_i^{(\kappa)}(\rikp_j^{(\kappa)})^2}{2\mathbf{r}_i^2\mathbf{p}_j^2}
\right]\nonumber\\
&\geq&\Lambda_i^{(\kappa)}(\mathbf{r}_i,\mathbf{p}_j,\mathbf{p}_i)\label{ob1}
\end{eqnarray}
for
\[
\begin{array}{lll}
\Lambda_i^{(\kappa)}(\mathbf{r}_i,\mathbf{p}_j,\mathbf{p}_i)
&=&\displaystyle\left(\ln(1+x_{ij}^{(\kappa)})+2y_{ij}^{(\kappa)}\right)\mathbf{p}_i-0.5y_i^{(\kappa)}\left(\frac{\sqrt{2}\mathbf{p}_i}{\sqrt{p_i^{(\kappa)}}}+
\frac{\sqrt{p_i^{(\kappa)}}\rikp_j^{(\kappa)}}{\sqrt{2}\mathbf{r}_i\mathbf{p}_j}\right)^2\\
&&- \displaystyle{\color{black}0.5y_i^{(\kappa)}}p_i^{(\kappa)}\left(\frac{\mathbf{r}_i}{r_i^{(\kappa)}}
+\frac{\mathbf{p}_j}{p_j^{(\kappa)}}-2.5\right).
\end{array}
\]
Note that in obtaining (\ref{ob1})  we also used the fact that function $\nu(\mathbf{r}_i,\mathbf{p}_j)\triangleq 1/\mathbf{r}_i^2\mathbf{p}_j^2$ is convex in the domain $\{ \mathbf{r}_i>0, \mathbf{p}_j>0\}$ and accordingly \cite{Tuybook}
$1/\mathbf{r}_i^2\mathbf{p}_j^2\geq \nu(r_i^{(\kappa)},p_j^{(\kappa)})+\langle \nabla \nu(r_i^{(\kappa)},p_j^{(\kappa)}), (\mathbf{r}_i,\mathbf{p}_j)-(r_i^{(\kappa)},p_j^{(\kappa)})\rangle=
[5-2(\mathbf{r}_i/r_i^{(\kappa)}+\mathbf{p}_j/p_j^{(\kappa)})]/(\rikp_j^{(\kappa)})^2$.

Initialized from a feasible point $(p^{(0)}, r^{(0)})$ for (\ref{sec5}) we solve the following convex program at the
$\kappa$-th iteration to generate $(p^{(\kappa+1)}, r_u^{(\kappa+1)})$:
\begin{subequations}\label{sec5k}
\begin{eqnarray}
\max_{\pmb{w},\mathbf{r}}\min_{i=1,...,M} [f_i^{(\kappa)}(\pmb{p})-a_i^{(\kappa)}(\mathbf{r}_i)]\quad\mbox{s.t}\quad (\ref{sec4b}), {\color{black}(\ref{sec5c})}, (\ref{tr1}), \label{sec5ka}\\
\pmb{p}_i\bar{h}_{ie}\ln(1-\epsilon_{EV})+\sigma_e^2\mathbf{r}_i+
\bar{h}_{ie}\sum_{j\neq i}^M\Lambda_{ij}^{(\kappa)}(\mathbf{r}_i,\pmb{p}_j,\pmb{p}_i){\color{black}\geq} 0,
\ i=1,...,M.
\label{sec5kb}
\end{eqnarray}
\end{subequations}
Then $r_i^{(\kappa+1)}$ is found from solving
\begin{eqnarray}
\psi_i(\mathbf{r}_i)\triangleq p_i^{(\kappa+1)}\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle\mathbf{r}_{i}\sigma_e^2 +
p_i^{(\kappa+1)}\bar{h}_{ie}\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}p_j^{(\kappa+1)}}{\bar{h}_{ie}p_i^{(\kappa+1)}}\right)= 0,\label{bi1}\\
\ i=1,...,M,\nonumber
\end{eqnarray}
by bisection on $[0,r_{u,i}^{(\kappa+1)}]$ such that
\begin{equation}\label{bi2}
0\leq \psi_i(r_i^{(\kappa+1)})\leq \epsilon_b\quad\mbox{(tolerance)}.
\end{equation}
A bisection on $[r_{l}, r_u]$
for solving $\psi_i(\mathbf{r}_i)=0$ where $\psi_i$ increases in $\mathbf{r_i}>0$ is implemented as
follows:
\begin{itemize}
\item Define $r_i=(r_l+r_u)/2$.  Reset  $r_l=r_i$
if $\psi_i(r_i)<0$. Otherwise reset $r_u=r_i$.
\item Terminate until $0\leq \psi_i(r_i)\leq \epsilon_b$.
\end{itemize}
In summary,  we propose in Algorithm \ref{alg2}
a path-following computational procedure for the maximin secrecy throughput optimization problem (\ref{sec5}),
which at least converges to its locally optimal solution satisfying the first order necessary optimality condition.
\begin{algorithm}
\caption{Path-following algorithm for maximin secrecy throughput optimization } \label{alg2}
\begin{algorithmic}
\STATE \textbf{Initialization}: Set $\kappa=0$. Choose an initial feasible point
$(p^{(0)}, r^{(0)})$ for (\ref{sec5}) and calculate $R_{\min}^{(0)}$ as the value of the objective function in
(\ref{sec5}) at $(p^{(0)}, r^{(0)})$.
\REPEAT \STATE $\bullet$ Set
$\kappa=\kappa+1$.
\STATE $\bullet$ Solve the
convex optimization problem \eqref{sec5k} to obtain the
solution $(p^{(\kappa)}, r_u^{(\kappa)})$.
\STATE $\bullet$ Solve the nonlinear equations (\ref{bi1}) to obtain the roots $r_i^{(\kappa)}$.
\STATE $\bullet$ Calculate  $R_{\min}^{(\kappa)}$ as the value of the objective function in (\ref{sec5}) at
$(p^{(\kappa)}, r^{(\kappa)})$.
 \UNTIL{$\frac{R_{\min}^{(\kappa)}-R_{\min}^{(\kappa-1)})}{R_{\min}^{(\kappa-1)}} \leq
 \epsilon_{\rm tol}$}.
\end{algorithmic}
\end{algorithm}

A feasible $(p^{(0)}, r^{(0)})$ is found as follows: taking $p^{(0)}$ feasible to the power constraint (\ref{sec4b})
and finding $r^{(0)}$ from solving
\[
\psi_i(\mathbf{r}_i)\triangleq p_i^{(0)}\bar{h}_{ie}\ln(1-\epsilon_{EV})+\displaystyle \mathbf{r}_{i}\sigma_e^2 +
\bar{h}_{ie}p_i^{(0)}\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}p_j^{(0)}}{\bar{h}_{ie}p_i^{(0)}}\right)= 0,
\ i=1,...,M,
\]
by  bisection on $[0,r_{u,i}^{(0)}]$ with $\psi_i(r_{u,i})>0$. Such $r^{(0)}_{u,i}$ can be easily found: from any $r_{u,i}>0$, if
$\psi_i(r_{u,i})\geq 0$ then we are done. Otherwise reset $r_{u,i}\leftarrow 2r_{u,i}$ and check $\psi_i(r_{u,i})$. Stop when $\psi(r_{u,i})>0$. Intuitively, taking $r_{u,i}^{(0)}=\bar{h}_{ie}p_i^{(0)}/\sigma_e^2$ will work.

Furthermore, the problem of SEE maximization can be formulated as
\begin{subequations}\label{esec55}
\begin{eqnarray}
\displaystyle\max_{\pmb{p},\mathbf{r}}\frac{\sum_{i=1}^M \left(f_i(\mathbf{p})-\ln(1+\mathbf{r}_i)\right)}{\pi(\mathbf{p})}\quad\mbox{s.t}\quad (\ref{sec4b}), (\ref{sec5c}), (\ref{sec5b}) \label{esec55a}\\
f_i(\mathbf{p})-\ln(1+\mathbf{r}_i)\geq c_i, i=1,\dots, M.\label{esec55b}
\end{eqnarray}
\end{subequations}
Using the inequality (\ref{nin1}) in Appendix II leads to
\[
\frac{-\ln(1+\mathbf{r}_i)}{\pi(\mathbf{p})}\geq \tilde{a}_i^{(\kappa)}(\mathbf{r}_i,\mathbf{p})
\]
for
\begin{eqnarray}
\tilde{a}_i^{(\kappa)}(\mathbf{r}_i,\mathbf{p})&\triangleq&2\displaystyle\frac{\alpha-\ln(1+r_i^{(\kappa)})}{\pi(p^{(\kappa)})}+
\frac{r_i^{(\kappa)}}{\pi(p^{(\kappa)})(1+r_i^{(\kappa)})}-\frac{\mathbf{r}_i}{\pi(p^{(\kappa)})(1+r_i^{(\kappa)})}\nonumber\\
&&-\displaystyle\frac{\alpha-\ln(1+r_i^{(\kappa)})}{\pi^2(p^{(\kappa)})}\pi(\mathbf{p})-\frac{\alpha}{\pi(\mathbf{p})}.\label{enin1}
\end{eqnarray}
Initialized by a feasible $(p^{(0)},r^{(0)})$, the following convex programm
is solved to generate $(p^{(\kappa+1)}, r^{(\kappa+1)})$ at the $\kappa$iteration:
\begin{subequations}\label{esec5k}
\begin{eqnarray}
\displaystyle\max_{\pmb{p},\mathbf{r}}\sum_{i=1}^M [F_i^{(\kappa)}(\pmb{p})+\tilde{a}_i^{(\kappa)}(\mathbf{r}_i,\mathbf{p})]\quad\mbox{s.t}\quad (\ref{sec4b}), {\color{black}(\ref{sec5c})}, (\ref{tr1}), (\ref{sec5kb}),\label{esec5ka}\\
f_i^{(\kappa)}(\pmb{p})-a_i^{(\kappa)}(\mathbf{r}_i)\geq c_i, i=1,\dots, M.
\end{eqnarray}
\end{subequations}
It can be shown that the computational procedure that invokes the convex program (\ref{esec5k}) to generate
the next iterative point, is path-following for (\ref{esec55}), which at least converges to its locally optimal solution satisfying the KKT conditions.\\
A point $(p^{(0)},r^{(0)})$ is feasible for (\ref{esec55}) if and only if
$\min_{i=1,\dots,M}[f_i(p^{(0)})-\ln(1+r_i^{(0)})]/c_i\geq 1$ and thus can be easily located by adapting Algorithm
\ref{alg2}.

Similarly, a path-following procedure for the following maximin SEE optimization problem can be proposed
\begin{equation}\label{msee}
\displaystyle\max_{\pmb{p},\mathbf{r}}\min_{i=1,\dots,M}\frac{f_i(\mathbf{p})-\ln(1+\mathbf{r}_i)}{\pi_i(\mathbf{p})}\quad\mbox{s.t}\quad (\ref{sec4b}), (\ref{sec5c}), (\ref{sec5b}), (\ref{esec55b}).
\end{equation}
\section{Robust Optimization}
Beside assuming that $h_{ie}\sim \bar{h}_{ie}\chi_{ie}$
with an exponential distribution $\chi_{ie}$ with the unit mean and deterministic $h_{ie}$,
we also assume that CSI of $h_{ji}$ is not known perfectly in the form
$h_{ji}=\bar{h}_{ji}(1+\delta \chi_{ji})$ with deterministic $\bar{h}_{ji}$
and $\delta$, and random  $\chi_{ji}$, which is an independent exponential distribution of the unit mean,
and $h_{ie}\sim \bar{h}_{ie}\chi_{ie}$ with exponential distributions  $\chi_{ji}$ and $\chi_{ie}$ of the unit mean.
Instead of (\ref{sec1}), the throughput at user $i$ is defined via the following outage probability
\begin{equation}\label{rotage1}
f_{i,o}(\mathbf{p})\triangleq \max \{ \ln(1+\mathbf{R}_i)\ :\ \mbox{Prob}\left(\frac{h_{ii}\mathbf{p}_i}{\sum_{j\neq i}^Mh_{ji}\mathbf{p}_j+\sigma_i^2}<\mathbf{R}_i\right)\leq
\epsilon_c\}
\end{equation}
for $0<\epsilon_c<<1$.\\
Using (\ref{root2}) in Appendix II, it follows that
\begin{equation}\label{rotage2}
f_{i,o}(\mathbf{p})=\ln(1+\mathbf{R}_i), i=1,\dots, M,
\end{equation}
where
\begin{eqnarray}
\pmb{p}_{i}\bar{h}_{ii}[\delta\ln(1-\epsilon_c)-1]+\displaystyle\mathbf{R}_{i}(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}\mathbf{p}_j) +
\delta\bar{h}_{ii}\pmb{p}_{i}\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}\pmb{p}_{j}}{\bar{h}_{ii}\pmb{p}_{i}}
\right)=0, \label{rotage3}\\
 i=1,\dots, M.\nonumber
\end{eqnarray}
Therefore, the problem of maximin secrecy throughput robust optimization is defined by
\begin{subequations}\label{rotage4}
\begin{eqnarray}
\max_{\pmb{p},\mathbf{R},\mathbf{r}}\min_{i=1,...,M} [\ln(1+\mathbf{R}_i)-\ln(1+\mathbf{r}_i)]
\quad\mbox{s.t}\quad (\ref{sec4b}),  {\color{black}(\ref{sec5c})}, (\ref{sec5b}), (\ref{rotage3}), \label{rotage4a}\\
{\color{black}\mathbf{R}_i>0, i=1,...,M.}\label{sec6c}
\end{eqnarray}
\end{subequations}
The following result unravels the computationally intractable nonlinear equality constraints in (\ref{rotage3}):
\begin{mypro}\label{prorob}
Problem (\ref{rotage4}) is equivalent to the following problem
\begin{subequations}\label{sec6}
\begin{eqnarray}
\max_{\pmb{p},\mathbf{R},\mathbf{r}}\min_{i=1,...,M} [\ln(1+\mathbf{R}_i)-\ln(1+\mathbf{r}_i)]
\quad\mbox{s.t}\quad (\ref{sec4b}), (\ref{sec5b}), {\color{black}(\ref{sec5c})}, (\ref{sec6c})\label{sec6a}\\
\pmb{p}_{i}\bar{h}_{ii}[\delta\ln(1-\epsilon_c)-1]+\displaystyle\mathbf{R}_{i}(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}\mathbf{p}_j) +
\delta\bar{h}_{ii}\pmb{p}_{i}\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}\pmb{p}_{j}}{\bar{h}_{ii}\pmb{p}_{i}}\right)\leq
0,\label{sec6be}\\
\ i=1,...,M. \nonumber
\end{eqnarray}
\end{subequations}
\end{mypro}
{\it Proof: \ } Again, it is obvious that
\[
\max\ (\ref{rotage4}) \leq (\ref{sec6}).
\]
Furthermore, at an optimal solution of (\ref{sec6}), if
\[
\pmb{p}_{i}\bar{h}_{ii}[\delta\ln(1-\epsilon_c)-1]+\displaystyle\mathbf{R}_{i}(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}\mathbf{p}_j) +
\delta\bar{h}_{ii}\pmb{p}_{i}\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}\pmb{p}_{j}}{\bar{h}_{ii}\pmb{p}_{i}}\right)<0,
\]
for some $i$ then there is $\gamma>1$ such that
\[
\pmb{p}_{i}\bar{h}_{ii}[\delta\ln(1-\epsilon_c)-1]+\displaystyle(\gamma\mathbf{R}_{i})(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}\mathbf{p}_j) +
\delta\bar{h}_{ii}\pmb{p}_{i}\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}(\gamma\mathbf{R}_{i})\pmb{p}_{j}}{\bar{h}_{ii}\pmb{p}_{i}}\right)=0,
\]
which results in $\ln(1+\gamma \mathbf{R}_i)>\ln(1+\mathbf{R}_i)$, implying that $\gamma \mathbf{R}_i$ is also an optimal solution
of (\ref{sec6}). We thus have proved that there is always an optimal solution of (\ref{sec6}) to satisfy
the equality constraints in (\ref{rotage3}), so
\[
\max\ (\ref{sec6}) \leq (\ref{rotage4}),
\]
completing the proof of Proposition \ref{prorob}.\qed
To address problem (\ref{sec6}), firstly we  provide a lower bounding approximation for the first term in
the objective function in (\ref{sec6a}) as follows
\[
\begin{array}{lll}
\ln(1+\mathbf{R}_i)&\geq& A^{(\kappa)}_i(\mathbf{R}_i)
\triangleq \displaystyle\ln(1+R_i^{(\kappa)})+\frac{R_i^{(\kappa)}}{R_i^{(\kappa)}+1}-\frac{(R_i^{(\kappa)})^2}{R_i^{(\kappa)}+1}\frac{1}{\mathbf{R}_i}.
\end{array}
\]
Next, to obtain an upper bounding approximation for the function in the left hand side of (\ref{sec6be}) and thus to provide an inner
approximation for constraint (\ref{sec6be}), we use the following inequality
\begin{eqnarray}
\mathbf{R}_{i}\pmb{p}_{j}&=&0.5(\mathbf{R}_{i}+\pmb{p}_{j})^2-0.5\mathbf{R}_{i}^2-0.5\pmb{p}_{j}^2\nonumber\\
&\leq&\Upsilon_{ij}^{(\kappa)}(\mathbf{R}_i,\pmb{p}_j)\nonumber\\
&\triangleq&0.5(\mathbf{R}_{i}+\pmb{p}_{j})^2-R_i^{(\kappa)}\mathbf{R}_i+0.5(R_i^{(\kappa)})^2
-p_j^{(\kappa)}\mathbf{p}_j+0.5(p_j^{(\kappa)})^2,\label{sec5.2R}
\end{eqnarray}
over the trust region
\begin{equation}\label{tr3}
2\mathbf{R}_i\geq R_i^{(\kappa)}, 2\mathbf{p}_j\geq p_j^{(\kappa)}.
\end{equation}
Then
\begin{eqnarray}
\displaystyle\mathbf{p}_i\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}\mathbf{p}_j}{\bar{h}_{ii}\mathbf{p}_i}\right)&\leq&
\displaystyle\mathbf{p}_i\left[\ln\left(1+\frac{\bar{h}_{ji}R_i^{(\kappa)} p_j^{(\kappa)}}{\bar{h}_{ii}p_i^{(\kappa)}}\right)
+\frac{1}{\frac{\bar{h}_{ii}}{\bar{h}_{ji}}+\frac{R_i^{(\kappa)} p_j^{(\kappa)}}{p_i^{(\kappa)}}}
(\frac{\mathbf{R}_{i}\mathbf{p}_{j}}{\pmb{p}_{i}}
-\frac{\Rikp_j^{(\kappa)}}{p_i^{(\kappa)}})\right]\nonumber\\
&\leq&\Phi_{ij}^{(\kappa)}\left(\mathbf{R}_i,\pmb{p}_j,\pmb{p}_i\right)\nonumber\\
&\triangleq&\mathbf{p}_i\ln\left(1+\frac{\bar{h}_{ji}R_i^{(\kappa)} p_j^{(\kappa)}}{\bar{h}_{ii}p_i^{(\kappa)}}\right)\\
&&\displaystyle+\frac{1}{\frac{\bar{h}_{ii}}{\bar{h}_{ji}}+\frac{R_i^{(\kappa)} p_j^{(\kappa)}}{p_i^{(\kappa)}}}
\left(\Upsilon_{ij}^{(\kappa)}(\mathbf{R}_i,\pmb{p}_j)
-\displaystyle\frac{R_i^{(\kappa)} p_j^{(\kappa)}}{p_i^{(\kappa)}}\mathbf{p}_i\right).\label{sec5.3R}
\end{eqnarray}
Initialized from a feasible $(p^{(0)}, R^{(0)}, r^{(0)})$ for (\ref{sec6}) we solve the following convex program at
the $\kappa$-th iteration to generate the next iterative point $(p^{(\kappa+1)},R_l^{(\kappa+1)}, r_u^{(\kappa+1)})$:
\begin{subequations}\label{sec5kR}
\begin{eqnarray}
\max_{\pmb{w},\mathbf{r}}\min_{i=1,...,M} [A_i^{(\kappa)}(\mathbf{R}_i)-a_i^{(\kappa)}(\mathbf{r}_i)]\quad\mbox{s.t}\quad (\ref{sec4b}), {\color{black}(\ref{sec5c})}, (\ref{tr1}),  (\ref{sec5kb}), (\ref{sec6c}),  (\ref{tr3})\label{sec5kRa}\\
\displaystyle\mathbf{p}_i\bar{h}_{ii}\left[\delta\ln(1-\epsilon_c)-1\right]+\sigma_i^2\mathbf{R}_i+\displaystyle\sum_{j\neq i}\bar{h}_{ji}\Upsilon_{ij}^{(\kappa)}(\mathbf{R}_i,\mathbf{p}_j)+
\delta\bar{h}_{ii}\sum_{j\neq i}^M\Phi_{ij}^{(\kappa)}(\mathbf{R}_i,\pmb{p}_j,\pmb{p}_i)\leq 0,
\label{sec5kRb}\\
\ i=1,...,M.\nonumber
\end{eqnarray}
\end{subequations}
At the same $\kappa$-th iteration, $r_i^{(\kappa+1)}$ is found from solving (\ref{bi1})
by bisection on $[0,r_{u,i}^{(\kappa+1)}]$ such that (\ref{bi2}),
while $R_i^{(\kappa+1)}$ is found from solving
\begin{equation}\label{nonlin1}
\zeta_i(\mathbf{R}_i)=0, i=1,\dots, M,
\end{equation}
for the increasing function
\[
\zeta_i(\mathbf{R}_i)\triangleq \delta\ln(1-\epsilon_c)-1+\displaystyle\frac{\mathbf{R}_{i}(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}p_j^{(\kappa+1)})}{\bar{h}_{ii}p _{i}^{(\kappa+1)}}+
\delta\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}p_{j}^{(\kappa+1)}}{\bar{h}_{ii}p_{i}^{(\kappa+1)}}\right),
\]
by bisection on $[R_{l,i}^{(\kappa+1)}, R_{u,i}]$ with  $\zeta_i(R_{u,i})>0$ such that
\begin{equation}\label{bi3}
-\epsilon_b\leq g_i(R_i^{(\kappa+1)})\leq 0.
\end{equation}
$R_{u,i}$ can be easily located: initialized by $R_i=2R_{l,i}^{(\kappa+1)}$ and check $\zeta_i(R_i)$.
If $\zeta_i(R_i)>0$ then we are done. Otherwise reset $R_i\leftarrow 2R_i$ until $\zeta_i(R_i)>0$.
Intuitively, taking $R_{u,i}=2\bar{h}_{ii}p_i^{(\kappa+1)}/(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}p_j^{(\kappa+1)})$
will work.

In summary,  we propose in Algorithm \ref{alg3}
a path-following computational procedure for the maximin secrecy throughput optimization problem (\ref{sec6}),
which at least converges to its locally optimal solution satisfying the first order necessary optimality condition.
\begin{algorithm}
\caption{Path-following algorithm for maximin secrecy throughput optimization } \label{alg3}
\begin{algorithmic}
\STATE \textbf{Initialization}: Set $\kappa=0$. Choose an initial feasible point
$(p^{(0)}, R^{(0)}, r^{(0)})$ for (\ref{sec6}) and calculate $R_{\min}^{(0)}$ as the value of the objective function in
(\ref{sec6}) at $(p^{(0)}, R^{(0)}, r^{(0)})$.
\REPEAT \STATE $\bullet$ Set
$\kappa=\kappa+1$.
\STATE $\bullet$ Solve the
convex optimization problem \eqref{sec5kR} to obtain the
solution $(p^{(\kappa)}, R_{l}^{(\kappa)}, r_u^{(\kappa)})$.
\STATE $\bullet$ Solve the nonlinear equations (\ref{bi1}) to obtain the roots $r_i^{(\kappa)}$.
\STATE $\bullet$ Solve the nonlinear equations (\ref{nonlin1}) to obtain the roots $R_i^{(\kappa)}$.
\STATE $\bullet$ Calculate  $R_{\min}^{(\kappa)}$ as the value of the objective function in (\ref{sec6}) at
$(p^{(\kappa)}, R^{(\kappa)}, r^{(\kappa)})$.
 \UNTIL{$\frac{R_{\min}^{(\kappa)}-R_{\min}^{(\kappa-1)})}{R_{\min}^{(\kappa-1)}} \leq
 \epsilon_{\rm tol}$}.
\end{algorithmic}
\end{algorithm}

An initial  feasible $(p^{(0)},R^{(0)}, r^{(0)})$ can be easily found as follows: taking any $p^{(0)}$ feasible
to the power constraint (\ref{sec4b}) to find $R^{(0)}$ and $r^{(0)}$ from solving
\[
\begin{array}{r}
\zeta_i(\mathbf{R}_i)\triangleq \delta\ln(1-\epsilon_c)-1+\displaystyle\frac{\mathbf{R}_{i}(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}p_j^{(0)})}{\bar{h}_{ii}p_{i}^{(0)}}+\delta
\sum_{j\neq i}^M\ln\left(1+\frac{\bar{h}_{ji}\mathbf{R}_{i}p_{j}^{(0)}}{\bar{h}_{ii}p_{i}^{(0)}}\right)= 0,\\
i=1,...,M,
\end{array}
\]
by bisection on $[0, 2\bar{h}_{ii}p_i^{(0)}/(\sigma_i^2+\sum_{j\neq i}\bar{h}_{ji}p_j^{(0)})]$,
and $r^{(0)}$ is found from solving
\[
\ln(1-\epsilon_c)+\displaystyle\frac{\mathbf{r}_{i}\sigma_e^2}{\bar{h}_{ie}p_i^{(0)}}+
\sum_{j\neq i}^M\ln\left(1+\frac{\mathbf{r}_{i}\bar{h}_{je}p_j^{(0)}}{\bar{h}_{ie}p_i^{(0)}}\right)= 0,
\ i=1,...,M,
\]
by bisection on $[0,\bar{h}_{ie}p_i^{(0)}/\sigma_e^2]$.

Lastly, the network secure energy efficiency problem is now formulated by
\begin{subequations}\label{esec6}
\begin{eqnarray}
\displaystyle\max_{\pmb{p},\mathbf{R},\mathbf{r}}\frac{\sum_{i=1}^{M} (\ln(1+\mathbf{R}_i)-\ln(1+\mathbf{r}_i))}{\pi(\mathbf{p})}
\quad\mbox{s.t}\quad (\ref{sec4b}),  {\color{black}(\ref{sec5c})}, (\ref{sec5b}),
(\ref{rotage3}), (\ref{sec6c}), \label{esec6a}\\
\ln(1+\mathbf{R}_i)-\ln(1+\mathbf{r}_i)\geq c_i, i=1,\dots,M.\label{esec6b}
\end{eqnarray}
\end{subequations}
To this end, we use inequality (\ref{esec5.2}) in Appendix II  to obtain
\begin{eqnarray}
\displaystyle\frac{\ln(1+\mathbf{R}_i)}{\pi(\mathbf{p})}&\geq&\tilde{A}^{(\kappa)}_i(\mathbf{R}_i,\mathbf{p})\nonumber\\
&\triangleq&\displaystyle\frac{2\ln(1+R_i^{(\kappa)})}{\pi(p^{(\kappa)})}+\frac{R_i^{(\kappa)}}{\pi(p^{(\kappa)})(1+R_i^{(\kappa)})}\left(1-\frac{R_i^{(\kappa)}}{\mathbf{R}_i}\right)-
\frac{\ln(1+R_i^{(\kappa)})}{\pi^2(p^{(\kappa)})}\pi(\mathbf{p})
\end{eqnarray}
Initialized by a feasible point $(R^{(0)}, r^{(0)}, p^{(0)})$,
at the $\kappa$-th iteration, the following convex programm is solved to generated $(R^{(\kappa+1)}, r^{(\kappa+1)}, p^{(\kappa_1)})$
\begin{subequations}\label{esec5kR}
\begin{eqnarray}
\max_{\pmb{w},\mathbf{r}}\sum_{i=1}^{M}\left[\tilde{A}^{(\kappa)}_i(\mathbf{R}_i,\mathbf{p})+\tilde{a}^{(\kappa)}_i(\mathbf{r}_i,\mathbf{p})\right] \quad\mbox{s.t}\quad (\ref{sec4b}), {\color{black}(\ref{sec5c})}, (\ref{tr1}),  (\ref{sec5kb}), (\ref{sec6c}),  (\ref{tr3}),
(\ref{sec5kRb}),\label{esec5kRa}\\
A_i^{(\kappa)}(\mathbf{R}_i)-a_i^{(\kappa)}(\mathbf{r}_i)\geq c_i, \ i=1,...,M.\label{esec5kRb}
\end{eqnarray}
\end{subequations}
It can be shown that the computational procedure that invokes the convex program (\ref{esec5kR}) to generate
the next iterative point, is path-following for (\ref{esec6}), which at least converges to its locally optimal solution satisfying the KKT conditions.\\
A point $(p^{(0)}, R^{(0)}, r^{(0)})$ is feasible for (\ref{esec6}) if and only if
$\min_{i=1,\dots,M}[f_i(R_i^{(0)})-\ln(1+r_i^{(0)})]/c_i\geq 1$ and thus can be easily located by adapting Algorithm
\ref{alg3}.

Similarly, a path-following procedure for the following maximin SEE optimization problem can be proposed
\begin{equation}\label{mseeR}
\displaystyle\max_{\pmb{p},\mathbf{r}}\min_{i=1,\dots,M}\frac{f_i(\mathbf{R}_i)-\ln(1+\mathbf{r}_i)}{\pi_i(\mathbf{p})}\quad\mbox{s.t}\quad
(\ref{sec4b}),  {\color{black}(\ref{sec5c})}, (\ref{sec5b}),
(\ref{rotage3}), (\ref{sec6c}), (\ref{esec6b}).
\end{equation}
\section{Simulation}
This section evaluates the performance of the proposed algorithms through extensive simulation. Considered in all simulation studies is a wireless network with $M=10$ transmitter-user communication pairs and noise variance $\sigma_i^2=\sigma_e^2=1$ mW \cite{WWN15}. All channels among each pair are i.i.d. complex normal random variable with zero mean and unit variance. The drain efficiency of power amplifier $1/\zeta$ is set
to be $40 \%$ and the circuit power of each transmitter is $P_c^i=5$ mW. Besides, we set
{\color{black}$\epsilon_c=0.1$ and $\epsilon_{EV}\in\{0.1, 0.6\}$}
and $\delta=0.1$.
The computation tolerance for terminating all proposed Algorithms is $\epsilon_{\rm tol}=10^{-4}$.
\vspace*{-0.3cm}
\subsection{Maximin secrecy throughput optimization}
\begin{figure}
\centering
\includegraphics[width=3.5in]{Rate_PI.eps}
\caption{Min secrecy throughput among users versus the transmit power budget}
\label{fig:Rate_PI}
\end{figure}
This subsection analyzes the secrecy throughput in the presence of eavesdropper. {\color{black} In what follows, we consider five cases, including ``Perfect CSI'', ``Partial CSI ($\epsilon_{EV}=0.1$)'', ``Partial CSI ($\epsilon_{EV}=0.6$)'', ``Robust Opt. ($\epsilon_{EV}=0.1$ $\epsilon_c=0.1$)'' and ``Robust Opt. ($\epsilon_{EV}=0.6$ $\epsilon_c=0.1$)''.
The terms ``Perfect CSI'', ``Partial CSI'' and ``Robust Opt.''  correspond to the scenarios discussed in Sections III, IV and V, respectively.}
Fig. \ref{fig:Rate_PI} plots the minimum secrecy throughput versus the transmit power budget $P_i$ varying from 10 to 50 mW.
As expected, it is seen that the secrecy throughput increase with the transmitted power budget $P_i$.
It is also observed that the secrecy throughput of ``Partial CSI'' with $\epsilon_{EV}=0.1$ is always better than the secrecy throughputs of others.
For $\epsilon_{EV}=0.1$, ``Partial CSI'' and ``Robust Opt.'' clearly and significantly outperform ``Perfect CSI''. However, the secrecy throughput of ``Perfect CSI'' is superior to the secrecy throughputs of ``Partial CSI'' and ``Robust Opt.'' with $\epsilon_{EV}=0.6$. This is not a surprise because according to the wiretapped throughput defined by (\ref{dwire1}) and
the throughput outage defined by (\ref{wire1})-(\ref{esec5b}), the former is seen higher than the later for small $\epsilon_{EV}$.

Table \ref{tab:rate_P_Ite} provides the average number of iterations required to solve maximin secrecy throughput optimization for the above three cases. As can be observed, our proposed algorithm converges in less than 14 iterations, on average, for all considered cases.

\begin{table}
   \centering
   \caption{Average number of iterations for maximin secrecy throughput optimization.}
   \begin{tabular}{ | c | c | c | c | c | c | }
    \hline
   $P_i$ (mW) & 10  &  20 &  30 &  40 &  50  \\
   \hline
   Perfect CSI & 8.12 & 7.63 &  7.61 &  7.71 &  8.56 \\
    \hline
   Partial CSI ($\epsilon_{EV}=0.1$) & 11.25 & 10.87 & 10.73 & 10.40 & 10.31 \\
  \hline
   Partial CSI ($\epsilon_{EV}=0.6$) & 13.12 & 12.18 & 14.92 & 12.60 & 11.68 \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.1$) & 4.20 & 4.33 & 4.20 & 3.52 & 3.35 \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.6$) & 5.18 & 4.96 & 4.82 & 4.14 & 4.90 \\
  \hline
\end{tabular}
\label{tab:rate_P_Ite}
\end{table}

\begin{figure}
\centering
\includegraphics[width=3.5in]{EE_c.eps}
\caption{Energy efficiency versus QoS constraint}
\label{fig:EE_c}
\end{figure}

\begin{figure*}[t]
    \centering
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Rate_c.eps}
  \caption{Sum throughput versus QoS constraint}
  \label{fig:Rate_c}
  \end{minipage}
    \hspace{0.1cm}
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Power_c.eps}
  \caption{Total power consumption versus QoS constraint}
  \label{fig:Power_c}
  \end{minipage}

\end{figure*}

\vspace*{-0.3cm}
\subsection{Energy efficiency maximization}
In this subsection, we first examine the performance of EE maximization algorithm versus the QoS constraint. The transmitted power $P_i$ is
fixed at 20 mW and QoS constraint $c_i$ varies from 0.1 to 0.5 bps/Hz. It can be observed from Fig. \ref{fig:EE_c}
that the EE performance degrades as the QoS constraint $c_i$ increases. Moreover, ``Partial CSI'' with $\epsilon_{EV}=0.1$ outperforms others in terms of EE performance. To find out the impact on the sum throughput and total power consumption in EE maximization algorithm, the achieved sum throughput and the total power consumed are illustrated in Fig. \ref{fig:Rate_c} and \ref{fig:Power_c}, respectively.
It can be seen that the total power consumption increases faster than the sum throughput, which explains why
EE degrades as $c_i$ increases in Fig. \ref{fig:EE_c}.
Although the sum throughput of ``Robust Opt.'' is slightly better than ``Partial CSI'', ``Partial CSI'' consumes less power
than ``Robust Opt.''.
Table \ref{tab:EE_c_Ite} shows that our proposed EE maximization algorithm converges in less than 35 iterations, on average, in all considered cases.

\begin{table}
   \centering
   \caption{Average number of iterations for energy efficiency maximization.}
   \begin{tabular}{ | c | c | c | c | c | c | }
    \hline
   $c_i$ (bps/Hz) & 0.1  &  0.2 &  0.3 &  0.4 &  0.5  \\
   \hline
   Perfect CSI & 32.21 &  29.62 &  24.26 &  21.23  & 15.33 \\
    \hline
   Partial CSI ($\epsilon_{EV}=0.1$) & 33.73 & 33.12 & 28.75 & 25.74 & 23.25 \\
  \hline
   Partial CSI ($\epsilon_{EV}=0.6$) & 35.82 & 30.56 & 34.22 & 22.26 & 18.34  \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.1$) & 24.25  & 27.41 & 25.53  &  30.06 & 31.75  \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.6$) & 29.02  & 23.76 & 26.80 & 29.32 & 23.46  \\
  \hline
\end{tabular}
\label{tab:EE_c_Ite}
\end{table}

\begin{figure}
\centering
\includegraphics[width=3.5in]{EE_P.eps}
\caption{Energy efficiency versus the transmit power budget}
\label{fig:EE_P}
\end{figure}

\begin{figure*}[t]
    \centering
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Rate_P.eps}
  \caption{Sum throughput versus the transmit power budget}
  \label{fig:Rate_P}
  \end{minipage}
    \hspace{0.1cm}
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Power_P.eps}
  \caption{Total power consumption versus the transmit power budget}
  \label{fig:Power_P}
  \end{minipage}

\end{figure*}

Next, we further investigate the performance of EE versus the transmit power budget. The QoS constraint $c_i$
is fixed at 0.4 bps/Hz and $P_i$ varies from 10 to 50 mW. {\color{black} As shown in Fig. \ref{fig:EE_P}, we observe that the EE performance of ``Partial CSI'' with $\epsilon_{EV}=0.1$ clearly and significantly outperforms the optimized EE of other cases.}
Furthermore, it can be seen that the EE performances saturate when the transmit power budget
exceeds the threshold.
That is because for small transmit power budget, the denominator of EE is dominated by the circuit power and
the EE is maximized by maximization of the sum throughput in the numerator.
However, for larger transmit power budget, the denominator of EE is dominated by the actual transmit power.
When the total power consumption saturates in Fig. \ref{fig:Power_P}, both the
EE and the sum throughput accordingly saturate in Figs. \ref{fig:EE_P} and \ref{fig:Rate_P}.

\begin{figure}
\centering
\includegraphics[width=3.5in]{Maxmin_EE_c.eps}
\caption{Minimum energy efficiency versus the QoS constraint}
\label{fig:Maxmin_EE_c}
\end{figure}

\begin{figure*}[t]
    \centering
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Maxmin_Rate_c.eps}
  \caption{Throughput versus the QoS constraint}
  \label{fig:Maxmin_Rate_c}
  \end{minipage}
    \hspace{0.1cm}
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Maxmin_Power_c.eps}
  \caption{Power consumption versus the QoS constraint}
  \label{fig:Maxmin_Power_c}
  \end{minipage}

\end{figure*}

\vspace*{-0.3cm}
\subsection{Maxmin energy efficiency optimization}
In this subsection, we aim to maximize the minimum EE performance.
Firstly, Fig. \ref{fig:Maxmin_EE_c} plots the maximized minimum EE versus QoS constraint. The transmitted power $P_i$ is
fixed at 20 mW and QoS constraint $c_i$ varies from 0.1 to 0.5 bps/Hz.
{\color{black}It can be seen that the optimized minimum EE decreases with increasing $c_i$ and the EE performance of ``Partial CSI'' with $\epsilon_{EV}=0.1$ is always better than the optimized EE of other cases.
Furthermore, it is also observed that for $\epsilon_{EV}=0.1$ ``Partial CSI'' and ``Robust Opt.'' outperform ``Perfect CSI'' in terms of EE performance, while ``Perfect CSI'' is superior to ``Partial CSI'' and ``Robust Opt.'' for $\epsilon_{EV}=0.6$.}
The corresponding throughput and power consumption
are plotted in Fig. \ref{fig:Maxmin_Rate_c} and \ref{fig:Maxmin_Power_c}, respectively.
Table \ref{tab:Maxmin_EE_c_Ite} shows that maximin EE optimization converges in less than 33 iterations, on average, in all considered cases.

\begin{table}
   \centering
   \caption{Average number of iterations for maximin energy efficiency optimization.}
   \begin{tabular}{ | c | c | c | c | c | c | }
    \hline
   $c_i$ (bps/Hz) & 0.1  &  0.2 &  0.3 &  0.4 &  0.5  \\
   \hline
   Perfect CSI & 32.42 &  30.35 & 31.61 &  29.23 & 22.25 \\
    \hline
   Partial CSI ($\epsilon_{EV}=0.1$) & 21.86 & 22.13 & 20.42 & 20.82 & 30.23 \\
  \hline
   Partial CSI ($\epsilon_{EV}=0.6$) & 23.66 & 25.02 & 22.68 & 33.30 & 29.34 \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.1$) & 16.05 &  23.27 & 23.36 & 31.15 & 18.62  \\
  \hline
  Robust Opt. ($\epsilon_{EV}=0.6$) & 20.78  &  26.12  & 31.24  & 27.46  &  23.92  \\
  \hline
\end{tabular}
\label{tab:Maxmin_EE_c_Ite}
\end{table}

\begin{figure}
\centering
\includegraphics[width=3.5in]{Maxmin_EE_P.eps}
\caption{Minimum energy efficiency versus the transmit power budget}
\label{fig:Maxmin_EE_P}
\end{figure}

\begin{figure*}[t]
    \centering
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Maxmin_Rate_P.eps}
  \caption{Throughput versus the transmit power budget}
  \label{fig:Maxmin_Rate_P}
  \end{minipage}
    \hspace{0.1cm}
    \begin{minipage}[h]{0.48\textwidth}
    \centering
    \includegraphics[width=1.05 \textwidth]{Maxmin_Power_P.eps}
  \caption{Power consumption versus the transmit power budget}
  \label{fig:Maxmin_Power_P}
  \end{minipage}

\end{figure*}

Next, we investigate the maximin EE performance versus the transmit power budget. The QoS constraint $c_i$
is fixed at 0.4 bps/Hz and $P_i$ varies from 10 to 50 mW.
The minimum EE performance versus the transmit power budget is illustrated in Fig. \ref{fig:Maxmin_EE_P}. Again, we observe that
the optimized minimum EE saturates when the transmit power is larger than some threshold. This is due to the fact that
under small transmit power regime, the EE is maximized by maximizing the throughput in the numerator.
When the transmit power is large enough to obtain the optimized EE, both throughput and power consumption accordingly saturate in Figs. \ref{fig:Maxmin_Rate_P} and \ref{fig:Maxmin_Power_P}.
\section{Conclusions}
We have considered the problem of power allocation to maximize the worst links's secrecy throughput or the network's
secure energy efficiency under various scenarios of available channel state information.
We have further proposed path-following algorithms tailored for each of the considered scenarios. Finally, we have provided
 simulations to show the efficiency of the proposed algorithms. Extensions to beamforming in multi-input single-output
 (MISO) interference networks are under current investigation.
\section*{Appendix I: outage probability fundamental}
Recall a probability distribution $\chi$ is called an {\it exponential distribution}
if its probability density function (pdf) is in form   $\lambda e^{-\lambda x}$ with
$\lambda >0$. It is immediate to check that $\mbox{Prob}(\chi\geq t)=e^{-\lambda t}$ and
$\mathbb{E}[\chi]=1/\lambda$.
Recall the following result \cite[(15)]{KB02}.
\begin{myth}\label{thop1} Suppose $z_1,\cdots, z_n$ are independent exponentially distributed random variables
with $\mathbb{E}(z_i)=1/\lambda_i$. Then for deterministic $p_i>0$, $i=1,\cdots, n$:
\begin{equation}\label{ap1}
\mbox{Prob}(p_1z_1\leq \sum_{i=2}^np_iz_i)=1-\prod_{i=2}^n\frac{1}{1+(\lambda_1/p_1)/(\lambda_i/p_i)}.
\end{equation}
\end{myth}
It follows from (\ref{ap1}) that
\begin{equation}\label{ap2}
\mbox{Prob}(p_1z_1> c+\sum_{i=2}^np_iz_i)=e^{-\lambda_1 c/p_1}\prod_{i=2}^n\frac{1}{1+(\lambda_1/p_1)/(\lambda_i/p_i)}
\end{equation}
and
\begin{eqnarray}
\displaystyle\mbox{Prob}(\frac{p_1z_1}{\displaystyle\sum_{i=2}^np_iz_i+\sigma}> \gamma)&=&
\displaystyle\mbox{Prob}(p_1z_1> \displaystyle\sum_{i=2}^n\gamma p_iz_i+\gamma\sigma)\\
&=&\displaystyle e^{-\lambda_1\gamma\sigma/p_1}\prod_{i=2}^n\frac{1}{1+\gamma(\lambda_1/p_1)/(\lambda_i/p_i)}\label{ap2.1}
\end{eqnarray}
Sometimes, it is also more convenient to write (\ref{ap1}), (\ref{ap2}) and (\ref{ap2.1}) in terms of
means $\bar{\lambda}_i=1/\lambda_i$ of $z_i$
as
\begin{eqnarray}
\mbox{Prob}(p_1z_1\leq \sum_{i=2}^np_iz_i)=1-\prod_{i=2}^n\frac{p_1\bar{\lambda}_1}{p_1\bar{\lambda}_1
+p_i\lambda_i},\label{ap1e}\\
\mbox{Prob}(p_1z_1> c+\sum_{i=2}^np_iz_i)=e^{-c/p_1\bar{\lambda}_1}\prod_{i=2}^n\frac{p_1\bar{\lambda}_1}{p_1\bar{\lambda}_1
+p_i\bar{\lambda}_i},\label{ap2e}\\
\displaystyle\mbox{Prob}(\frac{p_1z_1}{\sum_{i=2}^np_iz_i+\sigma}> \gamma)=
\displaystyle e^{-\gamma\sigma/p_1\bar{\lambda}_1}\prod_{i=2}^n\frac{p_1\bar{\lambda}_1}{p_1\bar{\lambda}_1
+\gamma p_i\bar{\lambda}_i}
\end{eqnarray}
\begin{myth}\label{thop2}
For given $\varepsilon>0$, define
\begin{equation}\label{ap5}
r_{\max}\triangleq \max \{ r\ :\ \mbox{Prob}(\frac{p_1z_1}{\sum_{i=2}^np_iz_i+\sigma^2})<r)\leq \varepsilon\}
\end{equation}
Then $r_{\max}$ is the unique positive root of the nonlinear equation
\begin{equation}\label{root1}
\ln(1-\varepsilon)+ \displaystyle\frac{r\sigma^2}{p_1\bar{\lambda}_1}+\sum_{i=2}^n\ln(1+
\frac{rp_i\bar{\lambda}_i}{p_1\bar{\lambda}_1})=0.
\end{equation}
\end{myth}
{\it Proof: \ } Applying (\ref{ap2e}) yields
\begin{eqnarray}
\displaystyle\mbox{Prob}(\frac{p_1z_1}{\sum_{i=2}^np_iz_i+\sigma^2}<r)&=&\displaystyle\mbox{Prob}(p_1z_1<r(\displaystyle\sum_{i=2}^np_iz_i+\sigma^2))\nonumber\\
&=&1-e^{-r\sigma^2/p_1\bar{\lambda}_1}\displaystyle\prod_{i=2}^n\frac{p_1\bar{\lambda}_1}{p_1\bar{\lambda}_1
+rp_i\bar{\lambda}_i}.\label{ap6}
\end{eqnarray}
Therefore,
\begin{eqnarray}
\mbox{Prob}(\frac{p_1z_1}{\sum_{i=2}^np_iz_i+\sigma^2})<r)\leq \varepsilon&\Leftrightarrow&1-e^{-r\sigma^2/p_1\bar{\lambda}_1}\displaystyle\prod_{i=2}^n\frac{p_1\bar{\lambda}_1}{p_1\bar{\lambda}_1
+rp_i\bar{\lambda}_i}\leq \varepsilon\nonumber\\
&\Leftrightarrow&\ln(1-\varepsilon)+ \displaystyle\frac{r\sigma^2}{p_1\bar{\lambda}_1}+\sum_{i=2}^n\ln(1+
\frac{rp_i\bar{\lambda}_i}{p_1\bar{\lambda}_1})\leq 0.\label{ap7}
\end{eqnarray}
By noticing that the function in the left hand side (LHS) of (\ref{ap7}) is increasing in $r$, we arrive at (\ref{root1}).\qed
\begin{myth}\label{thop3}
Suppose $\bar{z}_i>0$, $p_i>0$, $\delta>0$ and $\sigma>0$
are deterministic values, while $\tilde{z}_i$ are  independent exponential distributions. For $\varepsilon>0$, define
\begin{equation}\label{ap8}
r_{p}\triangleq \max\ \{r\ :\
\mbox{Prob}(\frac{p_1\bar{z}_1(1+\delta \tilde{z}_1)}{\sum_{i=2}^np_i\bar{z}_i(1+\delta \tilde{z}_i)+\sigma^2}<r)\leq \varepsilon
\}.
\end{equation}
Then $r_p$ is the unique positive root of the nonlinear equation
\begin{equation}\label{root2}
\delta \ln(1-\varepsilon)+\frac{r(\sigma^2+\sum_{i=2}^np_i\bar{z}_i)-p_1\bar{z}_1}{\bar{z}_1p_1}+\delta
\sum_{i=2}^n\ln(1+\frac{r\bar{z}_ip_i}{\bar{z}_1p_1})=0.
\end{equation}
\end{myth}
{\it Proof: \ } Using (\ref{ap7}) yields
\begin{eqnarray}
\mbox{Prob}(\frac{p_1\bar{z}_1(1+\delta \tilde{z}_1)}{\sum_{i=2}^np_i\bar{z}_i(1+\delta \tilde{z}_i)+\sigma^2}<r)\leq \varepsilon
&\Leftrightarrow&\nonumber\\
\ln(1-\varepsilon)+\frac{r(\sigma^2+\sum_{i=2}^np_i\bar{z}_i)-p_1\bar{z}_1}{\bar{z}_1p_1\delta}+
\sum_{i=2}^n\ln(1+\frac{r\bar{z}_ip_i}{\bar{z}_1p_1})\leq 0&\Leftrightarrow\label{ap9}\\
\delta \ln(1-\varepsilon)+\frac{r(\sigma^2+\sum_{i=2}^np_i\bar{z}_i)-p_1\bar{z}_1}{\bar{z}_1p_1}+\delta
\sum_{i=2}^n\ln(1+\frac{r\bar{z}_ip_i}{\bar{z}_1p_1})\leq
0.\label{ap10}&&
\end{eqnarray}
Again, by noticing that the function in the LHS of (\ref{ap10}) is increasing in $r$ we arrive at (\ref{root2}).
\qed

One can see that for  $\delta\rightarrow 0$ (less uncertainty), (\ref{ap10}) becomes
\[
\begin{array}{lll}
\displaystyle\frac{r(\sigma^2+\sum_{i=2}^np_i\bar{z}_i)-p_1\bar{z}_1}{\bar{z}_1p_1}\leq 0
&\Leftrightarrow&r(\sigma^2+\sum_{i=2}^np_i\bar{z}_i)-p_1\bar{z}_1\leq 0\\
&\Leftrightarrow&r \leq \displaystyle\frac{p_1\bar{z}_1}{\sigma^2+\sum_{i=2}^np_i\bar{z}_i},
\end{array}
\]
so $r_p$ is the standard ratio
\[
\displaystyle\frac{p_1\bar{z}_1}{\sigma^2+\sum_{i=2}^np_i\bar{z}_i}.
\]
\section*{Appendix II: fundamental inequalities}
\begin{mylem}\label{baslem}It is true that
\begin{equation}\label{lem1}
\ln(1+1/t)\geq 1/(t+1) \quad\forall\ t>0
\end{equation}
\end{mylem}
{\it Proof: \ }  One can easily  check $(t+1)\ln(1+1/t)\geq 1\ \forall\ t>0$ by
plotting the graph of function $(t+1)\ln(1+1/t)$ over $(0,+\infty)$.\qed

\begin{myth}\label{fundth} The function  $f(x,y,t)\triangleq \ln(1+1/xy)^{1/t}$ is convex in the domain
$\{ x>0, y>0, t>0\}$.
\end{myth}
{\it Proof: \ }  Writing $f(x,y,t)=(1/t)(\ln(xy+1)-\ln x-\ln y)$, it is ease to see that the Hessian $\nabla^2 f(x,y,t)$ is
\begin{eqnarray}
\nabla^2 f(x,y,t)&=&
\begin{bmatrix}
\displaystyle\frac{2xy+1}{x^2(xy+1)^2t}&\displaystyle\frac{1}{(xy+1)^2t}&\displaystyle\frac{1}{t^2(xy+1)x}\cr
\displaystyle\frac{1}{(xy+1)^2t}&\displaystyle\frac{2xy+1}{y^2(xy+1)^2t}&\displaystyle\frac{1}{t^2(xy+1)y}\cr
\displaystyle\frac{1}{t^2(xy+1)x}&\displaystyle\frac{1}{t^2(xy+1)y}&\displaystyle\frac{2\ln(1+1/xy)}{t^3}
\end{bmatrix}\nonumber\\
&\succeq&
(x^2y^2(xy+1)^2t^3)^{-1}\begin{bmatrix}
 (xy+1)y^2t^2&x^2y^2t^2&t(xy+1)xy^2\cr
x^2y^2t^2&(xy+1)x^2t^2&t(xy+1)x^2y\cr
t(xy+1)xy^2&t(xy+1)x^2y&2(xy+1)x^2y^2
\end{bmatrix},\label{mat1}
\end{eqnarray}
where the inequality (\ref{lem1}) has been applied to the $(3,3)$-th entry of $\nabla^2 f(x,y,t)$ to
arrive the matrix inequality in (\ref{mat1}).  Here and after,
${\cal A}\succeq {\cal B}$ for matrices ${\cal A}$ and ${\cal B}$ means that ${\cal A}-{\cal B}$ is
a positive definite matrix.
Then, calculating the subdeterminants of matrix in the right hand side (RHS) of (\ref{mat1}) yields
$(xy+1)y^2t^2>0$,
\[
\left|\begin{array}{cc}
\displaystyle\frac{1}{x^2(xy+1)t}&\displaystyle\frac{1}{(xy+1)^2t}\cr
\displaystyle\frac{1}{(xy+1)^2t}&\displaystyle\frac{1}{y^2(xy+1)t}
\end{array}\right|=x^2y^2t^4(2xy+1)>0
\]
and
\[
\left|\begin{array}{ccc}
(xy+1)y^2t^2&x^2y^2t^2&t(xy+1)xy^2\cr
x^2y^2t^2&(xy+1)x^2t^2&t(xy+1)x^2y\cr
t(xy+1)xy^2&t(xy+1)x^2y&2(xy+1)x^2y^2
\end{array}\right|=x^4y^4t^4(xy+1)[(xy+1)^3-1]>0.
\]
Therefore the matrix in the RHS of (\ref{mat1}) is positive definite, implying that the Hessian $\nabla^2 f(x,y,t)$
is positive definite, which is the necessary and sufficient condition for the convexity of $f$ \cite{Tuybook}.\qed

As the function $f(x,y)\triangleq \ln(1+1/xy)$ is convex in the domain $\{x>0, y>0\}$ it follows that \cite{Tuybook}
for every $x>0$, $y>0$, $\bar{x}>0$ and $\bar{y}>0$,
\begin{eqnarray}
\displaystyle\ln(1+1/xy)&=&f(x,y)\nonumber\\
&\geq&f(\bar{x},\bar{y})+\langle \nabla f(\bar{x},\bar{y}), (x,y)-(\bar{x},\bar{y})\rangle\nonumber\\
&=&\displaystyle\ln(1+1/\bar{x}\bar{y})+\displaystyle\frac{1/\bar{x}\bar{y}}{1+1/\bar{x}\bar{y}}(2-x/\bar{x}-
y/\bar{y}).
\label{sec5.2}
\end{eqnarray}
Similarly, for the convex function $f(x,y,t)\triangleq \ln(1+1/xy)^{1/t}$, one has the following
inequality for every $x>0$, $y>0$, $t>0$, $\bar{x}>0$, $\bar{y}>0$ and $\bar{t}>0$,
\begin{eqnarray}
\displaystyle\ds\frac{\ln(1+1/xy)}{t}&=&f(x,y,t)\nonumber\\
&\geq&f(\bar{x},\bar{y},\bar{t})+\langle \nabla f(\bar{x},\bar{y},\bar{t}), (x,y,t)-(\bar{x},\bar{y},\bar{t})\rangle\nonumber\\
&=&\displaystyle\ds\frac{2\ln(1+1/\bar{x}\bar{y})}{\bar{t}}
+\displaystyle\frac{1/\bar{x}\bar{y}}{\bar{t}(1+1/\bar{x}\bar{y})}(2-x/\bar{x}-y/\bar{y})
-\displaystyle\frac{\ln(1+1/\bar{x}\bar{y})}{\bar{t}^2}t
\label{esec5.2}
\end{eqnarray}
Analogously, the  inequality
\begin{eqnarray}
\displaystyle\frac{-\ln (1+x)}{t}\geq 2\displaystyle\frac{\alpha-\ln(1+\bar{x})}{\bar{t}}+\displaystyle\frac{\bar{x}}{\bar{t}(1+\bar{x})}-\displaystyle\frac{x}{\bar{t}(1+\bar{x})}
-\displaystyle\frac{\alpha-\ln(1+\bar{x})}{\bar{t}^2}t-\displaystyle\frac{\alpha}{t}\nonumber\\
\forall\ 0\leq x\leq M,  \alpha\geq \ln(1+M)+0.5 \label{nin1}
\end{eqnarray}
follows from the convexity of function $\displaystyle\frac{\alpha-\ln(1+x)}{t}$ over the trust region $0\leq x\leq M$.

Lastly, the inequality
\begin{equation}\label{conv1}
\displaystyle\ln (1+x/y)\leq \ln(1+\bar{x}/\bar{y})+\displaystyle\frac{1}{1+\bar{x}/\bar{y}}
(0.5(x^2/\bar{x}+\bar{x})/y-\bar{x}/\bar{y})
\end{equation}
follows from the concavity of function $\ln(1+z)$ and then the inequality
\begin{eqnarray}\label{conv3}
x&=&0.5(x^2/\bar{x}+\bar{x})-0.5(x-\bar{x})^2/\bar{x}\nonumber\\
&\leq& 0.5(x^2/\bar{x}+\bar{x}) \quad\forall x>0, \bar{x}>0.
\end{eqnarray}
\vspace*{-1.0cm}
\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv


"We don't want fire safety to take a back seat, this year's message is if you hear the beep, move your feet, you know, have a smoke detector in every bedroom." - Ken Fortier, Vestal Fire Prevention Officer. Ken Fortier is Vestal's Fire Safety officer, today he was at African Road Elementary School to teach students what to do in case of fire. A couple important tips that students learned were to always have a family meeting ground, always check the batteries in your smoke detectors and to never be afraid of fire fighters.
Students got to see fire fighting gear, trucks and even meet the department's mascot, spot. The firefighters do a great job at helping the kids to be engaged in the program, they really enjoy the day and now they're having a little trip on the fire buses with them so we're happy to have them."( - Megan Stenta, principal of African Road Elementary. Making for a fun day, and a valuable lesson.
"If there's a fire, call 9-1-1 if they're hurt." - Devan Uppal, student.


text:
Books in my library
Facilitation Resources
Open Space Resources
Planning an Open Space Technology Meeting
Books and Papers
CV and Client list
The synchronicity of old connections
March 4, 2008 By Chris Corrigan Being, CoHo, First Nations 5 Comments
I was listening to The Current on CBC Radio this morning and I caught an interview with Marlene Brant-Castellano on the newly announced Truth and Reconciliation Commission on Residential Schools (hear the interview here). Back in the mid 1980s when I was at Trent University, Marlene was a professor in the Native Studies Department. She was a beautiful teacher – quiet and inviting and embodying tremendous dignity and powerful conviction all at the same time. I connected with her quite deeply as I began to explore questions of culture and community. It was in her classes that I was introduced to systems theory, feedback loops and dynamic living communities and cultures. She intriduced me to indigenous philosophy and showed me that there was an incredibly deep a valid intrepretation of the world that had been born and nurtured in North America. Not every way of thinking about things needed to come from elsewhere.
In large part due to her unwavering commitment to this way of thinking, I wrote some amazing papers under her guidance and my learning opened up and expanded. Three major papers stand out: I wrote a paper on the dynamics of culture change in communities, I wrote one with my friend Gary Heuval on hunting as a negotiating strategy for the James Bay Cree in the 1970s and I wrote one looking at themes of connection and interdependence in Native American and First Nations poetry. That last one included a meditation on Duan BigEagle's poem "My Grandfather was a Quantum Physicist", interestingly enough. All of this led up to my developing a particular worldview that culminated in my honours thesis, a piece of original research under David Newhouse that developed a multi-disciplinary process for understanding organizational culture in Native organizations. That was what was possible at the undergraduate level at Trent in the 1980s – I think things have changed now. Marlene was an incredibly influential person on my life.
And so it was delightful to hear her this morning, even if it was only out of the side of one ear. What I did catch though was a short story she told about the phycisist David Bohm meeting with indigenous scientists in Banff in the early 1990s, shortly before his death. I had no idea that Bohm had done this. I knew of course of his meetings with Krishnamurti and his cross-cultural examinations of consciousness and dialogue, but I had no idea he had extended that inquiry into indigenous America.
So I Googled the event and came across a report from Dan Moonhawk Alford, a linguist who was at one of the meetings, and who subsequently participated in two other Bohmian dialogues. He did some interviews with some of the partiicpants, including Sakej Henderson, who I have since worked with. Here is a long quote from the report on the eight things everyone agreed on:
1. Everything that exists vibrates
This point of agreement is important because it moves beyond our usual 'thingy' or particle notion of existence based on raw sensory impressions, which is favored in the indo-european language family, and allows a justification on the part of Native Americans for the existence of spirits.
2. Everything is in flux
(Sa'ke'j:) The only constant is change–constant change, transformations; everything naturally friendly, trying to reach a more stable state instead of bullying each other around. That kind of process the English language doesn't allow you to talk about too much, but most Native American languages are based on capturing the motions of nature, the rhythms, the vibrations, the relationships, that you can form with all these elements, just like a periodic table in a different way: relationships rather than a game of billiards, where you only count the ones that go in–all of their motion doesn't count.
3. The Part Enfolds the Whole:
(not just whole is more than the sum of its parts)
(Sa'ke'j:) When we wear leathers and beads and eagle thongs and things like that, it's not seen as totally ludicrous, as decoration – it's seen as containing something you want to have a relationship with.
4. There is an implicate order to the universe
(Sa'ke'j:) This implicate order holds everything together whether we want it to or not, and exists independently of our beliefs, our perceptions, or our linguistic categories. It exists totally independently of the methods or rules that people use to arrive at what it is, and David Bohm's captured that with the great phrase the implicate order, versus the explicate order of things that they can explain quite concretely, such as a rock falling out of a window. This also agrees with the lakhota phrase 'skan skan,' which points to the motion behind the motion.
5. This ecosphere is basically friendly
Sa'ke'j maintains that the planet, and especially the Americas as well as the physical universe, are basically gentle and friendly: You don't have an electron jumping and bullying into other(s) unless it knows it's missing a stable state and knows it can reach that stable state and increase its own stability.
6. Nature can be taught new tricks
(Sa'ke'j:) We also agreed that that world out there that exists–that reality, not imaginality–can be taught new tricks with the cyclotron; and what was raised in the meeting was, are these new tricks beneficial, or will they create a hostile universe on their own, independent of scientists, once they teach electrons how to jump and how to amass the energy to jump, and it becomes a bullying, hostile biological world.Reminds me of Alan Watts talking about how the universe has had to learn how to get ever smaller and ever larger as we probe it with microscopes and telescopes, receding ever further in the distance as self observes itself.
7. Quantum Potential and Spirit
After listening to the physicists and American Indians talk for a few days, it struck me that the way physicists use the term potential, or quantum potential, is nearly identical to the way Native Americans use the term spirit. They all agreed there was something similar going on.
8. The principle of complementarity
Physicists for all this century have realized that our usual notion of bipolar or black & white opposites was insufficient when working with nature. The first clue came when they asked incoming light, 'Are you particle?' and it answered Yes; 'Are you wave?' and it answered Yes. This is equivalent to asking whether something is a noun or a verb and getting a yes answer to both–which is exactly how Native American language nouns are made up: as verbs with suffixes that make them temporarily into nouns for discussion sake. this yes-yes complementarity is foreign to Indo-European languages, but quite common in other language families (such as the Chinese notion of Yin-Yang), and represents a higher level of formal operations, in Piaget's terms, referred to by some as post-formal operations–that which lies beyond normal Western Indo-European development.
There is much more at Alford's archive of papers and notes. This is really a rather remarkable find for me – all the more so in that it came to me from the mouth of my first academic teacher 22 years after I first met her. I would love to be in touch with others who were at that meeting or who have more substantial artifacts of the gathering.
The mother map
Ontario Mininster of Aboriginal Affairs on YouTube
jeff aitken says:
schweeet. Moonhawk was on the committee for Nancy Maryboy and David Begay's collaborative dissertation (see my earlier comment.)
The dialogue that Moonhawk and Sakej and Bohm were involved in continues at SEED Graduate Institute in Albuquerque, every August. It's called The Language of Spirit Conference, and there are transcripts and tapes available thru SEED, where I worked in 2003-04.
Leroy Little Bear (facilitator) and Glenn Parry (president of SEED) collaborate to invite new people every year as well as many who have been involved for years. And it's open to the public (for conference fee) to sit around the edges of the room and listen in for three days.
see http://www.seedgraduateinstitute.org, or search
Chris Corrigan says:
Jeff…you a-rock my world friend. I thought you might visit in on this one. Thanks!
Linked at that site (within the invitation to this August's conference) is a chapter of Glenn Aparicio Parry's forthcoming dissertation about dialogue, with some of the story of the 1992 dialogue.
http://www.seedgraduateinstitute.org/conferences/language/ChapterFour-Dialogue.pdf
Table 1: (Participants in the 1992 Historic Dialogue)
Leroy Little Bear, Blood Indian Tribe, Blackfoot Confederacy
Amethyst First Rider, Blood Indian Tribe, Blackfoot Confederacy
David Bohm, physicist,
Saral Bohm, wife of David
Henry Bush – Potowatomi
Lou Ann Bush – Potowatomi
Joe Couture – Cree psychologist
Ruth Couture – Cree
Carol Hegedus – Fetzer Institute
Alan Ford, linguist studying Cree
Sam Kounosu, a Japanese physicist
Danny Musqua – Seauteau elder
Buff Parry – linguist
Lee Nichol – writer and editor of Bohm"™s work
Paul Grof – psychiatrist
Dick Katz – psychologist
Tobasonakwut Peter Kinew – Ojibway elder
Anab Whitehouse – Sufi philosopher
Joel Elkes – medical anthropologist
Therese Schroeder-Sheker – thanatologist/harpist – aka "The Angel."
merln wood says:
I say David Bohm's quantum potential definitely is the spirit in quantum physics.
Trouble is, it seems, most physicists either ignore it or try to interpret or calculate it out of existence.
I call this potential the universal action of a form conserving cause.
Pam Matchie says:
Having just finished Louise Erdrich's "Shadow Tag", I was struck with the concept of "Complementarity" (Concept #8 in this article) but explained to his sister by Irene & Gil's son, Florian.
Thank you for this posting. I knew our frequent lack of common understanding was really just a language barrier.
Find Interesting Things
The Art of Hosting Meaningful Conversations and Participatory Leadership, November 2020, Bowen Island, BC Canada with Caitlin Frost, Amanda Fenton, and Kelly Poirier
Complexity from the Inside Out with Caitlin Frost, June 4-7, 2020, Bowen Island, Canada (Preregistration now open)
Hosting Powerful Conversations: Introduction to World Cafe and Open Space, January 17, 2020, Vancouver, Canada
A list of books in my library
© 2015 Chris Corrigan. All rights reserved. | Site by Square Wave Studio

redpajama_set_name: RedPajamaCommonCrawl


Against the fading grandeur of contemporary Havana, Jesus (Héctor Medina), an impoverished, sensitive young make-up assistant escapes the grim reality of everyday life through his backstage work in a drag club. When his dream of performing finally comes true, a stranger emerges from the crowd and punches him in the face. The stranger is his father, Angel (Jorge Perugorría), a boxer, absent for many years and now in need of a place to live. They struggle to cohabit peacefully and understand each other's very different expectations.
for director Paddy Breathnach (I Went Down, Man About Dog) and writer Mark O'Halloran (Adam and Paul, Garage).
We are delighted to be joined by director Paddy Breathnach and writer Mark O'Halloran who will participate in a Q&A after the 18.30 screening on August 19th.


Our companion Adam Hincks, SJ, a Ph.D. in physics from prestigious Princeton University, New Jersey, is a fascinating and puzzling person. While he was destined for a brilliant career as an astrophysicist at a renowned university or astronomical observatory, he rather felt... a powerful call for religious life. He then embarked on a discernement process at the end of which he joined the Society of Jesus and the Jesuit Province of Canada. He then completed part of his philosophical and theological training at Regis College in Toronto, where he was particularly interested in the cosmology of another Jesuit and scientist : none other than Pierre Teilhard de Chardin, SJ. Since then, Adam has pursued a brilliant academic career at the no less prestigious Gregorian University in Rome.
On October 9th, he was visiting his alma mater Regis College, where he gave a lecture on the contribution of Jesuits to the development of science. He then granted an interview to journalist Michael Swan of Toronto's Catholic Register about the porous boundaries between faith and reason, as well as between his vocation as a Jesuit and his vocation as a scientist.
"I never considered that I left science to become a Jesuit," Hincks told about 40 people gathered at Toronto's Regis College for an evening talk about Jesuit contributions to science. More details in The Catholic Register.


Guide to and directory of, Al-Anon & Alateen, in Kentucky.
Area 21 (Kentucky & Southern Indian) : Al-Anon - Offers statewide meeting schedules.
Kentucky & Southern Indiana (Area 21) : Al-Anon - Offers statewide meeting schedules.
Lexington (Greater) : Al-Anon - Greater Lexington Al-Anon Info Service (859) 277-1877. Lexington, Fayette County, Kentucky.
Tri-State Convention - Annual Alcoholics Anonymous (AA) event with Al-Anon & Alateen participation. 53rd Annual. November 8, 9, & 10, 2013. Kentucky Dam Village State Resort Park. 113 Administration Drive. Gilbertsville, Marshall County, Kentucky.


text:
UCSD Announces Shift to Online Learning for First Two Weeks of Winter Quarter
January 2, 2022 January 2, 2022 Tianrui Huang
On Dec. 21, UC San Diego Chancellor Pradeep Khosla announced in a campus operations update that instruction will be remote-only from Jan. 3 to Jan. 17, 2022, in anticipation of a surge in COVID-19 cases that may coincide with the planned start of Winter Quarter. In-person instruction is scheduled to return on Tuesday. Jan.18 during Week 3, but is subject to change depending on COVID-19 infection rates.
The Omicron variant, first detected in South Africa, has swept through the United States. According to the New York Times, the number of new daily cases in New York was up to 55,000 as of Dec. 27.
A similar rise occurred in California, as new cases rose to a high of 20,000 on Dec. 26, a 5,000 person increase since Dec. 24. The average number of daily new cases rose to 1,400 in San Diego reported on Dec. 24, while on Dec. 14 the average case was only half that number, reaching a total of 700.
According to the Return to Learn webpage, faculty and teaching assistants may teach remotely from home or from campus during the online instruction period. Research will continue in person, but undergraduate and graduate students will not be required to report to campus research sites in person during the online instruction period.
During this time campus will be incrementally populated using a more comprehensive testing regime. Additional steps have been implemented to increase safety as students return to campus.
As a measure of caution against the new variant, UCSD is mandating that all students test on the first and the fifth day of check-in. Individuals who are not vaccinated and individuals who are vaccinated but have not received a vaccine booster will be required to test for SARS-CoV-2 two times per week, between three and five days apart.
Housing Dining and Hospitality (HDH) Executive Director Hemlata Jhaveri and Student Health and Well-Being Interim Executive Director Dr. Angela Scioscia sent a student residential update on Dec. 29, in which they strongly recommend testing within 48 hours of arrival on campus and outlined procedures for the return of students to on-campus housing.
Students are recommended to report their planned date of arrival via Housing Portal, so that Student Health Services can better deal with the repopulation process. UCSD has offered incentives encouraging residential students to put off their arrival to campus. Students who arrive between Jan. 8–14 will receive a $325 credit, while those who arrive between Jan. 15–22 will receive a $650 credit. Credits will be applied to resident housing accounts by the end of Winter Quarter.
24 hours prior to their arrival, residential students who return to campus after Jan. 7 are required to self-administer rapid antigen tests.. Students who traveled over the break within the United States can complete this form before Jan. 9 to request that a rapid test be mailed to them.
Students should take a picture of their negative results before moving into campus. If the test result is positive, students must complete their isolation before they are allowed to return to campus. These students would need to reach out to Student Health Services either through email ([email protected]) or sending messages through MyStudentChart), so that they can receive enough medical guidance and additional support from college.
All residential students are also required to complete a PCR test upon arrival. According to the email, returning residents would first need to check in at the PCR test vending machine located in their residential community. Triton Health Ambassadors (THAs), who will be present from 8 am to 5 pm, will make sure if resident students have arrived and completed the PCR test. They will also provide KN95 masks that resident students should wear until students receive their PCR test results.
These cautionary measures in campus operations and housing are subject to change if needed. If students have any questions regarding the test or process, they are encouraged to email UCSD Triton Health Ambassadors at [email protected] or visit the Return to Learn website.
Photo taken by August de Richelieu for Pexels
PCR testing remote learning Return to Learn Triton Health Ambassadors UCSD
A&E's 2021 Favorites
Film Review: "Don't Look Up"
2 thoughts on "UCSD Announces Shift to Online Learning for First Two Weeks of Winter Quarter"
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Katelin says:
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A Trellis Hoop For Any Planter!!
When these willow cages are set into terracotta pots and planted with colorful flowers, they provide a focal point all season long in your garden or patio areas. This cage takes about 2 hrs to make, and all the materials are easy to come by. Give it a go and let us know how you did.
Pencil thick willow switches make up the hoops. You will need two 42" long, four 36" long and four 26" long. The cage looks best when it is a bit taller than the pot it sits in. The measurements here are for a 11" terracotta clay azalea pot, which is wider than tall. It also has nearly vertical walls, which help the cage sit in it snugly.
Peel the leaves and snip off the side branches from the switches. Cut the two best looking 42" long for the main hoops. Cut four thick switches 36" for the middle and found 26" for the lower. Mark the inside of the pot at the 12 o'clock spot and again at 3,6 & 9 o'clock spots. Put the two main hoops inside the pot at right angels to each other so that each end sticks 3" into the pot. Use a twist tie to hold the marked centers together. You can tape the inside of the pots to hold them together.
Then curve one of the middle hoops and place the ends on each side of one of the main hoops. The top of the curve should rest about 1/3 of the way down outside the main hoop.
The lower hoops are placed in the same way. To make the cage more stable weave the lower hoops in front of and behind the middle and main hoops where you can. Fasten the remaining intersections with twist ties.
Traditionally these cages are planted with carnations but many different types of plant materials look good in them. Try to select plants that bloom for a long time or that have nice foliage over several seasons. You can always plant bulbs in them and they look fantastic.
Place the pots on a patio or deck, or even in a mixed garden border. Stick them right into the bed. These are easy to do and lovely once made.


Escape artist and illusionist Adam Cardone has always been a showman.
"I've been an extrovert in my home life," he says, "so why be any different in my professional life?"
Make that lives for this Queens multitasker.
Tomorrow, he's on the bill in "The Olde Time Coney Island Strongman Spectacular," a free event on W. 12th St. and the Bowery in Brooklyn. His act starts at 1:45 p.m. Inspired by Harry Houdini, he'll attempt to free himself from an old-fashioned milk can filled to the brim with water.
And through July 31, he's a featured performer in "House of Ghostly Haunts," a Tuesday night spook show at the Canal Park Playhouse (508 Canal St.). Tickets: $20; (866) 811-4111.
The offbeat shows are par for the course for Cardone, who studied acting at Carnegie Mellon University but found straight theatrical roles trying.
"For me, an artist takes reality and turns it into fantasy," he says. "A magician takes fantasy and turns it into reality.
"If you type 'milk can escape' into Google," he continues, "it immediately comes up with the secret to the trick. I start my show by telling the audience Houdini's way."
He goes on to explain that his act is no illusion and that the dangers of the escape are nothing if not genuine. The straitjacket is real, the locks are real, and the possibility of it all going very wrong is the most real of all.
No wonder his show comes with a tagline: "A claustrophobic hell. A watery grave awaits."


"I want to offer a few words regarding John William Pye, member of the International Catacomb Society Board of Directors, who was tragically and unexpectedly taken from us in the summer of 2016.
John was one of the best friends I've ever had. I met him at Trinity College when we were students. He was a year ahead of me. After he graduated in 1970, he attended what was then known as ETS, the Episcopal Theological Seminary, now the Episcopal Divinity School, where he received a Master's in Divinity (M.Div). He was very much devoted to his faith, and was deeply involved in all the church venues he attended, whether the Old North Church in Boston, or parishes in Weymouth, Quincy, or on Cape Cod. Originally, he was involved primarily in outreach work: in college, he ran a program called "Give a Hand to a Boy." Thereafter, for a number of years, he ran the Quincy Boys' Home, and stewarded many of his protégées out of the perilous and apparently hopeless environments which had previously fostered them, to placements in colleges or other professional training institutions, all done with a firm but generous and caring guidance, and with unique perception and skill which can only be classified as talent, even genius.
Ticknor and Fields payment to author Celia Thaxter (Trinity College Library Collection, gift of John W. Pye).
It was during this time relatively early in his adult career that John intensified his book collecting interest and in the fashion which was typical of what he did, built up out of a very tight budget the second largest collection of Ticknor and Fields imprints in the world. Ticknor and Fields as the Boston publishing house which published the first American editions of Charles Dickens, as well as, inter alia, Longfellow, Thoreau, Hawthorne, Alcott and Poe. John's collection included royalty checks signed by Hawthorne and Poe, galley sheets from Longfellow's "Evangeline" with the author's handwritten corrections, and first editions of everything from Thoreau's Walden to Hawthorne's The Scarlet Letter, wherein the author insisted that the title of the book be printed in red ink! John was also the foremost collector of Edward Arlington Robinson. That collection had started while he was still in college. With all these treasures in his possession, John became a major book dealer and vital presence in the annual Boston Antiquarian Book Fair, as well as a core member of such groups as the Massachusetts and Rhode Island Antiquarian Book Dealers Association.
From the International Catacomb Society perspective, we have lost a board member who brought with him a tremendous wealth of practical experience on diverse boards of both religious and lay outfits, of dealing with people of every background and persuasion, and, moreover, someone ready to lend his voice and strong opinions when needed during our meetings. From my own viewpoint, I have lost a dear friend, and ask for a moment of silence in his honor and memory.
John William Pye obituary (link).