\Huge Joint Optimization of User Association, Data Delivery Rate and Precoding for Cache-Enabled F-RANs
\IEEEauthorblockN{Tung T. Vu\IEEEauthorrefmark{1
%\IEEEauthorblockN{Author 1 and Author 2
%User-Base-Station (UE-BS) association, base-stations (BSs) on/off switching and precoder design are promising ways of reducing consumption power in wireless networks. %In this paper, we propose to combine those three key methods for a fog radio access network (F-RAN) where each cache-enabled BS is connected to a centralized base band unit (BBU) via a limited frouthaul capacity. %Each BS is equipped with multiple antenna and a cache with finite storage size. %When UEs request their files, they are assigned to a group of BSs and these BSs cooperatively transmit requested files, either already in the local caches or fetched from the BBU, to these UEs. %For given cache placement, this paper considers the joint design of precoder, BS switching and UE-BS association by formulating an optimization problem of maximizing the network object function under quality-of-service (QoS) constraints. %The network objective function includes both the sum data rate and the total power cost. %We model the power cost using $\ell_0$-norm of precoder matrices. %To solve this highly non-convex problem, we first split it into separate and more amenable subproblems using alternating method. %We then develop a successive convex quadratic programming (SCQP) algorithm to obtain suboptimal solutions iteratively. %Simulation results indicate the advantage of our proposed sheme and show the effects of different fronthaul capacities in F-RANs. %This paper investigates on the aspects of caching and fronthaul capacity of cache-enabled fog random access networks (Fog-RANs), namely: how much does the network performance gain? %We tackle this issue by first formulating a practical problem of jointly designing user association, base-station (BS) activation and precoders. %Optimization algorithms based on successive convex quadratic programming (SCQP) are then proposed without rank relaxation as the different convex functions (DC) algorithms used in previous works. %With our proposed design, the gains provided from caching and the effect of fronthaul capacity on the network system are confirmed via simulation results on a standard LTE model. %As an example, it is observed that the larger fronthaul capacity provides equal or greater throughput-power-ratio (TPR) and caching can provides TPR gains of approximate $25$ times compared to no caching at the same fronthaul capacity. %Fog radio access network (F-RAN) architecture is expected to significantly enhance both network throughput and energy performance by exploiting local caching and centralized signal processing functionalities. %This paper proposes a joint design of user association, DDR and precoding for the downlink of a cache-enabled fog radio access network with limited fronthaul capacity. The objective is to improve the network throughput while reducing the total power consumption, subject to predefined DDR requirements and maximum transmit power at each evolved remote radio head (eRRH). This paper considers the downlink of a cache-enabled fog radio access network (F-RAN) with limited fronthaul capacity, where user association (UA), data delivery rate (DDR) and signal precoding are jointly optimized. % which improves the network throughput and reduces the total power consumption. We formulate a mixed-integer nonlinear programming problem in which the weighted difference of network throughput and total power consumption {is maximized, subject to {the predefined DDR requirements and {the maximum transmit power at each eRRH. To address this challenging problem, we first apply the $\ell_0$-norm approximation and $\ell_1$-norm minimization techniques to deal with the UA. After this key step, we arrive at an approximated problem that only involves {the joint optimization of DDR and precoding. By using the alternating descent method, we further decompose this problem into a convex subproblem for DDR allocation and a nonconvex subproblem for precoding design. While the former is globally solved by the interior-point method, the latter is solved by a specifically tailored successive convex quadratic programming method. %The nonconvex precoding subproblem is then solved by successive convex quadratic programming. Finally, we propose an iterative algorithm for the original joint optimization that is guaranteed to converge. %Each iteration of the developed algorithm only involves solving simple convex programs. %Numerical examples demonstrate the significant advantage of the proposed joint design. Importantly, each iteration of the developed algorithm only involves solving simple convex problems. Numerical examples demonstrate that the proposed design significantly improves both throughput and power performances, especially in practical F-RANs with limited fronthaul capacity. Compared to the sole precoder design for a given cache placement, our joint design is shown to improve the throughput by $50\%$ while saving at least half of the total power consumption in the considered examples.
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\begin{document}
%
{
\title{\Huge Joint Optimization of User Association, Data Delivery Rate and Precoding for Cache-Enabled F-RANs}
}
%
\author{\IEEEauthorblockN{Tung T. Vu\IEEEauthorrefmark{1}, Duy T. Ngo\IEEEauthorrefmark{1}, Lawrence Ong\IEEEauthorrefmark{1}, Salman Durrani\IEEEauthorrefmark{2} and Richard H. Middleton\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\small\IEEEauthorrefmark{1}School of Electrical Engineering and Computing, The University of Newcastle, Callaghan NSW 2308, Australia
\\ Email: thanhtung.vu@uon.edu.au, \{duy.ngo, lawrence.ong, richard.middleton\}@newcastle.edu.au}
\IEEEauthorblockA{\small\IEEEauthorrefmark{2}Research School of Engineering, The Australian National University, Canberra ACT 2601, Australia
\\ Email: salman.durrani@anu.edu.au}
%\author{
%\IEEEauthorblockN{Author 1 and Author 2}
%\IEEEauthorblockA{\small\\
% \\
%    }
%\thanks{This work is supported in part by an ECR-HDR scholarship from The University of Newcastle and in part by an Australian Research Council Discovery Project grant DP170100939.}
}
\maketitle
%=====================================================
\begin{abstract}
%User-Base-Station (UE-BS) association, base-stations (BSs) on/off switching and precoder design are promising ways of reducing consumption power in wireless networks.
%In this paper, we propose to combine those three key methods for a fog radio access network (F-RAN) where each cache-enabled BS is connected to a centralized base band unit (BBU) via a limited frouthaul capacity.
%Each BS is equipped with multiple antenna and a cache with finite storage size.
%When UEs request their files, they are assigned to a group of BSs and these BSs cooperatively transmit requested files, either already in the local caches or fetched from the BBU, to these UEs.
%For given cache placement, this paper considers the joint design of precoder, BS switching and UE-BS association by formulating an optimization problem of maximizing the network object function under quality-of-service (QoS) constraints.
%The network objective function includes both the sum data rate and the total power cost.
%We model the power cost using $\ell_0$-norm of precoder matrices.
%To solve this highly non-convex problem, we first split it into separate and more amenable subproblems using alternating method.
%We then develop a successive convex quadratic programming (SCQP) algorithm to obtain suboptimal solutions iteratively.
%Simulation results indicate the advantage of our proposed sheme and show the effects of different fronthaul capacities in F-RANs.
%This paper investigates on the aspects of caching and fronthaul capacity of cache-enabled fog random access networks (Fog-RANs), namely: how much does the network performance gain?
%We tackle this issue by first formulating a practical problem of jointly designing user association, base-station (BS) activation and precoders.
%Optimization algorithms based on successive convex quadratic programming (SCQP) are then proposed without rank relaxation as the different convex functions (DC) algorithms used in previous works.
%With our proposed design, the gains provided from caching and the effect of fronthaul capacity on the network system are confirmed via simulation results on a standard LTE model.
%As an example, it is observed that the larger fronthaul capacity provides equal or greater throughput-power-ratio (TPR) and caching can provides TPR gains of approximate $25$ times compared to no caching at the same fronthaul capacity.

%Fog radio access network (F-RAN) architecture is expected to significantly enhance both network throughput and energy performance by exploiting local caching and centralized signal processing functionalities.
%This paper proposes a joint design of user association, DDR and precoding for the downlink of a cache-enabled fog radio access network with limited fronthaul capacity. The objective is to improve the network throughput while reducing the total power consumption, subject to predefined DDR requirements and maximum transmit power at each evolved remote radio head (eRRH).

This paper considers the downlink of a cache-enabled fog radio access network (F-RAN) with limited fronthaul capacity, where user association (UA), data delivery rate (DDR) and signal precoding are jointly optimized.
% which improves the network throughput and reduces the total power consumption.
We formulate a mixed-integer nonlinear programming problem in which the weighted difference of network throughput and total power consumption {is} maximized, subject to {the} predefined DDR requirements and {the} maximum transmit power at each eRRH.
To address this challenging problem, we first apply the $\ell_0$-norm approximation and $\ell_1$-norm minimization techniques to deal with the UA.
After this key step, we arrive at an approximated problem that only involves {the} joint optimization of DDR and precoding.
By using the alternating descent method, we further decompose this problem into a convex subproblem for DDR allocation and a nonconvex subproblem for precoding design.
While the former is globally solved by the interior-point method, the latter is solved by a specifically tailored successive convex quadratic programming method.
%The nonconvex precoding subproblem is then solved by successive convex quadratic programming.
Finally, we propose an iterative algorithm for the original joint optimization that is guaranteed to converge.
%Each iteration of the developed algorithm only involves solving simple convex programs.
%Numerical examples demonstrate the significant advantage of the proposed joint design.
Importantly, each iteration of the developed algorithm only involves solving simple convex problems.
Numerical examples demonstrate that the proposed design significantly improves both throughput and power performances, especially in practical F-RANs with limited fronthaul capacity.
Compared to the sole precoder design for a given cache placement, our joint design is shown to improve the throughput by $50\%$ while saving at least half of the total power consumption in the considered examples.
\end{abstract}
%
%\emph{\textbf{Index Terms}}--\textbf{Interference alignment, multiuser MIMO, wiretap channel, energy efficiency.}
%\begin{keywords}
%Interference alignment, multiuser MIMO, cognitive radio, energy efficiency.
%\end{keywords}
\IEEEpeerreviewmaketitle
%=====================================================
\balance
\section{Introduction}
\label{sec:introd}
A fog radio access network (F-RAN) is recently proposed as an alternative to the cloud radio access network (C-RAN) to support mobile edge computing \cite{liu2017,Chiang2016,shi2015}.
% and extends the traditional cloud computing paradigm to the network edge \cite{Chiang2016}.
Capable of exploiting the advantages of both local caching and centralized signal processing, this novel network architecture is expected to significantly improve both spectral and energy efficiencies of the fifth-generation (5G) of cellular systems \cite{Peng2016}.
In an F-RAN, traditional high-cost high-power base stations (BSs) are replaced by low-cost low-power enhanced remote radio heads (eRRHs).
Equipped with a finite-storage cache, each eRRH is connected to a central base band unit (BBU) via a fronthaul link.
%If the users' requested files are available at the eRRHs' caches, they can now be retrived directly from the local cache of the serving eRRHs instead of the BBU.
If a user (UE) requests a file that is available at the local caches of its serving eRRHs, the file can be directly retrieved from the caches.
Otherwise, the file will be fetched from the BBU to the serving eRRHs before being transferred the UEs via radio access links.
%In this paper, we consider the downlink of the F-RAN with a practical constraint of the finite-capacity fronthaul links.
%We formulate and solve a network optimization problem with a practical constraint of the finite-capacity fronthaul links.

%This general F-RAN architecture is commonly used and was proposed in \cite{Park2016}.

%{The BBU in an F-RAN allows for joint signal processing and resource allocation, which is similar to cloud-RANs \cite{shi2015,liu2017}}.
%However, the deployment of local caching at {eRRHs} and the limited fronthaul capacity in the F-RAN would change the way that user (UE) scheduling, interference management and data delivery are traditionally conducted.% In such a large-scale network, the energy consumed by fronthaul links also needs to be taken into account \cite{Tombaz2011}, the cache-aware scheduling in F-RANs shows its potentials in greatly reducing the overall power consumption of a network.
%User association (UA) problems in F-RANs remain open to the best of our knowledge. For given cache placements in eRRHs, a main challenge is to decide which eRRH to be assigned to which UE. The aim is to improve the throughput by reducing the bottleneck in the limited-capacity fronthaul links and exploiting the cache as well as to save energy by turning unassigned eRRHs to sleep mode (See \cite{liu2016} and reference therein).

%To address the issue of limited fronthaul capacity,
In practical settings, the performance of an F-RAN is constrained by the capacity of its fronthaul links.
User association (UA) can help reduce the fronthaul traffic by assigning UEs to appropriate eRRHs and save power by putting the unassigned eRRHs into sleep mode \cite{liu2016}.
To minimize the fronthaul traffic and transmission power, the work of \cite{Tao2016} considers {the} joint design of multicast beamforming and BS clustering (which is essentially {a} UA problem) in cache-enabled C-RANs, where the BSs are assigned to the groups of UEs requesting the same file.
%Nevertheless, it should be noted that the aim of a F-RAN is not only minimizing the deployment and operating cost but also maximizing the system throughput.
%content-centric eRRH clustering is a possible approach which assigns each group of UEs requesting the same files to a cluster of eRRHs \cite{Tao2016}.
%Consider the downlink of a general F-RAN in Fig. \ref{fig:FRANModel}.
%, where each file is assumed to be split into several subfiles due to the finite-storage cache at each RRH.
To maximize the throughput, {the} joint design of data delivery rate (DDR) and precoding has been studied in \cite{Park2016}.
%This approach is equivalent to choosing which missing subfile to be transferred to associated eRRHs.
However, since \cite{Park2016} assigns the UEs to the eRRHs heuristically, it may not exploit the full potential of UA to enhance the system performance.
%If these UA issue are also taken into account, the main optimization problem in \cite{Park2016} becomes significantly more difficult due to the the binary nature of UA variables.
%Different from \cite{Park2016}, we use a default scheme that all missing subfiles are delivered to eRRHs and focus on choosing which UE to be associated to which eRRH.
%This approach requires more data to be transferred via fronthaul links but can achieve more gain from coherence combing and make the problem more amenable.
\vspace{-0mm}


In this paper, we jointly design UA, DDR and signal precoding for the downlink of a cache-enabled F-RAN with limited fronthaul capacity.
% in order to balance between the two objectives of improving network throughput and reducing the consuming power.
%We also answer the question: \emph{how much do we gain from the joint design in terms of system throughput and total power consumption?}
We aim to maximize the weighted difference between the network throughput and the total power consumption, the latter of which consists of the operating power and the transmission power in both fronthaul and radio access links.
The formulated optimization problem is constrained on meeting the predefined DDR requirements, the limited fronthaul capacity and the maximum transmit power at each eRRH.
In this mixed-integer nonlinear program,
%that involves three types of variables: (i) UA, (ii) DDRs and (iii) precoding matrices.
%%(which are additional variables to make the problem more tractable without lost of optimality).
the strong coupling among the optimizing variables makes it even more challenging to be solved globally.

Here, we first express the UA variables as the functions of the $\ell_0$-norm of {the} precoding matrices.
%To deal with the nonconvex $\ell_0$-norm, one approach is to approximate the $\ell_0$-norm with its smooth function \cite{Tao2016} but its performance highly depends the smoothness factor.
%Therefore, we use another approach that
We approximate this $\ell_0$-norm with its weighted $\ell_1$-norm \cite{Candes2008} and {update} the weight factor iteratively.
We then obtain an approximated problem in the variables of DDRs and precoding matrices only.
%An approximated problem is then obtained with only {the} variables of DDRs and precoders.
Next, we apply the alternating descent method in \cite{Bertsekas1999} {to decompose the approximated problem into a convex subproblem {for the DDR allocation} and a nonconvex subproblem for {the precoder design}.
%The latter nonconvex subproblem is then solved by successive convex quadratic programming \cite{Tam2016}.
%To cope with such nonconvexity, we employ successive convex quadratic programming \cite{Tam2016}.
The former is readily solved by a convex solver, whereas the latter is dealt with by the successive convex quadratic programming framework of \cite{Tam2016}.
Finally, we propose an iterative algorithm that is proved to converge once initialized from a feasible point.
Each iteration of the algorithm corresponds to solving simple convex programs.

In the numerical examples with practical parameter settings, the proposed design demonstrates its capability in substantially improving both throughput and power {performances}.
Compared to solely designing the precoders for a given cache placement, the developed joint design offers $50\%$ throughput gain while consuming only $50\%$ of the total power otherwise required.
The performance enhancement is particularly pronounced in cases with limited-capacity fronthaul links.

%\begin{itemize}
%\item We formulate the optimization problem as a weighted sum of the total network throughput and the total power consumption. The design problem is constrained on meeting predefined DDR requitements, limited fronthaul capacity and maximum transmit power at each eRRH. It is a nonconvex nonsmooth problem {involving three types of variables: (i) UA variable, (ii) precoding matrices and (iii) delivery rate allocations}. The strong coupling among the optimizing variables even make it more challenging {and difficult to be solved globally}.
%\item To solve the problem, we first utilize the iterative weighted $\ell_1$-norm minimization technique in \cite{Candes2008} {to absorb the binary variable of UE-eRRH association into an approximated problem with only variables of precoders and delivery rate}. Next, we apply the alternating descent method in \cite{Bertsekas1999} {to decouple this approximated problem into two separate sub-problems of data delivery allocations and precoding design} and solve them one at a time. While the former problem is convex for a fixed precoder design, the latter problem is unfortunately not so for a given DDR design. To cope with such nonconvexity, we employ successive convex quadratic programming which avoids the rank relaxation in the concave-convex procedures in \cite{Park2016,Tao2016}. We then propose an iterative algorithm that is proved to converge once initialized from a feasible point. Each iteration of the algorithm corresponds to solving simple convex programs.
%\item Our numerical examples with practical parameter settings demonstrate the capability of the proposed algorithm to assign UE to appropriate eRRHs and exploit local caches at the eRRHs to substantially improve both throughput and power performance. Compared to solely designing the precoders with local caching, the developed joint design can achieve an improvement to $150\%$ in terms of throughput while saving at least $50\%$ total power consumption. The performance enhancement is particularly pronounced in cases with limited-capacity fronthaul links.
%\end{itemize}
%Here, we formulate a new optimization problem of {jointly} maximizing the total network throughput and minimizing the total power consumption of a cache-enabled F-RAN with limited fronthaul capacity. The design problem is also constrained on meeting predefined DDR requitements and maximum transmit power at each eRRH. It is a nonconvex nonsmooth problem {involving three types of variables: (i) eRRH-UE association (which is binary), (ii) precoding matrices and (iii) delivery rate allocations}. The strong coupling among the optimizing variables even makes it more challenging {and difficult to be solved globally}.

%To solve the problem, we first utilize the iterative weighted $\ell_1$ minimization technique in \cite{Candes2008} {to absorb the binary variable of UE-eRRH association into an approximated problem with only variables of precoders and delivery rate}. Next, we apply the alternating descent method in \cite{Bertsekas1999} {to decouple this approximated problem into two separate sub-problems of data delivery allocations and precoding design} and solve them one at a time. While the former problem is convex for a fixed precoder design, the latter problem is unfortunately not so for a given DDR design. To cope with such nonconvexity, we employ successive convex quadratic programming. We then propose an iterative algorithm that is proved to converge {at least to a sub-optimum} once initialized from a feasible point. Each iteration of the algorithm corresponds to solving simple convex programs.

%In the numerical examples with practical parameter settings, the proposed algorithm demonstrates its capability to exploit local caches at the eRRHs to substantially improve both throughput and power performance. Compared to solely designing the precoders with local caching, the developed joint design doubles the throughput while saving at least $50\%$ total power consumption. The performance enhancement is particularly pronounced in cases with limited-capacity fronthaul links.

%The advantages of F-RAN brings insight into aspects of caching and fronthaul capacity, namely how much do we gain from caching and what is the effect of limited fronthaul capacity on the F-RAN performance?
%Assuming fixed cache placement at eRRHs, this paper propose a energy-efficient view for F-RAN design. A network optimization problem are formulated and solved while taking the balance between energy saving and system rate enhancement into account.

%To investigate the issues of caching and fronthaul capacity, the previous work \cite{Lou2015} on cloud radio access networks (C-RANs) considers the problem of energy minimization by jointly design downlink and uplink user-access-point association and beamforming vectors while \cite{Dai2016} reduces the consuming energy by switching on/off RHHs under user target rate constraints.
%Furthermore, \cite{Li2016} considers a stochastic optimization problem of joint RRH activation and beamforming in the downlink in C-RANs.
%However, all above works concentrate on the ideal fronthaul link or introduce a user-centric design without considering the advantage of the local cache in F-RANs and have not answered how caching and fronthaul capacity affect the network performance.
%Storing the requested file at the local cache at each eRRH in advance can help provide fast content access at user (UE) and improve user-perceived quality of experience.
%This motivates us to consider the effect of local cache on the energy performance of FRANs.
%Compared with networks without caching, a cache-enable system design can not only reduce the peak traffic but also requires less energy usage.
%It should be emphasized that the aim of F-RANs is not only to minimize the energy but also maximize the system performance in terms of delivery rate \cite{Park2016}.
%Motivated by the previous works which focused only on sum rate maximization \cite{Park2016} or only on power minimization \cite{Lou2015,Dai2016,Li2016}, this paper focuses on the problem of balancing between both these issues.

%In this paper, we tackle these issues by studying a joint and practical design of user association, BS activation and precoding, which differs from the fixed RRH switching in a multicast network \cite{Tao2016} or fixed user association in the downlink C-RAN \cite{Dai2016}.
%With predefined caching placement, an optimization problem is formulated of maximizing the objective function including sum data rate and the total power consumption under the quality-of-service (QoS) constraints.
%The binary variables of user association and BS activation making the problem nonconvex are approximated by their $\ell_0$ and $\ell_1$ norms.
%Unlike previous works using the different convex function (DC) algorithms, we then apply successive convex quadratic programming (SCQP) algorithm \cite{Tam2016} to solve the problem without rank relaxation.
%The simulation results illustrate that caching and our proposed algorithm can reach a throughput-power-ratio (TPR) gain of $25$ times compared to no caching at the same fronthaul capacities and also demonstrate the fronthaul capacity's role in the TPR improvement.


\emph{Notation:}
%$\pmb{X}$ and $\pmb{x}$ are respectively denoted as a complex matrix and a vector.
Boldfaced symbols are used for vectors and capitalized boldfaced symbols for matrices. $\pmb{X}^H$ is the conjugate transposition of a matrix $\pmb{X}$. $\pmb{I}$ and $\pmb{0}$ are the identity and zero matrix with the appropriate dimensions respectively. $||.||_0$ denotes the $\ell_0$-norm. $\langle\X\rangle$ means the trace of a matrix $\X$.

\section{System Model and Problem Formulation}
\label{sec:systemmodel}
Consider the general F-RAN model {\cite{Park2016}} illustrated in Fig.~\ref{fig:FRANModel}, where there are $K_U$ UEs capable of establishing wireless connections with $K_R$ RRHs. Each UE $k\in\K_U\triangleq\{1,\dots,K_U\}$ is equipped with $N_u$ antennas, whereas each eRRH $i \in \K_R\triangleq\{1,\dots,K_R\}$ with $N_r$ antennas. The eRRH $i\in \K_R$ connects to the baseband unit (BBU) in the core network via a fronthaul link of capacity $C_i>0$ (b/s). \vspace{-2mm}
%{In the following, we briefly introduce the data request, pre-fetching phase which is similar as \cite{Park2016} and then discuss the data delivery phase which is the focus of this paper in more details.}
\begin{figure}[t!]
\centering
\includegraphics[width=0.35\textwidth]{Model.eps}
\caption{Illustration of a general F-RAN}
\label{fig:FRANModel}
\end{figure}

%Over each transmission interval??,
\subsection{Data Request and Pre-fetching}
First, in the data request phase each UE $k\in\K_U$ requests a {random} file $f_k$ from the library $\FF\triangleq\{1,...,F\}$ stored in the BBU. Without loss of generality, assume that all files in the library are of the same size $S$ bits; hence, the total size of the file library is $SF$ bits. Denote by $\FF^{*}\triangleq\bigcup_{k\in\K_U}\{f_k\}$ the set of all the files requested by all $K_U$ users. To bring data contents closer to the UEs, each eRRH $i\in\K_R$ is equipped with a local cache that can store $B_i>0$ bits.
%

%Due to the limited caching capacity, each eRRH needs to decide which files to store by estimating the probability of a requested file using a popularity distribution.
%In this paper, we assume that the global file popularity is known and modeled as the Zipf distribution \cite{Park2016,Tao2016}. Specifically, the probability of file $f$ in order of popularity for $f\in\FF\triangleq\{1,...,F\}$ can be defined as
%\begin{align}\label{Pf}
%P(f)=cf^{-\phi},
%\end{align}
%where $\phi$ is a given popularity parameter and $c\geq 0$ is a constant and set such that $\sum_{f\in\FF}P(f)=1$.
%It should be noted that the popularity distribution becomes more skewed toward the most popular files when increasing $\phi$.
It is practical to assume that the eRRHs have limited storage capacity and therefore only a subset of the file library is cached at each eRRH. In this paper, we adopt the fractional cache distinct strategy \cite{Park2016}. Each file in the library is split into $M$ subfiles of equal size $\bar{S}=S/M$ bits. Each eRRH $i\in\K_R$ then randomly selects the set $\MM^i$ of $\lfloor \frac{B_i}{\bar{S}}\rfloor$ subfiles from the BBU file library to store in its local cache. {The cache state information, which shows whether the subfile $(f_k,m)$ is cached at eRRH $i$ or not}, $\ f_k\in\FF^{*}, \ m\in \MM \triangleq\{1,\dots,M\}$  is summarized as \vspace{-4mm}

{
\small
\begin{align}\label{CDset}
c_{f_k,m}^i &\triangleq
\left\{
\begin{array}{ll}
  1, & \text{if $(f_k,m)\in\MM^i$} \\
  0, & \text{otherwise}
\end{array}.
\right.
\end{align}
}
\vspace{-4mm}
%where $\MM^i=\{(f,m)\}$ is a set of $\lfloor \frac{BM}{S}\rfloor$ random subfiles chosen randomly from the BBU and stored at eRRH $i$, $\forall f\in\FF,m\in\MM$, where and $\MM\triangleq\{1,...,M\}$.
%
%Before moving to the next section, it should be noted that the pre-fetching optimization problem could be formulated and solved by stochastic optimization techniques \cite{Razaviyayn2016} but this challenging and open problem is out of the scope of this paper.
%\section{Problem Formulation
%%: Delivery Phase Design with Data-Sharing Fronthaul Mode
%}
%For a chosen pre-fetching strategy, this section considers the delivery phase design under data-sharing fronthaul (DSF) mode where the subfiles which are not cached in eRRHs are transfered from the BBU via fronthaul links. This mode was also investigated in \cite{Park2016,Tao2016} (and reference therein). Different with \cite{Park2016,Tao2016}, we study the eRRH clustering and user association to maximize energy efficiency under fronthaul capacity constraints.
%
%The DSF mode

With data pre-fetching, the files requested by the UEs can now be retrieved directly from the local cache of the serving eRRHs instead of from the BBU.
%The traffic burden on the fronthaul links could therefore be reduced in the later data delivery phase.
If the subfile $(f_k,m)$ requested by UE $k\in\K_U$ is not available at eRRH $i$'s local cache, $(f_k,m)$ will be fetched from the BBU via the fronthaul link.

In each data-request duration, the requested files and local caches are known.
The cache state information $c_{f_k,m}^i \ (f_k\in\FF^{*}, \ m\in \MM, \ i\in\K_R)$ is thus available at the BBU.
In the following, we focus on the data delivery phase from the BBU and/or the eRRHs to the UEs, where each optimization and transmission block (OTB) interval is assumed to be shorter than the data-request duration.
We do not optimize across multiple OTBs but instead only in one OTB.

\vspace{-3mm}

\subsection{Data Delivery}\label{ssec:data_deliver}
In the considered F-RAN, the central BBU allows each eRRH to serve multiple UEs and each UE to be served by multiple eRRHs. We model the associations of eRRHs and UEs by the following binary variables: \vspace{-5mm}

{
\small
\begin{align}\label{aselect}
a_{k,i} \triangleq
\left\{
\begin{array}{ll}
  1, & \text{if eRRH $i$ serves UE $k$} \\
  0, & \text{otherwise}
\end{array}
\right.
\end{align}
}
\vspace{-4mm}

Let $R_{f_k,m} \leq \bar{S}$ be the data delivery rate (DDR) of subfile $(f_k,m)$. Here, we allow $R_{f_k,m}$ to be transferred to UE $k\in\K_U$ in the considered OTB interval and leave $\bar{S}-R_{f_k,m}$ bits for the next OTB.
%\cite{Park2016}.

First, let us consider the data transmission from the BBU to the eRRHs.
At this step, the BBU needs to decide the set of associated eRRHs to which each missing requested subfile $(f_k,m)$ is transferred via the fronthaul links.
There are trade-offs between transferring all missing requested subfiles and only transferring selected subfiles to the associated eRRHs.
While the former requires more data to be transferred via fronthaul links, it can give more throughput gain as a result of coherence combining.
%Furthermore, the eRRHs does not have the requested files in their local cache and does not receive those files from the BBU should not be associated to users.
In this paper, we adopt the former approach. The total data rate on the fronthaul link of eRRH $i\in\K_R$ can therefore be expressed as: \vspace{-5mm}

{
\small
\begin{align}\label{FH:cons}
R_i^{FH}\triangleq\sum_{f_k\in\FF^{*}}a_{k,i}\sum_{m\in\MM}(1-c_{f_k,m}^i)R_{f_k,m}.
\end{align}
}\vspace{-3mm}
%which is required to not exceed $C_i$.

Next, we consider the data transmission from the eRRHs to the UEs. Let $\s_{f_k,m}\in\C^{d\times 1}$ denote the encoded baseband signal of subfile $(f_k,m)$ with $d$ data streams. Assume that $\s_{f_k,m}\sim\CN(\pmb{0},\pmb{I})$. The eRRH $i$ precodes $\s_{f_k,m}$ by a matrix $\F_{f_k,m}^i\in\C^{N_{r}\times d}$ to obtain the transmitted signal $\x_i=\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\F_{f_k,m}^i\s_{f_k,m}$.
%\begin{align}\label{xi}
%\x_i =
%%\sum_{f\in\FF}\sum_{m\in\MM}(1-\bar{c}_{f,m}^i)\F_{f,m}^i\s_{f,m},
%\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\F_{f_k,m}^i\s_{f_k,m},
%\end{align}
%where $\F_{f_k,m}^i\in\C^{N_{r}\times d}$ is the corresponding precoding matrix of $\s_{f_k,m}\in\C^{d\times 1}$ which is the encoded baseband signal of subfile $(f_k,m)$ with $d$ data streams and is distributed as $\s_{f_k,m}\sim\CN(\pmb{0},\pmb{I})$.
%Here, we assumed that $d_{f_k,m}=d,\forall f_k\in\FF^{*}, m\in\MM$ for ease of presentation.
Denote by $\HH_{k,i}\in\C^{N_{u}\times N_{r}}$ the flat-fading channel matrix from eRRH $i$ to UE $k$ and by $\HH_k\triangleq[\HH_{k,1},...,\HH_{k,K_R}]\in\C^{N_{u}\times N_R}$ the channel matrix from all eRRHs to UE $k$, where $N_R \triangleq K_RN_{r}$. Assume that channel states $\HH_{k,i}, \ k\in\K_U, \ i\in\K_R$ remain unchanged during each OTB and are made available to the BBU and eRRHs.

Define % $\bar{\F}_{f_k,m}\in\C^{N_R\times d}$ is the common precoder of the networks to serve subfile $m$ of the $k$-th UE's requested file and defined as
$\bar{\F}_{f_k,m}\triangleq\left[(\F_{f_k,m}^1)^H,
  (\F_{f_k,m}^2)^H,
  \dots
  (\F_{f_k,m}^{K_R})^H\right]^H \in\C^{N_R\times d}$.
%\begin{align}\label{tildeF}
%\bar{\F}_{f_k,m}\triangleq
%\left[
%\begin{array}{c}
%  \F_{f_k,m}^1 \\
%  \F_{f_k,m}^2 \\
%  ... \\
%  \F_{f_k,m}^{K_R}
%\end{array}
%\right]
%\end{align}
Note that since $\F_{f_k,m}^i = \pmb{0}$ when eRRH $i$ does not serve UE $k$. Then, the received signal $\y_k\in\C^{N_{u}\times 1}$ at UE $k$ for the requested file $f_k$ is: \vspace{-5mm}

{
\small
\begin{align}\label{yk:full}
\nonumber
\y_k \triangleq
&\HH_k\bar{\F}_{f_k,m}\s_{f_k,m}+\sum_{q\in\MM\setminus \{m\}}\HH_k\bar{\F}_{f_k,q}\s_{f_k,q}
\\
&+\sum_{f_\ell\in\FF^{*}\setminus \{f_k\}}\sum_{m\in\MM}\HH_k\bar{\F}_{f_\ell,m}\s_{f_\ell,m}
%\\
%&
+ \n_k,
\end{align}
}where $\n_k\sim\CN(\pmb{0},\SIGMA_k)$ is the additive noise term.


{To deal with the interference expressed in the second and third terms on the right-hand side of \eqref{yk:full}}, we assume each UE $k$ performs the successive interference cancellation (SIC) decoding for the subfiles with the order $\s_{f_k,1}\rightarrow...\rightarrow\s_{f_k,M}$.
After applying the SIC scheme, {the achievable data rate $R_{f_k,m}$ can be bounded as \cite{Park2016,Park2016SPL}}: \vspace{-5mm}
%explain why "bounded"?
%\begin{align}\label{R:bound}
%\nonumber
%&R_{f_k,m}
%\leq g_{f_k,m}(\tilde{\F})
%\\
%\nonumber
%&\triangleq
%\log
%\left|
%\begin{array}{c}
%  \sum_{q=m}^M\HH_k\DIG_{f_k,q}\tilde{\F}_{f_k,q}\tilde{\F}_{f_k,q}^H\DIG_{f_k,q}^H\HH_k^H + \\
%  \sum_{f_\ell\in\FF^{*}\setminus \{f_k\}}\sum_{q\in\MM}\HH_k\DIG_{f_\ell,q}\tilde{\F}_{f_\ell,q}\tilde{\F}_{f_\ell,q}^H\DIG_{f_\ell,q}^H\HH_k^H \\
%   + \SIGMA_k
%\end{array}
%\right|
%\\
%&-
%\log
%\left|
%\begin{array}{c}
%  \sum_{q=m+1}^M\HH_k\DIG_{f_k,q}\tilde{\F}_{f_k,q}\tilde{\F}_{f_k,q}^H\DIG_{f_k,q}^H\HH_k^H + \\
%  \sum_{f_\ell\in\FF^{*}\setminus \{f_k\}}\sum_{q\in\MM}\HH_k\DIG_{f_\ell,q}\tilde{\F}_{f_\ell,q}\tilde{\F}_{f_\ell,q}^H\DIG_{f_\ell,q}^H\HH_k^H \\
%   + \SIGMA_k
%\end{array}
%\right|.
%\end{align}

{
\small
\begin{align}\label{eq:R_ub}
R_{f_k,m}
\leq g_{f_k,m}(\bar{\F})
\triangleq
\log
\left|
\pmb{I}_{N_{U,k}}+\PI_{f_k,m}\PI_{f_k,m}^H\XI_{f_k,m}^{-1}
\right|,
\end{align}
}\vspace{-3mm}

\noindent
where $\bar{\F}\triangleq\{\bar{\F}_{f_\ell,m}\}_{f_\ell\in\FF^{*},m\in\MM}$,
$\PI_{f_k,m}\triangleq\HH_k\bar{\F}_{f_k,m}$,
and \vspace{-0mm}
{
\small
\begin{align}\label{XI}
\nonumber
\XI_{f_k,m}\triangleq
&\sum_{q=m+1}^M\HH_k\bar{\F}_{f_k,q}\bar{\F}_{f_k,q}^H\HH_k^H +
\\
&\sum_{f_\ell\in\FF^{*}\setminus \{f_k\}}\sum_{q\in\MM}\HH_k\bar{\F}_{f_\ell,q}\bar{\F}_{f_\ell,q}^H\HH_k^H + \SIGMA_k.
\end{align}
}\vspace{-3mm}
%\\
%

\noindent
The network throughput is then defined as the following sum rate:\vspace{-4mm}

{
\small
\begin{align}\label{eq:sum_rate}
R_\text{sum}&\triangleq\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}R_{f_k,m}.
\end{align}
}\vspace{-4mm}

%$\RR\triangleq\{R_{f_k,m}\}_{f_k\in\FF^{*},m\in\MM}$

\subsection{Power Consumption Model}
%We adopt the following power consumption model \cite{Dai2016}.
%Assigning UEs to appropriate eRRHs and turning unassigned eRRHs into sleep mode can reduce the system's total power.
To see the effect of the joint design on the power performance, we adopt a practical power consumption model in \cite{Dai2016}.
%Here, we assume that each optimization block interval is shorter than each requested file duration and there is also no power allocation between these blocks.
Specifically, the per-OTB power consumption by eRRH $i\in\K_R$ is modeled as:\vspace{-5mm}

{
\small
\begin{align}\label{eRRH:Power}
P_i^{eRRH} \triangleq
\left\{
\begin{array}{ll}
  \beta_iP_i^{tx}+P_{i,a}, & \text{if $0<P_i^{tx}\leq P_i$} \\
  P_{i,s}, & \text{if $P_{i}^{tx} = 0$}
\end{array}
\right.,
\end{align}
}\vspace{-4mm}

\noindent
where constant $\beta_i>0, \ i\in\K_R$ reflects the power amplifier efficiency, feeder loss and other loss factors due to power supply and cooling for eRRH $i$ \cite{Dai2016};
$P_i^{tx}$ is the transmit power required to deliver all requested files from eRRH $i$ as \vspace{-4mm}

{
\small
\begin{align}\label{Pi:Tx}
P_i^{tx}&\triangleq\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle\bar{\E}_{i}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H\bar{\E}_{i}\rangle,
\end{align}
}in which  $\bar{\E}_i\in\C^{N_R\times N_{r}}$ is zero everywhere except an identity matrix of size $N_r$ from row $(i-1) N_r+1$ to row $i N_r$; $P_{i,a}$ is the power required to support eRRH $i$ in the active mode; and $P_{i,s} < P_{i,a}$ is the power consumption in the sleep mode.

%\subsubsection{Fronthaul Power Consumption}
The fronthaul link from the BBU to eRRH $i\in\K_R$ is modeled as a set of communication channels with a total capacity $C_i$ and total power dissipation $P_{i,\max}^{FH}$. Its power consumption is given by \vspace{-5mm}

{
\small
\begin{align}\label{FH:Power}
P_i^{FH} &\triangleq \frac{R_i^{FH}}{C_i}P_{i,\max}^{FH}=\alpha_iR_i^{FH},
\end{align}
}where $\alpha_i\triangleq{P_{i,\max}^{FH}}/{C_i}$ and $R_i^{FH}$ is defined in \eqref{FH:cons}.

From \eqref{eRRH:Power} and \eqref{FH:Power}, the total network power consumption is: \vspace{-4mm}

{
\small
\begin{align}\label{total:power}
\nonumber
P_\text{total}
&\triangleq \sum_{i\in\K_R}(P_i^{eRRH}+P_i^{FH})
\\
%\nonumber
%&= \sum_{i\in\K_R}\bigg(\beta_iP_i^{tx}+ \one_{\{P_i^{tx}\}}(P_{i,a}-P_{i,s})
%+ P_{i,s} + \alpha_iR_i^{FH}\bigg)
%\\
&
= \sum_{i\in\K_R}\big(\beta_iP_i^{tx}+ \one_{\{P_i^{tx}\}}P_{i,\Delta}+\alpha_iR_i^{FH}\big) + P_{s},
\end{align}
}
where $P_{i,\Delta} \triangleq P_{i,a}-P_{i,s}$, $P_{s}\triangleq\sum_{i\in\K_R}P_{i,s}$ and \vspace{-4mm}

{
\small
\begin{align}\label{indicator}
\one_{\{P_i^{tx}\}}&\triangleq
\left\{
\begin{array}{cc}
  1, & \text{if $P_i^{tx}>0$} \\
  0, & \text{otherwise}
\end{array}
\right..
\end{align}
} \vspace{-4mm}

\subsection{Problem Formulation}
%Joint UE-eRRH Association, eRRH Activation and Precoder Design Problem}
In this paper, we aim to jointly design the UA, DDR and precoding in order to improve the network sum rate in \eqref{eq:sum_rate} as well as to reduce the total power consumption in \eqref{total:power}.
Let us define $\aaa\triangleq\{a_{k,i}\}_{k\in\K_U,i\in\K_R}$ and $\RR\triangleq\{R_{f_k,m}\}_{f_k\in\FF^{*},m\in\MM}$. For a given cache state information $\{c_{f_k,m}^i\}_{i\in\K_R, f_k\in\FF^{*},m\in\MM}$, the design problem is formulated as:

\vspace{-3mm}
{
\small
\begin{subequations}\label{mainP:GEE}
\begin{align}
\underset{{\aaa}, \RR, \bar{\F}}{\max} \,\,
&\PPP_1(\RR,\bar{\F})\triangleq R_\text{sum}-\eta P_\text{total}
\\
\mathrm{s.t.}\,\,
%&R_{\min}\leq\sum_{m\in\MM}R_{f,m}, \forall f\in\FF^{*}
%\\
&R_\text{QoS}\leq R_{f_k,m}\leq \bar{S},\forall f_k\in\FF^{*},m\in\MM,\label{eq:R_qos}
\\
&\sum_{f_k\in\FF^{*}}a_{k,i}\sum_{m\in\MM}(1-c_{f_k,m}^i)R_{f_k,m}\leq C_i,\forall i\in\K_R\label{eq:C_FH}
\\
&R_{f_k,m}\leq g_{f_k,m}(\bar{\F}), \forall f_k\in\FF^{*},m\in\MM,,\label{eq:R_ubound}
\\
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle\bar{\E}_{i}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H\bar{\E}_{i}\rangle
\leq P_i,\forall i\in\K_R\label{eq:Pi_max}
\\
&\text{and} \ \eqref{aselect}, \ \forall k\in\K_U, \ i\in\K_R.\label{eq:a_ki}
%\\
%&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\forall i\in\K_R,
%\\
%\label{assign:cont:GEE}
%%&(1-a_{k,i})\E_i^H\bar{\F}_{f,m}=\pmb{0}, \forall i\in\K_R,k\in\K_U,m\in\MM,
%&a_{k,i} =
%\left\{
%\begin{array}{ll}
%  0, & \text{iff  $\bar{\E}_{i}^H\bar{\F}_{f_k,m} = \pmb{0}$},\forall m\in\MM %\\
%  1, & \text{otherwise}
%\end{array}
%\right.
%\\
\end{align}
\end{subequations}
}\vspace{-3mm}

\noindent
Here, the weight $\eta>0$ specifies the relative importance between the sum rate and the total power. Constraint \eqref{eq:R_qos} imposes the minimum rate $R_\text{QoS}\geq 0$ and maximum rate $\bar{S}$ for each subfile [see the second paragraph of Sec.~\ref{ssec:data_deliver}]. Constraint \eqref{eq:C_FH} expresses the bottleneck at fronthaul link $i\in\K_R$ with the limited backhaul capacity $C_i\geq0$ [see \eqref{FH:cons}]. Constraint \eqref{eq:R_ubound} is indeed \eqref{eq:R_ub}.
%{where the equality sign occurs at the optimal point}.
Finally, constraint \eqref{eq:Pi_max} requires that the total transmit power at each eRRH $i\in\K_R$ must not exceed the predefined budget $P_i\geq 0$. %constraint \eqref{assign:cont:GEE} is another expression of \eqref{aselect} and indicates the one-to-one matching between $a_{k,i}$ and the precoders.
{While problem \eqref{mainP:GEE} is already a difficult nonconvex nonsmooth optimization problem, the strong coupling among the optimizing variables makes it even more challenging to be solved globally.
%involving three types of variables: (i) $\aaa$ which is binary, (ii) $\RR$ and (iii) $\bar{\F}$.

%In what follows, we will attempt to resolve \eqref{mainP:GEE}.}

{
%\begin{remark}
%It should be noted that our joint problem is different from that of \cite{Park2016} in which the eRRH-UE association and power minimization issues were not taken into account.
%\end{remark}
}


\section{Proposed Joint Optimization Algorithm
%For Joint UE-eRRH Association, eRRH Switching on/off and Precoder Designs
}
\label{sec:proposedalgorithm}
%\subsubsection{Proposed algorithm for problem \eqref{mainP:GEE}}
%\subsection{Proposed Algorithm}
%To resolve \eqref{mainP:GEE}, we first approximate \eqref{mainP:GEE} using the $\ell_0$-norm transformation and reweighted $\ell_1$-norm minimization techniques \cite{Dai2016}. The approximated problem is then decomposed into a convex subproblem of DDR and a nonconvex subproblem of precoding design by using the alternating descent method \cite{Bertsekas1999}. The latter nonconvex subproblem is finally solved by using successive convex quadratic programming \cite{Tam2016}.
%%Each optimization technique in each step has been extensively studied and shown to work well in a wide range of different problems.
%The detailed steps are presented in the following.

First, we will deal with the nonsmooth nature of \eqref{eq:a_ki} and \eqref{indicator}.
%(which is part of $P_\text{total}$).
We begin by expressing \eqref{eq:a_ki} as: \vspace{-3mm}

{
\small
\begin{align}
\label{assign:cont:GEE}
%&(1-a_{k,i})\E_i^H\bar{\F}_{f,m}=\pmb{0}, \forall i\in\K_R,k\in\K_U,m\in\MM,
&a_{k,i} =
\left\{
\begin{array}{ll}
  0, & \text{iff  $\bar{\E}_{i}^H\bar{\F}_{f_k,m} = \pmb{0}$},\forall m\in\MM \\
  1, & \text{otherwise}
\end{array}
\right.
\end{align}
}\vspace{-3mm}

\noindent
$\forall k\in\K_U, i\in\K_R$. To see this, note that if eRRH $i$ does not serve UE $k$, all corresponding precoders $\F_{f_k,m}^i=\bar{\E}_{i}^H\bar{\F}_{f_k,m}$ must be $\pmb{0}$ and then $a_{k,i}=0$. Otherwise, $a_{k,i}=1$ and there exists at least one of the corresponding precoders $\F_{f_k,m}^i=\bar{\E}_{i}^H\bar{\F}_{f_k,m}\neq\pmb{0}$. Therefore, without loss of optimality, $a_{k,i}$ can be further rewritten as: \vspace{-3mm}
% an $\ell_0$-norm as:% of $\sum_{m\in\MM}\langle\bar{\E}_{i}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H\bar{\E}_{i}\rangle$ as

{
\small
\begin{align}\label{assignvar:apprxm}
a_{k,i} = \big\|\sum_{m\in\MM}\langle\bar{\E}_{i}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\big\|_0.
\end{align}
}\vspace{-3mm}

\noindent
Similarly, \eqref{indicator} can also be replaced by: \vspace{-3mm}
% \cite{Dai2016}:% by the $\ell_0$-norm of a scalar as

{
\small
\begin{align}\label{indicator:apprxm}
%\nonumber
\one_{\{P_i^{tx}\}}
=\big\|P_i^{tx}\big\|_0
%\\
%&
=\Big\|\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\Big\|_0.
\end{align}
}\vspace{-3mm}

\noindent
We respectively approximate the nonconvex $\ell_0$-norms \eqref{assignvar:apprxm} and \eqref{indicator:apprxm} by their reweighted $\ell_1$-norms as \cite{Dai2016} \vspace{-3mm}

{
\small
\begin{align}%\label{assignvar:l1norm}
%\nonumber
a_{k,i}
%&=\big\|\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\big\|_0
%\\
%&
&= \mu_{k,i}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle,\label{assignvar:l1norm}\\
%\end{align}
%\begin{align}\label{indicator:l1norm}
%\nonumber
\one_{\{P_i^{tx}\}}
%&=\Big\|\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\Big\|_0
%\\
%&
&=\theta_i\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle,\label{indicator:l1norm}
\end{align}
}\vspace{-3mm}

\noindent
where weights $\mu_{k,i}$ and $\theta_i$ are iteratively updated  according to \vspace{-3mm}

{
\small
\begin{align}
\label{mu:update}
\mu_{k,i}&=\frac{c_1}{\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle+\tau_1},
\\
\label{theta:update}
\theta_i&=\frac{c_2}{\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle+\tau_2}.
\end{align}
}\vspace{-3mm}

\noindent
Here, $c_1$ and $c_2$ are constant numbers whereas $\tau_1$ and $\tau_2$ are constant regularization factors. % The weight $\mu_{k,i}$ affects the transmit power of eRRH $i$ to UE and the weight $\theta_{i}$ has the effect of putting appropriate eRRHs to sleep mode if their powers are too small for the energy saving purpose.
With \eqref{assignvar:l1norm}, variable $\aaa$ can now be expressed in terms of $\bar{\F}$ and thus be removed from \eqref{mainP:GEE}. With \eqref{indicator:l1norm}, we can rewrite $P_{\text{total}}$ as: \vspace{-3mm}

%\begin{figure*}
{
\small
\begin{align}\label{Ptotal:apprxm}
%\nonumber
P_\text{total}
%=& P_{sleep}+\sum_{i\in\K_R}\Bigg(\upsilon_i\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\tr({\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}})
%\\
%\nonumber
%&+ \alpha_i\sum_{f_k\in\FF^{*}}\mu_{k,i}\sum_{m\in\MM}\tr({\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}})\sum_{m\in\MM}(1-c_{f_k,m}^i)R_{f_k,m}\Bigg)
%\\
%\nonumber
=& \sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\tau_{f_k}^i\sum_{m\in\MM}
\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
%\\
%&
 + P_{s}
\end{align}
}
%\hrulefill
%\end{figure*}

\noindent
where $\tau_{f_k}^i\triangleq\upsilon_i+\alpha_i\vartheta_{f_k}^i$, $\vartheta_{f_k}^i\triangleq\mu_{k,i}\sum_{m\in\MM}(1-c_{f_k,m}^i)R_{f_k,m}$ and
$\upsilon_i\triangleq\beta_i+\theta_iP_{i,\Delta}$.

The above result allows us to transform \eqref{mainP:GEE} to the following approximated problem: \vspace{-5mm}

{
\small
\begin{subequations}\label{mainP:GEE:equiv}
\begin{align}
\underset{\RR,\bar{\F}}{\max} \,\,
&\PPP_2(\RR,\bar{\F})\triangleq R_\text{sum}-\eta P_\text{total}
\\
\mathrm{s.t.}\,\,
%&R_{\min}\leq\sum_{m\in\MM}R_{f,m}, \forall f\in\FF^{*}
%\\
&R_{QoS} \leq R_{f_k,m}\leq \bar{S},\forall f_k\in\FF^{*},m\in\MM,
\\
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\vartheta_{f_k}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\leq C_i,\forall i\in\K_R
\\
&R_{f_k,m}\leq g_{f_k,m}(\bar{\F}),\forall f_k\in\FF^{*},m\in\MM,
\\
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\leq P_i,\forall i\in\K_R.
%\\
%\label{assign:cont:GEE:equiv}
%&(1-a_{k,i}){\bar{\E}_{i}}^H\bar{\F}_{f,m}=\pmb{0}, \forall i\in\K_R,k\in\K_U,m\in\MM,
\end{align}
\end{subequations}
}\vspace{-2mm}

\noindent
Problem \eqref{mainP:GEE:equiv} is still difficult due to the strong coupling between $\RR$ and $\bar{\F}$. As such, we propose using an alternating method \cite{Bertsekas1999} that deals with one variable at a time and repeats until convergence. Specifically,
%First, we denote $\bar{\V}_{f,m}=\bar{\F}_{f,m}\bar{\F}_{f,m}^H$ as the covariance matrix of $\bar{\F}_{f,m}$.
for a given $\bar{\F}$, we solve the following convex subproblem to update $\RR$: \vspace{-2mm}
{
\small
\begin{subequations}\label{mainP:GEE:equiv:updateR}
\begin{align}
%\nonumber
\underset{\RR}{\max} \,\,\,\,\,\,
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}R_{f_k,m}
%\\
%&
-\eta\left(b+\sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}q_{f_k,m}^iR_{f_k,m}\right)
\\
\mathrm{s.t.}\,\,\,\,\,\,
%&R_{\min}\leq\sum_{m\in\MM}R_{f,m}, \forall f\in\FF^{*}
%\\
&R_{QoS}\leq R_{f_k,m}\leq \bar{S},\forall f_k\in\FF^{*},m\in\MM,
\\
&\sum_{f_k\in\FF^{*}}q_{f_k}^i\sum_{m\in\MM}R_{f_k,m}\leq C_i,\forall i\in\K_R,
\\
&R_{f_k,m}\leq g_{f_k,m}(\bar{\F}), \forall f_k\in\FF^{*},m\in\MM,
\end{align}
\end{subequations}
}\vspace{-3mm}

\noindent
where
{
\small
$q_{f_k}^i\triangleq\alpha_i\mu_{k,i}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle(1-c_{f_k,m}^i)$
} and
{
\small
$b\triangleq P_{s}+\sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\upsilon_i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle$.
}
%\begin{align}
%\nonumber
%q_{f_k}^i=&\alpha_i\mu_{k,i}\sum_{m\in\MM}\tr({\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}})(1-c_{f_k,m}^i)
%\\
%\nonumber
%\label{b}
%b=&P_{sleep}
%\\
%\nonumber
%&+\sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\upsilon_i\tr({\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}})
%\end{align}
%It can be shown that problem \eqref{mainP:GEE:equiv:updateR} is convex and thus can be solved efficiently by a standard convex solver such as CVX \cite{cvx}.


Next, for a given $\RR$, we solve the following subproblem to update $\bar{\F}$: \vspace{-3mm}

{
\small
\begin{subequations}\label{mainP:GEE:equiv:updateF}
\begin{align}
\underset{\bar{\F}}{\min} \,
&\sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}
\eta\tau_{f_k}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\\
\mathrm{s.t.}\,
\label{fronthaul:cons}
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\vartheta_{f_k}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\leq C_i,\forall i\in\K_R,
\\
\label{rate:cons}
&R_{f_k,m}\leq g_{f_k,m}(\bar{\F}),\forall f_k\in\FF^{*},m\in\MM,,
\\
\label{power:cons}
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\leq P_i,\forall i\in\K_R.
\end{align}
\end{subequations}
} \vspace{-3mm}
%Here, we relax the rank constraints $\rank(\bar{\V}_{f,m})\leq 1$.
%where $d_{f,m}=d,\forall f\in\FF^{*},m\in\MM$.
\begin{figure*}
{
\small
\begin{align}
\label{Gamma}
\nonumber
\Gamma_{f_k,m}^{(n)}(\bar{\F})=
& g_{f_k,m}(\bar{\F}^{(n)})
+ 2\Re
\left\{
\Bigg\langle
\left(\left(\PHI_{f_k,m}^{(n)}-\PI_{f_k,m}^{(n)}(\PI_{f_k,m}^{(n)})^H\right)^{-1}\PI_{f_k,m}^{(n)}\right)^H(\PI_{f_k,m}(\bar{\F}_{f_k,m})-\PI_{f_k,m}^{(n)})
\Bigg\rangle
\right\}
\\
&-
\Bigg\langle
\left(\left(\PHI_{f_k,m}^{(n)}-\PI_{f_k,m}^{(n)}(\PI_{f_k,m}^{(n)})^H\right)^{-1}-(\PHI_{f_k,m}^{(n)})^{-1}\right)^H(\PHI_{f_k,m}(\bar{\F})-\PHI_{f_k,m}^{(n)})
\Bigg\rangle
\end{align}
}\vspace{+2mm}
\hrulefill
\end{figure*}
\begin{algorithm}[!t]
\caption{Joint design of UA, data delivery rate and precoding for F-RANs}
\begin{algorithmic}[1]\label{alg:GEEMax}
{
\small
\STATE \textbf{Initialization}:
Use Alg.~\ref{alg:IFPF} to find a feasible initial point $(\bar{\F}^{(1)}, \{\mu_{k,i},\theta_i\}, \RR^{(1)})$. Set error tolerances $\epsilon_1,\epsilon_2,\epsilon_3>0$. Set $p := 1$.
\REPEAT
\STATE Set $p:=p+1$ and $\kappa:=1$
\REPEAT
\STATE Set $\kappa:=\kappa+1$
\STATE For given $\RR^{(\kappa-1)}$, use $\bar{\F}^{(\kappa-1)}$ as the initial feasible point  and set $n:=1$
\REPEAT
\STATE Update $n:=n+1$
%\REPEAT
\STATE Find an optimal solution $\bar{\F}^*$ by solving convex problem \eqref{mainP:GEE:equiv:updateF:convexQP}
%\STATE Update $\{\Zeta,\Nu,\Rho\}$ as \eqref{zeta}, \eqref{nu} and \eqref{rho};
%\UNTIL{$|t_{\zeta_{f_k,m}}|\leq \epsilon_4$, $|t_{\nu_i}|\leq \epsilon_4$ and $|t_{\rho_i}|\leq \epsilon_4$ }
\STATE Update $\bar{\F}^{(n)}:=\bar{\F}^*$
\UNTIL{$\left|\frac{\PPP_3(\bar{\F}^{(n)})-\PPP_3(\bar{\F}^{(n-1)})}{\PPP_3(\bar{\F^{(n)}})}\right|\leq \epsilon_3$}
\STATE Update $\bar{\F}^{(\kappa)}:=\bar{\F}^*$
\STATE Find an optimal solution $\RR^*$ by solving convex problem \eqref{mainP:GEE:equiv:updateR}
\STATE Update $\RR^{(\kappa)}:=\RR^*$
\UNTIL{$\left|\frac{\PPP_2(\bar{\F}^{(\kappa)},\RR^{(\kappa)})-\PPP_2(\bar{\F}^{(\kappa-1)},\RR^{(\kappa-1)})}{\PPP_2(\bar{\F}^{(\kappa-1)},\RR^{(\kappa-1)})}\right|\leq \epsilon_2$}
\STATE Update $\bar{\F}^{(p)}:=\bar{\F}^*$ and $\RR^{(p)}:=\RR^*$
\STATE Update $\{\mu_{k,i},\theta_i\}$ according to \eqref{mu:update} and \eqref{theta:update}
\UNTIL{$\left|\frac{\PPP_1(\bar{\F}^{(p)},\RR^{(p)})-\PPP_1(\bar{\F}^{(p-1)},\RR^{(p-1)})}{\PPP_1(\bar{\F}^{(p-1)},\RR^{(p-1)})}\right|\leq \epsilon_1$}
}
\end{algorithmic}
\end{algorithm}

\noindent
The key difficulty with subproblem \eqref{mainP:GEE:equiv:updateF} is due to the nonconvex contraint \eqref{rate:cons}. Let $\PHI_{f_k,m}\triangleq\PI_{f_k,m}\PI_{f_k,m}^H+\XI_{f_k,m}$.
Using the first-order Taylor series expansion, we approximate the nonconcave part $g_{f_k,m}(\bar{\F})$ of \eqref{rate:cons} by
%in $\PI_{f_k,m}$ and $\XI_{f_k,m}\succ 0$ as follows:
%\begin{align}\label{g:apprxm}
%g_{f_k,m}(\bar{\F})\approx\Gamma_{f_k,m}^{(n)}(\bar{\F}).
%\end{align}
its concave lower bound $\Gamma_{f_k,m}^{(n)}(\bar{\F})$ defined in \eqref{Gamma}.
This fact can be proved similarly as in \cite{Tam2016}, and the detailed proof is omitted due to limited space.
The nonconvex subproblem \eqref{mainP:GEE:equiv:updateF} can thus be transformed into the following \emph{convex} quadratic program:
\vspace{+8mm}
\begin{algorithm}[h]
\caption{Finding an initial feasible point for Alg.~\ref{alg:GEEMax}}
\begin{algorithmic}[1]\label{alg:IFPF}
{
\small
\STATE \textbf{Initialization}: Set error tolerance $\epsilon_4>0$.
Find $\bar{\F}^{(1)}$ as $\bar{\F}_{f_k,m}^{(1)}:=\sqrt{\frac{\bar{P}}{\langle\bar{\E}_{7}^H\F_{\text{ran}}\F_{\text{ran}}^H\bar{\E}_{7}\rangle}}\F_{\text{ran}}, \forall f_k\in\FF^{*},m\in\MM$ with a random matrix $\F_{\text{ran}}\in\C^{K_RN_r\times d}$ and $\bar{P} = \frac{P_i}{MK_U}$ until \eqref{fronthaul:cons:intial} and \eqref{power:cons:intial} are satisfied $\forall i\in\K_R$. Set $n := 1$.
\STATE For given $\bar{\F}^{(1)}$, update $\{\mu_{k,i},\theta_{i}\}$ by \eqref{mu:update} and \eqref{theta:update}, and find $\RR$ by solving \eqref{mainP:GEE:equiv:updateR}
\REPEAT
\STATE Update $n:=n+1$
\STATE Find an optimal solution $\bar{\F}^*$ by solving \eqref{maxminP:CQP}
\STATE Update $\bar{\F}^{(n)}:=\bar{\F}^*$
\UNTIL{$\left|\frac{\PPP_4(\bar{\F}^{(n)})-\PPP_4(\bar{\F}^{(n-1)})}{\PPP_4(\bar{\F^{(n)}})}\right|\leq \epsilon_4$}
}
\end{algorithmic}
\end{algorithm}
%\footnote{\color{blue}DN: Should reset the counter for equations and put a label for (24)! Also in (24), \text{tr} has not been defined.}
%We state this fact by the following theorem.
%\begin{theorem}
%\label{Theorem:lowerbound}
%Function $\Gamma_{f_k,m}^{(n)}(\bar{\F})$ is the lower bound of $g_{f_k,m}(\bar{\F})$ and at a certain point $\bar{\F}^{(n)}$, we have
%\begin{align}\label{lowerbound}
%g_{f_k,m}(\bar{\F}^{(n)})=\Gamma_{f_k,m}(\bar{\F}^{(n)})
%\\
%\Gamma_{f_k,m}^{(n)}(\bar{\F})\leq g_{f_k,m}(\bar{\F}), \forall \bar{\F}
%\end{align}
%where $n$ is the iteration index and $\Gamma_{f_k,m}(\bar{\F})$ is concave.
%\end{theorem}
%\begin{IEEEproof}
%The theorem can be proved similarly as in \cite{Tam2016}. The detailed proof is omitted due to limited space.% The detailed proofs can be found in our technical report [11].
%\end{IEEEproof}
%By replacing $g_{f_k,m}(\bar{\F})$ with its concave lower bound $\Gamma_{f_k,m}^{(n)}(\bar{\F})$ at a given point $\bar{\F}^{(n)}$,
%Theorem~\ref{Theorem:lowerbound} allows

{
\small
\begin{subequations}\label{mainP:GEE:equiv:updateF:convexQP}
\begin{align}
\underset{\bar{\F}}{\min} \,\,
& \PPP_3(\bar{\F}) \triangleq
 \sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}
\eta\tau_{f_k}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\\
\mathrm{s.t.}\,\,
\label{fronthaul:cons:convexQP}
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\vartheta_{f_k}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\leq C_i,\forall i\in\K_R,
\\
\label{rate:cons:convexQP}
&R_{f_k,m}\leq \Gamma_{f_k,m}^{(n)}(\bar{\F}),\forall f_k\in\FF^{*},m\in\MM,
\\
\label{power:cons:convexQP}
&\sum_{f\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\leq P_i,\forall i\in\K_R.
\end{align}
\end{subequations}
}\vspace{-3mm}


%where
%\begin{align}\label{P3}
%\nonumber
%\PPP_3(\bar{\F})\triangleq\sum_{i\in\K_R}\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}
%\tau_{f_k}^i\tr({\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}})
%\end{align}

We are now ready to present Alg.~\ref{alg:GEEMax} to solve the original problem  \eqref{mainP:GEE}.
First, to deal with precoding design, the inner loop finds the locally optimal $\bar{\F}$ for a given $\RR$ by iteratively solving the convex problem \eqref{mainP:GEE:equiv:updateF:convexQP}. For the DDR design, the middle loop alternates between solving \eqref{mainP:GEE:equiv:updateF} to find a locally optimal $\bar{\F}$ for a given $\RR$ and solving \eqref{mainP:GEE:equiv:updateR} to find an optimal $\RR$ for a given $\bar{\F}$. Finally, the outer loop updates $\{\mu_{k,i},\theta_i\}$ according to \eqref{mu:update} and \eqref{theta:update} for UA. Alg.~\ref{alg:GEEMax} terminates when there is no improvement in the objective value $\PPP_1(\RR,\bar{\F})$ of \eqref{mainP:GEE}.
\begin{remark}
Let $N_p,N_\kappa$ and $N_n$ respectively be the numbers of iterations of the outer loop, the middle loop and the inner loop when Alg.~\ref{alg:GEEMax} converges. It can be observed that Alg.~\ref{alg:GEEMax} solves problem \eqref{mainP:GEE:equiv:updateF:convexQP} $N_p N_\kappa N_n$ times and problem \eqref{mainP:GEE:equiv:updateR} $N_p N_\kappa$ times.
\end{remark}

%\footnote{\color{blue}DN: In both algorithms, please check all the subscript and superscript of the variables in the iteration. Also carefully check through the algorithm commands to make sure they are executable with no error.}.
%\begin{remark}
%Computational complexity analysis is written here.
%\end{remark}
%
%\begin{proposition}
%Convergence analysis is written here.
%\end{proposition}
%\begin{proof}
%Please see Appendix C.
%\end{proof}


Alg.~\ref{alg:GEEMax} requires an initial feasible point that satisfies  constraints \eqref{fronthaul:cons}, \eqref{rate:cons} and \eqref{power:cons}.
For this, we consider the following problem: \vspace{-6mm}

{
\small
\begin{subequations}\label{maxminP}
\begin{align}
\underset{\bar{\F}}{\max} \,\,\,
&  \underset{m\in\MM,f_k\in\FF^{*}}{\min} \left\{\frac{g_{f_k,m}(\bar{\F})}{R_{f_k,m}}\right\}
\\
%\nonumber
\mathrm{s.t.}\,\,\,
%\nonumber
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\vartheta_{f_k,m}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\leq C_i,\forall i\in\K_R,
\\
%\nonumber
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\leq P_i,\forall i\in\K_R.
\end{align}
\end{subequations}
}\vspace{-3mm}
%Using the lower bound $\Gamma_{f_k,m}^{(n)}(\bar{\F})$ of $g_{f_k,m}(\bar{\F})$,

\noindent
Problem \eqref{maxminP} can further be transformed into the following \emph{concave} quadratic program: \vspace{-5mm}

{
\small
\begin{subequations}\label{maxminP:CQP}
\begin{align}
\underset{\bar{\F}}{\max} \,
& \PPP_4(\bar{\F})\triangleq \underset{m\in\MM,f_k\in\FF^{*}}{\min}\left\{\frac{\Gamma_{f_k,m}(\bar{\F})}{R_{f_k,m}}\right\}
\\
%\nonumber
\label{fronthaul:cons:intial}
\mathrm{s.t.}\,
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\vartheta_{f_k,m}^i\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle\leq C_i,\forall i\in\K_R,
\\
%\nonumber
\label{power:cons:intial}
&\sum_{f_k\in\FF^{*}}\sum_{m\in\MM}\langle{\bar{\E}_{i}}^H\bar{\F}_{f_k,m}\bar{\F}_{f_k,m}^H{\bar{\E}_{i}}\rangle
\leq P_i,\forall i\in\K_R,
\end{align}
\end{subequations}
}\vspace{-3mm}

\noindent
the solution of which can be found by Alg.~\ref{alg:IFPF}.
%\subsection{Convergence analysis}

\textbf{Convergence Analysis}:  For the inner loop of Alg.~\ref{alg:GEEMax}, the optimal solution $\bar{\F}^{(n)}$ of convex problem \eqref{mainP:GEE:equiv:updateF:convexQP} is feasible to the nonconvex problem \eqref{mainP:GEE:equiv:updateF} and is also better than $\bar{\F}^{(n-1)}$, i.e., {$\PPP_3(\bar{\F}^{(n)})\leq \PPP_3(\bar{\F}^{(n-1)})$}.
Once initialized from a feasible point $\bar{\F}^{(0)}$ by using Alg.~\ref{alg:IFPF}, the inner loop generates a sequence $\{\bar{\F}^{(n)}\}$ of improved feasible solutions for the nonconvex program \eqref{mainP:GEE:equiv:updateF}, and it will eventually converge to a {locally optimal} solution of \eqref{mainP:GEE:equiv:updateF}.
The middle loop utilizes the alternating optimization framework that solves a series of convex problems \eqref{mainP:GEE:equiv:updateR} and \eqref{mainP:GEE:equiv:updateF:convexQP}, and is guaranteed to converge \cite{Tam2016TWC}.
By choosing $c_1=\frac{1}{\ln(1+\tau_1^{-1})}$ and $c_2=\frac{1}{\ln(1+\tau_1^{-2})}$, the outer loop can be proven to be a special case of the majorization-minimization algorithm \cite{Dai2016} and is thus guaranteed to converge.
\vspace{-0mm}
%{Since each optimization techniques in each loop is proved to converge to a locally optimal point, Alg. \eqref{alg:GEEMax} is guaranteed to obtain at least a locally optimal solution of the main problem \eqref{mainP:GEE}.}

\section{Numerical Results}
\label{Simulation}
\begin{figure}[t!]
  \centering
  \includegraphics[width=0.4\textwidth]{Figure2.eps}
  \caption{Network scenario used in simulations}
  \label{Fig:2}
\end{figure}
We consider the network scenario in Fig.~\ref{Fig:2} where the locations of the $K_R=7$ eRRHs and $K_U=3$ UEs are fixed.
LTE parameters in Table~\ref{table:parameters} are used in our simulations \cite{3GPP-R9}. Assume that each eRRH is equipped with $N_r=5$ antennas and each UE with $N_u=2$ antennas. We set $P_i=P$, $C_i=C^{FH}$ $\forall i\in\K_R$ and $\SIGMA_k = \sigma^2\pmb{I},\forall k\in\K_U$. At each eRRH, the active mode and the sleep mode consume $84$W and $56$W, respectively \cite{Dai2016}. Constant regulation factors are set as $\tau_1=10^{-5}, \ \tau_2=10^{-3}$. The slope of transmit power is $\beta_i=\beta=2.8$ and $\alpha_i=\alpha=5$ $\forall i\in\K_R$ \cite{Dai2016}.
The error tolerances for the proposed algorithms are taken as $\epsilon_1=10^{-3}$ and $\epsilon_2=\epsilon_3=\epsilon_4=10^{-2}$.
We set $R_{\text{QoS}}=0.1$Mbps,
$\bar{S}=40$Mbps and $M=2$. The numerical result is obtained by averaging over 100 independent channel realizations.

\begin{table}[!h]
\caption{System parameters used in simulations}
\vspace{-2mm}
\label{table:parameters}
\begin{center}
\begin{tabular}{|c|c|}
\hline
%\textbf{Parameters}  & \textbf{Values}
%\\
%\hline
%\hline
Distance between adjacent eRRHs      & $0.3$ km\\
Total bandwidth        & $10$ MHz\\
Std. deviation of log-normal shadowing      & $10$ dB\\
Path loss at distance $d$ (km)
& $140.7+36.7\log_{10}(d)$ dB \\
Noise variance $\sigma_k^2=\sigma^2$
& $-174$ dBm/Hz\\
Maximum eRRH transmit power
& $24$ dBm\\
\hline
\end{tabular}%
\end{center}
\end{table}
\vspace{+0mm}
\begin{table}[!h]
{
\renewcommand{\arraystretch}{1.4}
\tiny
\caption{Example {cache state information} used in simulations}
\vspace{-6mm}
\label{table:cacheplacement}
\begin{center}
\resizebox{0.49\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$c_{f_k,m}^i$& $i=1$& $i=2$& $i=3$& $i=4$& $i=5$& $i=6$& $i=7$\\
\hline
$(f_{1},1)$, $(f_{1},2)$    &$1,1$  &$0,0$  &$0,1$  &$0,0$  &$1,0$  &$0,1$  &$0,0$\\
\hline
$(f_{2},1)$, $(f_{2},2)$    &$1,1$  &$0,1$  &$0,0$  &$1,0$  &$0,0$  &$0,1$  &$0,1$ \\
\hline
$(f_{3},1)$, $(f_{3},2)$    &$1,1$  &$0,0$  &$1,0$  &$0,0$  &$1,0$  &$1,0$  &$0,1$\\
\hline
\end{tabular}}
\end{center}
}
\end{table}
In the simulation, each eRRH's caching capacity is set as $B_i=B=\xi SF, \ \forall i\in\K_R$ where $\xi$ indicates the fractional caching capacity. Each eRRH can store a maximum of $\lfloor\xi FM\rfloor$ subfiles randomly chosen from the file library.
Each UE randomly and independently requests one file from a library of $F=6$ files.
{We pick one specific set $\FF^{*}$ of the requested files. The cache state information $\{c_{f_{k},m}^{i}\}$ in \eqref{CDset} are recorded in Table \ref{table:cacheplacement} by checking $\FF^{*}$ against the local caches of eRRHs.
Since we do not consider the caching problem,
%the locations of UEs are fixed and close to eRRH $1$ that stores all of the requested files.
we are allowed to selectively place the UEs close to eRRH $1$ where all the requested files are cached.
%This helps us see easier how much the system performance can be improved from the cache if the cache is placed intelligently.
This arrangement helps us reveal the potential gains resulting from an intelligent caching strategy.
%Each numerical result is obtained by averaging over $100$ independent simulation trials.
%In each simulation trial, we only generate channel realizations.}



%In Fig. \ref{Fig:2}, we examine the user association of our proposed algorithm in a F-RAN of  $C_i=C^{FH}=50$Mbps, $\eta=0.01$ and the cache placement is selected as stated in Table. \ref{table:cacheplacement}.
%Each UE is assigned to at least one eRHH based on the the caching placement and the channel state information.
%For example, it can be seen that all UEs are connected to eRRH $1$.
%It is reasonable because eRRH $1$ has requested files of UE $1$. UEs ${2,3,4}$ are still assigned to eRRH $1$ because the channel gains of links from them to eRHH $1$ are stronger than those to other eRRHs.
%It is confirmed by checking the norms of the channel matrices from UEs to eRHHs and the mutual comparison is omitted in this paper due to lack of space.


To demonstrate the advantages of the joint design offered by Alg.~\ref{alg:GEEMax}, we evaluate the performance of Alg.~\ref{alg:GEEMax} with local caching (referred to as Alg.~\ref{alg:GEEMax}-C) and Alg.~\ref{alg:GEEMax} without local caching (referred to as  Alg.~\ref{alg:GEEMax}-NC).
We also implement the sole precoder design with local caching (referred to as SPD-C) by modifying Alg.~\ref{alg:GEEMax} such that each UE is always connected to all eRRHs.

%The cache state information is presented according to Table~\ref{table:cacheplacement}.
%To capture both throughput gain and power saving, we define a new metric of throughput-to-power ratio $\text{TPR}=R_\text{sum}/(P_{\text{total}}-P_{s})$ to quantify the number of bits transmitted per Joule of energy where the fixed eRRH sleeping power is discounted.
{Fig.~\ref{Fig:3}(a) compares the throughput performance among Alg.~\ref{alg:GEEMax}-C, Alg.~\ref{alg:GEEMax}-NC and SPD-C. Attention should be paid to the region of small $\eta$ where problem \eqref{mainP:GEE} prioritizes throughput maximization.
When the fronthaul capacity is limited ($C^{FH}=50$Mbps), the sum rate by Alg.~\ref{alg:GEEMax}-C and Alg.~\ref{alg:GEEMax}-NC are approximately $1.5$ times higher than that by the SPD-C scheme. %Furthermore, Alg.~\ref{alg:GEEMax}-C outperforms Alg.~\ref{alg:GEEMax}-NC by at least $5$Mbps.
This is because the joint designs show their advantages over the SPD-C in terms of reducing the bottleneck in the fronthaul links [see \eqref{eq:C_FH}] to improve the throughput.
However, for the ample fronthaul capacity of $C^{FH}=1$Gbp, the local caching at the eRRHs or the fronthaul traffic offload via selective UE-eRRH associations offers almost no throughput advantage.
In this case, there is virtually no bottleneck in transferring data traffic from the BBU to each eRRH.
When $\eta=10^{-6}$, the SPD-C even offers higher throughput than that offered by the Alg.~\ref{alg:GEEMax}-C and Alg.~\ref{alg:GEEMax}-NC since it better exploits the coherence combining gain.}

Fig. \ref{Fig:3}(b) demonstrates the power consumption incurred by the three considered schemes. To have meaningful comparisons, we consider the ``busy" power by discounting the fixed eRRH sleeping power from the total power as $P_\text{busy}\triangleq P_{\text{total}}-P_{s}$. From the figure, Alg.~\ref{alg:GEEMax}-C and Alg.~\ref{alg:GEEMax}-NC are the two best performers, where they save approximate $50\%$ of the power consumption for large $\eta$ and more than $66\%$ for small $\eta$ over the SPD-C scheme.
This is because the proposed joint design can effectively save power by: (i) using fewer active eRRHs and less transmission power in the fronthaul links by assigning UEs to appropriate eRRHs, and (ii) reducing the transmission power in the access links by designing effective precoders.
% and reducing the transmission powers on both fronthaul and radio access links.
\begin{figure}[t!]
  \centering
  \subfigure[Average total throughput]{\includegraphics[width=0.5\textwidth]{Figure3a_100trials.eps}}\\
  \subfigure[Average total busy power]{\includegraphics[width=0.5\textwidth]{Figure3b_100trials.eps}}\\
  \caption{Effects of joint design, local caching and limited backhaul capacity on network performance}
  %\caption{Average TPR (a), Average Throughput (b) and Average Total Power (c) versus $\eta$ of joint designs with selective caching (SD)/ no caching (ND), and separate design (SD) in different fronthaul capacities $\text{C}^{\text{FH}}=\{\text{50Mbps,1Gbps}\}$.}
  \label{Fig:3}
%  \includegraphics[width=19cm]{Figure4.eps}
%  \caption{Convergence of proposed algorithm}
%  \label{Fig:4}
\end{figure}
%Alg.~\ref{alg:GEEMax}-C

%When $\eta$ is large, problem \eqref{mainP:GEE} reduces to one of power minimization subject to rate constraints. From Fig. \ref{Fig:3}(c), all the considered designs incur the same power consumption. Because it is the minimum power required to satisfy the QoS requirements, no further power saving is possible. For smaller values of $\eta$ and for the same $C^{FH}$, our joint design with local caching consumes the lowest power followed by the joint design without caching and lastly the sole precoder design. This can be explained by noting that the joint designs save more power by assigning users to appropriate eRRHs and putting the unassigned eRRHs into sleep mode. When the fronthaul capacity is ample (i.e., $C^{FH}=1$Gbps), these fronthaul links are more likely to be utilized for data transport instead of local caching. As the power consumption of the fronthaul links is typically large, a performance degradation is observed in Fig.~\ref{Fig:3}(c) for the joint designs.


%When comparing Algorithm \ref{alg:GEEMax} with selective caching to Algorithm \ref{alg:GEEMax} without caching, the former offers around $2$ times better TPR performance for the same fronthaul capacity and $\eta$ as can be observed from Fig. \ref{Fig:3}(a). This observation is expected because the selective caching scheme puts the requested file in eRRH $1$ [see Table \ref{table:cacheplacement}] which is nearest to all UEs, and hence, power is saved by not fetching requested files from the BBU while the throughput obtained from the selective scheme is still greater and equal to that of no caching scheme [see Fig. \ref{Fig:3}(b)]. Such these results verifies the significant advantage of caching in the network performance in terms of TPR.

%Fig. \ref{Fig:3}(a) shows that the joint design give TPRs approximately $4$ times higher over precoder design alone. This can be explained by noting that the joint design save more power [see Fig. \ref{Fig:3}(c)] by assigning users to appropriate eRRHs and turning unassigned eRRHs into sleep mode but achieves equal or greater throughput [see Fig. \ref{Fig:3}(b)] at the same fronthaul capacity and $\eta$.




%Furthermore, Fig. \ref{Fig:3}(b) shows that as we reduce the fronthaul capacity from $1$Gbps to $50$Mbps, the throughput of each joint design is degraded much less than that of the separate design. This is reasonable because the joint designs make the fronthaul capacities constraints [see \eqref{eq:C_FH}] less restricted by removing those of unassigned eRRHs and putting the requested files in the local cache of assigned eRRHs. It means that the bottleneck effect of fronthaul capacity on the network performance can be decreased by the joint design with caching.

\begin{figure}[t!]
  \centering
  \includegraphics[width=0.5\textwidth]{Figure4.eps}
  \caption{Convergence behavior of Alg.~\ref{alg:GEEMax}}
  \label{Fig:4}
\end{figure}
Fig.~\ref{Fig:4} illustrates the convergence behavior of Alg.~\ref{alg:GEEMax}-C with ${C}^{\text{FH}}=50$Mbps. Figs.~\ref{Fig:4}(a), \ref{Fig:4}(b) and \ref{Fig:4}(c) plot the convergence of the objective functions of the inner, the middle and the outer loops, respectively. In total, the proposed algorithm requires fewer than $50$ iterations to converge, where each iteration corresponds to solving at most two simple convex programs \eqref{mainP:GEE:equiv:updateR} and \eqref{mainP:GEE:equiv:updateF:convexQP}.



\section{Conclusions}
\label{sec:Conclusion}
This paper has studied the joint design of user association, data delivery rate and signal precoding in the downlink of a cache-enabled F-RAN with limited fronthaul capacity. An optimization problem has been formulated with the objective of maximizing the weighted difference of network throughput and total power consumption. The requirements on data delivery rates and maximum eRRH transmit powers have also been included in the design. Applying a range of optimization techniques, we have solved this challenging optimization problem and proposed an iterative algorithm that is guaranteed to converge to obtain a locally optimal solution. Numerical results have shown that our joint design markedly improves both network throughput and power consumption performances of the considered F-RAN.

%\section*{Appendix A\\Proof of Theorem \ref{Theorem:lowerbound}}
%Theorem \ref{Theorem:lowerbound} can be proved similarly as in \cite{Tam2016} and hence, omitted.
%The detailed proofs can be found in our technical report in [10].
%\begin{proof}
%First, we rewrite function $g_{f_k,m}^{(n)}$ in \eqref{R:bound} as follows\footnote{by using the transformation $\log|\pmb{I}+\A\B^{-1}|=-\log|\pmb{I}-\A(\A+\B)^{-1}|$.}
%\begin{align}
%\label{g:rewrite:1}
%g_{f_k,m}(\bar{\F})=-h_{f_k,m}(\bar{\F})
%\end{align}
%where
%\begin{align}
%h_{f_k,m}(\bar{\F})&=\log\left|\pmb{I}_{N_{U,k}}-\PI_{f_k,m}\PI_{f_k,m}^H\PHI_{f_k,m}^{-1}\right|
%\\
%\label{PHI}
%\PHI_{f_k,m} &= \PI_{f_k,m}\PI_{f_k,m}^H + \XI_{f_k,m} \succeq \PI_{f_k,m}\PI_{f_k,m}^H.
%\end{align}
%We then rewrite function $h_{f_k,m}(\bar{\F})$ as follows\footnote{by using the transformation $\log|\pmb{I}+\A\B|=\log|\pmb{I}+\B\A|$.}
%\begin{align}\label{h:rewrite:1}
%h_{f_k,m}(\PI_{f_k,m},\XI_{f_k,m}) = \log|\pmb{I}_{d}-\psi_{f_k,m}(\PI_{f_k,m},\XI_{f_k,m})|
%\end{align}
%where $\psi_{f_k,m}(\PI_{f_k,m},\XI_{f_k,m})\triangleq\PI_{f_k,m}^H\PHI_{f_k,m}^{-1}\PI_{f_k,m}$ is a matrix function.
%
%For ease of presentation, we denote $\PI_{f_k,m}$ as $\PI$, $\PHI_{f_k,m}$ as $\PHI$, $g_{f_k,m}$ as $g$ and $\psi_{f_k,m}$ as $\psi$.
%The property of $\psi(\PI,\PHI)$ is stated by the following lemma.
%\begin{lemma}
%\label{psi:inequality}
%For matrix function $\psi(\PI,\PHI)$, the following matrix inequality holds for all $\PI_1,\PI_2$ and $\PHI_1\succ 0,\PHI_2\succ 0$ and $\lambda\in\{0,1\}$
%\begin{align}
%\nonumber
%&\lambda\psi(\PI_1,\PHI_1)+(1-\lambda)\psi(\PI_2,\PHI_2)
%\\
%&\succeq \psi\big(\lambda(\PI_1,B_1)+(1-\lambda)(\PI_2,\PHI_2)\big)
%\end{align}
%\end{lemma}
%\begin{proof}
%The proof of lemma \eqref{psi:inequality} is able to be derived directly from \cite{Tam2016} and thus, omitted.
%%For all $\PI_i$ and $\PHI_i\succ 0$, by using Schur's complement \cite{BenTal2001}, we always have
%%\begin{align}
%%\left[
%%\begin{array}{cc}
%%  \PHI_i & \PI_i^H \\
%%  \PI_i & \PI_i^H\PHI_i^{-1}\PI_i
%%\end{array}
%%\right]
%%\succeq 0
%%\end{align}
%%Thus, for all $0\leq\lambda\leq 1, \PI_i$ and $\PHI_i\succ 0$, we have
%%\begin{align}
%%\nonumber
%%\label{SchurComplement}
%%&\lambda
%%\left[
%%\begin{array}{cc}
%%  \PHI_1 & \PI_1^H \\
%%  \PI_1 & \PI_1^H\PHI_1^{-1}\PI_1
%%\end{array}
%%\right]
%%+
%%(1-\lambda)
%%\left[
%%\begin{array}{cc}
%%  \PHI_2 & \PI_2^H \\
%%  \PI_2 & \PI_2^H\PHI_2^{-1}\PI_2
%%\end{array}
%%\right]
%%=
%%\\
%%&\left[
%%\begin{array}{cc}
%%  \lambda\PHI_1+(1-\lambda)\PHI_2 & \lambda\PI_1^H+(1-\lambda)\PI_2^H \\
%%  \lambda\PI_1+(1-\lambda)\PI_2 & \lambda\PI_1^H\PHI_1^{-1}\PI_1+(1-\lambda)\PI_2^H\PHI_2^{-1}\PI_2
%%\end{array}
%%\right]
%%\succeq 0.
%%\end{align}
%%Applying Schur's complement to \eqref{SchurComplement}, we have
%%\begin{align}
%%\nonumber
%%&\lambda\PI_1^H\PHI_1^{-1}\PI_1+(1-\lambda)\PI_2^H\PHI_2^{-1}\PI_2 \succeq
%%\\
%%&(\lambda\PI_1+(1-\lambda)\PI_2)(\lambda\PHI_1+(1-\lambda)\PHI_2)^{-1}(\lambda\PI_1^H+(1-\lambda)\PI_2^H)
%%\end{align}
%%which means that
%%\begin{align}
%%\nonumber
%%&\lambda\psi(\PI_1,\PHI_1)+(1-\lambda)\psi(\PI_2,\PHI_2)
%%\\
%%&\succeq \psi\big(\lambda(\PI_1,B_1)+(1-\lambda)(\PI_2,\PHI_2)\big)
%%\end{align}
%%The proof of lemma \ref{psi:inequality} is completed.
%\end{proof}
%
%The relationship between function $h(\psi(\PI,\PHI))$ and its variable $\psi(\PI,\PHI)$ can be expressed in the following lemma
%\begin{lemma}
%\label{psi:monotonic}
%$h(\psi(\PI,\PHI))$ is a concave and monotonically decreasing function in $\pmb{I}_d\succeq\psi\succeq0$, which means that
%\begin{align}
%h(\psi_2) \leq h(\psi_1), \forall \,\,0\preceq \psi_1 \preceq \psi_2 \preceq \pmb{I}_d.
%\end{align}
%\end{lemma}
%\begin{proof}
%$h(\psi(\PI,\PHI))$ is well known as a concave function.
%Since $\pmb{I}_d-\psi_2\preceq\pmb{I}_d-\psi_1$, we have $|\pmb{I}_d-\psi_2|\leq|\pmb{I}_d-\psi_1|$ and thus, $h(\psi_2) \leq h(\psi_1)$.
%The proof of lemma \ref{psi:monotonic} is completed.
%\end{proof}
%
%From lemma \ref{psi:inequality} and lemma \ref{psi:monotonic}, the property of $h$ in $\PI$ and $\PHI\succ 0$ is revealed in the following theorem.
%\begin{theorem}
%\label{Theorem:upperbound}
%Function $h(\PI,\PHI)$ is jointly concave in $(\PI,\PHI)$ and the following upper bound holds for all $\PI$ and $\PHI\succ 0$
%\begin{align}
%\label{upperbound}
%\nonumber
%&h(\PI,\PHI)\leq
%\\
%&h(\PI^{(n)},\PHI^{(n)}) + \tr\big(\nabla^H h(\PI^{(n)},\PHI^{(n)})((\PI,\PHI)-(\PI^{(n)},\PHI^{(n)}))\big)
%\end{align}
%where
%\begin{align}
%\label{diffential}
%\nonumber
%&\tr\left(\nabla^H h(\PI^{(n)},\PHI^{(n)})\left((\PI,\PHI)-(\PI^{(n)},\PHI^{(n)})\right)\right)=
%\\
%\nonumber
%&-2\Re
%\left\{
%\tr
%\left(
%\left(\left(\PHI^{(n)}-\PI^{(n)}(\PI^{(n)})^H\right)^{-1}\PI^{(n)}\right)^H(\PI-\PI^{(n)})
%\right)
%\right\}
%\\
%&+\tr
%\left(
%\left(\left(\PHI^{(n)}-\PI^{(n)}(\PI^{(n)})^H\right)^{-1}-(\PHI^{(n)})^{-1}\right)^H(\PHI-\PHI^{(n)})
%\right)
%\end{align}
%\end{theorem}
%\begin{proof}
%From lemma \ref{psi:inequality} and lemma \ref{psi:monotonic},
%the proof of Theorem \eqref{Theorem:upperbound} is obtained similarly as \cite{Tam2016}, and thus omitted.
%%for all $\lambda\in\{0,1\}$, we have
%%\begin{align}
%%\label{h:concave:proof}
%%\nonumber
%%&h(\lambda(\PI_1,\PHI_1)+(1-\lambda)(\PI_2,\PHI_2))
%%\\
%%\nonumber
%%&=h(\psi(\lambda(\PI_1,\PHI_1)+(1-\lambda)(\PI_2,\PHI_2)))
%%\\
%%\nonumber
%%&\geq h(\lambda\psi(\PI_1,\PHI_1)+(1-\lambda)\psi(\PI_2,\PHI_2))
%%\\
%%&\geq \lambda h(\psi(\PI_1,\PHI_1))+(1-\lambda)h(\psi(\PI_2,\PHI_2))
%%\end{align}
%%and \eqref{h:concave:proof} proves the concavity of $h$ in $\PI$ and $\PHI\succeq 0$.
%%The upper bound in \eqref{upperbound} is followed from the concavity of $h$ \cite{HTuy1998}.
%%To prove \eqref{diffential}, we rewrite function $h(\PI,\PHI)$ as follows
%%\begin{align}\label{h:rewrite:2}
%%h(\PI,\PHI) = \log|\PHI-\PI\PI^H| - \log|\PHI|,
%%\end{align}
%%and recall some useful results in the matrix differential calculus for given matrix function $\A$ as \cite{Petersen2012}
%%\begin{align}
%%\label{diff1}
%%\partial(\A^{-1})&=-\A^{-1}(\partial\A)\A^{-1}
%%\\
%%\label{diff2}
%%\partial(\log|\A|)&=\tr(\A^{-1}\partial\A)
%%\end{align}
%%
%%Applying \eqref{diff1} and \eqref{diff2} to the differential of $h$ in $\PI$ and $\PHI$, we have
%%\begin{align}
%%\nonumber
%%\partial h(\PI)
%%&=\partial(\log|\PHI-\PI\PI^H|)
%%\\
%%\nonumber
%%&=\tr
%%\left(
%%(\PHI-\PI\PI^H)^{-1}\partial(\PHI-\PI\PI^H)
%%\right)
%%\\
%%&=-2
%%\tr
%%\left(
%%(\PHI-\PI\PI^H)^{-1}\PI\partial\PI^H
%%\right)
%%\\
%%\nonumber
%%\partial h(\PHI) &= \partial(\log|\PHI-\PI\PI^H|) - \partial(\log|\PHI|)
%%\\
%%\nonumber
%%&=\tr\left((\PHI-\PI\PI^H)^{-1}\partial(\PHI-\PI\PI^H)\right)-\tr\left(\PHI^{-1}\partial\PHI\right)
%%\\
%%&=\tr\left(\left((\PHI-\PI\PI^H)^{-1}-\PHI^{-1}\right)\partial\PHI\right)
%%\end{align}
%%which means that
%%\begin{align}
%%\label{diff:PI}
%%\nabla_{\PI}h(\PI,\PHI) &= -2(\PHI-\PI\PI^H)^{-1}\PI,
%%\\
%%\label{diff:PHI}
%%\nabla_{\PHI}h(\PI,\PHI) &= (\PHI-\PI\PI^H)^{-1}-\PHI^{-1},
%%\end{align}
%%The differential of $h$ at the given point $(\PI^{(n)},\PHI^{(n)})$ can be approximated as
%%\begin{align}
%%\nonumber
%%&\tr\big(\nabla^H h(\PI^{(n)},\PHI^{(n)})((\PI,\PHI)-(\PI^{(n)},\PHI^{(n)}))\big)
%%\\
%%\nonumber
%%\label{diff:full}
%%=&\,\,\Re\left\{\tr\left(\nabla_{\PI}^Hh(\PI^{(n)},\PHI^{(n)})(\PI-\PI^{(n)})\right)\right\}
%%\\
%%&+ \tr\left(\nabla_{\PHI}^Hh(\PI^{(n)},\PHI^{(n)})(\PHI-\PHI^{(n)})\right)
%%\end{align}
%%Substituting \eqref{diff:PI} and \eqref{diff:PHI} to \eqref{diff:full}, the proof of theorem \ref{Theorem:upperbound} is completed.
%\end{proof}
%
%It should be noted that $\left(\PHI-\PI\PI^H\right)^{-1}-\PHI^{-1}\succeq 0$. Therefore, function $\Gamma_{f_k,m}(\bar{\F})$ in \eqref{Gamma} is a concave quadratic function. Finally, from \eqref{g:rewrite:1} and theorem \ref{Theorem:upperbound}, the proof of theorem \ref{Theorem:lowerbound} is completed.
%\end{proof}

\section*{Acknowledgment}
This work is supported in part by an ECR-HDR scholarship from The University of Newcastle and in part by the Australian Research Council Discovery Project grant DP170100939.


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\end{thebibliography}
