\section{Introduction}
Cell-free massive MIMO networks have attracted significant interest for their potential in enhancing the performance of future generation mobile access networks. The main focus is the evolution of known coordinated multi-point (CoMP) concepts towards practically attractive access solutions that combine the benefits of access point (AP) cooperation and ultra-dense deployments. To this end, considerable research effort has been devoted to the development of scalable and possibly user-centric system architectures and algorithms covering, for instance, power control, pilot-based channel estimation, joint processing such as precoding and combining, fronthaul overhead, network topology, and initial access \cite{ngo2017cellfree,nayebi2017precoding,interdonato2019scalability,bashar2019maxmin,bashar2019quantization,interdonato2019ubiquitous,buzzi2020,emil2020scalable,attarifar2020subset,liu2020analysis,italo2021ota,interdonato2021enhanced,dandrea2021soft,lancho2021short,du2021cellfree,shaik2021mmse,chen2021access,gottsch2022subspace}.      

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Against this background, in this study we address the open problem of optimal joint downlink precoding design in cell-free massive MIMO networks, by considering minimum quality-of-service requirements and by assuming that each AP is subject to an individual power constraint and an individual information constraint. The information constraints model limited AP cooperation capabilities, which are motivated by the need for realizing scalable cell-free architectures with reduced fronthaul and joint processing load. More specifically, the information constraints model:
\begin{itemize}
\item limited sharing of data bearing signals, for instance, only within user-centric cooperation clusters as in \cite{buzzi2020,emil2020scalable}; and
\item limited sharing of channel state information (CSI), covering, for instance, the local CSI model as in the original version of the cell-free massive MIMO paradigm \cite{ngo2017cellfree}.
\end{itemize}
Each AP must form its transmit signal as a function of the data bearing signals and CSI specified by the constraints, and no additional information exchange between the APs is allowed. This is in contrast to related works such as \cite{italo2021ota}, which covers iterative information exchange during precoding computation. In the cell-free massive MIMO literature, the best performing joint precoders are typically designed from joint uplink combiners, motivated by a known uplink-downlink duality principle for fading channels \cite[Ch.~4]{massivemimobook} \cite[Ch.~6]{demir2021foundations}. However, optimal joint precoders are generally unknown owing to the following two reasons. First, the known uplink-downlink duality principle for fading channels holds for a looser and somewhat less practical sum power constraint. Second, until very recently, optimal joint combiners were not known except for the case of full CSI sharing within each cooperation cluster, an information constraint leading to so-called \textit{centralized} combining.

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A partial solution to the first issue is given by the alternative uplink-downlink duality principle under per-antenna power constraints developed in \cite{yu2007transmitter}, applied for instance in \cite{emil2013optimal} in the context of CoMP. However, the method in \cite{yu2007transmitter} applies to deterministic channels, i.e., to fixed channel fading realizations, which imposes many limitations. A first limitation is that optimal schemes typically involve solving relatively complex optimization problems for each channel realization, which may be impractical in large systems. Perhaps the most important limitation is that addressing the second issue, i.e., designing optimal \textit{distributed} combiners under limited CSI sharing, becomes generally ill-posed for deterministic channels. This is because standard optimization problems taking as input a fixed channel realization produce solutions that may depend on fixed yet unknown channel variables, and hence violate the information constraints. Addressing the second issue is known to be quite challenging. In essence, depending on the information constraints, each AP may need to take combining decisions that are robust not only to channel estimation errors, but also to the possibly unknown combining decisions taken at the other APs. A novel method for optimally addressing this issue, called the \textit{team} minimum mean-square error (MMSE) method, has been recently proposed in \cite{miretti2021team}. Together with the known uplink-downlink duality principle for fading channels, this method is used in \cite{miretti2021team,miretti2021team2} to derive novel joint precoders under various relevant information constraints, including the aforementioned local CSI model. However, the obtained solutions are provenly optimal  under a sum power constraint only.

To address the above limitations, in this study\footnote{A part of the results in this study is presented in \cite{miretti2022duality} without proof. This study extends \cite{miretti2022duality} by providing complete derivations, the expressions for optimal centralized and local precoding, and additional details on the numerical implementation of the proposed algorithms.} we derive a novel uplink-downlink duality principle for fading channels under per-AP power constraints, which, as discussed in details throughout this study, significantly differs from \cite{yu2007transmitter}. Furthermore, building on this result, we show that optimal joint precoders are given by solutions to properly parametrized MMSE problems under per-AP information constraints, i.e., as properly parametrized variations of team MMSE precoders. In summary, this work can be interpreted as a nontrivial generalization of the method in \cite{yu2007transmitter} to fading channels, and of the method in \cite{miretti2021team} to per-AP power constraints. Moreover, we illustrate the application of our findings by deriving novel solutions to the problems of optimal centralized and local precoding design in user-centric cell-free massive MIMO networks under per-AP power constraints. 

The paper is organized as follows. Section~\ref{sec:math} and Section~\ref{sec:model} provide the main definitions, mathematical tools, and modeling assumptions. Section~\ref{sec:lagrange} presents and studies the main optimization problem using Lagrangian duality arguments. Building on the obtained insights, Section~\ref{sec:duality} derives the proposed uplink-downlink duality principle, which is then exploited in Section~\ref{sec:applications} to characterize the optimal solution structure. Simple applications of the main results are illustrated in Section~\ref{sec:sim} by means of numerical simulations. Finally, Section~\ref{sec:conclusion} summarizes the main results, and outlines some limitations and possible future directions.

\section{Mathematical preliminaries}
\label{sec:math}
\subsection{Notation and definitions}
We denote by $\stdset{R}_+$ and $\stdset{R}_{++}$ the sets of, respectively, nonnegative and positive reals. The Euclidean norm in $\stdset{C}^K$ is denoted by $\|\cdot\|$. Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space. We denote by $\set{H}^K$ the set of complex-valued random vectors, i.e., $K$-tuples of $\Sigma$-measurable functions $\Omega \to \stdset{C}$ satisfying $(\forall \rvec{h}\in \set{H}^K)$ $\E[\|\rvec{h}\|^2]<\infty$. Together with the standard operations of addition and real scalar multiplication, we recall that $\set{H}^K$ is a real vector space. Given a random variable $X\in \set{H}$, we denote by $\E[X]$ and $\V(X)$ its expected value and variance, respectively. Inequalities involving vectors in $\stdset{R}^K$ should be understood coordinate-wise. The $k$th column of the $K$-dimensional identity matrix $\vec{I}_K$ is denoted by $\vec{e}_k$. 

\subsection{Lagrangian duality in general vector spaces} 
The following key result is frequently invoked throughout our study, and can be found in \cite{zalinescu2002convex}.
\begin{proposition}\label{prop:duality}
Consider the functions $f: \set{X} \to \stdset{R}$ and $\vec{g}: \set{X} \to \stdset{R}^N$, where $\set{X}$ is a real vector space, and the optimization problem
\begin{equation*}
\begin{aligned}
\underset{X \in \set{X}} {\text{minimize}} \quad &  f(X)\\
\text{subject to} \quad & \vec{g}(X)\leq \vec{0}.
\end{aligned}
\end{equation*}
Define the primal optimum $p^\star := \inf\{f(X)~|~\vec{g}(X)\leq \vec{0}, X\in\set{X}\}$, and the dual optimum $ d^\star := \sup\{d(\vec{\lambda})~|~\vec{\lambda}\in \stdset{R}^N_+\}$, where $d(\vec{\lambda}):=\inf\{f(X)+\vec{\lambda}^\T\vec{g}(X)~|~ X\in\set{X}\}$ for $\vec{\lambda}\in \stdset{R}_+^N$. Each of the following holds: 
\begin{enumerate}[(i)]
\item $d^\star \leq p^\star$ (weak duality);
\item If $f$ and $\vec{g}$ are proper convex functions \cite[pp. 39]{zalinescu2002convex}, and $\{X\in \set{X} ~|~ \vec{g}(X) < \vec{0}\}\neq \emptyset$ (Slater's condition), then $d^\star = p^\star$ holds (strong duality). Furthermore, there exist Lagrangian multipliers $\vec{\lambda}^\star \in \stdset{R}_+^N$ such that $d(\vec{\lambda}^\star) = d^\star$.
\end{enumerate} 
\end{proposition} 
\begin{proof}
For statement (i), see \cite[Theorem 2.6.1(iii)]{zalinescu2002convex}. For statement (ii), see \cite[Theorem 2.9.3(ii)]{zalinescu2002convex}.
\end{proof}
For $\set{X} = \stdset{R}^K$, the above proposition corresponds to the familiar Lagrangian duality principle for optimization problems in Euclidean spaces \cite{boyd2004convex}. In this study, we exploit the more general result in Proposition~\ref{prop:duality} to address optimization problems in the real vector space of complex-valued random vectors $\set{X}=\set{H}^K$. 

\section{System model}
\label{sec:model}
\subsection{Downlink achievable rates}
Consider the downlink of a cell-free wireless network composed of $L$ APs indexed by $\mathcal{L}:=\{1,\ldots,L\}$, each of them equipped with $N$ antennas, and $K$ single-antenna UEs indexed by $\mathcal{K}:=\{1,\ldots,K\}$. By assuming  a standard synchronous and frequency-flat channel model governed by an ergodic and stationary fading process, and simple transmission techniques based on 
linear precoding and on treating interference as noise, we focus on simultaneously achievable ergodic rates in the classical Shannon sense, approximated by the popular \emph{hardening} inner bound \cite{marzetta2016fundamentals}. In more detail, we define the downlink rates achieved by each UE $k\in \set{K}$ for a given precoding design as  
\begin{equation}\label{eq:hardening_bound}
R_k^{\mathrm{DL}}(\rvec{T}) :=  \log\left(1+\mathrm{SINR}_k^{\mathrm{DL}}(\rvec{T})\right),
\end{equation}
\begin{equation}\mathrm{SINR}_k^{\mathrm{DL}}(\rvec{T}) := \dfrac{|\E[\rvec{h}_k^\herm\rvec{t}_k]|^2}{\V(\rvec{h}_k^\herm\rvec{t}_k) + \sum_{j\neq k}\E[|\rvec{h}_k^\herm\rvec{t}_j|^2]+1},
\end{equation}
where $\rvec{h}_k \in\set{H}^{NL}$ is a random channel vector modeling the fading state between UE $k$ and all APs, $\rvec{t}_k \in \set{H}^{NL}$ is a joint precoding vector applied by all APs to the coded and modulated data bearing signal for UE~$k$, and $\rvec{T} := [\rvec{t}_1,\ldots,\rvec{t}_K]\in \set{H}^{NL\times K}$ is the aggregate joint precoding matrix. We stress that precoders are defined and denoted as random quantities, since they may adapt to random fading realizations on the basis of the the available instantaneous CSI. This aspect is treated in detail in the next sections.  

\subsection{Per-AP power and information constraints}
\label{ssec:info}
In practical cell-free wireless networks, each AP must typically satisfy an individual power constraint. In addition, motivated by the need for realizing scalable cell-free architectures, each AP is also typically subject to an individual information constraint induced by limited data and instantaneous CSI sharing, which impair its cooperation capabilities. In this work, the above per-AP constraints are modelled as follows. Let $[\rvec{t}_{1,k}^\T,\ldots,\rvec{t}_{L,k}^\T]^\T := \rvec{t}_k$, where $\rvec{t}_{l,k} \in \set{H}^N$ denotes the portion of the precoder applied by AP~$l\in\set{L}$ to serve UE~$k\in\set{K}$. By assuming unitary power data bearing signals, we consider the average power constraints
\begin{equation}
(\forall l \in \set{L})~\sum_{k=1}^K\E[\|\rvec{t}_{l,k}\|^2]\leq P_l < \infty.
\end{equation}
For modeling impairments related to limited CSI sharing, we follow the recently proposed approach in \cite{miretti2021team} and let \begin{equation}\label{eq:CSI}
(\forall k \in \set{K})~\rvec{t}_k\in \set{T}_k := \set{H}_1^N \times \ldots \times \set{H}_L^N,
\end{equation}
where $\set{H}_l^N\subseteq \set{H}^N$ denotes the set of $N$-tuples of $\Sigma_l$-measurable functions $\Omega \to \stdset{C}$ satisfying $(\forall \rvec{h}\in \set{H}_l^N)$ $\E[\|\rvec{h}\|^2]<\infty$, and where $\Sigma_l \subseteq \Sigma$ is the sub-$\sigma$-algebra induced by the available CSI at AP $l\in \set{L}$, also called the \emph{information subfield} of AP $l$ \cite{yukselbook}. Informally, this constraint enforces the precoders of each AP to be functions of the available CSI only. From a mathematical point of view, $\set{H}_l^N$ is a subspace of the real vector space $\set{H}^N$, which in turn implies that the constraint in $\eqref{eq:CSI}$ models limited CSI sharing by constraining $\rvec{t}_k$ within a subspace $\set{T}_k$ of the real vector space $\set{H}^{LN}$ \cite{yukselbook}.  
\begin{remark}
\label{rem:CSI}
The constraint in \eqref{eq:CSI} is fairly general. In particular, it covers the case of local CSI \cite{ngo2017cellfree} (i.e., where each AP has information on only the channel between the UEs and itself), but also more advanced cooperation structures involving quantized or delayed CSI sharing, or exploiting the peculiarities of efficient fronthauls such as in the so-called radio stripes concept where the APs are daisy-chained \cite{miretti2021team}. To keep our results general, in most of this study we do not specify the CSI structure. However, examples are provided in Section~\ref{sec:applications}. 
\end{remark}
Furthermore, following the well-known user-centric network clustering approach \cite{buzzi2020,emil2020scalable}, we assume that each UE $k \in \set{K}$ is only served by a subset $\set{L}_k\subseteq \set{L}$ of APs. As shown in \cite{miretti2021team2}, this additional practical impairment can be straightforwardly included in \eqref{eq:CSI} by replacing $\set{H}_l^N$ with the set $\{\vec{0}_N~ (\mathrm{a.s.})\}$ for each AP~$l \notin \set{L}_k$, i.e., not serving UE~$k$. Since $\{\vec{0}_N~ (\mathrm{a.s.})\}$ is a (trivial) subspace of $\set{H}^N$, this replacement does not alter the property of $\set{T}_k$ being a real vector space (in fact, a subspace of $\set{H}^{NL}$). This key property will allow us to address the problem of optimal joint precoding design under per-AP information constraints using Lagrangian duality for real vector spaces (Proposition~\ref{prop:duality}). 

\subsection{Dual uplink rates under arbitrary noise powers}\label{ssec:dual_UL_model}
The main optimization approach developed in this work has a natural interpretation in terms of a virtual dual uplink channel with arbitrary noise powers. More specifically, similarly to the chosen downlink model, we consider virtual uplink ergodic rates given by the \textit{use-and-then-forget} inner bound \cite{marzetta2016fundamentals}  
\begin{equation}\label{eq:UatF_bound}
R_k^{\mathrm{UL}}(\rvec{v}_k,\vec{p},\vec{\sigma}) :=  \log\left(1+\mathrm{SINR}_k^{\mathrm{UL}}(\rvec{v}_k,\vec{p},\vec{\sigma}) \right),
\end{equation}
\begin{equation}
\begin{split}&\mathrm{SINR}_k^{\mathrm{UL}}(\rvec{v}_k,\vec{p},\vec{\sigma}) := \\ 
&\dfrac{p_k|\E[\rvec{h}_k^\herm\rvec{v}_k]|^2}{p_k\V(\rvec{h}_k^\herm\rvec{v}_k) + \sum_{j\neq k}p_j\E[|\rvec{h}_j^\herm\rvec{v}_k|^2]+\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2]},
\end{split}
\end{equation}
where $\rvec{v}_k=[\rvec{v}_{1,k}^\T,\ldots,\rvec{v}_{L,k}^\T]^\T\in \set{H}^{NL}$ is a joint combiner, $\vec{p} := [p_1,\ldots,p_K]^\T \in \stdset{R}_{+}^K$ is a vector of transmit powers, and where we define $(\forall \rvec{v}_k \in \set{H}^{NL})(\forall \vec{\sigma} \in \stdset{R}_{++}^L)$
\begin{equation}\label{eq:norm}
\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] := \sum_{l=1}^L\sigma_l\E[\|\rvec{v}_{l,k}\|^2]
\end{equation}
for given $\vec{\sigma}:=[\sigma_1,\ldots,\sigma_L]^\T \in \stdset{R}_{++}^L$. In the above expressions, $\vec{\sigma}$ can be interpreted as a vector collecting uplink noise powers for each AP. We remark that the term \textit{virtual} here refers to the fact that the above rates may not be achievable in the true uplink channel, since $(\vec{p},\vec{\sigma})$ may differ from the true uplink transmit and noise powers. The major difference between $R_k^{\mathrm{UL}}$ and $R_k^{\mathrm{DL}}$ is that the former depends only on the joint combiner $\rvec{v}_k$ for the signal of UE~$k\in\set{K}$, while the latter depends on the entire precoding matrix $\rvec{T}$. In fact, the uplink achievable rates are coupled only via the transmit and noise powers $\vec{p},\vec{\sigma}$. This known aspect makes optimization on the uplink channel generally easier than on the downlink channel. 

\section{Problem statement and Lagrangian duality}
\label{sec:lagrange}
To address the problem of optimal joint precoding design in cell-free networks, in this section we study a certain SINR feasibility problem under per-AP power and information constraints. In particular, given a tuple of power constraints $(P_1,\ldots,P_L)\in \stdset{R}_{++}^K$ and of SINR requirements $(\gamma_1,\ldots,\gamma_K)\in \stdset{R}_{++}^K$, we consider the following infinite dimensional optimization problem:
\begin{equation}\label{prob:DL_QoS}
\begin{aligned}
\underset{\rvec{T} \in \set{T}} {\text{minimize}} \quad &  \sum_{k=1}^K\E[\|\rvec{t}_k\|^2]\\
\text{subject to} \quad & (\forall k \in \set{K})~\mathrm{SINR}^{\mathrm{DL}}_k(\rvec{T}) \geq \gamma_k,\\
& (\forall l \in \set{L})~\sum_{k=1}^K\E[\|\rvec{t}_{l,k}\|^2]\leq P_l,
\end{aligned}
\end{equation}
where $\set{T}\subset \set{H}^{NL\times K}$ is a real vector space obtained by collecting all per-AP information constraints defined in Section~\ref{ssec:info}. We recall that these constraints accommodate both limited instantaneous CSI sharing and user-centric network clustering. 
In the following, to avoid technical digressions, we assume that strictly feasible joint precoders exist, i.e., 
\begin{equation*}
\left\{ \rvec{T}\in\set{T} \middle| \begin{array}{l}
(\forall k \in \set{K}) ~\mathrm{SINR}^{\mathrm{DL}}_k(\rvec{T}) > \gamma_k \\
(\forall l \in \set{L})~\sum_{k=1}^K\E[\|\rvec{t}_{l,k}\|^2]< P_l
\end{array} \right\} \neq \emptyset.
\end{equation*} 
Furthermore, due to both mathematical convenience and practical reasons, in Problem~\eqref{prob:DL_QoS} we focus on the subset of feasible joint precoders minimizing the total power consumption.

\subsection{Lagrangian dual problems}
Inspired by the related results in \cite{yu2007transmitter} based on Lagrangian duality for finite dimensional optimization problems, in this section we apply Lagrangian duality for infinite dimensional optimization problems to the study of Problem~\eqref{prob:DL_QoS}. For convenience, we adopt the compact notation 
\begin{equation}
\mathrm{SINR}_k^{\mathrm{DL}}(\rvec{T}) = \dfrac{|b_k(\rvec{T})|^2}{|c_k(\rvec{T})|^2-|b_k(\rvec{T})|^2},
\end{equation}
where $b_k(\rvec{T}) := \E[\rvec{h}_k^\herm\rvec{t}_k]$ is the useful signal term, and $|c_k(\rvec{T})|^2 := \sum_{j=1}^K\E[|\rvec{h}_k^\herm\rvec{t}_j|^2]+1$ is the interference plus noise power term. Furthermore, we rearrange the SINR constraints using simple algebraic manipulations as 
\begin{equation}\label{eq:SINR_rearranged}
(\forall k \in \set{K})~ |c_k(\rvec{T})|-\nu_k|b_k(\rvec{T})|\leq 0,
\end{equation}
where $\nu_k:= \sqrt{1+1/\gamma_k}$. More specifically, we have the following simple property, provided without proof:
\begin{lemma}\label{lem:SINR_rewriting}
For all $\rvec{T}\in\set{T}$ and $k\in\set{K}$,
\begin{equation*}
\mathrm{SINR}_k^{\mathrm{DL}}(\rvec{T})\geq \gamma_k \iff |c_k(\rvec{T})|-\nu_k|b_k(\rvec{T})|\leq 0.
\end{equation*}
\end{lemma}
A Lagrangian dual problem to \eqref{prob:DL_QoS} is then given by
\begin{equation}\label{prob:DL_QoS_dual}
\underset{(\vec{\lambda},\vec{\mu})\in \stdset{R}_+^L\times \stdset{R}_+^K} {\text{maximize}}~d(\vec{\lambda},\vec{\mu}),
\end{equation}
where (recalling \eqref{eq:norm}) we define the dual function $d(\vec{\lambda},\vec{\mu}) :=$
\begin{equation*}
\inf_{\rvec{T} \in \set{T}}  \sum_{k=1}^K \E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2] -\sum_{l=1}^L \lambda_l P_l + \sum_{k=1}^K\mu_k(|c_k(\rvec{T})|-\nu_k|b_k(\rvec{T})|).
\end{equation*}
Since the primal problem in \eqref{prob:DL_QoS} is nonconvex, by Proposition~\ref{prop:duality} we can only guarantee (a-priori) weak duality. Furthermore, guaranteeing existence of a solution is not immediate. However, the following important result holds.
\begin{proposition}\label{prop:strong_duality}
Problem~\eqref{prob:DL_QoS} admits a solution $\rvec{T}^\star\in \set{T}$. Furthermore, denote by $p^\star$ and $d^\star$ the optimum of the primal problem \eqref{prob:DL_QoS} and of the dual problem \eqref{prob:DL_QoS_dual}, respectively. Strong duality holds, i.e., $d^\star = p^\star$, and there exist Lagrangian multipliers $(\vec{\lambda}^\star,\vec{\mu}^\star)\in \stdset{R}_+^L\times \stdset{R}_+^K$ solving Problem~\eqref{prob:DL_QoS_dual}.
\end{proposition}
\begin{proof}
The proof follows a similar idea as in \cite{yu2007transmitter,wiesel2006linear}, and it is based on establishing connections between Problem~\eqref{prob:DL_QoS} and a convex reformulation obtained by replacing $|b_k(\rvec{T})|$ in \eqref{eq:SINR_rearranged} with $\Re(b_k(\rvec{T}))$. For additional details, see Appendix~\ref{app:convex_reformulation}. 
\end{proof}

We conclude this section by stating a useful consequence of Proposition~\ref{prop:strong_duality} that will be instrumental for proving our main results on uplink-downlink duality. In particular, we provide an alternative version of Proposition~\ref{prop:strong_duality} based on a \emph{partial} dual problem, obtained by keeping the SINR constraints implicit.
\begin{lemma}\label{lem:partial_dual}
Given the subset $\set{T}_{\vec{\gamma}} := \{\rvec{T}\in\set{T} ~|~ (\forall k \in \set{K})~\mathrm{SINR}_k^{\mathrm{DL}}(\rvec{T}) \geq \gamma_k\}$ of precoders satisfying the SINR constraints in Problem~\eqref{prob:DL_QoS}, define the \emph{partial} dual problem 
\begin{equation}\label{prob:partial_dual}
\underset{\vec{\lambda}\in \stdset{R}_+^L} {\text{maximize}}~\tilde{d}(\vec{\lambda}) := \inf_{\rvec{T} \in \set{T}_{\vec{\gamma}}} \sum_{k=1}^K \E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2] -\sum_{l=1}^L \lambda_l P_l.
\end{equation}
Strong duality holds, i.e., Problem~\eqref{prob:DL_QoS} and Problem~\eqref{prob:partial_dual} have the same optimum $p^\star$. Furthermore, there exist Lagrangian multipliers $\vec{\lambda}^\star$ solving Problem~\eqref{prob:partial_dual}.
\end{lemma}
\begin{proof}
Consider the alternative optimization problem 
\begin{equation}\label{prob:indicator}
\begin{aligned}
\underset{\rvec{T} \in \set{T}} {\text{minimize}} \quad &  \sum_{k=1}^K\E[\|\rvec{t}_{k}\|^2] + \gamma(\rvec{T})\\
\text{subject to} \quad & (\forall l \in \set{L})~\sum_{k=1}^K\E[\|\rvec{t}_{l,k}\|^2]\leq P_l,
\end{aligned}
\end{equation}
where $\gamma(\rvec{T}) = 0$ if $\rvec{T}$ belongs to the set $\set{T}_{\vec{\gamma}}$, and $\gamma(\rvec{T}) = +\infty$ otherwise. Problem~\eqref{prob:indicator} is equivalent to Problem~\eqref{prob:DL_QoS}, in the sense that it has the same optimum $p^\star$ and set of  optimal solutions. Its Lagrangian dual problem can be written as \eqref{prob:partial_dual}. By applying weak duality (see Proposition~\ref{prop:duality}) to the term $\inf_{\rvec{T} \in \set{T}_{\vec{\gamma}}} \sum_{k=1}^K \E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2]$ for any fixed $\vec{\lambda}$, and by rewriting $\set{T}_{\vec{\gamma}}$ according to Lemma~\ref{lem:SINR_rewriting}, we obtain
$\sup_{\vec{\mu}\in\stdset{R}_+^K}d(\vec{\lambda},\vec{\mu})\leq \tilde{d}(\vec{\lambda})$. Taking the supremum over $\vec{\lambda}$ on both sides gives
\begin{equation*}
p^\star = \sup_{\vec{\lambda}\in\stdset{R}_+^L}\sup_{\vec{\mu}\in\stdset{R}_+^K}d(\vec{\lambda},\vec{\mu}) \leq \sup_{\vec{\lambda}\in\stdset{R}_+^L} \tilde{d}(\vec{\lambda}) \leq p^\star,
\end{equation*}
where the first equality follows from Proposition~\ref{prop:strong_duality} and the last inequality follows from weak duality applied to Problem~\eqref{prob:indicator}. The second part of the statement follows from the existence of a solution $(\vec{\lambda}^\star,\vec{\mu}^\star)$ to Problem~\eqref{prob:DL_QoS_dual}, and $p^\star = \sup_{\vec{\mu}\in\stdset{R}_+^K}d(\vec{\lambda}^\star,\vec{\mu}) \leq  \tilde{d}(\vec{\lambda}^\star) \leq p^\star$.
\end{proof}

\subsection{Primal-dual solution methods}
A key aspect of Lagrangian optimization is the possibility of recovering a primal solution from a dual solution. However, we emphasize that this is not always possible even if strong duality holds. Nevertheless, the following proposition ensures that a primal solution $\rvec{T}^\star$ to Problem~\eqref{prob:DL_QoS} can be indeed recovered from a solution to the partial dual problem~\eqref{prob:partial_dual}.
\begin{proposition}\label{prop:primal-dual}
Let $\vec{\lambda}^\star$ be a solution to Problem~\eqref{prob:partial_dual}. Then, a solution to Problem~\eqref{prob:DL_QoS} is given by any solution to
\begin{equation}\label{prob:DL_QoS_augmented}
\underset{\rvec{T}\in\set{T}_{\vec{\gamma}}} {\text{minimize}}~\sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}^\star}^2].
\end{equation}
\end{proposition}
\begin{proof}
The proof is given in Appendix~\ref{app:primaldual}, and it is based on the same convex reformulation of the SINR constraints adopted in the proof of Proposition~\ref{prop:strong_duality}.
\end{proof}
Starting from Proposition~\ref{prop:primal-dual}, and in particular by studying and solving Problem~\eqref{prob:DL_QoS_augmented}, in the next section we derive structural properties for optimal joint precoding. However, before moving to the next section, we first complete the discussion on recovering a primal solution from a dual solution by illustrating a simple algorithm for solving Problem~\eqref{prob:partial_dual}, which is a concave maximization problem. In particular, we consider a standard primal-dual iterative algorithm based on the projected subgradient method \cite{polyak1969minimization,nesterov2003introductory}.
\begin{lemma}\label{lem:proj_grad}
Choose $\vec{\lambda}^{(1)} \in \stdset{R}_{+}^L$ and a sequence $\{\alpha^{(i)}\}_{i\in \stdset{N}}$ such that
\begin{equation*}
(\forall i \in \stdset{N})~\alpha^{(i)}\in\stdset{R}_{++}, \quad \lim_{i\to \infty} \alpha^{(i)} = 0, \quad \sum_{i\in \stdset{N}} \alpha^{(i)} = \infty.
\end{equation*}
Define the sequence $(\vec{\lambda}^{(i)})_{i\in\stdset{N}}$ generated via 
\begin{equation*}
(\forall i \in \stdset{N})~\vec{\lambda}^{(i+1)} = \max\left\{\vec{\lambda}^{(i)} + \frac{\alpha^{(i)}}{\|\vec{g}(\vec{\lambda}^{(i)})\|}\vec{g}(\vec{\lambda}^{(i)}) ,\vec{0}\right\},
\end{equation*}
where the $l$th entry of $\vec{g}(\vec{\lambda})$ is given by $(\forall \vec{\lambda}\in \stdset{R}_+^L)(\forall l\in \set{L})$,
\begin{equation*}\label{eq:supergradient}
g_l(\vec{\lambda}):=\sum_{k=1}^K\E[\|\rvec{t}_{l,k}\|^2]- P_l, \quad 
\rvec{T} \in \arg\min_{\set{T}_{\vec{\gamma}}}\sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2].
\end{equation*}
Then, the subsequence of $(\vec{\lambda}^{(i)})_{i\in\stdset{N}}$ corresponding to the best objective after $n$ iterations $\max_{i=1,\ldots,n}\tilde{d}(\vec{\lambda}^{(i)})$  converges to a solution $ \vec{\lambda}^\star$ to Problem~\eqref{prob:partial_dual}.
\end{lemma}
\begin{proof} The proof is given in Appendix~\ref{app:proof_subgradient}.
\end{proof}
Note that the above algorithm requires a method for solving Problem~\eqref{prob:DL_QoS_augmented} for arbitrary Lagrangian multipliers. This is provided in the following section.

\section{Uplink-downlink duality}
\label{sec:duality}
Building on the above analysis based on Lagrangian duality, in this section we present our main result, which states that the problem of optimal joint precoding design under per-AP power and information constraints can be reformulated as a joint combining design and long-term power control problem in a dual uplink channel with properly designed noise vector $\vec{\sigma}$ (see Section \ref{ssec:dual_UL_model}). More precisely, we show later in Proposition~\ref{prop:UL-DL} that an optimal solution to Problem~\eqref{prob:DL_QoS} can be recovered from a solution to 
\begin{equation}\label{prob:dualUL}
\begin{aligned}
\underset{\rvec{V} \in \set{T}, \vec{p} \in \stdset{R}_+^K} {\text{minimize}} \quad  &  \sum_{k=1}^Kp_k\\
\text{subject to} \quad & (\forall k \in \set{K})~\mathrm{SINR}^{\mathrm{UL}}_k(\rvec{v}_k,\vec{p},\vec{\sigma}) \geq \gamma_k,\\
& (\forall k \in \set{K})~\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] = 1,
\end{aligned}
\end{equation} 
for some $\vec{\sigma} \in \stdset{R}_{++}^L$, and where $\rvec{V} := [\rvec{v}_1,\ldots,\rvec{v}_k]$.
In addition, we present an efficient numerical method that solves the above problem.

\begin{remark} Our derivation differs significantly from the derivation of the related result in \cite{yu2007transmitter}. Specifically, \cite{yu2007transmitter} exploits the peculiar structure of optimal centralized precoding in deterministic channels, its relation to a certain Rayleigh quotient, and a series of properties from the theory of semidefinite programming. Unfortunately, these arguments do not seem applicable to our setup, which covers distributed precoding and random channels. To address this limitation, we follow a different path and replace the above arguments by a variation of well-known uplink-downlink duality results under a sum power constraint, reviewed, e.g., in \cite{schubert2004solution,massivemimobook}.
\end{remark}  

 

\subsection{Joint precoding optimization over a dual uplink channel}

The desired connection between the downlink channel and its dual uplink channel is established by studying Problem~\eqref{prob:DL_QoS_augmented}, the solutions of which are optimal joint precoders solving Problem~\eqref{prob:DL_QoS}. The key idea lies in interpreting $\sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}^\star}^2]$ as an unconventional weighted definition of the average sum transmit power. To keep the discussion general and, for instance, applicable to the algorithm given by Lemma~\ref{lem:proj_grad}, we consider arbitrary Lagrangian multipliers, i.e., we consider the following problem: $(\forall \vec{\sigma}\in \stdset{R}_{++}^L)$
\begin{equation}\label{prob:DL_QoS_augmented_general}
\begin{aligned}
\underset{\rvec{T} \in \set{T}} {\text{minimize}} \quad  & \sum_{k=1}^K \E[\|\rvec{t}_k\|_{\vec{\sigma}}^2]\\
\text{subject to} \quad & (\forall k \in \set{K})~\mathrm{SINR}^{\mathrm{DL}}_k(\rvec{T}) \geq \gamma_k.
\end{aligned}
\end{equation} 
Since the SINR constraints are feasible by assumption, following the same arguments as in the proof of Lemma~\ref{lem:proj_grad}, we observe that the above problem always admits a solution.

\begin{proposition}\label{prop:UL-DL}
For all $\vec{\sigma}\in \stdset{R}_{++}^L$, Problem~\eqref{prob:DL_QoS_augmented_general} and Problem~\eqref{prob:dualUL} have the same optimum.  Furthermore, a solution to Problem~\eqref{prob:DL_QoS_augmented_general} is given by
\begin{equation*}
(\forall k \in \set{K})~\rvec{t}_k^\star = \sqrt{q^\star_k}\rvec{v}_k^\star,
\end{equation*}
where $(\rvec{V}^\star,\vec{p}^\star)\in \set{T}\times\stdset{R}_{++}^K$ is a solution to Problem~\eqref{prob:dualUL}, and $\vec{q}^\star:= (q_1^\star,\ldots,q^\star_K) \in \stdset{R}_{++}^K$ is given by
\begin{equation*}
\vec{q}^\star = (\vec{D}-\vec{B})^{-1}(\vec{D}-\vec{B}^\T)\vec{p}^\star,
\end{equation*}
where 
\begin{equation*}
\vec{B}:= \begin{bmatrix}
\E[|\rvec{h}_1^\herm\rvec{v}^\star_1|^2] & \ldots & \E[|\rvec{h}_1^\herm\rvec{v}^\star_K|^2] \\
\vdots & \ddots & \vdots \\
\E[|\rvec{h}_K^\herm\rvec{v}^\star_1|^2] & \ldots & \E[|\rvec{h}_K^\herm\rvec{v}^\star_K|^2]
\end{bmatrix}, \quad \vec{D}:= \mathrm{diag}(\vec{d}),
\end{equation*}
\begin{equation*}
\vec{d}:= \left((1+\gamma_1^{-1})|\E[\rvec{h}^\herm_1\rvec{v}^\star_1]|^2,\ldots,(1+\gamma_K^{-1})|\E[\rvec{h}^\herm_K\rvec{v}^\star_K]|^2\right).
\end{equation*}

\end{proposition}
\begin{proof}
The function $\E[\|\cdot\|_{\vec{\sigma}}^2]$ is a valid norm in $\set{T}_k$. Hence, we can rewrite $\inf_{\rvec{T} \in \set{T}_{\vec{\gamma}}} \sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{\sigma}}^2]$ in a normalized form as the following optimization problem:
\begin{equation}
\begin{aligned}
\underset{\rvec{V} \in \set{T}, \vec{q} \in \stdset{R}_+^K} {\text{minimize}} \quad  &  \sum_{k=1}^Kq_k\\
\text{subject to} \quad & (\forall k \in \set{K})~\mathrm{SINR}^{\mathrm{DL}}_k(\rvec{V}\mathrm{diag}(\vec{q})^{\frac{1}{2}}) \geq \gamma_k,\\
& (\forall k \in \set{K})~\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] = 1,
\end{aligned}
\end{equation} 
where we used the change of variables $(\forall k \in \set{K})$ $\rvec{t}_k =: \sqrt{q_k}\rvec{v}_k$. The vector $\vec{q}$ can be interpreted as a downlink power control vector, by (unconventionally) measuring the power of each $\rvec{t}_k$ in terms of its norm $\E[\|\rvec{t}_k\|_{\vec{\sigma}}^2] = q_k$. For any choice of $\rvec{V}\in\set{T}$ with normalized columns, i.e., such that $(\forall k \in \set{K})~\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] = 1$, consider now the following downlink power control problem:
\begin{equation}\label{prob:DL_power_control}
\begin{aligned}
\underset{\vec{q} \in \stdset{R}_+^K} {\text{minimize}} \quad  &  \sum_{k=1}^Kq_k\\
\text{subject to} \quad & (\forall k \in \set{K})~ \mathrm{SINR}^{\mathrm{DL}}_k(\rvec{V}\mathrm{diag}(\vec{q})^{\frac{1}{2 }}) \geq \gamma_k.
\end{aligned}
\end{equation}
From known sum power duality arguments in the power control literature (reviewed, e.g., in \cite{schubert2004solution, massivemimobook}), it follows that Problem~\eqref{prob:DL_power_control} is feasible if and only if the following uplink power control problem is feasible, for the same choice of $\rvec{V}$:
\begin{equation}\label{prob:UL_power_control}
\begin{aligned}
\underset{\vec{p} \in \stdset{R}_+^K} {\text{minimize}} \quad  &  \sum_{k=1}^Kp_k\\
\text{subject to} \quad & (\forall k \in \set{K})~\mathrm{SINR}^{\mathrm{UL}}_k(\rvec{v}_k,\vec{p},\vec{\sigma}) \geq \gamma_k.
\end{aligned}
\end{equation}
When feasible, Problem~\eqref{prob:DL_power_control} and Problem~\eqref{prob:UL_power_control} are known to have unique and positive solutions meeting the SINR constraints with equality, and to attain the same optimum. The solutions are related by rearranging the constraints as full rank linear systems $(\vec{D}-\vec{B})\vec{q}^\star = \vec{1}$ and $(\vec{D}-\vec{B}^\T)\vec{p}^\star = \vec{1}$, respectively. When not feasible, we say that the two optima equal $\infty$. By taking the infimum of both optima over the set of $\rvec{V}\in\set{T}$ such that $(\forall k \in \set{K})~\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] = 1$, we obtain that Problem~\eqref{prob:dualUL} and Problem~\eqref{prob:DL_QoS_augmented_general} have the same optimum. Finally, the proof is completed by recalling that Problem~\eqref{prob:DL_QoS_augmented_general} always has a solution.
\end{proof}

The key message of the uplink-downlink duality principle in Proposition~\ref{prop:UL-DL} is that optimal joint precoders solving Problem~\eqref{prob:DL_QoS} are given by a scaled version of optimal joint combiners solving the dual uplink problem~\eqref{prob:dualUL} with noise vector $\vec{\sigma}= \vec{1}+\vec{\lambda}^\star$, where $\vec{\lambda}^\star$ are  Lagrangian multipliers solving Problem~\eqref{prob:partial_dual}.

\subsection{Dual uplink power control with implicit optimal combining}\label{ssec:inner}
We now focus on the solution to the dual uplink problem~\eqref{prob:dualUL}. By exploiting the property that the uplink SINR constraints are only coupled via the power vector $\vec{p}$, we observe that the optimum to Problem~\eqref{prob:dualUL} is equivalently given by the optimum to the following power control problem:
\begin{equation}\label{prob:implicit_power_control}
\underset{\vec{p} \in \stdset{R}_+^K} {\text{minimize}}\: \sum_{k=1}^Kp_k  
\text{ subject to }  (\forall k \in \set{K})~u_k(\vec{p},\vec{\sigma})\geq \gamma_k,
\end{equation}
where the optimization of the combiners is left implicit in the definition of $(\forall k \in \set{K})(\forall \vec{p} \in \stdset{R}_+^{K})(\forall \vec{\sigma} \in \stdset{R}_{++}^L)$
\begin{equation}\label{eq:maxSINR}
u_k(\vec{p},\vec{\sigma}) := \sup_{\substack{\rvec{v}_k \in \set{T}_k \\ \E[\|\rvec{v}_k\|_{\vec{\sigma}}^2] \neq 0 } }\mathrm{SINR}^{\mathrm{UL}}_k(\rvec{v}_k,\vec{p},\vec{\sigma}).
\end{equation}
A solution $(\rvec{V}^\star,\vec{p}^\star)$ to Problem~\eqref{prob:dualUL} is related to a solution to Problem~\eqref{prob:implicit_power_control} in the sense that $\vec{p}^\star$ is also a solution to Problem~\eqref{prob:implicit_power_control}, and that the columns of $\rvec{V}^\star$ attain the suprema in \eqref{eq:maxSINR} for $\vec{p}=\vec{p}^\star$. Note that, without loss of generality and for mathematical convenience in later steps, we removed in \eqref{eq:maxSINR} the unit-norm constraint, since the value of $\E[\|\rvec{v}_k\|_{\vec{\sigma}}^2]$ does not change the uplink SINR, as long as it is non-zero.

The main implication of the above discussion is that optimal joint combiners (and hence, by Propostion~\ref{prop:UL-DL}, optimal joint precoders) solving Problem~\eqref{prob:dualUL} can be obtained by solving a set of disjoint uplink SINR maximization problems under per-AP information constraints, i.e., by evaluating all $u_k(\vec{p},\vec{\sigma})$ in \eqref{eq:maxSINR} for some coefficients $(\vec{p},\vec{\sigma})\in \stdset{R}_{++}^K\times \stdset{R}_{++}^L$. The next sections discuss challenges and solution approaches related to this crucial step, and reveals a useful solution structure. However, before moving to the next section, we first present a numerical method for computing $\vec{p}^\star$, i.e., for solving Problem~\eqref{prob:implicit_power_control}, assuming that $u_k(\vec{p},\vec{\sigma})$ in \eqref{eq:maxSINR} can be indeed evaluated. Specifically, we apply the celebrated framework of interference calculus for power control \cite{yates1995power,martinbook}, and obtain:
\begin{lemma}\label{lem:fixed_point}
Fix $\vec{\sigma}\in \stdset{R}_{++}^L$. For every  $\vec{p}^{(1)} \in \stdset{R}_{++}^K$, the sequence $(\vec{p}^{(i)})_{i\in\stdset{N}}$ generated via the fixed-point iterations $
(\forall i \in \stdset{N})~\vec{p}^{(i+1)} = T_{\vec{\sigma}}(\vec{p}^{(i)})$, where
\begin{equation}\label{eq:int_map}
(\forall \vec{p}\in \stdset{R}_{++}^K)~T_{\vec{\sigma}}(\vec{p}) := \begin{bmatrix}
\frac{\gamma_1p_1}{u_1(\vec{p},\vec{\sigma})} & \ldots & \frac{\gamma_Kp_K}{u_K(\vec{p},\vec{\sigma})}
\end{bmatrix}^\T,
\end{equation}
converges in norm to the unique solution $\vec{p}^\star$ to Problem~\eqref{prob:implicit_power_control}.
\end{lemma}
\begin{proof} (Sketch) A simple contradiction argument proves that a solution $\vec{p}^\star$ must satisfy $(\forall k \in \set{K})~u_k(\vec{p}^\star,\vec{\sigma})= \gamma_k$. By trivially extending the arguments in \cite[Proposition~3]{miretti2022joint} for $\vec{\sigma}=\vec{1}$ to an arbitrary $\vec{\sigma}\in \stdset{R}_{++}^L$, it follows that this condition can be equivalently expressed as the fixed point equation $\vec{p}^\star = [\gamma_1f_{1,\vec{\sigma}}(\vec{p}^\star),\ldots,\gamma_Kf_{K,\vec{\sigma}}(\vec{p}^\star)]^\T$, where $(\forall k \in \set{K})~f_{k,\vec{\sigma}}: \stdset{R}_+^K\to \stdset{R}_{++}$ is a given \textit{standard interference function} (see, e.g.,  \cite[Definition~2]{miretti2022joint}) satisfying $(\forall k \in \set{K})(\forall \vec{p}\in \stdset{R}_+^K)~u_k(\vec{p},\vec{\sigma}) = p_k/f_{k,\vec{\sigma}}(\vec{p})$. For the above arguments to hold, an important requirement is the property $(\forall k \in \set{K})(\forall \vec{p}\in \stdset{R}_{++}^K)~u_k(\vec{p},\vec{\sigma}) > 0$. However, this property is guaranteed by recalling that Problem~\eqref{prob:dualUL} always has a solution, which implies the existence of $\rvec{V} \in \set{T}$ such that all uplink SINRs are strictly positive for any $\vec{p}\in\stdset{R}_{++}^K$. The proof is concluded by invoking known properties of fixed points of standard interference mappings \cite{yates1995power}. In particular, existence and uniqueness of a fixed point $\vec{p}^\star$ is guaranteed, and fixed-point iterations converge in norm to the unique solution.  
\end{proof} 
The above lemma shows that optimal joint combiners solving Problem~\eqref{prob:dualUL} can be obtained via the iterative evaluation of $T_{\vec{\sigma}}(\vec{p})$ in \eqref{eq:int_map}, which in turn involves solving the aforementioned uplink SINR maximization problems in \eqref{eq:maxSINR}.
 






 




\section{Optimal joint precoding structure}
\label{sec:applications}
By exploiting the obtained uplink-downlink duality principle, we now derive the structure of an optimal solution to Problem~\eqref{prob:DL_QoS}. Specifically, we show that it suffices to consider properly scaled and regularized variations of so-called \textit{team} MMSE precoders \cite{miretti2021team}, parametrized by a set of coefficients $(\vec{p},\vec{\sigma})\in \stdset{R}_{++}^K\times \stdset{R}_{++}^L$.

\subsection{MMSE precoding under information constraints}\label{ssec:TMMSE}
As discussed in the previous sections, an optimal solution to Problem~\eqref{prob:DL_QoS} can be obtained by solving a set of uplink SINR maximization problems of the type in \eqref{eq:maxSINR}. Solving these problems appears quite challenging for the following reasons: (i) the non-convex fractional utility involving expectations in both the numerator and denominator; (ii) the information constraints. However, the next proposition shows that (i) is not the main challenge, because we can consider an alternative and simpler convex utility, in the same spirit of the known relation between SINR and MMSE in deterministic channels \cite{shi2007duality}.
\begin{proposition}
\label{prop:MSE}
For given $k\in\set{K}$, $\vec{p}\in\stdset{R}_{++}^K$, and $\vec{\sigma}\in \stdset{R}_{++}^L$, consider the optimization problem 
\begin{equation}\label{eq:MSE}
\underset{\rvec{v}_k \in \set{T}_k}{\emph{minimize}}~\E\left[\|\vec{P}^{\frac{1}{2}}\rvec{H}^\herm\rvec{v}_k-\vec{e}_k\|^2\right] + \E\left[\|\rvec{v}_k\|_{\vec{\sigma}}^2\right],
\end{equation}
where $\vec{P}:= \mathrm{diag}(\vec{p})$, and $\rvec{H}:=[\rvec{h}_1,\ldots,\rvec{h}_K]$.
Problem~\eqref{eq:MSE} has a unique solution $\rvec{v}_k^\star\in \set{T}_k$, and this solution satisfies
\begin{equation*}
u_k(\vec{p},\vec{\sigma}) = \mathrm{SINR}_k\left(\rvec{v}_k^\star,\vec{p},\vec{\sigma}\right).
\end{equation*}
\end{proposition}
\begin{proof}
The proof follows readily by extending the arguments in \cite[Proposition~4]{miretti2022joint} for $\vec{\sigma}=\vec{1}$ to an arbitrary $\vec{\sigma}\in \stdset{R}_{++}^L$. Note that the assumption $u_k(\vec{p},\vec{\sigma})>0$, required for the proof, is always satisfied in this work, as already discussed in the proof of Lemma~\ref{lem:fixed_point}.
\end{proof}
\begin{remark}
Together with Proposition~\ref{prop:UL-DL} and the discussion in Section~\ref{ssec:inner}, Proposition~\ref{prop:MSE} shows that optimal joint precoders solving Problem~\eqref{prob:DL_QoS} can be obtained as solutions to MMSE problems under information constraints, i.e., they are given by (properly scaled) solutions to  Problem~\eqref{eq:MSE}, for some parameters $(\vec{p},\vec{\sigma})\in \stdset{R}_{++}^K\times\stdset{R}_{++}^L$
\end{remark}
In the particular case where all APs have complete knowledge of the channel $\rvec{H}$, and fully share UEs' data, a solution to Problem~\eqref{eq:MSE} is simply given by a variation of the well-known regularized zero-forcing solution
\begin{equation}\label{eq:MMSE}
\rvec{V} = \left(\rvec{H}\vec{P}\rvec{H}^\herm + \vec{\Sigma}\right)^{-1}\rvec{H}\vec{P}^{\frac{1}{2}},
\end{equation}
where, similar to \cite{yu2007transmitter}, the typical regularization of the matrix inversion stage via a scaled identity matrix is replaced by a more general diagonal regularization matrix $\vec{\Sigma} := \mathrm{diag}(\sigma_1\vec{I}_N,\ldots,\sigma_L\vec{I}_N)$ parametrized by $\vec{\sigma}$ which takes into account the per-AP power constraints. 
\begin{remark}
A major difference between the solution in \cite{yu2007transmitter} and \eqref{eq:MMSE} is that the former considers short-term optimization of its parameters, i.e.,  for every channel realization. In contrast, our work considers long-term optimization of the parameters $(\vec{p},\vec{\sigma})\in \stdset{R}_{++}^K\times \stdset{R}_{++}^L$ based on channel statistics.
\end{remark}
For more general and nontrivial information constraints, the solution to Problem~\eqref{eq:MSE} can be interpreted as the best distributed approximation of regularized channel inversion, and it can be obtained via a minor variation of the recently developed \emph{team} MMSE precoding method given by \cite{miretti2021team}. In the next sections we discuss this aspect in detail.

\subsection{Uplink channel estimation and CSI sharing}
\label{ssec:canonical_model}
To provide concrete examples of solutions to Problem~\eqref{eq:MSE}, we first slightly restrict the model in Section~\ref{sec:model}, while still covering most scenarios studied in the (cell-free) massive MIMO literature.  
For all AP $l\in \set{L}$ and UE $k\in\set{K}$, we let each sub-vector $\rvec{h}_{l,k}$ of $\rvec{h}_k^{\herm} =: [\rvec{h}_{1,k}^\herm, \ldots \rvec{h}_{L,k}^\herm]$ be independently distributed as $\rvec{h}_{l,k} \sim \CN\left(\vec{\mu}_{l,k}, \vec{K}_{l,k}\right)$ for some channel mean $\vec{\mu}_{l,k} \in \stdset{C}^N$ and covariance matrix $\vec{K}_{l,k}\in \stdset{C}^{N\times N}$. Independence can be easily motivated by assuming that UEs and APs are not colocated. As customary in the literature, we further assume that the APs (or, more generally, the processing units controlling the APs) perform pilot based over-the-uplink MMSE channel estimation, based on the channel reciprocity property of time division duplex systems. Specifically, we assume each AP~$l \in \set{L}$ to acquire local estimates $\hat{\rvec{H}}_l:=[\hat{\rvec{h}}_{l,1},\ldots,\hat{\rvec{h}}_{l,K}]$ of the local channel $\rvec{H}_l:=[\rvec{h}_{l,1},\ldots,\rvec{h}_{l,K}]$ with error $\rvec{Z}_l :=[\rvec{z}_{l,1},\ldots,\rvec{z}_{l,K}] :=  \rvec{H}_l-\hat{\rvec{H}}_l$ independent from $\hat{\rvec{H}}_l$, and satisfying $(\forall l \in \set{L})(\forall k \in \set{K})$ $\rvec{z}_{l,k} \sim \CN(\vec{0},\vec{\Psi}_{l,k})$ for some error covariance $\vec{\Psi}_{l,k} \in \stdset{C}^{N\times N}$. Moreover, again motivated by the geographical separation of the APs, we assume that $(\hat{\rvec{H}}_l,\rvec{Z}_l)$ is independent from $(\hat{\rvec{H}}_j,\rvec{Z}_j)$ for all $(l,j)\in \set{L}^2$ such that $l\neq j$.

After this local channel estimation step, we assume that the APs may acquire additional information of the global channel matrix $\rvec{H}$ via some CSI sharing step. More specifically, we assume that each AP $l\in \set{L}$ must form its precoders based on some side information $S_l := (\hat{\rvec{H}}_l, \bar{S}_l)$, where $\bar{S}_l$ denotes additional channel information defined by the chosen CSI sharing pattern. Overall, we model this two steps channel acquisition scheme (local channel estimation followed by CSI sharing) by assuming the following Markov chain $\rvec{H}_l \to \hat{\rvec{H}}_l \to S_l \to S_j \to \hat{\rvec{H}}_j \to \rvec{H}_j$ for all $(l,j)\in \set{L}^2$. Essentially, this Markov chain ensures that each AP $l\in \set{L}$ can only acquire a perfect or degraded version of the local estimate $\hat{\rvec{H}}_j$ of $\rvec{H}_j$ available at another AP $j\neq l$. We map the above assumptions to information constraints $\set{T}_1,\ldots,\set{T}_K$ in Problem~\eqref{prob:DL_QoS} by letting the information subfield $\Sigma_l$ of each AP $l\in \set{L}$, defining the subspace $\set{H}_l^N$ in \eqref{eq:CSI}, be the sub-$\sigma$-algebra generated by its available CSI $S_l$ on $\Omega$.

\subsection{Centralized precoding with per-AP power constraints}
A similar expression to \eqref{eq:MMSE} covering user-centric network clustering and channel estimation errors can be easily obtained, provided that imperfect channel estimates are perfectly shared within each cluster of AP. This corresponds to a variation of the known centralized MMSE precoding solution in \cite{emil2020scalable,demir2021foundations}. 
\begin{proposition}
\label{prop:CMMSE}
For given $k\in\set{K}$, $\vec{p}\in\stdset{R}_{++}^K$, and $\vec{\sigma}\in \stdset{R}_{++}^L$, and under the model in Section~\ref{ssec:canonical_model} with $(\forall l\in \set{L})~S_l = (\hat{\rvec{H}}_1,\ldots,\hat{\rvec{H}}_L)$ (centralized CSI), the unique solution to Problem~\eqref{eq:MSE} is given by 
\begin{equation}\label{eq:CMMSE}
\rvec{v}_k = \left(\vec{C}_k\hat{\rvec{H}}\vec{P}\hat{\rvec{H}}^\herm\vec{C}_k+\vec{C}_k\vec{\Psi}\vec{C}_k+\vec{\Sigma}\right)^{-1}\hat{\rvec{H}}^\herm\vec{C}_k\vec{P}^\frac{1}{2}, 
\end{equation}
where $\hat{\rvec{H}}^\herm :=[\hat{\rvec{H}}_1^\herm,\ldots,\hat{\rvec{H}}_L^\herm]$, $\vec{\Sigma}:= \mathrm{diag}(\sigma_1\vec{I}_N,\ldots,\sigma_L\vec{I}_N)$, $\vec{\Psi} := \sum_{k=1}^Kp_k \mathrm{diag}(\vec{\Psi}_{1,k},\ldots,\vec{\Psi}_{L,k})$, and $\vec{C}_k := \mathrm{diag}(\vec{C}_{1,k},\ldots,\vec{C}_{L,k})$ is a block-diagonal matrix satisfying
\begin{equation*}
(\forall l \in\set{L})~\vec{C}_{l,k} = \begin{cases}\vec{I}_N & \text{if } \in \set{L}_k,\\
\vec{0}_{N\times N} & \text{otherwise}.
\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}(Sketch) We equivalently model user-centric network clustering by replacing $\rvec{v}_k$ with $\vec{C}_k\rvec{v}_k$ as in \cite{emil2020scalable}, instead of modifying the sets $\set{T}_k$ in \eqref{eq:CSI} as described in Section~\ref{ssec:info}. Then, since all APs share the same CSI $\hat{\rvec{H}}$, optimal precoders can be obtained as solutions to disjoint, unconstrained, and finite dimensional conditional MMSE problems, one for each CSI realization:
\begin{equation}\label{eq:condMMSE}
\rvec{v}_k = \arg\min_{\vec{v}_k \in \stdset{C}^{LN}}\E\left[\|\vec{P}^{\frac{1}{2}}\rvec{H}^\herm\vec{C}_k\vec{v}_k-\vec{e}_k\|^2+\|\vec{C}_k\vec{v}_k\|_{\vec{\sigma}}^2 \middle|\hat{\rvec{H}}\right].
\end{equation}
The rest of the proof follows by evaluating the conditional expectations using the CSI error model, and by standard results on unconstrained minimization of quadratic forms.
\end{proof}

Although derived by assuming full CSI sharing, we observe that the computation of the optimal precoder \eqref{eq:CMMSE} for UE $k\in \set{K}$ only requires knowledge of $(\hat{\rvec{H}}_l)_{l\in \set{L}_k}$, i.e., only the channel estimates of the APs serving UE $k$. Furthermore, similarly to the discussion in \cite{demir2021foundations}, the computation of the inverse in \eqref{eq:CMMSE} only involves the inversion of a submatrix of size $N|\set{L}_k|$.

\begin{remark}
The special case of \eqref{eq:CMMSE} with $\vec{\sigma}=\vec{1}$ was already proposed in \cite{emil2020scalable,demir2021foundations} as a good heuristic under a sum power constraint. In particular, it was motivated by first observing that \eqref{eq:CMMSE} maximizes a coherent uplink rate bound, different than \eqref{eq:UatF_bound}, and then by invoking uplink-downlink duality between \eqref{eq:UatF_bound} and \eqref{eq:hardening_bound} under a sum power constraint. Although \cite{emil2020scalable,demir2021foundations} observed that \eqref{eq:CMMSE} also solves Problem~\eqref{eq:condMMSE}, the connection with the maximization of \eqref{eq:UatF_bound} given by Propostion~\ref{prop:MSE}, and hence the formal optimality of \eqref{eq:CMMSE} in terms of downlink rates in \eqref{eq:hardening_bound}, was not reported. 
\end{remark}

\subsection{Distributed precoding with per-AP power constraints}
When the APs have different CSI, it is not possible to decompose the problem into disjoint conditional MMSE problems as in the proof of \eqref{eq:CMMSE}, and more advanced methods must be used \cite{miretti2021team}. In this section we illustrate the extension of some key results in \cite{miretti2021team} to the case of per-AP power constraints. We start with the following general result:
\begin{proposition}
\label{prop:TMMSE}
For given $k\in\set{K}$, $\vec{p}\in\stdset{R}_{++}^K$, and $\vec{\sigma}\in \stdset{R}_{++}^L$, and under the model in Section~\ref{ssec:canonical_model}, the unique solution to Problem~\eqref{eq:MSE} is also the unique $\rvec{v}_k \in \set{T}_k$ satisfying $(\forall l \in \mathcal{L}_k)$
\begin{equation}\label{eq:TMMSE}
\rvec{v}_{l,k} =  \rmat{V}_l\left(\vec{e}_k-\sum_{j \in \set{L}_k\backslash  \{l\}} \vec{P}^{\frac{1}{2}}\E\left[\hat{\rmat{H}}_{j}^\herm\rvec{v}_{j,k}\Big|S_l\right] \right) \quad \mathrm{a.s.}, 
\end{equation} 
where $\rmat{V}_l:=\left(\hat{\rmat{H}}_{l}\vec{P}\hat{\rmat{H}}_{l}^\herm+\sum_{k\in\set{K}}p_k\vec{\Psi}_{l,k} + \sigma_l\vec{I}_N\right)^{-1}\hat{\rmat{H}}_{l}\vec{P}^{\frac{1}{2}}$. 
\end{proposition}
\begin{proof}
The proof follows readily by replacing $\vec{\sigma}=\vec{1}$ with an arbitrary $\vec{\sigma}\in \stdset{R}_{++}^L$ in the proofs of \cite[Lemma~2]{miretti2021team} and \cite[Lemma~1]{miretti2021team2}. Informally, \eqref{eq:TMMSE} is obtained by minimizing the objective in \eqref{eq:MSE} with respect to a subvector $\rvec{v}_{l,k}$, by conditioning on $S_l$, and by fixing the other subvectors $\rvec{v}_{j,k}$ for $j\neq l$. This readily gives a set of necessary optimality conditions. The key step of the proof shows that these conditions are also sufficient.
\end{proof}
Proposition~\ref{prop:TMMSE} shows that optimal distributed precoding is composed by a local MMSE precoding stage, followed by a corrective stage taking into account the possibly unknown effect of the other APs based on the available CSI. From an optimization point of view,  Proposition~\ref{prop:TMMSE} states that the solution to \eqref{eq:MSE} is the unique solution to an infinite dimensional linear feasibility problem, that can be solved in closed form for many nontrivial yet relevant setups, or via efficient approximation techniques when the expectations in \eqref{eq:TMMSE} cannot be easily evaluated. In particular, we remark that Proposition~\ref{prop:TMMSE} applies to fairly general CSI sharing patterns (see Remark~\ref{rem:CSI} and the discussions in \cite{miretti2021team,miretti2021team2}). Due to space limitations, in the remainder of this work we focus on the relatively simple case of local precoding with user-centric network clustering, and leave the study of more complex setups for future work.
\begin{proposition}\label{prop:LTMMSE}
For given $k\in\set{K}$, $\vec{p}\in\stdset{R}_{++}^K$, and $\vec{\sigma}\in \stdset{R}_{++}^L$, and under the model in Section~\ref{ssec:canonical_model} with  $(\forall l\in \set{L})~S_l = \hat{\rvec{H}}_l$ (local CSI), the unique solution to Problem~\eqref{eq:MSE} is given by 
\begin{equation}\label{eq:LTMMSE}
(\forall l\in \set{L})~\rvec{v}_{l,k} = \rvec{V}_l\vec{c}_{l,k}, 
\end{equation}
where $\rvec{V}_l$ is a local MMSE stage as in Proposition~\ref{prop:TMMSE}, and  $\vec{c}_{l,k}\in \stdset{C}^K$ is a statistical precoding stage given by the unique solution to the linear system of equations
\begin{equation*}
\begin{cases}\vec{c}_{l,k} + \sum_{j \in \set{L}_k \backslash \{l\}}\vec{\Pi}_j \vec{c}_{j,k} = \vec{e}_k & \forall l \in \set{L}_k, \\
\vec{c}_{l,k} = \vec{0}_{K\times 1} & \text{otherwise,}
\end{cases}
\end{equation*}
where $(\forall l\in \set{L})~\vec{\Pi}_l := \E\left[\vec{P}^{\frac{1}{2}}\hat{\rmat{H}}_l^\herm\rmat{V}_l\right]$.
\end{proposition}
\begin{proof}
The proof follows by verifying that \eqref{eq:LTMMSE} satisfies the optimality coditions \eqref{eq:TMMSE}. The details are similar to the full data sharing and sum power constraint case in \cite[Theorem~4]{miretti2021team}, hence they are omitted.
\end{proof}
The role of the statistical precoding stage is to optimally enhance the local MMSE precoding decisions by exploiting known statistical features of the channel and CSI of the other APs. Interestingly, we observe that the matrix $\vec{\Pi}_l$ for $l\in \set{L}$ takes the form of a $K\times K$ covariance matrix that can be locally estimated at the $l$th AP using local CSI only, and then shared on a long-term basis for the computation of the statistical precoding stages. As observed in \cite{miretti2021team} for the simpler case of a sum power constraint, \eqref{eq:LTMMSE} may provide significant performance gain over the standard local MMSE solution (where, for each $l\in \set{L}$ and $k\in \set{K}$, $\vec{c}_{l,k}$ is replaced by $c_{l,k}\vec{e}_k$ for a single power scaling coefficient $c_{l,k}\in \stdset{R}_+$ \cite{demir2021foundations}) for non-zero mean channel models and/or in the presence of pilot contamination.

\section{Numerical examples}
\label{sec:sim}
In this section we illustrate some applications of our results by simulating the performance of optimal joint precoding in a simple example of user-centric cell-free massive MIMO network under either centralized or local CSI. Note that due to the theoretical nature and space limitations of this paper, the numerical results that follow are for illustrative purposes only and should by no means be understood as an exhaustive evaluation of the performance of cell-free networks.

\subsection{Simulation setup}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{network}
\caption{Pictorial representation of the simulated setup: $K=16$ UEs uniformly distributed within a squared service area of size $1\times 1~\text{km}^2$, and $L=16$ regularly spaced APs with $N=4$ antennas each. Each UE is jointly served by a cluster of $Q=4$ APs offering the strongest channel gains.}
\label{fig:network}
\end{figure}
We consider the downlink of the network depicted in Figure~\ref{fig:network}, where $K=16$ UEs are independently and uniformly distributed within a squared service area of size $1\times 1~\text{km}^2$, and served by $L=16$ regularly spaced APs with $N=4$ antennas each. For all $(l,k)\in \set{L}\times \set{K}$, we assume for simplicity zero mean uncorrelated channels, i.e., $\vec{\mu}_{l,k} = \vec{0}$ and $\vec{K}_{l,k} = \kappa_{l,k}\vec{I}_N$, where $\kappa_{l,k}$ denotes the channel gain between AP~$l$ and UE~$k$. We adopt the following 3GPP-like channel gain model, suitable for an urban microcell scenario at $3.7$~GHz carrier frequency \cite[Table 7.4.1-1]{3GPP}
\begin{equation*}
\kappa_{l,k} = -35.3 \log_{10}\left(\Delta_{l,k}/1 \; \mathrm{m}\right) -34.5 + Z_{l,k} -P_{\mathrm{noise}} \quad \text{[dB]},
\end{equation*}
where $\Delta_{l,k}$ is the distance between AP $l$ and UE $k$ including a difference in height of $10$ m, and $Z_{l,k}\sim \CN(0,\rho^2)$ [dB] are shadow fading terms with deviation $\rho = 7.82$. The shadow fading is correlated as $\E[Z_{l,k}Z_{j,i}]=\rho^22^{-\frac{\delta_{k,i}}{13 \text{ [m]}}}$ for all $l=j$ and zero otherwise, where $\delta_{k,i}$ is the distance between UE $k$ and UE $i$. The noise power is 
$P_{\mathrm{noise}} = -174 + 10 \log_{10}(B) + F$ [dBm],
where $B = 100$ MHz is the bandwidth, and $F = 7$ dB is the noise figure. The per-AP power constraints are set to $(\forall l \in\set{L})~P_l = 30$ dBm.

We assume that each UE~$k\in\set{K}$ is served by its $Q=4$ strongest APs only, i.e., by the subset of APs indexed by $\set{L}_k\subseteq \set{L}$, where each set $\set{L}_k$ is formed by ordering $\set{L}$ w.r.t. decreasing $\kappa_{l,k}$ and by keeping only the first $Q$ elements. Finally, we assume that the APs estimate the small-scale fading channel coefficients of the served UEs only. In particular, we consider the following simple model where  the channel coefficients are either perfectly known or completely unknown:
\begin{equation*}
(\forall k \in\set{K})~\hat{\rvec{h}}_{l,k} := \begin{cases}
\rvec{h}_{l,k} & \text{if } l \in \set{L}_k,\\
\E[\rvec{h}_{l,k}] & \text{otherwise}.
\end{cases}
\end{equation*}

\subsection{Probability of feasibility}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{rate_feas}
\caption{Probability of feasibility of different minimum rate requirements, under different information constraints and a per-AP power constraint $P_l= 30$ dBm. Remarkably, our feasibility analysis covers optimal joint precoding under user-centric network clustering and either local CSI (local precoding) or centralized CSI (centralized precoding). The performance of the corresponding solutions under a sum power constraint $\sum_{l=1}^L P_l$ are also evaluated. As expected, due to the more restrictive information constraint, local precoding offers worse performance than centralized precoding. Similarly, a per-AP power constraints offers worse performance than a sum power constraint. Nevertheless, we observe that the local precoding with per-AP power constraint still robustly supports fairly high rates around $2.5$ b/s/Hz to all UEs.}
\label{fig:rate_feas}
\end{figure}
Figure~\ref{fig:rate_feas} plots the probability that the feasible set of Problem \eqref{prob:DL_QoS} is nonempty, by letting $(\forall k \in \set{K})~\gamma_k = \gamma$ for different choices of $\gamma$, corresponding to minimum rate requirements within $1$-$4$ b/s/Hz for all UEs.  We consider both centralized precoding as in Proposition~\ref{prop:CMMSE} and local precoding as in Proposition~\ref{prop:LTMMSE}. As a baseline, we also consider the  corresponding optimal precoders subject to a sum power constraint $P_{\text{sum}} = \sum_{l=1}^LP_l$. 

The probability of feasibility is approximated by solving $100$ instances of Problem~\eqref{prob:DL_QoS} for $100$ independent UE drops. Each instance is solved using the numerical methods given by Lemma~\ref{lem:proj_grad} and Lemma~\ref{lem:fixed_point}, combined as nested loops. In particular, building on Proposition~\ref{prop:UL-DL}, the subgradients $\vec{g}(\vec{\lambda})$ in Lemma~\ref{lem:proj_grad} are computed by solving the downlink problem $\inf_{\set{T}_{\vec{\gamma}}}\sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2]$ through its dual uplink problem \eqref{prob:dualUL} with $\vec{\sigma} = \vec{1}+\vec{\lambda}$, using the fixed-point algorithm in Lemma~\ref{lem:fixed_point}. For each UEs drop, we draw a sample set of $100$ independent CSI realizations for approximating the expectations via empirical averages. Although developed for producing a solution to Problem~\eqref{prob:DL_QoS} under the assumption of strict feasibility, we use the same algorithm for testing feasibility. Note that, for the considered setup, the event of having a feasible set with empty interior has zero probability mass, hence we test feasibility by detecting the events of strict feasibility and infeasibility only. Additional details on how the algorithm performs this test are given in Section~\ref{ssec:algo}.

\subsection{Comments on numerical implementation}
\label{ssec:algo}
In this section we give some additional details on the numerical implementation of the algorithm producing Figure~\ref{fig:rate_feas}.

We first test feasibility under a sum power constraint, i.e., we test if $\inf_{\set{T}_{\vec{\gamma}}}\sum_{k=1}^K\E[\|\rvec{t}_k\|^2]\leq P_{\text{sum}}$ holds. To this end, we initialize the outer loop with $\vec{\lambda}^{(1)} = \vec{0}$, and the inner loop with $(\forall k \in \set{K})~p^{(1)}_k = \gamma_k/\sum_{l\in\set{L}}\kappa_{l,k}$. This choice ensures that $\vec{p}^{(1)}\leq T_{\vec{1}}(\vec{p}^{(1)})$ holds, and hence, by known properties of the considered fixed-point algorithm \cite[Fact~4]{renato2016elementary}, $(\sum_{k\in\set{K}}p_k^{(i)})_{i\in\stdset{N}}$ is a monotonically increasing sequence  converging to $\inf_{\set{T}_{\vec{\gamma}}}\sum_{k=1}^K\E[\|\rvec{t}_k\|^2]$ if $\set{T}_{\vec{\gamma}}\neq \emptyset$, or diverging to $+\infty$ if $\set{T}_{\vec{\gamma}}=\emptyset$ (unfeasible SINR requirements). The inner loop is terminated at some step $i\in \stdset{N}$ if no significant progress is observed, in which case we declare the problem feasible under a sum power constraint, or if the early stop condition $\sum_{k\in\set{K}}p_k^{(i)} > P_{\mathrm{sum}}$ is met, in which case we declare infeasibility under a sum power constraint.

If this initial feasibility test is passed, we continue with the outer loop, using the step size rule $(\forall i\in \stdset{N})~\alpha^{(i)} = \alpha/\sqrt{i}$, $\alpha=10$. Since the condition $\set{T}_{\vec{\gamma}}\neq \emptyset$ was already detected, the inner loops are guaranteed to compute the partial dual function $\tilde{d}(\vec{\lambda})=\inf_{\set{T}_{\vec{\gamma}}}\sum_{k=1}^K\E[\|\rvec{t}_k\|_{\vec{1}+\vec{\lambda}}^2]-\sum_{l=1}^L\lambda_lP_l$ and its subgradient $\vec{g}(\vec{\lambda})$ for all $\vec{\lambda}\in\stdset{R}_{+}^L$, up to some numerical tolerance. The outer loop is terminated at some step $i\in \stdset{N}$ if no significant progress is observed, or if the early stop condition $\tilde{d}(\vec{\lambda}^{(i)})>P_{\mathrm{sum}}$ is met. In the former case, and if the obtained solution is feasible up to some numerical tolerance, we declare feasibility. In all other cases, we declare infeasibility. In fact, if feasible, the obtained precoders are also a solution to Problem~\eqref{prob:DL_QoS}, i.e., a minimum sum power solution. In practice, when the interest is to test only feasibility, we terminate the algorithm as soon as a feasible solution is detected, i.e., if the additional early stop condition $\vec{g}(\vec{\lambda}^{(i)})\leq \vec{0}$ is met at some step $i\in\stdset{N}$. Mostly because of the heuristic step size constant $\alpha$, we remark that the algorithm might have slow convergence for some UE drops (see Figure~\ref{fig:convergence} for an illustration of the convergence behavior for a particular UE drop). Hence, we also introduce a maximum number of iterations and a maximum number of algorithm restarts with different constants $\alpha$. If a conclusion is not reached within these thresholds, we remove the corresponding UEs drop.

\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[width=3.5in]{convergence_dual}}
\hfil
\subfloat[]{\includegraphics[width=3.5in]{convergence_power}}
\caption{Example of convergence behavior of the proposed algorithm for different step size constants $\alpha$. We consider an arbitrary UE drop, local precoding, and a minimum rate requirement of $3.5$ b/s/Hz. For each iteration $i\in \stdset{N}$ of the outer loop, we plot (a) the dual objective $\tilde{d}(\vec{\lambda}^{(i)})$, and (b) the maximum transmit power over all APs $\max_{l\in\set{L}}g_l(\vec{\lambda}^{(i)})+P_l$. The non-monotonic convergence is a common feature of projected subgradient methods. We observe that, despite a seemingly slower convergence in the first iterations, the more aggressive step size choice $\alpha = 17$ produces a feasible solution satisfying the per-AP power constraint $(\forall l\in\set{L})~P_l = 30$ dBm after $20$ iterations only. This is enough to declare feasibility, since, for each outer iteration, the inner loops ensure that the SINR constraints are always satisfied.}
\label{fig:convergence}
\end{figure}

\section{Conclusion and future directions}
\label{sec:conclusion}
This study marks a step forward in the process of extending known analytic tools for deterministic channels and instantaneous rate expressions to fading channels and ergodic rate expressions. Such extensions are often advocated in the cellular and cell-free massive MIMO literature because they  allow for a more refined analysis of modern networks covering practical aspects such as imperfect CSI and system optimization based on long-term channel statistics. Just as coding over fading realizations is an efficient way to manage fading dips, system optimization based on the long-term perspective typically lead, in our opinion, to more efficient solutions. 

As first main result, this study advances the current understanding of distributed cell-free networks by showing that the recently introduced team MMSE precoding method provides joint precoders that are optimal not only under a sum power constraint, as stated in \cite{miretti2021team}, but also under a per-AP power constraint. For illustration purposes, this study derives the structure of optimal local precoding under a per-AP power constraint. Although not explicitly covered in this study, an optimal structure can be derived also for other examples of distributed precoding such as the ones based on sequential information sharing over radio stripes \cite{miretti2021team,miretti2021team2}. An interesting future direction is thus to revisit the available studies on performance evaluation of distributed cell-free downlink implementations, in light of the above results.

As second main result, this study provides an alternative tool to \cite{yu2007transmitter} for designing centralized cell-free networks subject to per-AP power constraints, by demonstrating optimality of centralized MMSE precoding with parameters tuned only once for many channel realizations, instead of for each channel realization as in the studies based on \cite{yu2007transmitter}. This is a consequence of considering the hardening bound as figure of merit. Although the hardening bound generally gives more pessimistic capacity estimates than alternative ergodic rate bounds based on coherent or semi-coherent decoding \cite{caire2018ergodic}, in our opinion this drawback is counterbalanced by a significantly enhanced treatability for precoding optimization.

A key aspect of our results is the compact parametrization of optimal joint precoding in terms of a set of coefficients $(\vec{p},\vec{\sigma})$, interpreted as virtual uplink transmit and noise powers. One limitation of this study is that the presented algorithm for tuning these parameters can be quite slow, and it is designed to solve SINR feasibility problems only (more specifically, Problem~\eqref{prob:DL_QoS}). Thus, another interesting future direction is the development of efficient algorithms, perhaps based on heuristics such as neural networks or other statistical learning tools, for tuning $(\vec{p},\vec{\sigma})$ in network utility maximization problems such as sum rate or minimum rate maximization.

Another subtle limitation of this work lies in the assumption of strict feasibility for the constraints of Problem~\eqref{prob:DL_QoS}, which is used for ensuring Slater's condition to hold in Proposition~\ref{prop:duality}. Using an information-theoretical perspective, this means that our results do not cover optimal joint precoding for rate tuples lying on the boundary of the achivable rate region, but only on its interior. However, from an engineering perspective, this limitation is practically irrelevant, since rate tuples arbitrarily close to the boundary are still covered. Nevertheless, resolving this limitation is an interesting yet involved future direction for theoretical analysis.
  
