# Closed-form max-min power control for some cellular and cell-free massive
MIMO networks ††thanks: L. Miretti, R. L. G. Cavalcante, and S. Stańczak
acknowledge the financial support by the Federal Ministry of Education and
Research of Germany in the programme of “Souverän. Digital. Vernetzt.” Joint
project 6G-RIC, project identification number: 16KISK020K and 16KISK030.

Lorenzo Miretti, Renato L. G. Cavalcante, Sławomir Stańczak  Fraunhofer
Institute for Telecommunications Heinrich-Hertz-Institut
Technical University of Berlin
Berlin, Germany
<EMAIL_ADDRESS>Martin Schubert, Ronald Böhnke, Wen Xu  Huawei
Munich, Germany
<EMAIL_ADDRESS>
###### Abstract

Many common instances of power control problems for cellular and cell-free
massive MIMO networks can be interpreted as max-min utility optimization
problems involving affine interference mappings and polyhedral constraints. We
show that these problems admit a closed-form solution which depends on the
spectral radius of known matrices. In contrast, previous solutions in the
literature have been indirectly obtained using iterative algorithms based on
the bisection method, or on fixed-point iterations. Furthermore, we also show
an asymptotically tight bound for the optimal utility, which in turn provides
a simple rule of thumb for evaluating whether the network is operating in the
noise or interference limited regime. We finally illustrate our results by
focusing on classical max-min fair power control for cell-free massive MIMO
networks.

###### Index Terms:

power control, massive MIMO, cell-free, max-min fairness, interference
calculus, 6G

##  I Introduction

Weighted max-min utility optimization problems are common in power control
tasks in wireless networks because the solutions are known to promote fair
allocation of resources among users, and to span the boundary of the
achievable performance region [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. This is not
true, for instance, for weighted sum utility maximization problems: except for
the particular case of convex performance regions, their solutions may miss
some boundary points that are important under a user fairness perspective
[11], [2, Remark 1.3].

Many existing globally optimal solvers are based on a bisection method that
addresses a sequence of convex feasibility problems at each iteration [6,
4][10, Algorithm 1]. One of the main limitations of solutions of this type is
that the feasibility problems need to be solved with high numerical precision
[5, Sect. 3.4.1]. Therefore, depending on the numerical technique being
applied, the overarching bisection algorithm can become too slow in large
systems. As a means of addressing this potential drawback, the algorithm in
[5, Algorithm 3.1] considers a sequence of standard optimization problem for
which fast algorithms are readily available. Nevertheless, the resulting
scheme can still be too complex for real time implementation. To address this
issue, recently, some authors even started to consider suboptimal solutions
based on meta-heuristics [12].

Promising optimal alternatives to the bisection method are the iterative power
control algorithms in [5, Algorithm 3.2], [13, 14, 15, 16], which can be seen
as applications of the mathematical tools reviewed in [17, 18] and the
references therein. However, many of these algorithms are tailored to
challenging problems going beyond simple power control, and they do not fully
exploit the affine structure of some power control problems of practical
interest. Therefore, simple expressions for the optimal solution have not been
obtained.

More specifically, in some max-min utility optimization problems, which cover
the power control problems in cellular and cell-free networks described in [3,
Ch. 5.3.2], [4] [5, Ch. 7] [10, Theorem 7.1],[8, 9] as particular instances,
the optimal power allocation can be seen as the fixed point of an affine
mapping, and the power constraint is a polyhedral set. By exploiting this
structure and the results in [18], Section II presents a simple expression for
a solution to this class of problems. In particular, we show that the limit
point (i.e., the optimal power allocation) sought by some existing iterative
power control algorithms can be straightforwardly obtained by computing the
spectral radius of known matrices and by solving only one system of linear
equations that, unlike the approaches in [8, 9], do not require the bisection
method for its construction. We also specialize the bounds in [18] to gain
insights into the expression of the optimal utility. These bounds are useful
to identify whether a network is operating close to an interference limited or
noise limited regime.

Section III illustrates the practical application of the above results to
cell-free massive multiple-input multiple-output (MIMO) networks [5], by
revisiting common instances of uplink (UL) and downlink (DL) power control
problems. The main feature of these problems is that their solutions depend
only on long-term channel statistics. As a result, we open up the possibility
of performing real-time, large-scale, and globally optimal power control,
e.g., at a remote central controller, without requiring excessive fronthaul
and computational resources.

###  I-A Related results

Although somewhat unkown within the cellular and cell-free massive MIMO
literature, our solution to the max-min utility optimization problem in
Section II has been rediscovered many times in different contexts. For
instance, under slightly more restrictive conditions, a related solution can
be found in [1, Algorithm 5.3]. Furthermore, the same solution can also be
obtained from [19, Appendix A], where it was used as a basis for addressing a
more general problem. In this work we provide an alternative proof and show
that, in fact, the solution follows from a simple application of [17].
Moreover, from a more practical perspective, we show how this solution can be
directly applied to popular power control problems in the modern cellular and
cell-free massive MIMO frameworks.

###  I-B Notation and mathematical preliminaries

We denote by $\mathbbmss{R}_{+}$ and $\mathbbmss{R}_{++}$ the sets of,
respectively, nonnegative and positive reals. The spectral radius of a matrix
$\bm{M}\in\mathbbmss{R}^{K\times K}$ is denoted by $\rho(\bm{M})$. A norm
$\|\cdot\|$ on $\mathbbmss{R}^{K}$ is said to be _monotone_ (on the
nonnegative orthant) if
$(\forall\bm{x}\in\mathbbmss{R}_{+}^{K})(\forall\bm{y}\in\mathbbmss{R}_{+}^{K})~{}\bm{x}\leq\bm{y}\Rightarrow\|\bm{x}\|\leq\|\bm{y}\|$,
where inequalities involving vectors should be understood coordinate-wise. A
mapping $T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}^{K}_{++}$ is called a
standard interference mapping if the following properties hold [20]:

  * (i)

[monotonicity]
$(\forall\bm{x}\in\mathbbmss{R}^{K}_{+})(\forall\bm{y}\in\mathbbmss{R}^{K}_{+})~{}\bm{x}\geq\bm{y}\Rightarrow
T(\bm{x})\geq T(\bm{y})$

  * (ii)

[scalability] $(\forall\bm{x}\in\mathbbmss{R}^{K}_{+})$ $(\forall\alpha>1)$
$\alpha{T}(\bm{x})>T(\alpha\bm{x})$.

For later reference, given a standard interference mapping, we call each
coordinate function of the mapping a _standard interference function_. For
simplicity, in the remainder of this study we also require continuity for a
mapping to be called a standard interference function. A mapping
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}$ is said to be a _positive
concave mapping_ if each coordinate function is positive and concave, and we
recall that positivity and concavity imply properties (i) and (ii) above [21,
Proposition 1]. The set of fixed points of
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{+}^{K}$ is denoted by
$\mathrm{Fix}(T):=\\{\bm{x}\in\mathbbmss{R}_{+}^{K}~{}|~{}T(\bm{x})=\bm{x}\\}$,
and we note that standard interference mappings have at most one fixed point
[20].

######  Proposition 1.

[17, Theorem 3.2] Given a standard interference mapping
$T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}$ and a monotone norm
$\|\cdot\|$, the following _conditional eigenvalue_ problem is guaranteed to
have a unique solution:

$\begin{array}[]{rl}\mathrm{find}&(\lambda,\bm{x})\in\mathbbmss{R}_{++}\times\mathbbmss{R}_{++}^{K}\\\
\mathrm{such~{}that}&T(\bm{x})=\lambda\bm{x},\quad\|\bm{x}\|=1.\\\
\end{array}$ (1)

We finish this section with a known result that is used to keep the proof of
our main contributions self-contained.

######  Proposition 2.

Let $\bm{M}\in\mathbbmss{R}^{K\times K}_{+}$ and
$(\lambda,\bm{x})\in\mathbbmss{R}_{++}\times\mathbbmss{R}_{++}^{K}$ satisfy
the inequality $\bm{Mx}\leq\lambda\bm{x}$. Then the spectral radius
$\rho(\bm{M})$ of $\bm{M}$ is upper bounded by $\lambda$, i.e.,
$\rho(\bm{M})\leq\lambda$.

###### Proof.

From the classical Perron-Frobenius theorem [22, Theorem 2.4.1(i)], we known
that $\bm{M}$ has a maximal nonnegative eigenvalue $\rho(\bm{M})$, and there
exists $\bm{v}\in\mathbbmss{R}^{K}_{+}\backslash\\{\bm{0}\\}$ such that
$\bm{Mv}=\rho(\bm{M})\bm{v}$. Now, consider the following linear mapping
defined on the nonnegative cone $\mathbbmss{R}_{+}^{K}$:

$\displaystyle
T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{+}^{K}:\bm{u}\mapsto\bm{Mu}.$

Since $T(\bm{x})=\bm{Mx}\leq\lambda\bm{x}$ and
$\bm{x}\in\mathbbmss{R}_{++}^{K}$ by assumption, it follows from [23, Lemma
3.3] that $T(\bm{v})=\bm{Mv}=\rho(\bm{M})\bm{v}$ implies
$\rho(\bm{M})\leq\lambda$, and the proof is complete. ∎

##  II Max-min power control with polyhedral power constraints

###  II-A Problem statement

As discussed in [5, Ch. 3, Ch. 7], many existing uplink and downlink power
control algorithms for cellular and cell-free massive MIMO networks solve
particular instances of the following optimization problem, which can be seen
as a generalization of the power control problems in [1, Ch. 5.6.3] [3, Ch.
5.3.2] [10, Theorem 7.1][8, 9]:

$\begin{array}[]{rl}\underset{\bm{p}=[p_{1},\ldots,p_{K}]^{\mathsf{T}}\in\mathbbmss{R}^{K}}{\mathrm{maximize}}&\min_{k\in\\{1,\ldots,K\\}}\dfrac{b_{k}~{}p_{k}}{\bm{c}_{k}^{\mathsf{T}}\bm{p}+\sigma_{k}}\\\
\mathrm{subject~{}~{}to}&(\forall~{}n\in\\{1,\ldots,N\\})~{}\bm{a}_{n}^{\mathsf{T}}\bm{p}\leq
p_{\mathrm{max}}\\\ &\bm{p}\geq\bm{0},\end{array}$ (2)

where $K\in\mathbbmss{N}$, $N\in\mathbbmss{N}$,
$p_{\mathrm{max}}\in\mathbbmss{R}_{++}$,

$\displaystyle[\bm{a}_{1},\ldots,\bm{a}_{N}]$
$\displaystyle=:\bm{A}\in\mathbbmss{R}_{+}^{K\times N},$
$\displaystyle(b_{1},\ldots,b_{K})$
$\displaystyle=:\bm{b}\in\mathbbmss{R}_{++}^{K},$
$\displaystyle[\bm{c}_{1},\ldots,\bm{c}_{K}]$
$\displaystyle=:\bm{C}\in\mathbbmss{R}_{+}^{K\times K},\text{ and}$
$\displaystyle(\sigma_{1},\ldots,\sigma_{K})$
$\displaystyle=:\bm{\sigma}\in\mathbbmss{R}_{++}^{K}$

are problem parameters. For example, in uplink power control problems, $K$ is
the number of users in the network, $\bm{A}$ collects $N$ linear constraints
on the power vector $\bm{p}$ (the optimization variable), $p_{\max}$ is the
maximum allowed transmit power, $\bm{\sigma}$ is the vector of noise powers,
and $\bm{b}$ and $\bm{C}$ are parameters used to model the effective channel
for some fixed network configuration (e.g., the choice of beamformers). More
detailed examples related to cell-free massive MIMO are provided in Section
III; we simply anticipate that the cost function in Problem (2) can be
interpreted as the lowest signal-to-interference-plus-noise ratio (SINR) among
all users in the network. We emphasize that, in the above formulation, the
parameters $\bm{A}$, $\bm{b}$, $\bm{C}$, and $\bm{\sigma}$ must not depend on
$\bm{p}$.

###  II-B Optimal solution

The main contribution of this section is to show that a solution to Problem
(2) can be obtained by solving a single system of linear equations that can be
easily constructed with the problem parameters. The resulting system does not
require, for example, the bisection method for its construction, as done in
[8, 9] to solve a particular instance of Problem (2).

To derive this result, which is shown later in Proposition 4, we first need to
introduce a (mild) technical assumption. Denote by
$S\subset\mathbbmss{R}_{+}^{K}$ the set of power vectors satisfying all
constraints in Problem (2), which we call the _feasible set_. Hereafter, we
impose the following natural assumption on $S$, which implicitly restricts the
choice of the matrix $\bm{A}$.

######  Assumption 1.

The set of feasible power allocations $S$ is bounded, and
$S\cap\mathbbmss{R}_{++}^{K}\neq\emptyset$.

Boundedness of the set $S$ is expected because in any power control problem
the transmit power is limited by law or by hardware capabilities. In some
power control problems, existence of $\bm{p}\in\mathbbmss{R}_{++}^{K}\cap S$
can be intuitively understood as allowing every user to be served by the
network. With the above assumption, the set $S$ has the following additional
properties:

######  Remark 1.

The feasible set $S$ is a compact convex set with nonempty interior.
Furthermore, it is downward comprehensive on the nonnegative cone
$\mathbbmss{R}_{+}^{K}$, i.e.,
$(\forall\bm{q}\in\mathbbmss{R}_{+}^{K})(\forall\bm{p}\in
S)~{}\bm{q}\leq\bm{p}\Rightarrow\bm{q}\in S$.

###### Proof.

The set $S$ is closed and convex because it is the intersection of the closed
convex sets, or, more precisely, the cone $\mathbbmss{R}_{+}^{K}$ and the
closed half-spaces $(\forall
n\in\\{1,\ldots,N\\})~{}H_{n}:=\\{\bm{p}\in\mathbbmss{R}^{K}~{}|~{}\bm{a}_{n}^{\mathsf{T}}\bm{p}\leq
p_{\text{max}}\\}$. Being bounded by assumption (see Assumption 1) and closed,
$S$ is compact because the space $\mathbbmss{R}^{K}$ is finite dimensional.
The fact that $S$ is downward comprehensible on $\mathbbmss{R}_{+}^{K}$ is
immediate from nonnegativity of the vectors
$(\bm{a}_{n})_{n\in\\{1,\ldots,N\\}}$ and the definition of the half-spaces
$(H_{n})_{n\in\\{1,\ldots,N\\}}$. Downward comprehensibility of $S$ and the
existence of a vector $\bm{p}\in\mathbbmss{R}_{++}^{K}\cap S$ imply that $S$
has nonempty interior because, for any norm $\|\cdot\|$ on $\mathbbmss{R}^{K}$
and for $\epsilon>0$ sufficiently small, $\emptyset\neq
B:=\\{\bm{q}\in\mathbbmss{R}^{K}~{}|~{}\|(1/2)\bm{p}-\bm{q}\|<\epsilon\\}\subset
S.$ ∎

######  Remark 2.

Remark 1 and [14, Proposition 2] imply that there exists a monotone norm
$\|\cdot\|_{\star}$ on $\mathbbmss{R}^{K}$ such that the feasible set $S$ can
be equivalently written as
$S=\\{\bm{p}\in\mathbbmss{R}_{+}^{K}~{}|~{}\|\bm{p}\|_{\star}\leq 1\\}$. For
the particular setting in Problem (2), this monotone norm $\|\cdot\|_{\star}$
takes the form

$\displaystyle(\forall\bm{p}\in\mathbbmss{R}^{K})~{}\|\bm{p}\|_{\star}:=\dfrac{1}{p_{\max}}\max_{n\in\\{1,\ldots,N\\}}\bm{a}_{n}^{\mathsf{T}}|\bm{p}|,$
(3)

where $|\bm{p}|$ is the column vector obtained by taking the coordinate-wise
absolute value of a vector $\bm{p}$, i.e.,
$(\forall\bm{p}=[p_{1},\ldots,p_{K}]\in\mathbbmss{R}_{+}^{K})~{}|\bm{p}|:=[|p_{1}|,\dots,|p_{K}|]^{\mathsf{T}}$.

Below we further simplify $\|\cdot\|_{\star}$ in Remark 2 for two common
choices of $\bm{A}$:

######  Example 1.

If $K=N$ and $\bm{A}=\bm{I}_{K}$ is the $K$-dimensional identity matrix, then
$\|\cdot\|_{\star}$ reduces to a scaled $l_{\infty}$-norm:

$\|\bm{p}\|_{\star}=\dfrac{1}{p_{\mathrm{max}}}\max_{k\in\\{1,\ldots,K\\}}|p_{k}|=\dfrac{1}{p_{\mathrm{max}}}\|\bm{p}\|_{\infty}.$

######  Example 2.

If $N=1$ and $\bm{A}=\bm{1}_{K\times 1}$ is the $K$-dimensional vector of
ones, then $\|\cdot\|_{\star}$ reduces to a scaled $l_{1}$-norm:

$\displaystyle\|\bm{p}\|_{\star}=\dfrac{1}{p_{\mathrm{max}}}\sum_{k\in\\{1,\ldots,K\\}}|p_{k}|=\dfrac{1}{p_{\mathrm{max}}}\|\bm{p}\|_{1}.$

Now, consider the following optimization problem:

$\begin{array}[]{rl}\underset{(t,\bm{p})\in\mathbbmss{R}_{+}\times\mathbbmss{R}^{K}_{++}}{\mathrm{maximize}}&t\\\
\mathrm{subject~{}~{}to}&\bm{p}=t(\bm{Mp}+\bm{u}),\quad\|\bm{p}\|_{\star}\leq
1.\end{array}$ (4)

where $\|\cdot\|_{\star}$ is the monotone norm in (3) induced by the power
constraints $(\bm{A},p_{\max})$ satisfying Assumption 1, and where we define
the scaled coefficients

$\displaystyle\bm{M}$
$\displaystyle:=\mathrm{diag}(\bm{b})^{-1}\bm{C}^{\mathsf{T}},$
$\displaystyle\bm{u}$ $\displaystyle:=\mathrm{diag}(\bm{b})^{-1}\bm{\sigma}.$

Problems (2) and (4) are related in the next proposition, which can be proved
using standard arguments in the literature (see, for example, the discussion
below [5, Eq. (3.34)] and the references therein), so we omit the proof for
brevity.

######  Proposition 3.

If the tuple
$(t^{\star},\bm{p}^{\star})\in\mathbbmss{R}_{++}\times\mathbbmss{R}_{++}^{K}$
solves Problem (4), then
$\bm{p}^{\star}=:[p_{1}^{\star},\ldots,p_{K}^{\star}]^{\mathsf{T}}$ is a
solution to Problem (2), and the optimal value of (2) is

$\min_{k\in\\{1,\ldots,K\\}}\dfrac{b_{k}~{}p_{k}^{\star}}{\bm{c}_{k}^{\mathsf{T}}\bm{p}^{\star}+\sigma_{k}}=t^{\star}.$
(5)

In light of Proposition 3, we can obtain a solution to Problem (2) by solving
Problem (4), which has a unique solution with the simple expression derived
below.

######  Proposition 4.

Problem (4) has a unique solution
$(t^{\star},\bm{p}^{\star})\in\mathbbmss{R}_{++}\times\mathbbmss{R}_{++}^{K}$
given by

$\displaystyle
t^{\star}=\dfrac{1}{\underset{n\in\\{1,\ldots,N\\}}{\max}\rho(\bm{M}_{n})}$
(6)

and

$\displaystyle\bm{p}^{\star}=t^{\star}(\bm{I}-t^{\star}\bm{M})^{-1}\bm{u},$
(7)

where

$(\forall
n\in\\{1,\ldots,N\\})~{}\bm{M}_{n}:=\bm{M}+\dfrac{1}{p_{\mathrm{max}}}\bm{u}\bm{a}_{n}^{\mathsf{T}}.$
(8)

Furthermore, $\bm{p}^{\star}$ is also the eigenvector with norm
$\|\bm{p}^{\star}\|_{\star}=1$ associated with the largest eigenvalue of any
matrix $\bm{M}_{n}$ in (8) satisfying $\rho(\bm{M}_{n})=1/{t^{\star}}$ for
some $n\in\\{1,\ldots,N\\}$.

###### Proof.

It follows from Remark 2 and the results in [17] (see also the discussion in
[18]) that
$(t^{\star},\bm{p}^{\star})\in\mathbbmss{R}_{+}\times\mathbbmss{R}_{++}^{K}$
solves Problem (4) if and only if $(1/t^{\star},\bm{p}^{\star})$ is the unique
solution to the condition eigenvalue problem (1) with the monotone norm
$\|\cdot\|_{\star}$ in (3) and the mapping

$\displaystyle
T:\mathbbmss{R}_{+}^{K}\to\mathbbmss{R}_{++}^{K}:\bm{p}\mapsto\bm{Mp}+\bm{u}.$
(9)

As a result, from the definition of problem (1), we verify that

$\bm{p}^{\star}=t^{\star}(\bm{Mp}^{\star}+\bm{u}),\quad\|\bm{p}^{\star}\|_{\star}=1.$
(10)

Now let $i\in\\{1,\ldots,N\\}$ be any index satisfying

$\|\bm{p}^{\star}\|_{\star}=\frac{1}{p_{\max}}\bm{a}_{i}^{\mathsf{T}}\bm{p}^{\star}\geq\frac{1}{p_{\max}}\mathrm{max}_{n\in\\{1,\ldots,N\\}}\bm{a}_{n}^{\mathsf{T}}\bm{p}^{\star}.$
(11)

We then have $(\forall n\in\\{1,\ldots,N\\})$

$\begin{split}\bm{M}_{n}\bm{p}^{\star}&=\bm{Mp}^{\star}+~{}\dfrac{1}{p_{\mathrm{max}}}\bm{u}\bm{a}_{n}^{\mathsf{T}}\bm{p}^{\star}\\\
&\overset{\text{(a)}}{\leq}\bm{Mp}^{\star}+~{}\dfrac{1}{p_{\mathrm{max}}}\bm{u}\bm{a}_{i}^{\mathsf{T}}\bm{p}^{\star}\\\
&=\bm{M}_{i}\bm{p}^{\star}\overset{\text{(b)}}{=}\dfrac{1}{t^{\star}}\bm{p}^{\star},\end{split}$
(12)

where (a) follows from the inequality in (11) and positivity of $\bm{u}$, and
(b) follows from (10) and the equality in (11). Consequently, the equality (b)
in (12) shows that the solution $(t^{\star},\bm{p}^{\star})$ to problem (4)
can be obtained from an eigenpair
$(1/t^{\star},\bm{p}^{\star})\in\mathbbmss{R}_{++}\times\mathbbmss{R}^{K}_{++}$
of the matrix $\bm{M}_{i}$ with $\|\bm{p}^{\star}\|_{\star}=1$. Furthermore,
in light of Proposition 2, we verify from the equality (b) in (12) that
$1/t^{\star}>0$ is the spectral radius of $\bm{M}_{i}$ (i.e.,
$t^{\star}=1/\rho(\bm{M}_{i})$) and from the inequality (a) in (12) that

$\displaystyle(\forall
n\in\\{1,\ldots,N\\})~{}\rho(\bm{M}_{n})\leq\rho(\bm{M}_{i}).$ (13)

This completes the proof of (6) and the statement in the last sentence of the
proposition. The equality in (7) follows directly from the definition of the
equality constraint of Problem (4), and the proof is complete. (Note that the
matrix $(\bm{I}-t^{\star}\bm{M})$ is full rank because otherwise there would
exist infinitely many vectors $\bm{p}\in\mathbbmss{R}_{++}^{K}$ satisfying
$\bm{p}=t^{\star}(\bm{Mp}+\bm{u})$, thus contradicting uniqueness of the
solution to Problem (4), which has already been proved above.) ∎

###  II-C Bound on the optimal utility

The next corollary, which is immediate from [18, Proposition 3], shows a
simple bound for the optimal utility $t^{\star}$ in (6) as a function of
$p_{\mathrm{max}}$:

######  Corollary 1.

Assume that $\rho(\bm{M})>0$, and denote by
$(t^{\star}(p_{\mathrm{max}}),\bm{p}^{\star}(p_{\mathrm{max}}))$ the solution
in (6) and (7) to an instance of Problem (4), where we made explicit the
dependency of this solution on the choice of the maximum transmit power
$p_{\mathrm{max}}>0$. Then $(\forall p_{\mathrm{max}}\in\mathbbmss{R}_{++})$

$\displaystyle\quad
t^{\star}(p_{\mathrm{max}})~{}\leq\begin{cases}1/\rho(\bm{M})&\text{if
}p_{\mathrm{max}}\geq p_{\mathrm{T}},\\\
p_{\mathrm{max}}/\|\bm{u}\|&\text{otherwise},\end{cases}$ (14)

where $p_{\mathrm{T}}$ is the transition point defined by
$p_{\mathrm{T}}:=\|\bm{u}\|/\rho(\bm{M})$, and $\|\cdot\|$ is the monotone
norm

$\displaystyle(\forall\bm{p}\in\mathbbmss{R}^{K})~{}\|\bm{p}\|:=\max_{n\in\\{1,\ldots,N\\}}\bm{a}_{n}^{\mathsf{T}}|\bm{p}|.$

Furthermore, the bound in (14) is asymptotically tight as $p_{\mathrm{max}}\to
0^{+}$ and as $p_{\mathrm{max}}\to\infty$.

######  Remark 3.

As discussed in [18], in a power control problem, as the maximum transmit
power $p_{\mathrm{max}}$ increases, the network configuration obtained by
solving Problem (2) moves from a noise limited regime to an interference
limited regime around the transition point $p_{\mathrm{T}}$ defined in
Corollary 1.

##  III Applications to cell-free networks

###  III-A System model

We now apply the above results to a cell-free massive MIMO network, as defined
in [5], composed of $L$ access-points (APs) indexed by
$\mathcal{L}:=\\{1,\ldots,L\\}$, each of them equipped with $M$ antennas, and
$K$ single-antenna user equipments (UEs) indexed by
$\mathcal{K}:=\\{1,\ldots,K\\}$. The UL and DL ergodic rates simultaneously
achievable by each UE can be lower bounded by [3, 10] $(\forall
k\in\mathcal{K})$

$\displaystyle R_{k}^{\mathrm{UL}}(\bm{p})$
$\displaystyle:=\log_{2}(1+\mathrm{SINR}_{k}^{\mathrm{UL}}(\bm{p}))\quad\text{[bit/s/Hz]},$
$\displaystyle R_{k}^{\mathrm{DL}}(\bm{p})$
$\displaystyle:=\log_{2}(1+\mathrm{SINR}_{k}^{\mathrm{DL}}(\bm{p}))\quad\text{[bit/s/Hz]},$
$\displaystyle\mathrm{SINR}_{k}^{\mathrm{UL}}(\bm{p})$
$\displaystyle:=\dfrac{p_{k}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}}{p_{k}\mathsf{Var}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k})+\sum_{j\neq
k}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{v}_{k}|^{2}]+\sigma},$
$\displaystyle\mathrm{SINR}_{k}^{\mathrm{DL}}(\bm{p})$
$\displaystyle:=\dfrac{p_{k}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k}]|^{2}}{p_{k}\mathsf{Var}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{k})+\sum_{j\neq
k}p_{j}\mathsf{E}[|\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{v}_{j}|^{2}]+\sigma},$

where $\mathbbm{h}_{k}$ is a random vector taking values in
$\mathbbmss{C}^{ML}$, which models the fading channel between all APs and UE
$k$; $\mathbbm{v}_{k}$ is a random vector taking values in
$\mathbbmss{C}^{ML}$, which satisfies without loss of generality
$\mathsf{E}[\|\mathbbm{v}_{k}\|^{2}]=1$ (here, $\|\cdot\|$ denotes the
standard $l^{2}$ norm), and it is applied by all APs to filter jointly the
signal of UE $k$; $\sigma>0$ is the noise variance; and
$\bm{p}=[p_{1},\ldots,p_{K}]^{\mathsf{T}}\geq\bm{0}$ is a vector of
deterministic (i.e., long term) power scaling coefficients.

In the context of cell-free massive MIMO, $p_{k}$ should be interpreted as the
transmit power of UE $k$ for the UL case, or as the total transmit power used
by the network to serve UE $k$ for the DL case. Similarly, $\mathbbm{v}_{k}$
should be interpreted as a joint UL combiner for the message of UE $k$, or as
a joint DL precoder for the message of UE $k$. In practice, the filters
$\\{\mathbbm{v}_{k}\\}_{k\in\mathcal{K}}$ need to satisfy additional
constraints modeling limited APs cooperation, for instance, w.r.t. channel
state information sharing (CSI) and joint encoding/decoding capabilities.
Popular examples are the distributed or centralized user-centric clustered
models reviewed in [5], or the special case of cellular massive MIMO [10].
Since the focus of this article is on power control, we omit the details on
such constraints, and refer to [24, 25] for details. In the following, we
simply assume that $\\{\mathbbm{v}_{k}\\}_{k\in\mathcal{K}}$ are given, and
that the expectations in the following matrices and vector exist:

$\bm{G}:=\begin{bmatrix}\mathsf{E}[|\mathbbm{h}_{1}^{\mathsf{H}}\mathbbm{v}_{1}|^{2}]&\ldots&\mathsf{E}[|\mathbbm{h}_{1}^{\mathsf{H}}\mathbbm{v}_{K}|^{2}]\\\
\vdots&\ddots&\vdots\\\
\mathsf{E}[|\mathbbm{h}_{K}^{\mathsf{H}}\mathbbm{v}_{1}|^{2}]&\ldots&\mathsf{E}[|\mathbbm{h}_{K}^{\mathsf{H}}\mathbbm{v}_{K}|^{2}]\end{bmatrix},$
$\bm{d}:=\left(|\mathsf{E}[\mathbbm{h}_{1}^{\mathsf{H}}\mathbbm{v}_{1}]|^{2},\ldots,|\mathsf{E}[\mathbbm{h}_{K}^{\mathsf{H}}\mathbbm{v}_{K}]|^{2}\right),\quad\bm{D}:=\mathrm{diag}(\bm{d}).$

###  III-B Uplink power control with per-UE power constraint

By fixing an arbitrary combining design, and a per-UE power constraint
$(\forall k\in\mathcal{K})$ $p_{k}\leq p_{\max}$, each positive point on the
boundary of the UL rate region achieved by power control can be obtained as
the solution to problems of the type

$\underset{\bm{p}\geq\bm{0},~{}\|\bm{p}\|_{\infty}\leq
p_{\max}}{\mathrm{maximize}}\min_{k\in\mathcal{K}}\omega_{k}^{-1}\mathrm{SINR}^{\mathrm{UL}}_{k}(\bm{p})$
(15)

for some vector of weights
$\bm{\omega}:=(\omega_{1},\ldots,\omega_{K})\in\mathbbmss{R}_{++}^{K}$. The
solution to Problem (15) can be computed in closed form following the
methodology in Section II. Specifically, we can map Problem (15) to Problem
(2), and apply Proposition 3 and Proposition 4, after identifying

$\bm{A}=\bm{I}_{K},\quad\bm{b}=\mathrm{diag}(\bm{\omega})^{-1}\bm{d},\quad\bm{C}=\bm{G}-\bm{D},\quad\bm{\sigma}=\sigma\bm{1}.$

######  Remark 4.

An alternative way of finding boundary points on the rate region achieved by
power control is to consider problems of the type

$\underset{\bm{p}\geq\bm{0},~{}\|\bm{p}\|_{\infty}\leq
p_{\max}}{\mathrm{maximize}}\min_{k\in\mathcal{K}}\omega_{k}^{-1}R^{\mathrm{UL}}_{k}(\bm{p}).$
(16)

In general, using the same weights $\bm{\omega}$ in (16) and (15) produces
different boundary points. A notable exception is the choice
$\bm{\omega}=\bm{1}$, which produces in both cases the so-called _max-min
fair_ point. This boundary point is popular in the cell-free literature, where
the main motivation is indeed to improve user fairness w.r.t. cellular
networks [4].

###  III-C Downlink power control with sum power constraint

Similarly to the UL case, by fixing an arbitrary precoding design and a sum
power constraint $\sum_{k\in\mathcal{K}}p_{k}\leq p_{\max}$, each positive
point on the boundary of the DL rate region achieved by power control can be
obtained as the solution to problems of the type

$\underset{\bm{p}\geq\bm{0},~{}\|\bm{p}\|_{1}\leq
p_{\max}}{\mathrm{maximize}}\min_{k\in\mathcal{K}}\omega_{k}^{-1}\mathrm{SINR}^{\mathrm{DL}}_{k}(\bm{p}).$
(17)

for some vector of weights
$\bm{\omega}:=(\omega_{1},\ldots,\omega_{K})\in\mathbbmss{R}_{++}^{K}$. The
solution to Problem (17) can be similarly computed by applying Proposition 3
and Proposition 4 after identifying

$\bm{A}=\bm{1}_{K\times
1},\quad\bm{b}=\mathrm{diag}(\bm{\omega})^{-1}\bm{d},\quad\bm{C}=\bm{G}^{\mathsf{T}}-\bm{D},\quad\bm{\sigma}=\sigma\bm{1}.$

The main difference w.r.t. the UL case is the transpose operator in the
channel gain matrix $\bm{G}$, and the fact that using a single linear power
constraint removes the $\max$ operator in the expression of the optimal
solution (6).

######  Remark 5.

The considered DL sum power constraint refers to the power radiated by all APs
in the network. Although this is still an important metric for network
design/management, in practice, due to hardware and regulatory constraints,
each AP (or even antenna) is also typically subject to an individual power
constraint. We remark that imposing a per-AP power constraint is different
from constraining each $p_{k}$ as in the UL case. However, solving instances
of Problem (2) can still be useful for obtaining heuristic solutions, for
instance, based on a suitable design of
$\\{\mathbbm{v}_{k}\\}_{k\in\mathcal{K}}$ and $(\bm{A},p_{\max})$ such that
the feasibility under a per-AP power constraint is guaranteed.

######  Remark 6.

For the special case of a cellular network, where each AP serves a disjoint
set of UEs, a per-AP power constraint in the DL can be mapped to $L$ linear
constraints $\\{\bm{a}_{l}\\}_{l\in\mathcal{L}}$, each acting on a separate
subvector of $\bm{p}$ corresponding to the UEs served by AP $l$. Therefore,
the considered power control framework covers this case.

###  III-D Numerical examples

Figure 1: Pictorial representation of the simulated setup: $K=64$ UEs
uniformly distributed within a squared service area of size $1\times
1~{}\text{km}^{2}$, and $L=16$ regularly spaced APs with $M=8$ antennas each.
In the cell-free setup, each UE is jointly served by a cluster of $Q=4$ APs
offering the strongest channel gains. The cellular case is obtained as a
special case by letting $Q=1$.  Figure 2: Comparison of optimal max-min fair
UL power control versus full power transmission for a UE power budget
$p_{\max}=20$ dBm, under different AP cooperation regimes. Power control
significantly boosts user fairness, especially for regimes with lower
interference suppression capabilities.

(a)

(b)

(c)

Figure 3: Optimal max-min fair UL power control for different UE power budgets
$p_{\max}$, and related upper bound computed using (14), for the case of (a)
cellular network; (b) distributed cell-free network; (c) centralized cell-free
network. By computing the spectral radius of a single known matrix independent
from $p_{\max}$, the proposed bound provides accurate performance estimates
for the noise limited $(p_{\max}\to 0^{+})$ and interference limited
$(p_{\max}\to\infty)$ regimes. Furthermore, it also offers a useful rule-of-
thumb for estimating the transition point between the two regimes.

We consider the network depicted in Figure 1, where $K=64$ UEs are uniformly
distributed within a squared service area of size $1\times 1~{}\text{km}^{2}$,
and $L=16$ regularly spaced APs with $M=8$ antennas each. By neglecting for
simplicity channel correlation, we let each sub-vector $\mathbbm{h}_{l,k}$ of
$\mathbbm{h}_{k}^{\mathsf{H}}=:[\mathbbm{h}_{1,k}^{\mathsf{H}},\ldots\mathbbm{h}_{L,k}^{\mathsf{H}}]$
be independently distributed as
$\mathbbm{h}_{l,k}\sim\mathcal{CN}\left(\bm{0},\gamma_{l,k}\bm{I}_{M}\right)$,
where $\gamma_{l,k}>0$ denotes the channel gain between AP $l$ and UE $k$. We
follow the same 3GPP-like path-loss model adopted in [5] for a $2$ GHz carrier
frequency:

$\gamma_{l,k}=-21.9\log_{10}\left(\dfrac{D_{l,k}}{1\;\mathrm{m}}\right)-30.5+Z_{l,k}\quad\text{[dB]},$

where $D_{l,k}$ is the distance between AP $l$ and UE $k$ including a
difference in height of $10$ m, and $Z_{l,k}\sim\mathcal{CN}(0,\rho^{2})$ [dB]
are shadow fading terms with standard deviation $\rho=4$. The shadow fading is
correlated as
$\mathsf{E}[Z_{l,k}Z_{j,i}]=\rho^{2}2^{-\frac{\delta_{k,i}}{9\text{ [m]}}}$
for all $l=j$ and zero otherwise, where $\delta_{k,i}$ is the distance between
UE $k$ and UE $i$. The noise power is $\sigma=-174+10\log_{10}(B)+F$ [dBm],
where $B=20$ MHz is the system bandwidth, and $F=7$ dB is the noise figure.

We focus on UL power control as described in Section III-B. For the choice of
$\\{\mathbbm{v}_{k}\\}_{k\in\mathcal{K}}$, we consider optimal combiners
maximizing $R_{k}^{\mathrm{UL}}(p_{\max}\bm{1})$ under full power
transmission111We recall that the proposed power control framework requires
the combiners to be independent from the free variables $\bm{p}$.
$p_{\max}=20$ dBm, and under different APs cooperation constraints. In
particular, we assume that each UE $k$ is served only by its $Q$ strongest
APs, that is, by the the subset of APs indexed by
$\mathcal{L}_{k}\subseteq\mathcal{L}$, where each set $\mathcal{L}_{k}$ is
formed by ordering $\mathcal{L}$ w.r.t. decreasing $\gamma_{l,k}$ and by
keeping only the first $Q$ elements. Furthermore, we assume each AP $l$ to
acquire (and possibly exchange) local CSI
$\hat{\mathbbm{H}}_{l}:=[\hat{\mathbbm{h}}_{l,1},\ldots,\hat{\mathbbm{h}}_{l,K}]$,

$(\forall
k\in\mathcal{K})~{}\hat{\mathbbm{h}}_{l,k}:=\begin{cases}\mathbbm{h}_{l,k}&\text{if
}l\in\mathcal{L}_{k},\\\
\mathsf{E}[\mathbbm{h}_{l,k}]&\text{otherwise}.\end{cases}$

This model reflects the canonical cell-free massive MIMO implementation with
pilot-based local channel estimation, and possible CSI sharing through the
fronthaul; we neglect for simplicity estimation/quantization noise, and we
focus on a simple model where small-scale fading coefficients are either
perfectly known at some APs or completely unknown. In contrast, we assume that
the relevant long-term statistical information is perfectly shared within the
network. We then study the following AP cooperation regimes222To avoid
cumbersome notation, the combiners described below are given in a unnormalized
form. In the following simulations, a deterministic scaling factor is applied
to each $\\{\mathbbm{v}_{k}\\}_{k\in\mathcal{K}}$ s.t. $(\forall
k\in\mathcal{K})$ $\mathsf{E}[\|\mathbbm{v}_{k}\|^{2}]=1$ holds.

####  III-D1 Cellular network

We assume each UE to be served only by its strongest AP, i.e., $Q=1$, and no
CSI sharing. The $l$-th submatrix of the optimal combining matrix
$\mathbbm{V}:=[\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K}]$ corresponding to AP
$l$ is the known local MMSE solution [10]

$(\forall
l\in\mathcal{L})~{}\mathbbm{V}^{\mathrm{LMMSE}}_{l}:=\left(\hat{\mathbbm{H}}_{l}\hat{\mathbbm{H}}_{l}^{\mathsf{H}}+\bm{\Sigma}_{l}+\frac{\sigma}{p_{\max}}\bm{I}_{M}\right)^{-1}\hat{\mathbbm{H}}_{l},$

where
$\bm{\Sigma}_{l}:=\underset{i\in\\{k\in\mathcal{K}|l\notin\mathcal{L}_{k}\\}}{\sum}\gamma_{l,i}\bm{I}_{M}$
is the CSI error covariance.

####  III-D2 Distributed cell-free network

We assume each UE to be jointly served by its $Q=4$ strongest APs, and no CSI
sharing. The $l$-th submatrix of the optimal combining matrix
$\mathbbm{V}:=[\mathbbm{v}_{1},\ldots,\mathbbm{v}_{K}]$ corresponding to AP
$l$ is the so-called local team MMSE solution [24, 25]:

$(\forall
l\in\mathcal{L})~{}\mathbbm{V}^{\mathrm{LTMMSE}}_{l}:=\mathbbm{V}^{\mathrm{LMMSE}}_{l}\bm{W}_{l},$

where the columns of $\bm{W}_{l}=:[\bm{w}_{l,1},\ldots,\bm{w}_{l,K}]$ can be
computed from the knowledge of
$\bm{\Pi}_{l}:=\mathsf{E}[\hat{\mathbbm{H}}_{l}^{\mathsf{H}}\mathbbm{V}^{\mathrm{LMMSE}}_{l}]$
as the unique solution to the linear system of equations

$\begin{cases}\bm{w}_{l,k}+\sum_{j\neq
l}\bm{\Pi}_{j}\bm{w}_{j,k}=\bm{e}_{k}&\text{if }l\in\mathcal{L}_{k},\\\
\bm{w}_{l,k}=\bm{0}&\text{otherwise}.\end{cases}$

####  III-D3 Centralized cell-free network

We assume each UE to be jointly served by its $Q=4$ strongest APs, perfectly
sharing their CSI. For all $k\in\mathcal{K}$, the subvector
$\mathbbm{v}^{(k)}$ of the optimal joint combiner $\mathbbm{v}_{k}$
corresponding to the $Q$ APs serving UE $k$ is the $k$th column of the known
MMSE solution [5]

$\mathbbm{V}^{\mathrm{MMSE},(k)}:=\left(\hat{\mathbbm{H}}^{(k)}(\hat{\mathbbm{H}}^{(k)})^{\mathsf{H}}+\bm{\Sigma}^{(k)}+\frac{\sigma}{p_{\max}}\bm{I}_{QM}\right)^{-1}\hat{\mathbbm{H}}^{(k)}$

where $\hat{\mathbbm{H}}^{(k)}$ is obtained from the global CSI matrix
$\hat{\mathbbm{H}}^{\mathsf{H}}:=[\hat{\mathbbm{H}}_{1}^{\mathsf{H}},\ldots,\hat{\mathbbm{H}}_{L}^{\mathsf{H}}]$
by removing from $\hat{\mathbbm{H}}$ the rows corresponding to the channels of
all APs $\l\notin\mathcal{L}_{k}$, and $\bm{\Sigma}^{(k)}$ is the
corresponding CSI error covariance.

Figure 2 compares the performance of the considered cooperation regimes under
optimal max-min fair power control, that is, by letting $\bm{\omega}=\bm{1}$
in Problem (15), for a UE power budget $p_{\max}=20$ dBm. As a baseline, we
also show the achieved utility (i.e., the minimum rate) by assuming full power
transmission $\bm{p}=p_{\max}\bm{1}$. The results are averaged over
$N_{\text{setups}}=100$ i.i.d. realizations of the UE positions. In line with
related literature (see, e.g., [26]), Figure 2 highlights the importance of
optimal power control, especially for tighter AP cooperation constraints
impairing the spatial interference suppression capabilities.

Figure 3 illustrates the effectiveness of the proposed upper bound (14) in
predicting the performance of optimal max-min fair power control under
arbitrary UE power budget $p_{\max}$. We remark that the bound depends only on
fixed parameters $(\bm{G},\bm{d})$, i.e., on the choice of the combiners and
on channel statistics; here, we use the same combiners as in Figure 2, i.e.,
the optimal combiners for $(\forall k\in\mathcal{K})$ $p_{k}=20$ dBm, and we
focus on a single realization of the UE positions. The non-differentiable
point in the upper bound curve corresponds to the point $p_{T}$ after which
the system smoothly transitions from the noise limited to the interference
limited regime. As predicted by theory, the proposed bound becomes especially
tight as the system approaches these two extreme regimes. Furthermore, as
intuitively expected, the transition point $p_{T}$ is smaller for tighter AP
cooperation constraints.

##  IV Conclusion

We have presented a simple expression for the solution to a common class of
max-min power control problems, and illustrated its application to cellular
and cell-free massive MIMO networks. In previous studies this solution has
been characterized as the solution to a conditional eigenvalue problem, as the
limit of sequences produced via very specific iterative methods, or as the
solution to a system of linear equations that requires the bisection method
for its construction. Having a novel characterization of the solution, we now
open up the possibility of devising new scalable power control techniques for
large-scale problems. In particular, we showed that it suffices to implement
standard numerical routines for computing the largest eigenvalue (and the
corresponding eigenvector) among a set of matrices constructed from the
problem parameters. We also obtained a simple bound for the optimal utility,
and simulations have shown that the bound is tight asymptotically, as
predicted by theory. In addition, for a large interval of the maximum power
constraint, this bound may be also close to the optimum.

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