# The fixed point iteration of positive concave mappings converges
geometrically if a fixed point exists: implications to wireless systems

Tomasz Piotrowski† and Renato L. G. Cavalcante†,⋆
1 Faculty of Physics, Astronomy and Informatics,
Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń, Poland
2 Fraunhofer Heinrich Hertz Institute
Einsteinufer 37, 10587 Berlin, Germany

###### Abstract

We prove that the fixed point iteration of arbitrary positive concave mappings
with nonempty fixed point set converges geometrically for any starting point.
We also show that positivity is crucial for this result to hold, and the
concept of (nonlinear) spectral radius of asymptotic mappings provides us with
information about the convergence factor. As a practical implication of the
results shown here, we rigorously explain why some power control and load
estimation algorithms in wireless networks, which are particular instances of
a fixed point iteration, have shown geometric convergence in simulations.
These algorithms have been typically derived by considering fixed point
iterations of the general class of standard interference mappings, so the
possibility of sublinear convergence rate could not be ruled out in previous
studies, except in special cases that are often more restrictive than those
considered here.

###### Index Terms:

Positive concave mappings, geometric convergence, fixed point analysis,
nonlinear Perron-Frobenius theory.

$\dagger$ Both authors contributed equally.

$\star$ Corresponding author.

##  I Introduction

Many problems in economy [1], wireless communications [2, 3, 4, 5, 6, 7, 8, 9,
10], and machine learning [11], to cite a few fields, use the so-called
_standard interference mappings_ [2] in their formulations, and we note that
these mappings are also known as (positive) order-preserving (or monotone)
strictly subhomogeneous mappings in the mathematical literature [12, 13, 14].
If we restrict the attention to the wireless domain, problems such as load
estimation [3, 4, 15, 16, 17, 10], power control [5, 6, 7, 9], and signal-to-
interference-noise ratio (SINR) feasibility studies [8, 18] often involve
standard interference mappings. From a mathematical perspective, these
problems have in common that the objective is to answer whether a standard
interference mapping has a fixed point, and, if so, whether the fixed point,
which is unique if it exists [2], can be obtained with simple algorithms.

Regarding the first question, the existence of a fixed point, recently the
study in [4] has used the concept of spectral radius of asymptotic mappings to
obtain a necessary and sufficient condition for an arbitrary (possibly
nonlinear) standard interference mapping to have a fixed point (see Fact 2).
This condition is often easy to verify in practice, and it generalizes and
unifies many mathematical tools for feasibility studies in the wireless
literature. If a standard interference mapping has a fixed point, then its
usage with the fixed point iteration produces a sequence that converges to the
fixed point of the mapping [2], thus we have an answer to the second question
posed above; namely, how to obtain the fixed point.

Given the simplicity and widespread use of the fixed point iteration of
standard interference mappings in the wireless domain, a great deal of effort
has been devoted to understanding and improving its convergence rate [3, 19,
20, 21][5, Ch. 5.3]. From a theoretical perspective, an important but somewhat
disappointing result in this direction has been established in [20, Example
2], which shows that the fixed point iteration of standard interference
mappings can converge sublinearly. However, this bad performance does not
usually manifest in practice. A possible explanation is that very frequently
the standard interference mappings have additional structure that is not
exploited to prove convergence of the fixed point iteration. In particular,
positive concave mappings are common in applications [2, 21, 22, 5, 6, 3, 4,
9, 10, 23], and we recall that these mappings are a subclass of standard
interference mappings [3, Proposition 1]. Nevertheless, to the best our
knowledge, to date it is unknown whether this structure is sufficient to
guarantee geometric convergence in normed vector spaces except in special
cases. There are recent results proving geometric convergence of fixed point
iterations of a very general class of mappings [24, 25], but whether these
results give rise to easily identifiable subclasses of standard interference
mappings that guarantee geometric convergence of the fixed point iterations
(and obtaining easily computable bounds on the convergence rate) remains
largely unknown.

Geometric convergence in normed vector spaces has been established if the
fixed point iteration uses a concave mapping constructed by taking the
coordinate-wise minimum of _finitely_ many positive affine mappings [21],
provided that a fixed point exists. Nevertheless, extending the analysis in
that study to arbitrary positive concave mappings does not seem trivial, and
we emphasize that concave mappings without the particular structure considered
in [21] are very common in the wireless domain [3, 4, 9, 15, 16, 17, 26, 22].
In [22, Example 2], the authors show that the fixed point iterations of
positive affine mappings (a particular case of positive concave mappings)
cannot converge super-linearly, and in [22, Sect. VI] they illustrate the
convergence speed of a nonlinear positive concave mapping with figures.
However, that discussion is restricted to a very specific nonlinear mapping
related to a two-user power control problem with beamforming. Even in that
specific setting, no formal proof of geometric convergence is provided.

If a positive concave mapping has a fixed point, the study in [19] has shown
that, with a mild assumption, the convergence of the fixed point iteration
cannot be better than geometric, and, for specific starting points, the
convergence factor is bounded by the spectral radius of the asymptotic mapping
associated with the concave mapping. Nevertheless, those results do not rule
out the possibility of sublinear convergence. In this study, we give a
complete answer to this question. In more detail, our main contributions can
be summarized as follows:

  1. 1.

We prove that the fixed point iteration of an arbitrary (continuous)111The
assumption of continuity can be dropped in many of our results, but we only
consider mappings that are everywhere continuous on their domains to avoid
technical digressions. We recall that positive concave mappings can be
discontinuous on the boundary of their domains. See [3] for a simple example.
positive concave mapping with an arbitrary starting point is guaranteed to
converge geometrically in normed vector spaces if the mapping has a fixed
point (Proposition 3). We also construct an example illustrating that this
property does not necessarily hold if we drop the assumption of positivity
(Example 3).

  2. 2.

We establish useful connections between the convergence factor of the fixed
point iteration in Thompson’s metric space and normed vector spaces
(Proposition 2).

  3. 3.

We show that the concept of (nonlinear) spectral radius of the asymptotic
mapping associated with a continuous positive concave mapping provides us with
useful information about the convergence factor for any starting point of the
fixed point iteration (Proposition 4).

  4. 4.

We illustrate the main implications of the results derived here in power
control and load estimation problems in wireless networks (Sect. IV and Sect.
V)

To keep this study self-contained, we list in Appendix A definitions and facts
that are required in proofs but not for understanding the main results of this
study. Technical prerequisites that are crucial for setting the stage for the
main contributions are discussed in the next section. In a first reading, we
invite readers interested in problems in wireless networks to skip directly to
Sect. IV to understand the practical implications of the main technical
contributions in Sects. II and III.

##  II Preliminaries

Before introducing definitions and technical results required for the main
proofs, we need the following necessary notation on the nonnegative cone:

$\mathbb{R}^{k}_{+}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\\{(x_{1},\dots,x_{k})\in\mathbb{R}^{k}:\
x_{i}\geq 0\textrm{ for }1\leq i\leq k\\}.$

By $\operatorname{int}(\mathbb{R}^{k}_{+})$ we denote the set of (strictly)
positive vectors, i.e.,
$x=(x_{1},\dots,x_{k})\in\operatorname{int}(\mathbb{R}^{k}_{+})\iff x_{i}>0$
for $1\leq i\leq k.$ Let now $x,y\in\mathbb{R}^{k}_{+}$; then, $x\leq y$
denotes the partial ordering induced by the nonnegative cone, i.e., $x\leq
y\iff y-x\in\mathbb{R}^{k}_{+}$. Similarly, $x<y$ means that
$y-x\in\mathbb{R}^{k}_{+}$ and $x\neq y$, while $x\ll y$ means that
$y-x\in\operatorname{int}(\mathbb{R}^{k}_{+})$.

Let $x$ be an element of a topological space. We call $U$ a neighborhood of
$x$ if there exists an open set $V$ such that $x\in V\subseteq U.$ We denote
by $x[i]\in\mathbb{R}$ for $i\in\\{1,2,\dots,k\\}$ the $i$-th coefficient of a
vector $x\in\mathbb{R}^{k}$, and by $[X]_{i,j}\in\mathbb{R}$ the element of a
matrix $X\in\mathbb{R}^{m\times n}$ at the $i$-th row and $j$-th column with
$i\in\\{1,2,\dots,m\\}$ and $j\in\\{1,2,\dots,n\\}.$

Given a mapping $f:X\to Y$ for two sets $X$ and $Y$ satisfying $Y\subseteq
X\subseteq\mathbb{R}^{k}_{+}$, we denote by

$\mathrm{Fix}(f)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\\{x\in
X~{}|~{}f(x)=x\\}$

the set of fixed points of $f$. Moreover, for a metric space $(X,d_{1})$, a
sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ converges to $\widetilde{x}\in X$
if $\forall\epsilon>0$ $\exists n_{0}\in\mathbb{N}$ such that $\forall
n>n_{0}$ one has that $d_{1}(x_{n},\widetilde{x})<\epsilon.$ We also note that
a norm $\|\cdot\|$ of a normed vector space $(\mathbb{R}^{k}_{+},\|\cdot\|)$
induces a metric on $X$ by
$d_{2}(x,y)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\|x-y\|$
for $x,y\in X.$ In such a case, we say that $d_{2}$ is induced by a norm
$\|\cdot\|.$

Let $f\colon\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}$, and construct a sequence
$(x_{n})_{n\in\mathbb{N}}$ via

$\forall n\in\mathbb{N}\quad
x_{n+1}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}f(x_{n})\textrm{
with }x_{1}\in\mathbb{R}^{k}_{+}.$ (1)

We call the recursion in (1) a fixed point iteration of $f$ because (1) is a
classical algorithm to compute a point in $\mathrm{Fix}(f)$ provided that
$\mathrm{Fix}(f)\neq\emptyset$.

Under consideration in this study is the problem of establishing geometric
convergence, in the sense defined below, of a sequence constructed with the
iteration in (1) with a particular class of mappings described later in
Definition 5.

######  Definition 1.

Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{k}_{+}.$ We say
that $(x_{n})_{n\in\mathbb{N}}$ converges geometrically to
$\widetilde{x}\in\mathbb{R}^{k}_{+}$ with a factor $c\in[0,1)$ and a constant
$\gamma>0$ if

$\exists c\in[0,1)\quad\exists\gamma>0\quad\forall
n\in\mathbb{N}\quad\|x_{n+1}-\widetilde{x}\|\leq\gamma c^{n},$ (2)

where $\|\cdot\|$ is an arbitrary norm. Similarly, we say that
$(x_{n})_{n\in\mathbb{N}}$ converges linearly to
$\widetilde{x}\in\mathbb{R}^{k}_{+}$ with a factor $c\in[0,1)$ if

$\exists c\in[0,1)\quad\forall
n\in\mathbb{N}\quad\|x_{n+1}-\widetilde{x}\|\leq c\,\|x_{n}-\widetilde{x}\|,$
(3)

where $\|\cdot\|$ is an arbitrary norm. We verify that, if
$(x_{n})_{n\in\mathbb{N}}$ converges linearly to
$\widetilde{x}\in\mathbb{R}^{k}_{+}$ with a factor $c\in[0,1)$, then it
converges geometrically to $\widetilde{x}\in\mathbb{R}^{k}_{+}$ with the same
factor $c$ and a constant $\gamma=\|x_{1}-\widetilde{x}\|.$222Geometric
convergence is also known in literature as R-linear convergence, whereas
linear convergence is also known as Q-linear convergence [27, Chapter 9]. We
also note that the above definitions of geometric and linear convergence in
normed spaces can be naturally generalized to metric spaces, with the metric
induced by the norm in (2) and (3) replaced with a metric of a metric space.

Given an arbitrary mapping $f\colon\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}$,
in general there are no guarantees that the fixed point iteration converges,
even if $f$ has a fixed point. As a result, we need to impose additional
structure on $f$, and the structure should keep the class of mappings general
enough to be able to address as many real-world problems as possible. Of
particular interest in the wireless literature are the standard interference
mappings, which are known to produce fixed point iterations that converge if a
fixed point exists [2]. These mappings are defined below.

######  Definition 2.

A mapping $f\colon\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}$ is said to be a
standard interference mapping (SI mapping) if it is

  1. 1.

_monotonic_

$\forall x,y\in\mathbb{R}^{k}_{+}\quad x\leq y\implies f(x)\leq f(y),\textrm{
and }$ (4)

  2. 2.

_scalable_

$\forall x\in\mathbb{R}^{k}_{+}\quad\forall\lambda>1\quad f(\lambda
x)\ll\lambda f(x).$ (5)

######  Remark 1.

Functions satisfying the monotonicity property in Definition 2 receive
different names depending on the author. In particular, they are also known as
order-preserving. Hereafter, we simply call them monotone or monotonic
functions, as also done in some studies on mathematics [14, 28]. This
particular notion of monotonicity should not be confused with that used in
convex analysis [29], which is a different concept in general.

Note that some references explicitly add positivity to the definition of SI
mappings, but we recall that this requirement can be omitted because of the
next result.

######  Fact 1.

[30] Let $f\colon\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}$ be an SI mapping in
the sense of Definition 2. Then
$f(\mathbb{R}^{k}_{+})\subseteq\operatorname{int}({\mathbb{R}^{k}_{+}}).$

Before even trying to compute the fixed point of a SI mapping, we should first
answer the more fundamental question of whether the mapping has a fixed point.
The result shown in Fact 2 below can be used for this purpose, and note that
Fact 2 also establishes uniqueness of the fixed point. To understand the
statement, we first need to introduce the concepts of asymptotic mappings and
(nonlinear) spectral radius, defined as follows:

######  Definition 3.

[4] Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$
be a continuous SI mapping. The asymptotic mapping associated with $f$ is the
continuous mapping defined by

$f_{\infty}:\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}:x\mapsto\lim_{p\to\infty}\dfrac{1}{p}f(px).$
(6)

We recall that the above limit always exists in any normed vector space.

######  Definition 4.

The (nonlinear) spectral radius of a continuous SI mapping
$f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ is
given by the largest eigenvalue of the corresponding asymptotic mapping [13,
31, 4]:

$\rho(f_{\infty})=\\\ \max\\{\lambda\in\mathbb{R}_{+}~{}|~{}\ \exists
x\in\mathbb{R}^{k}_{+}\backslash\\{0\\}\textrm{ s.t. }f_{\infty}(x)=\lambda
x\\}\in\mathbb{R}_{+}.$

The existence of such an eigenvalue is established in [13], and we remark that
there exist simple iterative methods to compute $\rho(f_{\infty})$ [4, Remarks
2 and 3].

######  Fact 2.

If $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ is a
continuous SI mapping, then $f$ has a fixed point if and only if
$\rho(f_{\infty})<1$ [4]. Furthermore, if a fixed point exists, then it is
unique [2].

In many studies, convergence of fixed point iterations is proved using
arguments valid for any SI mapping [2, 21, 3, 4, 5, 6, 7, 8, 9, 10, 11].
Therefore, we are unable to establish fine properties of the fixed point
iteration, such as geometric convergence in normed vector spaces, because
properties of this type do not hold in general [20, Example 2]. However, the
problems being addressed in many studies typically involve only positive
concave mappings, which are described below in Definition 5. We remark that
these mappings can be seen as a further restriction imposed on SI mappings, as
shown in Fact 3. Nevertheless, positive concave mappings remain general enough
to address applications not only in the wireless domain, but in many other
fields, so we focus on (continuous) positive concave mappings in this study.

######  Definition 5.

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
concave with respect to (w.r.t.) the cone order:

$\forall x,y\in\mathbb{R}^{k}_{+}\quad\forall t\in(0,1)\\\ f(tx+(1-t)y)\geq
tf(x)+(1-t)f(y).$ (7)

Then $f$ is called a positive concave (PC) mapping.

######  Fact 3.

[3, 4] Let
$f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a PC
mapping. Then $f$ is an SI mapping (note: the converse does not hold in
general).

To analyze the convergence of fixed point iterations of monotone mappings on
positive or nonnegative cones, researchers in mathematics have extensively
used the metric space $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ [12,
14], where $d_{T}$ denotes Thompson’s metric defined below. Recalling that PC
mappings in Definition 5 are also monotone and defined on nonnegative cones,
we can build upon a huge body of the mathematical literature to prove the main
results in the next sections.

######  Definition 6.

We define Thompson’s metric by

$\begin{array}[]{rl}d_{T}:\operatorname{int}({\mathbb{R}^{k}_{+}})\times\operatorname{int}({\mathbb{R}^{k}_{+}})&\to[0,\infty)\\\
(x,y)&\mapsto\ln(\max\\{M(x,y),M(y,x)\\}),\end{array}$ (8)

where
$M(x,y)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\\{\beta>0~{}|~{}x\leq\beta
y\\}$ and analogously for $M(y,x).$

Hereafter, we call the metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ the _Thompson metric
space_. The chief results of this paper address properties of positive concave
(PC) mappings on compact sets. Therefore, we need to establish equivalence
results for compact sets between Thompson metric space and normed vector
spaces. To this end, we first state the following technical results in Remarks
2 and 3.

######  Remark 2.

The Thompson metric space $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$
is isometric (and, hence, also homeomorphic) to
$(\mathbb{R}^{k},\|\cdot\|_{\infty})$, with an isometry given by the
componentwise natural logarithm; namely

$L\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\mathbb{R}^{k}:x\mapsto(\ln
x[1],\ln x[2],\dots,\ln x[k]),$ (9)

where $x=(x[1],x[2],\dots,x[k])\in\operatorname{int}({\mathbb{R}^{k}_{+}})$
[12, Proposition 2.2.1]. This fact implies that
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ inherits all topological
and metric properties of $(\mathbb{R}^{k},\|\cdot\|_{\infty}).$ We also note
that the inverse isometry from $(\mathbb{R}^{k},\|\cdot\|_{\infty})$ to
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ is given as

$L^{-1}\colon\mathbb{R}^{k}\to\operatorname{int}({\mathbb{R}^{k}_{+}}):y\mapsto(\exp
y[1],\exp y[2],\dots,\exp y[k]),$ (10)

where $y=(y[1],y[2],\dots,y[k])\in\mathbb{R}^{k}.$

######  Remark 3.

From Remark 2 we obtain in particular that the Thompson metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ is complete (i.e., any
Cauchy sequence is convergent). Furthermore,
$U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ is a closed, bounded, or
compact set in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ if and only
if $L(U)$ is a closed, bounded, or compact set in
$(\mathbb{R}^{k},\|\cdot\|_{\infty})$, and by the norm equivalence in
$\mathbb{R}^{k}$, if and only if $L(U)$ is a closed, bounded, or compact set
in $(\mathbb{R}^{k},\|\cdot\|)$ for any norm $\|\cdot\|.$

A potential drawback of working in the metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$, which can be observed from
Remark 3, is that the topological properties of a set
$U\subseteq\operatorname{int}({\mathbb{R}^{k}_{+}})$ in Thompson metric space
are the same as of $L(U)\subseteq\mathbb{R}^{k}$ in a normed vector space, but
not necessarily of $U$ considered in this normed vector space. For example,
given $u\in\operatorname{int}({\mathbb{R}^{k}_{+}})$, the set
$U\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\\{x\in\operatorname{int}({\mathbb{R}^{k}_{+}})~{}|~{}x\leq
u\\}$ is _closed and unbounded_ in Thompson metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$, just like the set
$L(U)=\\{x\in\mathbb{R}^{k}~{}|~{}x\leq L(u)\\}$ in a normed vector space
$(\mathbb{R}^{k},\|\cdot\|).$ On the other hand, $U$ is clearly bounded and
neither closed nor open in a normed vector space. However, as Corollary 1
below shows, as far as compact sets are concerned, we do not have to worry
about the space in context [Thompson metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ or a normed vector space
$(\mathbb{R}^{k},\|\cdot\|)$].

######  Corollary 1.

A set $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ is compact in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ if and only if $U$ is
compact in $(\mathbb{R}^{k},\|\cdot\|)$ for any norm $\|\cdot\|.$

###### Proof.

$(\Rightarrow)$ Let $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
compact in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ From Remark 3,
$L(U)$ is compact in $(\mathbb{R}^{k},\|\cdot\|)$ for any norm $\|\cdot\|.$ As
$L^{-1}$ is continuous, $L^{-1}(L(U))=U$ is also compact in
$(\mathbb{R}^{k},\|\cdot\|).$

$(\Leftarrow)$ Let $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
compact in $(\mathbb{R}^{k},\|\cdot\|)$ for any norm $\|\cdot\|.$ As $L^{-1}$
is an isometry from $(\mathbb{R}^{k},\|\cdot\|_{\infty})$ to
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$, an analogous argument to
that used in Remark 3 allows us to conclude that $L^{-1}(U)$ is compact in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ As $L$ is continuous,
$L(L^{-1}(U))=U$ is also compact in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ ∎

The following corollary establishes further equivalent properties of compact
sets in Thompson metric space and in normed vector spaces. More precisely, it
provides an equivalent definition of a compact set in Thompson metric space,
and note that this definition corresponds to the well known Heine-Borel
theorem for finite-dimensional normed vector spaces.

######  Corollary 2.

A set $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ is compact in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ if and only if $U$ is
closed and bounded in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$

###### Proof.

$(\Rightarrow)$ Let $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
compact in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ In view of
Remark 3, $L(U)$ is compact in any normed vector space
$(\mathbb{R}^{k},\|\cdot\|).$ By the Heine-Borel Theorem, $L(U)$ is closed and
bounded in $(\mathbb{R}^{k},\|\cdot\|)$. From Remark 3, $U$ is closed and
bounded in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$

$(\Leftarrow)$ Let now $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a
closed and bounded set in $(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$
In view of Remark 3, $L(U)$ is closed and bounded in any normed vector space
$(\mathbb{R}^{k},\|\cdot\|).$ By the Heine-Borel Theorem, $L(U)$ is compact in
$(\mathbb{R}^{k},\|\cdot\|).$ From Remark 3, $U$ is compact in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ ∎

We finish this section with a list of standard definitions and results in
fixed point theory.

######  Definition 7.

Let
$f\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\operatorname{int}({\mathbb{R}^{k}_{+}}).$
We say that $f$ is a para-contraction (is strictly nonexpansive) w.r.t.
Thompson’s metric on $\operatorname{int}({\mathbb{R}^{k}_{+}})$ if

$\forall x,y\in\operatorname{int}({\mathbb{R}^{k}_{+}})\quad x\neq y\implies
d_{T}(f(x),f(y))<d_{T}(x,y).$ (11)

Moreover, we say that $f$ is a $c$-Lipschitz contraction w.r.t. Thompson’s
metric on $\operatorname{int}({\mathbb{R}^{k}_{+}})$ if

$\exists c\in[0,1)\quad\forall
x,y\in\operatorname{int}({\mathbb{R}^{k}_{+}})\quad d_{T}(f(x),f(y))\leq
c\,d_{T}(x,y).$ (12)

We call $c$ in (12) a contraction factor of $f.$ Finally, we say that $f$ is a
local $c$-Lipschitz contraction w.r.t. Thompson’s metric on a compact set
$U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ if

$\exists c\in[0,1)\quad\forall x,y\in U\quad d_{T}(f(x),f(y))\leq
c\,d_{T}(x,y).$ (13)

We call $c$ in (13) a local contraction factor of $f$ on $U.$

######  Remark 4.

Let
$f\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\operatorname{int}({\mathbb{R}^{k}_{+}})$
be a $c$-Lipschitz contraction w.r.t. Thompson’s metric on
$\operatorname{int}({\mathbb{R}^{k}_{+}}).$ Then, from the Banach fixed point
theorem, there exists a unique fixed point
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ of $f$ such that the
fixed point iteration in (1) converges both linearly and geometrically to
$x^{\star}$ with a (contraction) factor $c$, and, in the latter case, with
$\gamma=d_{T}(x_{2},x_{1})(1-c)^{-1}.$

######  Fact 4.

[12, Lemma 2.1.7] Let
$f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be an
SI mapping and let $f_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mu{\operatorname{int}({\mathbb{R}^{k}_{+}})}}$ be the restriction of $f$ to
$\operatorname{int}({\mathbb{R}^{k}_{+}}).$ Then $f_{\mkern 1.0mu\vrule
height=6.02777pt\mkern 2.0mu{\operatorname{int}({\mathbb{R}^{k}_{+}})}}$ is a
para-contraction w.r.t. Thompson’s metric on
$\operatorname{int}({\mathbb{R}^{k}_{+}}).$

##  III Convergence of the fixed point iteration of continuous PC mappings

The main objective of this study is to prove geometric convergence of the
fixed point iteration of an arbitrary continuous PC mapping with nonempty
fixed point set, and we recall that the fixed point set of these mappings is a
singleton as a consequence of Facts 2 and 3. To establish geometric
convergence, we start by proving that PC mappings
$f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ with
$\mathrm{Fix}(f)\neq\emptyset$ are local contractions w.r.t. Thompson’s metric
$d_{T}$ in (8) in a neighborhood $U$ of the fixed point
$x^{\star}\in\mathrm{Fix}(f)$. Therefore, since $(U,d_{T})$ is a complete
metric space, then standard arguments based on the Banach fixed point theorem
guarantees geometric convergence of the fixed point iteration in $(U,d_{T})$
for $x_{1}\in U$. These results are the main topic of Sect. III-A.

In engineering applications such as those in the wireless domain, we are
typically not interested in geometric convergence of the fixed point algorithm
in the metric space $(U,~{}d_{T})$ for an unspecified neighborhood $U$ of the
fixed point, but in a normed vector space $(\mathbb{R}^{k},~{}\|\cdot\|)$ (the
choice of the norm is irrelevant because of the equivalence of norms in finite
dimensional spaces). Unfortunately, the homeomorphism in (9) between
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),~{}d_{T})$ and
$(\mathbb{R}^{k},~{}\|\cdot\|_{\infty})$ is nonlinear, so it may be unclear
whether the results in Sect. III-A imply geometric convergence in a normed
vector space. This problem is addressed in Sect. III-B, where we prove that
geometric convergence of the fixed point iteration in $(U,~{}d_{T})$ with a
factor $c\in~{}[0,1)$ implies geometric convergence in
$(\mathbb{R}^{k},~{}\|\cdot\|)$ with the same factor $c$ for any starting
point $x_{1}\in{}_{+}^{k}$.

Since the contraction factor $c$ gives useful information about the
convergence speed of the fixed point iteration [see (2)], in Sect. III-C we
show that the concept of (nonlinear) spectral radius is useful not only for
establishing the existence of a fixed point as shown in Fact 2, but also for
bounding $c$.

###  III-A Local $c$-Lipschitz contractivity

Before delving into the convergence of the fixed point iteration of PC
mappings, we first derive properties required for a mapping to be a
contraction w.r.t. Thompson’s metric. We begin with the following lemma, which
is an extension of [12, Lemma 2.1.7] to $c\in[0,1).$

######  Lemma 1.

Let
$f\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\operatorname{int}({\mathbb{R}^{k}_{+}})$
be monotonic. Then $f$ is a $c$-Lipschitz contraction w.r.t. Thompson’s metric
if and only if

$\exists c\in[0,1)\quad\textrm{s.t.}\quad\forall
x\in\operatorname{int}({\mathbb{R}^{k}_{+}})\quad\forall\lambda>1\\\ f(\lambda
x)\leq\lambda^{c}f(x).$ (14)

###### Proof.

$(\Leftarrow)$ Let $x,y\in\operatorname{int}({\mathbb{R}^{k}_{+}})$, so that
$d_{T}(x,y)=\ln\lambda$ for some $\lambda\geq 1.$ If $x=y$, then
$d_{T}(x,y)=0$ and $\lambda=1$, in which case $d_{T}(f(x),f(y))\leq
c\,d_{T}(x,y)$ for any $c\in[0,1).$ Therefore, we assume below that
$\lambda>1.$

_The case $\lambda>1$ ($x\neq y$)_: From the definition of $d_{T}$, one has
that $x\leq\lambda y$ and $y\leq\lambda x.$ From monotonicity of $f$ we also
obtain that $f(x)\leq f(\lambda y)$ and $f(y)\leq f(\lambda x)$, and from (14)
one has that $f(\lambda y)\leq\lambda^{c}f(y)$ and $f(\lambda
x)\leq\lambda^{c}f(x)$ for some $c\in[0,1).$ As a result,

$f(x)\leq f(\lambda y)\leq\lambda^{c}f(y)\textrm{ and }f(y)\leq f(\lambda
x)\leq\lambda^{c}f(x).$

Thus,

$\beta_{1}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\big{\\{}\beta>0~{}|~{}f(x)\leq\beta
f(y)\big{\\}}\leq\lambda^{c}\textrm{ and }\\\
\beta_{2}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\big{\\{}\beta>0~{}|~{}f(y)\leq\beta
f(x)\big{\\}}\leq\lambda^{c},$

and hence also $\max\\{\beta_{1},\beta_{2}\\}\leq\lambda^{c}.$ We now conclude
that

$d_{T}(f(x),f(y))=\ln(\max\\{\beta_{1},\beta_{2}\\})\leq\ln\lambda^{c}=\\\
c\ln\lambda=c\,d_{T}(x,y).\ $

$(\Rightarrow)$ Conversely, assume that $f$ is a $c$-Lipschitz contraction
w.r.t. Thompson’s metric, and let
$x\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ and $\lambda>1$ be chosen
arbitrarily. We note that $d_{T}(x,\lambda x)=\ln\lambda$, and thus

$\displaystyle d_{T}(f(x),f(\lambda x))\leq c\,d_{T}(x,\lambda
x)=c\ln\lambda=\ln\lambda^{c}.$

Let

$\beta_{3}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\big{\\{}\beta>0~{}|~{}f(x)\leq\beta
f(\lambda x)\big{\\}}\textrm{ and }\\\
\beta_{4}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\big{\\{}\beta>0~{}|~{}f(\lambda
x)\leq\beta f(x)\big{\\}}.$

From monotonicity of $f$ we have that $f(x)\leq f(\lambda x)$ because
$\lambda>1$, thus $\beta_{3}\leq\beta_{4}.$ Therefore,

$\displaystyle d_{T}(f(x),f(\lambda x))=\ln\beta_{4}\leq\ln\lambda^{c}.$ (15)

From the definition of $\beta_{4}$, we obtain that $f(\lambda
x)\leq\beta_{4}f(x)$, and from (15) we obtain in particular that
$\beta_{4}\leq\lambda^{c}$. As a result, we also have
$\beta_{4}f(x)\leq\lambda^{c}f(x)$, and the proof of the desired result
$f(\lambda x)\leq\lambda^{c}f(x)$ is now complete. $\blacksquare$

######  Remark 5.

Mappings satisfying (14) are known as $c$-concave mappings in the mathematical
literature, where they are considered in the context of partially ordered
subsets (such as cones) of generic Banach spaces; see, for example, [32, 33,
34].

Continuous PC mappings (and, in particular, positive affine mappings
restricted to $\mathbb{R}^{k}_{+}$) do not satisfy the properties in Lemma 1
in general, even if the mappings have a fixed point. Therefore, these mappings
are not _global_ contractions in the metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),~{}d_{T})$, and thus Lemma 1 cannot
be directly used with the Banach fixed point theorem to deduce geometric
convergence of the fixed point iteration with PC mappings. Nevertheless, the
following corollary shows that a weaker condition than (14) is sufficient for
$f$ to be a _local_ $c$-Lipschitz contraction w.r.t. Thompson’s metric on a
compact set $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$, and in
Proposition 1 we show that this property is satisfied by PC mappings.

######  Corollary 3.

Let $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a compact set
containing a nonempty open set and let
$d_{0}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\max_{x,y\in
U}d_{T}(x,y)=\ln\lambda_{0}$ with $\lambda_{0}>1.$ Let
$f\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\operatorname{int}({\mathbb{R}^{k}_{+}})$
be monotonic. Then $f$ is a $c$-Lipschitz contraction w.r.t. Thompson’s metric
on $U$ if

$\exists c\in[0,1)\quad\textrm{s.t.}\quad\forall x\in
U\quad\forall\lambda\in(1,\lambda_{0}]\quad f(\lambda x)\leq\lambda^{c}f(x).$
(16)

###### Proof.

Let $x,y\in U.$ Then, $d_{T}(x,y)=\ln\lambda$ for some
$1\leq\lambda\leq\lambda_{0}.$ The proof now follows from the _if_ direction
$(\Leftarrow)$ of the proof of Lemma 1, with
$x,y\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ considered therein restricted
to $x,y\in U$, and $\lambda\geq 1$ considered therein restricted to
$\lambda\in[1,\lambda_{0}].$ $\blacksquare$

Thanks to Lemma 1 and Corollary 3, we are able to prove the following
proposition on local $c$-Lipschitz contractivity of continuous PC mappings.

######  Proposition 1.

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
a continuous and concave (w.r.t. to the cone order) mapping (i.e., a
continuous PC mapping), and let
$U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a compact set containing
a nonempty open set. Then $f$ is a local $c$-Lipschitz contraction w.r.t.
Thompson’s metric on $U.$

###### Proof.

For convenience, we divide the proof into the following parts.

_1) Global properties of $f$_: From concavity of $f$, one has in particular
that, for $t\in(0,1)$ and $x\in\mathbb{R}^{k}_{+}$,

$f(tx)=f(tx+(1-t)0)\geq tf(x)+(1-t)f(0).$

Furthermore,

$f(x)=f(t(x/t))\geq tf(x/t)+(1-t)f(0).$

Let $\lambda\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}t^{-1}>1.$
Then, the above inequality can be rewritten as

$f(x)\geq\lambda^{-1}f(\lambda x)+(1-\lambda^{-1})f(0),$

thus

$\lambda f(x)\geq f(\lambda x)+(\lambda-1)f(0),$

and consequently

$f(\lambda x)\leq\lambda f(x)+(1-\lambda)f(0),\quad x\in\mathbb{R}^{k}_{+}.$
(17)

_2) Local properties of $f$ on $U$_: By our assumptions, $f$ is continuous,
and $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ is closed and bounded
(see Corollary 2), hence $f(U)$ is also closed and bounded. In particular,
there exists $\mu>0$ such that $f(0)\geq\mu f(x)$ for all $x\in U.$ We note
that we can always select $\mu\in(0,1)$ because concavity and positivity of
$f$ on $\mathbb{R}^{k}_{+}$ implies its monotonicity on $\mathbb{R}^{k}_{+}$
[3] (see also Fact 3). Noticing that $1-\lambda<0$, we obtain from (17) that

$f(\lambda x)\leq\lambda f(x)+(1-\lambda)\mu f(x),\quad x\in U.$

Thus,

$f(\lambda x)\leq\lambda\Bigg{(}1+\frac{1-\lambda}{\lambda}\mu\Bigg{)}f(x)=\\\
\lambda\Bigg{(}\frac{(1-\mu)\lambda+\mu}{\lambda}\Bigg{)}f(x)=\lambda\eta
f(x),\quad x\in U,$ (18)

where
$\eta\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\frac{(1-\mu)\lambda+\mu}{\lambda}>0$,
because $\lambda>1$ and $\mu\in(0,1).$

_3) Exponential form of $(\ref{c2})$_: Using the _change of base_ rule of
logarithms, we note that
$\log_{\lambda}(\lambda\eta)=\frac{\ln(\lambda\eta)}{\ln\lambda}$, and hence
$\lambda\eta=\lambda^{\frac{\ln(\lambda\eta)}{\ln\lambda}}.$ Thus, we obtain
from (18) that

$f(\lambda
x)\leq\lambda^{\frac{\ln(\lambda\eta)}{\ln\lambda}}f(x)=\lambda^{c(\lambda)}f(x),\quad
x\in U,$ (19)

where

$c(\lambda)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\frac{\ln(\lambda\eta)}{\ln\lambda}=\frac{\ln((1-\mu)\lambda+\mu)}{\ln\lambda}$

is a continuous function of $\lambda>1.$

_4) Monotonic increase of $c(\lambda)$_: We show now that $c(\lambda)$ is
monotonically increasing for $\lambda>1.$ Denote

$s\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}(1-\mu)\lambda+\mu,\quad\lambda>1,$

and observe that $1<s<\lambda$ for any $\mu\in(0,1).$ Thus,

$c(\lambda)=\frac{\ln s}{\ln\lambda}<1,\quad\lambda>1.$

Moreover, we have

$c^{\prime}(\lambda)=\frac{\frac{(1-\mu)\ln\lambda}{s}-\frac{\ln
s}{\lambda}}{(\ln\lambda)^{2}},\quad\lambda>1.$ (20)

Define

$c_{2}(\lambda)\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\frac{(\ln
s)^{\prime}}{(\ln\lambda)^{\prime}}=\frac{(1-\mu)\lambda}{s},\quad\lambda>1.$

Then $c^{\prime}_{2}(\lambda)=\frac{(1-\mu)\mu}{s^{2}}>0$, hence
$c_{2}(\lambda)$ is increasing for $\lambda>1.$

We note that both $\lambda\mapsto\ln s$ and $\lambda\mapsto\ln\lambda$ are
continuous for $\lambda\geq 1$ and differentiable for $\lambda>1$, with
$1\mapsto 0$ in both cases. Therefore, using Cauchy’s (extended) mean value
theorem, we conclude that for any $\lambda_{1}>1$ there exists $\lambda_{2}$
such that $1<\lambda_{2}<\lambda_{1}$ with
$c_{2}(\lambda_{2})=c(\lambda_{1}).$ Thus, as $c_{2}(\lambda)$ is increasing
for $\lambda>1$, we obtain $c_{2}(\lambda_{1})>c(\lambda_{1})$ for
$\lambda_{1}>1.$ Therefore, one has

$c_{2}(\lambda)=\frac{(1-\mu)\lambda}{s}>\frac{\ln
s}{\ln\lambda}=c(\lambda),\quad\lambda>1,$

thus also $\frac{(1-\mu)\ln\lambda}{s}-\frac{\ln s}{\lambda}>0$ for
$\lambda>1.$ In view of (20), we conclude therefore that $c(\lambda)$ is
increasing for $\lambda>1$ and any $\mu\in(0,1).$ We also note that

$1-\mu<c(\lambda)<1,$

as $\lim_{\lambda\to 1^{+}}c(\lambda)=1-\mu$ and
$\lim_{\lambda\to\infty}c(\lambda)=1.$

_5) Explicit local contraction factor_ : Let now $\lambda_{0}>1$ satisfy
$\max_{x,y\in U}d_{T}(x,y)=\ln\lambda_{0}$, and let

$c\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}c(\lambda_{0})<1.$

Then, from (19) and the fact that $c(\lambda)\leq c$ for all
$1<\lambda\leq\lambda_{0}$, one has

$f(\lambda x)\leq\lambda^{c(\lambda)}f(x)\leq\lambda^{c}f(x),\quad x\in U,\
1<\lambda\leq\lambda_{0}.$ (21)

Therefore, we conclude that $f$ satisfies inequality (16) in Corollary 3,
implying that $f$ is a local $c$-Lipschitz contraction w.r.t. Thompson’s
metric on $U.$ $\blacksquare$

Interesting consequences of Proposition 1 are discussed below.

######  Remark 6.

If $f$ has a fixed point
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$, then $U$ can be
selected as a compact neighborhood of $x^{\star}.$ In this case, Proposition 1
provides a local contraction factor for $f$ on $U.$ We also note that the
convergence of a fixed point iteration of $f$ on $U$ is linear w.r.t.
Thompson’s metric in this case.

######  Example 1.

Let $k=1$ and define $f_{1}(x)=(1/2)x+1/2$ for $x\in\mathbb{R}_{+}.$ Let
$U=[1/2,3/2].$ Then, $(U,d_{T})$ is a complete metric space with
$f_{1}(U)\subseteq U$, and, in Proposition 1, we may set $\mu=1/3$ as
$f_{1}(0)=1/2$ and $f_{1}(3/2)=5/4.$ Moreover, we have $\lambda_{0}=3$, thus
$c\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}c(\lambda_{0})\approx
0.771$, which is a local contraction factor of $f_{1}$ on $U.$ Thus, by the
Banach fixed point theorem, $f_{1}$ admits a unique fixed point over $U$,
which is $x^{\star}=1$ in this case.

######  Example 2.

Let $k=1$ and define $f_{2}(x)=x+1$ for $x\in\mathbb{R}_{+}.$ As above, let
$U=[1/2,3/2]$, so that $(U,d_{T})$ is a complete metric space. Then, in
Proposition 1, we may choose $\mu=1/3$ because $f_{2}(0)=1$ and
$f_{2}(3/2)=5/2$, and, as above, we have $\lambda_{0}=3$, thus
$c\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}c(\lambda_{0})\approx
0.771$, which is a local contraction factor of $f_{2}$ on $U.$ We note that we
have the same values of $\mu,\lambda_{0}$, and $c$ as in the preceding
example. However, the Banach fixed point theorem cannot be applied in this
case because $f_{2}(U)\subsetneq U.$ Indeed, it is seen that $f_{2}$ does not
have a fixed point over $U.$

###  III-B Geometric convergence of fixed point iteration in normed vector
spaces

Having established linear, and, hence, geometric convergence of the fixed
point iteration with a continuous PC mapping in the metric space $(U,d_{T})$,
where $U$ is a neighborhood of the fixed point (which is assumed to exist), we
now proceed to prove that we also have geometric convergence in any normed
vector space $(\mathbb{R}^{k},\|\cdot\|)$ for any starting point in
$\mathbb{R}^{k}_{+}$, and the convergence factors in $(U,d_{T})$ and
$(\mathbb{R}^{k},\|\cdot\|)$ can be selected to be the same. To this end, we
need the next result.

######  Proposition 2.

Let
$f\colon\operatorname{int}({\mathbb{R}^{k}_{+}})\to\operatorname{int}({\mathbb{R}^{k}_{+}})$
be a $c$-Lipschitz contraction w.r.t. Thompson’s metric and let
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ be the unique fixed
point of $f.$ Then, the fixed point iteration of $f$ with
$x_{1}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ converges geometrically to
$x^{\star}$ with a factor $c\in[0,1)$ w.r.t. any metric induced by a norm in
$\mathbb{R}^{k}$.

###### Proof.

Let $x_{1}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ and let
$(x_{n})_{n\in\mathbb{N}}$ be the sequence defined through (1). We first note
that, from Remark 4, $(x_{n})_{n\in\mathbb{N}}$ is bounded in
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T}).$ Thus, from Corollary 1 we
conclude that the sequence $(x_{n})_{n\in\mathbb{N}}$ is also bounded in
$(\mathbb{R}^{k},\|\cdot\|)$ for any norm $\|\cdot\|.$ Let $b>0$ be such that
$\forall n\in\mathbb{N}\ \|x_{n}\|\leq b$ for a given norm $\|\cdot\|.$ We
also have $\|x^{\star}\|\leq b$ because $(x_{n})_{n\in\mathbb{N}}$ converges
to $x^{\star}.$ Now, let $\delta>0$ be the normality constant of
$\mathbb{R}^{k}_{+}$ for this norm (see Definition 8 in the Appendix). Then,
using the property that $\forall n\in\mathbb{N}\ d_{T}(x_{n+1},x^{\star})\leq
c^{n}\,d_{T}(x_{1},x^{\star})$ (as $f$ is a $c$-Lipschitz contraction w.r.t.
Thompson’s metric) and Fact 5 in the Appendix A, we deduce

$\forall n\in\mathbb{N}\quad\|x_{n+1}-x^{\star}\|\leq
b(1+2\delta)(e^{d_{T}(x_{n+1},x^{\star})}-1)\\\ \leq
b(1+2\delta)(e^{c^{n}d_{T}(x_{1},x^{\star})}-1)\\\
=b(1+2\delta){(e^{c^{n}d_{T}(x_{1},x^{\star})})}\left(1-\dfrac{1}{e^{c^{n}d_{T}(x_{1},x^{\star})}}\right).$
(22)

Then, noticing that $\forall n\in\mathbb{N}$ one has $c^{n}\in[0,1)$, we
obtain $e^{c^{n}d_{T}(x_{1},x^{\star})}\leq e^{d_{T}(x_{1},x^{\star})}$, and,
consequently,

$\forall n\in\mathbb{N}\quad\|x_{n+1}-x^{\star}\|\leq
s\left(1-\dfrac{1}{e^{c^{n}d_{T}(x_{1},x^{\star})}}\right),$ (23)

where
$s\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}b(1+2\delta){e^{d_{T}(x_{1},x^{\star})}}.$
Using the well-known bound $1-1/v\leq\ln v$ for $v>0$, we obtain

$\forall n\in\mathbb{N}\quad\|x_{n+1}-x^{\star}\|\leq
s\ln{e^{c^{n}d_{T}(x_{1},x^{\star})}}=\\\ sc^{n}d_{T}(x_{1},x^{\star})=\gamma
c^{n},$ (24)

where
$\gamma\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}s\,d_{T}(x_{1},x^{\star})$,
and the proof is complete. ∎

We now have all ingredients to establish geometric convergence of the fixed
point iteration of PC mappings:

######  Proposition 3.

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
a continuous and concave (w.r.t. to the cone order) mapping (i.e., a
continuous PC mapping) with a fixed point
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}}).$ Then, the fixed point
iteration of $f$ with $x_{1}\in{}_{+}^{k}$ converges geometrically to
$x^{\star}$ with a factor $c\in[0,1)$ w.r.t. any metric induced by a norm in
$\mathbb{R}^{k}.$

###### Proof.

Without any loss of generality, we can assume that
$x_{1}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ because, if
$x_{1}\in{}_{+}^{k}\backslash\operatorname{int}({\mathbb{R}^{k}_{+}})$, then
$f(x_{1})\in\operatorname{int}({\mathbb{R}^{k}_{+}})$, so we can consider
$f(x_{1})$ as the starting point of the iterations. We divide the proof into
two parts. We first prove geometric convergence of the fixed point iteration
of $f$ in a compact neighborhood $U$ of $x^{\star}$, and then we extend this
property to an arbitrary starting point
$x_{1}\in\operatorname{int}({\mathbb{R}^{k}_{+}}).$

_1) Geometric convergence on $U$_: We first note that the fixed point
$x^{\star}$ of $f$ satisfies $x^{\star}\geq f(x^{\star})$, thus $f$ is
feasible in the sense of [2]. Therefore, by Fact 6 in the Appendix A, the
fixed point iteration of $f$ with
$x_{1}\in\operatorname{int}({\mathbb{R}^{k}_{+}})$ produces a sequence
$(x_{n})_{n\in\mathbb{N}}$ converging to $x^{\star}$ w.r.t. any norm on
$\mathbb{R}^{k}_{+}.$ Let $U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$
be a compact neighborhood of $x^{\star}.$ Then, we can find
$n_{0}\in\mathbb{N}$ such that $\forall n>n_{0}$ one has $x_{n}\in U.$ Thus,
let $(y_{n})_{n\in\mathbb{N}}\subset U$ be a sequence defined as
$y_{n}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}x_{n+n_{0}}$ for
$n\in\mathbb{N}.$ Let us also note that $f$ is a local $c$-Lipschitz
contraction on $U$ w.r.t. Thompson’s metric in view of Proposition 1.
Moreover, using the isometry provided in (9) in Remark 2, we conclude that
$(U,d_{T})$ is a complete metric space isometric to a complete metric subspace
of $(\mathbb{R}^{k},\|\cdot\|_{\infty})$ [35]. Therefore, as
$(y_{n})_{n\in\mathbb{N}}\subset U$ is a fixed point iteration of $f$
converging to $x^{\star}$ in a complete metric space $(U,d_{T})$, and $f$ is a
local $c$-Lipschitz contraction on $U$ w.r.t. Thompson’s metric, we conclude
that $(y_{n})_{n\in\mathbb{N}}$ is obviously bounded and that it converges
linearly (and hence, also geometrically) to $x^{\star}$ w.r.t. Thompson’s
metric. Owing to this fact, we may reproduce now the proof of Proposition 2,
with the sequence
$(x_{n})_{n\in\mathbb{N}}\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$
considered therein replaced with the sequence $(y_{n})_{n\in\mathbb{N}}\subset
U$, and the complete metric space
$(\operatorname{int}({\mathbb{R}^{k}_{+}}),d_{T})$ considered therein replaced
with the complete metric space $(U,d_{T})$, to conclude that
$(y_{n})_{n\in\mathbb{N}}$ converges geometrically to $x^{\star}$ with a
factor $c\in[0,1)$ w.r.t. any metric induced by a norm in $\mathbb{R}^{k}$,
with a constant $\gamma_{y}>0$ determined through (22)-(24).

_2) Geometric convergence on $\operatorname{int}({\mathbb{R}^{k}_{+}})$_: If
$n_{0}>1$, then we need to show that the sequence $(x_{n})_{n\in\mathbb{N}}$
also has this property. To this end, let us define
$\eta\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\max_{n\in\\{1,\dots,n_{0}\\}}\|x_{n+1}-y_{n+1}\|$,
and assume that $c=0.$ Then, $f$ is constant on $U$, hence $\forall
n\in\mathbb{N}$ one has $y_{n}=x_{n_{0}+n}=x^{\star}.$ Thus, taking into
consideration that for any $v\in(0,1)$, we obtain

$\forall
n\in\\{1,\dots,n_{0}\\}\quad\|x_{n+1}-x^{\star}\|\leq\eta=\Big{(}\frac{\eta}{v^{n}}\Big{)}v^{n}\leq\Big{(}\frac{\eta}{v^{n_{0}}}\Big{)}v^{n},$

and we conclude that the sequence $(x_{n})_{n\in\mathbb{N}}$ converges
geometrically to $x^{\star}$ with a factor $v\in(0,1)$ w.r.t. any metric
induced by a norm in $\mathbb{R}^{k}$, with a constant $\eta/v^{n_{0}}.$
Assume now that $c\in(0,1).$ Then,

$\forall n\in\\{1,\dots,n_{0}\\}\quad\|x_{n+1}-x^{\star}\|\leq\\\
\|x_{n+1}-y_{n+1}\|+\|y_{n+1}-x^{\star}\|\leq\eta+\gamma_{y}c^{n}\\\
=\Big{(}\frac{\eta}{c^{n}}+\gamma_{y}\Big{)}c^{n}\leq\Big{(}\frac{\eta}{c^{n_{0}}}+\gamma_{y}\Big{)}c^{n}=\gamma_{x_{1}}c^{n},$

where
$\gamma_{x_{1}}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\eta
c^{-n_{0}}+\gamma_{y}.$ Moreover,

$\forall
n>n_{0}\quad\|x_{n+1}-x^{\star}\|=\|y_{n+1-n_{0}}-x^{\star}\|\leq\gamma_{y}c^{n+1-n_{0}}\\\
=c^{1-n_{0}}\gamma_{y}c^{n}=\gamma_{x_{2}}c^{n},$

where
$\gamma_{x_{2}}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}c^{1-n_{0}}\gamma_{y}.$
Let
$\gamma_{x}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\max\\{\gamma_{x_{1}},\gamma_{x_{2}}\\}.$
We have found $c\in(0,1)$ and $\gamma_{x}>0$ such that $\forall
n\in\mathbb{N}$ one has $\|x_{n+1}-x^{\star}\|\leq\gamma_{x}c^{n}.$ ∎

An interesting question is whether the fixed point iteration is guaranteed to
converge geometrically (in a normed vector space) with slightly more general
classes of concave mappings than those considered in Proposition 3. The next
example shows that this special property of PC mappings does not necessarily
hold even if we simply replace the assumption of positivity by nonnegativity.

(a)

(b)

Figure 1: Fixed point iteration of the mapping in Example 3. In this figure
$x^{\star}\in\mathrm{Fix}(g)$ is the limit of the sequence
$(x_{n})_{n\in\mathbb{N}}$, and note that the fixed point is not unique. (a)
Convergence rate. (b) Estimation error.

(a)

(b)

Figure 2: Fixed point iteration of the mapping in Example 4. (a) Convergence
rate. (b) Estimation error.

######  Example 3.

In [20], the authors consider the mapping

$f\colon\mathbb{R}_{+}\to\operatorname{int}({\mathbb{R}}):x\mapsto
4/(1+e^{2-x}),$

and they show that $f$ is an SI mapping with the property that the fixed point
iteration of $f$ converges sublinearly to its unique fixed point
$x^{\star}=2.$ In particular, the fixed point iteration of $f$ satisfies
neither (2) nor (3) for $\tilde{x}=x^{\star}$ and $x_{n+1}=f(x_{n})$ with
$x_{1}\in\mathbb{R}_{+}$, $n\in\mathbb{N}.$ This example motivates us to
consider the function $g\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that
$g(x)=x$ for $x\in[0,2]$ and $g(x)=f(x)$ for $x>2.$ Namely, let

$g(x)=\left\\{\begin{array}[]{ll}x&\textrm{ for }0\leq x\leq 2\\\
\frac{4}{(1+e^{2-x})}&\textrm{ for }x>2.\\\ \end{array}\right.$ (25)

Then, through algebraic manipulations, the following expressions of the first
and second derivatives of $g$ can be obtained:

$g^{\prime}(x)=\left\\{\begin{array}[]{ll}1&\textrm{ for }0\leq x\leq 2\\\
\frac{4e^{2+x}}{(e^{2}+e^{x})^{2}}&\textrm{ for }x>2,\\\ \end{array}\right.$

and

$g^{\prime\prime}(x)=\left\\{\begin{array}[]{ll}0&\textrm{ for }0\leq x\leq
2\\\ \frac{4e^{2+x}(e^{2}-e^{x})}{(e^{2}+e^{x})^{3}}&\textrm{ for }x>2.\\\
\end{array}\right.$

We verify that both $g^{\prime}$ and $g^{\prime\prime}$ are continuous on
$\mathbb{R}_{+}$ and that $g^{\prime\prime}(x)\leq 0$ for
$x\in\mathbb{R}_{+}$, implying that $g$ is a continuous and concave function.
Moreover, if $x_{1}>2$, then the fixed point iterations of $g$ and $f$
generate the same sequence; i.e., $\forall n\in\mathbb{N}$
$x_{n+1}=f(x_{n})=g(x_{n})$. Therefore, from [20, Example 2] we obtain that
the fixed point iteration of $g$ converges sublinearly to $x^{\star}=2$ for
$x_{1}>2$ irrespective of the selection of a compact neighborhood of
$x^{\star}=2$, for example, on the interval $U=[1,x_{1}].$ Fig. 1 illustrates
the convergence of the fixed point iteration of $g$ starting at $x_{1}=4$, and
we note that the ratio ${|x_{n+1}-x^{\star}|}/{|x_{n}-x^{\star}|}$ converges
to the value one, which is the expected behavior of sequences that converge
sublinearly. To highlight further the slow convergence of the algorithm, we
also remark that the error $|x_{n+1}-x^{\star}|$ remains greater than
$10^{-3}$ with $n=10^{5}$ iterations in this experiment. Thus, at first sight,
this example invalidates in particular the assertion of Proposition 3 on
geometric convergence of the fixed point iteration of $g.$ However, we note
that $g(0)=0$, so none of the assertions in Propositions 1-3 apply to $g$
because $g$ is not a PC mapping in the sense of Definition 5. Therefore, this
example underlines the importance of positivity of the mappings under
consideration for the results in the above propositions to hold.

######  Example 4.

By adding a small constant $\epsilon>0$ to the function $g$ in (25), we obtain
a continuous PC mapping with a unique fixed point denoted by $x^{\star}$, and
thus we re-establish geometric convergence of the fixed point iteration as an
implication of Proposition 3. This fact is illustrated in Fig. 2, where we set
$\epsilon=10^{-3}$. The result in this figure indicates linear convergence of
the fixed point iteration, and we recall that linear convergence implies
geometric convergence. In this experiment, the error $|x_{n+1}-x^{\star}|$ is
smaller than $10^{-3}$ already with $n=400$ iterations, and with around
$2,000$ iterations the error is reaching the machine precision of the computer
used for the simulation.

######  Remark 7.

A potential caveat of adding a small positive constant to a nonnegative
concave function is that we may lose its fixed points. This fact can be
verified for the function $g$ in (25) in Example 3. Its fixed point set
$\mathrm{Fix}(g)$ is the interval $[0,2]$, but $\mathrm{Fix}(g+\epsilon)$ is
only a singleton for any $\epsilon>0.$

###  III-C Spectral radius and the convergence rate

It follows from (2) that the convergence factor $c$ gives an indication on the
number of iterations required to obtain a good approximation of the fixed
point of a PC mapping. Therefore, the results derived so far should be
supplemented with a computationally simple means of obtaining information
about $c$. A result of this type is provided in this section. In particular,
we note that Proposition 4 proves that, under mild assumptions, the spectral
radius of a PC mapping is a lower bound on $c$, and we recall that simple
algorithms for computing the spectral radius are available in the literature
[4, Remarks 2 and 3].

######  Proposition 4.

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
a continuous and concave (w.r.t. to the cone order) mapping (i.e., a
continuous PC mapping) with a fixed point
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}}).$ Let
$U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a compact neighborhood
of $x^{\star}.$ Furthermore, let $c\in[0,1)$ be a local contraction factor of
$f$ on $U$, and denote by
$\rho\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\rho(f_{\infty})$
the spectral radius of the asymptotic mapping $f_{\infty}$ of $f$. Then
$c\geq\rho.$

###### Proof.

Denote by $v\in\mathbb{R}^{k}_{+}\backslash\\{0\\}$ a (nonlinear) eigenvector
associated with the spectral radius $\rho$; i.e., a vector
$v\in\mathbb{R}^{k}_{+}\backslash\\{0\\}$ satisfying $f_{\infty}(v)=\rho v$,
which is guaranteed to exist [36]. The result is trivial if $\rho=0$, so we
focus on the case $\rho>0$. Let $x^{\star}\in U$ be the unique fixed point of
$f$ (see Fact 2) and let $(x_{n})_{n\in\mathbb{N}}$ be constructed via the
fixed point iteration of $f$ with $x_{1}\in U$ such that $x_{1}\ll x^{\star}$
or $x_{1}\gg x^{\star}.$ Then, from Proposition 3, we can find a constant
$\gamma>0$ such that

$\displaystyle\forall n\in\mathbb{N}\quad\|x_{n+1}-x^{\star}\|\leq\gamma
c^{n}.$ (26)

Furthermore, there exists $\epsilon>0$ such that $x_{1}\leq x^{\star}-\epsilon
v$ or $x_{1}\geq x^{\star}+\epsilon v$, so [19, Proposition 5(iii)] asserts
that

$\displaystyle\forall
n\in\mathbb{N}\quad\rho^{n}\epsilon\|v\|\leq\|x_{n+1}-x^{\star}\|.$ (27)

By combining the inequalities in (26) and (27), we conclude that $\forall
n\in\mathbb{N}$ one has that $\rho^{n}\epsilon\|v\|\leq\gamma c^{n}$.
Therefore $0<\epsilon\|v\|\leq\gamma(c/\rho)^{n}$ for all $n\in\mathbb{N}.$ If
$c<\rho$, then $c/\rho<1$ and $\lim_{n\to\infty}(c/\rho)^{n}=0.$ As a result,
there exists $n_{0}\in\mathbb{N}$ such that
$\epsilon\|v\|>\gamma(c/\rho)^{n_{0}}.$ This contradiction implies that we
must have $c\geq\rho$, and the proof is complete. ∎

######  Remark 8.

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
a continuous PC mapping with a fixed point
$x^{\star}\in\operatorname{int}({\mathbb{R}^{k}_{+}}).$ Let
$U\subset\operatorname{int}({\mathbb{R}^{k}_{+}})$ be a compact neighborhood
of $x^{\star}.$ Then, it is important to highlight that the meaning of
$c\in[0,1)$ is consistent among

  * •

Proposition 1, where $c$ is used to denote the local contraction factor of $f$
on $U$,

  * •

Proposition 4, where $c$ is used in the sense of Proposition 1,

  * •

Proposition 3, where $c$ is used in the sense of Propositions 1 and 4, and it
is also recognized as a factor of geometric convergence of the fixed point
iteration of $f$ to the unique fixed point.

In particular, Part 5) of the proof of Proposition 1 provides a method to
determine the value of $c$ as
$c\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}c(\lambda_{0})=\frac{\ln((1-\mu)\lambda_{0}+\mu)}{\ln\lambda_{0}}$,
with $\lambda_{0}=e^{\max_{x,y\in U}d_{T}(x,y)}\in(1,\infty)$ and
$\mu\in(0,1)$ such that $f(0)\geq\mu f(x)$ for all $x\in U.$ Therefore, it is
seen that, if $f$ is increasing slowly in the sense that
$x^{\star}=f(x^{\star})$ is not much larger than $f(0)$, and the neighborhood
$U$ of $x^{\star}$ is sufficiently small, then the value of $\mu$ can be fixed
to a value close to 1. Similarly, the smaller the neighbourhood $U$ of
$x^{\star}$ is considered, the closer the value of $\lambda_{0}$ is to 1, with
$\lim_{\lambda\to 1^{+}}c(\lambda)=1-\mu.$ Thus, we obtain a useful insight
into the relationship between $f$ and $U$, and we also obtain an achievable
lower bound for $c$ alternative to the one provided in Proposition 4, where
$c$ is bounded below by the spectral radius of of the asymptotic mapping
$f_{\infty}$ of $f.$

##  IV Applications

We now discuss two common applications of the fixed point iteration of
positive concave mappings in wireless networks. We also show how the theory
developed in the previous sections is able to explain, with rigorous
mathematical statements, convergence properties of existing algorithms.

###  IV-A Joint optimization of transmit power, user-base station assignment,
and receive beamformers in massive MIMO systems

Consider the uplink of a wireless network with $k$ single antenna users
sending data to $m$ base stations, each equipped with $L$ antennas. The set of
users and base stations are given by $\mathcal{U}=\\{1,\ldots,k\\}$ and
$\mathcal{B}=\\{1,\dots,m\\}$, respectively. The set of base stations to which
user $u\in\mathcal{U}$ is allowed to connect is represented by the set
$\mathcal{B}_{u}\subseteq\mathcal{B}$. Denote by
$x=[x[1],\dots,x[k]]\in\mathbb{R}^{k}_{+}$ the transmit power vector, where
the $u$th coordinate $x[u]\geq 0$ of the vector $x$ is the transmit power of
user $u\in\mathcal{U}$. Let $h_{u,b}$, with
$(u,b)\in\mathcal{U}\times\mathcal{B}$, represent the channel between user $u$
and base station $b$. As common in the literature, for every
$(u,b)\in\mathcal{U}\times\mathcal{B}$, we assume $h_{u,b}$ to be a random
vector taking values in $\mathbb{C}^{L}$. For simplicity, we further assume
$h_{u,b}$ to be zero mean with positive definite channel covariance matrix
$R_{u,b}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}E[h_{u,b}h_{u,b}^{H}]\in\mathbb{C}^{L\times
L}$. In this setting, the signal-to-interference-noise ratio (SINR) of user
$u$ as a function of its receive beamformer $v\in\mathcal{V}_{u}$, base
station assignment $b$, and transmit power vector $x\in\mathbb{R}^{k}_{+}$ is
given by [5, Sect. 1.4.2]

$\displaystyle\begin{array}[]{rl}s_{u}:\mathbb{R}^{k}_{+}\times\mathcal{V}_{u}\times\mathcal{B}_{u}&\to{}_{+}\\\
(x,v,b)&\mapsto\dfrac{x[u]~{}v^{H}R_{u,b}v}{\sum_{j\in\mathcal{U}\backslash\\{u\\}}v^{H}(x[j]R_{j,b}+\sigma^{2}~{}I_{L})v},\end{array}$

where $\sigma^{2}>0$ denotes the noise power; $I_{L}\in\mathbb{R}^{L\times L}$
is the identity matrix; and $\mathcal{V}_{u}\subset\mathbb{C}^{L}$ is a set of
allowed beamformers for user $u$, typically chosen to be
$\mathcal{V}_{u}=\\{v\in\mathbb{C}^{L}~{}|~{}{v}^{H}{v}=1\\}$. As in [23],
users are free to choose the base station to be served.

A standard power control problem is to find the transmit power, the user-base
station assignments, and the beamformers that minimize the sum power while
satisfying given SINR requirements. Formally, denoting by $\gamma_{u}>0$ the
minimum SINR requirement of user $u\in\mathcal{U}$, we pose the optimization
problem as follows:

$\displaystyle\begin{array}[]{rl}\text{minimize}_{x\in\mathbb{R}^{k}_{+}}&\sum_{u\in\mathcal{U}}x[u]\\\
\text{subject to}&\forall
u\in\mathcal{U}\quad\max_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}s_{u}(x,v,b)\geq\gamma_{u}.\end{array}$
(30)

For later reference, we say that a vector $x\in{}_{+}^{k}$ is feasible to
problem (30) if it satisfies all constraints, even if $x$ is not necessarily a
solution. In the above formulation, the optimization of the beamformers and
user-base station assignments is implicit in the constraint. More precisely,
if a power allocation $x$ is given (e.g., an optimal power allocation
$x^{\star}$), then a beamformer $v^{\star}_{u}$ and a base station assignment
$b_{u}^{\star}$ maximizing the SINR for user $u\in\mathcal{U}$ can be
recovered by computing:

$\displaystyle(v^{\star}_{u},b_{u}^{\star})\in\mathrm{argmax}_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}s_{u}(x,v,b).$

In practice, given $x\in\mathbb{R}^{k}_{+}$ and user $u\in\mathcal{U}$, the
tuple $(v^{\star}_{u},b_{u}^{\star})$ is obtained by, first, computing the
beamformers maximizing the SINR for each of the $|\mathcal{B}_{u}|\leq m$
possible assignments, and by, second, checking the choice that leads to the
maximum SINR. The computational complexity of this approach is manageable if
for each user $u$ the number $|\mathcal{B}_{u}|$ of candidate assignments is
not too large, which is a typical property of cellular networks because base
stations that are too far away are unable to provide adequate SINR, so they
can be ruled out from consideration. We also note that, if fully digital
beamformers are used, in which case we can choose
$\mathcal{V}_{u}=\\{v\in\mathbb{C}^{L}~{}|~{}{v}^{H}{v}=1\\}$ for every
$u\in\mathcal{U}$, then the beamformer maximizing the SINR for a given user,
base station assignment, and power vector can be easily computed by maximizing
a generalized Rayleigh quotient. Nevertheless, we do not necessarily assume
digital beamformers in this section; we only require the following very mild
assumption:

######  Assumption 1.

The set
$C_{u,x}=\mathrm{argmax}_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}s_{u}(x,v,b)$
is nonempty for every $u\in\mathcal{U}$ and every $x\in\mathbb{R}^{k}_{+}$.

A well-known result in the literature shows that, if a vector $x^{\star}$ is a
solution to the optimization problem (30) (assuming that a solution exists),
then the constraints have to be satisfied with equality; i.e.,

$\displaystyle\forall
u\in\mathcal{U}~{}\max_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}s_{u}(x^{\star},v,b)=\gamma_{u}.$
(31)

This result can be easily proved by noticing that, if a strict SINR inequality
is obtained for some user, then we can decrease the transmit power of this
user (thus decreasing the cost function in $(\ref{problem.opt})$), until an
equality is achieved. By decreasing the transmit power of the user, the SINR
of other users cannot decrease, so the resulting power vector is also
feasible. Therefore, feasible power allocations satisfying all constraints
with at least one strict inequality cannot be optimal if problem
$(\ref{problem.opt})$ has a solution.

The above fact enables us to replace the inequalities in the constraints of
problem (30) with equalities. Therefore, using simple algebraic manipulations,
we can equivalently express the constraints in (30) as the fixed point set of
the following mapping
$f:\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$:

$\displaystyle\forall
u\in\mathcal{U}~{}\max_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}s_{u}(x^{\star},v,b)=\gamma_{u}\iff
x^{\star}\in\mathrm{Fix}(f),$ (32)

where

$\displaystyle
f:\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}}):x\mapsto[f_{1}(x),\ldots,f_{k}(x)]$
(33)

and $\forall u\in\mathcal{U}$

$\displaystyle\begin{array}[]{rl}f_{u}:\mathbb{R}^{k}_{+}&\to\operatorname{int}({\mathbb{R}_{+}})\\\
x\mapsto&\min\limits_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}\gamma_{u}\dfrac{\sum_{j\in\mathcal{U}\backslash\\{u\\}}v^{H}(x[j]R_{j,b}+\sigma^{2}~{}I_{L})v}{v^{H}R_{u,b}v}.\end{array}$

To avoid digressions of little practical relevance, assume that $k\geq 2$ and
$\forall u\in\mathcal{U}\quad 0\notin\mathcal{V}_{u}$, so that $\forall
v\in\mathcal{V}_{u}\quad v^{H}R_{u,b}v>0$ because every matrix $R_{u,b}$ is
positive definite. Since $\gamma_{u}>0$ and $\sigma^{2}>0$ by assumption, we
deduce $\forall x\in{}_{+}^{k}\quad\forall u\in\mathcal{U}$:

$f_{u}(x)=\min\limits_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}\gamma_{u}\dfrac{\sum_{j\in\mathcal{U}\backslash\\{u\\}}v^{H}(x[j]R_{j,b}+\sigma^{2}~{}I_{L})v}{v^{H}R_{u,b}v}\\\
\geq\min\limits_{(v,b)\in\mathcal{V}_{u}\times\mathcal{B}_{u}}\gamma_{u}\sigma^{2}(k-1)\dfrac{v^{H}v}{v^{H}R_{u,b}v}=f_{u}(0)>0.$
(34)

Furthermore, each function $f_{u}$ can be seen as the minimum of affine
functions, so $f_{u}$ is not only positive but also concave, and thus $f_{u}$
is a PC mapping. We can tacitly assume $f_{u}$ to be continuous in the whole
domain because, even if $f_{u}$ were not continuous on the boundary of its
domain, being concave, $f_{u}$ is continuous in
$\operatorname{int}({\mathbb{R}^{k}_{+}})$, and its restriction to
$\operatorname{int}({\mathbb{R}^{k}_{+}})$ can be continuously extended to the
whole domain $\mathbb{R}^{k}_{+}$ [12, Theorem 5.1.5]. Such an extension of
${f_{u}}_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mu{\operatorname{int}({\mathbb{R}^{k}_{+}})}}$ to the domain
$\mathbb{R}^{k}_{+}$ would preserve positivity because the range of
${f_{u}}_{\mkern 1.0mu\vrule height=6.02777pt\mkern
2.0mu{\operatorname{int}({\mathbb{R}^{k}_{+}})}}$ is bounded away from zero as
shown in (34).

Below we list some advantages of using the equivalence relation in (32) and
the result in (31) to replace the inequality constraints in problem (30) with
the fixed point set of the PC mapping in (33):

######  Remark 9.

  * (i)

We immediately deduce that problem (30) has at most one solution because the
set $\mathrm{Fix}(f)$ is a singleton if not empty (Fact 2);

  * (ii)

Problem (30) has a solution if and only if $\rho(f_{\infty})<1$ (Fact 2),
which is a condition that can be easily verified with the simple numerical
approaches described in [4, Remarks 2 and 3];

  * (iii)

If $x^{\star}$ is the solution to problem (30), then $x^{\star}$ is the limit
of the sequence $(x_{n})_{n\in\mathbb{N}}$ produced via the fixed point
iteration $x_{n+1}=f(x_{n})$, $x_{1}\in\mathbb{R}^{k}_{+}$ (Fact 6 in the
Appendix).

  * (iv)

The convergence of the fixed point iteration of $f$ is geometric (Proposition
3) if the fixed point exists.

In this particular application, the theory described in the previous sections
goes beyond results available in the wireless literature w.r.t. the following
aspects:

######  Remark 10.

  1. (i)

Assuming that the set $\mathcal{V}_{u}$ is a singleton for every
$u\in\mathcal{U}$, then the studies in [21, 20] show _sufficient_ conditions
for the mapping $f$ in (33) to have a fixed point. In contrast, Remark 9(ii)
describes a simple and more general _necessary and sufficient_ condition for
establishing the existence of the fixed point.

  2. (ii)

More importantly, if the set $\mathcal{V}_{u}$ contains uncountably many
beamformers for some user $u\in\mathcal{U}$ (e.g., if base stations are
equipped with fully digital beamformers, in which case
$\mathcal{V}_{u}=\\{v\in\mathbb{C}^{L}~{}|~{}{v}^{H}{v}=1\\}$ for every
$u\in\mathcal{U}$), then the mapping $f$ in (33) cannot be expressed as the
coordinate-wise minimum of a _finite_ number of affine functions in general,
so the theory developed in [21] cannot be used to establish geometric
convergence of the fixed point iteration of $f$. Furthermore, the study in
[20] does not describe any simple procedure to verify whether $f$ is a
$c$-contraction in the sense in [20, Definition 2], which would give us an
alternative proof of geometric convergence of the fixed point iteration of
$f$. In contrast to the results in [20, 21], we impose no additional structure
on $f$ other than positivity and concavity to establish geometric convergence
of the fixed point iteration (Proposition 3). These properties can be
immediately verified for the particular mapping in (33) as discussed above.

  3. (iii)

Sect. III-C shows simple techniques to obtain information about the
contraction factor of the fixed point iteration of $f$ (and hence the
convergence speed of the iteration) in normed vector spaces and in Thompson’s
metric space. Mathematical tools of this type are not available in the
wireless literature to the best of our knowledge.

We finish this section by showing that the theory developed in this study can
also explain the convergence speed of extensions of the power control
algorithm in Remark 9(iii).

######  Remark 11.

A possible drawback of the fixed point iteration in Remark 9(iii), which is a
consequence of the optimization problem posed in (30), is that no maximum
power constraints are imposed. To prevent users from transmitting with
excessive power, we can replace the mapping $f$ in the fixed point iteration
with

$\displaystyle\bar{f}:\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}}):x\mapsto[\min\\{f_{1}(x),\bar{p}\\},\ldots,\min\\{f_{u}(x),\bar{p}\\}],$

where $\bar{p}>0$ is the maximum allowed power. Note that $\bar{f}$ is also
concave because each coordinate function is the minimum of concave functions.
Well-known power control algorithms emerge as particular cases of the fixed
point iteration of $\bar{f}$. For example, by fixing the sets
$(\mathcal{V}_{u})_{u\in\mathcal{U}}$ to be singletons and the random channel
vectors to be almost surely constant, we obtain the power control algorithm
discussed in [37, Eq. (7)] as a special case of the fixed point iteration of
the resulting mapping $\bar{f}$. Whether such simplifications are used or not,
we always have $\rho(\bar{f}_{\infty})=0$ because each coordinate of $\bar{f}$
is upper bounded by $\bar{p}$, and thus $\forall
x\in\mathbb{R}^{k}_{+}\quad\bar{f}_{\infty}(x)=0$. As a result, Fact 2 shows
that $\mathrm{Fix}(\bar{f})\neq\emptyset$, and the fixed point iteration of
$\bar{f}$ converges geometrically to the unique vector in
$\mathrm{Fix}(\bar{f})$ as a consequence of Proposition 3.

###  IV-B Load estimation in orthogonal frequency division multiple access
(OFDMA) networks

As a second example of an application of the fixed point iteration of positive
concave mappings, we describe the problem of load estimation in wireless
networks [15, 16, 17, 10, 38, 3, 4, 9, 19]. We assume that a network uses the
OFDMA technology, and we consider the same downlink scenario (i.e., data
transmission is from base stations to users) used in the simulations in [19].
To pose the problem, we use the following definitions:

  * •

$\mathcal{B}=\\{1,\ldots,k\\}$ \- set of base stations;

  * •

$\mathcal{U}$ \- set of users;

  * •

$\mathcal{U}_{b}\neq\emptyset$ \- set of users connected to base station $b$;

  * •

$B$ \- bandwidth of each resource block;

  * •

$\forall u\in\mathcal{U}\quad d[u]$ \- traffic requested by the $u$th user;

  * •

$g[u,b]>0$ \- pathloss between base station $b$ and user $u$;

  * •

$R$ \- number of resource blocks in the OFDMA system;

  * •

$\forall b\in\mathcal{B}\quad p[b]$ \- transmit power per resource block of
base station $b$;

  * •

$\sigma^{2}$ \- noise power per resource block.

  * •

$x=[x[1],\ldots,x[k]]$ \- load vector. More precisely, the $b$th coordinate
$x[b]$ is the fraction of the total number of resource blocks that base
station $b$ requires to serve the traffic demand of its users in the set
$\mathcal{U}_{b}$.

For a given load $x$, the modified Shannon capacity
$r_{u,b}:\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}$, which has been
independently proposed in [16, 15] and defined below, is often used as a proxy
of the rate per resource block that a user $u\in\mathcal{U}$ can achieve if
connected to base station $b\in\mathcal{B}$ [15, 16, 17, 10, 38, 3, 4, 9, 19]:

$\displaystyle\begin{array}[]{rl}r_{u,b}:\mathbb{R}^{k}_{+}\to&\mathbb{R}^{k}_{+}\\\
x\mapsto&B\log_{2}\left(1+\dfrac{p[b]g[u,b]}{\sum\limits_{j\in\mathcal{B}\backslash\\{b\\}}x[j]p[j]g[u,j]+\sigma^{2}}\right).\end{array}$

The above model assumes no intracell interference; i.e., users connected to a
given base station use distinct resource blocks. However, there is intercell
interference, which corresponds to the term
$\sum\limits_{j\in\mathcal{B}\backslash\\{b\\}}x[j]p[j]g[u,j]$ in the
definition of $r_{u,b}$. As a result, if the load vector $x$ is given, and
user $u$ is connected to base station $b$, then this base station needs to
reserve $d[u]/r_{u,b}(x)$ resource blocks to satisfy the traffic demand of
user $u$. By definition, the $b$th coordinate $x[b]$ of the load vector $x$ is
the total number of resource blocks, normalized by number of resource blocks
$R$ in the system, that base station $b$ reserves for all its users; i.e.,

$\displaystyle\forall b\in\mathcal{B}\quad
x[b]=\dfrac{1}{R}\sum_{u\in\mathcal{U}_{b}}\dfrac{d[u]}{r_{u,b}(x)}.$

The above equations show that the load vector $x$ can be equivalently
expressed as the fixed point of the following mapping:

$\displaystyle
f:{}_{+}^{k}\to\operatorname{int}({\mathbb{R}^{k}_{+}}):{x}\mapsto[f_{1}({x}),\cdots,f_{k}({x})],$
(35)

where

$\displaystyle\forall
b\in\mathcal{B}\quad\begin{array}[]{rl}f_{b}:\mathbb{R}^{k}_{+}\to&\operatorname{int}({\mathbb{R}^{k}_{+}})\\\
x\mapsto&\dfrac{1}{R}\sum_{u\in\mathcal{U}_{b}}\dfrac{d[u]}{r_{u,b}(x)}.\end{array}$

Therefore, to estimate the load $x$ of a network, we need to solve the
following fixed point problem:

$\displaystyle\text{Find }x\in\mathbb{R}^{k}_{+}\text{ such that
}x\in\mathrm{Fix}(f).$ (36)

Standard arguments in convex analysis show that $f$ is a continuous positive
concave mapping [16, 15, 38]. Therefore, all the theory in the previous
sections can be applied to analyze and solve problem (36). In particular, we
deduce the following properties:

  1. (i)

Problem (36) has at most one solution (Fact 2);

  2. (ii)

The asymptotic mapping associated with $f$ is given by [4]

$f_{\infty}:\mathbb{R}^{k}_{+}\to\mathbb{R}^{k}_{+}:{x}\mapsto\mathrm{diag}({p})^{-1}{M}\mathrm{diag}({p}){x},$

where $\mathrm{diag}({p})$ is a diagonal matrix with diagonal elements given
by $\forall b\in\mathcal{B}\quad[\mathrm{diag}({p})]_{bb}=p[b]$, and
$M\in\mathbb{R}^{k\times k}_{+}$ is the matrix with its $i$th row and $b$th
column given by:

$\displaystyle[{M}]_{i,b}=\begin{cases}0,&\text{if }i=b\\\
\sum\limits_{u\in\mathcal{U}_{i}}\dfrac{\mathrm{ln}(2)d[u]g[u,b]}{{RBg[u,i]}}&\text{otherwise.}\end{cases}$

Note that the asymptotic mapping $f_{\infty}$ is linear, so
$\rho(f_{\infty})=\rho(\mathrm{diag}({p})^{-1}{M}\mathrm{diag}({p}))=\rho(M)$
is simply the spectral radius
$\rho(M)=\rho(\mathrm{diag}({p})^{-1}{M}\mathrm{diag}({p}))$ of the matrix
${M}$, where the spectral radius of matrices should be understood in the
conventional sense in linear algebra. It now follows from Fact 2 that

$\displaystyle\mathrm{Fix}(f)\neq\emptyset\iff\rho(M)<1.$

This criterion for checking the existence of a feasible load vector has been
first obtained in [10], and it can be seen as a particular application of Fact
2, as discussed in [4].

  3. (iii)

If $\mathrm{Fix}(f)\neq\emptyset$, then the unique fixed point of $f$ is the
limit of the sequence generated via the fixed point iteration of $f$ (Fact 6
in the Appendix).

  4. (iv)

The fixed point iteration of $f$ converges geometrically (Proposition 3), and
the convergence factor $c$ of the iteration is lower bounded by the spectral
radius of the matrix $M$ defined above (Proposition 4).

We emphasize that points (i)-(iii) are known, but point (iv) is a new result
that does not seem easy to derive from existing theory for reasons similar to
those given in Remark 10. More precisely, the mapping in (35) cannot be
written as the coordinate-wise minimum of a finite number of affine mappings,
as required in [21]. Furthermore, it is not clear whether that mapping
satisfies the conditions in [20, Definition 2]. Even if this last condition
could be established, the study in [20] does not discuss any simple
computational means, such as those described in Sect. III-C, to determine
bounds on the contraction factor.

Before we proceed with numerical examples to illustrate the main findings in
this study, we recall one potential caveat of the load model discussed here:

######  Remark 12.

By definition, the coordinates of the load vector $x$ cannot exceed the value
one, otherwise the corresponding base stations would be using more resource
blocks than available in the system, which is physically impossible.
Nevertheless, if any coordinate of the solution to problem (36) exceeds the
value one, we obtain useful information about the unserved traffic demand. We
refer the readers to [16] for more information on this point.

##  V Simulations

For brevity, we show numerical examples only for the load estimation problem
described in Sect. IV-B. In the simulation, the pathloss $g[u,b]$ between base
station $b\in\mathcal{B}$ and user $u\in\mathcal{U}$ is computed using the
standard Hata model for urban scenarios with the height of the antennas of the
users and the base stations set to, respectively, $1.5$m and $30$m. The number
of base stations is $k=25$, and the base stations are uniformly distributed in
a square of dimensions $2\text{km}\times 2\text{km}$. A total of $400$ users
are distributed uniformly at random in a square of dimensions
$2.5\text{km}\times 2.5\text{km}$, which is concentric to the square where
base stations are placed. Users are assigned to base stations with the lowest
pathloss. Other main parameters of the simulations are listed in Table I.

Parameter | Value
---|---
$R$ | 25
$B$ |  $2\cdot 10^{5}$ Hz
$\forall u\in\mathcal{U}\quad d[u]$ |  $10^{6}$ bits/second
$\forall b\in\mathcal{B}\quad p[b]$ | 1.6 W
$\sigma^{2}$ |  $6.2\cdot 10^{-18}$ W
TABLE I: Main parameters used in the simulation. Figure 3: Convergence of the
fixed point iteration as a function of the number of iterations. Figure 4:
Ratio ${\|x_{n+1}-x^{\star}\|_{2}}/{\|x_{n}-x^{\star}\|_{2}}$ as a function of
the iteration index $n$.

For the simulated scenario, we have $\rho(M)\approx 0.92,$ so the fixed point
exists, and it has been computed by constructing the sequence
$(x_{n})_{n\in\mathbb{N}}$ via the fixed point iteration $x_{n+1}=f(x_{n})$,
where $x_{1}=f(0)$. In Fig. 3 we show the convergence of the fixed point
iterations and also the lower bound derived in [19, Proposition 5(iii)]. In
Fig. 4 we show the ratio
${\|x_{n+1}-x^{\star}\|_{2}}/{\|x_{n}-x^{\star}\|_{2}}$ as a mapping of the
number of iterations. In these figures, $\|\cdot\|_{2}$ denotes the standard
Euclidean norm.

In light of Proposition 4, $\rho(M)$ is a lower bound for the local
contraction factor around the fixed point $x^{\star}$ of $f$, and this
property is indeed consistent with the result in Figs. 3 and 4. More
specifically, note that the bound has a slightly greater slope than the error
$\|x_{n}-x^{\star}\|_{2}$ for the iterations shown in Fig. 3. This property is
expected because the bound as a mapping of the iteration index $n$ takes the
form $y[n]=C_{1}\rho(M)^{n}$ for some $C_{1}>0$, whereas the error takes the
form $\|x_{n+1}-x^{\star}\|_{2}\leq C_{2}c^{n}$ for some $C_{2}>0$ and some
$c\in[\rho(M),1)$. Furthermore, the ratio in Fig. 4 is converging to a value
$\mu$ within the interval $(0,~{}1)$, which provides us with an indication of
the type of convergence described in Proposition 4 with $\mu\geq\rho(M)$ [24].
Note, however, that the results in that proposition are valid for any sequence
produced via the fixed point iteration, and this simulation shows the
convergence of a sequence produced by starting the iterations with a
particular vector.

##  VI Summary and final remarks

Previous studies in the wireless literature [3, 19, 20, 21][5, Ch. 5.3] have
devoted a great deal of attention to characterize the convergence speed of
fixed point iterations of mappings that are common in power control and load
estimation problems. In many of the these applications, the mappings are in
fact positive and concave. However, as discussed in Sect. IV, previous results
proving geometric convergence of the fixed point iteration of positive concave
mappings require additional assumptions that are often too strong to address
some problems of practical interest, or that are too difficult to verify in
practice, or both. In this study we have proved that these additional
assumptions are not required: we only need the existence of a fixed point for
the fixed point iteration of positive concave mappings to converge
geometrically. We have also shown that even by replacing the assumption of
positivity with nonnegativity we lose the guarantees of geometric convergence,
so the assumption of positivity is crucial. Finally, we have also derived
results that provide us with clear indication of the local contraction factor
of the mappings, which is useful information to assess whether many fixed
point iterations are required to obtain a good estimate of the fixed point.

##  Appendix A Definitions and Known Results

######  Definition 8.

([12, p. 45]) Let $\|\cdot\|$ be any norm on $\mathbb{R}^{k}.$ We call
$\delta>0$ the normality constant of the cone $\mathbb{R}^{k}_{+}$ for the
norm $\|\cdot\|$ if

$\delta\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\inf\\{q\in\mathbb{R}_{+}~{}|~{}\forall(x,y)\in
C\quad\|x\|\leq q\,\|y\|\\},$ (37)

where
$C\stackrel{{\scriptstyle\mathclap{\tiny\mbox{df.}}}}{{=}}\\{(x,y)\in\mathbb{R}^{k}_{+}\times\mathbb{R}^{k}_{+}~{}|~{}x\leq
y\\}.$

######  Fact 5 ([12, Lemma 2.5.1.]).

Let $K$ be a cone in a normed space $(V,\|\cdot\|),$333For example,
$V=\mathbb{R}^{k}$ with $K=\mathbb{R}^{k}_{+}.$ and let $\delta>0$ be the
normality constant of $K.$ For each $x,y\in K$ with $\|x\|\leq b$ and
$\|y\|\leq b$, we have that

$\|x-y\|\leq b(1+2\delta)(e^{d_{T}(x,y)}-1).$

######  Fact 6 ([2, Theorem 2]).

Let $f\colon\mathbb{R}^{k}_{+}\to\operatorname{int}({\mathbb{R}^{k}_{+}})$ be
a continuous SI mapping satisfying the feasibility condition $f(x)\leq x$ for
a vector $x\in\mathbb{R}^{k}_{+}.$ Then, for any
$x\in\operatorname{int}({\mathbb{R}^{k}_{+}})$, the fixed point iteration of
$f$ produces a sequence $(x_{n})_{n\in\mathbb{N}}$ converging in norm to the
unique vector $x^{\star}\in\mathrm{Fix}(f)$, i.e.,
$\lim_{n\to\infty}\|x_{n}-x^{\star}\|=0$ for an arbitrary norm $\|\cdot\|.$

## References

  * [1] M. Woodford, “Effective demand failures and the limits of monetary stabilization policy,” National Bureau of Economic Research, Tech. Rep., 2020\.
  * [2] R. D. Yates, “A framework for uplink power control in cellular radio systems,” _IEEE J. Sel. Areas Commun._ , vol. 13, no. 7, pp. pp. 1341–1348, Sept. 1995.
  * [3] R. L. G. Cavalcante, Y. Shen, and S. Stańczak, “Elementary properties of positive concave mappings with applications to network planning and optimization,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 7, pp. 1774–1783, 2016.
  * [4] R. L. G. Cavalcante, Q. Liao, and S. Stańczak, “Connections between spectral properties of asymptotic mappings and solutions to wireless network problems,” _IEEE Transactions on Signal Processing_ , vol. 67, no. 10, pp. 2747–2760, 2019.
  * [5] M. Schubert and H. Boche, _Interference Calculus - A General Framework for Interference Management and Network Utility Optimization_. Berlin: Springer, 2011.
  * [6] S. Stańczak, M. Wiczanowski, and H. Boche, _Fundamentals of Resource Allocation in Wireless Networks_ , 2nd ed., ser. Foundations in Signal Processing, Communications and Networking, W. Utschick, H. Boche, and R. Mathar, Eds. Berlin Heidelberg: Springer, 2009.
  * [7] L. You and D. Yuan, “A note on decoding order in user grouping and power optimization for multi-cell NOMA with load coupling,” _IEEE Transactions on Wireless Communications_ , 2020.
  * [8] S. Shindoh, “Some properties of SINR regions for standard interference mappings,” _SICE Journal of Control, Measurement, and System Integration_ , vol. 13, no. 3, pp. 50–56, 2020.
  * [9] R. L. G. Cavalcante, S. Stańczak, J. Zhang, and H. Zhuang, “Low complexity iterative algorithms for power estimation in ultra-dense load coupled networks,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 22, pp. 6058–6070, 2016.
  * [10] C. K. Ho, D. Yuan, and S. Sun, “Data offloading in load coupled networks: A utility maximization framework,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 4, pp. 1921–1931, 2014.
  * [11] T. Piotrowski and R. L. G. Cavalcante, “Fixed points of nonnegative neural networks,” _submitted, also available as arXiv preprint arXiv:2106.16239_ , 2021.
  * [12] B. Lemmens and R. Nussbaum, _Nonlinear Perron-Frobenius Theory_. Cambridge Univ. Press, 2012.
  * [13] Y. Oshime, “Perron-Frobenius problem for weakly sublinear maps in a Euclidean positive orthant,” _Japan Journal of Industrial and Applied Mathematics_ , vol. 9, no. 313, 1992.
  * [14] U. Krause, _Positive dynamical systems in discrete time: theory, models, and applications_. Walter de Gruyter GmbH & Co KG, 2015, vol. 62.
  * [15] K. Majewski and M. Koonert, “Conservative cell load approximation for radio networks with Shannon channels and its application to LTE network planning,” in _Telecommunications (AICT), 2010 Sixth Advanced International Conference on_ , May 2010, pp. 219 –225.
  * [16] I. Siomina and D. Yuan, “Analysis of cell load coupling for LTE network planning and optimization,” _IEEE Trans. Wireless Commun._ , vol. 11, no. 6, pp. 2287–2297, June 2012.
  * [17] A. Fehske, H. Klessig, J. Voigt, and G. Fettweis, “Concurrent load-aware adjustment of user association and antenna tilts in self-organizing radio networks,” _IEEE Trans. Veh. Technol._ , no. 5, June 2013.
  * [18] S. Shindoh, “The structures of SINR regions for standard interference mappings,” in _2019 58th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE)_. IEEE, 2019, pp. 1280–1285.
  * [19] R. L. G. Cavalcante and S. Stańczak, “Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 4509–4513.
  * [20] H. R. Feyzmahdavian, M. Johansson, and T. Charalambous, “Contractive interference functions and rates of convergence of distributed power control laws,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 12, pp. 4494–4502, 2012.
  * [21] C.-Y. Huang and R. D. Yates, “Rate of convergence for minimum power assignment algorithms in cellular radio systems,” _Wireless Networks_ , vol. 4, no. 4, pp. 223–231, 1998.
  * [22] H. Boche and M. Schubert, “A superlinearly and globally convergent algorithm for power control and resource allocation with general interference functions,” _IEEE/ACM Transactions on Networking_ , vol. 16, no. 2, pp. 383–395, 2008.
  * [23] S. V. Hanly, “An algorithm for combined cell-site selection and power control to maximize cellular spread spectrum capacity,” _IEEE Journal on selected areas in communications_ , vol. 13, no. 7, pp. 1332–1340, 1995.
  * [24] D. R. Luke, M. Teboulle, and N. H. Thao, “Necessary conditions for linear convergence of iterated expansive, set-valued mappings,” _Mathematical Programming_ , vol. 180, no. 1, pp. 1–31, 2020.
  * [25] F. Lauster and D. R. Luke, “Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond,” _Fixed Point Theory and Algorithms for Sciences and Engineering_ , vol. 2021, no. 13, 2021.
  * [26] H. Boche and M. Schubert, “Concave and convex interference functions – general characterizations and applications,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 10, pp. 4951–4965, 2008.
  * [27] J. M. Ortega and W. C. Rheinboldt, _Iterative Solution of Nonlinear Equations in Several Variables_. New York: Academic Press, 1970.
  * [28] S. Gaubert and J. Gunawardena, “The Perron-Frobenius theorem for homogeneous, monotone functions,” _Transactions of the American Mathematical Society_ , vol. 356, no. 12, pp. 4931–4950, 2004.
  * [29] H. H. Bauschke and P. L. Combettes, _Convex Analysis and Monotone Operator Theory in Hilbert Spaces_. New York: Springer, 2017.
  * [30] K. K. Leung, C. W. Sung, W. S. Wong, and T.-M. Lok, “Convergence theorem for a general class of power-control algorithms,” _IEEE Trans. Commun._ , vol. 52, no. 9, pp. 1566–1574, 2004.
  * [31] R. L. G. Cavalcante and S. Stańczak, “Weakly standard interference mappings: Existence of fixed points and applications to power control in wireless networks,” in _ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2019, pp. 4824–4828.
  * [32] A. J. B. Potter, “Applications of Hilbert’s projective metric to certain classes of non-homogeneous operators,” _The Quarterly Journal of Mathematics_ , vol. 28, no. 1, pp. 93–99, 1977.
  * [33] Z. Chengbo and G. Chunmei, “On -convex operators,” _Journal of Mathematical Analysis and Applications_ , vol. 316, pp. 556–565, 2005.
  * [34] Z. D. Liang, W. X. Wang, and S. J. Li, “On concave operators,” _Acta Mathematica Sinica, English Series_ , vol. 22, no. 2, pp. 577–582, 2006.
  * [35] W. Rudin, _Principles of Mathematical Analysis_. New York: McGraw-Hill, 1976.
  * [36] R. D. Nussbaum, “Convexity and log convexity for the spectral radius,” _Linear Algebra and its Applications_ , vol. 73, pp. 59–122, 1986.
  * [37] R. Sun and Z.-Q. Luo, “Globally optimal joint uplink base station association and power control for max-min fairness,” in _IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2014, pp. 454–458.
  * [38] R. L. G. Cavalcante, S. Stańczak, M. Schubert, A. Eisenbläter, and U. Türke, “Toward energy-efficient 5G wireless communication technologies,” _IEEE Signal Process. Mag._ , vol. 31, no. 6, pp. 24–34, Nov. 2014.