# Discerning and Enhancing the Weighted Sum-Rate Maximization Algorithms in
Communications

Zepeng Zhang, , Ziping Zhao, , Kaiming Shen, ,
Daniel P. Palomar, , and Wei Yu This paper was presented in part at the 48th
IEEE International Conference on Acoustics, Speech, and Signal Processing
(ICASSP), Rhodes Island, Greece, June 4th–10th, 2023 [1].

###### Abstract

Weighted sum-rate (WSR) maximization plays a critical role in communication
system design. This paper examines three optimization methods for WSR
maximization, which ensure convergence to stationary points: two block
coordinate ascent (BCA) algorithms, namely, weighted sum-minimum mean-square
error (WMMSE) and WSR maximization via fractional programming (WSR-FP), along
with a minorization-maximization (MM) algorithm, WSR maximization via MM (WSR-
MM). Our contributions are threefold. Firstly, we delineate the exact
relationships among WMMSE, WSR-FP, and WSR-MM, which, despite their extensive
use in the literature, lack a comprehensive comparative study. By probing the
theoretical underpinnings linking the BCA and MM algorithmic frameworks, we
reveal the direct correlations between the equivalent transformation
techniques, essential to the development of WMMSE and WSR-FP, and the
surrogate functions pivotal to WSR-MM. Secondly, we propose a novel algorithm,
WSR-MM+, harnessing the flexibility of selecting surrogate functions in MM
framework. By circumventing the repeated matrix inversions in the search for
optimal Lagrange multipliers in existing algorithms, WSR-MM+ significantly
reduces the computational load per iteration and accelerates convergence.
Thirdly, we reconceptualize WSR-MM+ within the BCA framework, introducing a
new equivalent transform, which gives rise to an enhanced version of WSR-FP,
named as WSR-FP+. We further demonstrate that WSR-MM+ can be construed as the
basic gradient projection method. This perspective yields a deeper
understanding into its computational intricacies. Numerical simulations
corroborate the connections between WMMSE, WSR-FP, and WSR-MM and confirm the
efficacy of the proposed WSR-MM+ and WSR-FP+ algorithms.

###### Index Terms:

Rate maximization, weighted sum-minimum mean-square error (WMMSE), fractional
programming (FP), block coordinate ascent (BCA), minorization-maximization
(MM), equivalent transform, surrogate function.

##  I Introduction

Weighted sum-rate (WSR) maximization plays a central role in a wide variety of
communication system design tasks [2], a typical case of which is to find the
optimal beamforming vectors for the multi-antenna channels [3, 4, 5, 6]. This
problem is proved to be NP-hard [7, 8]; the state-of-the-art for WSR
maximization is generally to attain a stationary point solution via iterative
algorithms, notably including the weighted sum-minimum mean-square error
(WMMSE) algorithms [3, 4], the WSR maximization via fractional programming
(WSR-FP) algorithms [5, 9], and the WSR maximization via minorization-
maximization (WSR-MM) algorithms [6], each offering a distinct strategy for
WSR beamforming problems.

As three common non-convex optimization approaches, WMMSE, WSR-FP, and WSR-MM
are driven by completely different motivations. The appeal of WMMSE originates
from a foundational principle in signal processing: maximizing the signal-to-
interference-plus-noise ratio (SINR) equates to minimizing the mean squared
error (MSE) of the received signal. The idea of casting the WSR maximization
problem into the weighted sum-MSE minimization problem to facilitate the
problem solving was first proposed in [3] for the multiple-input single-out
(MISO) channels and later extended by [4] to the multiple-input multiple-
output (MIMO) case. A recent progress in WMMSE aims to reduce the
computational complexity by using the range space of channels [10]. In
contrast, fractional programming approaches work towards a mathematical
goal—it reformulates a fractional optimization problem with one or more ratios
so as to decouple every ratio term. The classic methods, i.e., Dinkelbach’s
algorithm and Charnes-Cooper algorithm [11], have achieved this goal for a
single ratio. However, the use of the classic fractional programming
techniques in communication system design was typically restricted to the
single-ratio problem scenario such as the efficiency maximization [12, 13],
while the WSR maximization, a logarithmic fractional programming problem, does
not fall in this category. A new multi-ratio “equivalent transform” technique,
known as the quadratic transform, was developed in [5] to coordinate multiple
links (with multiple SINRs) in wireless networks. The capability of dealing
with multiple ratios enables the extensive applications of the quadratic
transform in communication system design, e.g., [14, 15, 16]. In particular,
aided by the quadratic transform and the Lagrangian dual transform for dealing
with logarithmic functions, WSR-FP was introduced for WSR maximization [5]. As
for the minorization-maximization (MM) method [17, 18], it involves two steps:
In the minorization step, a “surrogate function” that lower bounds the
objective function is constructed; subsequently, in the maximization step,
this surrogate function is maximized. The inherent flexibility in formulating
surrogate functions allows for MM algorithms that are tailored to exploit the
specific nuances of given problems. The MM-based method, WSR-MM [6], was
developed specifically to tackle the WSR maximization problems.

The trio of methodologies—WMMSE, WSR-FP, and WSR-MM—diverges in foundation and
approach. Both WSR-FP [5, 19] and WSR-MM [6] originate from a mathematical
algorithmic perspective. In contrast, WMMSE [3, 4], while later framed through
a mathematical lens [20], is anchored in the physical background of
communications, relying on transforming the WSR problem into a weighted sum-
MSE minimization problem. The WSR-MM method addresses the optimization problem
by iteratively constructing a surrogate function of the original objective,
which is then optimized in each iteration. In contrast, the WMMSE and WSR-FP
methods deploy equivalent transformations to the initial optimization problem.
They inadvertently increase the dimensionality of the optimization variables
by introducing auxiliary variables, thus leading to multi-block optimization
problems. In these scenarios, both the beamforming variables and the auxiliary
variables are optimized using block coordinate ascent (BCA) techniques [21].
Noteworthy is the non-unique decouple schemes when applying equivalent
transforms in WSR-FP and the non-unique variable block selection rules when
applying BCA, which makes them not two individual algorithms but two algorithm
categories. In this paper, we also provide novel insights and perspectives on
the WSR-FP algorithms, presenting findings previously unexplored in the
literature.

Although birthed from distinct technical lineages, there exists a strong
connection among the WMMSE, WSR-FP, and WSR-MM algorithms. While strides have
been made to elucidate the interrelationships among them [22, 19, 9], certain
elements remain sufficiently opaque. This paper endeavors to clarify these
connections through an exhaustive examination through two representative
communication systems: a MISO system with broadcast channels and a MIMO system
with interference channels. We demonstrate that WSR-FP [5, 19] subsumes WMMSE
[3, 4] which, in turn, subsumes WSR-MM [6]. Additionally, we establish that
not all instances of WSR-FP align with the MM methodology; rather, this
interpretation is contingent upon the fulfillment of specific criteria. We
also elucidate the parallelism between the equivalent transformations employed
in WMMSE and WSR-FP and the intricate surrogate function construction methods
pivotal in the WSR-MM strategy. By taking the minorization step in the WSR-MM
method as a bifurcated surrogate function development procedure [6], we
observe that these two steps have one-to-one correspondences to the two
equivalent transforms adopted in WMMSE and WSR-FP. By employing specific
variable update orders in WMMSE and particular decoupling schemes alongside
variable update orders in WSR-FP, the updates of auxiliary variables in both
algorithms can be viewed as the construction of surrogate functions within
WSR-MM. It is important to note that, unlike the methodologies in [9], which
retrospectively associate certain WSR-FP algorithms with the MM algorithmic
framework, our analysis employs a forward-looking, constructive approach. This
proactive method enhances our understanding of existing algorithms and
inspires the development of novel ones for WSR maximization, fully leveraging
the capabilities of the MM framework.

In the problem of WSR maximization subject to the total power budget
constraint, a particularly nuanced aspect of the prevailing WMMSE, WSR-FP, and
WSR-MM approaches is the calibration process for determining the optimal
Lagrange multiplier that satisfies the power constraint. This process is
computationally intensive due to the recurrent requirement for calculating
matrix (pseudo-)inversions. To forge a more efficient algorithm with lower
per-iteration computational complexity, we incorporate an extra minorization
phase in the application of the MM technique to the WSR maximization problem.
This leads to the development of a novel single-loop algorithm, termed WSR-
MM+, which offers analytical solutions and eliminates the need for adjusting
the Lagrange multiplier. Elaborating on this, we also interpret WSR-MM+
through the lens of BCA, which paves the way for an inventive equivalent
transform methodology. Capitalizing on this technique, we introduce a novel
class of single-loop algorithms, designated as WSR-FP+. In addition, we
establish that WSR-MM+ can be construed as a projected gradient ascent method,
which inherently incorporates an adaptive step size. This novel
conceptualization not only streamlines the computational mechanisms but also
sets the stage for further algorithmic developments.

To make it clear, main contributions of this paper are summarized as follows:

  * •

A thorough and systematic examination of the interconnections between WMMSE,
WSR-FP, and WSR-MM algorithms is presented. It is also established that the
equivalent transformations deployed in formulating WMMSE and WSR-FP are
intrinsically related to the surrogate function construction techniques
utilized in WSR-MM.

  * •

To spare the Lagrange multiplier tuning in existing iterative WSR maximization
algorithms, we introduce two novel single-loop algorithms: the MM-inspired
WSR-MM+ and its BCA counterpart WSR-FP+.

  * •

The proposed WSR-MM+ is characterized as a projected gradient ascent method
with an implicit step size.

  * •

Complexity analyses of existing and proposed algorithms are conducted. The
relationships between WMMSE, WSR-FP, and WSR-MM, as well as the superior
performances of WSR-MM+ and WSR-FP+, are substantiated through experiments
conducted in both MISO and MIMO system contexts.

The rest of this paper is organized as follows. Section II starts with a brief
introduction to the BCA and MM frameworks. To more easily grasp the concepts
underlying the WMMSE, WSR-FP, and WSR-MM methods, we will first review their
applications in the MISO case in Section III before progressing to the MIMO
scenario in Section IV. The intricate relationships among the WMMSE, WSR-FP,
and WSR-MM methods are dissected in Section V. The proposed WSR-MM+ algorithm,
along with its interpretations through BCA and projected gradient ascent, is
comprehensively introduced in Section VI. Algorithm complexity comparisons are
given in Section VII. Finally, Section VIII provides simulation results,
followed by conclusions and discussions in Section IX.

Notation: Italic letters, boldface lower-case letters, and boldface upper-case
letters denote scalars, column vectors, and matrices, respectively. We denote
by $\mathbf{I}$ the identity matrices. The real and complex numbers are
denoted by $\mathbb{R}$ and $\mathbb{C}$, respectively. $N$-dimensional
complex vectors are denoted by $\mathbb{C}^{N}$. $N\times M$-dimensional
complex matrices and Hermitian matrices are denoted by $\mathbb{C}^{N\times
M}$ and $\mathbb{H}^{N}$, respectively. Given $\mathbf{A}$,
$\mathbf{B}\in\mathbb{H}^{N}$, $\mathbf{A}\succ\mathbf{B}$ and
$\mathbf{A}\succeq\mathbf{B}$ stand for $\mathbf{A}-\mathbf{B}$ is positive
semidefinite and positive definite respectively. $(\cdot)^{*}$,
$(\cdot)^{\mathsf{H}}$, $(\cdot)^{-1}$, and $(\cdot)^{\dagger}$ are used to
denote the matrix conjugate, Hermitian, inverse, and pseudo-inverse,
respectively. $\mathrm{Re}(\cdot)$ denotes the real part of a scalar.
$\left\|\cdot\right\|$ denotes the vector $2$-norm or the matrix Frobenius
norm. $\mathrm{det}(\cdot)$, $\mathrm{tr}(\cdot)$, and $\lambda_{\max}(\cdot)$
denote the determinant, trace, and largest eigenvalue of a matrix,
respectively. $\frac{\partial f\left(x\right)}{\partial x}$ represents the
partial derivative of function $f$ at $x$. $f(n)={\cal O}(g(n))$ means there
are positive constants $c$ and $k$, such that $0\leq f(n)\leq cg(n)$ for all
$n\geq k$.

##  II Block Coordinate Ascent and Minorization-Maximization

We provide a brief overview of two classical iterative optimization
methods—block coordinate ascent (BCA) [21] and minorization-maximization (MM)
[17, 18]—for solving a maximization problem. To illustrate the fundamental
concept of BCA, we examine a problem with three variables:

$\displaystyle\underset{u,v,x}{\text{maximize}}$ $\displaystyle g(u,v,x)$ (1)
subject to $\displaystyle u\in\mathcal{U},v\in\mathcal{V},x\in\mathcal{X},$

where $\mathcal{U}$, $\mathcal{V}$, and $\mathcal{X}$ denote the feasible
domains for $u$, $v$, and $x$, respectively. Given initial feasible iterates,
BCA iteratively refines one block of variables at a time, holding the others
constant. This process is described by the following update rules for
$t=0,1,2,\ldots$:

$\begin{cases}u^{(t+1)}\in\arg\max_{u\in\mathcal{U}}g(u,v^{(t)},x^{(t)})\\\
v^{(t+1)}\in\arg\max_{v\in\mathcal{V}}g(u^{(t+1)},v,x^{(t)})\\\
x^{(t+1)}\in\arg\max_{x\in\mathcal{X}}g(u^{(t+1)},v^{(t+1)},x).\end{cases}$
(2)

The three variable blocks can be updated following a cyclic pattern or through
other block selection rules [22, 23]. Turning to MM, we consider the following
problem:

$\displaystyle\underset{x}{\text{maximize}}$ $\displaystyle f(x)$ (3) subject
to $\displaystyle x\in\mathcal{X},$

where ${\cal X}$ denotes the feasible set for $x$. MM, starting from a
feasible initial point, tackles this problem by solving a sequence of
surrogate optimization problems. Specifically, at $x^{(t)}$, the MM update
rule is:

$\mathbf{x}^{(t+1)}\in\arg\max_{x\in\mathcal{X}}\ell(x,x^{(t)}),$ (4)

where $\ell(x,x^{(t)})$ represents the surrogate function of $f(x)$ at
$x^{(t)}$ and must satisfy the conditions $\ell(x,x^{(t)})\leq f(x)$ and
$\ell(x^{(t)},x^{(t)})=f(x^{(t)})$ for all $x$ and $x^{(t)}$ within the set
$\mathcal{X}$. For discussions regarding the convergence properties of BCA and
MM, interested readers are directed to [24, 25, 22, 23].

##  III WSR Maximization Over MISO Broadcast Channel

###  III-A System Model and Problem Formulation

Consider a communication system of downlink transmission, where $K$ single-
antenna users is served by a base station with $M$ antennas. For each user $k$
$(k=1,\ldots,K)$, we define $\mathbf{w}_{k}\in\mathbb{C}^{M}$ as the transmit
beamforming vector and $\mathbf{h}_{k}\in\mathbb{C}^{M}$ as the channel from
the base station to the user. The signal received at the $k$-th user, $y_{k}$,
can be described as follows:

$y_{k}=\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}s_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}s_{j}+n_{k},$ (5)

where $s_{k}$ denotes the symbol intended for user $k$ with zero mean and unit
variance and $n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the additive white
Gaussian noise. The SINR for the $k$-th user is given by111Throughout this
paper, the set $\\{\mathbf{w}_{i}\\}$ is used to collectively represent the
beamforming vectors $\mathbf{w}_{1},\ldots,\mathbf{w}_{K}$, and similar
notations are adopted for other parameter sets.

$\mathsf{SINR}_{k}(\\{\mathbf{w}_{i}\\})=\frac{\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\bigr{|}^{2}}{\sum_{j=1,j\neq
k}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}},$
(6)

and the corresponding rate can be calculated as

$\mathsf{R}_{k}(\\{\mathbf{w}_{i}\\})=\log\left(1+\mathsf{SINR}_{k}(\\{\mathbf{w}_{i}\\})\right).$
(7)

Our objective is to maximize the overall WSR of the system by optimizing the
beamformer vectors, i.e.,

$\displaystyle\underset{\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle
f\left(\\{\mathbf{w}_{i}\\}\right)=\sum_{k=1}^{K}\omega_{k}\mathsf{R}_{k}(\\{\mathbf{w}_{i}\\}),$
(8)

where $\omega_{k}$ is a weighting coefficient reflecting the relative priority
of user $k$ in the system and the beamformer is subject to the following
transmit power limit constraint:

$\mathcal{W}=\Bigl{\\{}\\{\mathbf{w}_{i}\\}\mid\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P\Bigr{\\}},$ (9)

with $P$ the total available transmit power at the base station.

The WSR maximization problem (8) is non-convex. In the following, we will
first briefly review three prevalent methods for problem solving in the
literature, namely, WMMSE [4], WSR-FP [5], and WSR-MM [6].

###  III-B WMMSE Algorithms

Following the WMMSE method proposed in [3, 4], the WSR maximization problem
(8) is recast into an equivalent weighted sum-MSE minimization problem. Under
the independence assumption of $s_{k}$’s and $n_{k}$’s, the MSE between the
estimated signal and the original signal for user $k$ is given by

$\displaystyle e_{k}=$
$\displaystyle\mathbb{E}\bigl{[}\bigl{|}l_{k}^{*}(\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}s_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}s_{j}+n_{k})-s_{k}\bigr{|}^{2}\bigr{]}$
(10) $\displaystyle=$
$\displaystyle\left|l_{k}^{*}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}-1\right|^{2}+\sum_{j=1,j\neq
k}^{K}\left|l_{k}^{*}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\right|^{2}+\sigma_{k}^{2}\left|l_{k}^{*}\right|^{2},$

where $l_{k}$ denotes the receive beamformer for user $k$.

######  Proposition 1 ([4]).

Let $m_{1},\ldots,m_{K}\geq 0$ be the weights. The WSR maximization problem
(8) is equivalent to the following weighted sum-MSE minimization problem

$\displaystyle\underset{\\{l_{i}\\},\\{m_{i}\\},\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\left(\log m_{k}-m_{k}e_{k}\right),$
(11)

in the sense that they attain the identical optimal solution.

Based on Proposition 1, Problem (8) can be addressed via solving Problem (11),
a three-block optimization problem with beamforming variables
$\\{\mathbf{w}_{i}\\}$ and auxiliary variables $\\{l_{i}\\}$ and
$\\{m_{i}\\}$. Solving Problem (11) based on BCA leads to the WMMSE algorithms
[3, 4]. Given initial feasible iterates, LABEL:label:MISO-WMMSE summarizes the
variable update steps in each round.222In this paper, instead of using the
iteration index, we use $\underline{\mathbf{x}}$ to denote a variable
$\mathbf{x}$ with most recently updated value.

A1: A WMMSE algorithm for MISO beamforming [4] S1: $\displaystyle
l_{k}=\frac{\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}}{\sum_{j=1}^{K}\left|\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{j}\right|^{2}+\sigma_{k}^{2}},$
S2: $\displaystyle
m_{k}=1+\mathsf{SINR}_{k}(\\{\underline{\mathbf{w}}_{i}\\}),$ S3:
$\displaystyle\mathbf{w}_{k}=\bigl{(}\sum_{j=1}^{K}\omega_{j}\underline{m}_{j}\left|\underline{l}_{j}\right|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}+\mu\mathbf{I}\bigr{)}^{\dagger}\omega_{k}\underline{m}_{k}\underline{l}_{k}\mathbf{h}_{k}.$

Due to the block-wise convex nature of Problem (11), the subproblems
pertaining to $\\{l_{i}\\}$ and $\\{m_{i}\\}$ are efficiently resolved using
their respective first-order optimality conditions, whereas the subproblem
related to $\\{\mathbf{w}_{i}\\}$, which forms a quadratic constrained
quadratic program, is solved via Lagrangian multiplier method. In the update
of $\mathbf{w}_{k}$, we have employed the pseudo-inverse to generally address
the scenario where the matrix
$\sum_{j=1}^{K}\omega_{j}\underline{m}_{j}\left|\underline{l}_{j}\right|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}$
is of low rank, a particular case which has not been thoroughly considered in
the WSR maximization literature; $\mu$ is the Lagrangian multiplier for the
power constraint, ascertained by

$\mu=\min\Bigl{\\{}\mu\geq
0:\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\left(\mu\right)\right\|^{2}\leq
P\Bigr{\\}},$ (12)

which can be determined via one dimensional search [21].

######  Remark 2.

The variable update order in WMMSE is not unique. An alternative algorithm to
LABEL:label:MISO-WMMSE is given as follows:

$\dashrightarrow\underbrace{\text{\nameref{label:MISO-
WMMSE}-S2}\rightarrow\text{\nameref{label:MISO-
WMMSE}-S1}\rightarrow\text{\nameref{label:MISO-WMMSE}-S3}}_{\text{one round of
variable update}}\rightarrow\text{\nameref{label:MISO-
WMMSE}-S2}\dashrightarrow.$ (13)

###  III-C WSR-FP Algorithms

The WSR-FP algorithm, as described in [5], employs a similar strategy to WMMSE
by recasting (8) into a multi-block optimization problem. This is achieved
through two transformative steps: the Lagrangian dual transform and the
quadratic transform. Following these transformations, the BCA method is
applied to iteratively solve the resulting problem.

######  Proposition 3 (Lagrangian dual transform [19]).

Given ratios $\frac{c_{k}(\mathbf{x})}{d_{k}(\mathbf{x})}$ with
$c_{k}(\mathbf{x})\geq 0$ and $d_{k}(\mathbf{x})>0$ for $k=1,\ldots,K$, the
weighted sum-of-logarithmic-ratios maximization problem:

$\displaystyle\underset{\mathbf{x}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\log\left(1+\frac{c_{k}(\mathbf{x})}{d_{k}(\mathbf{x})}\right)$
(14)

is equivalent to

$\displaystyle\underset{\\{\gamma_{i}\\},\mathbf{x}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\left(\log\left(1+\gamma_{k}\right)-\gamma_{k}+\frac{\left(1+\gamma_{k}\right)c_{k}(\mathbf{x})}{n_{k}(\mathbf{x})+d_{k}(\mathbf{x})}\right),$
(15)

in the sense that they attain the identical optimal solution with the
identical optimal objective value.

######  Proposition 4 (Quadratic transform [5]).

Given nondecreasing functions $q_{k}(\cdot)$ and ratios
$\frac{\left|c_{k}(\mathbf{x})\right|^{2}}{d_{k}(\mathbf{x})}$ with
$c_{k}(\mathbf{x})\in\mathbb{C}$ and $d_{k}(\mathbf{x})>0$ for $k=1,\ldots,K$,
the following problem:

$\displaystyle\underset{\mathbf{x}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}q_{k}\left(\frac{\left|c_{k}(\mathbf{x})\right|^{2}}{d_{k}(\mathbf{x})}\right)$
(16)

is equivalent to

$\displaystyle\underset{\\{\phi_{i}\\},\mathbf{x}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}q_{k}\left(2\mathrm{Re}\left(\phi_{k}^{*}c_{k}(\mathbf{x})\right)-\left|\phi_{k}\right|^{2}d_{k}(\mathbf{x})\right),$
(17)

in the sense that they attain the identical optimal solution with the
identical optimal objective value.

Applying Proposition 3 to Problem (8) and setting
$c_{k}(\mathbf{x})=\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\bigr{|}^{2}$
and $d_{k}(\mathbf{x})=\sum_{j=1,j\neq
k}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}$,
we obtain

$\displaystyle\underset{\\{\gamma_{i}\\},\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\biggl{(}\log(1+\gamma_{k})-\gamma_{k}$
(18) $\displaystyle\hskip
42.67912pt+\frac{(1+\gamma_{k})|\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}|^{2}}{\sum_{j=1}^{K}|\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}|^{2}+\sigma_{k}^{2}}\biggr{)}.$

The ratios in the objective of Problem (18) can be further decoupled using
Proposition 4 with $q_{k}(x)=x$,
$c_{k}(\mathbf{x})=\sqrt{\omega_{k}\left(1+\gamma_{k}\right)}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}$,
and
$d_{k}(\mathbf{x})=\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}$,
leading to the following problem:

$\displaystyle\underset{\\{\gamma_{i}\\},\\{\phi_{i}\\},\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\Bigl{(}2\mathrm{Re}\bigl{(}\phi_{k}^{*}\sqrt{\omega_{k}\left(1+\gamma_{k}\right)}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\bigr{)}$
(19)
$\displaystyle-\left|\phi_{k}\right|^{2}\bigl{(}\sigma_{k}^{2}+\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}\bigr{)}+\omega_{k}\bigl{(}\log\left(1+\gamma_{k}\right)-\gamma_{k}\bigr{)}\Bigr{)}.$

Problem (19) is convex in each variable block—$\\{\gamma_{i}\\}$,
$\\{\phi_{i}\\}$, and $\\{\mathbf{w}_{i}\\}$, and can be solved by resorting
to the BCA method.

In [5], an “unconventional” type of BCA method was proposed, which is to
update $\\{\gamma_{i}\\}$ by solving Problem (18) and to update
$\\{\phi_{i}\\}$ and $\\{\mathbf{w}_{i}\\}$ by solving Problem (19). The
variable update steps are summarized in LABEL:label:MISO-FP with $\mu$
determined by (12).

A2: A WSR-FP algorithm for MISO beamforming [5] S1:
$\displaystyle\gamma_{k}=\mathsf{SINR}_{k}(\\{\underline{\mathbf{w}}_{i}\\}),$
S2:
$\displaystyle\phi_{k}=\frac{\sqrt{\omega_{k}\left(1+\underline{\gamma}_{k}\right)}\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}}{\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{j}\bigr{|}^{2}+\sigma_{k}^{2}},$
S3:
$\displaystyle\mathbf{w}_{k}=\Bigl{(}\sum_{j=1}^{K}|\underline{\phi}_{j}|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}+\mu\mathbf{I}\Bigr{)}^{\dagger}\sqrt{\omega_{k}(1+\underline{\gamma}_{k})}\underline{\phi}_{k}\mathbf{h}_{k}.$

Despite deviating from the traditional BCA, i.e., solving all variable blocks
based on the single problem (19), convergence of LABEL:label:MISO-FP to
stationary points is assured as demonstrated in [19].

######  Remark 5.

Employing the conventional BCA approach to addressing Problem (19) would
result in identical update procedures for the $\\{\phi_{i}\\}$ and
$\\{\mathbf{w}_{i}\\}$ variable blocks as delineated in LABEL:label:MISO-FP.
However, the update formula for $\gamma_{k}$ is modified to be

$\gamma_{k}=\frac{1}{2}\Biggl{(}\frac{\underline{\phi}_{k}^{2}|\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}|^{2}}{\omega_{k}}+\sqrt{\Bigl{(}\frac{\underline{\phi}_{k}^{2}|\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}|^{2}}{\omega_{k}}\Bigr{)}^{2}-4\frac{\underline{\phi}_{k}^{2}|\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}|^{2}}{\omega_{k}}}\Biggr{)}.$
(20)

Then, we can derive two variant WSR-FP algorithms characterized by the update
sequences:

$\dashrightarrow\underbrace{\text{Eq. \eqref{eq:conventional
gamma}}\rightarrow\text{\nameref{label:MISO-
FP}-S2}\rightarrow\text{\nameref{label:MISO-FP}-S3}}_{\text{one round of
variable update}}\rightarrow\text{Eq. \eqref{eq:conventional
gamma}}\dashrightarrow$ (21)

and

$\dashrightarrow\underbrace{\text{\nameref{label:MISO-
FP}-S2}\rightarrow\text{Eq. \eqref{eq:conventional
gamma}}\rightarrow\text{\nameref{label:MISO-FP}-S3}}_{\text{one round of
variable update}}\rightarrow\text{\nameref{label:MISO-FP}-S2}\dashrightarrow.$
(22)

For the aforementioned algorithms, it can be demonstrated that every limit
point of the sequence they generate is indeed a stationary point of Problem
(8), a result validated by the general BCA convergence theorem [24,
Proposition 5].

######  Remark 6.

Observe that if we substitute LABEL:label:MISO-FP-S2 into (20), we will obtain
LABEL:label:MISO-FP-S1. This indicates that the unconventional BCA update
strategy outlined in LABEL:label:MISO-FP can be construed as executing a
conventional BCA, that is,

$\dashrightarrow\underbrace{\text{\nameref{label:MISO-
FP}-S2}\rightarrow\text{Eq. \eqref{eq:conventional
gamma}}\rightarrow\text{\nameref{label:MISO-
FP}-S2}\rightarrow\text{\nameref{label:MISO-FP}-S3}}_{\text{one round of
variable update}}\rightarrow\text{\nameref{label:MISO-FP}-S2}\dashrightarrow.$
(23)

The algorithm in (23) can be viewed as an augmentation of the ones in (21) or
(22), enriched by an additional updating step LABEL:label:MISO-FP-S2. This
modification constitutes a BCA algorithm that updates variables following an
essentially cyclic rule.

To derive Problem (19), we have utilized a particular decoupling approach to
dealing with the ratio terms in the objective of Problem (18). However,
alternative decoupling techniques are also available as suggested in [9]. For
instance, Proposition 4 can be applied to Problem (18) by setting
$q_{k}(x)=\left(1+\gamma_{k}\right)x$,
$c_{k}(\mathbf{x})=\sqrt{\omega_{k}}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}$,
and
$d_{k}(\mathbf{x})=\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}$,
which is the particular formulation to be employed in Section IV-C for the
design of beamformers in MIMO systems. Additionally, by applying Proposition 4
to Problem (18) with $q_{k}(x)=\omega_{k}\left(1+\gamma_{k}\right)x$,
$c_{k}(\mathbf{x})=\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}$, and
$d_{k}(\mathbf{x})=\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}$,
we arrive at the well-known weighted sum-MSE minimization problem (11).
Overall, the WSR-FP framework exhibits versatility; by selecting from various
decoupling strategies and sequences for updating variables, we can develop
diverse variations of WSR-FP algorithms.

###  III-D WSR-MM Algorithms

In this section, we introduce the WSR-MM approach proposed in [6]. We first
introduce a pertinent result pivotal for constructing MM surrogate functions
for Problem (8).

######  Proposition 7 ([6]).

Given $\log(1+\frac{\left|x\right|^{2}}{z})$ with $x\in\mathbb{C}$ and $z>0$,
at $(\underline{x},\underline{z})$ we have

$\displaystyle\log\bigl{(}1+\frac{|x|^{2}}{z}\bigr{)}\geq$
$\displaystyle\log\bigl{(}1+\frac{|\underline{x}|^{2}}{\underline{z}}\bigr{)}-\frac{|\underline{x}|^{2}}{\underline{z}}+2\mathrm{Re}\bigl{(}\frac{\underline{x}^{*}}{\underline{z}}x\bigr{)}$
(24)
$\displaystyle-\frac{|\underline{x}|^{2}}{\underline{z}(\underline{z}+|\underline{x}|^{2})}(z+|x|^{2}),$

where the equality is attained when
$\left(x,z\right)=\left(\underline{x},\underline{z}\right)$.

By invoking Proposition 7 and applying it to
$f\left(\\{\mathbf{w}_{i}\\}\right)$, where
$x=\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}$ and $z=\sum_{j=1,j\neq
k}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}$,
we can construct a surrogate function at $\\{\underline{\mathbf{w}}_{i}\\}$ as
follows:

$\displaystyle\ell\left(\\{\mathbf{w}_{i}\\},\\{\underline{\mathbf{w}}_{i}\\}\right)=$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\Bigl{(}-a_{k}\sum_{j=1}^{K}|\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}|^{2}+2\mathrm{Re}(b_{k}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k})$
(25)
$\displaystyle+\mathsf{R}_{k}(\\{\underline{\mathbf{w}}_{i}\\})-\mathsf{SINR}_{k}(\\{\underline{\mathbf{w}}_{i}\\})-a_{k}\sigma_{k}^{2}\Bigr{)},$

where
$a_{k}=\frac{\mathsf{SINR}_{k}(\\{\underline{\mathbf{w}}_{i}\\})}{\sum_{j=1}^{K}|\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{j}|^{2}+\sigma_{k}^{2}}$
and
$b_{k}=\frac{\mathsf{SINR}_{k}(\\{\underline{\mathbf{w}}_{i}\\})}{\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}}$.
Disregarding constant terms in the surrogate function, we can solve Problem
(8) by iteratively solving the following problem:

$\displaystyle\underset{\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\Bigl{(}-a_{k}\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+2\mathrm{Re}\left(b_{k}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\right)\Bigr{)}.$
(26)

This approach not only simplifies the original problem but also ensures that
each step is tractable via convex optimization. Using the Lagrangian
multipliers method, we can obtain the optimal solution to (26) via (12). We
give the update step in WSR-MM in the following with $\mu$ determined by (12).

A3: The WSR-MM algorithm for MISO beamforming [6]
$\mathbf{w}_{k}=\Bigl{(}\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}+\mu\mathbf{I}\Bigr{)}^{\dagger}\omega_{k}b_{k}^{*}\mathbf{h}_{k}.$

##  IV WSR Maximization Over MIMO Interference Channel

###  IV-A System Model and Problem Formulation

We consider a device-to-device MIMO communication system with $K$ disjoint
links.333To streamline the discussion of MIMO system design alongside the MISO
scenario, certain notations will be abused in this section, though this
involves a slight repurposing of their original meanings. We assume, in the
$k$-th ($k=1,\ldots,K$) link, the transmitter $k$ is equipped with
$M_{k}^{\mathsf{t}}$ transmit antennas and the receiver $k$ has
$M_{k}^{\mathsf{r}}$ receive antennas. Denote by
$\mathbf{H}_{i,j}\in\mathbb{C}^{M_{i}^{\mathsf{r}}\times M_{j}^{\mathsf{t}}}$
the channel between receiver $i$ and transmitter $j$. Define
$\mathbf{W}_{k}\in\mathbb{C}^{M_{k}^{\mathsf{t}}\times M_{k}^{\mathsf{s}}}$
($M_{k}^{\mathsf{s}}$ is the number of parallel data streams transmitted on
the $k$-th link with
$M_{k}^{\mathsf{s}}\leq\min\\{M_{k}^{\mathsf{t}},M_{k}^{\mathsf{r}}\\}$) as
the beamforming matrix at the transmitter $k$,
$\mathbf{s}_{k}\in\mathbb{C}^{M_{k}^{\mathsf{s}}}$ as the corresponding symbol
vector, and
$\mathbf{n}_{k}\sim\mathcal{CN}(\mathbf{0},\sigma_{k}^{2}\mathbf{I})$ as the
noise. The received signal $\mathbf{y}_{k}$ at receiver $k$ is expressed as

$\mathbf{y}_{k}=\mathbf{H}_{k,k}\mathbf{W}_{k}\mathbf{s}_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{s}_{j}+\mathbf{n}_{k}.$ (27)

Accordingly, the data rate at the $k$-th receiver is defined by

$\mathsf{R}_{k}=\log\det\left(\mathbf{I}+\mathbf{W}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{F}_{k}^{-1}\mathbf{H}_{k,k}\mathbf{W}_{k}\right)$
(28)

where the interference-plus-noise matrix is denoted as

$\mathbf{F}_{k}=\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}+\sigma_{k}^{2}\mathbf{I}.$
(29)

Our goal is to maximize the WSR of the system by designing the beamformer
matrices $\\{\mathbf{W}_{i}\\}$, thereby solving the following maximization
problem:

$\displaystyle\underset{\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle
f\left(\\{\mathbf{W}_{i}\\}\right)=\sum_{k=1}^{K}\omega_{k}\mathsf{R}_{k},$
(30)

where the transmit power limit constraint is given by

$\mathcal{W}=\left\\{\\{\mathbf{W}_{i}\\}\mid\left\|\mathbf{W}_{i}\right\|_{\mathsf{}}^{2}\leq
P_{i},\ i=1,\ldots,K\right\\}$ (31)

with $P_{i}$ representing the power budget at the $i$-th transmitter.

In the following, we will introduce the WMMSE [4], WSR-FP [9], and WSR-MM [6]
methods for MIMO beamforming.

###  IV-B WMMSE Algorithms

We consider the WMMSE approach proposed in [4]. Let
$\mathbf{L}_{k}\in\mathbb{C}^{M_{k}^{\mathsf{r}}\times M_{k}^{\mathsf{s}}}$ be
the receive beamformer of receiver $k$, $k=1,\ldots,K$. Under the independence
assumption of $\mathbf{s}_{k}$’s and $\mathbf{n}_{k}$’s, the MSE matrix
between the estimated signal and the original signal at receiver $k$ is given
by

$\displaystyle\mathbf{E}_{k}=$
$\displaystyle\mathbb{E}\biggl{[}\Bigl{(}\mathbf{L}_{k}^{\mathsf{H}}\bigl{(}\mathbf{H}_{k,k}\mathbf{W}_{k}\mathbf{s}_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{s}_{j}+\mathbf{n}_{k}\bigr{)}-\mathbf{s}_{k}\Bigr{)}\cdot$
(32)
$\displaystyle\Bigl{(}\mathbf{L}_{k}^{\mathsf{H}}\bigl{(}\mathbf{H}_{k,k}\mathbf{W}_{k}\mathbf{s}_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{s}_{j}+\mathbf{n}_{k}\bigr{)}-\mathbf{s}_{k}\Bigr{)}^{\mathsf{H}}\biggr{]}$
$\displaystyle=$
$\displaystyle\left(\mathbf{L}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}\mathbf{W}_{k}-\mathbf{I}\right)\left(\mathbf{L}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}\mathbf{W}_{k}-\mathbf{I}\right)^{\mathsf{H}}$
$\displaystyle+\sum_{j=1,j\neq
k}^{K}\mathbf{L}_{k}^{\mathsf{H}}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}\mathbf{L}_{k}+\sigma_{k}^{2}\mathbf{L}_{k}^{\mathsf{H}}\mathbf{L}_{k}.$

######  Proposition 8 ([4]).

Let $\mathbf{M}_{1},\ldots,\mathbf{M}_{K}\succeq\mathbf{0}$ be the weight
matrices. The WSR maximization problem (30) is equivalent to the following
weighted sum-MSE minimization problem

$\displaystyle\underset{\\{\mathbf{L}_{i}\\},\\{\mathbf{M}_{i}\\},\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\left(\log\det\left(\mathbf{M}_{k}\right)-\mathrm{tr}\left(\mathbf{M}_{k}\mathbf{E}_{k}\right)\right),$
(33)

in the sense that they attain the identical optimal solution.

Similar to the MISO case, based on BCA, update rules of the WMMSE algorithm
for Problem (33) are summarized below.

A4: A WMMSE algorithm for MIMO beamforming [4] S1:
$\displaystyle\mathbf{L}_{k}=(\underline{\mathbf{F}}_{k}+\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}})^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}$
S2:
$\displaystyle\mathbf{M}_{k}=\bigl{(}\sum_{j=1}^{K}\underline{\mathbf{L}}_{k}^{\mathsf{H}}\mathbf{H}_{k,j}\underline{\mathbf{W}}_{j}\underline{\mathbf{W}}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}\underline{\mathbf{L}}_{k}-\underline{\mathbf{L}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}$
$\displaystyle\hskip
99.58464pt-\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{L}}_{k}+\sigma_{k}^{2}\underline{\mathbf{L}}_{k}^{\mathsf{H}}\underline{\mathbf{L}}_{k}+\mathbf{I}\bigr{)}^{-1}$
S3:
$\displaystyle\mathbf{W}_{k}=\bigl{(}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{k,j}^{\mathsf{H}}\underline{\mathbf{L}}_{j}\underline{\mathbf{M}}_{j}\underline{\mathbf{L}}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}+\mu_{k}\mathbf{I}\bigr{)}^{\dagger}\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{L}}_{k}\underline{\mathbf{M}}_{k}.$

In LABEL:label:MIMO-WMMSE, $\mu_{k}$ for $k=1,\ldots,K$ is optimally
determined by

$\mu_{k}=\min\bigl{\\{}\mu_{k}\geq
0:\left\|\mathbf{W}_{k}(\mu_{k})\right\|_{\mathsf{}}^{2}\leq P_{k}\bigr{\\}}.$
(34)

###  IV-C WSR-FP Algorithms

We discuss the WSR-FP approach proposed in [9]. To begin with, we first
present the matrix Lagrangian dual transform and the matrix quadratic
transform.

######  Proposition 9 (Matrix Lagrangian dual transform [9]).

Given matrix-ratios
$\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\mathbf{D}_{k}^{-1}(\mathbf{X})\sqrt{\mathbf{C}}_{k}(\mathbf{X})$
with $\mathbf{C}_{k}(\mathbf{X})\succeq\mathbf{0}$ and
$\mathbf{D}_{k}(\mathbf{X})\succ\mathbf{0}$ for $k=1,\ldots,K$, the following
problem:

$\displaystyle\underset{\mathbf{X}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\log\det\bigl{(}\mathbf{I}+\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\mathbf{D}_{k}^{-1}(\mathbf{X})\sqrt{\mathbf{C}}_{k}(\mathbf{X})\bigr{)},$
(35)

is equivalent to

$\begin{aligned}
&\negthickspace\negthickspace\underset{\\{\boldsymbol{\Gamma}_{i}\\},\mathbf{X}\in\mathcal{X}}{\text{maximize}}&&\negthickspace\negthickspace\sum_{k=1}^{K}\omega_{k}\Bigl{(}\log\det\left(\mathbf{I}+\boldsymbol{\Gamma}_{k}\right)-\mathrm{tr}\left(\boldsymbol{\Gamma}_{k}\right)+\mathrm{tr}\bigl{(}(\mathbf{I}\\\
&&&+\boldsymbol{\Gamma}_{k})\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\bigl{(}\mathbf{C}_{k}(\mathbf{X})+\mathbf{D}_{k}(\mathbf{X})\bigr{)}^{-1}\sqrt{\mathbf{C}}_{k}(\mathbf{X})\bigr{)}\Bigr{)},\end{aligned}\negthickspace$
(36)

in the sense that they attain the identical optimal solution with the
identical optimal objective value.

######  Proposition 10 (Matrix quadratic transform [9]).

Given nondecreasing matrix functions $q_{k}\left(\cdot\right)$ such that
$q_{k}\left(\mathbf{Z}_{1}\right)\geq q_{k}\left(\mathbf{Z}_{2}\right)$ if
$\mathbf{Z}_{1}\succeq\mathbf{Z}_{2}$ and ratios
$\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\mathbf{D}_{k}^{-1}(\mathbf{X})\sqrt{\mathbf{C}}_{k}(\mathbf{X})$
with $\mathbf{C}_{k}(\mathbf{X})\succeq\mathbf{0}$ and
$\mathbf{D}_{k}(\mathbf{X})\succ\mathbf{0}$ for $k=1,\ldots,K$, the following
problem:

$\begin{aligned}
&\underset{\mathbf{X}\in\mathcal{X}}{\text{maximize}}&&\sum_{k=1}^{K}q_{k}\bigl{(}\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\mathbf{D}_{k}^{-1}(\mathbf{X})\sqrt{\mathbf{C}}_{k}(\mathbf{X})\bigr{)}\end{aligned},$
(37)

is equivalent to

$\displaystyle\underset{\\{\boldsymbol{\Phi}_{i}\\},\mathbf{X}\in\mathcal{X}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}q_{k}\Bigl{(}2\mathrm{Re}\bigl{(}\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{X})\boldsymbol{\Phi}_{k}\bigr{)}-\boldsymbol{\Phi}_{k}^{\mathsf{H}}\mathbf{D}_{k}(\mathbf{X})\boldsymbol{\Phi}_{k}\Bigr{)},$
(38)

in the sense that they attain the identical optimal solution with the
identical optimal objective value.

By applying Proposition 9 to Problem (30), and considering
$\sqrt{\mathbf{C}}_{k}(\mathbf{X})=\mathbf{W}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}$
and $\mathbf{D}_{k}(\mathbf{X})=\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}+\sigma_{k}^{2}\mathbf{I}$,
we get the following problem with auxiliary variables
$\\{\boldsymbol{\Gamma}_{i}\\}$:

$\displaystyle\underset{\\{\boldsymbol{\Gamma}_{i}\\},\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\Bigl{(}\log\det(\mathbf{I}+\boldsymbol{\Gamma}_{k})-\mathrm{tr}(\boldsymbol{\Gamma}_{k})+\mathrm{tr}\bigl{(}(\mathbf{I}$
(39)
$\displaystyle+\boldsymbol{\Gamma}_{k})\mathbf{H}_{k,k}\mathbf{W}_{k}\bigl{(}\sum_{j=1}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}+\sigma_{k}^{2}\mathbf{I}\bigr{)}^{-1}\mathbf{W}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\bigr{)}\Bigr{)},$

The ratios in Problem (39) can be further decoupled by applying Proposition 10
with
$q_{k}(\mathbf{Z})=\mathrm{tr}\left(\left(\mathbf{I}+\boldsymbol{\Gamma}_{k}\right)\mathbf{Z}\right)$,
$\sqrt{\mathbf{C}}_{k}(\mathbf{x})=\sqrt{\omega_{k}}\mathbf{W}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}$,
and $\mathbf{D}_{k}(\mathbf{x})=\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}+\sigma_{k}^{2}\mathbf{I}$,
leading to an equivalent problem with further introduced auxiliary variables
$\\{\boldsymbol{\Phi}_{i}\\}$:

$\displaystyle\underset{\\{\boldsymbol{\Gamma}_{i}\\},\\{\boldsymbol{\Phi}_{i}\\},\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\biggl{(}\log\det(\mathbf{I}+\boldsymbol{\Gamma}_{k})$
(40)
$\displaystyle-\mathrm{tr}(\boldsymbol{\Gamma}_{k})+\mathrm{tr}\Bigl{(}(\mathbf{I}+\boldsymbol{\Gamma}_{k})\bigl{(}2\mathrm{Re}(\sqrt{\omega_{k}}\mathbf{H}_{k,k}\mathbf{W}_{k}\boldsymbol{\Phi}_{k})$
$\displaystyle-\boldsymbol{\Phi}_{k}^{\mathsf{H}}\bigl{(}\sum_{j=1}^{K}\mathbf{H}_{k,j}\mathbf{W}_{j}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}+\sigma_{k}^{2}\mathbf{I}\bigr{)}\boldsymbol{\Phi}_{k}\bigr{)}\Bigr{)}\biggr{)}.$

Then based on BCA, variable updates in a WSR-FP algorithm are summarized in
the following ($\mu_{k}$ is determined by (34)).

A5: A WSR-FP algorithm for MIMO beamforming [9] S1:
$\displaystyle\boldsymbol{\Phi}_{k}=\bigl{(}\underline{\mathbf{F}}_{k}+\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\bigr{)}^{-1}\sqrt{\omega_{k}}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k},$
S2:
$\displaystyle\boldsymbol{\Gamma}_{k}=\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k},$
S3:
$\displaystyle\mathbf{W}_{k}=\Bigl{(}\sum_{j=1}^{K}\mathbf{H}_{k,j}^{\mathsf{H}}\underline{\boldsymbol{\Phi}}_{j}(\mathbf{I}+\underline{\boldsymbol{\Gamma}}_{j})\underline{\boldsymbol{\Phi}}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}+\mu_{k}\mathbf{I}\Bigr{)}^{\dagger}\cdot$
$\displaystyle\hskip
28.45274pt\sqrt{\omega_{k}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\boldsymbol{\Phi}}_{k}(\mathbf{I}+\underline{\boldsymbol{\Gamma}}_{k}).$

###  IV-D WSR-MM Algorithms

In this section, we introduce the WSR-MM approach [6] for MIMO beamforming.

######  Proposition 11 ([6]).

Given
$\log\det\left(\mathbf{I}+\mathbf{X}^{\mathsf{H}}\mathbf{Z}^{-1}\mathbf{X}\right)$
with $\mathbf{X}\in\mathbb{C}^{M\times N}$ and $\mathbf{Z}\succ\mathbf{0}$, at
$\left(\underline{\mathbf{X}},\underline{\mathbf{Z}}\right)$ we have

$\displaystyle\log\det(\mathbf{I}+\mathbf{X}^{\mathsf{H}}\mathbf{Z}^{-1}\mathbf{X})\geq$
$\displaystyle\log\det(\mathbf{I}+\underline{\mathbf{X}}^{\mathsf{H}}\underline{\mathbf{Z}}^{-1}\underline{\mathbf{X}})$
(41) $\displaystyle-{\rm
tr}(\underline{\mathbf{X}}^{\mathsf{H}}\underline{\mathbf{Z}}^{-1}\underline{\mathbf{X}})+2{\rm{\rm
Re}}\bigl{(}{\rm
tr}\bigl{(}\underline{\mathbf{X}}^{\mathsf{H}}\underline{\mathbf{Z}}^{-1}\mathbf{X}\bigr{)}\bigr{)}$
$\displaystyle-\mathrm{tr}\bigl{(}(\underline{\mathbf{Y}}+\underline{\mathbf{X}}\underline{\mathbf{X}}^{\mathsf{H}})^{-1}\underline{\mathbf{X}}\underline{\mathbf{X}}^{\mathsf{H}}\underline{\mathbf{Z}}^{-1}(\mathbf{Z}+\mathbf{X}\mathbf{X}^{\mathsf{H}})\bigr{)},$

where the equality is attained when
$\left(\mathbf{X},\mathbf{Z}\right)=\left(\underline{\mathbf{X}},\underline{\mathbf{Z}}\right)$.

By applying Proposition 11 to $f\left(\\{\mathbf{W}_{i}\\}\right)$ with
$\mathbf{X}=\mathbf{H}_{k,k}\mathbf{W}_{k}$ and $\mathbf{Z}=\mathbf{F}_{k}$,
we construct a surrogate function at $\\{\underline{\mathbf{W}_{i}}\\}$ as
follows:

$\displaystyle\ell\left(\\{\mathbf{W}_{i}\\},\\{\underline{\mathbf{W}_{i}}\\}\right)$
(42) $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\omega_{k}\Bigl{(}-\mathrm{tr}\Bigl{(}\sum_{j=1}^{K}\mathbf{W}_{j}^{\mathsf{H}}\mathbf{H}_{k,j}^{\mathsf{H}}\mathbf{A}_{k}\mathbf{H}_{k,j}\mathbf{W}_{j}\Bigr{)}$
$\displaystyle+2{\rm{\rm Re}}\bigl{(}{\rm
tr}\bigl{(}\mathbf{B}_{k}\mathbf{H}_{k,k}\mathbf{W}_{k}\bigr{)}\bigr{)}$
$\displaystyle+\log\det(\mathbf{I}+\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k})$
$\displaystyle-{\rm
tr}(\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k})-\mathrm{tr}\Bigl{(}\sigma_{k}^{2}\mathbf{A}_{k}\Bigr{)}\Bigr{)},$

where
$\mathbf{A}_{k}=(\underline{\mathbf{F}}_{k}+\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}})^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\mathbf{B}_{k}$
and
$\mathbf{B}_{k}=\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}$.
Based on the principal of MM, we solve Problem (30) by successively solving
the following problem:

$\displaystyle\underset{\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{maximize}}$
$\displaystyle\sum_{k=1}^{K}\Bigl{(}-\mathrm{tr}\bigl{(}\mathbf{W}_{k}^{\mathsf{H}}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}\mathbf{W}_{k}\bigr{)}$
(43) $\displaystyle\hskip
56.9055pt+2\omega_{k}\mathrm{tr}\left(\mathbf{B}_{k}\mathbf{H}_{k,k}\mathbf{W}_{k}\right)\Bigr{)},$

whose solution can be obtained via the Lagrangian multipliers method (34).
WSR-MM is given in LABEL:label:MIMO-MM with $\mu_{k}$ defined in (34).

A6: The WSR-MM algorithm for MIMO beamforming [6]
$\mathbf{W}_{k}=\Bigl{(}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}+\mu_{k}\mathbf{I}\Bigr{)}^{\dagger}\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{B}_{k}^{\mathsf{H}}.$

##  V Connections Among WMMSE, WSR-FP, and WSR-MM

The connection between WMMSE and WSR-FP has been elucidated in [9]. Given that
the weighted sum-MSE minimization problem (33) emerges as a particular
instance of applying Proposition 10 to Problem (39), it is evident that WMMSE
algorithms represent specific cases within the broader scope of WSR-FP
algorithms. Our subsequent discussion mainly focus on the connections between
WSR-FP/WMMSE with WSR-MM algorithms. Our analysis will be centered on the MIMO
case, though it should be noted that the findings are equally applicable to
the MISO scenario. We will initiate by establishing a general result that
delineates the process of converting a three-block BCA into an MM algorithm,
laying the groundwork for the subsequent analysis.

###  V-A Mapping A Specific BCA Algorithm to An MM Algorithm

Consider the optimization variables $u$, $v$, and $x$ in Problem (1), which
are updated cyclically and non-repetitively according to (2). It is postulated
that $\max_{u\in\mathcal{U},v\in\mathcal{V}}g(u,v,x)=f(x)$, with the
subsequent assumption that the update for $u^{(t+1)}$ is solely dependent on
$x^{(t)}$ and is independent of $v^{(t)}$, and $v^{(t+1)}$ is updated based on
both $x^{(t)}$ and $u^{(t+1)}$. Under these premises, the triple update steps
in (2) can be reformulated into a dual-step process:

$\begin{cases}\left(u^{(t+1)},v^{(t+1)}\right)\in\arg\max_{u\in\mathcal{U},v\in\mathcal{V}}g(u,v,x^{(t)})\\\
x^{(t+1)}\in\arg\max_{x\in\mathcal{X}}g(u^{(t+1)},v^{(t+1)},x).\end{cases}$
(44)

This reformulation ensures that for all $(x,x^{(t)})\in{\cal X}$, the
following relations hold true:

$\displaystyle
g(u^{(t+1)},v^{(t+1)},x)\leq\max_{u\in\mathcal{U},v\in\mathcal{V}}g(u,v,x)=f(x),$

and

$g(u^{(t+1)},v^{(t+1)},x^{(t)})=\max_{u\in\mathcal{U},v\in\mathcal{V}}g(u,v,x^{(t)})=f(x^{(t)}),$

from which we deduce that $g(u^{(t+1)},v^{(t+1)},x)$ serves as a surrogate
function for $f(x)$ at iterate $x^{(t)}$. Consequently, (44) may be
interpreted as an MM algorithm step. It is noteworthy that (44) can be further
distilled into a fixed-point iteration:
$x^{(t+1)}\in\arg\max_{x\in\mathcal{X}}g(\arg\max_{u\in\mathcal{U},v\in\mathcal{V}}g(u,v,x^{(t)}),x)$.

###  V-B Connections Between WMMSE/WSR-FP and WSR-MM

Recall that both the WMMSE and WSR-FP are three-block BCA algorithms. The
result given in last section facilitates the comprehension of their
relationship with the WSR-MM approach. Considering the WMMSE as formulated in
LABEL:label:MIMO-WMMSE, and incorporating $\mathbf{L}_{k}$ into
$\mathbf{M}_{k}$, yields the following expression:

$\displaystyle\mathbf{M}_{k}$
$\displaystyle=\bigl{(}\mathbf{I}-\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}(\underline{\mathbf{F}}_{k}+\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}})^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\bigr{)}^{-1}$
(45)
$\displaystyle=\mathbf{I}+\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k},$

where the second line is due to the Woodbury matrix identity [26]. Define
$\mathbf{A}_{k}=\mathbf{L}_{k}\mathbf{M}_{k}\mathbf{L}_{k}^{\mathsf{H}}$ and
$\mathbf{B}_{k}=\mathbf{M}_{k}^{\mathsf{H}}\mathbf{L}_{k}^{\mathsf{H}}$. We
can recover LABEL:label:MIMO-MM from LABEL:label:MIMO-WMMSE, indicating that a
WMMSE algorithm can equate to the WSR-MM. However, the alternative formulation
of WMMSE presented in (13) defies this equivalence, as it does not conform to
the update rule in (44). The connection between WSR-FP and WSR-MM mirrors that
of WMMSE and WSR-MM. As demonstrated in [9], the update procedures in WSR-FP
in LABEL:label:MIMO-FP are profoundly linked to a surrogate function.
Advancing this concept, we show that the specialized WSR-FP algorithm
LABEL:label:MIMO-FP corresponds explicitly to the WSR-MM algorithm
LABEL:label:MIMO-MM. By substituting $\\{\boldsymbol{\Gamma}_{i}\\}$ and
$\\{\boldsymbol{\Phi}_{i}\\}$ into the update step of $\mathbf{W}_{k}$ in
LABEL:label:MIMO-FP and defining
$\mathbf{A}_{k}=\frac{1}{\omega_{k}}\boldsymbol{\Phi}_{k}(\mathbf{I}+\boldsymbol{\Gamma}_{k})\boldsymbol{\Phi}_{k}^{\mathsf{H}}$
and
$\mathbf{B}_{k}=\frac{1}{\sqrt{\omega_{k}}}(\mathbf{I}+\boldsymbol{\Gamma}_{k}^{\mathsf{H}})\boldsymbol{\Phi}_{k}^{\mathsf{H}}$,
we obtain LABEL:label:MIMO-MM. It is important to note, however, that certain
variants of WSR-FP do not align with the MM framework. For instance, replacing
S1 in LABEL:label:MISO-FP with (20), which entails a conventional BCA update
rule, results in an algorithm that diverges from the MM approach. The
relationships among WMMSE, WSR-FP, and WSR-MM are summarized in Fig. 1.

Figure 1: Relations among WMMSE, WSR-FP, and WSR-MM.

###  V-C Bridging Equivalent Transforms and Surrogate Functions

In this section, we elucidate that within the WSR-FP algorithm, the refinement
of auxiliary variables through both equivalent transforms—the Lagrangian dual
transform (39) and the quadratic transform (40)—can be interpreted as
methodologies for devising surrogate functions in the context of WSR-MM. Prior
to this, we introduce two pertinent lemmas.

######  Lemma 12 ([6]).

Given $\log\det(\mathbf{Z})$ with $\mathbf{Z}\succ\mathbf{0}$, at
$\underline{\mathbf{Z}}$ we have

$\log\det(\mathbf{Z})\geq\log\det(\underline{\mathbf{Z}})+{\rm
tr}\bigl{(}\mathbf{I}-\underline{\mathbf{Z}}\mathbf{Z}^{-1}\bigr{)},$ (46)

where the equality is attained when $\mathbf{Z}=\underline{\mathbf{Z}}$.

######  Lemma 13 ([6]).

Given a nondecreasing matrix function $\mathbf{Q}(\cdot)$ such that
$\mathbf{Q}\left(\mathbf{Z}_{1}\right)\succeq\mathbf{Q}\left(\mathbf{Z}_{2}\right)$
if $\mathbf{Z}_{1}\succeq\mathbf{Z}_{2}$, then for
$\mathbf{Q}\left(\mathbf{Z}_{1}^{\mathsf{H}}\mathbf{Z}_{2}^{-1}\mathbf{Z}_{1}\right)$
with $\mathbf{Z}_{1}\in\mathbb{C}^{n\times n}$ and
$\mathbf{Z}_{2}\succ\mathbf{0}$, at
$(\underline{\mathbf{Z}_{1}},\underline{\mathbf{Z}_{2}})$ we have

$\mathbf{Q}\left(\mathbf{Z}_{1}^{\mathsf{H}}\mathbf{Z}_{2}^{-1}\mathbf{Z}_{1}\right)\succeq\mathbf{Q}\left(2\mathrm{Re}\left(\underline{\mathbf{Z}_{1}^{\mathsf{H}}}\underline{\mathbf{Z}_{2}^{-1}}\mathbf{Z}_{1}\right)-\underline{\mathbf{Z}_{1}^{\mathsf{H}}}\underline{\mathbf{Z}_{2}^{-1}}\mathbf{Z}_{2}\underline{\mathbf{Z}_{2}^{-1}}\underline{\mathbf{Z}_{1}}\right),$
(47)

where the equality is attained when
$\left(\mathbf{Z}_{1},\mathbf{Z}_{2}\right)=\left(\underline{\mathbf{Z}_{1}},\underline{\mathbf{Z}_{2}}\right)$.

As shown in [6], the surrogate function $\ell$ in Section IV-D can be
conceptualized as being derived through two sequential operations anchored in
Lemmas 12 and 13. We present the following two propositions to demonstrate how
the updates of the auxiliary variables in the WSR-FP algorithm (refer to
LABEL:label:MIMO-FP) correlate with the aforementioned lemmas.

######  Proposition 14.

Updating variables $\\{\boldsymbol{\Gamma}_{i}\\}$ in Problem (36) by BCA can
be seen as constructing a surrogate function based on Lemma 12 for the
objective in Problem (35) by MM.

###### Proof:

With $\mathbf{X}$ fixed, $\\{\boldsymbol{\Gamma}_{i}\\}$ is optimally
determined by

$\boldsymbol{\Gamma}_{k}=\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\underline{\mathbf{X}})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}}),\
\ \ k=1,\ldots,K.$ (48)

Substituting $\\{\boldsymbol{\Gamma}_{i}\\}$ to the objective in problem (35),
we have

$\displaystyle\quad h\left(\mathbf{X},\\{\boldsymbol{\Gamma}_{i}\\}\right)$
(49)
$\displaystyle=\sum_{k=1}^{K}\omega_{k}\Biggl{(}\log\det\left(\mathbf{I}+\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\underline{\mathbf{X}})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}})\right)$
$\displaystyle\quad+\mathrm{tr}\Bigl{(}\mathbf{I}-\left(\mathbf{I}+\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\underline{\mathbf{X}})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}})\right)\cdot$
$\displaystyle\quad\left(\mathbf{I}-\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{x})\left(\mathbf{C}_{k}(\mathbf{x})+\mathbf{D}_{k}(\mathbf{x})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\mathbf{x})\right)\Bigr{)}\Biggr{)}.$

Applying Lemma 12 to the objective of (35) with
$\mathbf{Z}=\mathbf{I}+\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{x})\left(\mathbf{D}_{k}(\mathbf{x})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\mathbf{x})$,
we also obtain $h\left(\mathbf{X},\\{\boldsymbol{\Gamma}_{i}\\}\right)$.444The
following result would be necessary for the derivation: the term
$\left(\mathbf{I}-\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{x})\left(\mathbf{C}_{k}(\mathbf{x})+\mathbf{D}_{k}(\mathbf{x})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\mathbf{x})\right)$
in $h\left(\mathbf{X},\\{\boldsymbol{\Gamma}_{i}\\}\right)$ simplifies to
$\left(\mathbf{I}+\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{x})\left(\mathbf{D}_{k}(\mathbf{x})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\mathbf{x})\right)^{-1}$
by using the Woodbury identity. ∎

######  Proposition 15.

Updating variables $\\{\boldsymbol{\Phi}_{i}\\}$ in Problem (38) by BCA can be
seen as constructing a surrogate function based on Lemma 13 for the objective
in (37) by MM.

###### Proof:

With $\mathbf{X}$ fixed, $\\{\boldsymbol{\Phi}_{i}\\}$ is optimally determined
by

$\boldsymbol{\Phi}_{k}=\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}}),\
\ \ k=1,\ldots,K.$ (50)

Substituting $\\{\boldsymbol{\Phi}_{i}\\}$ to the objective in problem (38),
we have

$\displaystyle h$
$\displaystyle\left(\mathbf{X},\\{\boldsymbol{\Phi}_{i}\\}\right)=\sum_{k=1}^{K}f_{k}\Bigl{(}2\mathrm{Re}\left(\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\mathbf{x})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}})\right)$
(51)
$\displaystyle-\sqrt{\mathbf{C}}_{k}^{\mathsf{H}}(\underline{\mathbf{X}})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\mathbf{D}_{k}(\mathbf{x})\left(\mathbf{D}_{k}(\underline{\mathbf{X}})\right)^{-1}\sqrt{\mathbf{C}}_{k}(\underline{\mathbf{X}})\Bigr{)},$

which can also be obtained by applying Lemma 13 to the objective in (37) with
$\mathbf{Z}_{1}=\sqrt{\mathbf{C}}_{k}(\mathbf{x})$ and
$\mathbf{Z}_{2}=\mathbf{D}_{k}(\mathbf{x})$. ∎

##  VI A Novel Algorithm for WSR Maximization

The beamformer update procedures in WMMSE/WSR-FP and WSR-MM in Sections III
and IV necessitate the iterative adjustment of Lagrangian multipliers, an
operation inherently reliant on the frequent calculation of matrix pseudo-
inverses. This nested iteration scheme introduces a dual-layer loop structure
for the algorithms, imposing a computational complexity that scales cubically
with the number of transmit antennas—a considerable impediment for systems
with extensive antenna arrays. Furthermore, the convergence and monotonicity
traits of these algorithms are intimately tied to the accuracy of the
Lagrangian tuning process, presenting additional challenges. To overcome the
aforementioned issues, we introduce herein a streamlined, single-loop
algorithm with reduced per-iteration computational demands and assured
convergence—the WSR-MM+. Distinguished by its analytical update expressions
for beamformers that eschew matrix inversions, WSR-MM+ presents a substantial
computational advantage.

###  VI-A The Proposed WSR-MM+ Algorithm

We first discuss the algorithm for MIMO beamforming (30). To start with, we
introduce a useful result.

######  Proposition 16 ([18]).

Let $(\mathbf{L},\mathbf{M})\in\mathbb{H}^{n}$ such that
$\mathbf{M}\succeq\mathbf{L}$. Given
$\mathbf{X}^{\mathsf{H}}\mathbf{L}\mathbf{X}$ with
$\mathbf{X}\in\mathbb{C}^{n\times m}$, at $\underline{\mathbf{X}}$, we have

$\displaystyle\mathrm{tr}\Bigl{(}\mathbf{X}^{\mathsf{H}}\mathbf{M}\mathbf{X}\Bigr{)}$
$\displaystyle\geq\mathrm{tr}\left(\mathbf{X}^{\mathsf{H}}\mathbf{L}\mathbf{X}\right)+2\mathrm{Re}(\mathbf{X}^{\mathsf{H}}(\mathbf{M}-\mathbf{L})\underline{\mathbf{X}})$
(52)
$\displaystyle\quad+\mathrm{tr}\Bigl{(}\underline{\mathbf{X}}^{\mathsf{H}}(\mathbf{L}-\mathbf{M})\underline{\mathbf{X}}\Bigr{)}.$

Based on Proposition 16, we construct a novel surrogate function for problem
(30). This construction hinges on the application of the above result to the
quadratic term in $\mathbf{W}_{k}$, for $k=1,\ldots,K$, encapsulated within
$\ell$ (42), specifically,
$-\mathrm{tr}\bigl{(}\mathbf{W}_{k}^{\mathsf{H}}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}\mathbf{W}_{k}\bigr{)}$.
Setting
$\eta_{k}\geq\lambda_{\max}\left(\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}\right)$
we arrive at the surrogate function $\ell^{\prime}$ in (53).555Considering
that
$\lambda_{\max}\left(\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}\right)$
is bounded by the Frobenius norm of
$\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}$,
and by extension, a computationally efficient choice for $\eta_{k}$ is given
by
$\sum_{j=1}^{K}\omega_{j}\left\|\mathbf{A}_{j}\right\|\left\|\mathbf{H}_{j,k}\right\|^{2}$.

$\displaystyle\ell^{\prime}(\\{\mathbf{W}_{i}\\},\\{\underline{\mathbf{W}_{i}}\\})$
$\displaystyle=-\sum_{k=1}^{K}\mathrm{tr}\Bigl{(}\eta_{k}\mathbf{W}_{k}^{\mathsf{H}}\mathbf{W}_{k}+2\mathrm{Re}\Bigl{(}\mathbf{W}_{k}^{\mathsf{H}}(\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}-\eta_{k}\mathbf{I})\underline{\mathbf{W}}_{k}\Bigr{)}+\underline{\mathbf{W}}_{k}^{\mathsf{H}}(\eta_{k}\mathbf{I}-\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k})\underline{\mathbf{W}}_{k}\Bigr{)}$
(53) $\displaystyle\quad+\sum_{k=1}^{K}\omega_{k}2{\rm{\rm Re}}\bigl{(}{\rm
tr}(\mathbf{B}_{k}\mathbf{H}_{k,k}\mathbf{W}_{k})\bigr{)}+\sum_{k=1}^{K}\omega_{k}\left(\underline{\mathsf{R}}_{k}-{\rm
tr}(\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k})-\mathrm{tr}(\mathbf{A}_{k})\sigma_{k}^{2}\right).$

Given $\ell^{\prime}$ is a surrogate function of $\ell$ and hence a surrogate
of $f$ in problem (30), the MM principle dictates that we can iteratively
tackle the following optimization:

$\displaystyle\underset{\\{\mathbf{W}_{i}\\}\in\mathcal{W}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\eta_{k}\left\|\mathbf{W}_{k}-\mathbf{Q}_{k}\right\|^{2},$
(54)

with

$\mathbf{Q}_{k}=\eta_{k}^{-1}\Bigl{(}\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{B}_{k}^{\mathsf{H}}-\bigl{(}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}-\eta_{k}\mathbf{I}\bigr{)}\underline{\mathbf{W}}_{k}\Bigr{)}.$
(55)

Problem (54) has an analytical solution as follows:

A7: A WSR-MM+ algorithm for MIMO beamforming
$\mathbf{W}_{k}=\mathbf{Q}_{k}\min\left\\{\sqrt{\frac{P}{\left\|\mathbf{Q}_{k}\right\|_{\mathsf{}}^{2}}},1\right\\}.$

######  Remark 17.

It is noteworthy that WSR-MM+ is compatible with the prevalent per-antenna
power constraints [27], in which case analytical solutions are also attainable
in the subproblems. However, a detailed exposition is precluded due to space
limitations.

Compared to WSR-MM, WSR-MM+ obviates the need for Lagrange multiplier tuning
and repetitive matrix (pseudo-)inversions. Consequently, the efficiency of the
overall algorithm could be enhanced. WSR-MM+ for MISO beamforming (8) can be
derived similarly via Proposition 16 by restricting $\mathbf{X}$ to be a
column vector. Applying Proposition 16 to the quadratic term in $\ell$ (25),
with
$\eta\geq\lambda_{\max}\left(\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}\right)$
(A practical selection for $\eta$ mirrors that in the MIMO scenario, being
$\sum_{j=1}^{K}\omega_{j}a_{j}\left\|\mathbf{h}_{j}\right\|^{2}$.) gives rise
to the surrogate function $\ell^{\prime}$ in (56).

$\displaystyle\ell^{\prime}(\\{\mathbf{w}_{i}\\},\\{\underline{\mathbf{w}_{i}}\\})$
$\displaystyle=-\sum_{k=1}^{K}\Bigl{(}\phi\mathbf{w}_{k}^{\mathsf{H}}\mathbf{w}_{k}+2\mathrm{Re}\bigl{(}\underline{\mathbf{w}}_{k}^{\mathsf{H}}(\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}-\phi\mathbf{I})\mathbf{w}_{k}\bigr{)}+\underline{\mathbf{w}}_{k}^{\mathsf{H}}(\phi\mathbf{I}-\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}})\underline{\mathbf{w}}_{k}\Bigr{)}$
(56)
$\displaystyle\quad+\sum_{k=1}^{K}\omega_{k}2\mathrm{Re}(b_{k}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k})+\sum_{k=1}^{K}\omega_{k}(\underline{\mathsf{R}}_{k}-\underline{\mathsf{SINR}}_{k}-a_{k}\sigma_{k}^{2})$

Define

$\mathbf{q}_{k}=\eta^{-1}\Bigl{(}\omega_{k}b_{k}^{*}\mathbf{h}_{k}-(\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}-\phi\mathbf{I})\underline{\mathbf{w}}_{k}\Bigr{)},$
(57)

WSR-MM+ suffices to iteratively solve the following problem

$\displaystyle\underset{\\{\mathbf{w}_{i}\\}\in\mathcal{W}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\eta\left\|\mathbf{w}_{k}-\mathbf{q}_{k}\right\|^{2},$
(58)

which has a closed-form solution. WSR-MM+ in the MISO scenario is as follows:

A8: A WSR-MM+ algorithm for MISO beamforming
$\mathbf{w}_{k}=\mathbf{q}_{k}\min\left\\{\sqrt{\frac{P}{\sum_{k=1}^{K}\left\|\mathbf{q}_{k}\right\|^{2}}},1\right\\}.$

######  Remark 18.

The efficiency of the WSR-MM+ algorithm may be further improved through
methods including the extrapolation technique, as described in Section VII-E
of [6]. Although an extensive exploration of these techniques is beyond the
scope of this paper, readers interested in a comprehensive discussion are
encouraged to refer to [23].

###  VI-B Interpreting WSR-MM+ as BCA

In Section V, it is demonstrated that the updating of auxiliary variables
through equivalent transforms within WSR-FP can be seen as procedures for the
creation of surrogate functions in WSR-MM. Conversely, these equivalent
transforms also serve to parameterize certain intermediary variables within
WSR-MM. This perspective allows us to interpret the newly proposed WSR-MM+ as
a BCA algorithm. To substantiate this perspective, we introduce a novel
transform that aligns with the principles outlined in Proposition 16.

$\displaystyle\underset{\\{\boldsymbol{\Gamma}_{i}\\},\\{\boldsymbol{\Phi}_{i}\\},\left(\\{\mathbf{W}_{i}\\},\\{\mathbf{T}_{i}\\}\right)\in\mathcal{W}}{\text{maximize}}$
$\displaystyle-\sum_{k=1}^{K}\mathrm{tr}\biggl{(}\eta_{k}\mathbf{W}_{k}^{\mathsf{H}}\mathbf{W}_{k}+2\mathrm{Re}\Bigl{(}\mathbf{W}_{k}^{\mathsf{H}}\Bigl{(}\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\boldsymbol{\Phi}_{j}(\mathbf{I}+\boldsymbol{\Gamma}_{j})\boldsymbol{\Phi}_{j}^{\mathsf{H}}\mathbf{H}_{j,k}-\eta_{k}\mathbf{I}\Bigr{)}\mathbf{T}_{k}\Bigr{)}$
(59)
$\displaystyle+\mathbf{T}_{k}^{\mathsf{H}}\Bigl{(}\eta_{k}\mathbf{I}-\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\boldsymbol{\Phi}_{j}(\mathbf{I}+\boldsymbol{\Gamma}_{j})\boldsymbol{\Phi}_{j}^{\mathsf{H}}\mathbf{H}_{j,k}\Bigr{)}\mathbf{T}_{k}\biggr{)}+\sum_{k=1}^{K}\Biggl{(}2\mathrm{tr}\Bigl{(}(\mathbf{I}+\boldsymbol{\Gamma}_{k})\mathrm{Re}(\sqrt{\omega_{k}}\mathbf{H}_{k,k}\mathbf{W}_{k}\boldsymbol{\Phi}_{k})\Bigr{)}$
$\displaystyle+\sum_{k=1}^{K}\omega_{k}\bigl{(}\log\det(\mathbf{I}+\boldsymbol{\Gamma}_{k})-\mathrm{tr}(\boldsymbol{\Gamma}_{k})\bigr{)}\Biggr{)}-\sum_{k=1}^{K}\sigma_{k}^{2}\mathrm{tr}\bigl{(}(\mathbf{I}+\boldsymbol{\Gamma}_{k})\boldsymbol{\Phi}_{k}^{\mathsf{H}}\boldsymbol{\Phi}_{k}\bigr{)}.$

$\displaystyle\underset{\\{\gamma_{i}\\},\\{\phi_{i}\\},\left(\\{\mathbf{w}_{i}\\},\\{\mathbf{t}_{i}\\}\right)\in\mathcal{W}}{\text{maximize}}$
$\displaystyle-\sum_{k=1}^{K}\Bigl{(}\eta\mathbf{w}_{k}^{\mathsf{H}}\mathbf{w}_{k}+2\mathrm{Re}\bigl{(}\mathbf{w}_{k}^{\mathsf{H}}(\sum_{j=1}^{K}|\phi_{j}|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}-\eta\mathbf{I})\mathbf{t}_{k}\bigr{)}+\mathbf{t}_{k}^{\mathsf{H}}(\eta\mathbf{I}-\sum_{j=1}^{K}|\phi_{j}|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}})\mathbf{t}_{k}\Bigr{)}$
(60)
$\displaystyle+\sum_{k=1}^{K}2\mathrm{Re}\bigl{(}\phi_{k}^{*}\sqrt{\omega_{k}(1+\gamma_{k})}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\bigr{)}+\sum_{k=1}^{K}\omega_{k}\left(\log\left(1+\gamma_{k}\right)-\gamma_{k}\right)-\sum_{k=1}^{K}|\phi_{k}|^{2}\sigma_{k}^{2}.$

######  Corollary 19 (Matrix non-homogeneous transform).

Let $\mathbf{L},\mathbf{M}\in\mathbb{H}^{n}$ such that
$\mathbf{M}\succeq\mathbf{L}$. Problem

$\underset{\mathbf{X}\in\mathcal{X}}{\text{minimize}}\ \
\mathrm{tr}\left(\mathbf{X}^{\mathsf{H}}\mathbf{L}\mathbf{X}\right),$ (61)

where ${\cal X}$ denotes the constraint for $\mathbf{X}$, is equivalent to

$\underset{(\mathbf{X},\mathbf{Z})\in\mathcal{X}}{\text{minimize}}\ \
\mathrm{tr}\Bigl{(}\mathbf{X}^{\mathsf{H}}\mathbf{M}\mathbf{X}+2\mathrm{Re}(\mathbf{X}^{\mathsf{H}}(\mathbf{L}-\mathbf{M})\mathbf{Z})+\mathbf{Z}^{\mathsf{H}}(\mathbf{M}-\mathbf{L})\mathbf{Z}\Bigr{)},$
(62)

in the sense that they attain the identical optimal solution with the
identical optimal objective value.

###### Proof:

$\mathbf{Z}$ is optimally determined by deriving the first order optimality
condition of (62). Substituting the solution into the transformed objective
recovers the original one. ∎

Invoking Corollary 19, we obtain a BCA-based algorithm, termed WSR-FP+,
emerging as a counterpart of WSR-MM+. Applying Corollary 19 to Problem (40)
using
$\eta_{k}\geq\lambda_{\max}\left(\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\boldsymbol{\Phi}_{j}\left(\mathbf{I}+\boldsymbol{\Gamma}_{j}\right)\boldsymbol{\Phi}_{j}^{\mathsf{H}}\mathbf{H}_{j,k}\right)$,
we get an equivalent problem presented in (59). Solving (59) through BCA leads
the Algorithm LABEL:label:MIMO-FP+.

A9: A WSR-FP+ algorithm for MIMO beamforming S1:
$\displaystyle\mathbf{T}_{k}=\underline{\mathbf{W}}_{k},$ S2:
$\displaystyle\boldsymbol{\Phi}_{k}=\left(\underline{\mathbf{F}}_{k}+\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k}\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\right)^{-1}\sqrt{\omega_{k}}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k},$
S3:
$\displaystyle\boldsymbol{\Gamma}_{k}=\underline{\mathbf{W}}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\underline{\mathbf{F}}_{k}^{-1}\mathbf{H}_{k,k}\underline{\mathbf{W}}_{k},$
S4:
$\displaystyle\mathbf{W}_{k}=\mathbf{Q}_{k}\min\left\\{\sqrt{\frac{P}{\left\|\mathbf{Q}_{k}\right\|_{\mathsf{}}^{2}}},1\right\\}.$

It can be verified that $\mathbf{Q}_{k}$ in LABEL:label:MIMO-FP+-S4 is given
by (55). In a similar vein, WSR-FP+ can be used for MISO beamforming (8).
Based on Corollary 19, choosing
$\eta\geq\lambda_{\max}\left(\sum_{j=1}^{K}|\underline{y}_{j}|^{2}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}\right)$
leads to Problem (60). Using BCA to solve this problem, we obtain Algorithm
LABEL:label:MISO-FP+, where $\mathbf{q}_{k}$ appeared in LABEL:label:MISO-
FP+-S4 is given in (57).

A10: A WSR-FP+ algorithm for MISO beamforming S1:
$\displaystyle\mathbf{t}_{k}=\underline{\mathbf{w}}_{k},$ S2:
$\displaystyle\gamma_{k}=\underline{\mathsf{SINR}}_{k},$ S3: $\displaystyle
y_{k}=\frac{\sqrt{\omega_{k}\left(1+\underline{\gamma}_{k}\right)}\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{k}}{\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\underline{\mathbf{w}}_{j}\bigr{|}^{2}+\sigma^{2}},$
S4:
$\displaystyle\mathbf{w}_{k}=\mathbf{q}_{k}\min\left\\{\sqrt{\frac{P}{\sum_{k=1}^{K}\left\|\mathbf{q}_{k}\right\|^{2}}},1\right\\}.$

###  VI-C Interpreting WSR-MM+ as Projected Gradient Ascent

$\displaystyle\frac{\partial
f\left(\\{\mathbf{W}_{i}\\}\right)}{\partial\mathbf{W}_{k}}$
$\displaystyle=2\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{F}_{k}^{-1}\mathbf{H}_{k,k}\mathbf{W}_{k}\left(\mathbf{I}+\mathbf{W}_{k}^{\mathsf{H}}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{F}_{k}^{-1}\mathbf{H}_{k,k}\mathbf{W}_{k}\right)^{-1}$
(63) $\displaystyle\quad-2\sum_{i=1,i\neq
k}^{K}\omega_{i}\mathbf{H}_{i,k}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\left(\mathbf{I}+\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\right)^{-1}\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,k}\mathbf{W}_{k}$
$\displaystyle=2\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{F}_{k}^{-1}\mathbf{H}_{k,k}\mathbf{W}_{k}-2\sum_{i=1}^{K}\omega_{i}\mathbf{H}_{i,k}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\left(\mathbf{I}+\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\right)^{-1}\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,k}\mathbf{W}_{k}$
$\displaystyle=2\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{F}_{k}^{-1}\mathbf{H}_{k,k}\mathbf{W}_{k}-2\sum_{i=1}^{K}\omega_{i}\mathbf{H}_{i,k}^{\mathsf{H}}(\mathbf{F}_{i}+\mathbf{H}_{i,i}\mathbf{W}_{i}\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}})^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,k}\mathbf{W}_{k}.$

$\frac{\partial
f\left(\\{\mathbf{w}_{i}\\}\right)}{\partial\mathbf{w}_{k}}=\frac{2\omega_{k}\mathbf{h}_{k}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}}{\sum_{j=1,j\neq
k}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}}-\sum_{k=1}^{K}\frac{\omega_{k}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}\bigr{|}^{2}}{\sum_{j=1,j\neq
k}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}}\frac{2\mathbf{h}_{k}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{k}}{\sum_{j=1}^{K}\bigl{|}\mathbf{h}_{k}^{\mathsf{H}}\mathbf{w}_{j}\bigr{|}^{2}+\sigma_{k}^{2}}.$
(64)

In deriving WSR-MM+, the surrogate function $\ell^{\prime}$ is an isotropic
quadratic approximation of the original objective, which affords WSR-MM+ an
intriguing conceptual parallel to the projected gradient ascent method. Given
$f\left(\\{\mathbf{W}_{i}\\}\right)$ in problem formulation (30), the gradient
with respect to $\mathbf{W}_{k}$ is given in (63),666Wirtinger calculus is
adopted for differentials of complex variables [28]. where the last equality
is due to the positive definite identity [29]:
$\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\left(\mathbf{I}+\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\mathbf{F}_{i}^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}\right)^{-1}=\left(\mathbf{F}_{i}+\mathbf{H}_{i,i}\mathbf{W}_{i}\mathbf{W}_{i}^{\mathsf{H}}\mathbf{H}_{i,i}^{\mathsf{H}}\right)^{-1}\mathbf{H}_{i,i}\mathbf{W}_{i}$.
Then, $\mathbf{Q}_{k}$ in (55) can be rewritten as

$\displaystyle\mathbf{Q}_{k}$
$\displaystyle=\underline{\mathbf{W}}_{k}+\frac{1}{\eta_{k}}\Bigl{(}\omega_{k}\mathbf{H}_{k,k}^{\mathsf{H}}\mathbf{B}_{k}^{\mathsf{H}}-\sum_{j=1}^{K}\omega_{j}\mathbf{H}_{j,k}^{\mathsf{H}}\mathbf{A}_{j}\mathbf{H}_{j,k}\underline{\mathbf{W}}_{k}\Bigr{)}$
(65)
$\displaystyle=\underline{\mathbf{W}}_{k}+\frac{1}{2\eta_{k}}\frac{\partial
f\left(\\{\underline{\mathbf{W}}_{i}\\}\right)}{\partial\mathbf{W}_{k}}.$

Thus, each individual iteration within WSR-MM+, specifically solving problem
(58), can be interpreted as executing a step of projected gradient ascent with
step size $\frac{1}{2\eta_{k}}$ (note $\eta_{k}$ depends on
$\\{\underline{\mathbf{W}}_{i}\\}$), formally described by

$\mathbf{W}_{k}=\mathrm{Proj}_{{\cal
W}}(\underline{\mathbf{W}}_{k}+\frac{1}{2\eta_{k}}\frac{\partial
f\left(\\{\underline{\mathbf{W}}_{i}\\}\right)}{\partial\mathbf{W}_{k}}),$

where $\mathrm{Proj}_{{\cal W}}(\cdot)$ denotes the projection operator onto
set ${\cal W}$. Similar result applies to the MISO case. Considering the
objective function $f\left(\\{\mathbf{w}_{i}\\}\right)$ in problem formulation
(8), the gradient with respect to $\mathbf{w}_{k}$ is provided in (64). It
follows that $\mathbf{q}_{k}$ in (57) can be expressed as

$\displaystyle\mathbf{q}_{k}$
$\displaystyle=\underline{\mathbf{w}}_{k}+\frac{1}{\eta}\Bigl{(}\omega_{k}b_{k}^{*}\mathbf{h}_{k}-\sum_{j=1}^{K}\omega_{j}a_{j}\mathbf{h}_{j}\mathbf{h}_{j}^{\mathsf{H}}\underline{\mathbf{w}}_{k}\Bigr{)}$
(66) $\displaystyle=\underline{\mathbf{w}}_{k}+\frac{1}{2\eta}\frac{\partial
f\left(\\{\underline{\mathbf{w}}_{i}\\}\right)}{\partial\mathbf{w}_{k}}.$

Therefore, WSR-MM+ for MISO beamforming is equivalent to a projected gradient
ascent step with step size $\frac{1}{2\eta}$.

##  VII Per-Iteration Complexity Analysis

In this section, we analyze the complexity of WMMSE (LABEL:label:MISO-WMMSE
and LABEL:label:MIMO-WMMSE), WSR-FP (LABEL:label:MISO-FP and LABEL:label:MIMO-
FP), WSR-MM (LABEL:label:MISO-MM and LABEL:label:MIMO-MM), WSR-MM+
(LABEL:label:MISO-MM+ and LABEL:label:MIMO-MM+), and WSR-FP+
(LABEL:label:MISO-FP+ and LABEL:label:MIMO-FP+). For simplicity, we assume $M$
in the MISO case and $\\{M_{i}^{\mathsf{r}}\\}$, $\\{M_{i}^{\mathsf{t}}\\}$,
and $\\{M_{i}^{\mathsf{s}}\\}$ in the MIMO case are of the same order,
uniformly denoted as $M$. We denote by $I$ the iteration numbers for searching
the Lagrangian multiplier (i.e., (12) or (34)). In the following, we first
analyze the per-iteration computational cost of the WSR-MM and WSR-MM+.

For MISO case, the complexity of WSR-MM is dominated by the matrix pseudo-
inversions, the vector-vector outer products, and matrix-vector products. The
complexity of WSR-MM+ mainly comes from the multiplication between the vector-
vector outer products and matrix-vector products. The overall per-iteration
complexity of WSR-MM and WSR-MM+ are
$\mathcal{O}\left(I\left(KM^{2}+M^{3}\right)\right)$ and
$\mathcal{O}\left(KM^{2}\right)$ ($\eta$ is obtained independently of
eigenvalue computations), respectively. For MIMO case, the complexity of WSR-
MM and WSR-MM+ are both dominated by the matrix pseudo-inverse operation. The
overall per-iteration complexity of WSR-MM and WSR-MM+ are
$\mathcal{O}\left(I\left(KM^{3}+M^{3}\right)\right)$ and
$\mathcal{O}\left(KM^{3}\right)$, respectively.

For WMMSE, WSR-FP, and WSR-MM, as discussed in Section V, it becomes apparent
that these algorithms exhibit the same order of complexity. Nonetheless, WSR-
MM boasts a distinctive advantage in that it bypasses the computation of
auxiliary variables. Specifically, WSR-MM can eliminate certain operations
that, in the context of WMMSE and WSR-FP, carry a complexity of
$\mathcal{O}\left(K\right)$ for MISO and $\mathcal{O}\left(KM^{2}\right)$ for
MIMO beamforming problems, respectively. Correspondingly, the analysis extends
to WSR-MM+ and WSR-FP+, wherein both algorithms demonstrate the same order of
complexity, while WSR-MM+ offers additional advantage of obviating the need
for auxiliary variable computations.

##  VIII Numerical Experiments

In this section, we corroborate our theoretical findings with numerical
experiments performed in MATLAB on a personal computer with a 3.3 GHz Intel
Xeon W CPU.777Code available at https://github.com/zepengzhang/RateMax.

In our evaluation, we investigate both MISO and MIMO systems configured in a
three-dimensional Cartesian setting. For the MISO system, a base station is
placed at $(0,0,10)\mathsf{m}$ serving $K$ users randomly distributed in a
circle centered at $(200,30,0)\mathsf{m}$ with radius of $10\mathsf{m}$. We
assume that the channel fading is frequency flat and adopts the Rayleigh
fading model
$\mathbf{h}_{k}=\sqrt{\kappa(d)}\times\mathcal{CN}(\mathbf{0},\mathbf{I})$,
where $\kappa(x)$ represents the distance-dependent path loss. Specifically,
the path loss is computed as $\kappa(d)=T_{0}(\frac{d}{d_{0}})^{-\varrho}$
where the path loss at the reference distance $d_{0}=1\mathsf{m}$ is
$T_{0}=-30\mathsf{dB}$ and $\varrho=3.67$ is the path loss exponent. For the
MIMO system, the channels are modeled analogously to the MISO case. The
transmitters and the receivers are randomly distributed in two circles
centered at $(0,0,10)\mathsf{m}$ and $(200,30,0)\mathsf{m}$, respectively,
both with radius of $10\mathsf{m}$. For both systems, we have considered the
noise power spectrum density of $-169\mathsf{dBm/Hz}$ and the transmission
bandwidth of 240$\mathsf{kHz}$. We set $\sigma_{k}^{2}=1$ for $k=1,\ldots,K$,
and $P=0\mathsf{dBm}$. We set the algorithm convergence criterion to be the
objective value increment less than $10^{-6}$. All reported results are
averaged over 100 independent channel realizations.

We compare the WMMSE, WSR-FP, WSR-MM, WSR-MM+, and WSR-FP+ algorithms in both
a MISO system (with parameters $K=4$ and $M=4$), in which case we implement
LABEL:label:MISO-WMMSE, LABEL:label:MISO-FP, LABEL:label:MISO-MM,
LABEL:label:MISO-MM+, and LABEL:label:MISO-FP+, and a MIMO system (with
parameters $K=4$ and
$M_{k}^{\mathsf{r}}=M_{k}^{\mathsf{t}}=M_{k}^{\mathsf{s}}=4$ for
$k=1,\ldots,K$), in which case we implement LABEL:label:MIMO-WMMSE,
LABEL:label:MIMO-FP, LABEL:label:MIMO-MM, LABEL:label:MIMO-MM+, and
LABEL:label:MIMO-FP+. Figs. 2a and 2b illustrate the performance of these
algorithms in the MISO setting, based on the number of iterations and CPU
time, respectively. Correspondingly, Figs. 2c and 2d present the analogous
results for the MIMO configuration. From the simulation results, we can see
that the WMMSE, WSR-FP, and WSR-MM algorithms exhibit comparable performance
in both system configurations. This pattern of performance is consistent with
the results obtained for WSR-MM+ and WSR-FP+. Notably, WSR-MM+ and WSR-FP+
demonstrate a trade-off between the number of iterations required to achieve
convergence and the CPU time consumed. Despite a higher iteration count, these
algorithms benefit from reduced CPU time, suggesting that the per-iteration
analytical solution offers a significant computational advantage.

(a) Average WSR versus iteration in the MISO case. (b) Average WSR versus CPU
time in the MISO case. (c) Average WSR versus iteration in the MIMO case. (d)
Average WSR versus CPU time in the MISO case.

Figure 2: Performance comparison of different algorithms.

To more precisely quantify the performance enhancement offered by WSR-MM+ over
WSR-MM, we investigate their respective convergence times for WSR maximization
in MISO systems varying both number of users (i.e., $K$) in Fig. 3 and number
of antennas at the base station (i.e., $M$) in Fig. 4. From the results, we
can easily observe that WSR-MM+ significantly surpasses WSR-MM by a large
margin.

Figure 3: Average convergence time versus number of users. Figure 4: Average
convergence time versus number of antennas at the base station.

It might be posited that the augmented complexity inherent in WSR-MM relative
to WSR-MM+ is primarily attributed to the one-dimensional search procedures,
as given in (12) and (34). A conjecture may arise that by loosening the
stopping criteria for these searches, the total computation time for WSR-MM
may decrease, potentially resulting in a more efficient algorithm than WSR-
MM+. To explore this conjecture, Fig. 5 provides a numerical comparison
between WSR-MM+ and different versions of of WSR-MM that are subjected to
differing stopping criteria. Specifically, “WSR-MM$(i)$” in Fig. 5 denotes a
WSR-MM algorithm that terminates the one-dimensional search when the change in
the search variable falls below $2^{-i}$ across successive iterations.
Contrary to expectations, the results indicate that relaxing the stopping
criteria fails to expedite convergence; rather, it can result in a non-
monotonic convergence trajectory or cause the algorithm to settle at
suboptimal local minima.

Figure 5: Convergence behaviors of WSR-MM and WSR-MM+.

##  IX Conclusions and Discussions

In this work, we have explored the precise connections between the WMMSE, WSR-
FP, and WSR-MM algorithms in addressing the sum-rate maximization problems. We
have shown that specific variants of the WMMSE and WSR-FP algorithms can be
construed within the MM algorithmic paradigm. Furthermore, we have established
that the equivalent transforms employed in WMMSE and WSR-FP can be regarded as
methodologies to construct surrogate functions within the WSR-MM algorithm. We
have also introduced WSR-MM+, an enhanced algorithm characterized by its
analytical update steps. Complementing this, we have presented WSR-FP+, the
BCA equivalent of WSR-MM+, and revealed its relation to the projected gradient
ascent method. The superiority of WSR-MM+ and WSR-FP+ has been substantiated
through numerical simulations.

Several compelling directions for future research emerge from this study. A
promising one lies in examining the acceleration and convergence properties of
WSR-MM+. Another fertile area for exploration is the adaptation of WSR-MM+ to
more intricate system design challenges, such as multi-cell communication
systems, as well as to rate maximization problems that encompass multiple
design variables beyond just the transmit beamformers. Additionally, an
intriguing prospect involves the application of algorithm unrolling [30] to
transform WSR-MM+ into deep neural networks, thereby imbuing certain
algorithmic parameters with learnability. These neural networks after training
could potentially outstrip their iterative predecessors in terms of
convergence and generalization properties.

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