Robust Probabilistic-Constrained Optimization for IRS-Aided MISO Communication Systems

Abstract

Taking into account imperfect channel state information, this letter formulates and solves a joint active/passive beamforming optimization problem in multiple-input single-output systems with the support of an intelligent reflecting surface. In particular, we introduce an optimization problem to minimize the total transmit power subject to maintaining the users’ signal-to-interference-plus-noise-ratio coverage probability above a predefined target. Due to the presence of probabilistic constraints, the proposed optimization problem is non-convex. To circumvent this issue, we first recast the proposed problem in a convex form by adopting the Bernstein-type inequality, and we then introduce a converging alternating optimization approach to iteratively find the active/passive beamforming vectors. In particular, the transformed robust optimization problem can be effectively solved by using standard interior-point methods. Numerical results demonstrate the effectiveness of jointly optimizing the active/passive beamforming vectors.

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Le, Tuan Anh ORCID: https://orcid.org/0000-0003-0612-3717, Trinh, Van Chien and Di Renzo,
Marco (2020) Robust probabilistic-constrained optimization for IRS-aided MISO communication
systems. IEEE Wireless Communications Letters . ISSN 2162-2337 (Accepted/In press)
(doi:10.1109/LWC.2020.3016592)
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1
Robust Probabilistic-Constrained Optimization for
IRS-Aided MISO Communication Systems
Tuan Anh Le, Senior Member IEEE, Trinh Van Chien, and Marco Di Renzo, Fellow IEEE
Abstract—Taking into account imperfect channel state infor-
mation, this letter formulates and solves a joint active/passive
beamforming optimization problem in multiple-input single-
output systems with the support of an intelligent reflecting
surface. In particular, we introduce an optimization problem to
minimize the total transmit power subject to maintaining the
users’ signal-to-interference-plus-noise-ratio coverage probability
above a predefined target. Due to the presence of probabilistic
constraints, the proposed optimization problem is non-convex. To
circumvent this issue, we first recast the proposed problem in a
convex form by adopting the Bernstein-type inequality, and we
then introduce a converging alternating optimization approach
to iteratively find the active/passive beamforming vectors. In
particular, the transformed robust optimization problem can be
effectively solved by using standard interior-point methods. Nu-
merical results demonstrate the effectiveness of jointly optimizing
the active/passive beamforming vectors.
Index Terms—6G wireless, intelligent reflecting surface.
I. INTRODUCTION
Intelligent reflecting surface (IRS), i.e., a programmable pla-
nar array of passive reflecting elements, has been identified as
an energy-efficient technology for improving the performance
of wireless communication systems [1]–[4]. An IRS-aided
communication system comprises an IRS, a base station (BS)
and the mobile users. The location of the IRS is usually chosen
to assist the BS communicating with the users. An IRS can
also extend the communication range where there is no direct
link between the BS and the users. Each element of the IRS
can be independently controlled to produce a desired phase
shift on the impinging electromagnetic waves. A careful design
of the phase shifts of all the elements of the IRS helps to
improve the channel propagation conditions [5]. Moreover, the
combination of transmit beamforming and phase shift designs
has been proved to offer higher spectral and energy efficiencies
compared to conventional beamforming methods [6].
Manuscript received March 24, 2020; revised July 09, 2020 and August
07, 2020; accepted August 07, 2020. Date of publication XXX; date of
current version XXX. The work of M. Di Renzo was supported in part
by the European Commission through the H2020 ARIADNE Project under
Grant 871464. The associate editor coordinating the review of this letter and
approving it for publication was Dr. J. Xu. (Corresponding author: Trinh Van
Chien.)
T. A. Le is with the Department of Design Engineering & Mathematics,
Faculty of Science and Technology, Middlesex University, The Burroughs,
Hendon, London, NW4 4BT, U. K. Email: t.le@mdx.ac.uk.
T. V. Chien was with the School of Electronics and Telecommunications,
Hanoi University of Science and Technology, 100000 Hanoi, Vietnam. He
is now with the Interdisciplinary Centre for Security, Reliability and Trust
(SnT), University of Luxembourg, L-1855 Luxembourg, Luxembourg. Email:
vanchien.trinh@uni.lu.
M. Di Renzo is with Universit´
e Paris-Saclay, CNRS, CentraleSup´
elec,
Laboratoire des Signaux et Syst`
emes, 91192 Gif-sur-Yvette, France. Email:
marco.direnzo@centralesupelec.fr.
The performance of an IRS-aided communication system
heavily depends on the accuracy of the channel state informa-
tion (CSI), i.e., the CSI between the BS and the IRS, as well
as the CSI between the IRS and the mobile users. Most of
the works on IRS-aided systems are based on a perfect CSI
assumption, see e.g., [7] and references therein. Unfortunately,
due to the random nature of wireless systems, obtaining
accurate CSI is, in practice, a challenging task. Therefore, the
robust optimization of the active/passive beamforming against
imprecise estimates of the CSI is highly desirable in order
to fully realize the potential of IRS-aided communication
systems.
The first attempt to tackle imperfect estimates of the CSI
in IRS-aided systems can be found in [8]. In [8], the transmit
beamforming vectors at the BS and the phase shifts at the IRS
are jointly designed so as to minimize the total transmit power
subject to the worst-case quality of service (QoS) constraints,
i.e., maintaining the required achievable rate for every user
under all possible cases of CSI errors. To that end, an ellipsoid
approach was adopted to capture the channel uncertainties,
where the Frobenious norm of the channel error vector is
assumed to be confined within a given radius of the uncertainty
region. This method results in a conservative approach since it
allocates excessive system resources to satisfy rarely occurring
worst-case events.
In this letter, we tackle the problem of allocating excessive
system resources by allowing that the QoS constraints can be
violated with a certain probability. In particular, we formulate
an optimization problem to jointly design the active/passive
beamforming vectors at the BS and the IRS. The proposed
problem minimizes the total transmit power while maintain-
ing the users’ signal-to-interference-plus-noise-ratio (SINR)
coverage probability above a predefined value. Since the
considered probabilistic constraints are not convex, we adopt
the Bernstein-type inequality [9] to transform the formulated
problem into a linear matrix inequality (LMI) form, i.e., into
a convex optimization problem. In addition, we introduce
an alternating optimization approach to iteratively find the
active/passive beamforming vectors. Our proposed problem
differs from that in [8] for two main reasons. First, the QoS
constraints are formulated in a probabilistic form while those
of [8] are not. Second, unlike [8], the considered problem does
not rely on the assumption of bounded channel uncertainty
region.
The following notation is adopted in this letter. Bold
lower/upper case letters denote vectors/matrices; k·kand k·k𝐹
denote the Euclidean norm and the Frobenius norm, respec-
tively; (·)𝐻,(·)𝑇and (·)∗denote the complex conjugate
transpose operator, the transpose operator and the complex

2
conjugate operator, respectively; tr (·)denotes the trace of a
matrix; X0denotes the positive semidefinite condition; I𝑎
denotes an 𝑎×𝑎identity matrix; diag (x)denotes a diagonal
matrix whose diagonal entries are the elements of the vector
x; diag (X)denotes a vector comprising the diagonal elements
of matrix X;1𝑁denotes an 𝑁×1vector of all unity elements;
CN(·,·) denotes a circularly symmetric complex Gaussian
distribution; vec(·) denotes the vectorization operator; Pr (·)
denotes the probability of an event; ⊗denotes the Kronecker
product; |·|and arg(·) denote the absolute value and the
phase of a complex-valued scalar, respectively; E[·] denotes
the expectation of a random variable.
II. SY ST EM MO DE L
We consider an IRS-aided communication system that con-
sists of an 𝑀-antenna BS serving 𝐾single-antenna users. It is
assumed that there is no direct communication link from the
BS to the users due to the presence of blockages, e.g., high
buildings. To overcome this issue, an IRS with 𝑁reflective
elements is deployed to assist the communication between
the BS and the users. Let H=[h1, . . . , h𝑁] ∈ C𝑀×𝑁and
g𝑘=[𝑔𝑘1, . . . , 𝑔𝑘 𝑁 ]𝑇∈C𝑁×1denote the channel coefficients
between the BS and the IRS, as well as between the IRS and
the 𝑘-th user, respectively. We assume that the instantaneous
CSI is not known, and that the channel is estimated from the
uplink training data. Therefore, the system operates under a
time division duplexing protocol and by leveraging the channel
reciprocity of the downlink and uplink channels.1
A. Uplink training phase
In the uplink training phase, 𝑁time slots are dedicated to
the channel estimation by utilizing 𝐾orthogonal pilot signals
{𝜓
𝜓
𝜓1, . . . , 𝜓
𝜓
𝜓𝐾}, each comprising 𝜏𝑝symbols with 𝜏𝑝≥𝐾. In
each time slot, one reflective element is chosen to only reflect
the training pilot signal without introducing any phase shift
while the other 𝑁−1elements do not reflect the incident
signal.2The pilot signal reflected by the 𝑛-th reflective element
Y𝑝
𝑛∈C𝑀×𝜏𝑝received at the BS is
Y𝑝
𝑛=h𝑛
𝐾
Õ
𝑡=1
√𝜌𝑡𝑔𝑡𝑛 𝜓
𝜓
𝜓𝐻
𝑡+N𝑝
𝑛,(1)
where 𝜌𝑡is the pilot power of each symbol and N𝑝
𝑛∈
C𝑀×𝜏𝑝is the additive noise whose elements are distributed
as CN(0, 𝜎2). The channel between the 𝑘-th user and the
BS that receives the pilot via the 𝑛-th reflective element is
estimated from
y𝑝
𝑘𝑛 =Y𝑝
𝑛𝜓
𝜓
𝜓𝑘
=√𝜌𝑘𝜏𝑝h𝑛𝑔𝑘𝑛 +˜
n𝑘𝑛 ,(2)
1Although channel reciprocity is a widely adopted assumption in wireless
communications, it is affected by some practical realization difficulties, which
include the non-symmetric characteristics of the RF front-end circuitry at the
receiver and transmitter.
2The activation of all IRS elements by using reflection patterns based on
orthogonal designs, e.g., reflection patterns based on the discrete Fourier
transform [10], [11], can provide higher antenna gains and reduce channel
estimation errors.
where ˜
n𝑘𝑛 =N𝑝
𝑛𝜓
𝜓
𝜓𝑘∼ CN(0, 𝜏𝑝𝜎2I𝑀). By utilizing least
squares estimation methods [12], the estimated channel be-
tween the 𝑘-th user and the BS assisted by the 𝑛-th reflective
element can be formulated as
˜
g𝑘𝑛 =1
√𝜌𝑘𝜏𝑝
y𝑝
𝑘𝑛
=h𝑛𝑔𝑘𝑛 +1
√𝜌𝑘𝜏𝑝
˜
n𝑘𝑛 .
(3)
Given the matrix e
G𝑘=[˜
g𝑘1, . . . , ˜
g𝑘 𝑁 ] ∈ C𝑀×𝑁, the exact
channel G𝑘between the BS and the 𝑘-th user is
G𝑘=e
G𝑘+E𝑘,(4)
where E𝑘∈C𝑀×𝑁is the channel estimation error matrix
whose individual elements are distributed as CN(0,𝜎2
𝜌𝑘𝜏𝑝). In
the sequel, we use the channel estimates in (3) to evaluate
the impact of channel uncertainty on the spectral efficiency of
each user.
B. Downlink data transmission
Let w𝑘∈C𝑀×1and 𝑥𝑘, with E[|𝑥𝑘|2]=1, be the active
beamforming vector and the data symbol for the 𝑘-th user,
respectively. In the downlink phase, each reflective element
introduces a reflection coefficient to assist the communication
between the BS and the users. Let 𝜃
𝜃
𝜃=[𝜃1, 𝜃2, . . . , 𝜃 𝑁]𝑇
be the coefficients, i.e., the passive beamforming vector, intro-
duced by the IRS where |𝜃𝑛| ≤ 1and arg(𝜃𝑛) ∈ [−𝜋, 𝜋)are
the amplitude and the phase shift, respectively, ∀𝑛=1, . . . , 𝑁
[7].3The signal received by the 𝑘-th user can be written as
𝑦𝑘=g𝐻
𝑘diag(𝜃
𝜃
𝜃)𝐻H𝐻
𝐾
Õ
𝑡=1
w𝑡𝑥𝑡+𝑛𝑘,(5)
where 𝑛𝑘∼ CN(0, 𝜎2)is the additive noise at the 𝑘-th user.
Denoting G𝐻
𝑘=diag(g∗
𝑘)H𝐻∈C𝑁×𝑀, we can rewrite (5) as
𝑦𝑘=𝜃
𝜃
𝜃𝐻G𝐻
𝑘
𝐾
Õ
𝑡=1
w𝑡𝑥𝑡+𝑛𝑘.(6)
Using (4) and (6), the exact SINR at the 𝑘-th user, which
is denoted by Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃), can be written as4
Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)=|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑘|2
𝐾
Í
𝑡=1,𝑡 ≠𝑘|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑡|2+𝜎2
𝑘
,(7)
where {w𝑘}={w1,w2, . . . , w𝐾}is the set of active beam-
forming vectors.
3In this letter, the amplitude and phase of the reflection coefficient of each
reflecting element of the IRS can be independently optimized. Depending
on the practical implementation of the reflecting elements, it may not be
possible to optimize the amplitude and the phase of each reflecting element
independently of each other. This is shown in e.g., [7] for a specific
implementation of the reflecting elements of the IRS. The case study in [7]
constitutes a valuable and promising generalization of the problem formulation
analyzed in this letter, which is postponed to future research due to space
limitations.
4In (7), the exact SINR is considered. This implies that the estimation error
E𝐻
𝑘appears in both the numerator and denominator of (7). In [13], on the
other hand, the estimation error E𝐻
𝑘is regarded as an additional interference
term, and, therefore, it appears in the denominator of the corresponding SINR.

3
III. PROP OS ED OPTIMIZATION PROBLEM
We define the coverage probability of a user as the proba-
bility that its SINR is greater than a predefined threshold. Our
aim is to jointly design the active/passive beamforming vectors
so as to minimize the total transmit power while guaranteeing
that the coverage probability of each user is greater than a
predefined target. This problem can be formulated as follows
minimize
{w𝑘},𝜃
𝜃
𝜃
𝐾
Õ
𝑘=1
w𝐻
𝑘w𝑘
subject to Pr (Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)≥𝛾𝑘)≥1−𝜌𝑘,∀𝑘,
|𝜃𝑛| ≤ 1,∀𝑛,
(8)
where 𝛾𝑘and 𝜌𝑘are the required minimum SINR to be in
coverage and the predefined outage target, respectively, at the
𝑘-th user. Hereafter, unless otherwise stated, we assume 𝑘∈
{1, . . . , 𝐾 }and 𝑛∈ {1, . . . , 𝑁 }. Problem (8) is not convex
due to the robust probabilistic SINR constraints. To proceed
further, using (7) we recast the event Γ𝑘({w𝑘}, 𝜃
𝜃
𝜃)≥𝛾𝑘as
|𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑘|2
𝛾𝑘−
𝐾
Õ
𝑡=1,
𝑡≠𝑘𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘w𝑡
2
≥𝜎2
𝑘.(9)
Defining e𝑘=vec (E𝑘),K𝑘=𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻𝑇⊗B𝑘,W𝑘=w𝑘w𝐻
𝑘and
B𝑘=W𝑘
𝛾𝑘−Í𝐾
𝑡=1,𝑡 ≠𝑘W𝑡, we can rewrite (9) as
𝜃
𝜃
𝜃𝐻e
G𝐻
𝑘+E𝐻
𝑘B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃≥𝜎2
𝑘,(10)
⇔tr e
G𝐻
𝑘+E𝐻
𝑘B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻≥𝜎2
𝑘,(11)
⇔vec e
G𝑘+E𝑘𝐻
vec B𝑘e
G𝑘+E𝑘𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻≥𝜎2
𝑘,(12)
⇔vec e
G𝑘+E𝑘𝐻
K𝑘vec e
G𝑘+E𝑘≥𝜎2
𝑘,(13)
⇔e𝐻
𝑘K𝑘e𝑘+2Re ne𝐻
𝑘K𝑘vec e
G𝑘o≥𝑑𝑘.(14)
To obtain (11) from (10), the identity a𝐻Ba =Tr Baa𝐻is
used, and to obtain (13) from (12), the identity vec (AYB)=
B𝑇⊗Avec (Y)is used. By defining the following function
that accounts for the channel estimation error of the 𝑘-th user
𝑓(e𝑘)=e𝐻
𝑘K𝑘e𝑘+2Re ne𝐻
𝑘K𝑘vec e
G𝑘o,(15)
problem (8) can be equivalently formulated as
minimize
{W𝑘},𝜃
𝜃
𝜃tr 𝐾
Õ
𝑘=1
W𝑘!
subject to Pr (𝑓(e𝑘)≥𝑑𝑘)≥1−𝜌𝑘,∀𝑘,
rank(W𝑘)=1,∀𝑘 ,
|𝜃𝑛| ≤ 1,∀𝑛.
(16)
In (16), we have introduced the set of matrices {W𝑘}=
{W1,W2, . . . , W𝐾}and 𝑑𝑘=𝜎2
𝑘−vec e
G𝑘𝐻
K𝑘vec e
G𝑘.
We note that problem (16) is still non-convex due to the
inherent non-convexity of the probabilistic SINR and the rank
constraints. We first handle the probabilistic SINR constraints
by introducing the following lemma.
Lemma 1. For any 𝜂𝑘>0, the statement below holds true:
Pr (𝑓(e𝑘) ≥ Υ(𝜂𝑘))≥1−𝑒−𝜂𝑘,(17)
where Υ(𝜂𝑘)is defined as
Υ(𝜂𝑘)=−p2𝜂𝑘s𝜎4
𝜌2
𝑘𝜏2
𝑝kKkk2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2
+𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)−𝜂𝑘𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜆+(K𝑘)
(18)
and 𝜆+(K𝑘)=min{𝜆min (K𝑘),0}where 𝜆min (K𝑘)is the
minimum eigenvalue of K𝑘.
Proof. The proof is obtained by applying the Bernstein-
type inequality [9] to the Gaussian random variable e𝑘∼
CN 0,𝜎2
𝜌𝑘𝜏𝑝I𝑀 𝑁 with the positive definite matrix 𝜎2
𝜌𝑘𝜏𝑝K𝑘
and the vector 𝜎
√𝜌𝑘𝜏𝑝K𝑘vec(e
G𝑘)in a standard quadratic
form. 
By setting 𝜂𝑘=−ln 𝜌𝑘and using Lemma 1, the probabilis-
tic constraint in (16) can be rewritten as
−p2 ln(1/𝜌𝑘)s𝜎4
𝜌2
𝑘𝜏2
𝑝kK𝑘k2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2
+𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)−ln(1/𝜌𝑘)𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜆+(K𝑘) ≥ 𝑑𝑘.
(19)
We introduce two auxiliary variables 𝜇𝑘and 𝜈𝑘to equivalently
cast (19) as the following series of inequalities
𝜎2
𝜌𝑘𝜏𝑝
tr (K𝑘)−p2 ln(1/𝜌𝑘)𝜇𝑘−ln(1/𝜌𝑘)𝜎4
𝜌2
𝑘𝜏2
𝑝
𝜈𝑘≥𝑑𝑘,(20)
s𝜎4
𝜌2
𝑘𝜏2
𝑝kKkk2
𝐹+2𝜎2
𝜌𝑘𝜏𝑝kK𝑘vec(e
G𝑘)k2≤𝜇𝑘,(21)
𝜈𝑘I𝑀 𝑁 +K𝑘0,(22)
𝜈𝑘≥0.(23)
While the constraints (20), (22), and (23) adhere to the
standard form of a semi-definite program with respect to
K𝑘, we can reformulate (21) in a standard second-order-cone
(SOC) constraint as

√2𝜎
√𝜌𝑘𝜏𝑝K𝑘vec e
G𝑘
𝜎2
𝜌𝑘𝜏𝑝vec (K𝑘)≤𝜇𝑘.(24)
To proceed further, we introduce a new variable 𝚯=𝜃
𝜃
𝜃𝜃
𝜃
𝜃𝐻and
recast (16) as
minimize
{W𝑘},𝚯,{𝜇𝑘},{𝜈𝑘}tr 𝐾
Õ
𝑘=1
W𝑘!
subject to (20),(22),(23),(24),∀𝑘,
W𝑘0,∀𝑘,
rank(W𝑘)=1,∀𝑘 ,
diag(𝚯)  1𝑁,
𝚯0,
rank(𝚯)=1.
(25)

4
By analyzing the constraints (20), (22) and (24), we observe
that they are non-convex with respect to {W𝑘}and 𝚯. These
constraints, in fact, depend on K𝑘, which is a function of {W𝑘}
and 𝚯. To overcome this issue, we use alternating optimization
so as to obtain a fixed-point solution of problem (25). In partic-
ular, starting from an initial value of the reflection coefficients
Θ
Θ
Θ(0)by replacing W𝑘,𝜇𝑘, and 𝜈𝑘, respectively, with W(𝑖)
𝑘,
𝜇(𝑖)
𝑘, and 𝜈(𝑖)
𝑘,∀𝑘, we solve the following subproblem during
the 𝑖-th iteration
minimize
{W(𝑖)
𝑘},{𝜇(𝑖)
𝑘},{𝜈(𝑖)
𝑘}
tr 𝐾
Õ
𝑘=1
W(𝑖)
𝑘!
subject to (20),(22),(23),(24),∀𝑘,
W(𝑖)
𝑘0,∀𝑘,
rank(W(𝑖)
𝑘)=1,∀𝑘.
(26)
Problem (26) is still non-convex due to the rank-one constraint
on W(𝑖)
𝑘. However, problem (26) has a structure similar to that
in [14]. Hence, we can exploit the same methodology as in
[14, Theorem 2] to show that its optimal solution is rank-
one if problem (26) is feasible. Consequently, problem (26) is
equivalent to the following LMI form
minimize
{W(𝑖)
𝑘},{𝜇(𝑖)
𝑘},{𝜈(𝑖)
𝑘}
tr 𝐾
Õ
𝑘=1
W(𝑖)
𝑘!
subject to (20),(22),(23),(24),∀𝑘,
W(𝑖)
𝑘0,∀𝑘,
(27)
which yields the optimal solution {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}and {𝜈(𝑖)
𝑘}for
a given matrix Θ
Θ
Θ(𝑖−1). Since (27) is a semidefinite program,
its optimal solution is obtained in polynomial time by using
general purpose interior-point toolboxes, e.g., CVX [15]. After
obtaining {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}and {𝜈(𝑖)
𝑘}from the solution of (27),
the matrix Θ
Θ
Θ(𝑖)is attained by solving the following feasibility
problem
find 𝚯(𝑖)
subject to (20),(22),(24),∀𝑘,
diag(𝚯(𝑖))  1𝑁,
𝚯(𝑖)0,
rank 𝚯(𝑖)=1.
(28)
Problem (28) is non-convex due to the rank-one constraint on
𝚯(𝑖). To tackle this problem, we propose to solve the following
problem
minimize
𝚯(𝑖)tr(𝚯(𝑖))
subject to (20),(22),(24),∀𝑘,
diag(𝚯(𝑖))  1𝑁,
𝚯(𝑖)0,
(29)
whose global optimum can be obtained by using CVX.
Exploiting the same methodology as in [14, Theorem 2],
we can show that problem (29) yields a rank-one optimal
solution. Therefore, we can conclude that problem (29) is an
approximation of problem (28), i.e., every feasible solution of
Algorithm 1 Alternating optimization to solve (25)
Input: Channel estimate matrices e
G𝑘,∀𝑘; Initial phase shift
coefficients 𝚯(0); Tolerance value 𝛿; Set 𝑖=0and initial the
cost function 𝐶(0)=0.
1. 𝑖=𝑖+1; Update {W(𝑖)
𝑘},{𝜇(𝑖)
𝑘}, and {𝜈(𝑖)
𝑘}for all 𝑘 ,
by solving problem (27) where every 𝜃(𝑖−1)
𝑛is computed
from the previous iteration.
2. Update Θ
Θ
Θ(𝑖)
𝑘by solving problem (29) where {W(𝑖)
𝑘},
{𝜇(𝑖)
𝑘}, and {𝜈(𝑖)
𝑘}are obtained from Step 1.
3. Compute the cost function 𝐶(𝑖)=tr Í𝐾
𝑘=1W(𝑖)
𝑘.
4. Check the stopping criterion: If 𝐶(𝑖)−𝐶(𝑖−1)≤𝛿→
Stop. Otherwise, repeat Steps 1−3.
Output: The fixed point solution: {W(𝑖)
𝑘}and Θ
Θ
Θ(𝑖).
(29) is also feasible for (28), see e.g. [14, Section III.B] and
references therein.
By direct inspection, subproblems (27) and (29) are convex
and their feasible domains are convex sets. Therefore, the pro-
posed iterative algorithm converges to a fixed-point solution
[16] when the subproblems (27) and (29) are feasible. The
proposed alternating optimization method is summarized in
Algorithm 1.5
Finally, the active/passive beamforming vectors w𝑘and 𝜃
𝜃
𝜃
are, respectively, as follows q𝜆(𝑤)
𝑘z(𝑤)
𝑘and √𝜆(𝜃)z(𝜃), where
𝜆(𝑤)
𝑘and 𝜆(𝜃)are the non-zero eigenvalues, and z(𝑤)
𝑘and z(𝜃)
are the corresponding eigenvectors of the rank-one fixed-point
solution W(𝑖)
𝑘and Θ
Θ
Θ(𝑖), respectively [14, Section IV.A].
IV. NUMERICAL RES ULT S
We consider a MISO system in the absence of light-of-
sight so that there is no direct path from the BS to the users.
The 3GPP Urban Micro channel model is used [17]. In the
coverage area, the users are randomly distributed, but the
minimum distance to the IRS is 10 m. The distance between
the BS and the IRS is 80 m. The large-scale fading coefficient
of the point-to-point link between the BS and the IRS is
1. There are 15 elements at the IRS. The noise variance is
−96 dBm. The bandwidth is 10 MHz. The large-scale fading
coefficient between the user 𝑘and the IRS is defined as
𝛽𝑘[dB]=−15.1−26 log10 (𝑓𝑐) − 37.6 log10 (𝑑𝑘/1m),(30)
where 𝑓𝑐=3GHz is the carrier frequency, 𝑑𝑘is the distance
between the user 𝑘and the IRS (𝑑𝑘≥10 m). The SINR
requirement of each user is 4dB and the outage probability
is 0.1.
Fig. 1 illustrates the convergence of the proposed algorithm
as a function of the number of users. We observe that Algo-
rithm 1 converges to a fixed-point solution after less than 5
iterations in all tested cases. These numerical results confirm
the statement about the convergence of the proposed algorithm.
5Although global optimality can be obtained by solving subproblems (27)
and (29) in each iteration, the global optimal solution of the original problem
(25) may not be attained due to the inherent non-convexity of (25). In fact, the
proposed algorithm yields a suboptimal solution to the original non-convex
problem.

5
12345
0
2
4
6
8
10
12
14
Fig. 1: Convergence of Algorithm 1 for a different number of
users.
K=2
K=4
Fig. 2: Transmit power consumption per user [mW].
Fig. 2 illustrates the transmit power per user of the proposed
approach, i.e., Algorithm 1, which is denoted as “Robust
probabilistic-constrained”, and the benchmark method in [8],
which is denoted as “Robust beamforming design”. It can be
observed that the proposed approach consumes 66% and 28%
less power than the benchmark method in [8] when the system
serves 2users and 4users, respectively. This is due to the fact
that the benchmark method in [8] allocates extra resources to
protect rarely occurring worst-case events while the proposed
approach allows the QoS constraints to be violated with some
non-zero probabilities.
Due to the increase of the mutual interference, Fig. 2
shows that the system must allocate more power to each user
when the number of coexisting users increases. For example,
the proposed approach requires approximately 0.06 mW in
order to guarantee the SINR requirements with an outage
probability of 0.1when 2users are in the coverage area. On
the other hand, the power allocated to each user increases up
to 0.14 mW if 4users need to be served.
V. CONCLUSION
We have formulated and solved a robust probabilistic-
constrained optimization problem for IRS-aided MISO com-
munication systems in order to tackle imperfect estimates of
the CSI. The optimal beamforming vectors at the BS and the
reflecting elements at the IRS are iteratively computed via
a converging alternating optimization algorithm. Numerical
results reveal a fast convergent behavior of the proposed
algorithm, i.e., within a few iterations. The results confirm
the superior performance of the proposed approach compared
with a benchmark method.
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