Intelligent Reflecting Surface: A Programmable Wireless Environment for Physical Layer Security

# Intelligent Reflecting Surface: A Programmable Wireless Environment for Physical Layer Security

Jie Chen, , Ying-Chang Liang, , Yiyang Pei, ,
and Huayan Guo,
This work was supported in part by the National Natural Science Foundation of China under Grants 61631005, U1801261, and 61571100. J. Chen and H. Guo are with the National Key Laboratory of Science and Technology on Communications, and Center for Intelligent Networking and Communications (CINC), University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mails: chenjie.ay@gmail.com and guohuayan@uestc.edu.cn). Y.-C. Liang is with the Center for Intelligent Networking and Communications (CINC), University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mail:liangyc@ieee.org). Y. Pei is with the Singapore Institute of Technology, Infocomm Technology Cluster, Singapore 138683 (e-mail: yiyang.pei@singaporetech.edu.sg).
###### Abstract

In this paper, we introduce an intelligent reflecting surface (IRS) to provide a programmable wireless environment for physical layer security. By adjusting the reflecting coefficients, the IRS can change the attenuation and scattering of the incident electromagnetic wave so that it can propagate in a desired way toward the intended receiver. Specifically, we consider a downlink multiple-input single-output (MISO) broadcast system where the base station (BS) transmits independent data streams to multiple legitimate receivers and keeps them secret from multiple eavesdroppers. By jointly optimizing the beamformers at the BS and reflecting coefficients at the IRS, we formulate a minimum-secrecy-rate maximization problem under various practical constraints on the reflecting coefficients. The constraints capture the scenarios of both continuous and discrete reflecting coefficients of the reflecting elements. Due to the non-convexity of the formulated problem, we propose an efficient algorithm based on the alternating optimization and the path-following algorithm to solve it in an iterative manner. Besides, we show that the proposed algorithm can converge to a local (global) optimum. Furthermore, we develop two suboptimal algorithms with some forms of closed-form solutions to reduce the computational complexity. Finally, the simulation results validate the advantages of the introduced IRS and the effectiveness of the proposed algorithms.

Intelligent reflecting surface, programmable wireless environment, physical layer security, beamforming.

## I Introduction

A variety of wireless technologies have been proposed to enhance the spectrum- and energy-efficiency due to the tremendous growth in the number of communication devices, such as multiple-input multiple-output (MIMO)[1], cooperative communications [2], cognitive radio (CR) [3] and so on. However, these techniques only focus on the signal processing at the transceiver to adapt the changes of the wireless environment, but cannot eliminate the negative effects caused by the uncontrollable electromagnetic wave propagation environment [4, 5].

Recently, intelligent reflecting surface (IRS) has been proposed as a promising technique due to its capability to achieve high spectrum-/energy-efficiency through controlling the wireless propagation environment [6]. Specifically, IRS is a uniform planar array consisting of a large number of composite material elements, each of which can adjust the reflecting coefficients (i.e., phase or amplitude) of the incident electromagnetic wave and reflect it passively. Hence, by smartly adjusting the reflecting coefficients with a preprogrammed controller, the IRS can change the attenuation and scattering of the incident electromagnetic wave so that it can propagate in the desired way before reaching the intended receiver, which is called as programmable and controllable wireless environment. This also inspires us to design the communication systems by jointly considering the signal processing at the transceiver and the optimization of the electromagnetic wave propagation in the wireless environment.

Compared with the existing related techniques, i.e., traditional reflecting surfaces [7], amplify-and-forward (AF) relay [8], active intelligent surface [9], and backscatter communication [10, 11, 12], IRS has the following advantages. Firstly, IRS can reconfigure the reflecting coefficients in real time thanks to the recent breakthrough on micro-electrical-mechanical systems (MEMS) and composite material [6, 5] while the traditional reflecting surface only has fixed reflecting coefficients. Secondly, IRS is a green and energy-efficient technique which reflects the incident signal passively without additional energy consumption while the AF relay and the active intelligent surface require active radio frequency (RF) components. Thirdly, although both the IRS and the backscatter communication make use of passive communications, IRS can be equipped with a large number of reflecting elements while backscatter devices are usually equipped with a single/few antenna(s) due to the limitations of complexity and cost [13]. Besides, IRS only attempts to assist the transmission of the signals between the intended transmitter and receiver pair with no intention for its own information transmission while backscatter communication needs to support the information transmission of the backscatter device [14, 15].

Due to the significant advantages, the IRS has been introduced into various wireless communication systems. Specifically, [16, 17, 18, 19, 20] consider a downlink single user multiple-input single-output (MISO) system assisted by the IRS. In [16], both centralized and distributed algorithms were developed to maximize the signal-to-noise ratio (SNR) of the desired signals considering perfect channel state information (CSI). Then, in [17], the effect of the reflecting coefficients on the ergodic capacity was investigated by considering statistical CSI. Moreover, since achieving continuous reflecting coefficients on the reflecting elements is costly in practice due to the hardware limitation, the SNR maximization problem and transmitter power minimization problem were studied in [18, 19, 20] by considering discrete reflecting coefficients on the reflecting elements. As for a downlink multi-user MISO system [21, 22, 23], the spectrum-/energy-efficiency problem under the individual signal-to-interference-plus-noise ratio (SINR) constraints was investigated in [21] and [22] considering continuous or discrete reflecting coefficients on the reflecting elements. In addition, the minimum-SINR maximization problem was formulated in [23] by considering the two cases where the channel matrix between the transmitter and the IRS is of rank-one and of full-rank.

Furthermore, physical layer security is a fundamental issue in wireless communications [24]. The basic wiretap channel introduced by Wyner [25] consists of one transmitter, one legitimate receiver, and one eavesdropper. Then, the basic wiretap channel has been extended to broadcast channels [26], Gaussian channels [27], compound wiretap channels [28], and so on. It is worth noting that, in order to ensure secret communications, the transmission rate in the wiretap channel should be lower than the secrecy capacity of the channel. Thus, MIMO beamforming techniques were further introduced to improve the secrecy capacity (improving SNR of legitimate receivers and suppressing SNR of eavesdroppers) [29, 30, 31, 32]. Specifically, both power minimization and secrecy rate maximization were studied in [30] in a single user/eavesdropper MIMO systems considering both perfect and imperfect CSI. Then, the minimum-secrecy-rate of a single-cell multi-user MISO system was studied in [31] with a minimum harvested energy constraint, and it was further extended to a multi-cell network in [32].

However, consider the special case when the legitimate receivers and the eavesdroppers are in the same directions to the transmitter. In this case, the channel responses of the legitimate receivers will be highly correlated with those of the eavesdroppers. The beamformers proposed in [29, 30, 31, 32] to maximize the SNR of legitimate receivers will also maximize the SNR of eavesdroppers. Hence, it is intractable to guarantee the secret communications with the use of beamforming only at the transceivers. Hence, we want to explore the use of the IRS to provide additional communication links so as to increase the SNR at the legitimate receivers while suppressing the SNR at the eavesdroppers. Hopefully, this will create an effect as if the confidential data streams can bypass the eavesdroppers and reach the legitimate receivers, as shown in Fig. 1, and thus the secrecy rate will be improved.

Motivated by the above reasons, in this paper, we study a programmable wireless environment for physical layer security to achieve high-efficiency secret communication. Specifically, we consider a downlink MISO broadcast system where the base station (BS) transmits multiple independent confidential data streams to each legitimate receivers and keeps them secret from the eavesdroppers through the assistance of the IRS. The contributions of the paper are summarized as follows:

• To the best of our knowledge, this is the first work to explore the use of the IRS to enhance the physical layer secret communication. Particularly, we jointly optimize the beamformers at the BS and the reflecting coefficients at the IRS to maximize the minimum-secrecy-rate under various practical constraints on the reflection coefficients. The constraints capture both the continuous and discrete reflecting coefficients of the reflecting elements on the IRS. However, the objective function is not jointly concave with respect to both the beamformers and the reflecting coefficients, and even worse, they are coupled together. Hence, the formulated problem is non-convex, which is hard to solve and may require high complexity to obtain the optimal solutions.

• We solve the formulated problem efficiently in an iterative manner by developing alternating optimization based path-following algorithm [33, 34]. Specifically, we use the path-following algorithm to handle the non-concavity of the objective function and apply the alternating optimization to deal with the coupled optimization variables. Besides, we prove that the proposed algorithm is guaranteed to converge to a local (global) optimum and the corresponding solution will converge to a Karush-Kuhn-Tucker (KKT) point finally.

• To further reduce the computational complexity, we develop two suboptimal algorithms to solve the formulated problem for two cases. For the first case with one legitimate receiver and one eavesdropper, we develop an alternating optimization method to solve the formulated in an iterative manner, but in each iteration we provide the closed-form solutions, which leads the algorithm to be low complexity. For the second case with multiple legitimate receivers and eavesdroppers, we develop a heuristic closed-form solution based on zero-forcing (ZF) beamforming, which further reduces the computational complexity.

• Finally, the simulation results validate the advantages of the introduced IRS and also show the effectiveness of the proposed algorithms.

The rest of this paper is organized as follows: Section II introduces the system model of the downlink MISO broadcast system with multiple eavesdroppers. Section III formulates the minimum-secrecy-rate maximization problem. Section IV develops an efficient algorithm to solve the formulated problem and Section V provides two low-complexity suboptimal algorithms to solve it in two cases, respectively. Finally, Section VII concludes the paper.

The notations used in this paper are listed as follows. The scalar, vector, and matrix are lowercase, bold lowercase, and bold uppercase, i.e., , , and , respectively. , , , , and denote transpose, conjugate transpose, trace, expectation, and real dimension, respectively. denotes the distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean and variance . and denote the space of complex/real matrices. is the identify matrix, , and .

## Ii System Model

As shown in Fig. 1, we consider a programmable downlink MISO broadcast system which consists of one BS, one IRS, legitimate receivers, denoted as , and active eavesdroppers, denoted as . The BS and the IRS are equipped with antennas and reflecting elements, respectively, while the legitimate receivers and eavesdroppers are all equipped with a single antenna each. The BS sends independent confidential data streams with one stream for each of the legitimate receivers over the same frequency band, simultaneously. At the same time, the unauthorized eavesdroppers are trying to eavesdrop any of the data streams, independently.

Consider the special case when the legitimate receivers and the eavesdroppers are in the same directions to the BS. In this case, the channel responses of the legitimate receivers will be highly correlated with those of the eavesdroppers. As aforementioned, it is intractable to guarantee the secret communications with the use of beamforming only at the transceivers. Hence, we want to explore the use of the IRS to provide additional communication links so as to increase the SNR at the legitimate receivers while suppressing the SNR at the eavesdroppers. Hopefully, this will create an effect as if the confidential data streams can bypass the eavesdroppers and reach the legitimate receivers, and thus the secrecy rate will be improved. In this paper, we are interested in obtaining the performance limit of such a system. Hence, similarly to [16] and [21], we assume that the CSI of all the channels are perfectly known at the BS. In practical systems where such CSI cannot be obtained perfectly, the results derived in this paper can be considered as the performance upper bound. Note that the optimization (in terms of beamformers and reflecting coefficients) of the system to be presented in the subsequent sections is done at the BS and that the optimized reflecting coefficients are transmitted to the IRS to reconfigure the corresponding reflecting elements accordingly.

### Ii-a Channel Model

The baseband equivalent channel responses from the BS to the IRS, from the BS to , from the BS to , from the IRS to , and from the IRS to are denoted by , , , , and , respectively, with and . Specifically, without loss of generality, we adopt a Rician fading channel model, which consists of LoS and non-LoS (NLoS) components, i.e.,

 \boldmath{h}= ⎷κ\boldmath{h% }κ\boldmath{h}+1\boldmath{h}LoS+√1κ\boldmath{h}+1\boldmath{h% }NLoS, (1)

with , where , , and are the Rician factor, LoS components, and NLoS components of channel , respectively. The NLoS components are i.i.d. complex Gaussian distributed with zero mean and unit variance. We define a vector , where is the antenna element separation, is the carrier wavelength, is the dimension of the vector and is the angle, which can be interpreted as either angle of departure (AoD) or angle of arrival (AoA) depending on the context. We set for simplicity. Hence, the LoS components in (1) can be modeled as

 \boldmath{h}LoSd,k =\boldmath{a}M(ϑd,k)and\boldmath{h}LoSr,k=\boldmath{a}L(ϑr,k),for1≤k≤K, (2) \boldmath{g}LoSd,n =\boldmath{a}M(~ϑd,n)and\boldmath{h}LoSr,n=\boldmath{a}L(~ϑr,n),for1≤n≤N, (3) \boldmath{F}LoS =\boldmath{a}L(ϑAoA)\boldmath{a}HM(ϑAoA), (4)

where , , , are the AoA or AoD of a signal from the BS to , from the IRS to , from the BS to , and from the IRS to , respectively. and are the AoD from the BS and the AoA to the IRS, respectively.

### Ii-B Reflecting Coefficient Model

The reflecting coefficient channel of the IRS [16] is given by with and for , where denotes a diagonal matrix whose diagonal elements are given by the corresponding vector and denotes the set of reflecting coefficients of the IRS. In this paper, we consider the following three different sets of reflecting coefficients, which lead to three different constraints for the reflecting coefficients.

• Continuous Reflecting Coefficients: In this scenario, we further consider two detailed setups with the optimized or constant amplitude. Specifically, the reflecting coefficient set for the optimized amplitude with continuous phase-shift is denoted by

 (5)

and the reflecting coefficient set for the constant amplitude with continuous phase-shift is denoted by

 \boldmath{Φ}2={θn∣∣θn=ejφn,φn∈[0,2π)}. (6)
• Discrete Reflecting Coefficients: In this scenario, the reflecting coefficient set has constant amplitude and discrete phase-shift, which is given by

 \boldmath{Φ}3={θn∣∣θn=ejφn,φn∈{0,2πQ,⋯,2π(Q−1)Q}}, (7)

where is the number of reflecting coefficient values of the reflecting elements on the IRS.

Note that, it is costly in practice to achieve continuous reflecting coefficient on the reflecting elements due to the hardware limitation. Hence, applying the discrete reflecting coefficient on the reflecting elements, i.e., , is more practical than applying the continuous reflecting coefficients, i.e., and . But, it is also important to investigate the system performance with and since it serves as the upper bound to that with .

### Ii-C Signal Model

Let be the confidential message dedicated to . It is assumed that all messages transmitted are CSCG, i.e., for . Then, the signal transmitted from the BS can be expressed as

 \boldmath{x}=∑Kk=1\boldmath% {w}ksk, (8)

where is the downlink beamforming vector for . The received signals at and eavesdropped by can be expressed as

 yBk=[\boldmath{h}Hr,k% \boldmath{Θ}\boldmath{F}+\boldmath{h}Hd,k]K∑i=1\boldmath{w}ixi+uBk,1≤k≤K, (9) yEn=[\boldmath{g}Hr,n% \boldmath{Θ}\boldmath{F}+\boldmath{g}Hd,n]K∑i=1\boldmath{w}ixi+uEn,1≤n≤N, (10)

respectively, where and are denoted as the received noises at and , respectively. It is assumed that all noises are Guassian distributed with zero mean, i.e., and , respectively.

According to (9), the achievable transmission rate of the -th confidential message received at can be written as

 RBk=ln⎛⎜ ⎜⎝1+∣∣(\boldmath{h}% Hr,k\boldmath{Θ}\boldmath{F}+% \boldmath{h}Hd,k)\boldmath{w}k∣∣2∑Ki≠k∣∣(\boldmath{h}Hr,k% \boldmath{Θ}\boldmath{F}+\boldmath{h}Hd,k)\boldmath{w}i∣∣2+σ2k⎞⎟ ⎟⎠. (11)

According to (10), if attempts to eavesdrop the -th confidential message, the achievable wiretapped rate of the -th message received at can be written as

 REk,n=ln⎛⎜ ⎜⎝1+∣∣(\boldmath{g}Hr,n\boldmath{Θ}\boldmath{F}+% \boldmath{g}Hd,n)\boldmath{w}k∣∣2∑Ki≠k∣∣(\boldmath{g}Hr,n% \boldmath{Θ}\boldmath{F}+\boldmath{g}Hd,n)\boldmath{w}i∣∣2+δ2n⎞⎟ ⎟⎠. (12)

Since each eavesdropper can eavesdrop any of the confidential messages, the achievable secrecy rate (in nats/sec/Hz) for transmitting to and keeping it confidential from all the eavesdroppers should be the minimum-secrecy-rate among and for , which is given by [32]

 Ck=min∀n{RBk−REk,n}. (13)

## Iii Problem Statement

### Iii-a Problem Formulation

In this paper, we attempt to jointly optimize the beamfroming vector, i.e., , and reflecting coefficients, i.e., , to maximize the minimum-secrecy-rate among all the legitimate receivers. Mathematically, the optimized problem can be generally formulated as

 (P1):max\boldmath{W},\boldmath{θ}min∀k Ck s.t. ∑Kk=1∥∥\boldmath{w}k∥∥2≤P, (14a) θl∈\boldmath{Φ},1≤l≤L, (14b)

where denotes the maximum transmit power at the BS and may be set as , , and , respectively.

### Iii-B Problem Transformation

(P1) is hard to solve due to the non-concave objective function. In order to find the solution of (P1) efficiently, we will transform it into the following equivalent formulation.

To begin with, denoting , and , we have

 ∣∣(\boldmath{h}Hr,k\boldmath{Θ}\boldmath{F}+\boldmath{h}Hd,k)% \boldmath{w}k∣∣2=∣∣\boldmath{v}H% \boldmath{H}k\boldmath{w}k∣∣2, (15) ∣∣(\boldmath{g}Hr,n\boldmath{Θ}\boldmath{F}+\boldmath{g}Hd,n)% \boldmath{w}k∣∣2=∣∣\boldmath{v}H% \boldmath{G}n\boldmath{w}k∣∣2, (16)

where .

Then, in (11) and in (12) can be rewritten as

 RBk=ln⎛⎜ ⎜⎝1+∣∣\boldmath{v}H\boldmath{H}k\boldmath{w}k∣∣2bk(\boldmath{W},\boldmath{v})⎞⎟ ⎟⎠Δ=fBk(\boldmath{W},% \boldmath{v}), (17) REk,n=ln⎛⎜ ⎜⎝1+∣∣\boldmath{v}H\boldmath{G}n\boldmath{w}k∣∣2qk,n(\boldmath{W},\boldmath{v})⎞⎟ ⎟⎠Δ=fEk,n(\boldmath{W},% \boldmath{v}), (18)

where and . Thus, it is straightforward to know that (P1) can be transformed into the following equivalent form:

 (P2):max\boldmath{W},\boldmath{v} R(\boldmath{W},\boldmath{v})Δ=min∀k,∀n{fBk(\boldmath{W},\boldmath{v})−fEk,n(\boldmath{W},\boldmath{v})} s.t. vl∈\boldmath{Φ},1≤l≤L,vL+1=1, (19a)

However, the transformed problem (P2) is still hard to solve since is not jointly concave with respect to and , and even worse, they are coupled together. In the next section, we will develop an iterative algorithm to solve (P2) efficiently.

## Iv Minimum-Secrecy-Rate Maximization

In this section, we will propose two techniques to jointly solve the above challenging problem. Firstly, we apply the path-following algorithm to handle the non-concavity of the objective function. Then, we apply the alternating optimization technique to deal with the coupled optimization variables. Finally, we analyze the convergence of the proposed algorithm.

### Iv-a Path-Following Algorithm Development

In this part, we will develop path-following iterative algorithm to solve (P2) with the non-concave objective function, i.e., . In particular, the basic idea of the path-following is to follow a solution path of a family of the approximated problems of (P2), i.e., , is approximated by a concave lower bound function, which is obtained by applying linearly interpolating between the non-convex term and , respectively. Specifically, the approximated problem has a local (global) optimal value and can be increased in each iteration, which finally leads to a local (global) optimal solution of (P2) [33].

To begin with, let denote the solution of (P2) in the -th iteration. Then, in order to find the concave lower bound function of to develop path-following algorithm, we can fist find the lower bound function of and the upper bound function of at . The details are given in the following lemma.

###### Lemma IV.1

The lower bound function of and the upper bound function of at in the -th iteration of path-following algorithm are given by

 fBk(\boldmath{W},\boldmath{v})≥fBk(\boldmath{W}(t),% \boldmath{v}(t))+2R{(% \boldmath{w}(t)k)H\boldmath{H}Hk% \boldmath{v}(t)(\boldmath{v}H\boldmath{H}k\boldmath{w}k)}bk(% \boldmath{W}(t),\boldmath{v}(t)) −∣∣ ∣∣(\boldmath{v}(t))H\boldmath{H}k\boldmath{w}(t)k∣∣ ∣∣2bk(\boldmath{W}(t),\boldmath{v}% (t))(bk(\boldmath{W}(t),% \boldmath{v}(t))+∣∣ ∣∣(\boldmath{v}(t))H\boldmath{H}k\boldmath{w}(t)k∣∣ ∣∣2) Δ=fBk(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{% v}(t)), (20) fEk,n(\boldmath{W},\boldmath{θ})≤fEk,n(\boldmath{W}(t),\boldmath{v}(t))+(1+∣∣ ∣∣(\boldmath{v}(t))H\boldmath{G}n% \boldmath{w}(t)k∣∣ ∣∣2qk,n(\boldmath{W}(t),\boldmath{v}(t)))−1 ×(∣∣∣\boldmath{v}H\boldmath{G}n\boldmath{w}k∣∣∣2qk,n(\boldmath{W},\boldmath{v})−∣∣ ∣∣(\boldmath{v}(t))H\boldmath{G}n% \boldmath{w}(t)k∣∣ ∣∣2qk,n(\boldmath{W}(t),\boldmath{v}(t))) ≤fEk,n(\boldmath{W}(t),\boldmath{v}(t))+(1+∣∣ ∣∣(% \boldmath{v}(t))H\boldmath{G}n\boldmath{w}(t)k∣∣ ∣∣2qk,n(\boldmath{W}(t),\boldmath{v}(t)))−1 ×(∣∣∣\boldmath{v}H\boldmath{G}n\boldmath{w}k∣∣∣2qk,n(\boldmath{W},\boldmath{v};\boldmath{W}% (t),\boldmath{v}(t))−∣∣ ∣∣(\boldmath{v}(t))H\boldmath{G}n% \boldmath{w}(t)k∣∣ ∣∣2qk,n(\boldmath{W}(t),\boldmath{v}(t))) Δ=fEk,n(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath% {v}(t)), (21)

where

 qk,n(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t))=δ2n+ ∑Ki≠kR{(% \boldmath{w}(t)i)H\boldmath{G}Hn% \boldmath{v}(t)(2\boldmath{v}H\boldmath{G}n\boldmath{w}i−(\boldmath{v}H)(t)\boldmath{G}n\boldmath{w}(t)i)}, (22)
###### Proof:

Please refer to Appendix -A. \qed

Then, from (20) and (21), we know the lower bound of is given by

 R(\boldmath{W},\boldmath{v})=min∀k,∀n{fBk(% \boldmath{W},\boldmath{v})−fEk,n(%\boldmath$W$,\boldmath{v})} ≥min∀k,∀n{fBk(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t))−fEk,n(% \boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t))} Δ=Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),% \boldmath{v}(t)). (23)

Note that, according to (20) and (21), the equality in (23) holds when and .

Thus, a family of the approximated problems of (P2) is given as follows:

 (P2−t):max\boldmath{% W},\boldmath{v} Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t)) s.t. (???)and(???).

However, is still a non-convex problem due to the following reasons:

• First, and are coupled in the terms of and , which makes the objective function not jointly concave with respect to .

• Second, it is straightforward to know that (19a) with is a convex set but a non-convex set with and .

In subsection IV-B, we will first develop alternating optimization method to deal with the coupled optimization variables in with , and then we will extend it to the scenarios with and , respectively.

### Iv-B Alternating Optimization with Continuous and Discrete Reflecting Coefficients

#### Iv-B1 The Solution of (P2) with \boldmath{Φ}=\boldmath{Φ}1

In this part, we develop the alternating optimization to solve when in constraint (19a), which leads constraint (19a) to be a convex set. Hence, the non-convexity of only stems from the coupled optimization variables.

In fact, although the objective function is non-concave due to the coupled and , in (20) is biconcave in and , i.e., is concave both in with fixed and in with fixed . Similarly, for the domain , the function in (21) is a biconvex function with respect to and , which leads to a biconvex function with respect to and . Hence, is a biconcave function in and .

Therefore, we know with has convex constraints and concave objective function in with fixed and in with fixed . Hence, we can apply the alternating optimization method to solve in an alternating manner efficiently. Specifically, the alternating algorithm decouples into the following two subproblems for the optimization of and , respectively,

 (P3−A):max% \boldmath{W} Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t)) s.t.

and

 (P3−B):max% \boldmath{v} Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t)) s.t. (???)with\boldmath{Φ}=% \boldmath{Φ}1.

Note that (P3-A) is an optimization subproblem for solving with a given and (P3-B) is an optimization subproblem for solving with a given .

As aforementioned, we know both and are convex optimization problems, which can be solved optimally and efficiently by using CVX [35]. Thus, problem with can be solved efficiently by alternately solving and in an iterative manner of path-following algorithm. In particular, the algorithm steps of the alternating optimization based path-following algorithm are summarized in Algorithm 1.

#### Iv-B2 The Solutions of (P2) with \boldmath{Φ}=\boldmath{Φ}2

In this part, we extend the above alternating optimization to solve when in constraint (19a), which leads constraint (19a) to be a non-convex set. To handle this non-convex constraint, we propose the following two methods:

• In the first method, we introduce a positive constant relaxation factor to reformulate with as the following optimization problem,

 (P4−t):max\boldmath{% W},\boldmath{v} Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t))+λL+1∑l=1|vl|2 s.t. (???)and(???)with% \boldmath{Φ}=\boldmath{Φ}1.

Note that the added nonnegative quadratic term attempts to force the inequality holds for , i.e., .

However, the objective of is to maximize the summation of concave and convex functions, which belongs to a non-convex problem. To further deal with this challenge, we use the first-order Taylor series expansion to approximates the convex function as an affine function [36]. Then, we iteratively solve the approximated convex optimization problem until the convergence is met. Specifically, the approximated problem is

 (P4−A):max% \boldmath{W},\boldmath{v} Rlb(\boldmath{W},\boldmath{v};\boldmath{W}(t),\boldmath{v}(t))