Robust Beamforming Design for Sum Secrecy Rate Optimization in MU-MISO Networks

This paper studies the beamforming design problem of a multi-user downlink network, assuming imperfect channel state information known to the base station. In this scenario, the base station is equipped with multiple antennas, and each user is wiretapped by a specific eavesdropper where each user or eavesdropper is equipped with one antenna. It is supposed that the base station employs transmit beamforming with a given requirement on sum transmitting power. The objective is to maximize the sum secrecy rate of the network. Due to the uncertainty of the channel, it is difficult to calculate the exact sum secrecy rate of the system. Thus, the maximum of lower bound of sum secrecy rate is considered. The optimization of the lower bound of sum secrecy rate still makes the considered beamforming design problem difficult to handle. To solve this problem, a beamforming design scheme is proposed to transform the original problem into a convex approximation problem, by employing semidefinite relaxation and first-order approximation technique based on Taylor expansion. Besides, with the advantage of low complexity, a zero-forcing based beamforming method is presented in the case that base station is able to nullify the eavesdroppers' rate. When the base station doesn't have the ability, user selection algorithm would be in use. Numerical results show that the former strategy achieves better performance than the latter one, which is mainly due to the ability of optimizing beamforming direction, and both outperform the signal-to-leakage-and-noise ratio based algorithm.

There are no comments yet.

Authors

• 13 publications
• 21 publications
• 9 publications
• 5 publications
• 45 publications
• Robust Directional Modulation Design for Secrecy Rate Maximization in Multi-User Networks

In this paper, based on directional modulation (DM), robust beamforming ...
08/23/2018 ∙ by Linqing Gui, et al. ∙ 0

• Robust beamforming and jamming for enhancing the physical layer security of full duplex radios

In this paper, we investigate the physical layer security of a full-dupl...
03/26/2019 ∙ by Zhengmin Kong, et al. ∙ 0

• Sum-Rate Maximization of Multicell MISO Networks with Limited Information Exchange

Although there have been extensive studies on transmit beamforming in mu...
04/10/2019 ∙ by Youjin Kim, et al. ∙ 0

• Secrecy Beamforming for SWIPT MISO Heterogeneous Cellular Networks

In this paper, we consider the secure transmission design for a multiple...
06/28/2018 ∙ by Hui Ma, et al. ∙ 0

• Robust Energy Efficient Beamforming in MISOME-SWIPT Systems With Proportional Secrecy Rate

The joint design of beamforming vector and artificial noise covariance m...
09/08/2018 ∙ by Yanjie Dong, et al. ∙ 0

• Updating Beamformers to Respond to Changes in Users

We consider a multi-user multiple-input single-output downlink system th...
03/11/2018 ∙ by Mostafa Medra, et al. ∙ 0

• Deep Learning Enabled Optimization of Downlink Beamforming Under Per-Antenna Power Constraints: Algorithms and Experimental Demonstration

This paper studies fast downlink beamforming algorithms using deep learn...
02/28/2020 ∙ by Juping Zhang, et al. ∙ 0

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

Although the initial concept about secrecy transmission can be traced back to the 1970s [1], wireless transmission issues concerning physical layer security rate have attracted considerable attention in recent years. Traditional communication methods based on High-level encryption can hardly be used to improve physical layer security rate in practical situation, such as WLAN and Ad-hoc network. In WLAN scenario, the unpredictable random access and leave of users would lead to difficulties for the establishment of an appropriate and reasonable high-level encryption protocol. In addition, in Ad-hoc networks, a complete data transmission could go through several hops and other users may participate in relaying data, which would decrease secrecy rate.

One basic idea on the physical layer security is artificial noise. It adds artificial noise to the transmission signal expecting that the artificial noise would provide more negative effects for eavesdroppers than legitimate users. In [2] and [3], the authors analyze the effect of artificial noise on enhancing secrecy capacity. The difference is that in [2] the transmitter generates the artificial noise, while in [3] an external node helps to accomplish this work. In paper [4], the secure transmission in multiuser and multiple-eavesdropper systems is investigated. Several strategies are illustrated to suppress the channel interference. Numerical results demonstrate that adding artificial noise to the transmission message would improve the system secrecy rate. In paper [5], the authors investigate secure communication between two multi-antenna nodes with an undetected eavesdropper. Unlike previous work, no information regarding the eavesdropper is available. Artificial interference is applied to mask the desired signal. The authors maximize the power available to hide the desired signal from a potential eavesdropper, while maintaining a prespecified signal-to-interference-plus-noise-ratio (SINR) at the desired receiver. The case of the presence of imperfect channel state information (CSI) is also studied.

Another category to protect the physical layer security is based on beamforming. In [6], a relay assisted system is studied with the assumption that the relay is unreliable. Precoding designs of the base station (BS) and relay are provided to maximize the secrecy capacity. In addition, [7] takes the two-way relay scenario into consideration. In this situation, eavesdroppers would utilize the received information obtained from two transmission slots to get the desired eavesdropped users’ information. Optimal precoding design strategy in this scenario is presented. In [8], a joint beamforming design of the source and relay based on quality-of-service (QoS) requirements with presence of channel uncertainty is investigated. [9] studies the secrecy rate in decode-and-forward relay scenario with finite-alphabet input. A power control scheme based on semidefinite programming is presented for the purpose of maximizing secrecy rate with finite-alphabet input. [10] investigates the relay-eavesdropper network with two models of imperfect knowledge of the eavesdropper s channel. The approximation of the ergodic secrecy rate is studied under the Rician fading channel model and the worst-case secrecy rate is considered under the deterministic uncertainty model. Under both models, the optimal rank-1, match-and-forward (MF), and zero-forcing (ZF) beamformers are developed. The effectiveness of the proposed relay beamformers are verified by the numerical results. Paper [11] provides precoding strategies in a coordinated multi-point (CoMP) transmission system.

Robust design of beamforming and artificial noise has been investigated in multiple-input-single-output (MISO) networks. In [12], the authors address the physical layer security in MISO communication systems. The transmission covariance matrices of the steering information and the artificial noise are investigated to maximize the worst-case secrecy rate in a resource-constrained system and to minimize the use of resources to ensure an average secrecy rate. [12] investigates a three node network including only one user. Paper [13] considers a multiuser MISO downlink system with the presence of passive eavesdroppers and potential eavesdroppers. The problem of minimizing the total transmit power takes into account artificial noise and energy signal generation for protecting the transmitted information against both considered types of eavesdroppers. The semi-definite programming (SDP) relaxation approach is adopted to obtain the optimal solution. Both [12] and [13] use deterministic model for modelling the CSI uncertainty. Paper [14] proposes a linear precoder for a multiuser MIMO system in which multiusers potentially act as eavesdroppers. The proposed precoder is based on regularized channel inversion with a regularization parameter and power allocation vector. Then, an extension of the algorithm by jointly optimizing the regularization parameter and the power allocation vector is presented to maximize the secrecy sum-rate. Robust beamforming design in relay systems has also been investigated. [15] investigates the non-robust and robust cases of joint optimization in bidirectional multi-user multi-relay MIMO systems. The authors mainly concentrate on the sum MSE criterion as well as maximum user’s MSE. In [16], the relaying robust beamforming for device-to-device communication with channel uncertainty is considered.

In our paper, beamforming design of sum secrecy rate (SSR) optimization is investigated under sum power constraint for multiuser MISO channel. To solve the original complicated and nonconvex problem, an efficient approximation algorithm based on Taylor expansion is developed for the purpose of near-optimal solutions. In addition, a beamforming design scheme based on ZF with low complexity is presented. Numerical results demonstrate the better performance of the former algorithm. Both of the algorithms are compared to the SLNR algorithm, which mainly minimizes the power leaking to other user s channel space, and proved to be better.

The remaining part of this paper is organized as follows. The system model and the sum power constrained beamforming problem are presented in Section II. The solution based on Taylor expansion of the approximation problem under the assumption of imperfect CSI is discussed in detail in Section III. In Section IV, the beamforming design scheme based on ZF at the BS is demonstrated. Simulation results are presented in Section V. Finally, the conclusions are drawn in Section VI.

Notation: In this paper, we use bold uppercase and lowercase letters to denote matrices and vectors, respectively. and denote the transpose and the conjugate transpose of a matrix or a vector, respectively. denotes the conjugate of a matrix or a vector which means . is the trace of a matrix. denotes the rank of a matrix. We use the expression if

is complex Gaussian distributed with mean

and variance

. denotes the Frobenius norm. represents the property of semidefinite. . denotes the number of all combinations of different elements chosen from elements.

Ii System Model and Problem Formulation

Ii-a System Model

In this paper, we will investigate a network incorporating one BS and 2 users, or equivalently user-eaves pairs. In the user-eaves pair, one legitimate user is wiretapped by an eavesdropper as shown in Fig. 1. It is presumed that the BS is equipped with antennas where , and each user or eavesdropper with a single antenna. It is supposed that the BS serves the

wiretapped users while each eavesdropper attempts to wiretap the legitimate user in the same user-eaves pair. It is assumed that the BS only knows imperfect CSI of each user and eavesdropper due to limited feedback or other reasons, and the channel estimation error is norm-bounded. The BS employs transmit beamforming to communicate with

users. Let denote the information signal sent for the -th user at time t, and let be the corresponding beamforming vector. The received signal at the -th wiretapped user is given by

 xi(t)=hTiwisi(t)+K∑k=1,k≠ihTiwksk(t)+ni(t), (1)

where denotes the channel vector between the BS and the -th user, and is additive Gaussian noise at the -th user satisfying . As seen from (1), each eavesdropped user suffers from the intracell interference in addition to the noise. It is presumed that all receivers employ single-user detection where the intracell interference is simply treated as background noise.

Under the assumption that each user is wiretapped by the specific eavesdropper, let the -th eavesdropper be the one wiretapping the -th user, which means the -th eavesdropper has the knowledge of . The received signal at the -th eavesdropper can be written as

 zi(t)=gTiwisi(t)+K∑k=1,k≠igTiwksk(t)+mi(t), (2)

where denotes the channel vector between the BS and the -th eavesdropper, and is additive Gaussian noise at the -th eavesdropper satisfying .

The instantaneous achievable rate of the -th user can be transformed into the following

 ri=log2⎛⎝1+|hTiwi|2∑k≠i|hTiwk|2+σ2i⎞⎠, (3)

and the rate achieved by the -th eavesdropper can be expressed as

 si=log2⎛⎝1+|gTiwi|2∑k≠i|gTiwk|2+ς2i⎞⎠. (4)

Notice that in this paper, the channel is supposed to be imperfect. Under the assumption that the channel estimation error is norm-bounded, we have

 hi=¯¯¯¯hi+△hi,∥△hi∥≤εh, (5) gi=¯¯¯gi+△gi,∥△gi∥≤εg, (6)

where and are the estimated channel of users and eavesdroppers, while and are the channel estimation errors, respectively. and are the bounds of the norm of the channel estimation error. It is presumed that .

Based on the expressions above, the SSR is defined as

 K∑i=1(ri−si). (7)

Notice that in this paper, sum power constraint should be satisfied which is

 K∑i=1∥wi∥2≤P. (8)

Ii-B Problem Formulation

The objective is to maximize the SSR with the constraint of sum transmitting power. Thus, the optimization problem can be formulated as

 maxwi∈CNt×1i=1,…,KK∑i=1[log2⎛⎝1+|hTiwi|2∑k≠i|hTiwk|2+σ2i⎞⎠ maxwi∈CNti=1,…,K−log2⎛⎝1+|gTiwi|2∑k≠i|gTiwk|2+ς2i⎞⎠], (9a) s.t. K∑i=1∥wi∥2≤P. (9b)

However, the BS only has the knowledge of the estimated channel and the bound of the norm of the channel estimation error, which would bring difficulties to the modeling of the SSR problem. To deal with this, the lower bound of the objective in (9) would be taken into consideration merely. Lemma 1 would be introduced first to get the lower bound of the objective.

Lemma 1.

: For the two problems in the following,

 max∥x∥≤σϕ(x)=Re(xHy), (10) min∥x∥≤σφ(x)=Re(xHy), (11)

where and are given parameters, their solutions can be expressed as

 ϕ(σ∥y∥y)=σ∥y∥, (12) φ(−σ∥y∥y)=−σ∥y∥. (13)
Proof.

Apparently, we have

 −|xHy|≤Re(xHy)≤|xHy|, (14)

where and are vectors. The Cauchy-Schwarz inequality states that for all vectors x and y of an inner product space, it is true that , where is the inner product. According to Cauchy-Schwarz inequality,

 |xHy|≤∥x∥∥y∥≤σ∥y∥. (15)

Then we have

 −σ∥y∥≤Re(xHy)≤σ∥y∥. (16)

The inequality holds with equality when and are linearly dependent.

It can be concluded that the upper bound of is at the point , and the lower bound of is at the point . ∎

Then, based on Lemma 1, the upper bound of is given by

 |hTiwi|2 =hTiwiwHih∗i =(¯¯¯¯hTi+△hTi)wiwHi(¯¯¯¯h∗i+△h∗i) =¯¯¯¯hTiwiwHi¯¯¯¯h∗i+2Re{△hTiwiwHi¯¯¯¯h∗i} ≤¯¯¯¯hTiwiwHi¯¯¯¯h∗i+2ε∥wiwHi¯¯¯¯h∗i∥. (17)

Similarly, the lower bound of can be expressed as

 |hTiwi|2 =¯¯¯¯hTiwiwHi¯¯¯¯h∗i+2Re{△hTiwiwHi¯¯¯¯h∗i} ≥¯¯¯¯hTiwiwHi¯¯¯¯h∗i−2ε∥wiwHi¯¯¯¯h∗i∥. (18)

Notice that in (II-B) and (II-B), the second order error terms is omitted since it is quite small compared with other terms. In (II-B), is usually small and is larger than 0. When is large enough, the lower bound, , may be smaller than 0, which has no practical meaning. Due to , the problem what is large enough is dependent on the channels and the beamformers. The sum of several terms makes it more difficult to analyses the effect of large . From the simulation results, we find that if , we may not be able to get the beamformers or the sum secrecy rate will be quite low. In this case, the channel estimation error is intolerant and our method may not be able to solve this problem.

Thus, the lower bound of the objective in (9) is as follows.

 K∑i=1[log2⎛⎜⎝∑Kk=1(¯¯¯¯hTiwkwHk¯h∗i−2ε∥wkwHk¯¯¯¯h∗i∥)+σ2i∑k≠i(¯¯¯¯hTiwkwHk¯¯¯¯h∗i+2ε∥wkwHk¯¯¯¯h∗i∥)+σ2i⎞⎟⎠ −log2⎛⎜⎝∑Kk=1(¯¯¯gTiwkwHk¯¯¯g∗i+2ε∥wkwHk¯¯¯g∗i∥)+ς2i∑k≠i(¯¯¯gTiwkwHk¯¯¯g∗i−2ε∥wkwHk¯¯¯g∗i∥)+ς2i⎞⎟⎠].

Problem (20) can be formulated as the following

 maxwi∈CNt×1i=1,…,K K∑i=1[log2⎛⎜⎝∑Kk=1(¯¯¯¯hTiwkwHk¯h∗i−2ε∥wkwHk¯¯¯¯h∗i∥)+σ2i∑k≠i(¯¯¯¯hTiwkwHk¯¯¯¯h∗i+2ε∥wkwHk¯¯¯¯h∗i∥)+σ2i⎞⎟⎠ −log2⎛⎜⎝∑Kk=1(¯¯¯gTiwkwHk¯¯¯g∗i+2ε∥wkwHk¯¯¯g∗i∥)+ς2i∑k≠i(¯¯¯gTiwkwHk¯¯¯g∗i−2ε∥wkwHk¯¯¯g∗i∥)+ς2i⎞⎟⎠], (20a) s.t. K∑i=1∥wi∥2≤P. (20b)

In problem (9), the perfect CSI is not known to the BS. The BS only has the knowledge of the estimated channel and the bound of the norm of the channel estimation error. Under this condition, problem (9) can’t be solved since the parameters of the real channels in problem (9) are missing. Then, the problem to maximize the lower bound of SSR is investigated and problem (20) is formulated based on the estimated channel. If the BS can obtain perfect CSI, that means the bound of the norm of the channel estimation error is set to zero. Then problem (20) would be the same as problem (9).

The upper bound (17) and the lower bound (18) are important in the formulation of problem (20). It is not easy to estimate the tightness of the upper bound and the lower bound. In the simulation results, the solutions are obtained. Then we compute the lower bound of SSR through (20a) and the practical SSR through (9a). A comparison between them can illustrate that the gap between the lower bound of SSR and the practical SSR is small.

The traditional leakage-based beamforming scheme [17] might be applicable to such situation, which mainly minimizes the power leaking to other user s channel space. The solution is given by , where . The norm of is adjusted according to the transmit power. Besides the signal-to-leakage-and-noise ratio (SLNR) scheme, our proposed algorithms are illustrated in Section III and Section IV. The comparisons between the proposed algorithms and SLNR algorithm shown in section V will demonstrate that the proposed algorithms are better.

Iii Approximation Method Based on Taylor Expansion

In this section, an efficient approximation algorithm is developed for the purpose of local-optimal solutions of problem (20). Since the objective function (20a) is nonconvex and complicated, problem (20) is difficult to solve. A convex approximation method based on Taylor expansion will be presented to handle problem (20) efficiently in the following.

Iii-a Convex Approximation

To make the problem more tractable, new matrixs are introduced. Thus, we have . It should be noticed that if instead of is used as the variables to optimize, has to satisfy , which would violate the problem’s convexity. Then, SDR (semidefinite relaximation), a convex optimization based approximation technique, is applied to omit the rank-one constraint. The approximated problem can be transformed into the following

 maxWi∈CNt×Nti=1,…,K K∑i=1[log2⎛⎜⎝∑Kk=1(¯¯¯¯hTiWk¯¯¯¯hHi−2ε∥Wk¯¯¯¯h∗i∥)+σ2i∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i⎞⎟⎠ −log2⎛⎜⎝∑Kk=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i∑k≠i(¯¯¯gTiWk¯¯¯g∗i−2ε∥Wk¯¯¯g∗i∥)+ς2i⎞⎟⎠], (21a) s.t. K∑i=1Tr(Wi)≤P, (21b) s.t. Wi⪰0,i=1,…,K. (21c)

It should be mentioned that SDR has been widely used in various beamforming design problems. If the solution of the problem satisfies the rank-one constraints, then eigenvalue decomposition would be utilized to obtain the practical optimal solution; otherwise, randomization technique can be applied [18].

Problem (21), however, is still nonconvex yet since the objective function (21a) is nonconvex. Therefore, further approximations are needed. Let us consider the following change of variables,

 exi≜K∑k=1(¯¯¯¯hTiWk¯¯¯¯h∗i−2ε∥Wk¯¯¯¯h∗i∥)+σ2i, (22a) eyi≜∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i, (22b) epi≜K∑k=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i, (22c) eqi≜∑k≠i(¯¯¯gTiWk¯¯¯g∗i−2ε∥Wk¯¯¯g∗i∥)+ς2i, (22d)

for . Note that , which means the lower bound and upper bound of should be no less than 0. Then we have

 exi≥σ2i, eyi≥σ2i, epi≥ς2i, and eqi≥ς2i. (23)

Due to the sum power constraint, , , and won’t be infinity. It can be observed that , , and are bounded.

By substituting (22) into (21a), one can reformulate problem (21) as the following

 maxxi,yi,pi,qi∈RWi∈S,∀ilog2K∏i=1(e(xi−yi)−(pi−qi)), (24a) s.t.   K∑k=1(¯¯¯¯hTiWk¯¯¯¯h∗i−2ε∥Wk¯¯¯¯h∗i∥)+σ2i≥exi, (24b) s.t. ∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i≤eyi, (24c) s.t. K∑k=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i≤epi, (24d) s.t. ∑k≠i(¯¯¯gTiWk¯¯¯g∗i−2ε∥Wk¯¯¯g∗i∥)+ς2i≥eqi,

where

 S≜{W1,…,WK⪰0 ∣∣ Wi∈CNt×Nt,K∑i=1Tr(Wi)≤P}. (25)

The objective function can be transformed into . It can be seen that the objective function is convex. Notice that the equalities in (22) have been replaced by inequalities as in (24b) to (24e). It could be verified by the monotonicity of the objective function that all the inequalities in (24b) to (24e) hold with equalities at the optimal points. To be specific, in the process of solving problem (24), if the inequalities (24b) or (24e) doesn’t hold with equality, we can increase or until the equality holds. If the inequalities (24c) or (24d) doesn’t hold with equality, we can decrease or until the equality holds. In the mean time, the optimal value of problem (24) would also be improved. Thus, the inequalities from (24b) to (24e) would hold with equalities for the final solutions.

For the purpose of maximization of (24a), we maximize and which are the lower bound of and , respectively, while minimizing and which are the upper bound of and , as can be observed from (24b) to (24e). Thus, while solving problem (24), the lower bound of the numerator of the objective function (21a) are maximized and the upper bound of the denominator are minimized leading to the maximization of (21). As explained above, it can be seen that problem (24) is an appropriate approximation of problem (III-A).

It can be observed that constraints (24c) and (24d) are nonconvex resulting in difficulties for optimal solution. Let be a feasible point of problem (24). Define

 (26a) (26b)

for . Then and are feasible to problem (24). Aiming to make (24c) and (24d) convex, these constraints are conservatively approximated at the point based on Taylor expansion [19]. The Taylor series of a function that is infinitely differentiable at a number is the power series . Since both of and are convex, their first-order Taylor expansion at and are respectively given by

 e˜yi(yi−˜yi+1)and% e˜pi(pi−˜pi+1). (27)

Consequently, restrictive approximations for (24c) and (24d) are given by

 ∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i≤e˜yi(yi−˜yi+1), (28a) K∑k=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i≤e˜pi(pi−˜pi+1).

Through the first-order Taylor expansion of and , the original non-linear terms successfully turn out to be linear leading to convex constraints. By replacing (24c) and (24d) with (28a) and (28b), respectively, the following approximation of problem (24) can be obtained

 maxxi,yi,pi,qi∈RWi∈S,∀ilog2∏i(e(xi−yi)−(pi−qi)), (29a) s.t.   K∑k=1(¯¯¯¯hTiWk¯¯¯¯h∗i−2ε∥Wk¯¯¯¯h∗i∥)+σ2i≥exi, (29b) s.t. ∑k≠i(¯¯¯gTiWk¯¯¯g∗i−2ε∥Wk¯¯¯g∗i∥)+ς2i≥eqi, (29c) s.t. ∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i s.t.  ≤e˜yi(yi−˜yi+1), (29d) s.t. K∑k=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i s.t.  ≤e˜pi(pi−˜pi+1). (29e)

Problem (29) is a convex optimization problem which can be efficiently solved by CVX, a package for specifying and solving convex programs [20],[21].

In summary, the reformulation above consists of two approximation steps: a) the rank relaxation of to through SDR, and b) constraint restrictions of (24b) and (24e) to (29b) and (29e). Note that if problem (29) yields a rank-one optimal , a rank-one beamforming solution can be readily obtained by rank-one decomposition of for . It is then straightforward that this rank-one beamforming solution is also feasible to the original problem (20). Otherwise, randomization technique can be applied.

Iii-B Successive Convex Approximation

Formulation (29) is obtained by approximating problem (20) at the given feasible point , as described in (28). This approximation can be further improved by iterative procedure based on the optimal solution obtained through solving (29) in the previous approximation. Specifically, in the -th iteration, the following convex optimization problem is solved by CVX,

 maxxi,yi,pi,qi∈RWi∈S,∀ilog2∏i(e(xi−yi)−(pi−qi)), s.t.   K∑k=1(¯¯¯¯hTiWk¯¯¯¯h∗i−2ε∥Wk¯¯¯¯h∗i∥)+σ2i s.t.   ≥exi, (30a) s.t. ∑k≠i(¯¯¯gTiWk¯¯¯g∗i−2ε∥Wk¯¯¯g∗i∥)+ς2i s.t.   ≥eqi, (30b) s.t. ∑k≠i(¯¯¯¯hTiWk¯¯¯¯h∗i+2ε∥Wk¯¯¯¯h∗i∥)+σ2i s.t.  ≤e˜yi[n](yi−˜yi[n]+1), (30c) s.t. K∑k=1(¯¯¯gTiWk¯¯¯g∗i+2ε∥Wk¯¯¯g∗i∥)+ς2i s.t.  ≤e˜pi[n](pi−˜pi[n]+1). (30d)

The optimal solution of (30) is denoted as . Then, through (31),

 ˜yi[n+1]≜ln({∑k≠i(¯¯¯¯hTiˆWk[n]¯¯¯¯h∗i ˜yi[n−1]≜+2ε∥ˆWk[n]