Artificial noise-aided biobjective transmitter optimization for service integration in multi-user MIMO broadcast channel

This paper considers an artificial noise (AN)-aided transmit design for multi-user MIMO systems with integrated services. Specifically, two sorts of service messages are combined and served simultaneously: one multicast message intended for all receivers and one confidential message intended for only one receiver and required to be perfectly secure from other unauthorized receivers. Our interest lies in the joint design of input covariances of the multicast message, confidential message, and artificial noise (AN), such that the achievable secrecy rate and multicast rate are simultaneously maximized. This problem is identified as a secrecy rate region maximization (SRRM) problem in the context of physical-layer service integration. Since this biobjective optimization problem is inherently complex to solve, we put forward two different scalarization methods to convert it into a scalar optimization problem. First, we propose to prefix the multicast rate as a constant, and accordingly, the primal biobjective problem is converted into a secrecy rate maximization (SRM) problem with quality of multicast service (QoMS) constraint. By varying the constant, we can obtain different Pareto optimal points. The resulting SRM problem can be iteratively solved via a provably convergent difference-of-concave (DC) algorithm. In the second method, we aim to maximize the weighted sum of the secrecy rate and the multicast rate. Through varying the weighted vector, one can also obtain different Pareto optimal points. We show that this weighted sum rate maximization (WSRM) problem can be recast into a primal decomposable form, which is amenable to alternating optimization (AO). Then, we compare these two scalarization methods in terms of their overall performance and computational complexity via theoretical analysis as well as numerical simulation, based on which new insights can be drawn.


I. INTRODUCTION A. Background
Recently, physical-layer service integration (PHY-SI), a technique of combining multicast service and confidential service into one integrated service for one-time transmission at the physical layer, has received much attention in wireless communications.For one thing, PHY-SI caters to the demand for high transmission rate and secure communication, which has been identified as the key targets that need to be effectively addressed by fifth generation (5G) wireless systems [1].Besides, compared with the conventional upperlayer-based approach, PHY-SI enables coexisting services to share the same resources by solely exploiting the physical characteristics of wireless channels, thereby significantly increasing the spectral efficiency.This property makes PHY-SI a prominent approach to satisfy the ever-increasing need for radio spectrum.The technique of PHY-SI could also find a wide range of applications in the commercial and military areas.For example, many commercial applications, e.g., advertisement, digital television, Internet telephony, and so on, are supposed to provide personalized service customization.As a consequence, confidential service and public service are collectively provided to satisfy the demand of different user groups.In battlefield scenarios, it is essential to propagate commands with different security levels to the frontline.The public information should be distributed to all soldiers, while the confidential information can only be accessed by specific soldiers.Such emerging applications lead to a crucial problem: how to establish the security of confidential service while not compromising the quality of public service?

B. Related Works
The study of PHY-SI can be traced back to Csiszár and Körner's seminar work in [2].In the basic model of PHY-SI, a transmitter sends a common message to two receivers, and simultaneously, sends a confidential message intended only for one receiver and kept perfectly secure from the other one.Under discrete memoryless broadcast channel (DMBC) setup, Csiszár and Körner gave a closed-form expression of the maximum rate region that can be applied reliably under the secrecy constraint (i.e., the secrecy capacity region).In recent years, this kind of approach has gained renewed interest, especially that in various multi-antenna scenarios, such as Gaussian broadcast channels [3]- [6], and bidirectional relay channels [7], [8].In [3], the authors extended the results in [2] to a general MIMO Gaussian case by adopting the channel-enhancement argument.Further, the works [4], [5] considered the case with two confidential messages intended for two different receivers.The resulting secrecy capacity region is proved to be attainable by combining the secret dirtypaper coding (S-DPC) with Gaussian superposition coding.Furthermore, in [7] and [8], Wyrembelski and Boche amalgamated broadcast service, multicast service and confidential service in bidirectional relay networks, in which a relay adds an additional multicast message for all nodes and a confidential message for only one node besides establishing the conventional bidirectional communication.Nonetheless, the main goal of the aforementioned papers is just to obtain capacity results or to characterize coding strategies that lead to certain rate regions [9].For implementation efficiency, it is also important to treat physical layer service integration from a signal processing point of view.In particular, optimal or complexity-efficient transmit strategies have to be characterized, so that the achieved performance could reach/approach the boundary of the secrecy capacity region.Such strategies are usually given by optimization problems, which generally turn out to be nonconvex.Along with this comes the fact that most works on PHY-SI end once a certain characterization of a rate region is derived.
Recently, there is growing interest in analyzing PHY-SI from a signal processing point of view.In [3], the authors proposed a re-parameterizing method to devise transmit strategies for achieving the secrecy boundary performance.However, this method is only applicable to a very simple two-user MISO scenario.Besides, it involves solving a sequence of convex feasibility problems, which is computationally expensive.To improve it, the work [10] proposed a quality-of-service (QoS)based method to seek the boundary-achieving transmit strategies.Its basic idea is to establish the tradeoff between the secrecy rate and the multicast rate by maximizing the secrecy rate while ensuring the multicast rate above a given threshold.This method is demonstrated as effective in characterizing the secrecy boundary, and thus triggered research endeavors on extending the result to a more general and realistic setting.Notable results include the extension to the multi-user [11] and imperfect CSI [12], [13] settings.Even so, relatively less work focussed on the MIMO channel setup, due to the intractability of the associated optimization problems.In [14], the authors circumvented the intractability by proposing a generalized singular value decomposition (GSVD) based transmission scheme.Using GSVD, multicast message and confidential message can be perfectly decoupled and the resulting problem is easier to handle.However, this result is not applicable to the general multi-user MIMO case.In addition, it is also interesting to incorporate artificial noise (AN) into consideration, as such technique has been shown to be effective in enhancing transmission security [15]- [17].

C. Main Contributions
In this paper, we delve into the AN-aided transmit precoding design in PHY-SI under a general multi-user MIMO case.Specifically, two sorts of service messages are combined and promulgated at the same time: a multicast message intended for all receivers, and a confidential message intended for merely one authorized receiver.The confidential message must be kept perfectly secure from all other unauthorized receivers.Meanwhile, AN is employed to degrade the potential eavesdropping of the unauthorized receivers.This paper aims to jointly optimize the input covariance matrices of the multicast message, confidential message and AN, to maximize the achievable secrecy and multicast rates simultaneously, or equivalently, to maximize the achievable secrecy rate region.This secrecy rate region maximization (SRRM) problem turns out to be a biobjective optimization problem.To characterize its Pareto boundary, we develop two scalarization methods to convert it into an easier-to-handle scalar version.
1) In the first method, we propose to fix the multicast rate as a constant.Through varying the value of the constant, this method could yield different secrecy boundary points.Since the Pareto optimal points must reside on the boundary of the achievable rate region, this method is bound to provide a complete set of the Pareto optimal points.Though the resultant secrecy rate maximization (SRM) problem is nonconvex by nature, we show this problem actually falls into the context of differenceof-concave (DC) programming [18].Hence, it can be handled by classical DC algorithm with convergence guarantee.2) As for the second method, a weighted sum-based scalarization is introduced.The crux of this scalarization method is to optimize the weighted sum of the two objectives with different weight vectors.By varying the weight vector, this method gives rise to different Pareto optimal solution.To solve this weighted sum rate maximization (WSRM) problem, we reveal its hidden decomposability by recasting it as an equivalent form amenable to alternating optimization (AO).AO algorithm is naturally employed to solve the WSRM problem.It can be proved that this AO algorithm must converge to one stationary point of the WSRM problem.Then we compare these two sorts of scalarization methods in terms of their overall performance and computational complexity.The comparison results reveal that the first method is more efficacious in finding all Pareto optimal points than the second one.The advantage of the second method lies in its problem structure, which provides insights for the service provider, in how to maximize the overall revenue.Besides, we show that the DC algorithm is more time-efficient at low transmit power than the AO algorithm.Interestingly, the numerical results indicate that at high transmit power, the AO algorithm becomes the more time-efficient one.

D. Organization and Notations
This paper is organized as follows.Section II provides the system model description and problem formulation.The optimization aspects of our formulated problems are addressed in Section III and IV, corresponding to the first and the second scalarization methods, respectively.The comparison results are given in Sections V. Section VI presents simulation results to show the efficacy of our proposed methods.Finally, conclusions are drawn in Section VII.
The notation of this paper is as follows.Bold symbols in capital letter and small letter denote matrices and vectors, respectively.(•) H , rank(•) and Tr(•) represent conjugate transpose, rank and trace of a matrix, respectively.R + and H n + denote the set of nonnegative real numbers and of n-by-n Hermitian positive semidefinite (PSD) matrices.The n × n identity matrix is denoted by I n .x ∼ CN (µ, Ω) denotes that x is a complex circular Gaussian random vector with mean µ and covariance Ω.A 0 (A ≻ 0) implies that A is a Hermitian positive semidefinite (definite) matrix.• represents the vector Euclidean norm.K represents a proper cone, and K * represents a dual cone associated with K.

II. SYSTEM MODEL AND PROBLEM FORMULATION
We consider the downlink of a multiuser system in which a multi-antenna transmitter serves K receivers, and each receiver is equipped with multiple antennas.Assume that all receivers have ordered the multicast service and receiver 1 further ordered the confidential service.To enhance the security performance, the transmitter utilizes a fraction of its transmit power to send artificially generated noise to interfere the unauthorized receivers (eavesdroppers), i.e., receiver 2 to receiver K.
Remark 1: In this paper, we assume that only one receiver orders the confidential service within a single time slot.In practice, this assumption is valid under the cases where the confidential service is provided to all receivers in a round-robin manner to strengthen the security of confidential messages and to reduce the operational complexity at the transmitter.
The received signal at receiver k is modeled as where H k ∈ C N r,k ×Nt is the channel response between the transmitter and receiver k; and N t and N r,k are the number of transmit antennas employed by the transmitter and kth receiver, respectively.z k is independent identically distributed (i.i.d.) complex Gaussian noise with zero mean and unit variance.
x ∈ C Nt is the coded transmit message, which consists of three independent components, i.e., where x 0 is the multicast message intended for all receivers, x c is the confidential message intended for receiver 1, and x a is the artificial noise.We assume , where Q 0 and Q c are the transmit covariance matrices.The AN x a follows a distribution x a ∼ CN (0, Q a ), where Q a is the AN covariance.For ease of exposition, let us define K ∆ = {1, 2, ..., K} and K e ∆ = K/{1}, which denote the indices of all receivers and of all unauthorized receivers, respectively.Denote R 0 and R c as the achievable rates associated with the multicast and confidential messages, respectively.Then an achievable secrecy rate region R s ({H k } k∈K , P ) is given as the set of nonnegative rate pairs where and Tr(Q 0 +Q c +Q a ) ≤ P with P being total transmit power budget at the transmitter.
The secrecy rate region (3) implies that all receivers first decode their common multicast message by treating the confidential message as noise, and then receiver 1 acquires a clean link for the transmission of its exclusive confidential message, where there is no interference from the multicast message.This can be achieved by utilizing the encoding schemes proposed in [2].
To maximize this achievable secrecy rate region, our goal is to find the boundary-achieving Q 0 , Q a and Q c , which is also known as Pareto optimal solutions to this SRRM problem.Specifically, with perfect CSI available at the transmitter, we must first solve the following optimization problem, which is a biobjective maximization problem with cone where, with a slight abuse of notations but for notational simplicity, the explicit dependence of Since the SRRM problem is a biobjective maximization problem, it is necessary to harness some method of scalarization to convert it into an easier-tohandle scalar version.

Remark 2:
The perfect CSI assumption is valid in the context of PHY-SI, since all nodes are active in the network for ordering the multicast service so that their channels can be monitored.Besides, we remark that this assumption is commonly adopted in the area of physical layer security/multicasting, especially in MIMO channels [15]- [17], [19]- [23].

III. A DC-BASED APPROACH TO THE SRRM PROBLEM
In this section, we develop our first scalarization method to solve (5).The basic problem formulation is a secrecy rate maximization (SRM) with imposed quality of multicast service (QoMS) constraints.

A. Scalarization
In particular, our method is to move the multicast rate maximization part to the constraint, i.e., we fix at the time being the multicast rate as a constant τ ms ≥ 0. As a result, the biobjective SRRM problem (5) will be degraded into a scalar maximization problem, which is shown in (6).
In (6), R(τ ms ) is the optimal objective value, τ ms can be interpreted as the preset requirement on the multicast rate, and accordingly, the constraint (6a) can be interpreted as a QoMS constraint.To guarantee the feasibility of problem ( 6), τ ms cannot exceed a threshold τ max given by The value of τ max can be numerically obtained by solving (7) via the convex optimization solver CVX [24].This sort of problem formulation, in fact, enables us to find one boundary point (τ ms , R(τ ms )) of the secrecy rate region R s ({H k } k∈K , P ) by solving (6).All boundary points of R s ({H k } k∈K , P ) can be found if we traverse all possible τ ms 's lying within [0, τ max ] and store the corresponding optimal objective values.Since the Pareto optimal solution to (5) must reside on the boundary of R s ({H k } k∈K , P ), i.e., the Pareto optimal set of ( 5) is a subset of the boundary set of R s ({H k } k∈K , P ), all Pareto optimal solution to (6) can also be found by this means.
However, problem (6) is nonconvex.Especially, the determinant equality constraint (6a) is very difficult to handle.To circumvent this difficulty, we pay our attention to the following relaxed problem of (6), in which the equality constraint (6a) is replaced by the inequality constraint (8a).
Apparently, any optimal solution to ( 6) is feasible to (8) in the sense that replacing (6a) with (8a) yields a larger feasible solution set.Hence, problem (8) has R(τ ms ) ≤ R(τ ms ) in general.Interestingly, we show that R(τ ms ) = R(τ ms ) can always be achieved without loss of optimality to (8).Lemma 1: Problem ( 8) is a tight relaxation to problem (6).In other words, the rate pair (τ ms , R(τ ms )) must be a boundary point of R s ({H k } k∈K , P ).
Proof: The proof can be easily accomplished by construction.Suppose that the constraint (8a) is satisfied with strict inequality, we can always multiply Q 0 by a scalar ν (ν < 1) to make (8a) active, yet without decreasing the objective value of ( 8) and violating the total power constraint (8b).This fact implies that there always exists an optimal solution to (8) such that the constraint (8a) is satisfied with equality, and thus, accomplishes the proof.
Lemma 1 implies that problem (8) admits an optimal is also optimal to (6).The proof of Lemma 1 reveals that such an optimal can always be constructed algorithmically based on the following procedures: is an optimal solution returned by solving problem (8).If as an optimal solution of problem (6).Otherwise, solve the following equation with regard to ν, i.e., min as an optimal solution of problem (6).
Next, we will point out two special cases, under which problem ( 8) is equivalent to problem (6); or equivalently, any optimal solution to (8) is achieved with constraint (8a) active.This is described in the following proposition.
Proposition 1: Suppose that the system configurations satisfy either one of the following conditions: Condition 1: The number of antennas at the transmitter is larger than that at the authorized receiver, i.e., N t > N r,1 .
Condition 2: The number of antennas at the transmitter is larger than the sum of the antenna number at the unauthorized receivers, i.e., N t > k∈Ke N r,k .
Then the rate pair (τ ms , R(τ ms )) must be a Pareto optimal point of ( 5), and all Pareto optimal points of ( 5) can be obtained by solving (6) with different τ ms 's lying within the interval [0, τ max ].
Proof: The proof can be found in Appendix A. Remark 3: Proposition 1 bridges the Pareto optimal points of (5) to the boundary points of C s (H 1 , H 2 , P ).When either Condition 1 or Condition 2 is satisfied, all Pareto optimal points of (5) are also the boundary points of R s ({H k } k∈K , P ), and vice versa.

B. DC Iterative Algorithm
We now focus on solving the relaxed problem (8) derived in the last subsection.Problem (8) still remains nonconvex due to its objective function and constraint (8a).To deal with it, we first equivalently transform it into its epigraph form by introducing a slack variable η, i.e.,

R(τ
Next, we will show that problem (9) constitutes a DCtype programming problem, which can be iteratively solved by employing the DC algorithm.
To begin with, we reformulate the capacity function , one can easily notice that the objective function of (6) and constraints (12a) and (12b) are all in a difference-of-concave form.This property makes problem (5) fall into the context of DC program [18], which can be iteratively solved via DC algorithm.
Our next endeavor is to show the DC approach to (12) mathematically.Its basic idea is to locally linearize the nonconcave parts in (12) at some feasible point via Taylor series expansion (TSE), and then iteratively solve the linearized problem.To this end, we introduce the TSE via the following lemma.
Lemma 2 ( [25]): An affine Taylor series approximation of a function f (X) : R M×N → R can be expressed at X as below.
The TSE above enables us to reformulate the primal nonconcave parts of (12) into a linear form.In particular, by applying Lemma 2 and the fact in the objective function of (12), where Qa is a given transmit covariance matrix, and the equality (a) is due to the fact that Tr(A H B) = (vec(A)) H vec(B) for appropriate dimensions of A and B.
Likewise, φ k (Q c , Q a ), appearing in the constraints (12a) and (12b), can be approximated as in which U is determined by Based on the approximations above, the original QoMSconstrained SRM problem (12) can be reformulated as where R(τ ms ) is the optimal objective value of ( 9), serving as an approximation to R(τ ms ).According to the relationship between a concave function and its Taylor series expansion, it is immediate to get As a consequence, any feasible solution to (17) should also be feasible to (12), and R(τ ms ) ≤ R(τ ms ) must hold.This approximated problem ( 17) is convex with regard to (w.r.t.) (Q 0 , Q c , Q a ) and hence (Q 0 , Q c , Q a ) can be iteratively obtained by solving problem (17) via some off-theshelf interior-point algorithm, e.g., CVX.We summarize our proposed iterative algorithm for solving (12) in Algorithm 1.To acquire the secrecy rate region, we need to traverse τ ms lying within the interval [0, τ max ] and store the corresponding objective value of (17).

Remark 4:
In Algorithm 1, the initialization of ( Qc,0 , Qa,0 ) plays a crucial role in influencing the total iteration times.Let us define Q i c , Q i a as the output solution in ith traversal of τ ms .The following "warmstart operation" could be adopted to initialize ( Qc,0 , Qa,0 ) for achieving a fast convergence rate: Warmstart Operation: We start the traversal of τ ms from τ ms = τ max .In the first traversal, Qc,0 and Qa,0 are both initialized as 0. In the ith (i > 1) traversal, ( Qc,0 , Qa,0 ) is initialized as the solution output by Algorithm 1 in the (i−1)th traversal.

C. Convergence Analysis
As one can see, the basic merit of DC lies in its tractability, which caters to the numerical optimization using the parsersolver.As an additional merit, the proposed DC approach has a theoretically provable guarantee on its solution convergence, which will be demonstrated in the following proposition.
Proposition 2: Every limit point of ) is a stationary point of problem (6) Proof: The proof is a direct application of [18, Th 10], and thus omitted here for simplicity.

IV. AN AO-BASED APPROACH TO THE SRRM PROBLEM
In this section, we develop our another scalarization method, referred to as weighted-sum method, to solve (5).The basic problem formulation is a WSRM problem, which can be solved via an AO-based approach.

A. Scalarization
The basic idea of the weighted-sum method is to introduce a so-called weight vector [26] that is positive in the dual cone K * = R 2 + , and then to transform the primal vector optimization problem into a scalar optimization problem.By varying the vector, we can obtain different Pareto optimal solutions of (5).
To put into context, the Pareto boundary of (3) can be characterized by the solution of max Q0,Qc,Qa,R0,Rc (19) in which λ c ∈ [0, +∞) and λ = [1, λ c ] is our introduced weight vector.In general, the optimal (R 0 , R c ) to ( 6) is the point where a straight line with slope −1/λ c is tangent to the Pareto boundary.Before proceeding, let us first point out some special cases of problem (6).
1) When λ = [1, 1], the optimal (R 0 , R c ) turns out to be the so-called utilitarian point, also referred to as "sumrate" point in communications.
2) The single-service points are the two points where R 0 = 0 and where R c = 0, respectively.When R 0 = 0, problem ( 6) is degraded into a conventional AN-aided SRM problem in MIMO wiretap channel.When R c = 0, the maximum R 0 can be derived by solving the same convex optimization problem as (7).

B. AO Iterative Algorithm
We are now in a position to determine the tractable approaches to the WSRM problem (19).First, one can notice that by discarding R 0 and R c as slack variables, problem (19) is equivalent to the following optimization problem.

R(λ
The obstacle of solving (20) mainly lies in the nonsmoothness of its objective function, which negates the use of many derivative-related iterative algorithms.As a result, we next develop a derivative-free AO iterative algorithm to solve (20).To this end, we will first need to transform the WSRM problem (20) into a form amenable to AO.
Lemma 3 ( [16]): Let E ∈ C N ×N be any matrix satisfying and the optimal solution to the right-hand side (RHS) of ( 33) is Applying Lemma 3 to C b , C e,k and C m,k , one can obtain C e,k = min where we define in which {S k } k∈K and {U k } k∈K are slack variables satisfying S k 0 and U k 0 for ∀k ∈ K.
Following the matrix manipulations in [16], we have and Substituting (22a)-(22c) into (20) and making use of ( 24) and ( 25), one can check that problem ( 20) is equivalent to the following optimization problem.
in which we define The upshot of this reformation is that problem (20) becomes primal decomposable.Specifically, problem (26) Hence, AO is naturally employed to solve (26).With (Q 0 , Q c , Q a ) fixed, the optimal solution of ({S k } k∈K , {U k } k∈K ) admits an analytical expression, according to Lemma 3, given by in which we utilize the fact that {S k } k∈K and {U k } k∈K are decoupled among ϕ b , ϕ e,k and ϕ m,k .Comparatively, with ({S k } k∈K , {U k } k∈K ) fixed, the optimal solution of (Q 0 , Q c , Q a ) can be obtained by solving a convex optimization problem as below, i.e., where F denotes the feasible set of (20), which is convex.The whole AO process for solving (26) is given in Algorithm 2. In line 6 of Algorithm 2, the convex subproblem can be solved via CVX.Following the similar warmstart operation introduced in Remark 4, the iteration times of Algorithm 2 can be significantly decreased.

C. Convergence Analysis
It can be verified that the AO algorithm produces a nondecreasing objective value of (26).Besides, the following convergence result is always guaranteed.
Proposition 3: Suppose that is the solution generated by the AO algorithm in n th iteration, then the sequence } n must converge to one stationary point (i.e., Karush-Kuhn-Tucker (KKT) point) of the primal WSRM problem (20).
Proof: The proof can be found in Appendix B. Since the global optimal solutions to problem (20) hitherto remains inaccessible, our achieved secrecy rate region would serve as a lower bound on R s ({H k } k∈K , P ), which achieves KKT optimality.

V. COMPARISON OF THE PROPOSED METHODS
In the previous sections, we present two tractable convex formulations of the SRRM problem (5).This naturally leads to the question about the relative performance of the two formulations.In the following subsections, we address this question by comparing their performance and computational complexity in solving (5).

A. Performance Analysis
As introduced in the preceding sections, the QoMS-based scalarization can yield a complete set of boundary points of R s ({H k } k∈K , P ), which contains all Pareto optimal points of ( 5).The resulting scalar problem (8) aims to maximize the secrecy rate and meanwhile maintain the QoMS above a given threshold.Predictably, the use of AN should be effective merely at low QoMS region, since AN exerts a negative effect on the multicasting performance.To guarantee the high demand for QoMS, AN has to be prohibitive at high QoMS region.This QoMS-constrained SRM is a generalization of traditional SRM in physical-layer security, and provides the transmitter with some insights in how to tradeoff the security performance and the multicasting performance.
As for the weighted-sum scalarization method, the necessary condition for it to find all Pareto optimal points is that the secrecy rate region should be convex.Besides, its performance is also dependent on the precision of λ c .The traversal of λ c should span from zero to an extremely large number with appropriate step, so that each Pareto optimal points can be detected.Nonetheless, the weighted-sum problem structure has an interesting pricing interpretation from the field of economics.To elaborate a little further, let us define p 0 and p c as the unit price for the secrecy rate and the multicast rate, respectively, charged by the service provider.To maximize its revenue, the service provider should be concerned about how to solve the WSRM problem in (19) with setting λ c = p c /p 0 .The use of AN could also be explained in this context.It is evident to see when p 0 ≫ p c , the revenue from multicasting transmission would dominate the objective function of (19), and thus, eliminating AN would be helpful in increasing the overall revenue.
In all, these two scalarization methods are suitable for different application scenarios and provide different insights.Nonetheless, the QoMS-based scalarization could yield all Pareto optimal points, while the weighted-sum scalarization might only yield some of them, dependent on the shape of the secrecy rate region.

B. Complexity Analysis
The major computational complexity of the two scalarization methods comes from solving the problems ( 17) and (29).While both of problems ( 17) and ( 29) are convex, it is not in a standard semidefinite programming (SDP) form, owing to the logarithm functions therein.However, by utilizing the following approximation [27]: (30) all logarithm terms in problems ( 17) and ( 29) can be approximated by a trace function at low transmit power.This approximation further converts the convex problems ( 17) and ( 29) into SDP ones.Now, consider (17), which has three linear matrix inequality (LMI) constraints of size N t , and 2K LMI constraints of size 1 after introducing the approximation (30).Moreover, for (17), the number of decision variables is on the order n 1 = 3N 2 t + 1.Then, when a generic path-following interiorpoint method (IPM) is used to solve problem (17), the total arithmetic computation cost is on the order of [28] t + 1).On the other hand, for solving (29), we need to introduce two additional slack variables to move the maximum and minimum terms in the objective function of (29) to the constraints.Hence, the number of decision variables is on the order of n 2 = 3N 2 t + 2, and (29) also has three LMI constraints of size N t , and 2K LMI constraints of size 1.The total arithmetic computation cost for solving (29) is on the order of 31) and (32), one can note that the total arithmetic computation cost of solving the two problems is comparable, with T 1 slightly greater than T 2 due to n 1 > n 2 .This observation implies that the QoMS-based scalarization is more time-efficient at low transmit power.This is also consistent with our following simulation results, as we shall see in Section VI.

VI. NUMERICAL RESULTS
In this section, we provide numerical results to illustrate the secrecy rate region derived from the two proposed methods, compared with two other existing strategies.The first one is the no-AN transmission, i.e., prefixing Q a as 0 in problem (5).Thus, its achieved secrecy rate region can also be derived via the DC and AO algorithms.Another one is the traditional service integration using time division multiple address (TDMA), which assigns the confidential message and multicast message to two orthogonal time slots.Its maximum secrecy rate and multicast rate can be obtained by seeking the single-service points of R s ({H k } k∈K , P ).For the fairness of comparison, the secrecy rate and multicast rate achieved by this TDMAbased strategy should be halved [8].
In the first subsection, the convergence results of both algorithms are presented.The second subsection gives the comparison between these two algorithms in terms of achievable performance and computational complexity.

A. Convergence Results
In this subsection, we assume N t = 5, N r,k = 3 for all k ∈ K, and K = 4.The channel matrices are randomly generated from an i.i.d.complex Gaussian distribution with zero mean and unit variance.According to Proposition 1, since N t > N r,1 , the optimal solution to (8) is attained when the constraint (8a) holds with equality.
First, we evaluate the convergence of the DC algorithm.Especially, we are concerned about whether the primal constraint (8a) is violated by our approximation.Setting τ ms as 2 bps/Hz, Fig. 1 shows the convergence of the multicast rate in the iteration with different transmit power.Qc,0 and Qa,0 Fig. 2. DC algorithm: Convergence of the secrecy rate are both initiated as 0. The algorithm stops iterating when the difference between two successive values of R(τ ms ) returned by the algorithm is less than or equal to 10 −4 .One can observe that, the multicast rates ultimately converge to our predefined multicast rate with a limited number of iterations in all tested transmit powers.This observation indicates the efficacy of TSE in approximating the multicast rate.Then we also plot the achieved secrecy rates and the approximated secrecy rates in Fig. 2. The general observation of Fig. 1 is also applicable to Fig. 2.
The convergence results of the AO algorithm are presented in Fig. 3.In Fig. 3, we set λ c = 1 to seek the sum-rate point.Q 0 c and Q 0 a are both initialized as (P/(2N t ))I Nt .The algorithm stops iterating when the difference between two successive values of R(λ c ) is less than or equal to 10 −4 .As one can observe from Fig. 3, the achieved weighted sum rate is monotonically increasing and finally converges with a limited number of iterations in all tested transmit powers.

B. Performance Comparison
In this subsection, we focus on two sorts of system configuration.The first one is the same as that in the last subsection.Besides, we consider another sort of system configuration: N t = N r,1 = 4, N r,k = 5 for all k ∈ K e , and K = 4.Under the second system configuration, neither Condition 1 nor Condition 2 is satisfied.First, we will show the secrecy rate regions achieved by the first system configuration.Overall results are shown in Fig. 4, with P set as 10dB and 20dB, respectively.Fig. 4 reveals two general trends.First, our AN-aided scheme achieves a secrecy rate region larger than the no-AN one.The striking gap indicates the efficacy of AN in expanding the secrecy rate region.However, the gap between these two strategies dramatically reduces when R 0 increases.This phenomenon agrees with our conjecture in Section V-A.The second observation is that our proposed strategies, though only attain a lower bound on R s ({H k } k∈K , P ), is sufficient to achieve significantly larger secrecy rate regions than the TDMAbased one.This observation also implies that PHY-SI is an effective approach to improve the spectral efficiency.Then let us compare the achievable performance of the two proposed scalarization methods.One can notice that the performance gap between these two methods is negligible in the tested system configuration, especially when P = 10dB.
Fig. 5 plots the secrecy rate regions achieved by the second system configuration.Still, the secrecy rate region with AN is larger than the one without AN and the one achieved by TDMA.Besides, we can observe two very interesting phenomena.First, when we increase the transmit power from 10dB to 20dB, the secrecy rate regions practically expand in the horizontal direction.That is, under the second system configuration, the increasing transmit power mainly contributes to the multicast message transmission, rather than the confidential message transmission.This can be interpreted from the transmit degree of freedom (d.o.f.).The total d.o.f. of unauthorized receivers is K−1 k=2 N r,k = 15, much higher than the transmit d.o.f.N t = 4.The high d.o.f. at the unauthorized receivers leads to the d.o.f.bottleneck at the transmitter and thus compromises the overall secrecy performance.Second, one can notice that when P = 20dB: 1) there exist some boundary points residing on a line, marked by the red dashed lines, that are not Pareto optimal to (5).Apparently, these points cannot be detected by the weighted-sum scalarization, but can be easily detected by the QoMS-based scalarization; 2) the QoMS-based scalarization detects more Pareto optimal points than the weighted-sum scalarization.This is attributed to the insensitivity of the weighted-sum scalarization to the points residing on an approximately horizontal boundary.Finally, we tabulated the averaged running times of DC and AO for obtaining a boundary point in Table I under the same setting as Fig. 4. As seen, the DC algorithm runs faster than the AO algorithm when the transmit power is low.This phenomenon is consistent with our preceding analysis in Section V-B.However, at high transmit power, the DC algorithm scales nearly exponentially with P and gradually spends more time converging in each iteration than the AO algorithm.This observation indicates that the two proposed scalarization methods might exhibit a performance-complexity tradeoff at high transmit power.

VII. CONCLUSION
In this paper, we considered the AN-aided transmit design for multiuser MIMO broadcast channel with confidential service and multicast service.The transmit covariance matrices of confidential message, multicast message and AN were designed to maximize the achievable secrecy rate and achievable multicast rate simultaneously.To deal with this biobjective optimization problem, two different sorts of scalarization were introduced to transform this SRRM problem into a scalar optimization problem.In the QoMS-based scalarization, the scalar problem is an SRM problem with QoMS constraints, while in the weighted-sum scalarization, the scalar problem is a WSRM problem.DC and AO algorithms were utilized to solve the QoMS-constrained SRM problem and the WSRM problem, respectively.Both algorithms can converge to a stationary point of the primal problems.Further, we gave a detailed comparison between the two proposed scalarization methods.The comparison results indicated that at low transmit power, the QoMS-based scalarization is superior to the weighted-sum one in terms of achievable performance and computational complexity.On the other hand, at high transmit power, these two methods exhibit a tradeoff between achievable performance and computational complexity.Numerical results also confirmed the effectiveness of AN in expanding the secrecy rate region.

A. Proof of Proposition 1
First, we claim that problem (8) has a following interesting property provided that Condition 1 or Condition 2 is satisfied.
Property 1: The maximum objective value of problem (8), R(τ ms ), is obtained only when the equality in (8a) holds.
Proof: The proof of Property 1 can be accomplished by contradiction.Assume that the maximum value of problem ( 8) is obtained at the solution ( Q0 , Qc , Qa ) and the equality in (8a) does not hold, i.e., min . Our next step is to construct a new solution ( Q0 , Qc , Qa ) from ( Q0 , Qc , Qa ), which achieves a larger objective value and satisfies the constraint (8a) with equality.Let us first elaborate upon the construction method under Condition 1.
1) Case for Condition 1: Specifically, we multiply Q0 by a scaling factor ξ (0 < ξ < 1), add a positive semidefinite (PSD) matrix E = ρI − ρH H 1 (H 1 H H 1 ) −1 H 1 to Qa and keep Qc constant, i.e., Q0 = ξ Q0 , Qa = Qa + E and Qc = Qc , where the coefficient ρ controls the power of E. Note that E is the orthogonal complement projector of H H 1 , and its existence is guaranteed by Condition 1.To keep the total transmit power constant, the coefficient ρ should be chosen to satisfy Nt−Nr,1 .To proceed, we need the following lemma.Lemma 4 ( [29]): For matrices A, ∆ 0 and B ≻ 0, the following inequality hold: Then, by applying Lemma 4, one can obtain for any k ∈ K. Thus, by adjusting the value of ξ, the equality in ( 8) could be achieved.
To proceed, we will show that a larger objective value could always be achieved by ( Q0 , Qc , Qa ).By reapplying Lemma 4, it is easy to get Meanwhile, due to Combining ( 35) with (36), we obtain i.e., a larger objective value can be found with ( Q0 , Qc , Qa ).This fact is contrary to the primal assumption.

2) Case for Condition 2:
The only difference between the proof for Condition 1 and Condition 2 lies in the construction method of ( Q0 , Qc , Qa ).To begin with, let us first define a matrix , which stacks all of the unauthorized receivers' channel matrices.Then, we multiply Q0 by a scaling factor ξ (0 < ξ < 1), add a PSD matrix E = ρI − ρH H ua (H ua H H ua ) −1 H ua to Qc and keep Qa constant, i.e., Q0 = ξ Q0 , Qc = Qc + E and Qa = Qa , where the coefficient ρ controls the power of E. E is the orthogonal complement projector of H H ua , the existence of which is guaranteed by Condition 2. The coefficient ρ should be chosen to satisfy ρ = (1−ξ)Tr( Q0) Nt− k∈Ke N r,k to keep the total transmit power constant.
Again, by exploiting Lemma 4 and carrying out some matrix manipulations, one can verify that ( Q0 , Qc , Qa ) can achieve a larger objective value than ( Q0 , Qc , Qa ) with the constraint (8a) active.This fact contradicts the primal assumption.
Summarizing the conclusions drawn from the two cases above, we have accomplished the proof of Property 1.
Property 1 makes the proof of [10, Theorem 1] fully applicable to the proposition here.The remaining parts of the proof can be found in [10] and are omitted here for simplicity.

S1 = arg max
Second, we will check the KKT conditions of problems (39a)-(39d) to build a bridge between (39) and the KKT conditions of problem (38).
Step 1.By noting that S n l k = arg min U n l k = arg max we have and where the second inequality of (42) holds for the reason that AO algorithm yields non-descending objective values.Then letting l → ∞ in (41) and (42) will lead to (39a)-(39d).

Fig. 5 .
Fig. 5. Secrecy rate regions with and without AN.