Secure Beamforming For MIMO Broadcasting With Wireless Information And Power Transfer
Abstract
This paper considers a basic MIMO informationenergy broadcast system, where a multiantenna transmitter transmits information and energy simultaneously to a multiantenna information receiver and a dualfunctional multiantenna energy receiver which is also capable of decoding information. Due to the open nature of wireless medium and the dual purpose of information and energy transmission, secure information transmission while ensuring efficient energy harvesting is a critical issue for such a broadcast system. Assuming that physical layer security techniques are adopted for secure transmission, we study beamforming design to maximize the achievable secrecy rate subject to a total power constraint and an energy harvesting constraint. First, based on semidefinite relaxation, we propose global optimal solutions to the secrecy rate maximization (SRM) problem in the singlestream case and a specific fullstream case. Then, we propose inexact block coordinate descent (IBCD) algorithm to tackle the SRM problem of general case with arbitrary number of streams. We proves that the IBCD algorithm can monotonically converge to a KarushKuhnTucker (KKT) solution to the SRM problem. Furthermore, we extend the IBCD algorithm to the joint beamforming and artificial noise design problem. Finally, simulations are performed to validate the effectiveness of the proposed beamforming algorithms.
I Introduction
Since battery technologies have not yet matched advances in hardware and software technologies, conventional batterypowered wireless systems suffer from short lifetime and require frequent recharging in order to maintain system operation. On the other hand, the rapid development of information and communication technologies demands a huge amount of energy consumption and thus notably contributes to global warming and environmental pollution. As a result, energy harvesting from the environment has recently drawn a lot of interest in both industria and academia[1]. Among the common environmental energy resources, radio signal is particular due to its conventional role of information carrier. Recent research results have shown that the functions of wireless communications and radiobased energy harvesting could be attained simultaneously, which have been termed as (simultaneous) wireless information and power transfer (WIPT)[2, 3, 4].
WIPT has been studied for various communication systems in different context. For example, Zhang and Ho[2] considered a MIMO broadcast system made up of a transmitter, one information receiver (IR) and one energy receiver (ER), and investigated the relevant rateenergy region and optimal transmission schemes. Xu et. al. [3] investigated the optimal information/energy beamforming strategy to achieve the maximum harvested energy for multiuser MISO WIPT system with separated information/energy receivers. Two practical receiver designs for WIPT were proposed in [2, 4], namely, time switching (TS) and power splitting (PS). Based on the PS scheme, Shi et. al. studied the optimal joint beamforming and power splitting (JBPS) to achieve the minimum transmission power of a multiuser MISO downlink system subject to both signaltointerferenceplusnoise (SINR) constraints and energy harvesting constraints. The JBPS problem for MISO interference channel (IFC) was studied in [9]. The works [6, 7, 8] also considered interference channel with WIPT. Shen et. al.[6] studied transmitter design for sumrate maximization with energy harvesting constraints in MISO IFC, while Park and Clerckx [7, 8] investigated transmission strategy for MIMO IFC with energy harvesting. Furthermore, WIPT has been investigated in other channel setups such as relay channels [10, 11, 13, 12] and OFDM channels [14, 15].
The above research works have not considered security issues in WIPT. However, due to the open nature of wireless medium and the dual purpose of information and energy transmission, the wireless information in WIPT systems is more susceptible to eavesdropping. As one of the examples with security concerns, a dualfunctional energy harvester, which is capable of both information decoding (ID) and energy harvesting (EH), may be a potential eavesdropper. Hence, security is an important issue in WIPT. Recently, physical layer security (PLS) technologies[16] have attracted a lot of attentions due to the potentials to ensure highly secure communications by exploiting some physical properties of wireless channels. Based on the PLS technologies, a very limited number of research works have considered secure communication in WIPT[17, 20, 21, 22, 18, 19]. Liu et. al. [17] studied both the secrecy rate maximization problem and sumharvestedenergy maximization problem for a multiuser MISO WIPT system where one transmitter sends information and energy to one IR and multiple ERs. They proposed global optimal solutions to both problems by using semidefinite relaxation (SDR)[23] and onedimensional search. Considering conservative secrecy rate constraints, the works [19, 18] investigated secure transmission in PSbased multiuser MISO WIPT systems and studied transceiver design to achieve the minimum transmission power. Ng. et. al. [20] extended the work [18] and considered secure layered video transmission for PSbased downlink multicast systems using both information and energy beamforming. A chance constraint was introduced in [20] to guarantee a minimum secrecy rate with a given probability while achieving the minimum transmission power. Furthermore, Ng. et. al. [21] advocated the dual use of both artificial noise[24] and energy signals to provide both secure communication and efficient wireless energy transfer in a multiuser MISO WIPT system, and investigated Quality of Service (QoS)constrained robust beamforming to achieve the minimum transmission power. In addition, Ng. et. al. [22] also proposed a multiobjective approach to joint maximizing the energy harvesting efficiency and minimizing the total transmission power while ensuring secure communication in cognitive radio networks with WIPT.
Note that, none of existing works have investigated secure communications in MIMO WIPT systems. This paper considers a basic MIMO informationenergy broadcast system as shown in Fig. 1, where a multiantenna transmitter transmits information and energy simultaneously to a multiantenna information receiver and a multiantenna energy receiver (or called energy harvester). We assume that the energy receiver is a dualfunctional receiver which can also decode information from the received signal by switching its working mode from the EH mode to ID mode. Thus the energy receiver may eavesdrop the information intended for the information receiver only. By considering physicallayer security techniques, we study beamforming design to maximize the achievable secrecy rate subject to a total transmission power constraint and an energy harvesting constraint. The resulting secrecy rate maximization (SRM) problem is hard to solve due to not only the generally nonconcave secrecy rate function but also the nonconvex EH constraint. First, we deal with the SRM problem by considering two special cases—the singlestream case and a specific fullstream case where the difference of Gram matrices of the channel matrices is positive semidefinite. For the two special cases, we propose global optimal solutions to the SRM problem based on semidefinite relaxation[23]. Then, we treat the SRM problem of general case with arbitrary number of streams. We reformulate the SRM problem as another equivalent problem and propose inexact block coordinate descent (IBCD) algorithm to tackle the resulting problem. The convergence of the IBCD algorithm is studied in details. Furthermore, we extend the IBCD algorithm to the joint beamforming and artificial noise (AN) design problem. Finally, we evaluate the effectiveness of the proposed beamforming algorithms by simulations.
The remainder of this paper is organized as follows. In the next section, we describe the problem formulation. Section III presents global solutions to the singlestream case and a specific fullstream case, while Section IV proposes the IBCD algorithm to tackle the general case with an extension to joint beamforming and artificial noise design. In Section V we provide some numerical examples. Section VI concludes the paper.
Notations: Throughout this paper, we use uppercase bold type for matrices, lower–case bold type for column vectors, and regular type for scalars. For a square matrix , denotes its Hermitian transpose, () denotes its maximum (minimum) eigenvalue, () represents that is (isn’t) positive semidefinite, and denotes that is positive definite. denotes the identity matrix whose dimension will be clear from the context. The notations , and represent trace, rank and determinant operator, respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean and covariance matrix is denoted by , and ‘’ stands for ‘distributed as’. denotes the space of complex matrices. denotes the real part of a complex number .
Ii System Model And Problem Formulation
Consider an IE broadcast system (see Fig. 1) where one transmitter sends signal over the same spectrum to one IR and one ER with simultaneous information and power transfer. We assume that the transmitter is equipped with antennas while the IR and ER are equipped with and antennas, respectively. Assuming a narrowband transmission over the IE broadcast system, the equivalent baseband channels from the transmitter to both receivers are modeled by
(1)  
(2) 
where and denote the received signal at the IR and ER, respectively, and denote the channel matrices from the transmitter to the IR and ER, respectively, denotes the transmitted signal, is the transmit beamforming matrix employed by the transmitter, denotes the transmitted symbols (a stream of length ) intended for the IR, and denote the additive white Gaussian noise (AWGN).
Furthermore, we assume that the ER can work in dual functions of information decoding and energy harvesting (i.e., either in ID mode or EH mode). In this scenario, the ER may potentially eavesdrop the information of the IR by switching its working mode to ID. To guarantee secure transmission from the transmitter to the IR (no matter which mode the ER works in), the attractive physical layer security technique is assumed to be employed by the transmitter. Therefore, the achievable secrecy rate is given by[25]
(3) 
On the other hand, the ER captures energy from the received signal . By neglecting the noise power, the harvested power at the ER is given by
(4) 
where denotes the energy conversion efficiency.
In this paper, we are interested in beamforming design with the goal of maximizing the secrecy rate subject to both the harvested power constraint and the total transmission power constraint , where is the power budget for the transmitter and is the EH target required by the ER. For notational simplicity, we define , . The secrecy rate maximization problem can be stated as follows:
(5) 
Problem (5) is feasible if and only if . Furthermore, since the objective function of problem (5) is generally not concave and the EH constraint is not convex, problem (5) is nonconvex and hard to solve. If we remove the EH constraint in problem (5), the resulting problem, denoted by , is the beamforming design formulation of the wellknown powerconstrained SRM problem for Gaussian MIMO wiretap channel[32, 31, 30]. It is known that[31] problem must have positive maximum secrecy rate when . However, this is not the case for problem (5). For example, consider the singlestream case with . When , problem (5) has a unique feasible solution (up to phase rotation), for which, there exists some that achieves negative secrecy rate (given ) while satsifying . Hence, we may obtain a negative maximum secrecy rate under the EH constraint even if , which is not physically interesting. In this paper, we assume^{1}^{1}1If problem (5) has a negative maximum secrecy rate, we may consider only secure information transmission by neglecting the EH constraint in practical implementation of the studied IE broadcast system. that problem (5) has a positive optimal value, and focus our efforts on algorithm design to tackle problem (5). It is worth mentioning that, to the best of our knowledge, problem with has not yet considered in the literature. Moreover, the existing algorithms[32, 31, 30] developed for problem with or don’t apply to problem (5) due to the nonconvex EH constraint.
Iii Secure Beamforming Design: Global Solution To Two Special Cases
In this section, we investigate problem (5) by considering two special cases: singlestream case and fullstream case with . We propose global solutions to these two special cases.
Iiia Singlestream case:
For the singlestream case, we below propose an optimal solution to problem (5).
In the singlestream case, the beamforming matrix reduces to a vector. For notational simplicity and clearance, we use to denote when . Using the identity[35] , we can transform problem (5) equivalently to
(6) 
Problem (6) is a quadratically constrained quadratic fractional programming. By directly applying CharnesCooper transformation[26, 27] and semidefinite relaxation[23] to problem (6), we can turn the problem into a semidefinite programming (SDP) with three linear constraints and one additional variable (except the matrix variable). We below propose a more efficient solution to problem (6) by transforming the problem into a SDP with only two linear constraints.
Define , , . We have Lemma III.1.
Lemma III.1
The proof is relegated to Appendix A. Lemma III.1 shows that the involving fractional form of the objective function of problem (6) can be removed without introducing extra quadratic constraints. Moreover, the optimal solution to problem (6) can be easily obtained as long as problem (7) is solved.
Now let us consider how to solve problem (7). We note that the second quadratic constraint of problem (7) must be satisfied for any when . In this case, problem (7) simplifies to
(8) 
It is readily known[31] that the eigenvector (with proper scaling) of the matrix corresponding to the minimum eigenvalue is an optimal solution to problem (8).
When , problem (7) is a quadratic optimization problem with two quadratic constraints. It is wellknown[28] that such a quadratic optimization problem can be globally solved using semidefinite relaxation. Defining and ignoring the rankone constraint, we obtain the SDR of problem (7) as follows
(9) 
Problem (9) is a SDP which can be efficiently solved using interiorpoint algorithm[36]. If the optimal solution to problem (9), denoted by , satisfies , then the optimal solution to problem (7) can be obtained from the eigendecomposition of ; otherwise, if , we run rank reduction procedure[28] to to get a rankone solution to problem (9) and then perform eigendecomposition on the rankone solution to obtain the optimal solution to problem (7).
IiiB Fullstream case: and
When , is a square matrix. In this case, by using SDR, i.e., defining and dropping the rank constraint, we obtain the SDR of problem (5)
(10) 
It is easily seen that the SDR is tight when is a square matrix and thus problem (10) is equivalent to problem (5). Moreover, it can be shown that the objective function of problem (10) is concave when [32, 25]. To facilitate using some offtheshelf convex optimization tools (e.g., CVX[37]), we below reformulate problem (10) as an explicit convex problem with linear matrix inequality when .
Proposition III.1
Iv Secure Beamforming Design: KKT Solution To General Case
In this section, we consider problem (5) in the general case with arbitrary number of streams . We first propose inexact block coordinate descent (IBCD) algorithm to tackle problem (5) and then extend the IBCD algorithm to a more general case where artificial noise is employed to jam the energy harvester.
Iva Inexact Block Coordinate Algorithm For Problem (5)
Problem (5) is generally much harder to solve than problem (10) due to both the highly nonconcave objective function and the nonconvex constraint . To deal with the difficulties arising from the objective function and the nonconvex constraint, we first derive an equivalent problem of problem (5) and then propose inexact block coordinate descent algorithm for the resulting problem.
IvA1 Reformulation of problem (5)
To tackle the difficulty arising from the Shannon capacity expression in the objective function of problem (5), we extend the key idea of the popular WMMSE algorithm[32, 33], which is commonly used to address rate/sumrate maximization problems, to reformulating problem (5). The key idea behind the WMMSE algorithm is transforming a rate or sumrate maximization problem to another equivalent problem (by introducing auxiliary variables) which allows using simple block coordinate decent method[34]. Such an idea is based on three important facts which are summarized in Lemma IV.1.
Lemma IV.1
Define an by matrix function
where is any positive definite matrix. The following three facts hold true.

For any positive definite matrix , we have
(12) and
(13) 
For any positive definite matrix , we have
(14) and
(15) 
We have
(16)
Facts 1) and 2) can be proven by simply using the firstorder optimality condition, while Fact 3) directly follows from Facts 1) and 2) and the identity . We refer readers to [32, 33] for more detailed proof.
Next, using Lemma IV.1, we derive an equivalent problem of problem (5) by introducing some auxiliary variables. Define
(17) 
Then we have from Fact 3) that
(18) 
Furthermore, from Fact 1), we have
(19) 
Since the objective function of problem (5) is equivalent to the sum of the righthandside (RHS) of (18) and (19), problem (5) is equivalent to
(20) 
IvA2 Inexact block coordinate descent algorithm for problem (20)
Although problem (20) has more variables than problem (5), the former allows using simple block coordinate decent method, which optimizes the objective function over one variable (or one group of variables) while keeping all the others fixed at a time. In the BCD method applied to problem (20), it is required to iteratively solve three (or four) subproblems, among which, the most difficult one is
(21) 
which is obtained from (20) by fixing , and . Note that, problem (21) is a nonconvex problem due to the nonconvex EH constraint. To globally solve it, we need to use SDR and rankone reduction technique^{2}^{2}2Denote by the vectorization of the matrix variable . We first reformulate problem (21) as a quadratic optimization problem with respect to variable . Then, by defining and relaxing the rankone constraint, we can relax the resulting quadratic optimization problem as an SDP. Finally, from the SDR solution, we can find the optimal solution to problem (21) by performing eigendecomposition and rankone reduction[28].. As a result, the BCD method applied to problem (20) requires solving a number of semidefinite programmings as the iterations proceed, which makes the algorithm less efficient. For better efficiency and also ease of implementation, we propose inexact block coordinate descent (IBCD) method to tackle problem (20).
Similar to the BCD method, the IBCD method iteratively updates one (or one group of) variable while fixing the others. However, in the IBCD method, it is not required to globally solve all the subproblems; instead, we only find an inexact solution to some subproblems while keeping the objective function nondescending. Specifically, each iteration of the IBCD method consists of the following three subiterations.
Subiteration 1: Solve (20) for while fixing , and . This is equivalent to minimizing over . According to Fact 2) in Lemma IV.1, we obtain the optimal given as follows
(22) 
Subiteration 2: Solve (20) for and while fixing and . Note that the objective function of problem (20) is separable over and . Hence, Using Fact 1) in Lemma IV.1 twice, we can easily obtain the optimal and given and as follows
(23)  
(24) 
Subiteration 3: To update while fixing , , , we solve the following subproblem (instead of problem (21) in the BCD method):
(25) 
Problem (25) is obtained by replacing the quadratic function in the EH constraint of problem (21) with its firstorder approximation at , where is the update of obtained in the last iteration. In contrast to problem (21), problem (25) admits an efficient solution. As it will be shown later, although the solution to problem (25) is just a feasible solution to problem (21), it can keep the objective function of problem (20) nondescending.
Solution to problem (25): we here show how problem (25) can be solved efficiently. Note that problem (25) is a convex problem. Thus, it can be solved by dealing with its dual problem. To this end, by introducing Lagrange multiplier for the first constraint of problem (25), we define the partial Lagrangian associated with problem (25) as
(26) 
Furthermore, we define . Thus the second constraint of problem (25) can be compactly written as
and the dual problem of problem (21) is
(27) 
where is the dual function given by
(28) 
Note that problem (28) is a linearly constrained convex quadratic optimization problem. It can be solved in closedform by using Lagrange multiplier method. The solution to problem (28) given^{3}^{3}3If the solution to problem (28) with satisfies the total power constraint, then the optimal is zero. is summarized in Proposition IV.1.
Proposition IV.1
Let be the eigendecomposition of the matrix where consists of the orthonormal eigenvectors and is a diagonal matrix with each diagonal entry being the corresponding eigenvalue. Define . Given , the optimal solution to problem (28) can be expressed as
(29) 
where
(30) 
Moreover, is the derivative of .
The proof is easy and the details are omitted for brevity. Eq. (29) is obtained by using Lagrange multiplier method with being the optimal Lagrange multiplier associated with the linear constraint. Note that corresponds to the case when the solution to the unconstrained version of problem (28) satisfies the linear constraint. Furthermore, since the objective function of problem (28) given is strictly convex, problem (28) has a unique solution. It follows that is differentiable for and its derivative is simply . With this analytic form derivative, the dual problem (equivalently, problem (25)) can be efficiently solved using Bisection method[36], which is summarized in TABLE I.


Finally, we summarize the proposed algorithm^{4}^{4}4It is readily known that the complexity of the proposed algorithm is dominated by the eigendecomposition operation. Assuming , it can be shown that each iteration of the proposed IBCD method has complexity of . However, if we use SDR to directly solve problem (21), the complexity is at least [36]. for problem (5) in TABLE I, where Steps 47 correspond to the three subiterations of the IBCD method. Note that Step 5 follows from (23) and (15). The following proposition summarizes the convergence property of the IBCD method.
Proposition IV.2
The proof of Proposition (IV.2) is relegated to Appendix C. It indicates that the proposed algorithm monotonically converges to a stationary point of problem (5). The monotonic convergence is attractive since it guarantees an improved objective value with arbitrary random initialization. The convergence performance of the IBCD method is further explored later with numerical examples.
IvB Extension To Joint Artificial Noise and Beamforming Design
We here consider an extension of the IBCD algorithm to the case where the transmitter also sends artificial noise^{5}^{5}5The IBCD algorithm can be also extended to the AN plus energy beamforming case. (AN) to jam the energy harvester in order to achieve better secrecy rate[24, 30]. In this case, the transmitted signal is expressed as where represents the artificial noise with zero mean and covariance matrix . The achievable secrecy rate is given by
(31) 
The corresponding secrecy rate maximization problem is stated as
(32) 
By variable substitution with , problem (32) is equivalent to
(33) 
Next, we derive an equivalent problem of problem (33). First, we have
(34) 
Furthermore, according to Lemma IV.1, we have
(35)  
(36)  
(37) 
where
(38)  
(39) 
Therefore, the secrecy rate maximization problem in the AN case is equivalent to
(40) 
The IBCD algorithm can be generalized to tackle problem (40). In each iteration, given and , we can update , , , , in closedform, respectively, while, to update and given , we can linearize the EH constraint and solve the resulting problem using Bisection method. Furthermore, we can similarly prove that the algorithm can monotonically converge to a KKT point of problem (32).
V Numerical Examples
In this section, we provide numerical examples to illustrate the performance of the proposed beamforming algorithms. In all our simulations, we assume that both the IR and ER are equipped with two antennas. Moreover, we set dBm and . It is further assumed that the signal attenuation from the transmitter to both receivers is dB corresponding to an identical distance of about meters. The channels from the transmitter to both receivers are randomly generated from i.i.d Rayleigh fading with the average power specified as above (i.e., 1e5). It should be noted that, since the IBCD algorithm requires feasible initialization, we run a warmstart procedure to obtain an efficient feasible initial point. The warmstart procedure consists of the following three steps: 1) randomly generate ; 2) update , and as in Steps 46 in TABLE II; 3) obtain a feasible by solving (21) using SDR as argued in footnote 3. A similar warmstart procedure is also performed for the beamforming algorithm with artificial noise.
Va Convergence performance
First, we investigate the convergence performance of the IBCD algorithm for problem (5) by comparing with the global solutions in two special cases. We first consider the singlestream case with dBm, dBm, and . An example of convergence behavior of the IBCD algorithm is shown in Fig. 2, where circles represent different initializations and the dotted horizontal line denotes the optimal value obtained by the SDR method in Section III.A. It is observed that the IBCD algorithm can converge to the global optimal solution irrespective of initial points. We then consider the fullstream case with dBm, dBm, and
It can be easily verified that the matrix is positive semidefinite. Hence, the optimal value of problem (5) in this case can be obtained by the proposed method in Section III.B. Figure 3 shows the corresponding convergence performance of the IBCD algorithm, where the dotted horizontal line represents the optimal value. As in the singlestream case, it is observed that the IBCD algorithm has global convergence.
Then, we demonstrate the convergence performance of the generalized IBCD algorithm for problem (32). Figure 4 shows an example of convergence behavior of the generalized IBCD algorithm with dBm, dBm, and . It is seen that the generalized IBCD algorithm finally reaches the same objective value of problem (32) regardless of initial points.
VB Secrecy rate performance
In this set of simulations, we investigate the secrecy rate performance of the proposed beamforming algorithms with/without artificial noise. We set the number of streams to be . For both the (generalized) IBCD algorithm and the Bisection algorithm, we set to achieve a good accuracy.
First, we investigate the achieved secrecy rate versus the total transmission power, with the harvested power target, , being fixed as dBm. It is assumed that the transmitter is equipped with antennas. Figure 5 shows the achieved secrecy rate of the beamforming algorithms with and without artificial noise, where each data point is averaged over 100 random channel realizations. It is observed that the achieved secrecy rate increases with the total transmission power. Furthermore, it is seen that better secrecy rate can be achieved with the aid of artificial noise.
Next, we show in Fig. 6 the achieved secrecy rate versus the harvested power with fixed dBm and . It is observed that, for both artificial noise case and no artificial noise case, the secrecy rate decreases as the harvested power target increases. Moreover, similarly as in Fig. 5, it is seen that the ANaided beamforming design method outperforms the beamforming design method without AN in terms of the achieved secrecy rate.
At last, we plot the secrecy rate achieved by the proposed beamforming algorithms versus the number of transmit antennas in Fig. 7 with fixed dBm and dBm. Again, it is observed that the ANaided beamforming design method achieves better secrecy rate performance. However, the gap between the secrecy rate of the beamforming algorithms with and without artificial noise increases with the number of transmit antennas. This indicates that the artificial noise could impose more positive impact on the secrecy rate when the number of transmit antennas is large.
Vi Conclusions
This paper has studied secure beamforming design for a twouser MIMO informationenergy broadcast system. The problem of secrecy rate maximization subject to an energy harvesting constraint and a total power constraint is investigated. First, global optimal beamforming solutions are proposed for both the singlestream case and the fullstream case with channels satisfying positive semidefiniteness. Then, by developing the IBCD algorithm, a simple iterative beamforming solution is proposed for the general case with arbitrary number of streams. It is proven that the IBCD algorithm has monotonic convergence and any limit point of the IBCD algorithm is a KKT solution to the studied secrecy rate maximization problem. Furthermore, the IBCD algorithm is generalized to joint beamforming and artificial noise design. Finally, simulation results show that better secrecy rate is achieved with the aid of artificial noise.
Appendix A The Proof of Lemma iii.1
By assumption (i.e., problem (5) has positive optimal value), we have at the optimality of problem (6). On the other hand, it is known that the objective function of problem (6) is an increasing function in if . Hence, the total power constraint must be active at the optimality of problem (6). It follows that problem (6) has the same optimal solution set as that of the following problem
(41) 
Next, we reformulate problem (41) as problem (7). With variable substitution , problem (41) is equivalent to
(42) 
Note that the variable only appears in the two constraints of problem (42). By eliminating and combining the two constraints of problem (42), we obtain the following problem
(43) 
It is readily known that any feasible solution to problem (42) is feasible to problem (43). Moreover, given any feasible solution to problem (43), with is feasible to problem (42). Hence, problems (42) and (43) have the same feasible solution set regarding and thus have the same optimal solution set. Further, since scaling with any constant will not change the objective value while satisfying the constraint of problem (43), we can restrict to be equal to and rewrite problem (43) equivalently as (7).
Appendix B The Proof of Proposition iii.1
By noting that , we have
(44) 
where we use the identity in the three equalities. By replacing the objective of problem (10) with (44) and introducing the auxiliary variable , we rewrite problem (10) equivalently as
(45) 
Next, we prove that problem (45) is equivalent to problem (11). The proof is divided into two parts. The first part is to show that problem (45) is equivalent to
(46) 
while the second part is to prove that problem (46) can be recast as problem (11).
First, we prove the first part by showing that problems (45) and (46) have the same optimal value. Let be the optimal value of problem (45) and be an optimal solution to problem (46). Since problem (46) is a relaxation of problem (45), it follows that
(47) 
On the other hand, we have
(48) 
where the first inequality follows from the fact that, if , and the second inequality is due to the fact that is feasible to problem (45). Combining (47) and (48), we have . This implies that problems (45) and (46) are equivalent.
Next we prove the second part by showing that the first constraint of problem (46) can be recast as a linear matrix inequality (LMI). Since , we have
(49) 
By using the identity in (49), we obtain
(50) 
Leftmultiplying on both sides of (50) yields
(51) 
Using the identity [35, Sec. 3.2.4] in the RHS of Eq. (51), we obtain
(52) 
It follows that the first constraint of problem (46) is equivalent to