Multi-Objective Resource Allocation for Secure Communication in Cognitive Radio Networks with Wireless Information and Power Transfer
In this paper, we study resource allocation for multiuser multiple-input single-output (MISO) secondary communication systems with multiple system design objectives. We consider cognitive radio (CR) networks where the secondary receivers are able to harvest energy from the radio frequency when they are idle. The secondary system provides simultaneous wireless power and secure information transfer to the secondary receivers. We propose a multi-objective optimization framework for the design of a Pareto optimal resource allocation algorithm based on the weighted Tchebycheff approach. In particular, the algorithm design incorporates three important system design objectives: total transmit power minimization, energy harvesting efficiency maximization, and interference power leakage-to-transmit power ratio minimization. The proposed framework takes into account a quality of service (QoS) requirement regarding communication secrecy in the secondary system and the imperfection of the channel state information (CSI) of potential eavesdroppers (idle secondary receivers and primary receivers) at the secondary transmitter. The proposed framework includes total harvested power maximization and interference power leakage minimization as special cases. The adopted multi-objective optimization problem is non-convex and is recast as a convex optimization problem via semidefinite programming (SDP) relaxation. It is shown that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP relaxed problem. Besides, two suboptimal resource allocation schemes for the case when the solution of the dual problem is unavailable for constructing the optimal solution are proposed. Numerical results not only demonstrate the close-to-optimal performance of the proposed suboptimal schemes, but also unveil an interesting trade-off between the considered conflicting system design objectives.
Physical (PHY) layer security, cognitive radio (CR), wireless information and power transfer, robust beamforming.
The explosive growth of the demand for ubiquitous, secure, and high data rate wireless communication services has led to a tremendous solicitation of limited radio resources such as bandwidth and energy. In practice, fixed spectrum allocation has been implemented for resource sharing in traditional wireless communication systems. Although interference can be avoided by assigning different wireless services to different licensed frequency bands, such a fixed spectrum allocation strategy may result in spectrum under utilization. In fact, the Federal Communications Commission (FCC) has reported that percent of the allocated spectrum in the United States is not fully utilized, cf. . As a result, cognitive radio (CR) has emerged as one of the most promising solutions to improve spectrum efficiency . In particular, CR enables a secondary system to access the spectrum of a primary system as long as the interference from the secondary system does not severely degrade the quality of service (QoS) of the primary system –. CR is not only applicable to traditional cellular networks, but also has the potential to improve the performance of wireless sensor networks [5, 6]. In  and , cooperative spectrum sensing and the sensing-throughput trade-off were studied for single antenna systems, respectively. In , joint beamforming and power control was studied for transmit power minimization in multiple-transmit-antenna CR downlink systems. In  and , by taking into account the imperfectness of channel state information, robust beamforming designs were proposed for CR networks with single and multiple secondary users, respectively. Furthermore, a detailed performance analysis of transmit antenna selection in multiple-antenna networks was presented in  for multi-relay networks and then extended to CR relay networks in . However, since the transmit precoding strategies in  and  are not optimized, they do not fully exploit the available degrees of freedom in the network for maximizing the system performance.
Although the current spectrum scarcity may be partially overcome by CR technology, wireless communication devices, such as wireless sensors, are often powered by batteries with limited energy storage capacity. This constitutes another major bottleneck for providing communication services and extending the lifetime of networks. On the other hand, energy harvesting is envisioned to provide a perpetual energy source to facilitate self-sustainability of power-constrained communication devices –. In addition to conventional renewable energy sources such as biomass, wind, and solar, wireless power transfer has emerged as a new option for prolonging the lifetime of battery-powered wireless devices. Specifically, the transmitter can transfer energy to the receivers via electromagnetic waves in radio frequency (RF). Nowadays, energy harvesting circuits are able to harvest microwatt to milliwatt of power over the range of several meters for a transmit power of Watt and a carrier frequency of less than GHz . Thus, RF energy can be a viable energy source for devices with low-power consumption, e.g. wireless sensors [16, 17]. The integration of RF energy harvesting capabilities into communication systems provides the possibility of simultaneous wireless information and power transfer (SWIPT) –. As a result, in addition to the traditional QoS constraints such as communication reliability, efficient energy transfer is expected to play an important role as a new QoS requirement. This new requirement introduces a paradigm shift in the design of both resource allocation algorithms and transceiver signal processing. In  and , the fundamental trade-off between the maximum achievable data rate and energy transfer was studied for a noisy single-user communication channel and a pair of noisy coupled-inductor circuits, respectively. Then, in , the authors extended the trade-off study to a two-user multiple-antenna transceiver system. In , the authors proposed separated receivers for SWIPT to facilitate low-complexity receiver design; these receivers can be built by using off-the-shelf components. In , different resource allocation algorithms were designed for broadband far field wireless systems with SWIPT. In , the authors showed that the energy efficiency of a communication system can be improved by RF energy harvesting at the receivers. Nevertheless, resource allocation algorithms maximizing the energy harvesting efficiency of SWIPT CR systems have not been reported in the literature yet. Besides, two conflicting system design objectives arise naturally for a CR network providing SWIPT service to the secondary receivers in practice. On the one hand, the secondary transmitter should transmit with high power to facilitate energy transfer to the energy harvesting receivers. On the other hand, the secondary transmitter should transmit with low power to cause minimal interference at the primary receivers. Thus, considering these conflicting system design objectives, the single objective resource allocation algorithms proposed in – may not be applicable in SWIPT CR networks. Furthermore, transmitting with high signal power may also cause substantial information leakage and high vulnerability to eavesdropping.
Recently, physical (PHY) layer security has attracted much attention in the research community for preventing eavesdropping –. In , the authors proposed a beamforming scheme for maximization of the energy efficiency of secure communication systems. In  and , the spatial degrees of freedom offered by multiple antennas were used to degrade the channel of the eavesdroppers deliberately via artificial noise transmission. Thereby, communication secrecy was guaranteed at the expense of allocating a large portion of the transmit power to artificial noise generation. In , the authors addressed the power allocation problem in CR secondary systems with PHY layer security provisioning. However, the resource allocation algorithm designs in – cannot be directly extended to the case of of RF energy harvesting due to the differences in the underlying system models. On the other hand,  and  studied different resource allocation algorithms for providing secure communication in systems with separated information and energy harvesting receivers. Yet, the assumption of having perfect channel state information (CSI) of the energy harvesting receivers in  and  may be too optimistic if the energy harvesting receivers do not interact with the transmitter periodically. In , the case where the transmitter has only imperfect CSI of the energy harvesting receivers was considered and a robust beamforming design was proposed to minimize the total transmit power of a system with simultaneous energy and secure information transfer. In , the authors studied resource allocation algorithm design for secure information and renewable green energy transfer to mobile receivers in distributed antenna communication systems. In  and , beamforming algorithm design and secrecy outage capacity was studied for multiple-antenna potential eavesdropper and passive eavesdroppers, respectively. However, the beamforming algorithms developed in – may not be applicable in CR networks. Furthermore, in , the secrecy outage probability of CR networks was investigated in the presence of a passive eavesdropper.
Form the above discussions, we conclude that for CR communication systems providing simultaneous wireless energy transfer and secure communication services, conflicting system design objectives such as total transmit power minimization, energy harvesting efficiency maximization, and interference power leakage-to-transmit power ratio minimization play an important role for resource allocation. However, the problem formulations in – focus on a single system design objective and cannot be used to study the trade-off between the aforementioned conflicting design goals. In this paper, we address the above issues and the contributions of the paper are summarized as follows:
Different from our previous work in , in this paper, we propose a new non-convex multi-objective optimization problem with the aim to jointly minimize the total transmit power, maximize the energy harvesting efficiency, and minimize the interference power leakage-to-transmit power ratio for CR networks with SWIPT. The problem formulation takes into account the imperfectness of the CSI of potential eavesdroppers (idle secondary receivers) and primary receivers in secondary multiuser multiple-input single-output (MISO) systems with RF energy harvesting receivers. The solution of the optimization problem leads to a set of Pareto optimal resource allocation policies.
The considered non-convex optimization problem is recast as a convex optimization problem via semidefinite programming (SDP) relaxation. We show that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP relaxed problem.
The obtained solution structure is also applicable to the multi-objective optimization of the total harvested power, the interference power leakage, and the total transmit power.
Two suboptimal resource allocation schemes are proposed for the case when the solution of the dual problem of the SDP relaxed problem is unavailable for construction of the optimal solution.
Our results unveil a non-trivial trade-off between the considered system design objectives which can be summarized as follows: (1) A resource allocation policy minimizing the total transmit power also leads to a low total interference power leakage in general; (2) energy harvesting efficiency maximization and transmit power minimization are conflicting system design objectives; (3) the maximum energy harvesting efficiency is achieved at the expense of high interference power leakage and high transmit power.
Ii System Model
In this section, we first introduce the notation used in this paper. Then, we present the adopted CR downlink channel model for secure communication with SWIPT.
We use boldface capital and lower case letters to denote matrices and vectors, respectively. For a square-matrix , denotes the trace of matrix . and indicate that is a positive definite and a positive semidefinite matrix, respectively. and denote the conjugate transpose and the rank of matrix , respectively. denotes an identity matrix. and denote the space of matrices with complex and real entries, respectively. represents the set of all -by- complex Hermitian matrices. and denote the absolute value of a complex scalar and the Euclidean norm of a matrix/vector, respectively. denotes a diagonal matrix with the diagonal elements given by . extracts the real part of a complex-valued input. The distribution of a circularly symmetric complex Gaussian (CSCG) vector with mean vector and covariance matrix is denoted by , and means “distributed as”. represents statistical expectation. For a real valued continuous function , denotes the gradient of with respect to matrix . stands for .
Ii-B Downlink Channel Model
We consider a CR secondary network for short distance downlink communication. There are one secondary transmitter equipped with antennas, secondary receivers, one primary transmitter111We note that the considered system model can be extended to include multiple primary transmitters at the expense of a more involved notation., and primary receivers. The primary transmitter, primary receivers, and secondary receivers are single-antenna devices that share the same spectrum, cf. Figure 1. We assume to enable efficient communication in the CR secondary network. The secondary transmitter provides SWIPT services to the secondary receivers while the primary transmitter provides broadcast services to the primary receivers. In practice, the CR secondary operator may rent spectrum from the primary operator under the condition that the interference leakage from the secondary system to the primary system is properly controlled. We assume that the secondary receivers are ultra-low power devices, such as wireless sensors222 The power consumption of typical sensor micro-controllers, such as the Texas Instruments micro-controller: MSP430F2274 , is in the order of microwatt in the idle mode. As a result, wireless power transfer is a viable option for the energy supply of wireless sensors., which either harvest energy or decode information from the received radio signals in each time instant, but are not able to perform both concurrently due to hardware limitations [22, 24]. In each scheduling slot, the secondary transmitter not only conveys information to a given secondary receiver, but also transfers energy333We adopt the normalized energy unit Joule-per-second in this paper. Therefore, the terms “power” and “energy” are used interchangeably. to the remaining idle secondary receivers to extend their lifetimes.
We note that only one secondary receiver is selected for information transfer to reduce the multiple access interference leakage to the primary receivers . On the other hand, the information signal of the desired secondary receiver is overheard by both the idle secondary receivers and the primary receivers. Hence, if the idle secondary receivers and the primary receivers are malicious, they may eavesdrop the signal of the selected secondary receiver, which has to be taken into account for resource allocation design to provide communication secrecy in the secondary network. Thus, for guaranteeing communication security, the secondary transmitter has to employ a resource allocation algorithm that accounts for this unfavourable scenario and treat both idle secondary receivers and primary receivers as potential eavesdroppers. We assume a frequency flat slow fading channel. The received signals at the desired secondary receiver, idle secondary receiver , and primary receiver are given by, respectively,
Here, denotes the symbol vector transmitted by the secondary transmitter. , , and are the channel vectors between the secondary transmitter and the desired secondary receiver, idle receiver (potential eavesdropper) , and primary receiver (potential eavesdropper) , respectively. and are the transmit power of the primary transmitter and the information signal intended for the primary receivers, respectively. , , and are the communication channels between the primary transmitter and desired secondary receiver, idle secondary receiver , and primary receiver , respectively. includes the joint effects of the thermal noise and the signal processing noise at primary receiver and is modelled as additive white Gaussian noise (AWGN) with zero mean and variance444We assume that the noise characteristics are identical for all primary receivers due to similar hardware architectures. . and include the joint effects of thermal noise and signal processing noise at the desired secondary receiver and idle secondary receiver , respectively, and are modeled as AWGN. Besides, the equivalent noises at the desired and idle secondary receivers, which capture the joint effect of the received interference from the primary transmitter, i.e., and , thermal noise, and signal processing noise, are also modeled as AWGN with zero mean and variances and , respectively.
In this paper, we assume that the primary network is a legacy system and the primary transmitter does not actively participate in transmit power control. Furthermore, we assume that the primary transmitter transmits a Gaussian signal and we focus on quasi-static fading channels such that all channel gains remain constant within the coherence time of the secondary system. These assumptions justify modelling the interference from the primary transmitter to the secondary receivers as additive white Gaussian noise with different powers for different secondary receivers. This model has been commonly adopted in the literature for resource allocation algorithm design [10, 28, 36].
To guarantee secure communication and to facilitate an efficient power transfer in the secondary system, artificial noise is generated at the secondary transmitter and is transmitted concurrently with the information signal. In particular, the transmit signal vector
is adopted at the secondary transmitter, where and are the information bearing signal for the desired receiver and the corresponding beamforming vector, respectively. We assume without loss of generality that . is the artificial noise vector generated by the secondary transmitter to combat the potential eavesdroppers. Specifically, is modeled as a complex Gaussian random vector with mean and covariance matrix . We note that and have to be optimized such that the transmit signal of the secondary transmitter does not interfere severely with the primary users.
Iii Resource Allocation Problem Formulation
In this section, we define different quality of service (QoS) measures for the secondary CR network for providing wireless power transfer and secure communication to the secondary receivers while protecting the primary receivers. Then, we formulate three resource allocation problems reflecting three different system design objectives. For convenience, we define the following matrices: , , and .
Iii-a System Achievable Rate and Secrecy Rate
Given perfect CSI at the receiver, the achievable rate (bit/s/Hz) between the secondary transmitter and the desired secondary receiver is given by
where is the received signal-to-interference-plus-noise ratio (SINR) at the desired secondary receiver. On the other hand, the achievable rates between the secondary transmitter and idle secondary receiver and primary receiver are given by
respectively, where and are the received SINRs at idle secondary receiver and primary receiver , respectively. Since both the idle secondary receivers and the primary receivers are potential eavesdroppers, the maximum achievable secrecy rate between the secondary transmitter and the desired receiver is given by
In the literature, the secrecy rate, i.e., (8), is commonly adopted as a QoS requirement for system design to ensure secure communication [26, 27]. In particular, quantifies the maximum achievable data rate at which a transmitter can reliably send secret information to the intended receiver such that the eavesdroppers are unable to decode the received signal  even if the eavesdroppers have unbounded computational capability555 We note that, in practice, the malicious secondary idle receivers and primary receivers do not have to decode the eavesdropped information in real time. They can act as information collectors to sample the received signals and store them for future decoding by other energy unlimited and powerful computational devices..
Iii-B Energy Harvesting Efficiency
In the considered CR system, the secondary receivers harvest energy from the RF when they are idle to extend their lifetimes666 In fact, nowadays many sensors are equipped with hybrid energy harvesters for harvesting energy from different energy sources such as solar and thermal-energy [38, 39]. Thus, the harvested energy from the radio frequency may be used as a supplement for supporting the energy consumption of the secondary receivers.. The energy harvesting efficiency plays an important role in the system design of such secondary networks and has to be considered in the problem formulation. To this end, we define the energy harvesting efficiency in the secondary system as the ratio of the total power harvested at the idle secondary receivers and the total power radiated by the secondary transmitter. The total amount of energy harvested by the idle secondary receivers is modeled as
where is a constant, , which represents the RF energy conversion efficiency of idle secondary receiver in converting the received radio signal to electrical energy. We note that the power received at the secondary receivers from the primary transmitter and the AWGN power are neglected in (9) as we focus on the worst-case scenario for robust energy harvesting system design.
On the other hand, the power radiated by the transmitter can be expressed as
Thus, the energy harvesting efficiency of the considered secondary CR system is given by
Iii-C Interference Power Leakage-to-Transmit Power Ratio
In the considered CR network, the secondary receivers and the primary receivers share the same spectrum resource. However, the primary receivers are licensed users and thus the secondary transmitter is required to ensure the QoS of the primary receivers via a careful resource allocation design. Strong interference may impair the primary network when the secondary transmitter increases its transmit power for providing SWIPT services to the secondary receivers. As a result, the interference power leakage-to-transmit power ratio (IPTR) is an important performance measure for designing the secondary CR network and should be captured in the resource allocation algorithm design. To this end, we first define the total interference power received by the primary receivers as
Thus, the IPTR of the considered secondary CR network is defined as
Iii-D Channel State Information
In this paper, we focus on a Time Division Duplex (TDD) communication system with slowly time-varying channels. In practice, handshaking777The legitimate receivers can either take turns to send the handshaking signals or transmit simultaneously with orthogonal pilot sequences. is performed between the secondary transmitter and the secondary receivers at the beginning of each scheduling slot. This allows the secondary transmitter to obtain the statuses and the QoS requirements of the secondary receivers. As a result, by exploiting the channel reciprocity, the downlink CSI of the secondary transmitter to the secondary receivers can be obtained by measuring the uplink training sequences embedded in the handshaking signals. Thus, we assume that the secondary-transmitter-to-secondary-receiver fading gains, and , can be reliably estimated at the secondary transmitter at the beginning of each scheduling slot with negligible estimation error. Then, during the transmission, the desired secondary receiver is required to send positive acknowledgement (ACK) packets to inform the secondary transmitter of successful reception of the information packets. Hence, the transmitter is able to update the CSI estimate of the desired receiver frequently via the training sequences in each ACK packet. Therefore, perfect CSI for the secondary-transmitter-to-desired-secondary-receiver link, i.e., , is assumed over the entire transmission period. However, the remaining secondary receivers are idle and there is no interaction between them and the secondary transmitter after handshaking. As a result, the CSI of the idle secondary receivers becomes outdated during transmission. To capture the impact of the CSI imperfection and to isolate specific channel estimation methods from the resource allocation algorithm design, we adopt a deterministic model – for the resulting CSI uncertainty. In particular, the CSI of the link between the secondary transmitter and idle secondary receiver is modeled as
where is the CSI estimate available at the secondary transmitter at the beginning of a scheduling slot and represents the unknown channel uncertainty due to the time varying nature of the channel during transmission. The continuous set in (15) defines a Euclidean sphere and contains all possible channel uncertainties. Specifically, the radius represents the size of the sphere and defines the uncertainty region of the CSI of idle secondary receiver (potential eavesdropper) . In practice, the value of depends on the coherence time of the associated channel and the duration of transmission.
Furthermore, to capture the imperfectness of the CSI of the primary receiver channels at the secondary transmitter, we adopt the same CSI error model as for the idle secondary receivers. In fact, the primary receivers are not directly interacting with the secondary transmitter. Besides, the primary receivers may be silent for non-negligible periods of time due to bursty data communication. As a result, the CSI of the primary receivers can be obtained only occasionally at the secondary transmitter when the primary receivers communicate with a primary transmitter. Hence, we model the CSI of the link between the secondary transmitter and primary receiver as
where is the estimate of the channel of primary receiver at the secondary transmitter and denotes the associated channel uncertainty. and in (17) define the continuous set of all possible channel uncertainties and the size of the uncertainty region of the estimated CSI of primary receiver , respectively. We note that, in practice, the channel estimation qualities for primary receivers and secondary receivers at the secondary transmitter may be different which leads to different values for and .
Iii-E Optimization Problem Formulations
We first propose three problem formulations for single-objective system design for secure communication in the secondary CR network. In particular, each single-objective problem formulation considers one aspect of the system design. Then, we consider the three system design objectives jointly under the framework of multi-objective optimization. In particular, the adopted multi-objective optimization enables the design of a set of Pareto optimal resource allocation policies. The first problem formulation aims at maximizing the energy harvesting efficiency while providing secure communication in the secondary CR network. The problem formulation is as follows:
Energy Harvesting Efficiency Maximization:
The system objective in (III-E) is to maximize the worst case energy harvesting efficiency of the system for channel estimation errors belonging to set . Constant in C1 specifies the minimum required received SINR of the desired secondary receiver for information decoding. , and , in C2 and C3, respectively, are given system parameters which denote the maximum tolerable received SINRs at the potential eavesdroppers in the secondary network and the primary network, respectively. In practice, depending on the considered application, the system operator chooses the values of , , and , such that and . In other words, the secrecy rate of the system is bounded below by . We note that although , , and in C1, C2, and C3, respectively, are not optimization variables in this paper, a balance between secrecy rate and system achievable rate can be struck by varying their values. in C4 specifies the maximum transmit power in the power amplifier of the analog front-end of the secondary transmitter. C5 and are imposed since covariance matrix has to be a positive semidefinite Hermitian matrix.
The second system design objective is the minimization of the total transmit power of the secondary transmitter and can be mathematically formulated as:
Total Transmit Power Minimization:
Problem 2 yields the minimum total transmit power of the secondary transmitter while ensuring that the QoS requirement on secure communication is satisfied. We note that Problem 2 does not take into account the energy harvesting capability of the idle secondary receivers and focuses only on the requirement of secure communication via constraints C1, C2, and C3. Besides, although transmit power minimization has been studied in the literature in different contexts [1, 45, 46], combining Problem 2 with the new Problems 1 and 3 (see below) offers new insights for the design of CR networks providing secure wireless information and power transfer to secondary receivers.
The third system design objective concerns the minimization of the worst case IPTR while providing secure communication in the secondary CR network. The problem formulation is given as:
Interference Power Leakage-to-Transmit Power Ratio Minimization:
In (III-E) and (III-E), the maximization of the energy harvesting efficiency and the minimization of the IPTR are chosen as design objectives, respectively. Alternative design objectives are the maximization of the total harvested power, , and the minimization of the total interference power leakage, . We will show later that the maximization of the energy harvesting efficiency in (III-E) and the minimization of the IPTR in (III-E) subsume the total harvested power maximization and the total interference power leakage minimization as special cases, respectively. Please refer to Remark 7 for the solution of the total interference power leakage minimization and total harvested power maximization problems.
In fact, the optimization problem in (III-E) can be extended to the minimization of the maximum received interference leakage per primary receiver. However, such problem formulation does not facilitate the study of the trade-off between interference leakage, energy harvesting, and total transmit power as the system performance is always limited by those primary users which have strong channels with respect to the secondary transmitter.
In practice, the system design objectives in Problems – are all desirable for the system operators of secondary CR networks in providing simultaneous power and secure information transfer. Yet, theses objectives are usually conflicting with each other and each objective focuses on only one aspect of the system. In the literature, multi-objective optimization has been proposed for studying the trade-off between conflicting system design objectives via the concept of Pareto optimality. For facilitating the following exposition, we denote the objective function and the optimal objective value for problem formulation as and , respectively. We define a resource allocation policy which is Pareto optimal as:
Definition : A resource allocation policy, , is Pareto optimal if and only if there does not exist another policy, , such that , and for at least one index .
The set of all Pareto optimal resource allocation polices is called the Pareto frontier or the Pareto optimal set. In this paper, we adopt the weighted Tchebycheff method  for investigating the trade-off between objective functions 1, 2, and 3. In particular, the weighted Tchebycheff method can provide the complete Pareto optimal set despite the non-convexity (if any) of the considered problems888In the literature, different scalarization methods have been proposed for achieving the points of the complete Pareto set for multi-objective optimization [47, 48]. However, the weighted Tchebycheff method requires a lower computational complexity compared to other methods such as the weighted product method and the exponentially weighted criterion.; it provides a necessary condition for Pareto optimality. The complete Pareto optimal set can be achieved by solving the following multi-objective problem:
Multi-Objective Optimization – Weighted Tchebycheff Method:
In fact, by varying the values of , Problem 4 yields the complete Pareto optimal set [47, 48]. Besides, Problem 4 is a generalization of Problems 1, 2, and 3. In particular, Problem 4 is equivalent999Here, “equivalent” means that the considered problems share the same optimal resource allocation solution(s). to Problem when and . For instance, if the secondary energy harvesting receivers do not require wireless power transfer from the secondary transmitter, without loss of generality, we can set in Problem 4 to study the tradeoff between the remaining two system design objectives. In addition, this commonly adopted approach also provides a non-dimensional objective function, i.e., the unit of the objective function is normalized.
Finding the Pareto optimal set of the multi-objective optimization problem provides a set of Pareto optimal resource allocation policies. Then, depending on the preference of the system operator, a proper resource allocation policy can be selected from the set for implementation. We note that the resource allocation algorithm in  cannot be directly applied to the problems considered in this paper since it was designed for single-objective optimization, namely for the for minimization of the total transmit power.
Another possible problem formulation for the considered system model is to move some of the objective functions in (III-E), (III-E), and (III-E) to the set of constraints and constrain each of them by some constant. Then, by varying the constants, trade-offs between different objectives can be struck. However, in general, such a problem formulation does not reveal the Pareto optimal set due to the non-convexity of the problem.
Iv Solution of the Optimization Problems
The optimization problems in (III-E), (III-E), and (III-E) are non-convex with respect to the optimization variables. In particular, the non-convexity arises from objective function 1, objective function 3, and constraint C1. In order to obtain tractable solutions for the problems, we recast Problems 1, 2, 3, and 4 as convex optimization problems by semidefinite programming (SDP) relaxation [49, 50] and study the tightness of the adopted relaxation in this section.
Iv-a Semidefinite Programming Relaxation
To facilitate the SDP relaxation, we define
and rewrite Problems 1 – 4 in terms of new optimization variables , , and .
Transformed Problem 1
Energy Harvesting Efficiency Maximization:
where , , and in (IV-A) are imposed to guarantee that after optimizing .
Transformed Problem 2
Total Transmit Power Minimization:
Transformed Problem 3
Interference Power Leakage-to-Transmit Power Ratio Minimization:
Transformed Problem 4
Please refer to Appendix A. Since transformed Problem 4 is a generalization of transformed Problems 1, 2, and 3, we focus on the methodology for solving transformed Problem101010In studying the solution structure of transformed Problem 4, we assume that the optimal objective values of transformed Problems 1–3 are given constants, i.e., , are known. Once the structure of the optimal resource allocation scheme of transformed Problem 4 is obtained, it can be exploited to obtain the optimal solution of transformed Problems 1–3. 4. In practice, the considered problems may be infeasible when the channels are in unfavourable conditions and/or the QoS requirements are too stringent. However, in the sequel, for studying the trade-off between different system design objectives and the design of different resource allocation schemes, we assume that the problem is always feasible111111We note that multiple optimal solutions may exist for the considered problems and the proposed optimal resource allocation scheme is able to find at least one of the global optimal solutions..
First, we address constraints , , and . We note that although these constraints are convex with respect to the optimization variables, they are semi-infinite constraints which are generally intractable. For facilitating the design of a tractable resource allocation algorithm, we introduce two auxiliary optimization variables and and rewrite transformed Problem 4 in (IV-A) as
In fact, the introduced auxiliary variables and decouple the original two nested semi-infinite constraints into two semi-infinite constraints and two affine constraints, i.e., , and a, c, respectively. It can be verified that (IV-A) is equivalent to (26), i.e., constraints and are satisfied with equality for the optimal solution. Next, we transform constraints , , , and into linear matrix inequalities (LMIs) using the following lemma:
Lemma 1 (S-Procedure )
Let a function be defined as
where , , and . Then, the implication holds if and only if there exists a such that
provided that there exists a point such that .
Now, we apply Lemma 1 to constraint . In particular, we substitute into constraint . Therefore, the implication,
holds if and only if there exists a such that the following LMI constraint holds:
for where .
Similarly, we rewrite constraints , , and in the form of (29) which leads to
By using Lemma 1, constraint , , and can be equivalently written as
respectively, with and new auxiliary optimization variables and . We note that now constraints , and involve only a finite number of convex constraints which facilitates an efficient resource allocation algorithm design. As a result, we obtain the following equivalent optimization problem on the top of this page in (IV-A), where denotes the set of optimization variables after transformation; and are auxiliary variable vectors with elements and , respectively; , and are auxiliary optimization variable vectors with elements , and connected to the constraints in (32)–(IV-A), respectively.
The remaining non-convexity of problem (IV-A) is due to the combinatorial rank constraint in on the beamforming matrix . In fact, by relaxing constraint , i.e., removing it from (IV-A), the considered problem is a convex SDP and can be solved efficiently by numerical solvers such as SeDuMi  and CVX . Besides, if the obtained solution for the relaxed SDP problem admits a rank-one matrix , i.e., , then it is the optimal solution of the original problem. In general, the adopted SDP relaxation may not yield a rank-one solution and the result of the relaxed problem serves as a performance upper bound for the original problem. Nevertheless, in the following, we show that there always exists an optimal solution for the relaxed problem with . In particular, the optimal solution of the relaxed version of (IV-A) with