Efficiency Maximization for UAV-Enabled Mobile Relaying Systems with Laser Charging

This work studies the joint problem of power and trajectory optimization in an unmanned aerial vehicle (UAV)-enabled mobile relaying system. In the considered system, in order to provide convenient and sustainable energy supply to the UAV relay, we consider the deployment of a power beacon (PB) which can wirelessly charge the UAV and it is realized by a properly designed laser charging system. To this end, we propose an efficiency (the weighted sum of the energy efficiency during information transmission and wireless power transmission efficiency) maximization problem by optimizing the source/UAV/PB transmit powers along with the UAV's trajectory. This optimization problem is also subject to practical mobility constraints, as well as the information-causality constraint and energy-causality constraint at the UAV. Different from the commonly used alternating optimization (AO) algorithm, two joint design algorithms, namely: the concave-convex procedure (CCCP) and penalty dual decomposition (PDD)-based algorithms, are presented to address the resulting non-convex problem, which features complex objective function with multiple-ratio terms and coupling constraints. These two very different algorithms are both able to achieve a stationary solution of the original efficiency maximization problem. Simulation results validate the effectiveness of the proposed algorithms.

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I Introduction

Thanks to the continuous cost reduction and device miniaturization in unmanned aerial vehicles (UAVs), wireless communications equipped and enabled by UAVs have attracted a lot of attentions recently, such as relaying, data gathering, secure transmission and information dissemination, etc [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In order to provide wireless data service for devices without infrastructure coverage due to, e.g., severe blocking by urban or mountainous terrain, communications infrastructure failure caused by natural disasters, etc., UAV-enabled wireless communication exhibits great potential in providing throughput/reliability improvement and coverage extension. Among the various applications enabled by UAVs, the use of UAVs as relay nodes for achieving high-speed and reliable wireless communications between two or more distant users whose direct communication links are blocked or corrupted, is expected to play an important role in future communication systems [1, 3].

I-a Related Works and Motivation

UAV relays can be generally categorized into two types, i.e., statistic relaying and mobile relaying. The researches on statistic UAV relaying usually aim to find the best UAV position that maximizes the performance of the wireless network, along with the corresponding resource allocation strategy [12, 13, 14, 15, 16, 17, 18, 19]. Specifically, in [12], an algorithm was proposed to find the optimal position of the UAV based on the fine-grained line-of-sight (LoS) information. The work [13] investigated the optimum placement of UAV, where the total power loss, the overall outage and bit error rate were derived as reliability measures. The work [14] studied the optimal placement problem of a UAV relay without the need of any prior knowledge on the user locations and the underlying wireless channel pathloss parameters. In [15], a system of multiple communication pairs with one UAV relay was considered, the node placement and resource allocation was jointly optimized. In [16], joint 3D location and power optimization was investigated. Placement of multiple UAVs was considered in [17], where the cases that multiple UAVs form either a single multi-hop link or multiple dual-hop links were analyzed. The work [18] proposed to use UAVs as floating relaying nodes in order to resolve the problem of undesirable channel conditions of indoor users. The work [19] considered a UAV-enabled two-way relaying system, where the joint optimization of UAV positioning and transmit powers was studied.

Compared to the statistic relaying scheme, the deployment of UAVs which serve as mobile relaying nodes is a more cost-effective solution to extend the wireless communication range and offer more reliable connectivities. Generally, two distinct advantages can be achieved by UAV-enabled mobile relaying systems: 1) enhanced performance brought up by the dynamic adjustment of relay locations to better coordinate with the environment; 2) the high mobility of UAVs enables the system to provide more flexible and responsive serves. As a result, the exploitation and exploration of UAV-enabled mobile relaying for more efficient physical layer designs have received a lot of attention recently [20, 3, 21, 22, 23, 24]. In particular, the work [20]

proposed to use a mobile relay to carry data for several isolated communities and a genetic algorithm was designed where the trajectories of the mobile relay were represented by chromosomes that evolve to approximate the optimal solution. In

[3], the throughput maximization problem in a decode-and-forward (DF) mobile relaying system was studied by jointly optimizing the source/relay transmit powers and the relay trajectory. An alternating optimization (AO)-based algorithm was proposed to optimize the power allocation and relay trajectory in a sequential manner. The work [22] extended that of [3] to the multi-hop scenario, where a single multi-hop link was considered. The works [21] and [24] investigated the use of amplify-and-forward (AF) relay strategy. In [23], the spectrum efficiency and energy efficiency were optimized by assuming that the circular trajectory and time-division duplexing (TDD) were adopted. Furthermore, UAV-enabled mobile relaying can also be utilized to facilitate secure transmissions [25, 26, 27, 28, 29], full-duplex communications [30] and wireless power transfer (WPT) [31], etc.

Despite the various benefits brought about by UAV-enabled mobile relaying, the UAV’s operations are usually restricted by many energy-consuming factors, such as the propulsion power to support its mobility, communication with the ground devices, etc. Therefore, many of the advantages of UAV-enabled wireless communication systems would be untouchable if the UAV’s battery capacity is limited and no additional power supply is available. Recently, laser power is becoming a viable solution to prolong the flight time of UAVs [32, 33]. Compared to other WPT techniques enabled by wind, sunlight, or radio frequency (RF) signals, the laser-beamed power supply is more stable and it can deliver much larger energy amounts. It is regarded as an important technique for emergency responses, military operations, and also to accelerate the pace of implementing 5G-oriented UAV networks [34]. Moreover, the field tests conducted in [35] have validated the feasibility of laser-powered UAVs. Therefore, in order to provide convenient and sustainable energy supply to the UAV, we consider the employment of a laser power beacon (PB), which is able to send laser beams to charge the UAV in flight. As a result, in the considered mobile relaying system, we need to take the energy-causality constraint at the UAV relay into consideration, i.e., the total energy consumption of the UAV relay at the current time slot cannot exceed its remaining battery storage, in order to maintain its sustainable operations.

I-B Our Contributions

To this end, we propose an efficiency maximization problem, where the energy efficiency during information transmission and the laser power transmission efficiency are both taken into consideration by adding an adjustable weighting factor between them. In the considered problem, the UAV’s trajectory and the transmit powers of the source, UAV and laser PB are jointly optimized under the mobility constraints, information-causality and energy-causality constraints at the UAV. This joint design problem is very challenging due to the facts that the objective function is in a multiple-ratio form, the constraints are highly non-convex and the optimization variables are tightly coupled both in the objective and constraints. By taking advantage of the problem structure, we propose two algorithms which can both converge to the set of stationary solutions. The first algorithm, i.e., the concave-convex procedure (CCCP)-based algorithm, is designed by carefully introducing auxiliary variables and approximating the underlying non-convex components in the considered problem by convex ones. To derive the second algorithm, we employ the penalty dual decomposition (PDD) framework [36] and demonstrate that the optimization variables as well as the introduced auxiliary variables can be decoupled into several separate blocks. Then, the joint design problem can be addressed by iterating over a sequence of simple and efficient updates in each block of variables. These two algorithms exhibit similar performance in simulations, but they are essentially very different and each of them offers different advantages, i.e., the CCCP-based algorithm is able to converge within fewer iterations, while the PDD-based algorithm is more implementation-friendly.

The main contributions of this work can be summarized as follows:

1) A general optimization framework for joint power allocation and trajectory design in a UAV-enabled mobile relaying system with laser charging is proposed. In particular, the weighted sum of the information transmission efficiency and power transmission efficiency is proposed as the objective function, the source/UAV/PB transmit powers and the relay trajectory are jointly optimized under the mobility, information-causality and energy-causality constraints.

2) Despite the highly non-convexity of the considered problem and the intrinsic coupling in the optimization variables, two joint design algorithms, i.e., the CCCP and PDD-based algorithms, are proposed which are both guaranteed to converge to the set of stationary solutions.

3) In order to validate the effectiveness of the proposed algorithms, computer simulations are conducted and the performance of the AO-based algorithm is also investigated for comparison. We demonstrate that the proposed joint design algorithms are able to outperform the commonly used AO-based algorithm. Furthermore, the impacts of different laser wavelengths and weather conditions are shown, as well as the tradeoff between the information/power transmission efficiencies.

I-C Organization of the Paper and Notations

The rest of the paper is organized as follows. In Section II, we present the considered UAV-enabled mobile relaying system model and the corresponding problem formulation. In Section III and IV, the proposed CCCP and PDD-based algorithms are developed, respectively, along with their complexity analysis. In Section V, simulations are conducted to characterize the performance of the proposed algorithms and Section VI concludes the paper.

Notations:

Scalars, vectors and matrices are respectively denoted by lower case, boldface lower case and boldface upper case letters. For a matrix

, and denote its transpose and conjugate transpose, respectively. represents the dot product between the vectors and . denotes the Euclidean norm of a complex vector, represents the projection operator onto the interval and denotes the Hadamard product. The set difference is defined as .

Ii System Model and the Relay Problem

In this work, we consider a UAV-enabled mobile relaying system which contains a source node, a destination node, a UAV and a laser PB, as shown in Fig. 1. We assume that the direct link between the source and the destination is sufficiently weak and hence can be ignored due to e.g., severe blockage, and the UAV serves as a mobile relay node to assist their communications [3]. Furthermore, we assume that the UAV is wireless-powered by a PB which is realized by a properly designed laser charging system [32].

We consider a Cartesian coordinate system without loss of generality, where the source, the destination and the PB are located at

, and respectively. For simplicity, we assume that the UAV is flying at a fixed altitude and could be chosen to be the minimum altitude that is required for terrain or building avoidance without frequent aircraft ascending or descending.111Note that the proposed algorithms can be extended to the case where the UAV’s altitude is also a design variable without much difficulty. Moreover, we focus on the UAV’s operation during flight and ignore its take-off and landing phases. We discretize the time interval into equally spaced time slots, i.e., , where denotes the elemental slot length, which is chosen to be sufficiently small. Thus, the trajectory of the UAV over can be approximated by the -length sequences , where denotes the UAV’s coordinate at slot . Let and denote the initial and final locations of the UAV relay, which are given depend on various factors [3]. Furthermore, let denote the maximum UAV speed, then we assume is always satisfied such that there exists at least one feasible trajectory. With regards to the mobility constraints of the UAV [37], we have222For a fixed-wing UAV, the mobility constraints should further include and , where denotes the stall speed and represents the maximum angular turn rate in . However, in order to better focus on laser charging, we only consider constraint (1) when dealing with the UAV’s mobility. Further investigation into more sophisticated UAV controls is left for future work.

 q1=qI,qN=qF, (1a) ∥vn∥≜∥qn+1−qn∥/δt≤vmax,∀n∈N∖{N}. (1b)

Ii-a Information Transmission Model

We assume that LoS links dominate the wireless channels from the source to the UAV and that from the UAV to the destination, and the Doppler effect due to the mobility of the UAV can be perfectly compensated [3]. Therefore, at slot , the channel power from the source to the UAV follows the free-space path loss model, which can be expressed as , where denotes the channel power at the reference distance meter (m), whose value depends on the carrier frequency, antenna gain, etc., and is the link distance between the source and the UAV at slot . Similarly, the channel power from the UAV to the destination at slot can be expressed as .

Let and denote the transmit powers of the source and the UAV at slot , then the maximum transmission rate from the source to the UAV and from the UAV to the destination in bits/second/Hz (bps/Hz) at slot can be expressed as and , where is the noise power and

denotes the reference signal-to-noise ratio (SNR).

Ii-B Wireless Power Transmission Model

In this work, we model the PB as a laser charging system which was proposed in [32], where the optical components are divided into two separate parts, the transmitter and the receiver, respectively. Consequently, the received power of the UAV at slot can be expressed as , where , and denote the electricity-to-laser conversion efficiency, the laser transmission efficiency and the laser-to-electricity conversion efficiency, respectively [32], represents the transmit power of the PB at slot . Furthermore, can be modeled as [38], where denotes the laser attenuation coefficient and is the distance between the UAV and the PB at slot . can be further depicted as , where and are two constants, , and denote the visibility, wavelength and size distribution of the scattering particles, respectively.

Employing the approximation method in [32], we can alternatively model the received power as follows:

 Prn={a1a2ηltnPsn+a2b1ηltn+b2,Psn≥Psmin,0,0≤Psn

where denotes the minimum supply power that is required to activate the corresponding circuits of the laser transceiver, and the involved parameters are listed in Table I. Note that is a non-convex function with respect to the UAV’s trajectory .

Ii-C Energy Consumption Model

Note that the energy consumption of the UAV is dominated by the propulsion power for maintaining the UAV aloft and supporting its mobility, which is usually much higher than the communication power consumption (e.g., hundreds of watts versus a few watts or even mW) [1]. As a result, we consider the model in [37] and [39] to characterize the energy consumption of the UAV due to flying, which postulates the flying energy at each slot to depend only on the velocity vector as

 EFn(vn)=ω∥vn∥2, (3)

where and is the UAV’s mass, including its payload.333 There are more practical models which assume that the energy also depends on the acceleration vector [40, 41]. Furthermore, for rotary-wing aircrafts, there would be energy consumption when the UAV is in hover state [42]. However, in order to illustrate the merits of the proposed algorithms and to simplify derivations, we focus on model (3) in this work.

Ii-D Problem Formulation

In this work, we aim to maximize the information transmission efficiency of the UAV-enabled relay system and the laser power transmission efficiency simultaneously subject to the information/energy-causality constraints, the power budget constraints and the UAV’s mobility constraints (1). Specifically, the information-causality constraints mean that the UAV can only forward the data that has already been received from the source at each slot and by assuming that the processing delay at the UAV is one slot, we have

 m∑n=2Rrn≤m−1∑n=1Rsn,m∈N∖{1}. (4)

It is obvious that the source should not transmit at the last slot and thus we can see that should be satisfied (and hence ) without loss of optimality. For simplicity, we assume that the UAV is equipped with a data buffer with sufficiently large storage size. Similarly, in order to guarantee that the UAV can safely reach the final location with enough battery level in case of emergence and to avoid overcharging, the following energy-causality constraint should also be satisfied:

 θ≤E−m∑n=1EFn(vn)+m∑n=1Prnδt≤E,m∈N, (5)

where represents the UAV’s energy budget (i.e., the maximum energy storage capacity of the UAV’s battery if we assume that the UAV is fully charged before taking off) and is a predefined threshold which characterizes the minimum energy storage during the flight.

Furthermore, the energy efficiency of the UAV during information transmission can be expressed as

 fEE({qn,psn,prn})≜N∑n=2Rrn/(υsN−1∑n=1psn+υrN∑n=2prn+NPon), (6)

where and are the power inefficiencies of the amplifiers in the source and the UAV, respectively, denotes the constant link on-power induced mainly by signal processing (it will be elaborated in Section V). The laser power transmission efficiency is given by

 fPE({qn,Psn})≜N∑n=1Prn/(N∑n=1Psn). (7)

Therefore, the considered optimization problem can be formulated as

 max{qn,psn,prn,Psn}fEE({qn,psn,prn})+γfPE({qn,Psn}) (8a) s.t.0≤psn≤psmax,n∈N∖{N},0≤prn≤prmax,n∈N∖{1}, (8b) Psmin≤Psn≤Psmax,n∈N, (8c) N∑n=2Rrn≥Rsum, (8d) (???),(???)% and(???),

where denotes a weighting factor that accounts for the priority of over ; (8b) and (8c) denote the transmit power constraints of the UAV and the PB, respectively; represents the minimum sum-rate that should be achieved during the flight. Note that in order to maximize , the UAV should fly close to the source and destination, however for the maximization of , the UAV should be close to the PB instead. Since the source, the destination and the PB are not co-located in general, these two efficiencies are usually conflict with each other and there exists a tradeoff between them. Throughout this paper, we assume that the flight duration is sufficiently long such that the UAV must harvest energy from the PB otherwise its battery would be drained out.444Note that if the UAV has enough energy during the whole flight, the considered problem would reduce to the conventional UAV-enabled relay system, a similar problem has been considered in [3] and it is out of the scope of this paper.

Remark 1.

Problem (8) is highly non-convex, which involves multiple fractional terms in the objective function and the optimization variables are coupled in the constraints. It cannot be directly solved by standard convex optimization techniques. Moreover, neither the Dinkelbach’s transformation [43] nor the fractional programming technique [44] can be directly applied to solve this problem, since the former cannot deal with objective functions with multiple-ratio terms and the latter is not designed to handle coupling constraints. A feasible approach for problem (8) is the AO-based algorithm, which alternating between power optimization and trajectory optimization, however, no optimality (e.g., to stationary solutions) can be theoretically declared for such an algorithm as has been shown in [3, 45, 46], etc. To tackle this difficulty, in this work, we propose two algorithms to address problem (8) with different design techniques and both of them are guaranteed to achieve stationary solutions of problem (8).

Remark 2.

In this work, we assume that the energy supplys of the communication and propulsion systems of the UAV are independent for emergency purposes, e.g., sending localization signals when the UAV does not have enough power to maintain aloft, etc. As a result, in (6), the denominator does not contain the propulsion power (3). Besides, the PB’s location will affect the overall performance, however, it is regarded as a fixed infrastructure in this work and its location is considered to be a predefined parameter that cannot be optimized. The case that propulsion power dominates the denominator of (6) and the placement of the PB are left for future work. Moreover, the efficiencies of the communication and propulsion systems are formulated and optimized as two separate terms, i.e., and . Otherwise, the objective function would become and due to the fact that is not considered in this case, the laser power transmission efficiency would be ignored.

Iii The Proposed CCCP-based Algorithm

In this section, in order to make problem (8) more tractable, we propose to first transform it into an equivalent form by properly introducing auxiliary variables; we then present a CCCP-based algorithm to address the resulting problem. The proposed algorithm is motivated by the observation that by some skillful mathematical manipulations, the objective function with multiple-ratio terms (8a), the pivotal coupling constraint (4) and (5) can be expressed as difference of convex (DC) functions. Thus, we can use the CCCP technique [47] to iteratively solve problem (8), where in each iteration only a convex subproblem is needed to be solved.

Iii-a Problem Transformation

We first introduce auxiliary variables and , which satisfy

 prnγ0/(H2+∥qn−qD∥2)≥srn,∀n∈N∖{1}, (9a) psnγ0/(H2+∥qn−qS∥2)≥ssn,∀n∈N∖{N}. (9b)

It can be seen that constraints (9) must be satisfied with equality at optimality. If either of these two inequalities are satisfied with strict inequality, we can always decrease or , such that a higher objective value can be achieved without violating any constraints. As a result, problem (8) can be transformed into

 max{qn,psn,prn,Psn,srn,ssn}¯fEE({qn,psn,prn,srn})+γfPE({qn,Psn}) (10a) s.t.m∑n=2log2(1+srn)≤m−1∑n=1log2(1+ssn),m∈N∖{1}, (10b) N∑n=2log2(1+srn)≥Rsum, (10c) (???),(???),(???),(???)and(???), (10d)

where

 ¯fEE({qn,psn,prn,srn})≜N∑n=2log2(1+srn)/(υsN−1∑n=1psn+υrN∑n=2prn+NPon), (11)

and we can see that problem (10) is equivalent to (8).

Then, we proceed to handle the objective function which is in a multiple-ratio form and the main idea is also to introduce some auxiliary variables. Specifically, for the information transmission efficiency part, i.e., , we resort to the employment of auxiliary variables , and , which satisfy

 N∑n=2log2(1+srn)≥~R,υsN−1∑n=1psn+υrN∑n=2prn+NPon≤~p, (12a) ~R≥~pEi. (12b)

With the help of these variables, we can observe that can be replaced by a simple scalar variable and three additional inequality constraints in (12). It can be shown that this transformation incurs no loss of optimality by a similar argument as for constraints (9). The power transmission efficiency part, i.e., , can also be transformed into its equivalent form in a similar vein. To be specific, introduce auxiliary variables and which satisfy

 e−α√H2+∥qn−qP∥2≥tn,(usuallytn<1) (13)
 tnPsn≥^tn, (14)

respectively, then can be rewritten as . As a result, can be equivalently expressed as , with the help of the following constraints:

 N∑n=1a1a2^tn+a2b1tn+b2≥~t,N∑n=1Psn≤~P, (15a) ~t≥~PEe, (15b)

where , and are the introduced auxiliary variables. Therefore, we can see that the original objective function (8a), which is very difficult to handle, can now be equivalently transformed into the weighted sum of two scalar variables, i.e., . However, as a cost for this simple representation, we have to deal with the additional constraints (12), (13), (14) and (15), which will be detailed in the next subsection.

Next, we focus on constraints (9), which are also difficult to address due to the fact that is non-convex. To tackle this difficulty, we resort to the help of two auxiliary variables and , which measure the upper bounds of the squared distances from the UAV to the source and destination. Accordingly, constraints (9) can be decomposed into

 H2+∥qn−qD∥2≤dDn, (16a) srndDn−prnγ0≤0, (16b) H2+∥qn−qS∥2≤dSn, (16c) ssndSn−psnγ0≤0. (16d)

Note that constraint (16a) must be satisfied with equality at optimality, otherwise we can always decrease , increase and , and then properly adjust to increase the objective function. A similar argument also holds for constraint (16c), therefore we omit the details for brevity.

To summarize, we conclude that problem (8) can be equivalently transformed into the following problem:

 maxXEi+γEe (17a) s.t.θ≤E−m∑n=1EFn(vn)+m∑n=1(a1a2^tn+a2b1tn+b2)δt≤E,m∈N, (17b)

where . Although problem (17) is now in a much simpler form than that of (8), it is still highly non-convex and difficult to address. In the following, we present the design methodology to iteratively solve problem (17) by the concept of CCCP.

Iii-B Algorithm Design

Non-convex constraints are generally difficult to handle, e.g., (10b), (12b), (13), (14), (15), (16b), (16d) and (17b) etc. Among them, constraints (10b) and (13) are more difficult since the logarithm and exponential functions are involved. In the following, we show that these constraints can be expressed in DC forms by proper transformations and then by employing the CCCP concept, problem (17) can be iteratively solved to stationary solutions. Unless otherwise stated, we use subscript to indicate the variables obtained in the -th iteration.

Firstly, let us focus on constraints (10b) and (13). Since the function is concave, (10b) can be readily viewed as a DC function. By approximating the convex function in the -th iteration by its first order Taylor expansion around the current point , we can obtain

 −m−1∑n=1log2(1+ssn)+m∑n=2(log2(1+srn,l)+1(1+srn,l)ln(2)(srn−srn,l))≤0,m∈N∖{1}. (18)

As for constraint (13), the following equivalent form can be obtained: , and since the function is also concave and the left hand side is a second order cone (SOC) which is convex, this equivalent inequality is also in DC form and can be approximated by the following convex constraint:

 √H2+∥qn−qP∥2+(lntn,l)/α+(tn−tn,l)/(αtn,l)≤0,n∈N. (19)

Secondly, we consider constraints (12b), (14), (15), (16b) and (16d). It can be observed that these constraints are all in the form of or , which can be further expressed as or . They are also DC functions, and by using the CCCP concept, they can be approximated by convex function without any difficulty. The detailed expressions of these approximations will be given below.

Finally, it can be easily seen that (17b) can be decomposed into one convex constraint and one DC constraint, which can be handled in a similar way. Therefore, in the -th iteration of the proposed CCCP-based algorithm, we have the following convex problem:

 max{X}Ei+γEe (20a) s.t.(???),(???),(???),(???),(???),(???),(???),(???),(???),(???), (ssn+dSn)2+(ssn,l)2+(dSn,l)2−2ssn,lssn−2dSn,ldSn−2psnγ0≤0, (20b) (srn+dDn)2+(srn,l)2+(dDn,l)2−2srn,lsrn−2dDn,ldDn−2prnγ0≤0, (20c) (~p+Ei)2+~p2l+E2i,l−2~pl~p−2Ei,lEi−2~R≤0, (20d) (~P+Ee)2+~Pl2+E2e,l−2~Pl~P−2Ee,lEe−2~t≤0, (20e) 2^tn+t2n+(Psn)2+(tn,l+Psn,l)2−2(tn,l+Psn,l)(tn+Psn)≤0, (20f) θ−E−m∑n=1(a1a2^tn+a2b1tn+b2)+m∑n=1κ∥vn∥2≤0, (20g) m∑n=1κ(−∥qn+1,l−qn,l∥2+2(qn+1,l−qn,l)T(qn+1−qn)δ2t)−m∑n=1(a1a2^tn+a2b1tn+b2)δt≥0, (20h)

which is a second-order cone program (SOCP) and it can be solved by some off-the-shelf solvers, such as CVX [48]. The proposed CCCP-based algorithm to solve problem (8) is summarized in Algorithm 1 and we have the following proposition regarding its convergence property:

Proposition 1.

Every limit point of the sequence generated by Algorithm 1 is a stationary solution of problem (8).

Proof.

Please refer to reference [47] for the detailed proof. ∎

Furthermore, the computational complexity of Algorithm 1 is dominated by solving problem (20) times, where denotes the total iteration number. Since problem (20) involves linear constraints, SOCs with dimension , SOCs with dimension and the number of variables is on the order of , we can see that the complexity of Algorithm 1 is on the order of according to the basic elements of complexity analysis as used in [49]. Therefore, by letting , the worst-case asymptotic complexity of Algorithm 1 can be evaluated as .

Iv The Proposed PDD-based Algorithm

In the previous section, we proposed the CCCP-based algorithm (i.e., Algorithm 1), where in each iteration, an SOCP problem is required to be solved. The main idea is to replace the complex objective function and constraints with simpler ones and possibly with some linear approximations, thus employing convex solvers is inevitable. However, since the intrinsic structure of problem (8) may not be fully exploited, off-the-shelf software solvers might be inefficient in many scenarios. In this section, we take an alternative by embracing the PDD framework and present a PDD-based algorithm. Specifically, we first transform problem (8) into an equivalent form by introducing auxiliary variables and some additional equality constraints. Different from Algorithm 1, in this case, our aim is to make this problem fully decomposable, i.e., to relief the coupling of the constraints. Then, instead of directly handling the equivalent problem with many constraints, we focus on its augmented Lagrangian (AL) problem, where the equality constrains are augmented onto the objective function with certain dual variables and a penalty parameter. As a result, we obtain a twin-loop PDD-based algorithm, where the inner loop seeks to (approximately) solve the AL problem using a block minimization technique, while the outer loop updates the dual variables and the penalty parameter. Especially, we show that each subproblem can be solved either in closed-form or by the bisection method.

Iv-a Problem Transformation

Firstly, we introduce the following variable substitutions:

 (21a) prnγ0=srndDn,psnγ0=ssndSn, (21b) tnPsn=^tn, (21c)

where the purposes of , , , , and are similar to those in Section III-A, only in this case, we prefer to directly introduce equality constraints such that the PDD framework can be naturally blended in.

Next, since the trajectory variables are coupled in the velocity vectors and appear multiple times in (21a), in order to break these couplings, we further introduce four redundancy copies, i.e., , , , . Let and represent the squared velocity at slot and the sum of squared velocity from slot to and introduce , (due to the same reason with that of ). Then, it can be seen that holds.

Finally, in order to decompose the information-causality and energy-causality constraints, the following auxiliary variables are employed:

 log2(1+srn)=¯srn,log2(1+ssn)=¯ssn, (22a) m∑n=2¯srn−m−1∑n=1¯ssn=~sm, (22b) ln(tn)/α=tLn, (22c) ˘tn=a1a2^tn+a2b1tn, (22d) −m∑n=1κ¯vn+(m∑i=1˘ti+mb2)δt=em, (22e)

where the main motivation is to make these coupling constraints separable among each other and among different time slots. Therefore, we have the following optimization problem:

 maxY^fEE({psn,prn,¯srn})+γ^fPE({tn,^tn,Psn}) (23a) s.t.~sm≤0,m∈N∖{1}, (23b) N∑n=2¯srn≥Rsum, (23c) E≥E+em≥θ,m∈N, (23d) (23e) ∥~qn+1−qn∥2/δ2t=˘vn+1−~vn,n∈N, (23f) ˘vn+1−~vn=¯vn,˘vn=˙vn,~vn=˙vn,n∈N, (23g) ¯vn≤v2max, (23h) ˙qn=qn,¯qn=qn,^qn=qn,~qn=¯qn,n∈N, (23i) (???),(???),(???)powerconsoriPB,(???),(???),(???),

where ,

 ^fEE({psn,prn,¯srn})≜N∑n=2¯srn/(υsN−1∑n=1psn+υrN∑n=2prn+NPon), (24)
 ^fPE({tn,^tn,Psn})≜(N∑n=1a1a2^tn+a2b1tn+b2)/N∑n=1Psn. (25)

Note that problem (23) and (8) are equivalent, since to this end, we are basically introducing equality constraints. The roles and necessities of these additional variables and constraints would be clear in the next subsection.

Iv-B Algorithm Design

In this subsection, our aim is to solve problem (23) by proposing an efficient PDD-based algorithm. We first formulate the AL problem of (23) as follows:

 maxY^fEE({psn,prn,¯srn})+γ^fPE({tn,^tn,Psn})−fAL(Y,Λ) (26a) s.t.(???),(???),(???),(???)−(???), (26b)

where represents the AL part which is obtained by augmenting the equality constraints with certain dual variables and penalty functions.555For example, consider an equality constraint , the corresponding AL part can be expressed as , where denotes the dual variable and is the penalty parameter. In this work, since the exact expression of is kind of tedious, we omit it for brevity but its components will be presented in the following. denotes the collection of all dual variables, which is listed in Table II with their corresponding equality constraints.