I Introduction
THE Internet of Things (IoT) has been widely developed with the unprecedented proliferation of mobile devices, such as smart phones, cloudbased mobile sensors, tablet computers and wearable devices, which facilitates the realization of smart environment (e.g. smart city, smart home, smart transportation, etc.) [1]
. IoT enables mobile users to experience intelligent applications (e.g., automatic navigation, face recognition, unmanned driving, etc.) and to enjoy diverse services with high quality of service (QoS) such as mobile online gaming, augmented reality, etc. These services normally require a massive number of sizeconstrained and lowpower mobile devices to perform computationintensive and latencysensitive tasks
[2]. However, it is challenging for mobile devices to perform these services due to their low computing capability and finite battery lifetime.Mobile edge computing (MEC) and wireless power transfer (WPT) have been deemed two promising technologies to tackle the above mentioned challenges [2][4]. Recently, MEC has received an everincreasing level of attention from industry and academia since it can significantly improve the computation capability of mobile devices in a costeffective and energysaving manner [2]. It enables mobile devices to offload partial or all of their computationintensive tasks to MEC servers that locate at the edge of the wireless network, such as cellular base stations (BSs) and access points (APs). Different from the conventional cloud computing, MEC servers are deployed in a close proximity to end users. Thus, MEC has the potential to provide lowlatency services, to save energy for mobile users, and to achieve high security [2]. Up to now, there are a number of leading companies (e.g., IBM, Intel, and Huawei) that have identified MEC as a promising technique for the future wireless communication networks. In general, MEC has two operation modes, namely, partial and binary computation offloading. In the first mode, the computation task can be partitioned into two parts, and one part is locally executed while the other part is offloaded to the MEC servers for computing [5][9]. For the second mode, computation tasks cannot be partitioned. Thus they can be either executed locally or completely offloaded [10].
On the other hand, WPT can provide lowpower mobile devices with sustainable and costeffective energy supply by using radiofrequency (RF) signals [3]. It facilitates a perpetual operation and enables users to have high QoE, especially in the case that mobile devices do not have sufficient battery energy for offloading task or taking the services when the battery energy is exhausted. Compared to the conventional energy harvesting techniques, such as solar or wind charging, WPT is more attractive since it can provide a controllable and stable power supply [4]. It is envisioned that the computation performance can be significantly improved by integrating WPT into MEC networks [11][16]. However, the harvested power level can be significantly degraded by the severe propagation loss. Recently, an unmanned aerial vehicle (UAV)enabled WPT architecture has been proposed to improve the energy transfer efficiency [17][20]. It utilizes an unmanned aerial vehicle (UAV) as an energy transmitter for powering the ground mobile users. It was shown that the harvested power level can be greatly improved due to the fact that there is a high possibility that shortdistance lineofsight (LoS) energy transmit links exist [17][20]. Moreover, the computation performance can also be improved by using the UAVassisted MEC architecture [21][25]. Furthermore, UAVassisted architectures can provide flexible deployment and low operational costs, and are particularly helpful in the situations that the conventional communication systems are destroyed by natural disasters [26][32].
Motivated by the above mentioned reasons, a UAVenabled and wireless powered MEC network is studied in this paper. In order to maximize the achievable computation rate, the communication and computation resources and the trajectory of the UAV are jointly optimized under both partial and binary computation offloading modes. To the authors’ best knowledge, this is the first work that considers the UAVenabled wireless powered MEC network and studies the computation rate maximization problems in this type of network.
Ia Related Work and Motivation
In wireless powered MEC systems, it is of great importance to design resource allocation schemes so as to efficiently exploit energy, communication, and computation resources and improve the computation performance. Resource allocation problems have been extensively investigated in the conventional MEC networks [5][10] and also in MEC networks relying on energy harvesting [11][16]. Recently, efforts have also been dedicated to designing resource allocation and trajectory schemes in UAVenabled wireless powered communications network [17][20] and UAVassisted MEC networks [21][25]. These contributions are summarized as follows.
In MEC networks, the communication and computation resources and the selection of the offloading mode were jointly optimized to achieve the objective of the system design, e.g., the users’ consumption energy minimization [5], [6], the revenue maximization [7], the maximum cost minimization [8], etc. Specifically, in [5], the total energy of all users in a multicell MEC network was minimized by jointly optimizing the user transmit precoding matrices and the central processing unit (CPU) frequencies of the MEC server allocated to each user. It was shown that the performance achieved by jointly optimizing the communication and computation resources is superior to that obtained by optimizing these resources separately. The authors in [6] extended the energy minimization problem into the multiuser MEC systems with timedivision multiple access (TDMA) and orthogonal frequencydivision multiple access (OFDMA), respectively. It was proved that the optimal offloading policy has a thresholdbased structure, which is related to the channel state information (CSI) [6]. Particularly, mobile users offload their computation tasks when the channel condition is strong; otherwise, they can locally execute the computation tasks. In [7], the revenue of the wireless cellular networks with MEC was maximized by jointly designing the computation offloading decision, resource allocation, and content caching strategy. The works in [5][7] focused on optimizing a single objective, which overemphasizes the importance of one metric and may not achieve a good tradeoff among multiple metrics. Recently, the authors in [8] and [9] studied the fairness and multiobjective optimization problem in MEC networks. It was shown that there exist multiple tradeoffs in MEC systems, such as the tradeoff between the total computation rate and the fairness among users. Different from the works in [5][9], MEC systems with the binary computation offloading mode were considered and the optimal resource allocation strategy was designed to minimize the consumption energy in [10].
Energy harvesting was not considered in the MEC systems [5][10]. Recently, the authors in [11][16] have studied the resource allocation problem in various MEC systems relying on energy harvesting. In [11] and [12]
, The reinforcement learning and Lyapunov optimization theory were used to design resource allocation schemes in MEC systems relying on the conventional energy harvesting techniques. Different from
[11] and [12], the resource allocation problems were studied in wireless powered MEC systems [13][16]. Specifically, the authors in [13] proposed an energyefficient computing framework in which the energy consumed for local computing and task offloading is from the harvested energy. The consumed energy was minimized by jointly optimizing the CPU frequency and the mode selection. In [14], the energy minimization problem was extended into a multiinput singleout wireless powered MEC system, and the offloading time, the offloading bits, the CPU frequency and the energy beamforming were jointly optimized. Unlike [14], energy efficiency was defined and maximized in a fullduplex wireless powered MEC system by jointly optimizing the transmission power, offloaded bits, computation energy consumption, time slots for computation offloading and energy transfer [15]. In contrast to the work in [13][15], the computation bits were maximized in a wireless powered MEC system under the binary computation offloading mode [16]. Two suboptimal algorithms based on the alternating direction method were proposed to solve the combinatorial programming problem. The proposed algorithms actually did not provide the optimal selection scheme for the user operation mode.Although WPT has been exploited to improve the computation performance of MEC systems [13][16], the energy harvested by using WPT can be significantly degraded by the severe propagation loss. The energy conversion efficiency is low when the distance between the energy transmitter and the harvesting users is large. In order to tackle this challenge, the authors in [17][20] proposed a UAVenabled wireless powered architecture where a UAV transmits energy to the harvesting users. Due to the high possibility of having lineofsight (LoS) airtoground energy harvesting links, the harvesting energy can be significantly improved by using this architecture. Moreover, it was shown that the harvesting energy can be further improved by optimizing the trajectory of the UAV [18][20]. Thus, it is envisioned that the application of the UAVenabled architecture into wireless powered MEC systems is promising and valuable to be studied [26]. However, to the authors’ best knowledge, few investigations have focused on this area.
Recently, the UAVenabled MEC systems have been studied and their resource allocation schemes have been proposed [21][25]. In [21], the UAVenabled MEC architecture was first proposed and the computation performance was improved by using UAV. The authors in [22] proposed a new caching UAV framework to help small cells to offload traffic. It was shown that the throughput can be greatly improved while the overload of wireless backhaul can be significantly reduced. In order to further improve the computation performance, the authors in [23] and [24] designed a resource allocation scheme that jointly optimizes the CPU frequency and the trajectory of the UAV. In [25], a theoretical game method was applied to design a resource allocation scheme for the UAVenabled MEC system and the existence of Nash Equilibrium was demonstrated.
Although resource allocation problems have been well studied in MEC systems [5][10], MEC systems relying on energy harvesting [11][16] and UAVenabled MEC systems [21][25], few investigations have been conducted for designing resource allocation schemes in the UAVenabled wireless powered MEC systems. Moreover, resource allocation schemes proposed in the abovementioned works are inappropriate to UAVenabled MEC wireless powered systems since the computation performance not only depends on the optimization of energy, communication and computation resources, but also relies on the design of the UAV trajectory. Furthermore, the application of UAV into wireless powered MEC systems has the potential to enhance the user computation capability since it can improve the energy conversion efficiency and task offloading efficiency [33], [34]
. Thus, in order to improve the computation performance and provide mobile users with high QoE, it is of great importance and worthiness to study resource allocation problems in UAVenabled wireless powered MEC systems. However, these problems are indeed challenging to tackle. The reasons are from two aspects. On one hand, there exists dependence among different variables (e.g., the CPU frequency, the task offloading time and the variables related to the trajectory of the UAV), which makes the problems nonconvex. On the other hand, when the binary computation offloading mode is applied, the resource allocation problems in UAVenabled wireless powered MEC systems have binary variables related to the selection of either local computation or offloading tasks. It makes the problem a mixed integer nonconvex optimization problem.
IB Contributions and Organization
In contrast to [5][16], this paper studies the resource allocation problem in UAVenabled wireless powered MEC systems, where a UAV transmits energy signals to charge multiple mobile users and provides computation services for them. Although the computation performance is limited by the flight time of the UAV, it is worth studying UAVenabled wireless powered MEC systems since these systems are promising in environments such as mountains and desert areas, where no terrestrial wireless infrastructures exist, and in environments where the terrestrial wireless infrastructures are destroyed due to the natural disasters [33], [34]. Thus, in this paper, the weighted sum computation bits of all users are maximized under both partial and binary computation offloading modes. The main contributions of this work are summarized as follows:

It is the first time that the resource allocation framework is formulated in UAVenabled MEC wireless powered systems under both partial and binary computation offloading modes. The weighted sum computation bits are maximized by jointly optimizing the CPU frequencies, the offloading times and the transmit powers of users as well as the UAV trajectory. Under the partial computation offloading mode, a twostage alternative algorithm is proposed to solve the nonconvex and challenging computation bits maximization problem. The closedform expressions for the optimal CPU frequencies, the offloading times and the transmit powers of users are derived for any given trajectories.

Under the binary computation offloading mode, the weighted sum computation bits maximization problem is a mixed integer nonconvex optimization problem, for which a threestage alternative algorithm is proposed. The optimal selection scheme on whether users choose to locally compute or offload tasks is derived in a closedform expression for a given trajectory. The structure for the optimal selection scheme shows that whether users choose to locally compute or offload their tasks to the UAV for computing depends on the tradeoff between the achievable computation rate and the operation cost. Moreover, the trajectory of the UAV is optimized by using the successive convex approximation (SCA) method under both partial and binary computation offloading modes.

The simulation results show that the computation performance obtained by using the proposed resource allocation scheme is better than these achieved by using the disjoint optimization schemes. Moreover, it only takes several iterations for the proposed alternative algorithms to converge. Furthermore, simulation results verify that the priority and fairness of users can be improved by using the weight vector. Additionally, it is shown that the total computation bits increase with the number of users.
The remainder of this paper is organized as follows. Section II gives the system model. The resource allocation problem is formulated under the partial computation offloading mode in Section III. Section IV formulates the resource allocation problem under the binary computation offloading mode. Simulation results are presented in Section V. Finally, our paper is concluded in Section VI.
Ii System Model
A UAVenabled wireless powered MEC system is considered in Fig. 1, where an RF energy transmitter and an MEC server are implemented in UAV. The UAV transmits energy to users and provides MEC services for these users. Each user has an energy harvesting circuit and can store energy for its operation. The UAV has an onboard communication circuit and an onboard computing processor. So does each user. The computing processor of each user is an onchip microprocessor that has low computing capability and can locally execute simple tasks. The UAV has a powerful processor that can perform computationintensive tasks [21][25]. Similar to [13][16], each user can simultaneously perform energy harvesting, local computing and computation offloading while the UAV can simultaneously transmit energy and perform computation. In this paper, all devices are equipped with a single antenna.
Without loss of generality, a threedimensional (3D) Euclidean coordinate is adopted. Each user’s location is fixed on the ground. The location of the th ground user is denoted by , where , and . Boldface lower case letters represent vectors and boldface upper case letters represent matrices. and are the horizontal plane coordinates of the th ground user. It is assumed that user positions are known to the UAV for designing the trajectory [18][20]. A finite time horizon with duration is considered. During , the UAV flies at the same altitude level denoted by (). In practice, the fixed altitude is the minimum altitude that is appropriate to the work terrain and can avoid building without the requirement of frequent aircraft descending and ascending. A block fading channel model is applied, i.e., during each , the channel remains static.
For the ease of exposition, the finite time is discretized into equal time slots, denoted by . At the th slot, it is assumed that the horizontal plane coordinate of the UAV is . Similar to [27][32], it is assumed that the wireless channel between the UAV and each user is dominated by LOS. Thus, the channel power gain between the UAV and the th user, denoted by , can be given as
(1) 
where is the channel power gain at a reference distance m; is the horizontal plane distance between the UAV and the th user at the th slot, , ; denotes its Euclidean norm. The details for the UAVenabled wireless powered MEC system are presented under partial and binary computation offloading modes in the following, respectively.
Iia Partial Computation Offloading Mode
Under the partial computation offloading mode, the computation task of each user can be partitioned into two parts, one for local computing and one for offloading to the UAV. The energy consumed for local computing and task offloading comes from the harvested energy. In this paper, in order to shed meaningful insights into the design of a UAVenabled wireless powered MEC system, similar to [4], [13][16], the linear energy harvesting model is applied. Thus, the harvested energy at the th user during time slots is given as
(2) 
where denotes the energy conservation efficiency, and is the transmit power of the UAV. In this paper, the UAV employs a constant power transmission [18][20]. The details for the operation of each user under the partial computation offloading mode are presented as follows.
IiA1 Local Computation
Similar to [14][16], the energy harvesting circuit, the communication circuit, and the computation unit are all separate. Thus, each user can simultaneously perform energy harvesting, local computing, and computation offloading. Let denote the number of CPU cycles required for computing one bit of raw data at each user. In order to efficiently use the harvested energy, each user adopts a dynamic voltage and frequency scaling technique and then can adaptively control the energy consumed for performing local computation by adjusting the CPU frequency during each time slot [14][16]. The CPU frequency of the th user during the th slot is denoted by with a unit of cycles per second. Thus, the total computation bits executed at the th user during slots and the total consumed energy at the th user during slots are respectively given as and [14][16], where is the effective capacitance coefficient of the processor’s chip at the th user, , . Note that is dependent of the chip architecture of the th user.
IiA2 Computation Offloading
In order to avoid interference among users during the offloading process, a TDMA protocol shown in Fig. 2 is applied. Specifically, each time slot consists of three stages, namely, the offloading stage, the computation stage, and the downloading stage. In the offloading stage, users offload their respective computation task one by one during each slot. Let denote the duration in which the th user offloads its computation task to the UAV at the th slot, , . Similar to [16], the computation task of the th user to be offload is composed of raw data and communication overhead, such as the encryption and packer header. Let denote the total number of bits that the th user offloads to the UAV during the th slot, where is the number of raw data to be computed at the UAV and indicates the communication overhead included in the offloading task. Thus, one has
(3) 
where is the communication bandwidth; is the transmit power of the th user at the th slot and denotes the noise power at the th user.
After all users offload their computation tasks at the th slot, the UAV performs computing task and sends the computing results back to all the users. Similar to [14][16], the computation time and the downloading time of the UAV are neglected since the UAV has a much stronger computation capability than the users and the number of the bits related to the computation result is very small. Since the total offloading time of all users does not exceed the duration of one time slot, one has
(4) 
Since the energy consumed for local computing and task offloading comes from the harvested energy, the following energy harvesting causal constraint should be satisfied.
(5) 
Under the partial computation offloading mode, the total computation bits of the th user is given as
(6) 
IiB Binary Computation Offloading Mode
Under the binary computation offloading mode, the computation task cannot be partitioned. All the users need to choose to either locally compute the task completely or offload the entire task. This case can be widely experienced in practice. For example, in order to improve the estimation accuracy, the raw data samples that are correlated need to be jointly computed altogether
[10], [16]. Let and denote the set of users that choose to perform local computation and the set of users that choose to perform task offloading, respectively. Thus, and , where denotes the null set.IiB1 Users Choosing to Perform Local Computing
In this case, a user in exploits all the harvested energy to perform local computing. Thus, the total computation rate of the th user denoted by can be given as
(7) 
And the energy harvesting causal constraint for a user in can be given as
(8) 
IiB2 Users Choosing to Perform Task Offloading
Each user in exploits all the harvested energy to perform task offloading. The TDMA protocol is applied to avoid interference among these users during the offloading process. Since the total offloading time of all users in at the th slot cannot exceed the duration of a time slot, one has
(9) 
Let denote the total computation rate of the th user in the set . Then, one has
(10) 
The energy harvesting causal constraint for a user in can be given as
(11) 
Sections III and IV will respectively formulate the computation rate maximization problem for the partial and binary computation offloading modes.
Iii Resource Allocation Under The Partial Computation Offloading Mode
In this section, the resource allocation problem is studied under the partial computation offloading mode. The weighted sum computation bits are maximized by jointly optimizing the CPU frequencies, the offloading times and the transmit powers of users as well as the trajectory of the UAV. In order to tackle this nonconvex problem, a twostage alternative algorithm is proposed.
Iiia Resource Allocation Problem Formulation
Under the partial computation offloading mode, the weighted sum computation bits maximization problem in the UAVenabled wireless powered MEC system is formulated as ,
(12a)  
(12b)  
(12c)  
(12d)  
(12e)  
(12f) 
where denotes the maximum speed of the UAV in the unit of meter per second; and are the initial and final horizontal locations of the UAV, respectively. In , denotes the weight of the th user, which takes the priority and the fairness among users into consideration. is the CPU frequency constraint and the computation offloading power constraint imposed on each user; represents the energy harvesting causal constraint; is the time constraint that the total time of all users offloading the computation bits cannot exceed the duration of each time slot; and are the speed constraint and the initial and final horizontal location constraint of the UAV, respectively. is nonconvex since there exist nonlinear couplings among the variables, , ,, and the objective function is nonconcave with respect to the trajectory of the UAV. In order to solve it, a twostage alternative optimization algorithm is proposed. The details for the algorithm are presented as follows.
IiiB TwoStage Alternative Optimization Algorithm
Let . For a given trajectory, can be transformed into .
(13a)  
s.t.  (13b)  
(13c) 
It is easy to prove that is convex and can be solved by using the Lagrange duality method [35], based on which the optimal solutions for the CPU frequency and the transmit power can be derived. Let and denote the optimal CPU frequency and transmit power of the th user at the th time slot, respectively, where and . By solving , Theorem 1 can be stated as follows.
Theorem 1
For a given trajectory , the optimal CPU frequency and transmit power of users can be respectively expressed as
(14a)  
(14b) 
where is the dual variable associated with the constraint ; and denotes the bigger value of and .
Proof:
See Appendix A.
Remark 1
It can be seen from Theorem 1 that users choose to offload their computation tasks only when the channel state information between users and the UAV is stronger than a threshold, namely, . This indicates that the user chooses to perform local computation when the horizontal distance between the user and the UAV is larger than . Moreover, it can be seen that the larger the weight is, the higher the chance for the user to chooses to offload its computation task. Furthermore, users prefers to offload their computation task when the local computation frequency is very large, namely, .
Theorem 2
If there exists a time slot that , the equation must hold, .
Proof:
Since is the dual variable and , from Theorem 1 increases with . Thus, if there exists a time slot so that , one must have , for . Theorem 2 is proved.
Remark 2
Theorem 2 indicates that the user CPU frequency increases with the time slot index. This means that the number of computation bits obtained by local computing increases with the time slot index. Moreover, the user CPU frequency increases with the weight assigned to that user since more resources are allocated to the user with a higher weight.
Theorem 3
For a given trajectory , the optimal user offloading time can be obtained by solving the following equation.
(15) 
Remark 3
Theorem 3 can be readily proved based on the proof for Theorem 1. Thus this proof is omitted for the sake of saving space. Moreover, can be solved by using the bisection method [35].
The values of the dual variables are needed in order to obtain the optimal CPU frequency, the optimal transmit power and the optimal offloading time for all users. The subgradient method in Lemma 1 can be used to tackle this problem [36].
Lemma 1
The subgradient method for obtaining the dual variables is given as
(16a)  
(16b) 
where denotes the iteration index; and represent the iterative steps at the th iteration. In , and are the corresponding subgradients, given as
(17a)  
(17b) 
where , , and denote the optimal solutions at the th iterations. According to [35], the subgradient guarantees to converge to the optimal value with a very small error range.
IiiC Trajectory Optimization
For any given CPU frequency, transmit power, and offloading time of users, the trajectory optimization problem can be formulated as .
(18a)  
(18b)  
(18c) 
Since is nonconvex and the objective function is nonconcave with respect to , is nonconvex and we use the SCA technique to solve the optimization problem. The obtained solutions can be guaranteed to satisfy the KarushKuhnTucker (KKT) conditions of [27]. By using the SCA technique, Theorem 4 is given as follows.
Theorem 4
For any local trajectory at the th iteration, one has
(19a)  
(19b) 
where the equality holds when .
Proof:
Let , where and are positive constants, and . Since is convex with respect to , the following inequality can be obtained:
(20) 
where is a given local point. By using , Theorem 4 is proved.
In order to tackle the objective function of , Lemma 2 is given as follows.
Lemma 2
[27] Using the SCA method, the following inequality can be obtained,
(21a)  
(21b) 
where the equality holds when .
Algorithm 1: The twostage alternative optimization algorithm 
1: Setting: 
, , , , , , and the tolerance errors , ; 
2: Initialization: 
The iterative number , , and ; 
3: Repeat 1: 
calculate and using Theorem 1 
for given ; 
use the bisection method to solve and obtain ; 
update and using the subgradient algorithm; 
initialize the iterative number ; 
Repeat 2: 
solve by using CVX for the given , 
and ; 
update , and ; 
if 
; 
break; 
end 
end Repeat 2 
update the iterative number ; 
if 
break; 
end 
end Repeat 1 
4: Obtain solutions: 
, and and . 
Using Theorem 4 and Lemma 2, can be solved by iteratively solving the approximate problem , given as
(22a)  
(22b)  
(22c) 
It can be seen that is convex and can be readily solved by using CVX [4]. By solving and , a twostage alternative optimization algorithm denoted by Algorithm 1 is further developed to solve . The details for Algorithm 1 can be found in Table I. In Table I, denotes the value of the objective function of at the th iteration.
Iv Resource Allocation in Binary Computation Offloading Mode
In this section, the weighted sum computation bits maximization problem is studied in the UAVenabled wireless powered MEC system under the binary computation offloading mode. The CPU frequencies of the users that choose to perform local computation, the offloading times, the transmit powers of users that choose to perform task offloading, the trajectory of the UAV, and the mode selection are jointly optimized to maximize the weighted sum computation bits of all users. The formulated problem is a mixed integer nonconvex optimization problem, for which a threestage alternative optimization problem is proposed.
Iva Resource Allocation Problem Formulation
Under the binary computation offloading mode, the weighted sum computation bit maximization problem subject to the energy harvesting causal constraints, the UAV speed and position constraints is formulated as ,
(23a)  
(23b)  
(23c)  
(23d)  
(23e)  
(23f)  
(23g) 
and are the energy harvesting causal constraints imposed on these users who choose to perform local computation and on these users who choose to perform task offloading, respectively; is the offloading time constraint during each slot and is the user operation selection constraint. In there exist close couplings among different optimization variables. Furthermore, the binary user operation mode selection makes a mixed integer programming problem. The exhaustive search method leads to a prohibitively high computational complexity, especially when there exist a large number of users. Motivated by how we solve , has a similar structure as when the operation modes of users are determined. Thus, the optimal CPU frequency, transmit power, and offloading time of users can be obtained by using the same method as the one used for and the trajectory optimization for the UAV can also be achieved by using the SCA method. As such, a threestage alternative optimization algorithm is proposed based on the twostage Algorithm 1. The details for the algorithm are presented as follows.
IvB ThreeStage Alternative Optimization Algorithm
In order to efficiently solve , a binary variable denoted by is introduced, where and . indicates that the th user performs local computation mode while means that the th user performs task offloading. Moreover, the user operation selection indicator variable is relaxed as a sharing factor . Thus, can be rewritten as
(24a)  
(24b)  
(24c)  
(24d)  
(24e) 
Even by relaxing the binary variable , is still difficult to solve as there exist couplings among different variables. For any given and the trajectory of the UAV, has a similar structure as . Thus, using the same techniques applied to , the optimal CPU frequency, transmit power and offloading time of users for a given and the UAV trajectory can be obtained. It is easy to verify that the optimal CPU frequency, transmit power and offloading time of users for a given trajectory have the same forms given by Theorem 1 and Theorem 3.
Theorem 5
For any given , , and , the user operation selection scheme can be obtained by
(25a)  
(25b)  
(25c) 
where and are the dual variables associated with the constraints given by and , respectively.
Proof:
See Appendix B.
Remark 4
Theorem 5 indicates that the user operation selection scheme depends on the tradeoff between the achievable computation rate and the operation cost. If the tradeoff of the user achieved by local computing is better than that obtained by task offloading, the user chooses to perform local computing; otherwise, the user chooses to offload its computation tasks to the UAV for computing.
Finally, the trajectory optimization for any given , , and can be obtained by solving , given as
(26a)  
(26b)  
(26c) 
where and are given by and , respectively. is convex and can be efficiently solved by using CVX [4]. Based on Theorem 1, Theorem 5 and the solutions of , a threestage alternative optimization algorithm denoted by Algorithm 2 is proposed to solve . The details for Algorithm 2 are presented in Table 2. In Table 2, and denote the value of the objective function of at the th and iteration, respectively.
Algorithm 2: The threestage alternative optimization algorithm 
1: Setting: 
, , , , , , and the tolerance errors , and ; 
2: Initialization: 
The iterative number , and , and ; 
3: Repeat 1: 
initialize the iterative number and ; 
Repeat 2: 
calculate and using Theorem 1 
for given and ; 
use the bisection method to solve and obtain ; 
update and using the subgradient algorithm; 
calculate using Theorem 5 and update ; 
if 
break; 
end 
initialize the iterative number ; 
Repeat 3: 
solve by using CVX for the given , , 
and ; 
update , and ; 
if 
; 
break; 
end 
end Repeat 3 
update the iterative number ; 
if 
break; 
end 
end Repeat 2 
end Repeat 1 
4: Obtain solutions: 
, and , 
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