Joint Unmanned Aerial Vehicle Location and Beamforming and Caching Optimization for Cache-Enabled Multi-Unmanned-Aerial-Vehicle Networks

Abstract

Due to the advantages such as high flexibility, low cost and easy implementation offered by unmanned aerial vehicles (UAVs), a UAV-assisted network is regard as an appealing solution to a seamless coverage, high disaster-tolerant and on-demand wireless system. In this paper, we focus on the downlink transmission in a cache-enabled UAV-assisted wireless communication network, where UAVs cache popular content from a macro base station in advance and cooperatively transfer the content to users. We aim to minimize the average transmission latency of the system and to formulate an optimization problem that jointly optimizes the UAV location, beamforming and caching strategy. However, the formulated problem is very challenging because of its non-convexity and the highly coupled optimization variables. To solve this resulting problem efficiently, we decompose it into two subproblems, namely UAV location and beamforming optimization, and UAV caching strategy optimization. The first subproblem is an NP-hard joint optimization problem, while the second one is a linear programing problem. By adopting the first-order Taylor expansion, we propose a convex optimization algorithm based on the difference-of-convex (DC) method. Specifically, we bring out a method to apply linear approximation in the DC-based algorithm, which is particularly suitable to the problems involving complicated summations. The numerical results demonstrate that the proposed DC-based iterative optimization algorithm can efficiently reduce the average transmission latency of the system.

1. Introduction

In recent years, unmanned aerial vehicles (UAVs) have been increasingly employed in a variety of applications [1,2,3]. Due to many attractive advantages such as swift mobility, high flexibility, low cost and quick deployment, UAVs have been used to perform various tasks like photo taking, video recording, delivery and communication relay.

In particular, the feature of line-of-sight (LoS) channels makes it quite suitable for UAVs to provide wireless relay, such that the multi-UAV-assisted wireless network is supposed to be a powerful complement to conventional wireless cellular networks. For example, UAV-assisted communication could help deal with disasters as an alternative mode of emergency communication. Another important application is using UAVs to offload data traffic from crowded areas. In scenarios which temporarily involve large crowds, such as festivals, concerts and sporting events, UAVs could help mitigate the lack of capacity [2]. The ability to provide flexible and effective capacity expansion for existing network makes UAVs potential tools to improve the wireless coverage. Due to its promising performance, UAV-assisted communication has recently become an emerging research field. The Third Generation Partnership Project (3GPP) has submitted several technical specifications related to UAV-assisted communication systems, in which the requirements [4] and architecture [5] of Uncrewed Aerial Systems are defined.

1.1. Background of Cache-Enabled UAV-Assisted Network

Wireless caching has turned out to be an effective solution to meet the demand for low latency data traffic [6,7,8]. Studies have shown that in cellular networks, there is usually some popular content that is repeatedly requested by users [6]. This popular content would be requested and transferred multiple times in a period of time, which would result in a decrease in the data-transferring efficiency. In order to save on network resources, forwarding caching of some popular content at edge nodes is regarded as a promising solution. During idle periods of the network, edge nodes could cache popular content from a macro base station (MBS) via backhaul networks. Once the cached content is requested, it can be delivered directly to the user through edge nodes, and the user would enjoy low communication latency.

By employing cache-enabled UAVs as edge nodes, a wireless network could provide high-quality, high-efficiency and on-demand wireless coverage [9,10,11,12,13]. Consider UAV-assisted communication systems with cache-enabled multi-antenna UAVs; there are mainly three key issues in improving the system performance: UAV deployment, beamforming scheme and caching strategy. First, it is easy to understand that the locations of UAVs determine the distances from the served users, thus affecting the channel quality [14,15,16,17,18]. Second, multiple antennae enable UAVs to utilize beamforming to obtain higher SINRs; accordingly, an appropriate beamforming scheme is required [19,20,21,22]. At the same time, the caching strategy, or content placement, has been extensively researched in traditional cellular networks [6,7,8,19,20], and it is also important in UAV-assisted networks.

1.2. Related Work

The impact of UAV deployment has been intensively investigated in [14,15,16,17,18]. To optimize the 3D deployment of UAVs, the authors in [14] establish an integer linear programming model, while the authors in [15] approximate the rate expression to identify the concave regions and to apply alternating optimization. Particle swarm optimization [16] and game-theoretic framework [17] are also studied to optimize UAV deployment. Differently, ref. [18] not only determines the 3D coordinates but also minimizes the number of UAVs, in which a K-Means algorithm is proposed.

The work in [23,24] focuses on the caching problems in UAV-assisted networks. In [23], the caching strategy is optimized by investigating the user request preference with the aid of latent Dirichlet allocation, while [24] proposes a dual dynamic adaptive caching algorithm. Refs. [9,10,11] consider the optimization of UAV location and caching strategy. Refs. [19,20] focus on beamforming and caching optimization in cellular networks, in which the locations of base stations are fixed, different from UAV scenarios. Ref. [21] optimizes UAV location and beamforming jointly, where a framework based on difference of convex (DC) is proposed. Moreover, UAV trajectory is considered in [25,26], while resource is considered in [27,28]. Ref. [12] jointly optimizes resource, deployment and caching; ref. [22] jointly optimizes UAV location, beamforming, and power control; and ref. [13] optimizes UAV location, association, caching and computing resource. Much research has been carried out in the field of cache-enable UAV-assisted communications. However, we find that existing work has not sufficiently considered the impact of the joint design of UAV caching, beamforming and location, and the optimization in this scenario remains an open problem.

1.3. Contributions

The main contributions of this paper are enumerated as follows.

• We consider the downlink transmission in a cache-enabled UAV-assisted wireless communication system, which consists of UAVs, users, and an MBS. The design of UAV deployment, caching and beamforming is investigated. As far as we are concerned, this is a positive attempt to jointly optimize UAV location, caching and beamforming in a UAV-assisted network.
• We decompose the formulated problem into two subproblems and solve them iteratively. Inspired by [21], we use the DC-based method to convert the original problem into a convex form. Specifically, we bring out a method to apply linear approximation in the DC-based algorithm, which is particularly suitable to the problems involving complicated summations.
• Finally, via numerical results, we show that the proposed DC-based algorithm can efficiently reduce transmission latency. The DC-based algorithm outperforms the benchmark algorithms, especially in the case of the UAV caching capacity being small or the Zipf parameter of content popularity being large.

The rest of this paper is organized as follows. Section 2 introduces the system model. Then, in Section 3, the problem formulation is presented and a DC-based joint optimization algorithm is proposed. Section 4 presents and analyzes the numerical results. Finally, the conclusion of this paper is drawn in Section 5.

2. System Model

2.1. System Description

As shown in Figure 1, we consider the downlink of a distributed cache network consisting of one MBS, UAVs and ground users. The set of UAVs is denoted by $\mathcal{M}=\left\{1,2,\dots ,M\right\}$, and the set of users is denoted by $\mathcal{N}=\left\{1,2,\dots ,N\right\}$. Assume that the number of antennas for each UAV is the same, and the number is $N_T$. Consider that there are K popular content, denoted by set $\mathcal{K}=\left\{1,2,\dots ,K\right\}$.

Compared with MBS, UAVs are usually closer to users, so they provide higher transmission rates for related users. To simplify the discussion, it is assumed that the service rate of an MBS for each user is the same. The service rate of an MBS is set as a constant, which is lower than the average of UAVs.

2.2. UAVs and User Distribution Model

Without loss of generality, we assume all UAVs always hover at a constant height H. During the interested time slot, the positions of users are considered to be quasistatic, while all the horizontal positions of UAVs can be changed over time.

We denote the $i\mathrm{th}$ UAV as ${\mathrm{UAV}}_i$, $i\in \mathcal{M}$ and the $j\mathrm{th}$ user as $\mathrm{U}{\mathrm{E}}_j$, $j\in \mathcal{N}$. The position vectors of ${\mathrm{UAV}}_i$ and $\mathrm{U}{\mathrm{E}}_j$ are denoted by ${\mathbf{v}}_i=(x_i^{\mathrm{v}},y_i^{\mathrm{v}},H)$ and ${\mathbf{u}}_j=(x_j^{\mathrm{u}},y_j^{\mathrm{u}},0)$, respectively. For convenience of the subsequent formula expressions, we denote the position vector of the ${\mathrm{n}}_t\mathrm{th}$ antenna of ${\mathrm{UAV}}_i$ as ${\mathbf{v}}_{i{n}_t}=(x_{i{n}_t}^{\mathrm{v}},y_{i{n}_t}^{\mathrm{v}},H)$, $n_t=1,2,\dots ,N_T$, and if $\mathrm{U}{\mathrm{E}}_j$ is required for the $k\mathrm{th}$ content, denote the position of $\mathrm{U}{\mathrm{E}}_j$ as ${\mathbf{u}}_{jk}=(x_{jk}^{\mathrm{v}},y_{jk}^{\mathrm{v}},0)$, where $k\in \mathcal{K}$.

We assume that each UAV serves several nearest users and each user could be served by several UAVs. For simplicity, the user association is considered to be fixed, which means the connection between UAVs and users would be unchanged during the interested time. The connection relationship between ${\mathrm{UAV}}_i$ and $\mathrm{U}{\mathrm{E}}_j$ is denoted by $q_{ij}$. $q_{ij}=1$ represents that a connection exists, and $q_{ij}=0$ represents that a connection does not exist. That is,

$$\begin{array}{c}\hfill q_{ij}=\left\{\begin{array}{cc}& 1,\mathrm{connection}\mathrm{exists}\\ & 0,\mathrm{connection}\mathrm{does}\mathrm{not}\mathrm{exist}.\end{array}\right.\end{array}$$

(1)

The set of users served by ${\mathrm{UAV}}_i$ is denoted as $A_i$, $i\in \mathcal{M}$, and $A_i\subseteq \mathcal{N}$.

2.3. Content Request, Caching and Fetching

In the considered system, each user randomly requests content following a certain probability distribution. We denote the probability of content k being requested as $p_k$, $k\in \mathcal{K}$ and ${\displaystyle \sum _{k=1}^K}{p}_k=1$. For convenience, it is assumed that the content has been sorted by popularity, which means that content k is the $k\mathrm{th}$ most popular content of all content. Content popularity is modeled using a Zipf distribution, and the probabilities of popular content being requested by a user can be described as $P_K\sim \mathrm{Zipf}(\alpha ,K)$, and the probability of requesting the $k\mathrm{th}$ most popular content is

$$p_k=\frac{1}{k^{\alpha }{{\sum }_{n=1}^{K}n}^{-\alpha }}.$$

(2)

In (2), $\alpha$ is called the Zipf parameter, and $\alpha \in \left[0,1\right]$. Zipf distribution indicates the popularity of the content, and $\alpha$ affects the differences in content popularity. The smaller the parameter $\alpha$ is, the smaller the popularity differences among content will be, and the converse is also true. In extreme cases, if $\alpha =0$, all content will have the same probability of being requested, or if $\alpha =1$, the probability of content being requested will be in an inverse ratio to the ranking of the content.

The MBS is considered to be available for all users in the system and cached all contents. Differently, due to the limited power and capacity, each UAV can only connect some of the users and cache some of the content. Every UAV could choose to cache a complete content or just a part of it. The size of the $k\mathrm{th}$ content is denoted by $L_k$, and proportion of the $k\mathrm{th}$ content cached in ${\mathrm{UAV}}_i$ is denoted by $z_{ik}$, $z_{ik}\in \left[0,1\right]$, $\forall i\in \mathcal{M}$, $\forall k\in \mathcal{K}$. UAVs are able to implement the caching strategy through the high-capacity backhaul link with the MBS, which means that $z_{ik}$ is changeable and that limited storage of UAV could be reasonably allocated.

As for content fetching, UAVs and MBS cooperatively provide content for users. When the user initiates a data file request, the UAV serving the user only transmits the part of the file cached by the BTS, and the user reconstructs the file data from every connected UAV to obtain a complete file. If the requested file is not cached in the related UAVs, the missing file or part will be provided by MBS.

2.4. Signal, Channel and Noise Model

The number of antennas of each UAV is $N_T$; thus, the transmit signals could be denoted by $N_T$-dimension vectors. We denote the beamforming vector of the transmit signal ${\mathrm{UAV}}_i$ sent to $\mathrm{U}{\mathrm{E}}_j$ as

$${\mathbf{w}}_{ij}={(w_{ij1},w_{ij2},\dots ,w_{ij{n}_t},\dots ,w_{ij{N}_T})}^T,i\in \mathcal{M},j\in \mathcal{N},$$

(3)

where ${\mathbf{w}}_{ij}$ is a complex vector.

The channel coefficient between $\mathrm{U}{\mathrm{E}}_j$ and the $n_t\mathrm{th}$ antenna of ${\mathrm{UAV}}_i$ is denoted by $h_{ij{n}_t}$, which consists of two parts: path loss and small-scale fading. We denote that $h_{ij{n}_t}\left({\theta }_{ij{n}_t}\right)={\theta }_{ij{n}_t}{\tilde{h}}_{ij{n}_t}$, where ${\theta }_{ij{n}_t}$ reflects the pass loss, while ${\tilde{h}}_{ij{n}_t}$ represents the small-scale fading. ${\theta }_{ij{n}_t}$ is a function of ${\mathbf{u}}_{jk}$ and ${\mathbf{v}}_{i{n}_t}$, described as

$${\theta }_{ij{n}_t}={[{(x_{i{n}_t}^{\mathrm{v}}-x_{jk}^{\mathrm{u}})}^2+{(y_{i{n}_t}^{\mathrm{v}}-y_{jk}^{\mathrm{u}})}^2+H^2]}^{-0.5}.$$

(4)

${\tilde{h}}_{ij{n}_t}$ is considered to follow a Rician distribution. We adopt a Rician channel in our system model because a line of sight (LoS) signal usually exists in a UAV-assisted network. A Rician channel can be decomposed into a LoS propagation channel and a scattering channel, as shown in Equation (5).

$${\tilde{h}}_{ij{n}_t}=\sqrt{\frac{K_R}{1+K_R}}{\overline{h}}_{ij{n}_t}+\sqrt{\frac{1}{1+K_R}}{\widehat{h}}_{ij{n}_t},$$

(5)

where ${\overline{h}}_{ij{n}_t}$ and ${\widehat{h}}_{ij{n}_t}$ represent the LoS component and NLoS component, respectively, and ${\widehat{h}}_{ij{n}_t}$ usually follows a Rayleigh distribution. $K_R$ is the parameter of the Rician distribution, which determines the proportion of LoS and NLoS in the channel. These parameters are considered to be constant during the interested time slot. Let

$${\tilde{\mathbf{h}}}_{ij}={({\tilde{h}}_{ij1},{\tilde{h}}_{ij2},\dots \dots ,{\tilde{h}}_{ij{N}_T})}^T,$$

(6)

and

$${\theta }_{ij}={({\theta }_{ij1},{\theta }_{ij2},\dots \dots ,{\theta }_{ij{N}_T})}^T.$$

(7)

Then, we can denote the channel between ${\mathrm{UAV}}_i$ and $\mathrm{U}{\mathrm{E}}_j$ as ${\mathbf{h}}_{ij}={\theta }_{ij}\circ {\tilde{\mathbf{h}}}_{ij}$, where ∘ is the operator of the Hadamard product. Here, ${\tilde{\mathbf{h}}}_{ij}$ is a constant complex vector, and ${\theta }_{ij}$ is a complex vector variable. ${\mathbf{h}}_{ij}$ is a function of variable ${\theta }_{ij}$, so it is also denoted as ${\mathbf{h}}_{ij}\left({\theta }_{ij}\right)$.

It is assumed that MBS uses different frequency resources from UAVs, so there would be no interference between MBS and UAVs. Additive white gaussian noise (AWGN) and the interferences among UAVs are both considered. The signal to interference plus noise ratio (SINR) of the downlink transmission from ${\mathrm{UAV}}_i$ to $\mathrm{U}{\mathrm{E}}_j$ could be denoted as

$$SIN{R}_{ij}=\frac{|{\mathbf{h}}_{ij}^H{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^H{\mathbf{w}}_{mn}{|}^2+{\sigma }^2},$$

where ${\sigma }^2$ represents the power of AWGN. The transfer data rate from ${\mathrm{UAV}}_i$ to $\mathrm{U}{\mathrm{E}}_j$ could be calculated using

$$R_{ij}={log}_2(1+SIN{R}_{ij})={log}_2\left(1+\frac{|{\mathbf{h}}_{ij}^H{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^H{\mathbf{w}}_{mn}{|}^2+{\sigma }^2}\right),i\in \mathcal{M},j\in \mathcal{N}.$$

(8)

Specially, we denote the downlink data rate between MBS and $\mathrm{U}{\mathrm{E}}_j$ as $R_{0j}$. To simplify the discussion, $R_{0j}$ is set as constant $R_0$ for $\forall j\in \mathcal{N}$, where the link details of MBS are not considered. Generally, we set $R_0$ as a lower value compared with $R_{ij}$ because UAVs could usually achieve higher data rate by getting closer to users.

3. Problem Formulation and Solution

We minimize the average transmission latency of the network, which is related to locations, beamforming and caching strategies of UAVs. Specifically, the calculation of average latency is

$$\tau (\mathbf{w},\mathbf{u},\mathbf{z})=\sum _{k=1}^K\sum _{j=1}^N{p}_k\left(\frac{z_{0k}{L}_k}{R_0}+\sum _{i=1}^M{q}_{ij}\frac{z_{ik}{L}_k}{R_{ij}({\mathbf{w}}_{ij},{\theta }_{ij})}\right),$$

(9)

where R is the data rate, which is related to the beamforming weight $\mathbf{w}$ and path loss $\theta$, and $\theta$ is mapped to UAV locations $\mathbf{v}$. The system latency consists of two parts: MBS transmission latency and UAV transmission latency.

The related latency minimization problem is formulated as follows:

$$\begin{array}{cc}\hfill \underset{\mathbf{w},\theta ,\mathbf{v},\mathbf{z}}{min}& \sum _{k=1}^K\sum _{j=1}^N{p}_k\left(\frac{z_{0k}{L}_k}{R_0}+\sum _{i=1}^M{q}_{ij}\frac{z_{ik}{L}_k}{R_{ij}({\mathbf{w}}_{ij},{\theta }_{ij})}\right),\hfill \end{array}$$

(10a)

where $R_{ij}={log}_2\left(1+\frac{|{\mathbf{h}}_{ij}^H{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^H{\mathbf{w}}_{mn}{|}^2+{\sigma }^2}\right).$

$$\begin{array}{cc}\hfill \mathit{s}.\mathit{t}.& \sum _{j=1}^N\left|\right|{\mathbf{w}}_{ij}|{|}^2\le P,\forall i\in \mathcal{M},\hfill \end{array}$$

(10b)

$$\begin{array}{cc}\hfill & {\theta }_{ij{n}_t}^{-1}=\sqrt{{(x_{i{n}_t}-x_{jk})}^2+{(y_{i{n}_t}-y_{jk})}^2+H^2},\forall i\in \mathcal{M},\forall j\in \mathcal{N},\hfill \end{array}$$

(10c)

$$\begin{array}{cc}\hfill & \sum _{k=1}^M{z}_{ik}{L}_k\le L_{max},\forall i\in \mathcal{M},\hfill \end{array}$$

(10d)

$$\begin{array}{cc}\hfill & z_{0k}+\sum _{i=1}^M{z}_{ik}\ge 1,\forall k\in \mathcal{K},\hfill \end{array}$$

(10e)

$$\begin{array}{cc}\hfill & 0\le z_{ik}\le 1,0\le z_{0k}\le 1,\forall i\in \mathcal{M},\forall k\in \mathcal{K}.\hfill \end{array}$$

(10f)

Here, inequality (10b) is the power constraint. The transmit power of UAV is limited, and we assume that the maximum power budget of every UAV is P. Equation constraint (10c) reflects the relationship between path loss $\theta$ and UAV location $\mathbf{v}$. Inequality (10d) is the capacity constraint of every UAV, ensuring the cached contents are not larger than the capacity $L_{max}$. Inequality (10e) represents the content integrity constraint. For all popular content, the data transmitted by MBS should be able to be combined into complete content with the data transmitted by UAVs, as is shown in (10e). Furthermore, (10f) reflects the physical meaning of $\mathbf{z}$ as the caching proportions.

As shown in (10a), caching variable $\mathbf{z}$, UAV location variable $\mathbf{v}$ and beamforming variable $\mathbf{w}$ in the objective function are coupled, which means that it would be quite difficult to solve the problem directly. In order to reduce the complexity of the solution, a step-by-step iterative optimization method is adopted. We fix one of the variables, solve the remaining variables and iteratively solve the original problem.

We fix the caching variable $\mathbf{z}$ first, and the proportion of the content k cached in ${\mathrm{UAV}}_i$ is set as $z_{ik}^{*}$. Then, the optimization problem becomes

$$\begin{array}{cc}\hfill \underset{\mathbf{w},\theta ,\mathbf{R}}{min}& \sum _{k=1}^K\sum _{j=1}^N\frac{p_k{q}_{ij}{z}_{\mathrm{ik}}^{*}{L}_k}{{\tilde{R}}_{ij}},\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill \mathit{s}.\mathit{t}.& \sum _{j=1}^N\left|\right|{\mathbf{w}}_{ij}|{|}^2\le P,\forall i\in \mathcal{M},\hfill \end{array}$$

(11b)

$$\begin{array}{cc}\hfill & {\theta }_{ij{n}_t}^{-1}\ge \sqrt{{(x_{i{n}_t}-x_{jk})}^2+{(y_{i{n}_t}-y_{jk})}^2+H^2},\forall i\in \mathcal{M},\forall j\in \mathcal{N},\hfill \end{array}$$

(11c)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{ij}\le {log}_2\left(1+\frac{|{\mathbf{h}}_{ij}^H{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^H{\mathbf{w}}_{mn}{|}^2+{\sigma }^2}\right),\forall i\in \mathcal{M},\forall j\in \mathcal{N}.\hfill \end{array}$$

(11d)

Here, inequality (11c) is the relaxed version of the equality constraint (10c). Since the optimality always occurs at the equality, the relax of the constraint would not cause any loss of generality, and this treatment could reduce the complexity of the problem.

Due to the complex form of the denominator, the whole function is non-convex in (10a). We introduce variable ${\tilde{R}}_{ij}$ and its constraint, so that the objective function (11a) is convex now. Note that the constraint of ${\tilde{R}}_{ij}$ is also a relaxed version, and it is easy to prove that constraint (11d) turns into the tight constraint when the problem (11) obtains the optimal solution, which means that ${\tilde{R}}_{ij}$ is equal to $R_{ij}$ when the optimal problem is solved.

Problem (11) is still not a convex problem due to constraints (11c) and (11d) being neither concave nor convex. For (11c), we can make the left-hand side linearized via the first-order Taylor expansion, as shown in (12).

$$\frac{2{\theta }_{ij}^{[\ell ]}-{\theta }_{ij}}{{\left({\theta }_{ij}^{[\ell ]}\right)}^2}\ge \sqrt{{(x_i-x_{jk})}^2+{(y_i-y_{jk})}^2+H^2},\forall i\in \mathcal{M},\forall j\in \mathcal{N},$$

(12)

where ${\theta }_{ij}^{[\ell ]}$ represents the Taylor expansion point.

We now consider transforming inequality (11d) into a convex constraint via DC method. The basic idea of DC method is, first, to transform the original problem into the difference of two convex functions, and, then, to apply an approximation method to eliminate the non-convexity.

For inequality (11d), we introduce a strongly convex function, which is added to and then taken away from the right-hand side of the original inequality, as shown in inequality (13). Inspired by [21], we choose a quadratic form as the introduced strongly convex function. We write inequality (11d) as

$$\begin{array}{cc}\hfill {\tilde{R}}_{ij}\le & {log}_2\left(1+\frac{|{\mathbf{h}}_{ij}^{\mathrm{H}}{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\in \mathcal{M},m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^{\mathrm{H}}{\mathbf{w}}_{mn}{|}^2+{\sigma }^2}\right)+{\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2)\hfill \\ \hfill & -{\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2),\forall i\in \mathcal{M},\forall j\in \mathcal{N}.\hfill \end{array}$$

(13)

In inequality (13), ${\xi }_{ij}$ is a positive constant we set. It is proved in [21] that, if ${\xi }_{ij}$ is set to be positive and large enough, function ${\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2)$ will have strong convexity. Although the form of ${\tilde{R}}_{ij}$ is non-convex and non-concave, the sum of ${\tilde{R}}_{ij}$ and the introduced function would be a convex function, as long as the convexity of the introduced function is strong enough. In this way, a form of difference between two convex functions is constructed. Let

$$\begin{array}{cc}\hfill f_{ij}(\mathbf{w},\theta )=& {log}_2\left(1+\frac{|{\mathbf{h}}_{ij}^H{\mathbf{w}}_{ij}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\mathbf{h}}_{mj}^H{\mathbf{w}}_{mn}{|}^2+{\sigma }^2}\right)+{\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2)\hfill \\ \hfill =& R_{ij}(\mathbf{w},\theta )+{\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2),\hfill \end{array}$$

(14)

then constraint (13) is changed into

$${\tilde{R}}_{ij}\le f_{ij}(\mathbf{w},\theta )-{\xi }_{ij}{\left(\right|\left|\mathbf{w}\right||}^2+{\left|\right|\theta \left|\right|}^2),\forall i\in \mathcal{M},\forall j\in \mathcal{N}.$$

(15)

Constraint (15) is in the form of the difference of two convex functions. To actually eliminate the non-convexity, we need to apply an approximation method, as it is noted earlier. The factor leading constraint (15) to be non-convex is $f_{ij}(\mathbf{w},\theta )$ because it is a convex function on the right-hand side of “≤”. We tighten constraint (15) by replacing $f_{ij}(\mathbf{w},\theta )$ with its lower-bound Taylor linearization. The first-order Taylor expansion for $f_{ij}({\mathbf{w}}_{ij},{\theta }_{ij})$ can change the original formula into a convex constraint. The gradient of $R_{ij}(\theta ,\mathbf{w})$ is defined as $\nabla R_{ij}(\theta ,\mathbf{w})=[{\nabla }_{\theta }{R}_{ij}(\theta ,\mathbf{w});{\nabla }_w{R}_{ij}(\theta ,\mathbf{w})]$.

As shown in Equation (8), the transmission rate $R_{ij}(\theta ,\mathbf{w})$ has a complicated form, which is mainly caused by the SINR part. There are a series of summations of variables, making the derivative of $R_{ij}(\theta ,\mathbf{w})$ hard to express. To deal with this, we bring out a method to convert the summations into matrix multiplications and subsequently derive the gradient for applying the linear approximation. We reconstruct the optimization variables by introducing a series of vectors and matrices. Let

$$\mathbf{w}={\left[\underset{M·N·N_T}{\underbrace{\underset{N·N_T}{\underbrace{{\mathbf{w}}_{11},{\mathbf{w}}_{12},\dots \dots ,{\mathbf{w}}_{1j},\dots \dots ,{\mathbf{w}}_{1N}}},\underset{N·N_T}{\underbrace{{\mathbf{w}}_{21},\dots \dots ,{\mathbf{w}}_{2N}}},\dots \dots ,{\mathbf{w}}_{ij},\dots \dots ,{\mathbf{w}}_{MN}}}\right]}^T.$$

(16)

As one of optimization variables in this problem, $\mathbf{w}$ is a vector variable with $M·N·N_T$ dimensions, contains all beamforming weight information. The expression of SINR in matrix form can be constructed as follows:

Denote ${\widehat{\mathbf{H}}}_{ij}={\mathbf{h}}_{ij}{\mathbf{h}}_{ij}^H$, $i\in \mathcal{M}$, $j\in \mathcal{N}$, and then, let

$$\begin{array}{c}\hfill {\tilde{\mathbf{H}}}_{ij}=Bdiag\left(\underset{M·N\mathrm{blocks}}{\underbrace{\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}j\mathrm{th}\mathrm{block}\mathrm{is}\mathbf{0}}{{\widehat{\mathbf{H}}}_{1j},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{1j}}}},\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}(N+j)\mathrm{th}\mathrm{block}\mathrm{is}\mathbf{0}}{{\widehat{\mathbf{H}}}_{2j},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{2j}}}},\dots ,\stackrel{\begin{array}{c}\mathrm{blocks}\mathrm{in}\\ \mathrm{the}i\mathrm{th}\mathrm{group}\\ \mathrm{are}\mathrm{all}\mathbf{0}\end{array}}{\underset{N\mathrm{blocks}}{\underbrace{\mathbf{0},\mathbf{0},\dots ,\mathbf{0}}}},\dots ,\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\begin{array}{c}\mathrm{the}\left(\right(M-1)N+j)\mathrm{th}\\ \mathrm{block}\mathrm{is}\mathbf{0}\end{array}}{{\widehat{\mathbf{H}}}_{Mj},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{Mj}}}}}}\right)\end{array}.$$

(17)

Note that every ${\widehat{\mathbf{H}}}_{ij}$ is a $N_T$-dimension square matrix and that ${\tilde{\mathbf{H}}}_{ij}$ is a block diagonal matrix with $M·N$ blocks. Specifically, the diagonal blocks of ${\tilde{\mathbf{H}}}_{ij}$ are divided into M groups, and each group contains N blocks. Every diagonal block is a $N_T\times N_T$ matrix, so we can see that each ${\tilde{\mathbf{H}}}_{ij}$ is a square matrix with $M·N·N_T$ dimensions. Actually, ${\tilde{\mathbf{H}}}_{ij}$ consists of a series of ${\widehat{\mathbf{H}}}_{ij}$ and zero matrices. The N blocks of a group are all nearly the same, except the $j\mathrm{th}$ block of every group is a zero matrix. Furthermore, the N blocks of the $i\mathrm{th}$ group are all zero matrices. Except those above blocks, the blocks in the $i^{*}\mathrm{th}$ group are set as ${\widehat{\mathbf{H}}}_{i^{*}j}$. The blocks of the $i^{*}\mathrm{th}$ group roughly represent the channels between interference source ${\mathrm{UAV}}_{i^{*}}$ and $\mathrm{U}{\mathrm{E}}_j$. For the downlink channel from ${\mathrm{UAV}}_i$ to $\mathrm{U}{\mathrm{E}}_j$, the interference caused by signals sent from other UAVs to other users can be represented by ${\mathbf{w}}_{ij}^H{\tilde{\mathbf{H}}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}$. Let

$$\begin{array}{cc}\hfill & {\mathbf{H}}_{ij}=\hfill \\ \hfill & Bdiag\left(\underset{M·N\mathrm{blocks}}{\underbrace{\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}j\mathrm{th}\mathrm{block}\mathrm{is}\mathbf{0}}{{\widehat{\mathbf{H}}}_{1j},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{1j}}}},\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}(N+j)\mathrm{th}\mathrm{block}\mathrm{is}\mathbf{0}}{{\widehat{\mathbf{H}}}_{2j},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{2j}}}},\dots ,\stackrel{\begin{array}{c}\mathrm{the}i\mathrm{th}\mathrm{group}\mathrm{are}\\ \mathrm{all}\mathbf{0},\mathrm{except}\mathrm{the}\\ \left(\right(i-1)N+j)\mathrm{th}\mathrm{block}\end{array}}{\underset{N\mathrm{blocks}}{\underbrace{\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{ij},\dots ,\mathbf{0}}}},\dots ,\underset{N\mathrm{blocks}}{\underbrace{\stackrel{\begin{array}{c}\mathrm{the}\left(\right(M-1)N+j)\mathrm{th}\\ \mathrm{block}\mathrm{is}\mathbf{0}\end{array}}{{\widehat{\mathbf{H}}}_{Mj},\dots ,\mathbf{0},\dots ,{\widehat{\mathbf{H}}}_{Mj}}}}}}\right)\hfill \end{array}.$$

(18)

${\mathbf{H}}_{ij}$ is almost the same as ${\tilde{\mathbf{H}}}_{ij}$; the only difference is the $j\mathrm{th}$ block in the $i\mathrm{th}$ group. To be specific, the $\left(\right(i-1)N+j)\mathrm{th}$ block of ${\mathbf{H}}_{ij}$ is ${\widehat{\mathbf{H}}}_{ij}$ instead of $\mathbf{0}$. Then, we can give a concise form of the channel data rate as follows:

$$\begin{array}{cc}\hfill R_{ij}=& {log}_2\left(\frac{{\mathbf{w}}_{ij}^{\mathrm{H}}{\mathbf{H}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2}{{\mathbf{w}}_{ij}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2}\right)\hfill \\ \hfill =& {log}_2\left({\mathbf{w}}_{ij}^{\mathrm{H}}{\mathbf{H}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2\right)-{log}_2\left({\mathbf{w}}_{ij}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2\right),\forall i\in \mathcal{M},\forall j\in \mathcal{N}.\hfill \end{array}$$

(19)

The partial difference of $R_{ij}$ for $\mathbf{w}$ can be calculated as

$${\nabla }_{\mathbf{w}}{R}_{ij}(\theta ,\mathbf{w})=\left(\frac{2{\mathbf{w}}_{ij}^{\mathrm{H}}{\mathbf{H}}_{ij}\left(\theta \right)}{{\mathbf{w}}_{ij}^{\mathrm{H}}{\mathbf{H}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2}-\frac{2{\mathbf{w}}_{ij}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left(\theta \right)}{{\mathbf{w}}_{ij}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left(\theta \right){\mathbf{w}}_{ij}+{\sigma }^2}\right)/ln2,\forall i\in \mathcal{M},\forall j\in \mathcal{N}.$$

(20)

At this point, we have reconstructed the expression of $R_{ij}$, and it is much easier to seek the partial difference of $R_{ij}$ in Equation (19), compared with Equation (8). The main idea behind this proposed reconstruction method is to specially design the variables as structured vectors and to construct the form of matrix multiplications according to the summations in the original expression. This method turned out to be effective in dealing with complicated summations, such as the interference among UAVs in this study.

Equation (19) is a concise form with respect to variable $\mathbf{w}$. To obtain the partial difference for $\theta$, we need to construct a new form of $R_{ij}$. Let ${\omega }_{i{j}_1{j}_2}\left(\mathbf{w}\right)={\tilde{\mathbf{h}}}_{i{j}_1}\circ {\mathbf{w}}_{i{j}_2}$, ${\widehat{\Omega }}_{i{j}_1{j}_2}\left(\mathbf{w}\right)={\omega }_{i{j}_1{j}_2}{\omega }_{i{j}_1{j}_2}^{\mathrm{H}}$, $i\in \mathcal{M}$, $j_1,j_2\in \mathcal{N}$. The dimension of vector ${\omega }_{i{j}_1{j}_2}$ is $N_T$, and every ${\widehat{\Omega }}_{i{j}_1{j}_2}$ is a $N_T$-dimension square matrix.

We define

$${\theta }_j={\left[\underset{M·N_T}{\underbrace{{\theta }_{1j},{\theta }_{2j},\dots \dots ,{\theta }_{Mj}}}\right]}^T,\forall j\in \mathcal{N}.$$

(21)

and

$$\theta ={\left[\underset{M·N·N_T}{\underbrace{{\theta }_1,{\theta }_2,\dots \dots ,{\theta }_N}}\right]}^T.$$

(22)

where ${\theta }_{ij}={({\theta }_{ij1},{\theta }_{ij2},\dots \dots ,{\theta }_{ij{N}_T})}^T$, as defined in (7). Every ${\theta }_j$ is a vector with $M·N_T$ dimensions, and $\theta$ is a vector with $M·N·N_T$ dimensions, which consists of ${\theta }_j$ for all $j\in \mathcal{N}$. We use ${\theta }_j$ as one of the optimization variables and seek the partial difference for it. Let

$$\begin{array}{cc}\hfill {\ddot{\Omega }}_{i{j}_1{j}_2}\left(\mathbf{w}\right)=& Bdiag{\left[\underset{M\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}i\mathrm{th}\mathrm{block}\mathrm{is}\mathbf{0}}{{\widehat{\Omega }}_{i{j}_1{j}_2},{\widehat{\Omega }}_{2j_1{j}_2},\dots ,{\widehat{\Omega }}_{\left(i-1\right)j_1{j}_2},\mathbf{0},{\widehat{\Omega }}_{\left(i+1\right)j_1{j}_2},\dots ,{\widehat{\Omega }}_{M{j}_1{j}_2}}}}\right]}^T,\hfill \\ \hfill & i\in \mathcal{M},j_1,j_2\in \mathcal{N}.\hfill \end{array}$$

(23)

and

$${\dot{\Omega }}_{ij}\left(\mathbf{w}\right)=Bdiag{\left[\underset{M\mathrm{blocks}}{\underbrace{\stackrel{\mathrm{the}i\mathrm{th}\mathrm{block}\mathrm{is}\widehat{\Omega }}{\mathbf{0},\mathbf{0},\dots \mathbf{0},{\widehat{\Omega }}_{ijj},\mathbf{0},\dots ,\mathbf{0}}}}\right]}^T,i\in \mathcal{M},j\in \mathcal{N}.$$

(24)

Then, we define

$${\tilde{\Omega }}_{ij}\left(\mathbf{w}\right)=\sum _{n\in A_m,n\ne j}^N{\ddot{\Omega }}_{ijn},i\in \mathcal{M},j\in \mathcal{N},$$

(25)

and

$${\Omega }_{ij}\left(\mathbf{w}\right)=\left(\sum _{n\in A_m,n\ne j}^N{\ddot{\Omega }}_{ijn}+{\dot{\Omega }}_{ij}\right),i\in \mathcal{M},j\in \mathcal{N}.$$

(26)

Note that the only difference between ${\tilde{\Omega }}_{ij}\left(\mathbf{w}\right)$ and ${\Omega }_{ij}\left(\mathbf{w}\right)$ is that ${\Omega }_{ij}\left(\mathbf{w}\right)$ has one more summation term than ${\tilde{\Omega }}_{ij}\left(\mathbf{w}\right)$. Similar to (19), we can transform (8) into

$$\begin{array}{cc}\hfill R_{ij}=& {log}_2\left(\frac{{\theta }_j^{\mathrm{H}}{\Omega }_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2}{{\theta }_j^{\mathrm{H}}{\tilde{\Omega }}_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2}\right)\hfill \\ \hfill =& {log}_2\left({\theta }_j^{\mathrm{H}}{\Omega }_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2\right)-{log}_2\left({\theta }_j^{\mathrm{H}}{\tilde{\Omega }}_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2\right).\hfill \\ \hfill & i\in \mathcal{M},j_1,j_2\in \mathcal{N}.\hfill \end{array}$$

(27)

The partial difference of $R_{ij}$ for $\theta$ can be calculated as

$${\nabla }_{\theta }{R}_{ij}(\theta ,\mathbf{w})=\left(\frac{2{\theta }_j^{\mathrm{H}}{\Omega }_{ij}\left(\mathbf{w}\right)}{{\theta }_j^{\mathrm{H}}{\Omega }_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2}-\frac{2{\theta }_j^{\mathrm{H}}{\tilde{\Omega }}_{ij}\left(\mathbf{w}\right)}{{\theta }_j^{\mathrm{H}}{\tilde{\Omega }}_{ij}\left(\mathbf{w}\right){\theta }_j+{\sigma }^2}\right)/ln2,i\in \mathcal{M},j\in \mathcal{N}.$$

(28)

The first-order Taylor expansion for $f_{ij}({\mathbf{w}}_{ij},{\theta }_{ij})$ at ${\mathbf{w}}^{[\ell ]}$,${\theta }^{[\ell ]}$ is denoted as $F_{ij}(\mathbf{w},\theta ;{\mathbf{w}}^{[\ell ]},{\theta }^{[\ell ]})$:

$$\begin{array}{cc}\hfill & F_{ij}(\mathbf{w},\theta ;{\mathbf{w}}^{[\ell ]},{\theta }^{[\ell ]})=f_{ij}({\mathbf{w}}^{[\ell ]},{\theta }^{[\ell ]})+\frac{2Re\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\mathbf{H}}_{ij}\left({\theta }^{[\ell ]}\right)\mathbf{w}-{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}{\mathbf{H}}_{ij}\left({\theta }^{[\ell ]}\right){\mathbf{w}}^{[\ell ]}\right]}{{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}{\mathbf{H}}_{ij}\left({\theta }^{[\ell ]}\right){\mathbf{w}}^{[\ell ]}+{\sigma }^2}-\hfill \\ \hfill & \frac{2Re\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left({\theta }^{[\ell ]}\right)\mathbf{w}-{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left({\theta }^{[\ell ]}\right){\mathbf{w}}^{[\ell ]}\right]}{{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}{\tilde{\mathbf{H}}}_{ij}\left({\theta }^{[\ell ]}\right){\mathbf{w}}^{[\ell ]}+{\sigma }^2}+\frac{2Re\left[{\left({\theta }_j^{[\ell ]}\right)}^{\mathrm{H}}\Omega \left({\mathbf{w}}^{[\ell ]}\right){\theta }_j-{\left({\theta }_j^{[\ell ]}\right)}^H\Omega \left({\mathbf{w}}^{[\ell ]}\right){\theta }_j^{[\ell ]}\right]}{{\left({\theta }_j^{[\ell ]}\right)}^{\mathrm{H}}\Omega \left({\mathbf{w}}^{[\ell ]}\right){\theta }_j^{[\ell ]}+{\sigma }^2}-\hfill \\ \hfill & \frac{2Re\left[{\left({\theta }_j^{[\ell ]}\right)}^{\mathrm{H}}\tilde{\Omega }\left({\mathbf{w}}^{[\ell ]}\right){\theta }_j-{\left({\theta }_j^{[\ell ]}\right)}^H\tilde{\Omega }\left({\mathbf{w}}^{[\ell ]}\right){\theta }_j^{[\ell ]}\right]}{{\left({\theta }_j^{[\ell ]}\right)}^{\mathrm{H}}\tilde{\Omega }\left({\mathbf{w}}^{[\ell ]}\right){\theta }_j^{[\ell ]}+{\sigma }^2}-2{\xi }_{ij}Re\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^{\mathrm{H}}\mathbf{w}-{\left|\right|{\mathbf{w}}^{[\ell ]}\left|\right|}^2+{\left({\theta }^{[\ell ]}\right)}^{\mathrm{H}}\theta -{\left|\right|{\theta }^{[\ell ]}\left|\right|}^2\right]\hfill \\ \hfill & i\in \mathcal{M},j_1,j_2\in \mathcal{N}.\hfill \end{array}$$

(29)

Until now, the optimization subproblem (11) has been converted as

$$\begin{array}{cc}\hfill \underset{\mathbf{w},\theta ,\mathbf{R}}{min}& \sum _{k=1}^K\sum _{j=1}^N\frac{p_k{q}_{ij}{z}_{\mathrm{ik}}^{*}{L}_k}{{\tilde{R}}_{ij}},\hfill \end{array}$$

(30a)

$$\begin{array}{cc}\hfill \mathit{s}.\mathit{t}.& \sum _{j=1}^N{\left|\right|{\mathbf{w}}_{ij}\left|\right|}^2\le P,\forall i\in \mathcal{M},\hfill \end{array}$$

(30b)

$$\begin{array}{cc}\hfill & \frac{2{\theta }_{ij}^{[\ell ]}-{\theta }_{ij}}{{\left({\theta }_{ij}^{[\ell ]}\right)}^2}\ge \sqrt{{(x_i-x_{jk})}^2+{(y_i-y_{jk})}^2+H^2},\forall i\in \mathcal{M},\forall j\in \mathcal{N},\hfill \end{array}$$

(30c)

$$\begin{array}{cc}\hfill & {\tilde{R}}_{ij}\le F_{ij}(\mathbf{w},\theta ;{\mathbf{w}}^{[\ell ]},{\theta }^{[\ell ]})-{\xi }_{ij}({\left|\right|\mathbf{w}\left|\right|}^2+{\left|\right|\theta \left|\right|}^2),\forall i\in \mathcal{M},\forall j\in \mathcal{N}.\hfill \end{array}$$

(30d)

Problem (30) is a convex optimization problem, and it is a transformation of problem (11), which corresponds to the optimization of UAV location and beamforming variables with caching variable fixed.

Then, we oppositely consider optimizing a caching variable with the UAV location and beamforming variables fixed. UAV caching optimization subproblem is formulated as   

$$\begin{array}{cc}\hfill \underset{\mathbf{z}}{min}& \sum _{k=1}^K\sum _{j=1}^N{p}_k\left(\frac{z_{0k}{L}_k}{R_0}+\sum _{i=1}^M{q}_{ij}\frac{z_{ik}{L}_k}{R_{ij}^{*}({\mathbf{w}}_{ij},{\theta }_{ij})}\right),\hfill \end{array}$$

(31a)

$$\begin{array}{cc}\hfill \mathit{s}.\mathit{t}.& \sum _{k=1}^K{z}_{ik}{L}_{ik}\le L_{max},\forall i\in \mathcal{M},\hfill \end{array}$$

(31b)

$$\begin{array}{cc}\hfill & \sum _{i=0}^M{z}_{ik}\ge 1,\forall k\in \mathcal{K},\hfill \end{array}$$

(31c)

$$\begin{array}{cc}\hfill & 0\le z_{ik},z_{0k}\le 1,\forall i\in \mathcal{M},\forall k\in \mathcal{K}.\hfill \end{array}$$

(31d)

It is obvious that problem (31) is a linear programming problem.

Algorithm 1 outlines the DC-based method to solve (10).   

Algorithm 1: The DC-based joint optimization algorithm.

Initiation: 1. Set user positions $\mathbf{u}$ and connection relationship $q_{ij}$. 2. Set the iteration number $\ell =0$, and initialize start values of ${\mathbf{w}}^{[\ell ]}$, ${\mathbf{v}}^{[\ell ]}$ and ${\mathbf{z}}^{[\ell ]}$.  Calculate ${\theta }^{[\ell ]}$ from ${\mathbf{v}}^{[\ell ]}$ and $\mathbf{u}$. Repeat: 3. Solve problem (30) to obtain the optimal solutions ${\mathbf{w}}^{*}$, ${\mathbf{v}}^{*}$.  Calculate ${\theta }^{*}$ from ${\mathbf{v}}^{*}$ and $\mathbf{u}$. 4. Solve problem (31) to obtain the optimal solution ${\mathbf{z}}^{*}$. 5. $\ell =\ell +1$. 6. ${\mathbf{w}}^{[\ell ]}={\mathbf{w}}^{*}$, ${\mathbf{v}}^{[\ell ]}={\mathbf{v}}^{*}$, ${\mathbf{z}}^{[\ell ]}={\mathbf{z}}^{*}$. $\mathbf{Until}$ convergence of the objective function in (30). 7. Output ${\mathbf{w}}^{[\ell ]}$, ${\mathbf{v}}^{[\ell ]}$ and ${\mathbf{z}}^{[\ell ]}$ as the optimal solution.

4. Simulation Results

In this section, Matlab-based simulations are presented, and we evaluate the performance of the proposed joint optimization algorithm through numerical results. In our simulations, we randomly distribute ground users in a $500\mathrm{m}\times 500\mathrm{m}$ square area, and the number of users is $N=9$. We set the number of UAVs as $M=3$, and the initial locations of the UAVs are roughly uniformly distributed. Specifically, we decided the connection relationships first, in which each user is served by two UAVs. The initial locations of UAVs are set as the average location of served users. We also set the UAV altitude as $H=100\mathrm{m}$, the antenna number as $N_T=16$, and the popular content amount as $K=12$. As for channels, we set the $K_R=2$, ${\overline{h}}_{ij{n}_t}=1$, and ${\widehat{h}}_{ij{n}_t}$ as a complex random number with an average modulus of 1 in Equation (5). Moreover, we define the ratio of total capacity of UAVs to the total size of popular content as the UAV capacity coefficient, denoted by $r_{cc}$, which represents the capacity-to-content ratio. Specifically, we decide the capacity limit of UAVs as $L_{max}=\frac{r_{cc}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{k=1}^K}{z}_{ik}{L}_k}{M}$ in constraint (31b).

To show the effectiveness of the DC-based optimization algorithm, we present two benchmark algorithms, denoted by Algorithms 2 and 3, respectively. Algorithm 2 optimizes the beamforming and caching strategy with the UAV locations fixed, while Algorithm 3 optimizes the beamforming and UAV locations with cache fixed. These two benchmark algorithms are both variants of the proposed DC-based algorithm, and they are supposed to help evaluate the impact of different optimization variables in this model. Furthermore, we change the parameters in simulations to observe the performance changes in the algorithms.

The optimization problem of Algorithm 2 is shown in (32).

$$\begin{array}{cc}\hfill \underset{\mathbf{w},\mathbf{R}}{min}& \sum _{k=1}^K\sum _{j=1}^N\frac{p_k{q}_{ij}{z}_{\mathrm{ik}}^{*}{L}_k}{{\tilde{R}}_{ij}},\hfill \end{array}$$

(32a)

$$\begin{array}{cc}\hfill \mathit{s}.\mathit{t}.& \sum _{j=1}^N\left|\right|{\mathbf{w}}_{ij}{\left|\right|}^2\le P,\forall i\in \mathcal{M},\hfill \end{array}$$

(32b)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\tilde{R}}_{ij}\le & {log}_2\left(1+\frac{|{\left({\mathbf{h}}_{ij}^{*}\right)}^H{\mathbf{w}}_{ij}^{[\ell ]}{|}^2}{{\displaystyle \sum _{m\ne i}}{\displaystyle \sum _{n\in A_m,n\ne j}}|{\left({\mathbf{h}}_{mj}^{*}\right)}^H{\mathbf{w}}_{mn}^{[\ell ]}{|}^2+{\sigma }^2}\right)+{\xi }_{ij}\left|\right|{\mathbf{w}}^{[\ell ]}{\left|\right|}^2+\hfill \\ \hfill & \frac{2\mathrm{Re}\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\mathbf{H}}_{ij}^{*}\mathbf{w}-{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\mathbf{H}}_{ij}^{*}{\mathbf{w}}^{[\ell ]}\right]}{{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\mathbf{H}}_{ij}^{*}{\mathbf{w}}^{[\ell ]}+{\sigma }^2}-\frac{2\mathrm{Re}\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\tilde{\mathbf{H}}}_{ij}^{*}\mathbf{w}-{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\tilde{\mathbf{H}}}_{ij}^{*}{\mathbf{w}}^{[\ell ]}\right]}{{\left({\mathbf{w}}^{[\ell ]}\right)}^H{\tilde{\mathbf{H}}}_{ij}^{*}{\mathbf{w}}^{[\ell ]}+{\sigma }^2}-\hfill \\ \hfill & 2{\xi }_{ij}\mathrm{Re}\left[{\left({\mathbf{w}}^{[\ell ]}\right)}^H\mathbf{w}-\left|\right|{\mathbf{w}}^{[\ell ]}{\left|\right|}^2\right]-{\xi }_{ij}{\left|\right|\mathbf{w}\left|\right|}^2,\forall i\in \mathcal{M},\forall j\in \mathcal{N}.\hfill \end{array}\end{array}$$

(32c)

Algorithm 2 optimizes the beamforming and caching strategy iteratively, and the UAV locations are fixed as the average position of the connected users. In the case of Algorithm 2, the system is similar to a traditional cache network, with several small cellular base stations in it. To ensure the rigor of the conclusions, Algorithm 2 adopts the same DC-based method and reconstruction method as Algorithm 1, although the form is simplified because of the reduction in optimization variables.

Algorithm 3 is a variant with cache fixed, which jointly optimizes UAV location and beamforming. For reasonableness, we consider the cache occupied by each content being proportional to its size in Algorithm 3, where the proportion of every content cached in each UAV is $\frac{r_{cc}}{M}$. Compared with Algorithm 1, Algorithm 3 eliminates the step of cache iterative optimization, and the other steps are almost the same.

Compared with Algorithm 1, the benchmark Algorithm 2 indicates the impact of UAV location optimization, while Algorithm 3 reflects the effect of caching optimization. The two benchmark algorithms are outlined in Algorithms 2 and 3.    

Algorithm 2: The beamforming and caching iterative optimization algorithm.

Initiation: 1. Set user positions $\mathbf{u}$ and connection relationship $q_{ij}$. Set $\mathbf{v}$ to the average of  connected users’ positions. Calculate ${\mathbf{h}}_{ij}^{*}$ from $\mathbf{u}$ and $\mathbf{v}$. 2. Set the iteration number $\ell =0$, and initialize start values of ${\mathbf{w}}^{[\ell ]}$ and ${\mathbf{z}}^{[\ell ]}$. Repeat: 3. Solve problem (32) to obtain the optimal solutions ${\mathbf{w}}^{*}$. 4. Solve problem (31) to obtain the optimal solution ${\mathbf{z}}^{*}$. 5. $\ell =\ell +1$. 6. ${\mathbf{w}}^{[\ell ]}={\mathbf{w}}^{*}$, ${\mathbf{z}}^{[\ell ]}={\mathbf{z}}^{*}$. $\mathbf{Until}$ convergence of the objective function in (32). 7. Output ${\mathbf{w}}^{[\ell ]}$ and ${\mathbf{z}}^{[\ell ]}$ as the optimal solution.

Algorithm 3: The joint beamforming and location optimization algorithm.

Initiation: 1. Set user positions $\mathbf{u}$ and connection relationship $q_{ij}$. Set every elements of $\mathbf{z}$  to $\frac{r_{cc}}{M}$. 2. Set the iteration number $\ell =0$, and initialize start values of ${\mathbf{w}}^{[\ell ]}$, ${\mathbf{v}}^{[\ell ]}$.  Calculate ${\theta }^{[\ell ]}$ from ${\mathbf{v}}^{[\ell ]}$ and $\mathbf{u}$. Repeat: 3. Solve problem (30) to obtain the optimal solutions ${\mathbf{w}}^{*}$, ${\mathbf{v}}^{*}$.  Calculate ${\theta }^{*}$ from ${\mathbf{v}}^{*}$ and $\mathbf{u}$. 4. $\ell =\ell +1$. 5. ${\mathbf{w}}^{[\ell ]}={\mathbf{w}}^{*}$, ${\mathbf{v}}^{[\ell ]}={\mathbf{v}}^{*}$. $\mathbf{Until}$ convergence of the objective function in (30). 6. Output ${\mathbf{w}}^{[\ell ]}$ and ${\mathbf{v}}^{[\ell ]}$ as the optimal solution.

Firstly, we observe the impact of the UAV capacity coefficient $r_{cc}$ on the system latency. We set the Zipf parameter $\alpha =0.5$. As it is shown in Figure 2, the latency performance of the proposed algorithm is better than those of the two benchmark algorithms. The delay decreases with the increase in UAV coefficient, which means expanding the cache capacity of UAVs would achieve a higher network efficiency. And, note that the average transmission latency of Algorithm 3 gets closer and closer to that of proposed Algorithm 1, while $r_{cc}$ changes from 0.7 to 1. When $r_{cc}=1$, the latency of Algorithm 3 is almost the same as that of Algorithm 1 because if $r_{cc}\ge 1$, which means UAVs obtain enough capacity to cache all the contents, users do not need to request content from MBS. In this case, the content placement in UAVs has little impact on the average transmission latency.

Then, in Figure 3, we consider the impact of Zipf parameter $\alpha$. The UAV capacity coefficient is set as $r_{cc}=0.8$. We increase $\alpha$ from 0.3 to 1, and it is observed that the latency of Algorithm 1 is decreased. Predictably, the performance of Algorithm 2 is relatively poor compared with Algorithm 1, and the trend of its curve is consistent with that of Algorithm 1. Note that the performance of Algorithm 3 is much worse and that the trend of its curve is different from those of Algorithms 1 and 2. Via an analysis, the reason behind the difference is whether the caching strategy is optimized. Content placement is fixed in Algorithm 3, and the trend of its curve is related to the sizes of the files. The latency of Algorithm 3 is directly proportional to the expectation of the content size. If the size of content with a high popularity is relatively larger, the expectation of the content size will increase with the increase in the Zipf parameter; otherwise, the opposite happens. Especially, if all the contents are the same size, no matter how the request probability changes, the expectation of the content size will not change, and the curve mentioned above tends to be a horizontal straight line. In general, we can observe that the optimization of the caching strategy is more effective when the Zipf parameter is larger.

Furthermore, the impact of UAV altitude is considered. We vary the height of UAVs from 100 m to 500 m, and the latency performance is shown in Figure 4. The curves of three algorithms are generally flat and very slowly ascending. We consider that all the interference between UAVs exists simultaneously in the system, and accordingly, the interference has a significant impact on the system latency. As the UAV height increases, the received signal weakens and the interference also decreases; as a result, the overall transmission delay remains almost stable. Note that the curves show a slight upward trend because the AWGN will not decrease as the height increase, which causes a decrease in SINR. The proposed Algorithm 1 performs better than the other two in different UAV height situations.

After a comparison and analysis, we found that the performance of Algorithm 2 is usually better than Algorithm 3, excluding the situation that the UAV capacity is extremely large. This indicates that the caching strategy plays a more important role than UAV locations in this system model. Moreover, the proposed DC-based algorithm performs better in situations where the Zipf parameter of content popularity is large or the UAV caching capacity is limited.

5. Conclusions

In this paper, we investigated UAV deployment, and a beamforming scheme and caching strategy in a cache-enabled UAV-assisted wireless network. We formulated a joint optimization problem to minimize the average transformation latency. The problem was decomposed into two subproblems, a caching optimization and a location beamforming joint optimization, which were addressed in an iterative manner. We proposed a DC-based algorithm to convert the joint optimization subproblem into a convex one. Firstly, we introduced a quadratic form function to transform the original problem into the difference of two convex functions. Secondly, we adopted linear approximation to eliminate the non-convexity. The second step was the major difficulty when applying the DC-based method because the complicated form of the transmission rate function made it hard to obtain the derivative. Thus, we brought out a method to reconstruct the optimization variables and the functions, which simplified the linear approximate process effectively. Finally, the numerical results illustrated the benefits brought by optimizing the UAV deployment and the caching strategy. The proposed DC-based algorithm could efficiently reduce the transmission latency, especially when the Zipf parameter of content popularity was large as well as when the UAV caching capacity was relatively limited. Combined with the method of reconstruction we proposed, the DC-based method is supposed to be a promising approach to solving a series of complicated optimization problems.