1. Introduction

2. System Model

3. Analysis of Sum-Rate Optimization Problem

3.1. UAV Trajectory Optimization

In this subsection, we use the SCO method to optimize the trajectory of R for the OFDMA-UAVR and NOMA-UAVR protocols.

3.1.1. OFDMA-Based UAV Relaying Protocol

Substituting (1) into (3) and (5) yields

$$\begin{array}{cc}\hfill R_{R,s_i}^{\mathrm{ofdma}}\left[j\right]=& a_i\left[j\right]{log}_2\left(1+\frac{{\mathbb{A}}_1^{\mathrm{ofdma}}\left[j\right]}{H^2+{∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill R_{U_i,s_i}^{\mathrm{ofdma}}\left[j\right]=& a_i\left[j\right]{log}_2\left(1+\frac{{\mathbb{A}}_2^{\mathrm{ofdma}}\left[j\right]}{H^2+{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_i∥}^2}\right),\hfill \end{array}$$

(25)

where ${\mathbb{A}}_1^{\mathrm{ofdma}}[j]=\frac{P_S\left[j\right]{\beta }_0}{N_0}$ and ${\mathbb{A}}_2^{\mathrm{ofdma}}\left[j\right]=\frac{P_R\left[j\right]{\beta }_0}{N_0}$ .

It is seen that although (24) and (25) lack convexity concerning $\mathbf{w}\left[j\right]$, they exhibit convexity in relation to ${∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2$ and ${∥\mathbf{w}\left[j\right]-{\mathbf{U}}_i∥}^2$, respectively. This attribute allows us to derive lower bounds for $R_{R,s_i}^{\mathrm{ofdma}}\left[j\right]$ and $R_{U_i,s_i}^{\mathrm{ofdma}}\left[j\right]$ that are convex with respect to $\mathbf{w}\left[j\right]$. Specifically, at a given point ${\mathbf{W}}^l\triangleq \left[{\mathbf{w}}^l\left[1\right],\dots ,{\mathbf{w}}^l\left[j\right]\right]$ (we assume that ${\mathbf{W}}^l$ is the optimal UAV’s flight trajectory obtained after the $\mathit{l}$-th iteration), the following lower bounds can be obtained using the first-order Taylor expansion [17].

$$\begin{array}{cc}\hfill & R_{R,s_i}^{\mathrm{ofdma}}\left[j\right]\ge {\widehat{R}}_{R,s_i}^{\mathrm{ofdma}}\left[j\right]\hfill \\ \hfill & =a_i\left[j\right]\left({\mathcal{A}}_1^{\mathrm{ofdma}}\left[j\right]{∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2+{\mathcal{B}}_1^{\mathrm{ofdma}}\left[j\right]\right),\hfill \\ \hfill & R_{U_i,s_i}^{\mathrm{ofdma}}\left[j\right]\ge {\widehat{R}}_{U_i,s_i}^{\mathrm{ofdma}}\left[j\right]\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & =a_i\left[j\right]\left({\mathcal{A}}_{i+1}^{\mathrm{ofdma}}\left[j\right]{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_i∥}^2+{\mathcal{B}}_{i+1}^{\mathrm{ofdma}}\left[j\right]\right),\hfill \end{array}$$

(27)

where ${\mathcal{A}}_1^{\mathrm{ofdma}}\left[j\right],{\mathcal{A}}_{i+1}^{\mathrm{ofdma}}\left[j\right],{\mathcal{B}}_1^{\mathrm{ofdma}}\left[j\right]$ and ${\mathcal{B}}_{i+1}^{\mathrm{ofdma}}\left[j\right]$ are given by

$$\begin{array}{cc}\hfill {\mathcal{A}}_1^{\mathrm{ofdma}}\left[j\right]=& {\left({\mathbb{A}}_1^{\mathrm{ofdma}}\left[j\right]+H^2+{∥{\mathbf{w}}^l\left[j\right]-\mathbf{S}∥}^2\right)}^{-1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\left(H^2+{∥{\mathbf{w}}^l\left[j\right]-\mathbf{S}∥}^2\right)}^{-1},\hfill \\ \hfill {\mathcal{A}}_{i+1}^{\mathrm{ofdma}}\left[j\right]=& {\left({\mathbb{A}}_2^{\mathrm{ofdma}}\left[j\right]+H^2+{∥{\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_i∥}^2\right)}^{-1}\hfill \end{array}$$

(28a)

$$\begin{array}{cc}\hfill & -{\left(H^2+{∥{\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_i∥}^2\right)}^{-1},\hfill \\ \hfill {\mathcal{B}}_1^{\mathrm{ofdma}}\left[j\right]=& ln\left(1+\frac{{\mathbb{A}}_1^{\mathrm{ofdma}}\left[j\right]}{H^2+{∥{\mathbf{w}}^l\left[j\right]-\mathbf{S}∥}^2}\right)\hfill \end{array}$$

(28b)

$$\begin{array}{cc}\hfill & -{\mathcal{A}}_1^{\mathrm{ofdma}}\left[j\right]{∥{\mathbf{w}}^l\left[j\right]-\mathbf{S}∥}^2,\hfill \\ \hfill {\mathcal{B}}_{i+1}^{\mathrm{ofdma}}\left[j\right]=& ln\left(1+\frac{{\mathbb{A}}_2^{\mathrm{ofdma}}\left[j\right]}{H^2+{∥{\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_i∥}^2}\right)\hfill \end{array}$$

(28c)

$$\begin{array}{cc}\hfill & -{\mathcal{A}}_{i+1}^{\mathrm{ofdma}}\left[j\right]{∥{\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_i∥}^2.\hfill \end{array}$$

(28d)

Using (26) and (27), at any given $\mathbf{a}$, $\mathbf{P}$ and ${\mathbf{W}}^l$, (P1) is approximated by (P1.1) (or (29)) shown at the top of the next page.

It is seen that (P1.1) is a convex optimization problem that can be efficiently solved by standard convex optimization solvers (such as CVX implemented in Matlab).

3.1.2. NOMA-Based UAV Relaying Protocol

Substituting (1) into (8), (9), (11), () and (13) and after some manipulations, we have

$$\begin{array}{cc}\hfill R_{R,s_1}^{\mathrm{noma}}=& {log}_2\left(1+\frac{{\mathbb{A}}_1^{\mathrm{noma}}\left[j\right]}{H^2+{∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2}\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill R_{R,s_2}^{\mathrm{noma}}=& {log}_2\left(1+\frac{{\mathbb{A}}_0^{\mathrm{noma}}\left[j\right]{\mathbb{A}}_1^{\mathrm{noma}}\left[j\right]}{{\mathbb{A}}_1^{\mathrm{noma}}\left[j\right]+H^2+{∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2}\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill R_{U_1,s_1}^{\mathrm{noma}}=& {log}_2\left(1+\frac{{\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]}{H^2+{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_1∥}^2}\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill R_{U_1,s_2}^{\mathrm{noma}}=& {log}_2\left(1+\frac{{\mathbb{A}}_0^{\mathrm{noma}}\left[j\right]{\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]}{{\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]+H^2+{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_1∥}^2}\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill R_{U_2,s_2}^{\mathrm{noma}}=& {log}_2\left(1+\frac{{\mathbb{A}}_0^{\mathrm{noma}}\left[j\right]{\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]}{{\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]+H^2+{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_2∥}^2}\right),\hfill \end{array}$$

(34)

where ${\mathbb{A}}_0^{\mathrm{noma}}\left[j\right]=\frac{b_2\left[j\right]}{b_1\left[j\right]},{\mathbb{A}}_1^{\mathrm{noma}}\left[j\right]=\frac{b_1\left[j\right]P_S\left[j\right]{\beta }_0}{N_0}$ and ${\mathbb{A}}_2^{\mathrm{noma}}\left[j\right]=\frac{b_1\left[j\right]P_R\left[j\right]{\beta }_0}{N_0}$ .

Following the similar approach as in Section 3.1.1, we can obtain the following inequalities using the first-order Taylor expansion.

$$\begin{array}{cc}\hfill R_{R,s_k}^{\mathrm{noma}}\left[j\right]& \ge {\widehat{R}}_{R,s_k}^{\mathrm{noma}}\left[j\right]\hfill \\ \hfill & \triangleq {\mathcal{A}}_k^{\mathrm{noma}}\left[j\right]{∥\mathbf{w}\left[j\right]-\mathbf{S}∥}^2+{\mathcal{B}}_k^{\mathrm{noma}}\left[j\right],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill R_{U_k,s_k}^{\mathrm{noma}}\left[j\right]& \ge {\widehat{R}}_{U_k,s_k}^{\mathrm{noma}}\left[j\right]\hfill \\ & \triangleq {\mathcal{A}}_{k+2}^{\mathrm{noma}}\left[j\right]{∥\mathbf{w}\left[j\right]-{\mathbf{U}}_k∥}^2+{\mathcal{B}}_{k+2}^{\mathrm{noma}}\left[j\right],\hfill \end{array}$$

(36

where ${\mathcal{A}}_1^{\mathrm{noma}}\left[j\right],{\mathcal{A}}_2^{\mathrm{noma}}\left[j\right],{\mathcal{A}}_3^{\mathrm{noma}}\left[j\right],{\mathcal{A}}_4^{\mathrm{noma}}\left[j\right],{\mathcal{B}}_1^{\mathrm{noma}}\left[j\right],$${\mathcal{B}}_2^{\mathrm{noma}}\left[j\right],{\mathcal{B}}_3^{\mathrm{noma}}\left[j\right]$ and ${\mathcal{B}}_4^{\mathrm{noma}}\left[j\right]$ are given in (37) shown at the top of the next page.

Next, using (33) and (34), Constraint (23d) can be rewritten as

$$\begin{array}{c}\hfill {∥\mathbf{w}\left[j\right]-{\mathbf{U}}_1∥}^2\le {∥\mathbf{w}\left[j\right]-{\mathbf{U}}_2∥}^2.\end{array}$$

(38)

It is seen that (38) is not a convex constraint. Since the right-hand side of (38) is convex with respect to $\mathbf{w}\left[j\right]$, (38) can be rewritten applying the first-order Taylor expansion as follows.

$$\begin{array}{cc}\hfill {∥\mathbf{w}\left[j\right]-{\mathbf{U}}_1∥}^2\le & {\left({\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_2\right)}^{\top }\left(\mathbf{w}\left[j\right]-{\mathbf{U}}_2\right)\hfill \\ \hfill & -{∥{\mathbf{w}}^l\left[j\right]-{\mathbf{U}}_2∥}^2.\hfill \end{array}$$

(39)

Using (35), (36) and (39), for any given $\mathbf{a}$, $\mathbf{P}$ and ${\mathbf{W}}^l$, (P2) is approximated by (P2.1) (or (40)) shown at the top of the next page.

It is seen that (P2.1) is a convex optimization problem, and we can apply standard convex optimization techniques to address this efficiently.

3.2. Transmit Power Optimization

In this subsection, we optimize the transmit powers of S and R for the OFDMA-UAVR and NOMA-UAVR protocols.

3.2.1. OFDMA-Based UAV Relaying Protocol

Substituting (6) into (22), at a given $\mathbf{a}$ and ${\mathbf{W}}^l$, (P1) can be expressed as (P1.2) (or (41)) shown at the top of the next page

where ${\mathbb{B}}_1^{\mathrm{ofdma}}=\frac{{\left|h_{RS}\left[j\right]\right|}^2}{N_0},{\mathbb{B}}_2^{\mathrm{ofdma}}=\frac{{\left|h_{R{U}_1}\left[j\right]\right|}^2}{N_0}$ and ${\mathbb{B}}_3^{\mathrm{ofdma}}=\frac{{\left|h_{R{U}_2}\left[j\right]\right|}^2}{N_0}$ .

It is seen that (P1.2) represents a convex optimization problem amenable to efficient resolution using standard convex optimization solvers.

3.2.2. NOMA-based UAV relaying protocol

For any given $\mathbf{a}$ and ${\mathbf{W}}^l$, (9) and (13) can be rewritten as

$$\begin{array}{c}\hfill R_{R,s_2}^{\mathrm{noma}}={log}_2\left(1+{\mathbb{B}}_4^{\mathrm{noma}}{P}_S\left[j\right]\right)-{log}_2\left(1+{\mathbb{B}}_1^{\mathrm{noma}}{P}_S\left[j\right]\right),\end{array}$$

(42)

$$\begin{array}{c}\hfill R_{U_2,s_2}^{\mathrm{noma}}={log}_2\left(1+{\mathbb{B}}_5^{\mathrm{noma}}{P}_R\left[j\right]\right)-{log}_2\left(1+{\mathbb{B}}_2^{\mathrm{noma}}{P}_R\left[j\right]\right),\end{array}$$

(43)

where ${\mathbb{B}}_1^{\mathrm{noma}}=\frac{b_1\left[j\right]{\left|h_{SR}\left[j\right]\right|}^2}{N_0},{\mathbb{B}}_2^{\mathrm{noma}}=\frac{b_1\left[j\right]{\left|h_{R{U}_2}\left[j\right]\right|}^2}{N_0},{\mathbb{B}}_4^{\mathrm{noma}}=\frac{{\left|h_{SR}\left[j\right]\right|}^2}{N_0}$ and ${\mathbb{B}}_5^{\mathrm{noma}}=\frac{{\left|h_{R{U}_2}\left[j\right]\right|}^2}{N_0}$ .

Let ${\mathbf{P}}_S^l\stackrel{\mathsf{\Delta }}{=}\left\{{\mathbf{P}}_S^l,{\mathbf{P}}_R^l\right\}$ , where ${\mathbf{P}}_S^l\stackrel{\mathsf{\Delta }}{=}\left[P_S^l\left[1\right],\dots ,P_S^l\left[j\right]\right]$ and ${\mathbf{P}}_R^l\stackrel{\mathsf{\Delta }}{=}\left[P_R^l\left[1\right],\dots ,P_R^l\left[j\right]\right]$ are the optimal transmit powers of S and R, respectively, obtained after the l-th iteration. The application of the first-order Taylor expansion allows us to derive the following inequalities.

$$\begin{array}{c}{R}_{R,s_2}^{\mathrm{noma}}\ge {\widehat{R}}_{R,s_2}^{\mathrm{noma}}\hfill \\ =\frac{1}{ln\left(2\right)}\left(ln\left(1+{\mathbb{B}}_4^{\mathrm{noma}}{P}_S\left[j\right]\right)-{\mathcal{C}}_{1a}^{\mathrm{no}}{P}_S\left[j\right]-{\mathcal{C}}_{1b}^{\mathrm{no}}\right),\hfill \\ R_{U_2,x_2}^{\mathrm{noma}}\ge {\widehat{R}}_{U_2,x_2}^{\mathrm{noma}}\hfill \end{array}$$

(44)

$$\begin{array}{c}=\frac{1}{ln\left(2\right)}\left(ln\left(1+{\mathbb{B}}_5^{\mathrm{noma}}{P}_R\left[j\right]\right)-{\mathcal{C}}_{2a}^{\mathrm{no}}{P}_R\left[j\right]-{\mathcal{C}}_{2b}^{\mathrm{no}}\right),\hfill \end{array}$$

(45)

where ${\mathcal{C}}_{1a}^{\mathrm{noma}},{\mathcal{C}}_{1b}^{\mathrm{noma}},{\mathcal{C}}_{2a}^{\mathrm{noma}}$ and ${\mathcal{C}}_{2b}^{\mathrm{noma}}$ are given by

$$\begin{array}{cc}\hfill {\mathcal{C}}_{1a}^{\mathrm{noma}}=& \frac{{\mathbb{B}}_1^{\mathrm{noma}}}{1+{\mathbb{B}}_1^{\mathrm{noma}}{P}_S^l\left[j\right]},\hfill \end{array}$$

(46a)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{1b}^{\mathrm{noma}}=& ln\left(1+{\mathbb{B}}_1^{\mathrm{noma}}{P}_S^l\left[j\right]\right)-{\mathcal{C}}_{1a}^{\mathrm{noma}}{P}_S^l\left[j\right],\hfill \end{array}$$

(46b)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{2a}^{\mathrm{noma}}=& \frac{{\mathbb{B}}_2^{\mathrm{noma}}}{1+{\mathbb{B}}_2^{\mathrm{noma}}{P}_R^l\left[j\right]},\hfill \end{array}$$

(46c)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{2b}^{\mathrm{noma}}=& ln\left(1+{\mathbb{B}}_2^{\mathrm{noma}}{P}_R^l\left[j\right]\right)-{\mathcal{C}}_{2a}^{\mathrm{noma}}{P}_R^l\left[j\right].\hfill \end{array}$$

(46d)

Substituting (8), (11), (44) and (45) into (23), (P2) can be approximated by (P2.2) (or (47)) shown at the top of the next page

where ${\mathbb{B}}_3^{\mathrm{no}}=\frac{b_1\left[j\right]{\left|h_{R{U}_1}\left[j\right]\right|}^2}{N_0}$. It is seen that (P2.2) is a convex problem that can be efficiently addressed by using conventional convex optimization solvers.

3.3. Optimizing the Resource Allocation

In this subsection, we optimize the bandwidth allocation factor $\mathbf{a}$ and power allocation factor $\mathbf{b}$ for the OFDMA-UAVR and NOMA-UAVR protocols.

3.3.1. OFDMA-based UAV relaying protocol

For any given $\mathbf{P}$ and $\mathbf{W}$, (P1) can be rewritten as (P1.3) (or (48)) shown at the top of the next page.

It is seen that (P1.3) is a convex optimization problem that can be efficiently addressed by using conventional convex optimization solvers.

3.3.2. NOMA-based UAV relaying protocol

For any given $\mathbf{P}$ and $\mathbf{W}$, (9) and (13) are rewritten as

$$\begin{array}{cc}\hfill R_{R,s_2}^{\mathrm{noma}}& ={log}_2\left(\frac{1+{\mathbb{C}}_1^{\mathrm{noma}}}{b_1\left[j\right]+{\mathbb{C}}_1^{\mathrm{noma}}}\right)\ge R_{R,s_2}^{\mathrm{noma}}\hfill \\ \hfill & \triangleq {log}_2\left(1+{\mathbb{C}}_1^{\mathrm{noma}}\right)-{\mathcal{D}}_{1a}{b}_1\left[j\right]-{\mathcal{D}}_{1b},\hfill \\ \hfill R_{U_2,s_2}^{\mathrm{noma}}& ={log}_2\left(\frac{1+{\mathbb{C}}_3^{\mathrm{noma}}}{b_1\left[j\right]+{\mathbb{C}}_3^{\mathrm{noma}}}\right)\ge R_{U_2,s_2}^{\mathrm{noma}}\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \triangleq {log}_2\left(1+{\mathbb{C}}_3^{\mathrm{noma}}\right)-{\mathcal{D}}_{2a}{b}_1\left[j\right]-{\mathcal{D}}_{2b},\hfill \end{array}$$

(50)

where ${\mathbb{C}}_1^{\mathrm{noma}}=\frac{N_0}{P_S\left[j\right]{\left|h_{SR}\left[j\right]\right|}^2},{\mathbb{C}}_2^{\mathrm{noma}}=\frac{N_0}{P_R\left[j\right]{\left|h_{R{U}_1}\left[j\right]\right|}^2}$ and ${\mathbb{C}}_3^{\mathrm{noma}}=\frac{N_0}{P_R\left[j\right]{\left|h_{R{U}_2}\left[j\right]\right|}^2}$ .

Substituting (8), (11), (49) and (50) into (23), (P2) can be approximated by (P2.3) (or (51)) shown at the top of the next page.

We can see that (P2.3) is also a convex optimization problem that can be ably addressed by standard convex optimization methods.

4. Proposed Comprehensive Algorithm

4.1. Proposed Algorithm

In this part, we introduce the comprehensive algorithms for obtaining efficient approximate solutions for (P1) and (P2) through the utilization of BCGD method. We also present the outcomes of suboptimal problems, namely (P1.1), (P1.2), (P1.3), (P2.1), (P2.2) and (P2.3). More precisely, the optimization variables consist of three blocks $\left\{\mathbf{W},\mathbf{P},\mathit{\theta }\right\}$ where $\mathit{\theta }$ is the resource allocation factor, i.e., $\mathit{\theta }\equiv \mathbf{a}$ for the OFDMA-UAVR protocol and $\mathit{\theta }\equiv \mathbf{b}$ for the NOMA-UAVR protocol. Each block of $\left\{\mathbf{W},\mathbf{P},\mathit{\theta }\right\}$ is optimized via addressing (P1.1), (P1.2) and (P1.3) (or (P2.1), (P2.2) and (P2.3)) correspondingly while fixing the values of the rest blocks. The obtained solution after optimizing each block is updated to $\left\{\mathbf{P},\mathbf{W},\mathit{\theta }\right\}$ correspondingly. This process is repeated until a certain condition is met. The details of this algorithm are summarized in Algorithm 1.

Table 3. Algorithm 1: The Sum-Rate Maximization Algorithm for OFDMA/NOMA-UAVR Protocols.

Table 3. Algorithm 1: The Sum-Rate Maximization Algorithm for OFDMA/NOMA-UAVR Protocols.

1.	Initialize $\{{\mathit{W}}^0,{\mathit{P}}^0,{\mathit{\theta }}^0\}$. Let $l=0$.
2.	Repeat
3.	With given $\{{\mathbf{P}}^l,{\mathit{\theta }}^l\}$, solve (P1.1) or P(2.1) to find and then update
	the optimal UAV’s trajectory to ${\mathbf{W}}^{l+1}$.
4.	With given $\{{\mathbf{W}}^{l+1},{\mathit{\theta }}^l\}$, solve (P1.2) or P(2.2) to find and then
	update the optimal transmit powers to ${\mathbf{P}}^{l+1}$.
5.	With given $\{{\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1}\}$, solve (P1.3) or P(2.3) to find and then
	update the optimal PS ratio to ${\mathit{\theta }}^{l+1}$.
6.	Update $l=l+1$.
7.	Until The fractional increase of the objective value is below a small
	threshold $\epsilon$.

In the following, we show the convergence of Algorithm 1. Let $\eta \left(\mathbf{W},\mathbf{P},\mathit{\theta }\right)$, ${\eta }_{\mathbf{W}}^{\mathrm{lb},l}\left(\mathbf{W},\mathbf{P},\mathit{\theta }\right)$, ${\eta }_{\mathbf{P}}^{\mathrm{lb},l}\left(\mathbf{W},\mathbf{P},\mathit{\theta }\right)$ and ${\eta }_{\theta }^{\mathrm{lb},l}\left(\mathbf{W},\mathbf{P},\mathit{\theta }\right)$ be respectively the objective functions of either (P1), (P1.1), (P1.2) and (P1.3) for the OFDMA-UAVR protocol or of (P2), (P2.1), (P2.2) and (P2.3) for the NOMA-UAVR protocol. First, at any provided point $\left\{{\mathbf{P}}^l,{\mathbf{W}}^l,{\mathit{\theta }}^l\right\}$ , the subsequent inequalities are derived through the execution of Step (3) in Algorithm 1:

$$\begin{array}{cc}\hfill \eta \left({\mathbf{W}}^l,{\mathbf{P}}^l,{\mathit{\theta }}^l\right)& \stackrel{\left(\mathrm{a}\right)}{=}{\eta }_{\mathbf{W}}^{\mathrm{lb},l}\left({\mathbf{W}}^l,{\mathbf{P}}^l,{\mathit{\theta }}^l\right),\hfill \end{array}$$

(52a)

$$\begin{array}{cc}\hfill & \stackrel{\left(\mathrm{b}\right)}{\le }{\eta }_{\mathbf{W}}^{\mathrm{lb},l}\left({\mathbf{W}}^{l+1},{\mathbf{P}}^l,{\mathit{\theta }}^l\right),\hfill \end{array}$$

(52b)

$$\begin{array}{cc}\hfill & \stackrel{\left(\mathrm{c}\right)}{\le }\eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^l,{\mathit{\theta }}^l\right),\hfill \end{array}$$

(52c)

where (a) holds, since the first-order Taylor expansions at (26), (27), (35), (36) are performed at the point $\left\{{\mathbf{W}}^l,{\mathbf{P}}^l,{\mathit{\theta }}^l\right\}$ ; (b) holds, since ${\mathbf{W}}^{l+1}$ is the optimal solution of (P1.1) (or P(2.1)); and (c) holds, since the objective functions of (P1.1) and (P2.1) are the lower bounds of those of (P1) and (P2), respectively.

Using similar explanations for (P1.2), (P1.3), (P2.2), and (P2.3), the inequalities in (53) and (54) can be proven as follows. Then, at any provided point $\left\{{\mathbf{W}}^{l+1},{\mathbf{P}}^l,{\mathit{\theta }}^l\right\}$, we can get the following inequalities via Step (4) of Algorithm 1:

$$\begin{array}{cc}\hfill \eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^l,{\mathit{\theta }}^l\right)& ={\eta }_{\mathbf{P}}^{\mathrm{lb},l}\left({\mathbf{W}}^{l+1},{\mathbf{P}}^l,{\mathit{\theta }}^l\right),\hfill \end{array}$$

(53a)

$$\begin{array}{cc}\hfill & \le {\eta }_{\mathbf{P}}^{\mathrm{lb},l}\left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^l\right),\hfill \end{array}$$

(53b)

$$\begin{array}{cc}\hfill & \le \eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^l\right).\hfill \end{array}$$

(53c)

Next, at any provided point $\left\{{\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^l\right\}$ , we also get the following inequalities via Step (5) of Algorithm 1:

$$\begin{array}{cc}\hfill \eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^l\right)& ={\eta }_{\theta }^{\mathrm{lb},l}\left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^l\right),\hfill \end{array}$$

(54a)

$$\begin{array}{cc}\hfill & \le {\eta }_{\theta }^{\mathrm{lb},l}\left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^{l+1}\right),\hfill \end{array}$$

(54b)

$$\begin{array}{cc}\hfill & \le \eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^{l+1}\right).\hfill \end{array}$$

(54c)

Finally, we have

$$\begin{array}{c}\hfill \eta \left({\mathbf{P}}^l,{\mathbf{W}}^l,{\mathit{\theta }}^l\right)\le \eta \left({\mathbf{W}}^{l+1},{\mathbf{P}}^{l+1},{\mathit{\theta }}^{l+1}\right).\end{array}$$

(46)

This observation reveals that the objective values of (P1) and (P2) are non-decreasing trend throughout iterations. Furthermore, the optimized values of (P1) and (P2) are finite, ensuring the guaranteed convergence of Algorithm 1.

4.2. Initialization Scheme

In this part, we introduce the Algorithm 2 which aims to discover an attainable initial variable block $\left\{{\mathbf{W}}^0,{\mathbf{P}}^0,{\mathit{\theta }}^0\right\}$ for Algorithm 1. Due to the constraint of the sum data rate of $U_2$, the key concept behind Algorithm 2 that is maximizing the sum-rate of $U_2$ for the OFDMA-UAVR and NOMA-UAVR protocols, respectively defined as (P3) and (P4). Note that the values of $a_2\left[n\right]$ and $b_2\left[n\right]$ are set at their highest values; hence, ${\mathit{\theta }}_{\mathrm{InitS}}=\left(1-a_{min}\right){\left[\underset{N\mathrm{elements}}{\underbrace{1,\dots ,1}}\right]}^{\top }$ for the OFDMA-UAVR protocol and ${\mathit{\theta }}_{\mathrm{InitS}}=\left(1-b_{min}\right){\left[\underset{N\mathrm{elements}}{\underbrace{1,\dots ,1}}\right]}^{\top }$ for the NOMA-UAVR protocol. If the maximum achievable sum rate with Algorithm 2 surpasses that of $R_{U_2}^{\mathrm{th}}$, the current optimal flight trajectory and transmit powers are ${\mathbf{W}}^0$ and ${\mathbf{P}}^0$, respectively; otherwise, (P1) and (P2) are infeasible optimization problems. By modifying (P1) and (P2), we can obtain (P3) and (P4) as

Similarly, the problems of (P3) and (P4) are not convex optimization. Therefore, we have introduced alternative solutions for (P3) and (P4)employing the BCGD and SCO approaches. Concretely, the optimization variables consist of two blocks $\left\{\mathbf{W},\mathbf{P}\right\}$ . Subsequently, the BCGD technique is employed to perform optimization for each of these variable blocks within the context of (P3) (or (P4)), while holding the other variable blocks constant. With transmit powers $\mathbf{P}\stackrel{\mathsf{\Delta }}{=}\left\{{\mathbf{P}}_S,{\mathbf{P}}_R\right\}$, we optimize $\mathbf{W}$ (defined as (P3.1) and (P4.1) for the OFDMA-UAVR and NOMA-UAVR protocols, respectively); and for a given UAV’s trajectory $\mathbf{W}$, we optimize $\mathbf{P}$ (defined as (P3.2) and (P4.2) for the OFDMA-UAVR and NOMA-UAVR protocols, respectively). The result achieved through the optimization of each block is then adjusted in accordance with $\left\{\mathbf{W},\mathbf{P}\right\}$. This iteration continues until a specific criterion is satisfied. Furthermore, the objective functions with non-convex characteristics are addressed by applying the SCO technique.

Table 4. Algorithm 2: The Initialization Scheme for Algorithm 1.

Table 4. Algorithm 2: The Initialization Scheme for Algorithm 1.

1.	Initialize $\{{\mathit{W}}_{\mathrm{InitS}}^0,{\mathit{P}}_{\mathrm{InitS}}^0\}$. Let $l=0$.
2.	Repeat
3.	With given $\{{\mathbf{W}}_{\mathrm{InitS}}^l,{\mathbf{P}}_{\mathrm{InitS}}^l\}$, solve (P3.1) or P(4.1) to find and then
	update the optimal UAV’s trajectory to ${\mathbf{W}}_{\mathrm{InitS}}^{l+1}$.
4.	With given $\{{\mathbf{W}}_{\mathrm{InitS}}^{l+1},{\mathbf{P}}_{\mathrm{InitS}}^l\}$, solve (P3.2) or P(4.2) to find and then
	update the optimal transmit powers to ${\mathbf{P}}_{\mathrm{InitS}}^{l+1}$.
5.	Update $l=l+1$.
6.	Until The objective value ${\eta }_{\mathrm{InitS}}$ is higher than $R_{U_2}^{\mathrm{th}}$ or $l\ge L_{max}$.
7.	If ${\eta }_{\mathrm{InitS}}\ge R_{U_2}^{\mathrm{th}}$
8.	$\left\{{\mathbf{W}}^0,{\mathbf{P}}^0,{\mathit{\theta }}^0\right\}\leftarrow \left\{{\mathbf{W}}_{\mathrm{InitS}}^{l+1},{\mathbf{P}}_{\mathrm{InitS}}^{l+1},{\mathit{\theta }}_{\mathrm{InitS}}\right\}$
9.	Else
10.	(P1) and (P2) are infeasible optimization problems.
11.	End if

By applying (29) and (40), the optimization problems of (P3.1) and (P4.1) at any given $\mathbf{P}$ and ${\mathbf{W}}^l$ are respectively given by (58) and (59), shown in the top of the next page.

Using (41), (47), the optimization problems of (P3.2) and (P4.2) at any given ${\mathbf{P}}^l$ and $\mathbf{W}$ are respectively given by (60) and (61), shown in the top of the next page.

Ultimately, a summary of Algorithm 2 can be located in Table 2, where $L_{max}$ is the utmost limit of iterations.

Algorithm 2 begins with an initial flight path of the UAV , which consists of a straight line connecting points $\mathbf{R}I$ and $\mathbf{R}F$ and maintains a constant velocity of $V0$, where $V0=\frac{\left|{\mathbf{R}}_I-{\mathbf{R}}_F\right|}{T}$. Additionally, the initial transmission power settings are established ${\mathbf{P}}_{S,\mathrm{InitS}}^0\left[j\right]=P_{S,\mathsf{\Sigma }}/\phantom{P_{S,\mathsf{\Sigma }}N}N$ and ${\mathbf{P}}_{R,\mathrm{InitS}}^0\left[j\right]=P_{R,\mathsf{\Sigma }}/\phantom{P_{R,\mathsf{\Sigma }}N}N$ .

5. Simulation Results

6. Conclusions

Abbreviations

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