1. Introduction

2. Related Work

3. System Model and Problem Formulation

4. Algorithm Design

4.1. UAV Set Covering Probelm

Since the user devices within a single coverage circle can concurrently receive task offloading services provided by the UAV, to enhance system efficiency, we aim to minimize the number of coverage areas, thus minimizing the time spent by UAVs in position shifting. This problem can be described as a fixed-radius circular coverage problem or an UAV set covering problem, which aims to cover all UDs within the system area using multiple circles with a fixed radius r, while minimizing the number of circles used. This is a typical NP-hard problem, for which there is no polynomial-time exact algorithm. In this section, we propose the Welzl-based UAV set covering (WUSC) algorithm, which provides an approximate solution to the coverage circle set in polynomial time.

The Welzl algorithm is an incremental algorithm used to solve the minimum enclosing circle problem. It takes all points within a circular coverage area and returns the minimum radius of the circle. We consider iteratively invoking this algorithm to solve the fixed-radius circular coverage problem. As shown in Figure 3, the boundary of the system area is continuously reduced until there exists a device exactly on the boundary. Then, starting from a point i on the boundary and initializing a coverage area ${\mathcal{I}}_p=\left\{i\right\}$, the device closest to this point within the region is added to the set ${\mathcal{I}}_p$, and it is determined using the Welzl algorithm whether the set can be completely covered by a circle with a radius not exceeding r. This process is repeated until all user devices are included in some coverage set. Algorithm 1 describes the detailed procedure of the WUSC algorithm.

Algorithm 1: WUSC algorithm

Additionally, once the coverage strategy $\mathcal{P}$ is obtained, the UAV trajectory $\mathit{w}$ can be obtained by a standard TSP model based on the partitions. Then, P1 can be reformulated as:

$$\begin{array}{cc}\hfill \left(\mathrm{P}2\right){\displaystyle \underset{\mathit{d},\mathit{W},\mathit{f}}{min}}& \mathcal{S}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(2\right),\left(3\right),\left(7\right),\left(8\right),\left(14\right),\left(18\right)\hfill \end{array}$$

(19)

Note that P2 is still a MINLP problem. However, since the hover time is constant, there is no coupling between multiple partitions, and thus, the problem can be divided into multiple smaller sub-problems for parallel solution based on the partition dimension. We will present the algorithms for P2 as follows.

4.2. Computation Mode Selection and Resource Allocation

We can decompose P2 into P independent sub-problems in the following form according to the coverage areas:

$$\begin{array}{cc}\hfill \left(\mathrm{P}2.1\right){\displaystyle \underset{{\mathit{d}}_{\mathit{p}},{\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}}}{min}}& {\displaystyle \sum _{i\in {\mathcal{I}}_{w_p}}}\frac{d_i·\left(T_{w_p}^w+T_i^o+T_i^e\right)+(1-d_i)·T_i^l}{T_i^l}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(2\right),\left(3\right),\left(7\right),\left(8\right),\left(14\right),\left(18\right)\hfill \end{array}$$

(20)

For problem P2.1, we will propose two different algorithms to solve it. The first is a low-complexity CD-based algorithm, suitable for smaller-scale IoT networks. The second is a joint optimization algorithm based on ADMM technique, designed for larger-scale networks that require higher solution precision.

4.2.1. Alternating Optimization using CD Method

In this section, we consider alternating iterative optimization of discrete and continuous variables. Given the computation mode $\mathit{d}$, let ${\mathcal{I}}_p\left(0\right)$ represent the set of UDs choosing local computation mode in coverage area p, and ${\mathcal{I}}_p\left(1\right)$ represent the set of users choosing edge computation mode. Then, P2.1 can be equivalently transformed into the following form:

$$\begin{array}{cc}\hfill \left(\mathrm{P}2.1\text{’}\right){\displaystyle \underset{{\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}}}{min}}& {\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}\frac{T_i^o+T_i^e}{T_i^l}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(2\right),\left(7\right),\left(18\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & W_i,f_i>0,\forall i\in {\mathcal{I}}_p\left(1\right)\hfill \end{array}$$

(22)

P2.1’ is a convex problem, which can obtain the optimal closed-form solution for the problem using the KKT conditions. By introducing Lagrange multipliers $\lambda =\{{\lambda }_i,\forall i\in \mathcal{I}\},{\nu }_1,{\nu }_2$ for the inequality constraints (18), (2), (7), we can obtain the Lagrangian function as:

$$\begin{array}{cc}\hfill \mathcal{L}({\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}};& {\lambda }_{\mathit{p}},{\nu }_1,{\nu }_2)={\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}\left\{\frac{u_i}{W_i·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)·T_i^l}+\frac{c_i}{f_i·T_i^l}+{\lambda }_i·\left[-T^h+\frac{c_i}{f_i}\right.\right.\hfill \\ \hfill & \left.\left.+\frac{u_i}{W_i·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)}\right]\right\}+{\nu }_1·\left({\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{W}_i-W\right)+{\nu }_2·\left({\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{f}_i-f^{edge}\right)\hfill \end{array}$$

(23)

The corresponding KKT conditions can be derived as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}1+{\lambda }_i-{\nu }_1·\frac{W_i^2·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)·T_i^l}{u_i}=0,\forall i\in {\mathcal{I}}_p\left(1\right)\hfill & \left(a\right)\hfill \\ 1+{\lambda }_i-{\nu }_2·\frac{f_i^2·T_i^l}{c_i}=0,\forall i\in {\mathcal{I}}_p\left(1\right)\hfill & \left(b\right)\hfill \\ {\nu }_1·\left({\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{W}_i-W\right)=0\hfill & \left(c\right)\hfill \\ {\nu }_2·\left({\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{f}_i-f^{edge}\right)=0\hfill & \left(d\right)\hfill \\ {\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{W}_i-W\le 0\hfill & \left(e\right)\hfill \\ {\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{f}_i-f^{edge}\le 0\hfill & \left(f\right)\hfill \\ {\lambda }_i,{\nu }_1,{\nu }_2\ge 0\hfill & \left(g\right)\hfill \\ {\lambda }_i·\left[\frac{u_i}{W_i·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)}+\frac{c_i}{f_i}-T^h\right]=0,\forall i\in {\mathcal{I}}_p\left(1\right)\hfill & \left(h\right)\hfill \\ \frac{u_i}{W_i·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)}+\frac{c_i}{f_i}-T^h\le 0,\forall i\in {\mathcal{I}}_p\left(1\right)\hfill & \left(i\right)\hfill \end{array}\right.\end{array}$$

(24)

Based on the KKT system of equations (24), we can derive the following lemma:

Lemma 1.

When the optimal solution of P2.1’ is obtained, the resources are fully allocated to the users, i.e., ${\sum }_{i\in {\mathcal{I}}_p\left(1\right)}{W}_i=W,{\sum }_{i\in {\mathcal{I}}_p\left(1\right)}{f}_i=f^{edge}$ hold strictly.

Proof. 

From (a), (b), (c), (d), (g), it can be inferred that ${\nu }_1,{\nu }_2>0$. Substituting (c) and (d) into the equations yields ${\sum }_{i\in {\mathcal{I}}_p\left(1\right)}{W}_i=W$ and ${\sum }_{i\in {\mathcal{I}}_p\left(1\right)}{f}_i=f^{edge}$, which means that the bandwidth and computational resources should be allocated entirely to the users when the optimal solution is achieved.    □

We refer to the resource allocation strategy obtained without considering constraints (18) as the optimal solution ${\mathcal{S}}^{OPT}$, and the optimal solution of P2.1’ as a suboptimal solution ${\mathcal{S}}^{*}$. Based on whether ${\mathcal{S}}^{OPT}={\mathcal{S}}^{*}$, we can analyze P2.1’ in the two different cases as follows.

1) ${\mathcal{S}}^{OPT}={\mathcal{S}}^{*}$, meaning that there is no UDs violate the hovering constraint under ${\mathcal{S}}^{*}$. In this case, we have the following lemma for this optimal solution.

Lemma 2.

When ${\mathcal{S}}^{OPT}={\mathcal{S}}^{*}$ holds, both ${\mathit{W}}_{\mathit{p}}^{*}$ and ${\mathit{f}}_{\mathit{p}}^{*}$ have corresponding closed-form solutions.

Proof. 

We have ${\lambda }_{\mathit{p}}=0$ when hovering constraints are not in effect. Based on (a) and (b), we have $W_i^2=\frac{u_i}{{\nu }_1·{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)·T_i^l}$ and $f_i^2=\frac{c_i}{{\nu }_2·T_i^l}$. Then we have $\sqrt{{\nu }_1}=\frac{{\sum }_{i\in {\mathcal{I}}_p\left(1\right)}\sqrt{\frac{u_i}{{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)·T_i^l}}}{W}$ and $\sqrt{{\nu }_2}=\frac{{\sum }_{i\in {\mathcal{I}}_p\left(1\right)}\sqrt{\frac{c_i}{T_i^l}}}{f^{edge}}$. Finally, substituting ${\nu }_1$ and ${\nu }_2$ into (a) and (b) yields the following closed-form solutions for $W_i,f_i,\forall i\in {\mathcal{I}}_p\left(1\right)$:

$$\begin{array}{c}\hfill W_i=\frac{\sqrt{\frac{u_i}{{log}_2\left(1+\frac{P_i·h_i}{N_0^2}\right)·T_i^l}}}{{\sum }_{j\in {\mathcal{I}}_p\left(1\right)}\sqrt{\frac{u_j}{{log}_2\left(1+\frac{P_j·h_j}{N_0^2}\right)·T_j^l}}}·W,f_i=\frac{\sqrt{\frac{c_i}{T_i^l}}}{{\sum }_{j\in {\mathcal{I}}_p\left(1\right)}\sqrt{\frac{c_j}{T_j^l}}}·f^{edge}\end{array}$$

(25)

   □

2) ${\mathcal{S}}^{OPT}\ne {\mathcal{S}}^{*}$, which means that some UDs violate the hover constraint under ${\mathcal{S}}^{OPT}$. If the ${\mathcal{S}}^{*}$ has a solution, it indicates that the suboptimal strategy assigns additional bandwidth or computation resources to those UDs that exceed the hover time, allowing them to complete the computation within $T^h$. However, this approach simultaneously increases the time consumed for other non-violating devices. Let ${\mathcal{I}}_p^A=\{i\mid i\in {\mathcal{I}}_p\left(1\right),T_i^o+T_i^e>T^h\}$ denote the set of UDs which choose the edge computing mode and violate the hover constraint, and ${\mathcal{I}}_p^B={\mathcal{I}}_p\left(1\right)\setminus {\mathcal{I}}_p^A$ represents the others. According to (24), a complex system of equations with $\left(3\right|{\mathcal{I}}_p\left(1\right)|+2)$ variables and equalities can be derived. Enumerating $2^I$ possible closed-form solutions is impractical. Moreover, if the ${\mathcal{S}}^{*}$ has no solution, it implies the bandwidth and computation resource of the UAV is insufficient to meet the computational demands of all selected offloading UDs within the given hover time $T^h$. Therefore, these two parts $\left\{\mathit{d}\right\}$ and $\{\mathit{W},\mathit{f}\}$ cannot be fully decoupled by the CD method.

In summary, we can obtain the optimal $\{\mathit{W},\mathit{f}\}$ in $\mathcal{O}\left(1\right)$ using (25) for the first type of problem. For the second type, the corresponding KKT equations either have no solution or are challenging to solve directly. To minimize coupling and simplify the solution of the KKT equations, a greedy strategy can be adopted by converting the second type problem approximately to the first type. Additionally, for the computation mode $\left\{\mathit{d}\right\}$, we consider obtaining a locally optimal solution through a simple one-dimensional linear search. For convenience, let ${\mathit{d}}_p=(d_1,d_2,...,d_{\left|p\right|})$ represent the computation modes of all UDs within coverage area p. At the beginning of the $(l-1)$-th iteration, the initial value is defined as ${\mathit{d}}_p^{l-1}$. Each user attempts to change their computation mode based on ${\mathit{d}}_p^{l-1}$:

$$\begin{array}{c}\hfill {\mathit{d}}_p^{l-1}\left(i\right)=\left(d_1^{l-1},...,d_{i-1}^{l-1},\oplus \left(d_i^{l-1}\right),d_{i+1}^{l-1},...,d_{\left|p\right|}^{l-1}\right)\end{array}$$

(26)

where $\oplus (·)$ denotes the negation operation for binary variables, e.g., $\oplus \left(1\right)=0$, $\oplus \left(0\right)=1$. This generates $\left|p\right|$ different computation modes, and the one with the smallest shrinkage rate is selected as the return value for the $(l-1)$-th iteration. A locally optimal solution is achieved when no user can further reduce the global shrinkage rate by changing their computation mode. Algorithm 2 describes the detailed procedure of the CD-based greedy (CD-Greedy) algorithm for P2.1.

Algorithm 2: CD-Greedy Algorithm for P2.1

4.2.2. Joint Optimization Using ADMM Method

The main advantage of the CD-based algorithm lies in its simplicity and relatively high efficiency due to the closed-form resource allocation strategy. However, alternating optimization is prone to falling into local optima, and the one-dimensional search causes the number of iterations required for convergence to increase with the network scale. In this section, we will propose an ADMM-based algorithm to jointly optimize computation mode and resource allocation. The main idea is to decompose the coupled problem P2.1 into $|{\mathcal{I}}_p|$ parallel small-scale MINLP problems. First, to eliminate the coupled inequality constraints, we introduce auxiliary variables to reformulate P2.1 into the following equivalent form:

$$\begin{array}{cc}\hfill \left(\mathrm{P}2.2\right){\displaystyle \underset{{\mathit{d}}_{\mathit{p}},{\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}},{\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}}}{min}}& {\displaystyle \sum _{i\in {\mathcal{I}}_p}}q(d_i,W_i,f_i)+g({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(14\right),\left(22\right)\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & x_i,y_i\ge 0,\forall i\in {\mathcal{I}}_p\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & x_i=W_i,y_i=f_i,\forall i\in {\mathcal{I}}_p\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i\in {\mathcal{I}}_p}}{x}_i\le W,{\displaystyle \sum _{i\in {\mathcal{I}}_p}}{y}_i\le f^{edge}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \frac{u_i}{x_i·{log}_2\left(1+P_i·h_i/N_0^2\right)}+\frac{c_i}{y_i}\le T^h,\forall i\in {\mathcal{I}}_p\hfill \end{array}$$

(31)

where $q(d_i,W_i,f_i)$ is the simplified form of the original objective function:

$$\begin{array}{c}\hfill q(d_i,W_i,f_i)=\frac{d_i·\left(T_i^o+T_i^e\right)+(1-d_i)·T_i^l}{T_i^l}\end{array}$$

(32)

$g({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})$ is the equivalent form of the inequality constraints in the original problem:

$$\begin{array}{cc}\hfill g({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})& =\left\{\begin{array}{cc}0& \mathrm{if}({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})\in {\mathcal{G}}_p\\ +\infty & otherwise\end{array}\right.\hfill \end{array}$$

(33)

where ${\mathcal{G}}_p$ denotes the feasible set of $g({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})$:

$${\mathcal{G}}_p=\left\{({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})\mid {\displaystyle \sum _{i\in {\mathcal{I}}_p}}{x}_i\le W,{\displaystyle \sum _{i\in {\mathcal{I}}_p}}{y}_i\le f^{edge};\frac{u_i}{x_i·{log}_2\left(1+P_i·h_i/N_0^2\right)}+\frac{c_i}{y_i}\le T^h,\forall i\in {\mathcal{I}}_p\right\}$$

(34)

Problem P2.2 can be effectively decomposed using the ADMM method to find the optimal solution to the dual problem. By introducing Lagrange multipliers for the equality constraint (29), we can represent the augmented Lagrangian function of P2.2 as:

$$\begin{array}{cc}\hfill \mathit{L}(\mathit{u},\mathit{v},\theta )& ={\displaystyle \sum _{i\in {\mathcal{I}}_p}}q\left(\mathit{u}\right)+g\left(\mathit{v}\right)+{\alpha }_i·(W_i-x_i)+{\beta }_i·(f_i-y_i)\hfill \\ \hfill & -\frac{c}{2}·{\displaystyle \sum _{i\in {\mathcal{I}}_p}}{(W_i-x_i)}^2-\frac{c}{2}·{\displaystyle \sum _{i\in {\mathcal{I}}_p}}{(f_i-y_i)}^2\hfill \end{array}$$

(35)

where $\mathit{u}=\{{\mathit{d}}_{\mathit{p}},{\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}}\}$, $\mathit{v}=\{{\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}}\}$, $\theta =\{{\alpha }_{\mathit{p}},{\beta }_{\mathit{p}}\}$, and $c>0$ represents the step size. The corresponding dual problem can be formalized as:

$$\begin{array}{c}\hfill \left(\mathrm{P}2.3\right){\displaystyle \underset{\theta }{max}}d\left(\theta \right)\end{array}$$

(36)

where $d\left(\theta \right)={min}_{\mathit{u},\mathit{v}}\left\{\mathcal{L}(\mathit{u},\mathit{v},\theta )\mid \mathit{u}\in \mathcal{U},\mathit{v}\in \mathcal{V}\right\}$ represents the Lagrangian dual function of $\mathcal{L}$, $\mathcal{U}=\{({\mathit{d}}_{\mathit{p}},{\mathit{W}}_{\mathit{p}},{\mathit{f}}_{\mathit{p}})\mid W_i,f_i\ge 0,d_i\in \{0,1\},\forall i\in {\mathcal{I}}_p\}$, $\mathcal{V}=\{({\mathit{x}}_{\mathit{p}},{\mathit{y}}_{\mathit{p}})\mid x_i,y_i\ge 0,\forall i\in {\mathcal{I}}_p\}$.

The ADMM method solves the dual problem P2.3 by iteratively optimizing $\{\mathit{u},\mathit{v},\theta \}$. Let the solution obtained at the l-th iteration be $\{{\mathit{u}}^l,{\mathit{v}}^l,{\theta }^l\}$. The operations required in the $(l+1)$-th iteration can be described as the following three steps:

• Step 1: Given $\{{\mathit{v}}^l,{\theta }^l\}$, minimize $\mathcal{L}$ by finding suitable $\left\{{\mathit{u}}^l\right\}$.

$$\begin{array}{c}\hfill {\mathit{u}}^{l+1}=arg{\displaystyle \underset{\mathit{u}}{min}}\mathcal{L}(\mathit{u},{\mathit{v}}^l,{\theta }^l)\end{array}$$

(37)

Since there is no coupling among multiple users, (37) can be decomposed into $|{\mathcal{I}}_p|$ identical subproblems solved in parallel. Each subproblem can be formulated as:

$$\begin{array}{cc}\hfill {\mathit{u}}_i^{l+1}& =arg{\displaystyle \underset{\mathit{u}\in \mathcal{U}}{min}}q(d_i,W_i,f_i)+{\alpha }_i·(W_i-x_i)+{\beta }_i·(f_i-y_i)-\frac{c}{2}·\left[{(W_i-x_i)}^2+{(f_i-y_i)}^2\right]\hfill \end{array}$$

(38)

When $d_i=0$, we have $B_i=f_i=0$. In this case, ${\mathit{u}}_i^{l+1}$ can be directly computed using (38). When $d_i=1$, (38) becomes a standard convex optimization problem and yields a closed-form optimal solution. For example, by separating $W_i$ and $f_i$ into two independent parts, setting the first derivative to zero and combining with the Cardano formula, we can obtain a unique optimal solution. Therefore, we consider the two possible values of $d_i$ separately. After obtaining the corresponding $W_i$ and $f_i$, they are substituted into (38), and the smaller value is chosen as the optimal solution for ${\mathit{u}}_i^{l+1}$. Note that the time complexity for solving each subproblem ${\mathit{u}}_i^{l+1}$ is $\mathcal{O}\left(1\right)$.

• Step 2: Given $\{{\mathit{u}}^{l+1},{\theta }^l\}$, minimize $\mathcal{L}$ by finding suitable $\left\{{\mathit{v}}^{l+1}\right\}$.

From (33), it is evident that to ensure the problem’s solvability, it is necessary to ensure ${\mathit{v}}^{l+1}\in {\mathcal{G}}_p$. Since the offloading decisions are known, here we only need to consider the auxiliary variables corresponding to the edge computing mode UDs. The solution for ${\mathit{v}}^{l+1}$ can be formulated as a convex problem in the following form:

$$\begin{array}{cc}\hfill \left(\mathrm{P}2.4\right)arg{\displaystyle \underset{\mathit{v}}{min}}& {\displaystyle \sum _{i\in {\mathcal{I}}_p\left(1\right)}}{\alpha }_i·(W_i-x_i)+{\beta }_i·(f_i-y_i)-\frac{c}{2}·\left[{\left(W_i-x_i\right)}^2+{\left(f_i-y_i\right)}^2\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(28\right),\left(29\right),\left(30\right),\left(31\right)\hfill \end{array}$$

(39)

To obtain the optimal solution to P2.4, we consider using convex optimization solver, e.g., CPLEX, CVX. Solvers based on standard convex optimization methods like interior point methods ensure that the obtained solution is the optimal solution, thereby preventing the problem P2.4 from being infeasible due to $g\left(\mathit{v}\right)$ taking on positive infinity.

• Step 3: Given $\{{\mathit{u}}^{l+1},{\mathit{v}}^{l+1}\}$, minimize $\mathcal{L}$ by finding suitable $\left\{\theta \right\}$.

This can be achieved by updating the Lagrange multipliers ${\theta }_m^l$ according to the following rules:

$$\begin{array}{cc}\hfill & {\alpha }_i^{l+1}={\alpha }_i^l-c·(W_i^{l+1}-x_i^{l+1})\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & {\beta }_i^{l+1}={\beta }_i^l-c·(f_i^{l+1}-y_i^{l+1})\hfill \end{array}$$

(41)

Obviously, the time complexity for updating the Lagrange multipliers is $\mathcal{O}\left(I\right)$.

Repeat the above three steps until the absolute errors of $\mathit{u}$ and $\mathit{v}$ reach the desired accuracy. The absolute errors of both can be measured by the following equation:

$$\begin{array}{c}\hfill {\displaystyle \sum _{i\in {\mathcal{I}}_p}}\left(|W_i^l-x_i^l|+|f_i^l-y_i^l|\right)\le 2\delta \end{array}$$

(42)

where $\delta$ is a predefined error bound, typically a very small positive number. Due to the presence of duality gap for P2.1, the ADMM algorithm may not converge to the optimal solution. Therefore, when the algorithm terminates, the dual optimal solution $\{{\mathit{d}}^l,{\mathit{W}}^l,{\mathit{f}}^l\}$ is an approximate solution, and its performance gap will be evaluated through simulation experiments. Algorithm 3 describes the detailed procedure of the ADMM-based joint optimization algorithm for P2.1.

Algorithm 3: ADMM-based Algorithm for P2.1

5. Simulation Results

6. Conclusions and Future Work

References

1. Feng, C.; Han, P.; Zhang, X.; Yang, B.; Liu, Y.; Guo, L. Computation offloading in mobile edge computing networks: A survey. Journal of Network and Computer Applications 2022, 202, 103366. [Google Scholar] [CrossRef]
2. Wang, Z.; Sun, G.; Su, H.; Yu, H.; Lei, B.; Guizani, M. Low-Latency Scheduling Approach for Dependent Tasks in MEC-Enabled 5G Vehicular Networks. IEEE Internet of Things Journal 2024, 11, 6278–6289. [Google Scholar] [CrossRef]
3. Xu, J.; Ota, K.; Dong, M. Big Data on the Fly: UAV-Mounted Mobile Edge Computing for Disaster Management. IEEE Transactions on Network Science and Engineering 2020, 7, 2620–2630. [Google Scholar] [CrossRef]
4. Nasir, A.A. Latency Optimization of UAV-Enabled MEC System for Virtual Reality Applications Under Rician Fading Channels. IEEE Wireless Communications Letters 2021, 10, 1633–1637. [Google Scholar] [CrossRef]
5. Liu, W.; Li, B.; Xie, W.; Dai, Y.; Fei, Z. Energy Efficient Computation Offloading in Aerial Edge Networks With Multi-Agent Cooperation. IEEE Transactions on Wireless Communications 2023. [Google Scholar] [CrossRef]
6. Michailidis, E.T.; Miridakis, N.I.; Michalas, A.; Skondras, E.; Vergados, D.J.; Vergados, D.D. Energy Optimization in Massive MIMO UAV-Aided MEC-Enabled Vehicular Networks. IEEE Access 2021, 9, 117388–117403. [Google Scholar] [CrossRef]
7. Pervez, F.; Sultana, A.; Yang, C.; Zhao, L. Energy and Latency Efficient Joint Communication and Computation Optimization in a Multi-UAV Assisted MEC Network. IEEE Transactions on Wireless Communications.
8. Hua, M.; Wang, Y.; Zhang, Z.; Li, C.; Huang, Y.; Yang, L. Optimal Resource Partitioning and Bit Allocation for UAV-Enabled Mobile Edge Computing. 2018 IEEE 88th Vehicular Technology Conference (VTC-Fall), 2018, pp. 1–6.
9. Zhao, N.; Ye, Z.; Pei, Y.; Liang, Y.C.; Niyato, D. Multi-Agent Deep Reinforcement Learning for Task Offloading in UAV-Assisted Mobile Edge Computing. IEEE Transactions on Wireless Communications 2022, 21, 6949–6960. [Google Scholar] [CrossRef]
10. Xu, Y.; Zhang, T.; Liu, Y.; Yang, D.; Xiao, L.; Tao, M. UAV-Assisted MEC Networks With Aerial and Ground Cooperation. IEEE Transactions on Wireless Communications 2021, 20, 7712–7727. [Google Scholar] [CrossRef]
11. Deng, X.; Li, J.; Shi, L.; Wei, Z.; Zhou, X.; Yuan, J. Wireless Powered Mobile Edge Computing: Dynamic Resource Allocation and Throughput Maximization. IEEE Transactions on Mobile Computing 2022, 21, 2271–2288. [Google Scholar] [CrossRef]
12. Zhan, C.; Hu, H.; Sui, X.; Liu, Z.; Niyato, D. Completion Time and Energy Optimization in the UAV-Enabled Mobile-Edge Computing System. IEEE Internet of Things Journal 2020, 7, 7808–7822. [Google Scholar] [CrossRef]
13. Zhang, L.; Ansari, N. Latency-Aware IoT Service Provisioning in UAV-Aided Mobile-Edge Computing Networks. IEEE Internet of Things Journal 2020, 7, 10573–10580. [Google Scholar] [CrossRef]
14. Chen, Z.; Zheng, H.; Zhang, J.; Zheng, X.; Rong, C. Joint computation offloading and deployment optimization in multi-UAV-enabled MEC systems. Peer-to-Peer Networking and Applications 2022, 1–12. [Google Scholar]
15. Shen, L. User Experience Oriented Task Computation for UAV-Assisted MEC System. IEEE INFOCOM 2022 - IEEE Conference on Computer Communications, 2022, pp. 1549–1558.
16. Liu, W.; Xu, Y.; Wu, D.; Wang, H.; Chu, X.; Xu, Y. QoE-aware Data Aggregation in MEC-enabled UAV Systems: A Matching Game Approach. 2022 IEEE 8th International Conference on Computer and Communications (ICCC), 2022, pp. 725–730.
17. Xiang, K.; He, Y. UAV-Assisted MEC System Considering UAV Trajectory and Task Offloading Strategy. ICC 2023 - IEEE International Conference on Communications, 2023, pp. 4677–4682.
18. He, Y.; Gan, Y.; Cui, H.; Guizani, M. Fairness-Based 3-D Multi-UAV Trajectory Optimization in Multi-UAV-Assisted MEC System. IEEE Internet of Things Journal 2023, 10, 11383–11395. [Google Scholar] [CrossRef]
19. Zhang, S.; Zhang, H.; He, Q.; Bian, K.; Song, L. Joint Trajectory and Power Optimization for UAV Relay Networks. IEEE Communications Letters 2018, 22, 161–164. [Google Scholar] [CrossRef]
20. Gao, Y.; Lu, F.; Wang, P.; Lu, W.; Ding, Y.; Cao, J. Resource Optimization of Secure Data Transmission for UAV-Relay Assisted Maritime MEC System. ICC 2023 - IEEE International Conference on Communications, 2023, pp. 3345–3350.
21. Jiao, S.; Fang, F.; Zhou, X.; Zhang, H. Joint Beamforming and Phase Shift Design in Downlink UAV Networks with IRS-Assisted NOMA. Journal of Communications and Information Networks 2020, 5, 138–149. [Google Scholar] [CrossRef]
22. Wang, F.; Zhang, X. IRS/UAV-Based Edge-Computing and Traffic-Offioading Over 6G THz Mobile Wireless Networks. ICC 2023 - IEEE International Conference on Communications, 2023, pp. 6480–6485.
23. Baker Siddiki Abir, M.A.; Zaman Chowdhury, M. Digital Twin-based Aerial Mobile Edge Computing System for Next Generation 6G Networks. 2023 6th International Conference on Electrical Information and Communication Technology (EICT), 2023, pp. 1–5.
24. Liu, Y.; Peng, M.; Shou, G.; Chen, Y.; Chen, S. Toward Edge Intelligence: Multiaccess Edge Computing for 5G and Internet of Things. IEEE Internet of Things Journal 2020, 7, 6722–6747. [Google Scholar] [CrossRef]
25. Li, Y.; Li, H.; Xu, G.; Xiang, T.; Lu, R. Practical Privacy-Preserving Federated Learning in Vehicular Fog Computing. IEEE Transactions on Vehicular Technology 2022, 71, 4692–4705. [Google Scholar] [CrossRef]
26. Xu, W.; Sun, Y.; Zou, R.; Liang, W.; Xia, Q.; Shan, F.; Wang, T.; Jia, X.; Li, Z. Throughput Maximization of UAV Networks. IEEE/ACM Transactions on Networking 2022, 30, 881–895. [Google Scholar] [CrossRef]
27. Tian, S.; Chi, C.; Long, S.; Oh, S.; Li, Z.; Long, J. User preference-based hierarchical offloading for collaborative cloud-edge computing. IEEE Transactions on Services Computing 2021. [Google Scholar] [CrossRef]
28. Park, Y.; Nielsen, P.; Moon, I. Unmanned aerial vehicle set covering problem considering fixed-radius coverage constraint. Computers & Operations Research 2020, 119, 104936. [Google Scholar] [CrossRef]
29. Alzenad, M.; El-Keyi, A.; Lagum, F.; Yanikomeroglu, H. 3-D Placement of an Unmanned Aerial Vehicle Base Station (UAV-BS) for Energy-Efficient Maximal Coverage. IEEE Wireless Communications Letters 2017, 6, 434–437. [Google Scholar] [CrossRef]
30. Du, Y.; Yang, K.; Wang, K.; Zhang, G.; Zhao, Y.; Chen, D. Joint Resources and Workflow Scheduling in UAV-Enabled Wirelessly-Powered MEC for IoT Systems. IEEE Transactions on Vehicular Technology 2019, 68, 10187–10200. [Google Scholar] [CrossRef]
31. Ji, J.; Zhu, K.; Niyato, D.; Wang, R. Joint Cache Placement, Flight Trajectory, and Transmission Power Optimization for Multi-UAV Assisted Wireless Networks. IEEE Transactions on Wireless Communications 2020, 19, 5389–5403. [Google Scholar] [CrossRef]
32. Dai, B.; Niu, J.; Ren, T.; Hu, Z.; Atiquzzaman, M. Towards Energy-Efficient Scheduling of UAV and Base Station Hybrid Enabled Mobile Edge Computing. IEEE Transactions on Vehicular Technology 2022, 71, 915–930. [Google Scholar] [CrossRef]
33. Welzl, E. Smallest enclosing disks (balls and ellipsoids). New Results and New Trends in Computer Science: Graz, Austria, 20–21 June 1991 Proceedings. Springer, 2005, pp. 359–370.