A Method for Maximizing UAV Deployment and Reducing Energy Consumption Based on Strong Weiszfeld and Steepest Descent with Goldstein Algorithms

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The proposed SW + SDG algorithm provides an efficient solution for emergency response teams to deploy UAV networks in disaster zones. This method enables (1) real-time optimization of multi-UAV placement for maximum area coverage during critical first-response operations and (2) significant energy conservation for prolonged mission duration in power-constrained environments. The balanced coverage–energy solution is particularly valuable for natural disaster monitoring, search-and-rescue missions, and post-disaster damage assessment scenarios where both surveillance range and operational endurance are crucial.

Abstract

Unmanned Aerial Vehicles (UAVs) play a crucial role in the continuous monitoring and coordination of disaster relief operations, providing real-time information that aids in decision-making and resource allocation. However, optimizing the deployment of multi-UAV systems in disaster-stricken areas presents a significant challenge. This challenge arises due to conflicting objectives, such as maximizing coverage while minimizing energy consumption, critical to ensuring prolonged operational capability in dynamic and unpredictable environments. To address these challenges, this paper proposes a novel successive deployment method specifically designed for optimizing UAV placements in complex disaster relief scenarios. The overall optimization problem is decomposed into two NP-hard subproblems: the coverage problem and the Energy Consumption (EC) problem. To achieve maximum coverage of the affected area, we employ the Strong Weiszfeld (SW) algorithm to determine optimal UAV placement. Simultaneously, to minimize energy consumption while maintaining optimal coverage performance, we utilize the Steepest Descent with Goldstein (SDG) algorithm. This dual-algorithmic approach is tailored to balance the trade-offs between wide-area coverage and energy efficiency. We validate the effectiveness of the proposed SW + SDG method by comparing its performance against traditional deployment strategies across multiple scenarios. Experimental results demonstrate that our approach significantly reduces energy consumption while maintaining extensive coverage, and outperforms conventional algorithms. This not only ensures a more sustainable and long-lasting operational network but also enhances deployment efficiency and stability. These findings suggest that the SW + SDG algorithm is a robust and versatile solution for optimizing multi-UAV deployments in dynamic, resource-constrained environments, providing a balanced approach to coverage and energy efficiency.

1. Introduction

In recent years, UAVs have attracted extensive attention and have been applied across diverse sectors, attributed to their notable benefits such as exceptional mobility, adaptable deployment, robust adaptability, and relatively low operational costs [1]. Their applications encompass data collection and transmission [2], network deployment and maintenance [3], environmental monitoring [4], disaster management [5], communication relays [6], and critical tasks like real-time video surveillance. Moreover, UAVs provide an efficient means of disaster relief in emergencies.

UAV systems deployed for emergency response necessitate rapid and effective deployment methods to ensure comprehensive coverage of the target area. Efficient deployment directly influences coverage performance, and deployed UAVs can better monitor and reach all required locations within a specified time. In response to this challenge, scholars have devised various optimization strategies, including the Genetic Algorithm (GA) [7], Particle Swarm Optimization (PSO) [8], Grey Wolf Optimizer (GWO) [9], Mixed Integer Linear Programming (MILP) [10], centralized deployment algorithms [11], and centralized greedy search algorithms [12]. While these methods have yielded positive outcomes in enhancing the response speed and deployment efficiency of UAV systems, there remains ample scope for research aimed at optimizing UAV deployment while minimizing EC.

In recent years, Deep Learning (DL) techniques have demonstrated significant potential in UAVs applications. Numerous DL-based approaches, such as Deep Reinforcement Learning (DRL) and Deep Neural Networks (DNNs), have been extensively investigated and applied. In [13], the authors proposed a Multi-Agent Reinforcement Learning (MARL)-based framework for UAV–ground vehicle cooperative control, leveraging DL-optimized trajectory planning to address the dual challenges of real-time data collection for IoT devices and UAV sendurance. In [14], the authors introduced a Peer-to-Peer (P2P) networking solution and a collaborative UAV–satellite system to enhance communication reliability and extend coverage in disaster zones, ensuring uninterrupted operation under harsh conditions. In [15], the authors developed a reinforcement learning (RL)-based adaptive scheduling algorithm, providing a robust framework for optimizing energy efficiency and system resilience in dynamic environments. These methods have achieved remarkable success in complex environmental perception and policy learning, enabling data-driven optimization of UAV deployment strategies, with promising performance in both simulated and real-world scenarios. However, deploying DL in disaster response still faces critical challenges, including real-time computational constraints on edge devices, the lack of interpretability in safety-critical decision-making, and insufficient adaptability to dynamic disaster data patterns.

To address the issues discussed above, this study proposes a novel decomposition-based collaborative optimization framework (SW + SDG), which aims to achieve the widest possible coverage while reducing the number of required UAVs and enabling rapid deployment to optimal positions. First, the SW algorithm is employed to ensure precise UAV positioning, followed by the SDG algorithm to optimize EC and path planning. The main contributions of this paper are summarized as follows: (1) An improved SW algorithm is proposed for the first time to achieve accurate UAV positioning, delivering extensive and stable network coverage to users within the area, thus solving the challenge of optimal UAV deployment; (2) The SDG algorithm is adopted for the first time to devise energy-efficient adjustment strategies and movement trajectories for UAVs. This facilitates the maximization of network coverage while minimizing energy expenditure; (3) Simulation results indicate that the refined SW algorithm decreases the UAV count required and boosts deployment precision. The SDG algorithm, in turn, diminishes EC, prolongs UAV flight duration, and enhances the network system’s operational efficiency overall.

The layout of this paper is organized as follows: Section 2 surveys the pertinent literature. Section 3 formulates and expounds the deployment model for multi-UAV systems, detailing its architecture, states, assumptions, coverage prerequisites, and key notations. Section 4 delivers a comprehensive exposition of the problem statement and algorithmic design. Section 5 exhibits and scrutinizes the outcomes via experimental simulations. It concludes with Section 6, which encapsulates the research findings and proposes avenues for future investigation.

2. Related Work

As shown in

Table 1. Relevant references.

Reference	Coverage	Energy	Path	Method
[16]	✓	×	×	Relative Positioning Coverage Algorithm
[17]	✓	×	×	Game Theory Decision Method
[18]	✓	×	×	MBS Minimization
[9]	✓	×	×	Space-Adaptive Game Algorithm
[7]	✓	×	×	Efficient Deployment Algorithm
[19]	✓	×	×	DACAMD-DDQN Algorithm
[20]	✓	×	×	VBCA Clustering Algorithm
[21]	✓	×	×	Particle Swarm and Other Algorithms
[22]	✓	×	×	Directional Antenna Optimization
[23]	✓	✓	×	MBS Minimization
[24]	✓	✓	×	Joint Parameter Optimization
[25]	✓	×	✓	Joint Parameter Optimization
[26]	✓	×	✓	Joint Model Optimization
[27]	×	✓	✓	Joint Model Optimization
[28]	×	✓	✓	Joint Model Optimization
[29]	×	✓	✓	Minimum Spanning Tree Recursive Algorithm
Our work	✓	✓	✓	SW + SDG Algorithm and Joint Parameter Optimization

“✓” indicates [supported]; “×” indicates [not supported].

, numerous studies have conducted in-depth analyses of UAV deployment strategies, including expanding network coverage, reducing Energy Consumption (EC), and planning efficient movement trajectories. However, an integrated solution encompassing all aspects has not yet been fully realized. Although these studies provide valuable strategies in individual areas, they typically optimize only one or two aspects and do not delve into the interdependencies and balance among all three factors.

To address the multi-UAV collaborative coverage problem, the authors in [16] proposed a coverage algorithm for multi-UAV systems based on relative positioning, optimizing deployment to ensure comprehensive coverage of the target area. In [17], the authors utilized a game theory-based autonomous decision-making method for efficient UAV deployment in a multi-level, multi-dimensional auxiliary network, which improved area coverage and reduced computation time. In [18], the authors introduced a spiral algorithm designed to minimize the number of UAVs required to cover all Ground Terminals (GTs) by sequentially deploying UAVs in a spiral pattern from the boundary toward the center of the target GT coverage area, prioritizing the coverage of edge GTs to reduce redundancy. In [9], an optimization scheme for UAV networks was proposed and designed, considering antenna direction and height shadowing, optimizing antenna parameters to enhance low-altitude coverage probability. In [7], a UAV base station deployment strategy leveraging the Grey Wolf Optimizer (GWO) algorithm was presented, finding the optimal UAV base station location under coverage probability constraints using stochastic geometry analysis to improve coverage and capacity in 5G communication systems.

Ref. [19] proposed a 3D Virtual force-Based Clustering Algorithm (VBCA) grounded in the VSEPR model, which significantly enhances coverage efficiency. Ref. [20] introduced a Dynamic Discrete Pigeon-Inspired Optimization (D²PIO) algorithm to address search and coverage problems, demonstrating its feasibility and superiority across multiple scenarios. In [21], the distribution of UAV base stations was modeled as a Poisson Point Process (PPP), effectively overcoming limitations in existing studies such as insufficient consideration of UAV network dynamics, key channel characteristics, and the complexity of coverage probability computation. Ref. [22] presented a multi-UAV 3D deployment method based on circle packing theory, achieving both maximal total coverage area and prolonged coverage lifetime. Ref. [23] developed a multi-UAV coverage model using Adaptive Virtual Force-directed Particle Swarm Optimization (AVF-PSO), leading to a notable improvement in coverage efficiency.

Although studies such as those in [19,20,21,22,23] have achieved coverage optimization, they generally presuppose ample energy and focus on solving the coverage problem without considering EC. These studies, however, have not delved into strategies for reducing system EC while maximizing coverage. To effectively reduce EC in multi-UAV systems, refs [24,25] introduced innovative algorithms: one based on space-adaptive game theory and the other on an adaptive virtual force particle swarm optimization algorithm, both aimed at minimizing EC while maximizing coverage. Nevertheless, despite these strides in energy efficiency, the significance of devising efficient movement paths for UAVs has been largely overlooked in these studies.

Thus, we continued to explore the issue of path planning for UAVs. In [26], the authors proposed a recursive algorithm leveraging the minimum spanning tree and an autonomous path planning strategy for persistent coverage, providing intelligent movement schemes for UAVs. In [27], the authors introduced and designed a UAV-based legal surveillance framework that combines trajectory design and energy management. This framework capitalizes on the rapid deployment and high maneuverability of UAVs to effectively eavesdrop on suspicious communication links by optimizing their positions and transmitting jamming signals while minimizing overall interference EC. In [28], the authors presented and designed an algorithm that jointly optimizes UAV trajectories, task completion time, and sensor node wake-up scheduling. This algorithm facilitates efficient data collection in UAV-assisted wireless sensor networks by optimizing the combined EC of UAVs and sensor nodes, achieving a balance and flexible trade-off between the EC of UAVs and sensor nodes. In [29], a strategy integrating resource allocation and trajectory planning was proposed to manage interference during wireless data collection from distributed sensors in multi-UAV scenarios. Experimental results demonstrated that this approach reduces the total data collection time compared to baseline methods. However, to date, no studies have undertaken an integrated approach to network coverage, energy efficiency, and path planning.

Therefore, after a detailed analysis of the existing literature, we adopt an integrated approach to problem-solving, simultaneously assessing the three core dimensions mentioned above. Furthermore, in contrast to prior works that often rely on a single objective or a unified optimization framework, this paper proposes a novel decomposed methodology. We decouple the problem into two specialized subproblems: coverage-driven placement and energy-driven movement. Each is solved with a tailored algorithm (enhanced SW and enhanced SDG, respectively). This dedicated two-stage approach (SW + SDG) leverages domain-specific optimization, yielding more robust, efficient, and balanced performance than conventional strategies.

3. System Model

The specific model of the network deployment and operation scenario is shown in Figure 1. It is assumed that there are m users in the disaster-stricken area where the base stations are damaged, and the users are fixed in their locations within the area network coverage area. These users rely on UAVs for data collection and transmission. The system includes n available UAVs to optimize coverage and flight paths to maintain reliable network communication.

3.1. UAV System Model

In this system, users are denoted as $o_j$ and indexed by $j\in \{1,2,\dots ,m\}$, and are assumed to be uniformly and randomly distributed within the network area. The set of UAVs is represented as $u_i$, indexed by $i\in \{1,2,\dots ,n\}$. Each UAV flies at a fixed altitude H, with an effective coverage area represented as a circular region centered at position $P\left(u_i\right)$ and with a radius $R_{ci}$, defining the UAV’s maximum communication or sensing range based on a Line-of-Sight (LoS) binary disk model [29]. To enhance the generality and scalability of the model, both UAVs and users are assumed to be uniformly and randomly distributed across the network area [30]. $o_j$ is considered to be effectively covered only when it is covered by at least one $u_i$, as shown in (1):

$$\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}\le R_v$$

(1)

where $R_V=\sqrt{R_{ci}^2-H^2}$ is the horizontal coverage radius of the UAV.

3.2. Coverage Constraints

To represent the coverage of UAVs on users, the coverage relationship between UAV $u_i$ and user $o_j$ is defined as follows:

$$f_j=\left\{\begin{array}{c}10\le R(u_i,o_j)\le R_v\hfill \\ 0R(u_i,o_j)>R_v\hfill \end{array}\right.$$

(2)

$$R(u_i,o_j)=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$$

(3)

where $f_j$ represents the coverage status of $o_j$ by $u_i$: 1 indicates that the user is covered, and 0 indicates that the user is not covered. $R(u_i,o_j)$ represents the Euclidean distance between $u_i$ and $o_j$.

Each UAV $u_i$ has a defined coverage set $C_i$, which includes all users $o_j$ covered by $u_i$, so that $C_i=\left\{o_j|f_j=1\right\}$. The combined coverage of the UAVs in the network can then be represented as $C={\bigcup }_{i=1}^n{C}_i$ [31].

We define the coverage of each UAV for the users as follows:

$$BC\left(C_i\right)=\sum _{j=0}^m{f}_j$$

(4)

In this model, the intersection of any two UAV subsets $C_i$ and $C_k$ (where $i\ne k$) is allowed to be non-empty, i.e., $C_i\bigcap C_k\ne \varnothing$. This means that a target user may be covered by multiple UAVs. Therefore, the coverage of the total set C for the users is defined as follows:

$$Cover\left(C\right)=\bigcup _{i=1}^{n}BC\left(C_i\right)$$

(5)

3.3. Energy Constraints

For UAVs, Energy Consumption (EC) is an essential factor in coverage and communication. Each UAV incurs energy costs, including distance loss $l_d$ between the assigned users and the UAV, and information loss $l_c$ due to lack of communication between UAVs. For the energy consumption analysis of $u_i$, let $d_{ij}$ denote the Euclidean distance between user $o_j$ and UAV $u_i$. Typically, the distance-dependent path loss $l_d$ can be modeled as a function of the user-UAV distance: $l_d=d_{ij}=\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2+{(z_i-z_j)}^2}$. Let $m_{uj}$ denote the number of UAVs deployed around user $o_j$. The information loss $l_c$ can be quantified as a function of the UAV deployment density per user: $l_c=m_{uj}$. These constraints are addressed in more detail in Section 4.2.

3.4. UAV Flight Trajectories and Obstacle Avoidance

Regarding their flight trajectories, UAVs follow a pre-established route designed to avoid obstacles while maintaining continuous communication links. By dynamically adjusting their velocity in response to terrain, wind conditions, and battery levels, they ensure optimal coverage efficiency and minimize the risk of collisions. The flight paths are refined in real time using onboard sensors’ data and collaboration with other UAVs. Advanced sensors like LiDAR and cameras enable precise obstacle detection, allowing the UAVs to autonomously reroute and avoid potential impediments.

4. Problem Definition and Optimization of UAV Deployment

From a broader perspective, multi-objective optimization strategies can generally be categorized into three classes: a priori methods (where decision-maker preferences are predefined before optimization), a posteriori methods (which seek Pareto-optimal solutions to reflect trade-offs among objectives), and decomposition-based methods (which break the problem into multiple single-objective subproblems for collaborative solving). The SW + SDG framework proposed in this paper falls into the category of a decomposition-based a priori method: we express a priori preference through sequential optimization—first maximizing coverage, then minimizing energy consumption—and further decompose the NP-hard problem of jointly optimizing coverage and energy consumption into two subproblems. For each subproblem, dedicated and efficient algorithms (SW and SDG) are designed, thereby circumventing the difficulties faced by traditional single-algorithm approaches in handling multi-objective trade-offs.

4.1. Formulation of the Target Coverage Optimization Problem for Multi-UAV Systems

Maximizing user coverage is framed as a set cover problem, where the objective is to cover the set of users with a subset of UAVs at minimal cost. Specifically, given the constraints on the number of UAVs and their limited range of movement, it is necessary to ensure that at least one UAV covers each user.

To ensure maximum coverage of nodes, it is necessary to find an optimal total set $C^{\ast }$ that covers as many target nodes as possible. We employ the binary coverage model Equation (6) to rigorously achieve maximal coverage while maintaining computational efficiency and theoretical tractability:

$$Cover\left(C^{\ast }\right)=max\bigcup _{i=1}^{n}BC\left(C_i\right)$$

(6)

To achieve optimal coverage, it is necessary to minimize the total Euclidean distance to multiple points by leveraging the geometric median:

$$E\left(p_i\right)=min\sum _{j\in O}{\displaystyle \frac{1}{{∥p_i-P\left(o_j\right)∥}^2}}$$

(7)

where $p_i$ is the deployment location of the $u_i$ and $P\left(o_j\right)$ is the spatial location of the $o_j$. A smaller sum of the relative distances between $P\left(o_j\right)$ and $p_i$ within each UAV’s communication coverage indicates a larger number of users covered by the UAV. Since the UAV is closer to the user group, it can cover more users, meaning that the number of elements included in the coverage set $C_i$ is larger.

To address this problem, we will use the improved SW algorithm from Section 4.3 to obtain the optimal coverage set $C^{\ast }$ for the UAVs. We use target coverage rate as the evaluation metric [32]:

$$CR={\displaystyle \frac{Cover\left(C\right)}{m}}$$

(8)

To further minimize energy consumption during UAV swarm deployment, the optimal coverage set $C^{\ast }$ generated in Section 4.2 will be utilized as terminal positions for swarm mobility, upon which the energy model will be optimized.

4.2. Formulation of the Energy Optimization Problem for Multi-UAV Systems

During task execution, UAVs experience EC, which includes distance loss $l_d$ between the assigned users and the UAV, and information loss $l_c$ due to lack of communication between UAVs. The loss function for UAVs is expressed as [33]:

$$L(D,N)={\mu }_{d}D\left(C\right)+{\mu }_{c}N\left(C\right)={\mu }_d{l}_d+{\mu }_c{l}_c$$

(9)

$${\mu }_d+{\mu }_c=1$$

(10)

where L represents the loss of the UAV, D is the sum of the distances between the UAV and the assigned location points, and N is the total number of UAV resources surrounding each user. The coefficients ${\mu }_d$ and ${\mu }_c$ are the weighting factors for balancing distance loss and information loss and are positive real numbers.

To minimize the impact between coverage and EC, after solving the problem proposed in Section 4.1 and obtaining the optimal total set $C^{\ast }$ for the UAV swarm, we further optimize EC constraints. To achieve this, we propose the following optimization model:

$$minf\left(z\right)=L\left(D\left(C^{\ast }\right)\right),\left(N\left(C^{\ast }\right)\right)$$

(11)

The model introduces two key performance indicators: $D\left(C^{\ast }\right)$ and $N\left(C^{\ast }\right)$. $D\left(C^{\ast }\right)=(l_d^1,l_d^2,\dots ,l_d^m)$ represents the sum of the straight-line distances between all users and their corresponding UAVs when applying the optimal coverage set strategy. This indicator reflects the proximity of the UAVs to the ground targets and is a key factor in measuring coverage efficiency. $N\left(C^{\ast }\right)=(m_u^1,m_u^2,\dots ,m_u^m)$ represents the sum of the number of UAVs deployed around each user in the optimal coverage set. This indicator is used to evaluate the balance of resource allocation and its ability to support task execution.

To solve this optimization problem, it is first simplified to:

$${\phantom{,}}_z^{min}\left\{f\right(z),z\in Z\}$$

(12)

Here, $f\left(z\right)$ is a continuously differentiable function, $Rn\to R$, and Z is a subset of $Rn$, $Z\subset Rn$, where $Rn$ is an n-dimensional Euclidean space.

4.3. Maximizing UAV Coverage Using the Strong Weiszfeld (SW) Algorithm

The proposed SW algorithm addresses the UAV positioning optimization problem by leveraging the geometric median and Weiszfeld algorithm [34], as detailed in Algorithm 1. The geometric median minimizes the total Euclidean distance to multiple points, which is crucial for maximizing UAV coverage and minimizing travel distance. The strength of the SW algorithm lies in its ability to handle non-convex optimization problems commonly encountered in UAV deployment, where factors such as dynamic user distributions complicate the solution space. The calculation of the geometric median in Equation (7) employs the conventional Weiszfeld algorithm, an iterative numerical procedure that converges to the geometric median by successively minimizing the specified objective function.

To find the optimal deployment location for each UAV $u_i$, the Weiszfeld algorithm iteratively updates the UAV’s position $p_i^{k+1}$ to minimize $E\left(p_i\right)$:

$$p_i^{k+1}={\displaystyle \frac{{\sum }_{j\in O}\frac{P\left(o_j\right)}{{∥p_i^k-P\left(o_j\right)∥}^2}}{{\sum }_{j\in O}\frac{1}{{∥p_i^k-P\left(o_j\right)∥}^2}}}$$

(13)

where $p_i^{k+1}$ represents the new position of the UAV in the $k+1$-th iteration. The Weiszfeld algorithm seeks to minimize $E\left(p_i\right)$ by iteratively updating $p_i$, enabling each UAV to cover as many users as possible. During the iterative computation in Equation (13), when $∥p_i^k-P\left(o_j\right)∥$ becomes extremely small, the term ${\displaystyle \frac{1}{{∥p_i^k-P\left(o_j\right)∥}^2}}$ tends to infinity. This situation may cause severe numerical instability, potentially leading to computational overflow, failure of algorithm convergence, or erroneous iterative results, ultimately compromising the accurate solution of the geometric median. To address this stability issue, we implement a dynamic adjustment strategy. In the initial iteration phase where data points are relatively dispersed, the occurrence of extremely small $∥p_i^k-P\left(o_j\right)∥$ values is unlikely.

Thus, we employ the original Equation (13) for computation during this stage. As iterations progress, we continuously monitor the magnitude of $∥p_i^k-P\left(o_j\right)∥$. Specifically, when $∥p_i^k-P\left(o_j\right)∥\le \tau$ (where $\tau$ is a predefined threshold), the algorithm automatically switches to a regularized computation scheme: $p_i^{k+1}={\displaystyle \frac{{\sum }_{j\in O}\frac{P\left(o_j\right)}{max({∥p_i^k-P\left(o_j\right)∥}^2,\tau )}}{{\sum }_{j\in O}\frac{1}{max({∥p_i^k-P\left(o_j\right)∥}^2,\tau )}}}$.

Here, $\tau$ is an infinitesimal positive constant that prevents denominator vanishing while minimally affecting the original algorithm’s weight distribution. This dynamic adjustment strategy allows full utilization of the original algorithm’s advantages during early iterations while ensuring numerical stability when potential risks emerge, thereby guaranteeing robust algorithm performance.

The algorithm selects the optimal center of coverage during the iteration process as the new deployment location for the UAV, denoted as $BEST{p}_i$. To avoid falling into local optima, we introduce multiple perturbations to improve the Weiszfeld algorithm, forming the SW algorithm:

$$BEST{p}_i^{n+1}=BEST{p}_i^n+{\delta }_k·{\gamma }_i$$

(14)

$$S_j\left(BEST{p}_i^{n+1}\right)=\sum _{i=1}^n{o}_j$$

(15)

where ${\delta }_k$ represents the pre-set small perturbation range at the k-th iteration and ${\gamma }_i$ is a random vector drawn from a probability distribution (e.g., normal distribution). It is worth noting that ${\delta }_k$ is not constant across all UAVs and during the iterative process. We adopt a dynamic adjustment mechanism, specifically an exponential decay strategy: ${\delta }_{k+1}=0.5{\delta }_k$. In the initial stage of the iteration, a relatively large value of ${\delta }_k$ helps the algorithm to extensively explore the search space, increasing the probability of escaping from local optimal solutions. As the iteration progresses, the gradually decreasing value of ${\delta }_k$ enables the algorithm to converge stably and precisely.

Algorithm 1: Strong Weiszfeld Algorithm
	Input: Set of targets O, maximum number of UAVs n
	Output: Optimized UAV positions set $C^{\ast }$
1	Initialize parameters;
2	Set maximum iterations $maxIter$;
3	while (length($C^{\ast }$) $<n$ and length(T) $<m$) do
4	for each target in O do
5	Find the target with maximum coverage density $p_i$;
6	Perform Weiszfeld algorithm to find new center by (13);
7	if new coverage is better than current best then
8	Update $C^{\ast }$;
9	end
10	Perform perturbation on the new circle within a defined range $\delta ·{\gamma }_i$;
11	if coverage increases after perturbation then
12	Update the circle position with perturbed position by (14);
13	Update $C^{\ast }$;
14	end
15	if all targets are covered or maximum UAVs reached then
16	Break
17	end
18	end
19	Update the set of targets T by (15);
20	Update the best solution $C^{\ast }$ if a better solution is found;
21	end
22	return $C^{\ast }$

If the number of users covered by the updated UAV location $S_j\left(BEST{p}_i^{n+1}\right)>S_j\left(BEST{p}_i^n\right)$ after a previous perturbation, the algorithm will perform another position perturbation. This iterative process continues in search of the optimal deployment position for the UAV, denoted as $BEST{p}_i^{\ast }$. After the perturbation ends, the algorithm uses the aforementioned optimization method to iteratively calculate the precise optimal deployment point for each UAV based on the total number of UAVs. The final result is a set of optimal deployment locations for the UAVs $C^{\ast }$:

$$C^{\ast }=\{BEST{p}_1^{\ast },BEST{p}_2^{\ast },\dots ,BEST{p}_n^{\ast }\}$$

(16)

After the initial iterative search is completed, the algorithm introduces strategic small-scale perturbations to fine-tune the solution, avoiding local optima during the optimization process and achieving precise optimization of UAV deployment.

4.4. Enhancing System Energy Efficiency Using the Steepest Descent with Goldstein (SDG) Algorithm

In Section 4.2, the SDG algorithm is selected for its effectiveness in addressing unconstrained optimization problems, particularly in the context of complex multi-UAV energy optimization. By incorporating gradient-based optimization techniques, the SDG algorithm enhances the traditional Steepest Descent (SD) method, improving both convergence speed and accuracy. Transforming the original problem into an unconstrained form simplifies the solution space, enabling the SDG algorithm to achieve reliable convergence due to the smooth gradient structure of the problem. Furthermore, its capability to handle large-scale and dynamic variables makes it highly suitable for multi-UAV systems. The detailed procedure of the SDG algorithm is presented in Algorithm 2. The SDG algorithm is particularly suitable for multi-UAV systems due to its ability to manage large-scale and dynamic variables [35].

To optimize the objective function defined in Equation (12), we employ an iterative algorithm characterized by the update process:

$$z_{k+1}=z_k+\Delta z_k=z_k+{\alpha }_k{p}_k$$

(17)

where $z_{k+1}$ represents the new position of the UAV after $k+1$ iterations and $z_k$ represents the current position after k iterations. In each iteration of the algorithm, $\Delta z_k={\alpha }_k{p}_k$ represents the increment generated by the k-th iteration. Here, $\alpha$ is a positive scalar that represents the step size, determining the magnitude of the increment in each iterative update; $p_k$ is a vector that represents the direction of the descent increment.

To ensure the convergence of this process, a direction search method is usually used first to determine the descent direction, followed by a line search method to obtain the optimal step size. The direction search algorithm uses the vector $p_k$ to represent the descent direction of the objective function $f\left(z\right)$ at $z_k$, which is a crucial step in finding the optimal point. This can be demonstrated from a mathematical perspective:

First, calculate the gradient vector $\nabla f\left(z_k\right)$ of the objective function $f\left(z\right)$ at the current iteration point $z_k$:

$$g_k=\nabla f\left(z_k\right)$$

(18)

The gradient vector $\nabla f\left(z_k\right)$ points in the direction of the fastest increase of the function, so its opposite direction is the descent direction. That is, if $g_k^T{p}_k<0$, then $p_k$ is the descent direction of $f\left(z\right)$, where $g_k^T$ represents the transpose of the gradient vector $\nabla f\left(z_k\right)$. Then, the search direction $p_k$ is calculated:

$$p_k=-D_k{g}_k$$

(19)

where $D_k$ is the identity matrix, ensuring the unit property of the search direction. Therefore, the search direction can be more concisely expressed as $p_k=-g_k$. Once the search direction is determined, the next step is to select an appropriate step size ${\alpha }_k$. The step size ${\alpha }_k$ determines the distance moved along the search direction in each iteration.

The iterative Equation (17) is used to find the optimal value ${\alpha }_k={\phantom{,}}_{{\alpha }_k}^{min}f(z_k+{\alpha }_k{p}_k)$, which is known as exact search. If the goal is only to find an ${\alpha }_k$ such that $f(z_k+{\alpha }_k{p}_k)$ has a sufficient decrease relative to $f\left(z_k\right)$, then this process is known as an inexact search. Because the problem this study faces is high-dimensional and large-scale and requires high computational efficiency, an inexact search method based on the improved Goldstein rule is adopted. The improved Goldstein line search condition is defined to ensure sufficient function decrease:

$$\begin{array}{cc}\hfill f\left(z_k\right)& +c_2{\alpha }_k\nabla f{\left(z_k\right)}^T{p}_k\le f(z_k+{\alpha }_k{p}_k)\le f\left(z_k\right)+c_1{\alpha }_k\nabla f{\left(z_k\right)}^T{p}_k\hfill \end{array}$$

(20)

where $f\left(z\right)={\mu }_{d}D\left(C^{\ast }\right)+{\mu }_{c}N\left(C^{\ast }\right)={\mu }_{d}d(z,u^{\ast })+{\mu }_c{\sum }_{j=1}^m{m}_{uj}$ is the objective function and $u^{\ast }\in C^{\ast }$ represents the unmanned terminal route point. $\nabla f\left(z_k\right)$ is the gradient of the objective function at $z_k$, $p_k$ is the search direction, and ${\alpha }_k$ is the step size at step k. Parameters $c_1$ and $c_2$ are used to control the selection of the step size, satisfying $0<c_1<c_2<0.5$. The right-hand inequality is the Armijo condition, ensuring that ${\alpha }_k$ is not too large, thus avoiding stepping out of the neighborhood of the optimal solution. The left-hand inequality prevents ${\alpha }_k$ from being too small, which would slow down the algorithm’s computational speed. Specifically, if $f(z_k+{\alpha }_k{p}_k)\le f\left(z_k\right)+c_1{\alpha }_k\nabla f{\left(z_k\right)}^T{p}_k$, it indicates that the step size ${\alpha }_k$ is excessively large. In this case, we reduce ${\alpha }_k$ by a factor of ${\sigma }_1$ (i.e., ${\alpha }_k={\sigma }_1{\alpha }_k$). Notably, to ensure algorithmic convergence, we enforce the upper bound constraint ${\alpha }_k\le {\alpha }_{max}$. Conversely, if $f\left(z_k\right)+c_2{\alpha }_k\nabla f{\left(z_k\right)}^T{p}_k\le f(z_k+{\alpha }_k{p}_k)$, this signifies an insufficient step size. Consequently, we expand ${\alpha }_k$ by a factor of ${\sigma }_2$ (i.e., ${\alpha }_k={\sigma }_2{\alpha }_k$). When neither of these two conditions is met, the current ${\alpha }_k$ value is deemed appropriate for this iteration and is retained as the selected step size.

The inexact search method not only enables quick convergence to an acceptable solution but also exhibits robustness across a range of initial conditions and parameter selections, making it versatile for various problem instances. The SDG algorithm effectively integrates the inexact search approach with the traditional SD algorithm to achieve a satisfactory optimization solution while ensuring algorithm efficiency. This ensures quick response and cost-effectiveness in practical applications.

In terms of complexity, the SDG algorithm maintains a computational complexity comparable to the traditional SD method, approximately $O\left(n\right)$ per iteration, where n represents the dimensionality of the problem. This linear complexity ensures that the algorithm is suitable for large-scale problems. As for convergence, by incorporating the inexact search approach, the SDG algorithm guarantees convergence to a local minimum under standard assumptions of smoothness and continuity of the objective function. The convergence rate is linear, similar to the traditional SD algorithm, but with enhanced practical performance due to its robustness in handling various initial conditions.

Algorithm 2: Steepest Descent with Goldstein Algorithm
	Input: Set of UAVs position P,Set of maxmize coverage $C^{\ast }$
	Output: System energy consumption f, Optimal deployment points set V
1	Initialize parameters;
2	Set maximum iterations $maxIter$;
3	while (length(V) $<n$) do
4	for $k=0$ to $maxIter$ do
5	if $L\left(z\right)=L\left(z_{best}\right)$ then
6	Break;
7	end
8	Compute the gradient of the objective function $g_k^T$ by (18);
9	Choose a descent direction $p_k$ by (19);
10	Initialize ${\alpha }_k={\alpha }_k^0;0<c<0.5$;
11	whlie $j=0,1,2,\dots$ do
12	Determine the step size ${\alpha }_k$ using a line search method by (20);
13	end
14	Update the solution by (17);
15	Check for convergence;
16	if converged then
17	Break;
18	end
19	end
20	end
21	return f

5. Experimental Results and Analysis

The simulation first compares the performance of the SDG algorithm against the traditional SD algorithm. The scenario, as depicted in Figure 2, is set in a 400 m × 400 m area. The algorithms were implemented using MATLAB R2023a as the simulation platform. To ensure robustness and reliability, each experiment was conducted more than 500 times with varying initial conditions, and the results were averaged to obtain the final performance metrics. The detailed experimental parameters are listed in

Table 2. Network configuration parameter settings.

Parameter	Range	Initial Settings
UAV Density	[1, 12]	10
Crowd Density	[30, 200]	100
Coverage Radius	[20, 60]	50 m
Area Width	400 m	400 m
Area Height	400 m	400 m

.

5.1. Algorithm Performance Comparison

5.1.1. Superiority of the SDG Algorithm

Through extensive experimentation, we systematically optimized the parameters ($c_1,c_2,{\sigma }_1,{\sigma }_2,{\alpha }_{max}$) in Equation (20). Initial parameter ranges were set as $0<c_1<c_2<0.5,$ $0<{\sigma }_1<{\sigma }_2<2,$ $0<{\alpha }_{max}<10$. Multiple parameter combinations were evaluated across diverse experimental scenarios with varying scales and characteristics to assess their impact on algorithmic convergence and computational efficiency. After rigorous comparative analysis of experimental results, we observed that excessively large values of $c_1,c_2,{\sigma }_1,{\sigma }_2,{\alpha }_{max}$ adversely affected convergence properties. The optimal parameter configuration was identified as $c_1=0.01, c_2=0.08, {\sigma }_1=0.5, {\sigma }_2=1.2, {\alpha }_{max}=5,$ which demonstrated superior performance in terms of both convergence rate and numerical stability. Consequently, this optimal parameter set was adopted for all subsequent experiments in our study. A detailed parameter optimization experiment for the SDG algorithm is presented in Appendix A, including a systematic comparison of multiple parameter sets and the identification of the optimal configuration.

Figure 3 illustrates the performance comparison of the SDG algorithm against several other algorithms: SD-Armijo (SDA), SD-Armijo1 (SDA1), SD-Wlof (SDW) [35], SD-Wolfe-Powell (SDWP) [36], SD-Strong-Wolfe (SDSW) [37], and SD-Backtracking (SDB) [38].

The figure indicates that all algorithms show notable performance gains during the initial iterations, with the improvement rate decelerating as iterations continue, leading to stabilization. However, SDA, SDA1, SDW, SDWP, SDSW, and SDB among the compared algorithms exhibit a propensity for premature convergence, elevating the likelihood of these algorithms converging to local minima instead of the global minimum. In contrast, the SDG algorithm, as introduced in this study, exhibits significant superiority in terms of convergence performance. After a relatively small number of iterations, the SDG algorithm not only hastens convergence but also reliably achieves a superior fitness value compared to other algorithms. Furthermore, the SDG algorithm consistently converges to the optimal solution before the 200-iteration mark. This result highlights the SDG algorithm’s high efficiency in pursuing the global optimum and validates its robustness in algorithm design.

The convergence performance and robustness of the SDG algorithm under different parameter configurations are further validated through confidence interval analysis in Appendix B.1 and Appendix B.2.

5.1.2. Superiority of the SW Algorithm

Through comprehensive testing and multi-faceted analysis of the algorithm’s performance in this study, we have determined the optimal parameter configuration as follows: the threshold $\tau$ is set to ${10}^{-6}$, while the initial step size is established as $0.2R_c$.

Figure 2 illustrates the UAV coverage scenario, showing the coverage of UAV users under four different algorithms given the initial experimental parameters. The figure provides an intuitive and clear view of the different coverage situations exhibited by each algorithm. Running time as a key indicator of algorithm performance directly impacts the real-time applicability and practicality of UAV deployment strategies.

Table 3. Running time of each algorithm.

Algorithm Name	Running Time (Seconds)
KED	0.9531
GREEDY	0.2500
MEAN	0.4219
SW	0.2031

reveals the differences in run time among the four algorithms: KED [39], GREEDY [40], MEAN, and SW. The SW algorithm demonstrates remarkable efficiency with a run time of 0.2031 s, whereas the KED algorithm shows a relatively longer run time of 0.9531 s, and the MEAN algorithm has an intermediate run time of 0.4219 s. The shorter run time of the SW algorithm indicates that it can quickly respond to and adapt to dynamically changing environments, thereby enhancing the timeliness of UAV deployment and the overall efficiency of mission execution. According to (8), Figure 4 illustrates the trend of network coverage rate versus the number of UAVs for different algorithms. As shown, the coverage rate of all algorithms increases monotonically with the number of UAVs, which aligns with general expectations. However, the SW algorithm consistently demonstrates superior or near-optimal performance. Specifically, when the number of UAVs is small (fewer than 8), the SW algorithm performs comparably to the MEAN algorithm, both significantly outperforming KED and GREEDY. This confirms the efficiency of the SW algorithm even under resource-constrained conditions. When the number of UAVs exceeds eight, the advantage of the SW algorithm becomes more pronounced, maintaining a leading coverage rate over all other algorithms. This result strongly indicates that the SW algorithm more intelligently computes optimal UAV placements, minimizes overlapping coverage, and makes efficient use of additional UAVs to cover more user nodes, thereby achieving higher coverage gain under the same resource constraints. The computational efficiency and coverage performance of the SW algorithm are statistically confirmed using 95% confidence intervals, as detailed in Appendix B.3.

5.2. System Simulation Analysis

Figure 5 analyzes the impact of the number of user nodes on the average network coverage rate. The results indicate that the average coverage rate decreases rapidly at first and then stabilizes as user density increases. When the number of users is small (below 60), the network coverage remains high since limited UAV resources can easily cover all users. As the number of users increases to between 60 and 180, the coverage rate begins to decline significantly, due to the more sparse and widespread user distribution, which makes it difficult for a fixed number of UAVs to maintain comprehensive coverage. When the number of users exceeds 180, the average coverage rate gradually stabilizes to around 75%. This suggests that the network coverage capacity has reached saturation at this density; newly added users are mostly located within already covered areas or on the edges, and thus do not cause a further sharp decline in coverage. This phenomenon reveals the practical capacity limit of the deployed network and provides important insights for planning the number of UAVs required in emergency scenarios.

Figure 6 vividly demonstrates the significant positive impact of expanding the UAV coverage radius on enhancing network coverage effectiveness. As the coverage radius gradually expands from 20 m to 60 m, the average coverage rate steadily increases, approaching a high coverage level of 0.9. This result visually demonstrates the importance and effectiveness of expanding the coverage radius to enhance the network coverage range. By incrementally increasing the coverage radius, the UAV network can more effectively serve a wider area, significantly improving overall coverage performance.

In the previous sections, we determined the optimal deployment positions for maximizing UAV coverage. We now apply the SDG algorithm to iteratively optimize the loss function, precisely planning the optimal path for each UAV, thereby significantly reducing system EC. As per Equation (11), the simulation energy consumption results are depicted in Figure 7 and detailed in

Table 4. Energy optimization results.

Number of UAVs	EC
	SW	SW + SDG	Reduction Rate
4	472.83	342.83	$27.5\%$
5	954.67	498.62	$47.8\%$
10	1329.60	490.47	$63.1\%$

. The introduction of the SDG algorithm results in a significant reduction in EC values. Specifically, when deploying four UAVs, the energy loss decreased by $27.5\%$. With five UAVs, the EC value dropped by $47.8\%$. In a complex environment with ten UAVs, the reduction in EC value reached as high as $63.1\%$. These data strongly demonstrate that under the guidance of the energy loss function, the EC value has been substantially reduced, indicating a highly effective optimization outcome.

Figure 8 clearly shows the comparison of system energy consumption before and after the application of the SDG algorithm across the four methods. It is evident from the figure that as the number of UAVs increases, the value of EC rises for all methods. This trend indicates that adding each additional UAV requires more energy to move it to its optimal coverage location. A detailed comparison was conducted between the KED, GREEDY, MEAN, and SW methods, with and without the integration of the SDG algorithm. The results show that incorporating the SDG algorithm effectively reduces the EC. This underscores that the SDG strategy, via its sophisticated path planning optimization, markedly reduces the energy expenditure of UAVs throughout their mission execution. This improvement is evident across all algorithms, particularly for the SW + SDG combination. Compared to the original SW algorithm, the reduction in EC is the most pronounced, substantially enhancing the overall system performance.

To highlight the superior performance of the proposed algorithm in this study, Figure 9 compares both coverage rate and EC. The SW + SDG algorithm demonstrates the most outstanding performance in terms of coverage rate, while the KED + SDG algorithm excels in minimizing EC. Although the SW + SDG algorithm has a marginally higher energy consumption compared to KED + SDG, it significantly outperforms KED + SDG in terms of coverage rate. Therefore, the overall comparison demonstrates that the SW + SDG algorithm provides the optimal balance of coverage rate and energy efficiency, showing the best comprehensive performance among the four algorithms.

To validate the effectiveness of the proposed SW + SDG method, we compared its performance with three common optimization algorithms (GREEDY, PSO, and GA). The experimental setup was as follows: the number of UAVs was set to 12, the number of target users was 100, the coverage radius was 50 m, and the area size was 400 × 400 m. The experimental results are shown in

Table 5. Performance comparison of different algorithms.

Algorithm	Average Coverage	Average Energy Consumption	Average Running Time (Seconds)
SW + SDG	0.79	615.3	0.21
GREEDY	0.74	1386.4	0.25
PSO	0.79	1467.7	0.53
GA	0.72	1335.8	0.98

.

The SW + SDG algorithm demonstrates the best performance in terms of coverage, energy consumption, and running time. Its average coverage rate reaches 0.79, significantly outperforming GREEDY (0.74) and GA (0.72), and is comparable to PSO (0.79). At the same time, SW + SDG achieves the lowest energy consumption (615.3), which is far lower than GREEDY (1386.4), PSO (1467.7), and GA (1335.8). Additionally, SW + SDG has the shortest running time (0.21 s), outperforming GREEDY (0.25 s), PSO (0.53 s), and GA (0.98 s). The GREEDY algorithm has a relatively short running time but lower coverage and higher energy consumption, making it suitable for simple scenarios. The PSO algorithm achieves coverage comparable to SW + SDG but with higher energy consumption and longer running time, making it suitable for offline optimization scenarios. The GA algorithm has the lowest coverage, the longest running time, and higher energy consumption, making it suitable for complex scenarios with low real-time requirements. Overall, SW + SDG performs best in terms of coverage, energy consumption, and running time, making it an efficient and practical optimization method for UAV deployment, particularly suitable for scenarios with high real-time demands.

6. Conclusions

This study focuses on the deployment optimization of multi-UAV systems, addressing the dual objectives of maximizing network coverage and minimizing system EC. An innovative optimization strategy is proposed, meticulously designed and validated through a series of rigorous experiments. Building on this foundation, the SW algorithm was developed to minimize the number of UAVs required while ensuring comprehensive coverage of disaster-stricken areas. Furthermore, the SDG algorithm enabled precise adjustments to UAV flight paths and energy efficiency, significantly reducing overall system energy consumption. Experimental results demonstrate that the proposed SW + SDG framework achieves extensive coverage of affected populations and remarkable energy savings, outperforming traditional methods such as GREEDY, PSO, and GA in terms of coverage, energy efficiency, and computational speed.

Overall, SW + SDG performs best in terms of coverage, energy consumption, and running time, making it an efficient and practical optimization method for UAV deployment, particularly suitable for scenarios with high real-time demands. The proposed method demonstrates significant application potential beyond theoretical optimization. In disaster emergency response, it can enable the rapid establishment of communication coverage and real-time situational awareness in damaged areas [41]; in precision agriculture, it facilitates efficient and energy-conscious monitoring of large-scale crops; and in regional security surveillance, it supports the deployment of persistent and wide-ranging monitoring networks. Building on current findings, our future research will focus on three key directions: (1) Algorithmic enhancement through hybrid optimization strategies combining DRL (TD3/PPO) with our SW + SDG framework, including formal convergence analysis and MARL-based collaborative deployment; (2) Environmental adaptation by extending to 3D positioning with probabilistic coverage models, obstacle formulations, and environmental factor integration (wind, terrain); (3) Practical validation via large-scale field trials incorporating real-world data, with comprehensive benchmarking against state-of-the-art methods across key metrics (convergence speed, energy efficiency, computational requirements).