In recent years, rapid technological development has increased the complexity of real-life problems and heightened the need for more effective metaheuristic techniques to find optimal solutions for complex issues [1,2]. The metaheuristic is a process that produces heuristic approaches to determine the most optimal combination of variables to provide the fitting solution for complex problems in the optimization algorithm [3]. The popularity of these algorithms increased dramatically due to their performance and complexity in solving large-scale problems compared to other classical techniques [4]. In general, metaheuristic techniques could be categorized into four classes: evolutionary-based, human-based, physics-based, and swarm-based algorithms [5].

Evolutionary-based algorithms are primarily inspired by the theory of natural selection. In these algorithms, the initial population evolves based on evolutionary processes to enhance the population’s fitness. This process produces a new population with inherent characteristics and features derived from a set of randomly selected parents from the current population, using them to produce offspring for the new generation [6]. Recent population-based optimizers, such as the Chaotic Arithmetic Optimization Algorithm (CAOA) [7], Crested Porcupine Optimizer (CPO) [8], and Hippopotamus Optimization (HO) [9], have further advanced population evolution strategies on complex benchmark suites. The most common operators used in these processes are selection, crossover, and mutation. Some of the widely used evolutionary-based optimization algorithms are Genetic Algorithms (GA) [10], Differential Evolution (DE) [11], Forest Optimization Algorithm (FOA) [12], Firefly Algorithm (FA) [13], Evolution Strategy (ES) [14], and Biogeography-Based Optimizer (BBO) [15]. Evolutionary algorithms mostly provide near-optimal solutions for many problems since they do not assume the adaptive landscape or essential fitness.

The human-based methods are algorithms that are explicitly inspired by the theory of the social life behavior of humans or concepts humans have developed [16]. In addition, novel human-based methods like the Preschool Education Optimization Algorithm (PEOA) [17] and the Dragon Boat Optimization (DBO) [18] have recently been proposed to emulate teacher-student dynamics and coordinated team-paddling behavior. Some of the popular human-based techniques are Poor and Rich Optimization (PRO) [19], Group Teaching Optimization Algorithm (GTOA) [20], Teaching-Learning-Based Optimization (TLBO) [21], Queuing Search Algorithm (QSA) [22], and Group Counseling Optimization (GCO) [23], which are proposed based of this theory.

Physics-based approaches encompass any algorithm that is inspired by perceived physical phenomena or chemistry. These methods operate based on physical rules or theories and direct search agents using formulae derived from physical laws. Recent physics-driven optimizers, including the Energy Valley Optimizer (EVO) [24] and the Snow Ablation Optimizer (SAO) [25], have demonstrated effective search behaviors by modeling particle decay processes and heat-induced phase changes. Some of the optimization algorithms that are recently introduced based on this theory are Galactic Swarm Optimization (GSO) [26], Henry Gas Solubility Optimization (HGSO) [27], Atom Search Optimization (ASO) [28], Electro-Search algorithm (ES) [29], Optics Inspired Optimization (OIO) [30], Water Evaporation Optimization (WEO) [31], Equilibrium Optimizer (EO) [32], Lévy flight distribution (LFD) [33], Multi-Verse Optimizer (MVO) [34], Water Cycle Algorithm (WCA) [35], Thermal Exchange Optimization (TEO) [36], and Gravitational Search Algorithm (GSA) [37].

The swarm intelligence-based algorithms emulate social behavior, interactions, and communications between a set of plants, varieties of animals, or different creatures. The most popular social behaviors studied in this approach are hunting, exploring for food, memorizing, and mating [38]. Because communication and sharing of information are vital factors in interactions between animals and different species, swarm-based algorithms share information between multiple agents during iterations to perform efficient search processes [39]. New swarm-based approaches such as the Special Forces Algorithm (SFA) [40], the Squid Game Optimizer (SGO) [41], the Artificial Satellite Search Algorithm (ASSA) [42], the Dhole Optimization Algorithm (DOA) [43], and the Chaotic Evolution Optimization (CEO) [44] have recently been introduced, drawing on tactical, game-based, and chaotic group behaviors. In recent years, the popularity of swarm-based methods has increased in both algorithm development and application. There are various optimization algorithms introduced based on this approach, such as Particle Swarm Optimization (PSO) [45], Squirrel Search Algorithm (SSA) [46], Harris Hawks Optimization (HHO) [47], Sine Cosine Algorithm (SCA) [48], Seagull Optimization Algorithm (SOA) [49], Whale Optimization Algorithm (WOA) [50], Moth-flame optimization (MFO) [51], Pathfinder algorithm (PFA) [52], Gray Wolf Optimizer (GWO) [53], and Emperor Penguin Optimizer (EPO) [54].

Furthermore, metaheuristic algorithms can be categorized by the number of agents they employ in the search process [55]. Accordingly, population-based algorithms utilize multiple search agents, whereas individualist methods employ a single search agent in each iteration. A population-based method can provide more information using multiple agents and better explore the search space to find the optimum solution. However, it can have massive increases in the number of evaluations on the objective function, making computationally intensive objective function problems. Some of the well-known individualist metaheuristics methods are Tabu Search (TS) [56], Hill Climbing [57], and Simulated Annealing (SA) [58]. The metaheuristic method is widely applied by researchers to solve various optimization problems across several domains; however, no single algorithm can provide the optimal solution for all cases. Some metaheuristic methods are particularly suited to specific types of issues but may be ineffective for others [59]. Therefore, there is a continuous need for the development of new optimization techniques that can more effectively address a broad spectrum of current and future challenges that are complex to solve using existing algorithms. This paper introduces a new swarm-based metaheuristic algorithm, named Painted Wolf Optimization (PWO), inspired by the social hierarchy and hunting behaviors of painted wolves, also known as African wild dogs. The Painted Wolf Optimization algorithm draws inspiration from the intelligent behaviors of painted wolves in hunting and exploring as a pack, as well as their unique pre-hunt voting system, which contributes to their success as hunters in the natural world. It is essential to note that, although several wolf-inspired algorithms exist in the literature, particularly the Grey Wolf Optimizer (GWO) [53], the proposed PWO algorithm is fundamentally distinct. The painted wolf (Lycaon pictus) is a biologically separate species from the grey wolf (Canis lupus), exhibiting unique social behaviors including consensus-based decision-making through sneezing rallies. PWO introduces novel algorithmic mechanisms, including a rally-based exploration, exploitation switching mechanism, dynamic alpha influence modulation, and dual probabilistic exploration strategies that are not present in GWO or its variants.

The main contributions of this work are outlined as follows:

A novel optimization algorithm named Painted Wolf Optimization is proposed for solving a diverse array of optimization problems.

The remainder of this paper is organized as follows: Section 2 introduces the biological inspiration behind the proposed PWO algorithm, including the pre-hunting rally, exploration, and exploitation phases, as well as the pseudocode and flowchart of PWO. Section 3 presents the experimental results and discussion, detailing the benchmark set, compared algorithms, experimental setup, and performance analyses (exploitation, exploration, local minima avoidance, convergence, scalability, and stagnation). This section also provides a summary of the performance comparisons. Section 4 explores the application of the PWO algorithm to engineering benchmark problems, while Section 5 focuses on its utility in a cybersecurity context, covering data collection, model training, experiment setup, and evaluation. Finally, Section 6 concludes the paper by summarizing the key findings and contributions.

This section presents the system model, optimization framework, and working mechanism of the proposed Painted Wolf Optimization algorithm. The mathematical formulation translates the unique biological behaviors of painted wolves into computational procedures for optimization.

The painted wolf (Lycaon pictus), also known as the African wild dog or Cape hunting dog, is an endangered species residing in various African countries [60]. Renowned for their intricate social structures, painted wolves live in packs of 6 to 20 individuals. This social living confers significant advantages, particularly in hunting, where the success of the pack is largely dependent on the cumulative endurance of its members [61]. Unlike predators relying on stealth, speed, and strength, painted wolves utilize exhaustive predation strategies to pursue and capture prey [62]. They typically hunt larger prey as a pack, while solitary individuals may hunt smaller animals such as hares and rodents. Additionally, painted wolves exhibit a nomadic lifestyle, often covering distances of up to 50 km per day [63]. Fig. 1 showcases photographs of painted wolves in their natural habitat.

A sophisticated behavior observed in painted wolves is their consensus-driven decision-making process for hunting. Before a hunt, they engage in a pre-hunting ritual known as a rally, during which each pack member may express their willingness to participate by sneezing [64]. This process does not limit the number of sneezes per individual, allowing for a form of voting that is influenced by the social hierarchy within the pack. When an alpha painted wolf initiates the rally, a smaller number of affirmative signals (often just three sneezes) are needed for the pack to proceed with the hunt. In contrast, if a lower-ranking member starts the rally, the minimum for the hunt’s commencement typically requires ten sneezes [64]. This demonstrates that while dominant individuals can sway the decision-making process, they are not essential for initiating a rally.

The intricate group behavior and consensus decision-making of painted wolf packs, particularly their pre-hunting rally ritual, are the primary inspirations for the proposed Painted Wolf Optimization (PWO) algorithm. This algorithm draws on the complex hunting behaviors and strategies of painted wolves, translating these into a mathematical model that underpins the development of a novel optimization approach. In the following subsections, we detail how the natural behaviors of painted wolves are mathematically modeled and how this model informs the structure and function of the PWO algorithm.

This section includes the experimental setup and the investigation of the experiment outcomes of the proposed algorithm. The effectiveness of the PWO algorithm is evaluated on 33 standard test functions. Section 3.1 describes the benchmark test functions used, Section 3.2 explains the experimental preparation and parameters, and Section 3.3 reports the analysis of the results achieved by the proposed PWO algorithm and compares it with well-known metaheuristic algorithms.

In this section, in order to evaluate the performance of the PWO algorithm in solving real-world engineering problems, the proposed algorithm is used to tackle three well-known engineering design problems. Solving engineering design problems using metaheuristic optimization algorithms has been widely used in previous studies. This is because of the nonlinear characteristics of engineering design problems, making metaheuristic algorithms a proper candidate for tackling these problems. In this study, the PWO algorithm was applied in solving the following engineering benchmark problems: welded beam design, tension/compression spring design, and pressure vessel design. In all experiments, the PWO provided results compared with the other well-known optimization techniques utilized in previous sections.

In this section, in order to evaluate the performance of the PWO algorithm in solving cyber security issues, the proposed algorithm is used to tackle training issues in Machine Learning Intrusion Detection Systems (ML-IDS). Solving training issues in ML-IDS using metaheuristic optimization algorithms can overcome the limitations of current training methods and improve the performance of ML models [67]. This is because they offer effective and efficient optimization techniques for finding reasonable solutions in complex and high-dimensional search spaces. Furthermore, ANN model training involves adjusting the weights and biases of the network to minimize a defined objective function, typically the error between the network’s output and the desired output, making metaheuristic algorithms a proper candidate for tackling these problems. This study applied the PWO algorithm to train the ML-IDS weights and biases on an NSL_KDD dataset. The PWO provided results in all experiments compared with the other well-known optimization techniques utilized in previous sections.

In this study, we proposed a new nature-inspired metaheuristic, Painted Wolf Optimization (PWO), for solving a broad range of optimization problems. PWO is motivated by the social organization and cooperative hunting behavior of painted wolves, and these behaviors are translated into mathematical operators that guide the search process. The performance of PWO was evaluated on 33 benchmark functions, including the CEC2017 suite, which spans unimodal, multimodal, and composite scenarios, to examine exploration, exploitation, and the ability to avoid local optima. Additionally, PWO was validated in a cybersecurity application by optimizing a machine-learning intrusion detection system (ML-IDS) for accurate detection of abnormal traffic. Overall, the experimental results indicate that PWO achieves competitive and often superior performance compared with several recent, well-established metaheuristics across the tested benchmarks. In the ML-IDS task, PWO demonstrated a strong detection capability, achieving high accuracy and F-measure compared to the other methods. Furthermore, experiments on three standard real-world engineering design problems showed that PWO can produce high-quality solutions and compares favorably with existing optimization techniques.

Future research directions include: (1) extending PWO to large-scale and multi-objective optimization problems; (2) hybridizing PWO with other metaheuristics to further enhance performance; (3) evaluating PWO on additional real-world domains such as healthcare, finance, and smart grid optimization; (4) conducting systematic parameter sensitivity and robustness analyses; and (5) benchmarking against more recently proposed optimization algorithms.