## 1 Introduction
There are various hybrid techniques that incorporate PSO with other optimization algorithms, with the aim of improving global search ability and convergence speed. This has been demonstrated in hybrids that involve PSO with genetic algorithms (GAs), ant colony optimization, and differential evolution. Each of these hybrids showed superior performance compared to the standard PSO, finding wide applications in fields such as engineering, finance, and image processing [25; 26]. Specific strategies have been adopted to enhance the social and cooperative aspects of PSO. For instance, the integration of GA with PSO has been implemented in a few distinct ways. One approach involves using GA and PSO either sequentially or in parallel, while another strategy involves the exchange of the fittest particles between GA and PSO when run concurrently [27; 28]. Some studies have even employed a two-phase mechanism, where PSO accelerates evolution, and GA ensures diversity [29]. Several innovative hybrid approaches have also been proposed. These include a fuzzy approach that utilizes simple rules for decision making [30], a hybrid algorithm known as FAPSO that combines PSO with the firefly algorithm [31], and a heuristic based on PSO and simulated annealing for resolving multi-objective problems in network-based models [32]. A unique hybrid PSO method focuses specifically on estimating the parameters of photovoltaic (PV) cells. This comes in response to the escalating demand for renewable energy and the need for accurate PV cell models [33]. In a similar vein, an adaptive simulated annealing-parallel particle swarm optimization (ASA-PPSO) approach was developed, incorporating PSO with an infix condition that applies simulated annealing (SA). Moreover, hybridizing differential evolution (DE) with PSO has been explored in various ways, one of which involves using DE to maneuver particles and thereby enhance the convergence rate of PSO [26; 34]. Hybrid optimization techniques that combine the strengths of gradient-based and evolutionary algorithms have demonstrated excellent performance in finding satisfactory solutions. One such method is the PSO-descent hybrid, developed by [PERSON] and [PERSON] [35], which combines the global search ability of particle swarm optimization (PSO) with the local search capability of the quasi-Newton method. The PSO algorithm runs until convergence, and then the quasi-Newton method is applied to the best solution found. This hybrid approach has proven highly effective in solving the economic dispatch problem. Another hybrid approach, GRPSO, developed by [PERSON] et al. [36], merges the quasi-Newton method and PSO to leverage the strengths of both algorithms. The hybrid begins by running the quasi-Newton method from a random starting point until a local minimum is achieved, then initiates a population of random particles, and runs the PSO algorithm until the global best particle surpasses the quasi-Newton method result. The PSO algorithm is then stopped, and the global best particle is used to restart the quasi-Newton method. This hybrid approach benefits from the local search capability of the quasi-Newton method and the global search ability of the PSO algorithm, enabling it to escape local minima and explore different regions of the search space. Results show that GRPSO is more efficient and reliable than other methods compared to it [36]. While global search methods, such as PSO, are effective at finding the global optimum in large search spaces, they can be less efficient when applied to local search. Gradient-based methods, on the other hand, can be faster and more efficient in finding the optimal solution in a local search space. However, gradient-based methods can struggle when there are multiple local minima or the function being optimized is non-convex. As a result, finding the global optimum using only gradient-based methods can be challenging. To address these issues, a hybrid approach that combines the strengths of both global and local search methods is often employed.