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Particle swarm optimization is a population-based stochastic optimization algorithm that searches for optimal solutions by simulating the social behavior of a population of candidate solutions, called particles, moving through a multidimensional search space. Each particle maintains a position vector representing a candidate solution, a velocity vector determining its direction and step size, and memory of the best fitness position it has personally encountered and the best position discovered by any particle in its social neighborhood ^[1]. At each iteration, particle velocities are updated through a combination of inertial momentum, cognitive attraction toward the particle's own historical best position, and social attraction toward the global or neighborhood best position, weighted by acceleration coefficients and random stochastic factors that introduce exploration diversity. The position is then updated by integrating the new velocity, and the process repeats until a convergence criterion is met. The algorithm does not require gradient information, making it applicable to non-differentiable, discontinuous, or noisy objective functions, and its convergence behavior is governed by the balance between exploration (driven by random factors and inertia weight) and exploitation (driven by cognitive and social attraction to known good regions) ^[2]. The theoretical foundation treats the swarm as a dynamical system whose stability conditions depend on the constriction factor or inertia weight, and the algorithm's convergence to a local optimum is guaranteed under specific parameter regimes ^[3].