Symmetry-Based Convergence Theory for Particle Swarm Optimization: From Heuristic to Provably Convergent Optimization

This study establishes a rigorous theoretical framework for Particle Swarm Optimization (PSO) convergence by introducing a novel symmetry assumption governing the algorithm’s stochastic components and a monotonicity condition between function values and Euclidean distance to the global optimum. Under this assumption, we prove linear convergence in expectation and almost sure linear convergence for a modified PSO algorithm with symmetric zero-mean random coefficients when parameters satisfy the explicit condition $w+{\displaystyle \frac{8(c_1^2+c_2^2){\sigma }_r^2}{1-w}}<1$. This provides the first closed-form relationship between inertia weight w, learning factors $c_1,c_2$, and random variance ${\sigma }_r^2$ that guarantees convergence. Building on this theoretical foundation, we develop three hierarchical applications: (1) static parameter design that replaces empirical tuning with theoretical calculation from desired convergence rates; (2) symmetric random factor optimization that eliminates directional bias and stabilizes velocity dynamics while preserving exploration variance; and (3) dynamic adaptive strategies that adjust parameters in real-time based on particle dispersion feedback. By bridging the gap between empirical performance and theoretical guarantees, this work transforms PSO from an empirically driven heuristic into a provably convergent optimization tool with rigorous performance guarantees for objective functions satisfying strict monotonicity between fitness and distance to the optimum (e.g., strictly convex functions).