Particle Swarm Optimization on Sphere Function and Benchmark Analysis

Particle Swarm Optimization (PSO), as a swarm intelligence optimization algorithm, is widely used to solve various complex function optimization problems. This article analyzes PSO's performance characteristics on standard benchmark functions including Sphere, Rosenbrock, Ackley, and Griewank.

### Sphere Function Optimization The Sphere function represents one of the simplest convex optimization problems, with its global optimum located at the origin. PSO typically demonstrates stable performance on Sphere function since the gradient direction directly points toward the optimal solution, enabling rapid particle convergence. Algorithm parameters (such as inertia weight and learning factors) significantly impact convergence speed. Smaller inertia weights facilitate local search, while larger values promote global exploration. In code implementation, setting w=0.7 for inertia weight with c1=c2=1.5 as learning factors often yields good balance.

### Rosenbrock Function Optimization The Rosenbrock function, often called the "banana function," has its optimum located in a long, narrow, flat valley. Optimizing Rosenbrock presents greater challenges for PSO as particles tend to oscillate within the valley, hindering rapid convergence. Implementation typically requires balancing global and local search through dynamic inertia weights (e.g., linearly decreasing from 0.9 to 0.4) or adaptive learning strategies to improve optimization efficiency.

### Ackley Function Optimization Ackley is a multimodal function containing numerous local minima that can trap optimization algorithms in suboptimal solutions. When handling Ackley function, PSO requires strong global exploration capabilities. Code implementations often employ larger initial particle velocities and diversity maintenance strategies (such as random reinitialization or velocity clamping) to prevent premature convergence.

### Griewank Function Optimization The Griewank function features numerous local minima but relatively smooth regions near the global optimum. Optimizing Griewank requires PSO to balance global exploration for escaping local optima with fine-tuning near the solution. Incorporating local search strategies through neighborhood topology optimization (like ring or von Neumann structures) can significantly enhance PSO performance.

### Conclusion PSO demonstrates varying performance across different test functions, requiring problem-specific parameter and strategy adjustments. The Sphere function serves well for validating basic algorithm performance, while Rosenbrock, Ackley and Griewank functions better test algorithm robustness and adaptability. Proper adjustment of inertia weights, learning factors, and swarm topology through systematic parameter tuning in code implementations can effectively improve PSO's optimization effectiveness.