Testing Four Standard Benchmark Functions

Particle Swarm Optimization (PSO) is a classic swarm intelligence algorithm widely used for solving various optimization problems. To validate the optimization performance of PSO, standard benchmark functions are typically employed in experiments. Below are four commonly used benchmark functions along with analysis methods using convergence curves to evaluate algorithm effectiveness.

Sphere Function: This is the simplest convex function used to test algorithm convergence speed and precision. Its global optimum is located at the origin, making it suitable for verifying PSO's ability to quickly locate optimal solutions. In code implementation, the function can be defined as f(x) = Σx_i², where the optimal value is 0 at position (0,...,0). Rastrigin Function: Characterized by numerous local optima, this function tests an algorithm's ability to escape local minima. Its high optimization difficulty makes it ideal for evaluating global search capabilities. The mathematical formulation f(x) = 10n + Σ[x_i² - 10cos(2πx_i)] requires careful parameter tuning in PSO implementation. Ackley Function: Featuring flat regions and local minima, this function is commonly used to test algorithm adaptability in complex environments. The implementation involves handling exponential terms and cosine oscillations in f(x) = -20exp(-0.2√(1/nΣx_i²)) - exp(1/nΣcos(2πx_i)) + e + 20. Rosenbrock Function: With a narrow, curved valley containing the global optimum, this function tests local search capabilities. The banana-shaped valley in f(x) = Σ[100(x_{i+1} - x_i²)² + (x_i - 1)²] requires precise step-size control in PSO velocity updates.

Experimental verification of PSO performance typically involves these steps: Parameter Configuration: Set swarm size, inertia weight, learning factors, and velocity limits to ensure reasonable search space exploration. Code implementation often uses population sizes of 20-50 particles and adaptive inertia weights. Iterative Computation: Execute PSO iterations while recording best fitness values per generation to observe optimization progress. The algorithm update equations include velocity calculation: v_i(t+1) = w*v_i(t) + c1*r1*(pbest_i - x_i(t)) + c2*r2*(gbest - x_i(t)). Convergence Curve Plotting: Visualize convergence curves for different test functions to compare optimization speed and solution quality. MATLAB or Python implementations typically use plot() functions with generation counts on x-axis and fitness values on y-axis. Performance Analysis: Compare optimization results across functions to evaluate PSO's stability and adaptability. Statistical measures like mean best fitness and standard deviation over multiple runs provide quantitative assessments.

This methodology provides intuitive demonstration of PSO performance across various optimization problems, offering valuable references for practical applications.