Several Classic Test Functions for Optimization Algorithms

This section introduces several classic test functions widely used in optimization algorithm research, including the Sphere function, Schaffer function, Ackley function, Griewank function, and Rosenbrock function. These benchmark functions serve as standard tools for evaluating algorithm performance. The Sphere function, typically implemented as f(x) = Σx_i², represents a fundamental test for continuous optimization problems with a single global minimum. The Schaffer function features multiple local minima and a global minimum, making it ideal for testing an algorithm's ability to escape local optima. The Ackley function, characterized by its large flat region and narrow global minimum basin, is particularly useful for evaluating convergence speed and premature convergence prevention. The Griewank function, with its complex landscape of regularly distributed local minima, challenges an algorithm's search capability and dimensional scalability. The Rosenbrock function (also known as the banana function), characterized by a curved parabolic valley, is extensively used for testing nonlinear optimization algorithms' performance on non-convex problems. These classic test functions form an essential part of algorithm research, providing critical insights for understanding and assessing optimization algorithm capabilities through standardized mathematical formulations and implementation frameworks.