Griewankfüggvény

The Griewank function is a non-convex, multimodal function commonly used for testing the performance of optimization algorithms. It is defined as:

f(x) = sum(x_i^2 / 4000) - product(cos(x_i / sqrt(i))) + 1

where x is a vector of n variables, and the summation and product are taken from i = 1 to n. The Griewank function has a global minimum at the origin (0, 0, ..., 0) where f(x) = 0. However, it possesses a large number of local minima, making it a challenging benchmark for optimization.

The function's behavior is characterized by its highly oscillatory nature. As the number of dimensions increases, the number of local minima grows exponentially. This complexity arises from the interplay between the quadratic term, which tends to pull the function towards the origin, and the cosine term, which introduces numerous local optima across the search space.

Many optimization algorithms, such as genetic algorithms, particle swarm optimization, and simulated annealing, have been tested on the Griewank function. Its ability to expose the weaknesses of algorithms that struggle with multimodal landscapes has made it a standard testbed for evaluating and comparing optimization techniques. Researchers often use the Griewank function to demonstrate the robustness and effectiveness of novel optimization approaches.