A derivation of the number of minima of the Griewank function

The Griewank function is commonly used to test the ability of different solution procedures to find local optima. It is important to know the exact number of minima of the function to support its use as a test function. However, to the best of our knowledge, no attempts have been made to analytically derive the number of minima. Because of the complex nature of the function surface, a numerical method is developed to restrict domain spaces to hyper-rectangles satisfying certain conditions. Within these domain spaces, an analytical method to count the number of minima is derived and proposed as a recursive functional form. The numbers of minima for two search spaces are provided as a reference. The Griewank function [1] has been widely used to test the convergence of optimization algorithms [2–15] because its number of minima grows exponentially as its number of dimensions increases [7,14]. The function is defined as follows: f n ð~ xÞ ¼ 1 4000 X n i¼1 x 2 i À Y n i¼1 cos x i ffi ffi i p þ 1; within ½À600; 600Š n where n is the number of dimensions of the function. The global minimum is located at ~ 0 with a value of 0. The actual number of minima may not be important when global optimization is performed, but it needs to be known to test techniques that search for local optima. Most studies vaguely mention the number of minima of the Griewank function [7–9], and, to the best of our knowledge, no analytical derivation to determine it has been given in the literature. Knowing the number of minima is critical if the Griewank function serves as the basis for evaluating algorithms designed to find local minima as well as global ones (i.e., multi-modal optimization). In some cases [14], the number of solutions given is inconsistent with analytical results. For example, [14] compared the ability of NichePSO, nbest PSO, lbest PSO, sequential niching, and deterministic crowding based on the number of minima found through numerical searches. However, further work with another algorithm has found a different number of solutions than found by [14]. In order to address this issue and provide a consistent basis for comparing algorithms, this paper analytically derives the number of minima of the Griewank function. We develop an approach in three basic steps. First, we restrict the search space to a hyperrectangle. Second, we show …