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 hyperrectangles 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. (C) 2008 Elsevier Inc. All rights reserved.