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.

[1]  Andries Petrus Engelbrecht,et al.  Locating multiple optima using particle swarm optimization , 2007, Appl. Math. Comput..

[2]  Christian Jacob,et al.  Exploratory Toolkit for Evolutionary and Swarm-Based Optimization , 2010 .

[3]  Marco Locatelli,et al.  A Note on the Griewank Test Function , 2003, J. Glob. Optim..

[4]  Kevin D. Seppi,et al.  Adaptive diversity in PSO , 2006, GECCO '06.

[5]  Jacques Riget,et al.  A Diversity-Guided Particle Swarm Optimizer - the ARPSO , 2002 .

[6]  Kevin D. Seppi,et al.  Bayesian optimization models for particle swarms , 2005, GECCO '05.

[7]  J R Saunders,et al.  A particle swarm optimizer with passive congregation. , 2004, Bio Systems.

[8]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[9]  Andries Petrus Engelbrecht,et al.  Scalability of niche PSO , 2003, Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706).

[10]  Xiao Zhi Gao,et al.  A Hybrid Optimization Algorithm Based on Ant Colony and Immune Principles , 2007, Int. J. Comput. Sci. Appl..

[11]  Adnan Acan,et al.  Enhanced particle swarm optimization through external memory support , 2005, 2005 IEEE Congress on Evolutionary Computation.

[12]  T. Krink,et al.  Particle swarm optimisation with spatial particle extension , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[13]  Gisbert Schneider,et al.  Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training , 2006, BMC Bioinformatics.

[14]  A. Griewank.  Generalized descent for global optimization , 1981 .

[15]  J. Kennedy,et al.  Stereotyping: improving particle swarm performance with cluster analysis , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[16]  Wenjun Zhang,et al.  Dissipative particle swarm optimization , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).