Abstract: The Griewank function is a typical multimodal benchmark function, composed of a quadratic convex function and an oscillatory nonconvex function. The comparative importance of Griewank’s two major parts alters in different dimensions. Different from most test functions, an unusual phenomenon appears when optimizing the Griewank function. The Griewank function first becomes more difficult and then becomes easier to optimize with the increase of dimension. In this study, from the methodology perspective, this phenomenon is explained by structural, mathematical, and quantum analyses. Furthermore, frequency transformation and amplitude transformation are implemented on the Griewank function to make a generalization. The multi-scale quantum harmonic oscillator algorithm (MQHOA) with quantum tunnel effect is used to verify its characteristics. Experimental results indicate that the Griewank function’s two-scale structure is the main reason for this phenomenon. The quantum tunneling mechanism mentioned in this paper is an effective method which can be generalized to analyze the generation and variation of solutions for numerous swarm optimization algorithms.

Key words: Griewank, Two-scale structure, Multi-scale quantum harmonic oscillator algorithm, Quantum tunnel effect

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