Using Double Well Function as a Benchmark Function for Optimization Algorithm

The double well function (DWF) is an important theoretical model originating from quantum physics and has been used to describe the energy constraint problem in quantum mechanics and structural chemistry. Although its form may vary, the DWF has two different local minima in the one-dimensional case, and the number of local minima increases as its dimension grows. As a multi-stable function, the DWF is assumed to be a potential candidate for testing the performance of the heuristic optimization algorithm, which aims to seek the global minimum. To verify this idea, a typical DWF is employed in this paper, and a mathematical analysis is presented herein, and its properties as a benchmark function are discussed in different cases. In addition, we conducted a set of experiments utilizing a few optimization algorithms, such as the multi-scale quantum harmonic oscillator algorithm and covariance matrix adaptation evolution strategy; thus the analysis has been illustrated by the results of the numerical experiment. Moreover, to guide the design of an ideal DWF used as a benchmark function in experiment, different values of the decisive parameters were tested corresponding to our analysis, and some useful rules were given based on the discussion of the results.