A Region Convergence Analysis for Multi-mode Stochastic Optimization Based on Double-Well Function.

In multi-mode heuristic optimization, the output fitness of an algorithm cannot converge to the global optimal value if its search points have not converged to the region with optimal solution. Generally, more samplings and more converged points in this optimal region may result in a higher probability of fitness convergence toward the optimal value. However, studies focus mainly on fitness convergence rather than region convergence (RC) of search points. This is partly because, for most objective functions, it is usually hard to track the region of search points in dynamic optimization. To remedy this, a novel analysis method is proposed using the double-well function (DWF), since it has a unique fitness landscape that makes it convenient to trace these points. First, a mathematical analysis of the DWF is given to explore its landscape. Then, RC is defined and discussed using DWF. On these bases, experiments are conducted and analyzed using Particle Swarm Optimization (PSO), and much useful information about its RC is revealed. Besides, this method can be used to analyze the RC of similar optimization algorithms as well.