Convergence analysis for a class of continuous single-step meta-heuristic algorithms based on Harris chain

As one of the most important properties to measure the optimization performance of a meta-heuristic algorithm (MA), the convergence property has been widely concerned and studied. So far, most theoretical research in this field has mainly focused on specific MAs, and the corresponding Markov chain theory has also been mainly utilized to analyze the MAs with discrete finite states. How to further investigate the convergence of a class of continuous single-step MAs from the perspective of theoretical analysis still needs in-depth and detailed exploration. In this paper, for a class of single-step MAs, the sampling distribution convergence and global convergence are elaborately analyzed based on Harris chain. Firstly, based on the similarities of the formulation of solution update operator, a class of single-step MAs are defined. Then, the corresponding transition kernel of each search agent position is derived, based on which a sufficient condition for sampling distribution convergence and global convergence is proposed and rigorously proved through Harris chain theory. Finally, some case studies are performed to verify the rationality and effectiveness of the proposed definitions, conditions, and theorems. On the basis of these cases, some meaningful conclusions are drawn to provide guidance for leveraging existing single-step MAs and designing efficient single-step MAs.