A New Scope and Domain Measure Comparison Method for Global Convergence Analysis in Evolutionary Computation

Convergence analysis is a fundamental research topic in evolutionary computation (EC). The commonly used analysis method models the EC algorithm as a homogeneous Markov chain for analysis, which is not always suitable for different EC variants, and also sometimes causes misuse and confusion due to their complex process. In this article, we categorize the existing researches on convergence analysis in EC algorithms into stable convergence and global convergence, and then prove that the conditions for these two convergence properties are somehow mutually exclusive. Inspired by this proof, we propose a new scope and domain measure comparison (SDMC) method for analyzing the global convergence of EC algorithms and provide a rigorous proof of its necessity and sufficiency as an alternative condition. Unlike traditional methods, the SDMC method is straightforward, bypasses Markov chain modeling, and minimizes errors from misapplication as it only focuses on the measure of the algorithm's search scope. We apply SDMC to two algorithm types that are unsuitable for traditional methods, confirming its effectiveness in global convergence analysis. Furthermore, we apply the SDMC method to explore the gene targeting mechanism's impact on the global convergence in large-scale global optimization, deriving insights into how to design EC algorithms that guarantee global convergence and exploring how theoretical analysis can guide EC algorithm design.