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.

翻译：收敛性分析是进化计算领域的基础研究课题。常用的分析方法将进化算法建模为齐次马尔可夫链进行分析，这种方法并不总是适用于不同的进化算法变体，且其复杂过程有时会导致误用和混淆。本文将现有关于进化算法收敛性分析的研究归类为稳定收敛与全局收敛，并证明这两种收敛性质的条件在某种程度上是互斥的。受此证明启发，我们提出一种新的范围与域测度比较方法，用于分析进化算法的全局收敛性，并严格证明了其作为替代条件的必要性与充分性。与传统方法不同，SDMC方法直接简洁，绕过了马尔可夫链建模，并且由于仅关注算法搜索范围的测度，从而最大限度地减少了误用带来的误差。我们将SDMC应用于两类不适合传统方法的算法，验证了其在全局收敛性分析中的有效性。此外，我们应用SDMC方法探究基因靶向机制在大规模全局优化中对全局收敛性的影响，从而获得关于如何设计能保证全局收敛的进化算法的见解，并探索理论分析如何指导进化算法的设计。