In probabilistic structural mechanics, the Hasofer-Lind reliability index problem is a paradigmatic equality constrained problem of searching for the minimum distance from a point to a surface. In practical engineering problems, such surface is defined implicitly, requiring the solution of a boundary-value problem. Recently, it was proposed in the literature a hybrid micro-genetic algorithm (HmGA), with mixed real-binary genotype and novel deterministic operators for equality-constraint handling, namely the Genetic Repair and Region Zooming mechanisms (G. das Neves Carneiro and C. Concei\c{c}\~ao Ant\'onio, "Global optimal reliability index of implicit composite laminate structures by evolutionary algorithms", Struct Saf, vol. 79, pp. 54-65, 2019). We investigate the limit-behavior of the HmGA and present the convergence theorems for the algorithm. It is proven that Genetic Repair is a conditionally stable mechanism, and its modes of convergence are discussed. Based on a Markov chain analysis, the conditions for the convergence with probability 1 of the HmGA are given and discussed.

翻译：在概率结构力学中，Hasofer-Lind可靠度指标问题是一个典型的等式约束优化问题，旨在寻找从一点到曲面的最小距离。在实际工程问题中，该曲面通常隐式定义，需要求解边值问题。近期，文献提出了一种混合微遗传算法（HmGA），采用实值-二进制混合基因型及新型确定性算子处理等式约束，即遗传修复与区域缩放机制（G. das Neves Carneiro and C. Conceição António, "Global optimal reliability index of implicit composite laminate structures by evolutionary algorithms", Struct Saf, vol. 79, pp. 54-65, 2019）。本文研究了HmGA的极限行为，给出了算法的收敛性定理。证明了遗传修复是一种条件稳定机制，并讨论了其收敛模式。基于马尔可夫链分析，给出了HmGA以概率1收敛的条件并进行了讨论。