  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.

Neural and Evolutionary Computing