A Markov chain analysis on simple genetic algorithms

Conditions for the convergence of evolutionary algorithms

“…In the past years, good results have been obtained about the convergence of EAs in finite space and discrete time, including both time-homogeneous transitions and time-inhomogeneous transitions [3], [4], [5], [6], [7]. The convergence theory of EAs beyond the finite space and discrete time has also been discussed [7].…”

Stochastic Optimization

“…It is possible to construct an N P × N P Markov transition matrix P, where the ijth element is the probability of transitioning from the ith population of N chromosomes to the jth population of the same size. These elements depend in a nontrivial way on N, the crossover rate, and the mutation rate; the number of elite chromosomes is assumed to be N e = 1 (Suzuki, 1995). Let p k be an N P × 1 vector having jth component p k (j) equal to the probability that the kth generation will result in population j, j = 1, 2, … , N P .…”

“…Consider a binary bit-coded GA with a population size of N and a string length of B bits per chromosome. Then the total number of possible unique populations is: Suzuki, 1995). It is possible to construct an N P × N P Markov transition matrix P, where the ijth element is the probability of transitioning from the ith population of N chromosomes to the jth population of the same size.…”

A Genetic Algorithm for Searching the Shortest Lattice Vector of SVP Challenge

“…Actually, earlier elitist strategies [39][10] are adopted to meet conditions, under which GAs are guaranteed to converge. Among the analysis schemes, the Markov chain analysis, which models the process of genetic algorithms as a finite Markov chain, is an effective tool to prove rigorously the convergence of GAs as in [9,4,38,31]. For a survey of genetic algorithms, you can refer to [36,8].…”