On the importance of the second largest eigenvalue on the convergence rate of genetic algorithms

Genetic algorithms are sometimes disparagingly denoted as just a fancier form of a plain, stupid heuristic. One of the main reasons for this kind of critique is that users believed a GA could not guarantee global convergence in a certain amount of time. Because the proof of global convergence of GAs using elitism has been performed elsewhere (13), in this work we want to extend previous work by J. Suzuki (15) and focus on the identification of the determinants that influence the convergence rate of genetic algorithms. The convergence rate of genetic algorithms is addressed using Markov chain analysis. Therefore, we could describe an elitist GA using mutation, recombination and selection as a discrete stochastic process. Evaluating the eigenvalues of the transition matrix of the Markov chain we can prove that the convergence rate of a GA is determined by the second largest eigenvalue of the transition matrix. The proof is first performed for diagonalizable transition matrices and then transferred to matrices in Jordan normal form. The presented proof allows a more detailed and deeper understanding of the principles of evolutionary search. As an extension to this work we want to encourage researchers to work on proper estimations of the second largest eigenvalue of the transition matrix. With a good approximation, the convergence behavior of GAs could be described more exactly and GAs would be one step ahead on the road to a fast, reliable and widely accepted optimization method.