An analysis of the behavior of simplified evolutionary algorithms on trap functions

Methods are developed to numerically analyse an evolutionary algorithm that applies mutation and selection on a bit-string representation to find the optimum for a bimodal unitation function called a trap function. This research bridges part of the gap between the existing convergence velocity analysis of strictly unimodal functions and global convergence results assuming the limit of infinite time. As a main result of this analysis, a new so-called (1: )evolutionary algorithm is proposed, which generates offspring using individual mutation rates . While a more traditional evolutionary algorithm using only one mutation rate is not able to find the global optimum of the trap function within an acceptable (non-exponential) time, our numerical investigations provide evidence that the new algorithm overcomes these limitations. The analysis tools used for the analysis, based on absorbing Markov chains and the calculation of transition probabilities, are demonstrated to provide an intuitive and useful method for investigating the capabilities of evolutionary algorithms to bridge the gap between a local and a global optimum in bimodal search spaces. Index Terms Evolutionary algorithm, genetic algorithm, convergence velocity, trap functions, absorption time, mutation. Siegfried Nijssen is with the Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, NL-2333 CA, The Netherlands. E-mail: [email protected] Thomas Bäck is with the LIACS and NuTech Solutions GmbH, Martin-Schmeisser-Weg 15, D-44227, Dortmund, Germany. E-mail: [email protected] 2