We present four abstract evolutionary algorithms for multiobjective optimization and theoretical results that characterize their convergence behavior. Thanks to these results it is easy to verify whether or not a particular instantiation of these abstract evolutionary algorithms offers the desired limit behavior. Several examples are given.
Evolutionary algorithms (EAs) are random optimization methods inspired by genetics and natural selection, resembling simulated annealing. We develop a method that can be used to find a meaningful tradeoff between the difficulty of the analysis and the algorithms' efficiency. Since the case of a discrete search space has been studied extensively, we develop a new stochastic model for the continuous n-dimensional case. Our model uses renewal processes to find global convergence conditions. A second goal of the paper is the analytical estimation of the computation time of EA with uniform mutation inside the (hyper)-sphere of volume 1, minimizing a quadratic function.
Adaptive evolutionary algorithms require a more sophisticated modeling than their static-parameter counterparts. Taking into account the current population is not enough when implementing parameter-adaptation rules based on success rates (evolution strategies) or on premature convergence (genetic algorithms). Instead of Markov chains, we use random systems with complete connections - accounting for a complete, rather than recent, history of the algorithm's evolution. Under the new paradigm, we analyze the convergence of several mutation-adaptive algorithms: a binary genetic algorithm, the 1/5 success rule evolution strategy, a continuous, respectively a dynamic (1+1) evolutionary algorithm.
We consider the finite homogeneous Markov chain induced by a class of one-dimensional asynchronous cellular automata-automata that are allowed to change only one cell per iteration. Furthermore, we confine to totalistic automata, where transitions depend only on the number of Is in the neighborhood of the current cell. We consider three different cases: (i) size of neighborhood equals length of the automaton; (ii) size of neighborhood two, length of automaton arbitrary; and (iii) size of neighborhood three, length of automaton arbitrary. For each case, the associated Markov chain proves to be ergodic. We derive simple-form stationary distributions, in case (i) by lumping states with respect to the number of Is in the automaton, and in cases (ii) and (iii) by considering the number of 0 1 borders within the automaton configuration. For the three-neighborhood automaton, we analyze also the Markov chain at the boundary of the parameter domain, and the symmetry of the entropy. Finally, we show that if the local transition rule is exponential, the stationary probability is the Boltzmann distribution of the Ising model
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.