A Bound on GA Convergence Using Lyapunov-like Functions

We extend the work done in modeling the Genetic Algorithm (GA) from fundamental principles [8] by calculating a bound for convergence time for two selection schemes. When a Lyapunov function is used to show convergence, we demonstrate a method whereby the same Lyapunov function can give convergence time. We then calculate that bound for proportional selection and ranking selection. Our result is important because to date there has only been a described expected convergence for the GA given that the assumptions for the Schema Theorem [3] hold [1]. Key-Words: Complexity, Convergence Bound, Genetic Algorithm, Evolutionary Computation 1 Background Prior work that described the expected convergence rate of the GA was based on the many assumptions underlying the Schema Theorem [3], [1]. These assumptions are not universally agreed upon. Consequently, attempts at modeling the GA from fundamental principles have been undertaken. Using techniques from dynamical systems Vose showed that certain genetic algorithms are focused [8]. We use the same techniques to give actual bounds on convergence times. Vose showed that if a Lyapunov function is monotone, then an algorithm is focused. We show that if the rate of change of the Lyapunov function is monotone then convergence time can be bounded. And, we show that the rate of change of the Lyapunov function is indeed monotone for two specific selection schemes. 2 Convergence Time Bounded We assume a simple genetic algorithm with no crossover and no mutation. The genetic algorithm is modeled by specifying a search space Ω and a simplex Λ of population vectors, the vertices of Λ being the points of Ω. The algorithm is a map G from Λ to Λ. G is focused if it is continuously differentiable and for every p contained in Λ and for every p contained in Λ the following sequence converges: p, G(p), G(p), ... G will be broken down into a composition of two functions F and M, F being the selection scheme and M being a combination of mutation and crossover. For our results, we assume M is the identity function and then G is the same as F. If G is continuously differentiable and has finitely many fixed points, and if there is a continuous function φ satisfying x ≠ G(x) ⇒ φ(x) > φ(G(x)) then G is focused [8]. In this setting, φ is called a Lyapunov function. Theorem 1: Bounded Convergence Time If F is a continuously differentiable function from Λ to Λ, and F has finitely many fixed points, and φ is a Lyapunov function for F, then if φ(F(p))/φ(F(p)) ≥ 1 + k (except when F(p) = F(p)) where k > 0 does not depend on m, then the convergence time of F is bounded.