Convergence Velocity Of Evolutionary Algorithm With Self-adaptation

A stochastic Lyapunov function was used to assess the convergence velocity of a simple evolutionary algorithm with self-adaptation, which searches for a maximum of a “fitness” function. This algorithm uses two types of parameters: “fitness” parameters belonging to the domain of the function, and strategy parameters, which control changes of fitness parameters. It was shown that the convergence velocity of the evolutionary algorithm with selfadaptation is exponential, similar to the convergence velocity of the optimal deterministic algorithm, the Fibonacci search, on the class of unimodal functions. 1 OPTIMAL DETERMINISTIC SEARCH ALGORITHM Let [ ] , K a b be a class of unimodal functions [ ] : , f a b → . Let be a set of n-point sequential deterministic algorithms n P { } n p . A n-point algorithm n p searches for maximum of a function [ ] , f K a b ∈ by sequentially selecting k x , based on calculation of values ( ) 1 , , ( ) 1 k f x f ... x − , where . For any k n ≤ [ ] , and n f K a ∈ n b p ∀ let the error of an algorithm p on a function f be defined as ( ) , n n f p f x x δ = − where f x is a value where the function f has maximum ( ) [ ] ( ) , max f y a b f x f ∈ = n y . A guaranteed error of the algorithm p on the class of functions [ ] , K a b is defined as ( ) [ ] , sup f K a b p δ ∈ Λ =