Analysis of Selection Algorithms: A Markov Chain Approach

Abstract

A Markov chain framework is developed for analyzing a wide variety of selection techniques used in genetic algorithms (GAs) and evolution strategies (ESs). Specifically, we consider linear ranking selection, probabilistic binary tournament selection, deterministic s-ary (s = 3,4, …) tournament selection, fitness-proportionate selection, selection in Whitley's GENITOR, selection in (μ, λ)-ES, selection in (μ + λ)-ES, (μ, λ)-linear ranking selection in GAs, (μ + λ)-linear ranking selection in GAs, and selection in Eshelman's CHC algorithm. The analysis enables us to compare and contrast the various selection algorithms with respect to several performance measures based on the probability of takeover. Our analysis is exact—we do not make any assumptions or approximations. Finite population sizes are considered. Our approach is perfectly general, and following the methods of this paper, it is possible to analyze any selection strategy in evolutionary algorithms.

Implementation of a genetic algorithm for energy design optimization of livestock housing using a dynamic thermal simulator

“…Therefore 8 is the length of the population (L). The simulations were conducted varying the number of the population N (4,8,12,16,20,30, 60, 100, 150) and for various combinations of mutation probability (Pm) and crossover probability (Pc) respectively: 1/L -0; 0.5/L -0 ; 0 -0.9; 1 / L -0.9 ; 0.5 / L -0.9 and 0.1 / L -0.9. The combinations were selected so as to investigate the effects of the application of only crossover (for example the 0-0.9 combination), mutation alone (for example the 1/L -0 combination), or combinations of the two.…”

“…In some cases, empirical studies have been conducted (Wu. A. et al, 1997) or complex stochastic models like the Markov chains (Chakraborty U. et al 1996; Chakraborty U and Janikow C.Z., 2003) have been used.In this paper, which is inspired by a work of Kalyanmoy Deb (1998), we addressed the effects induced by five parameters of the GA: population size, crossover probability, mutation probability, encoding and reproduction strategy.…”

Some modifications of low-dimensional simplex evolution and their convergence

“…Markov chain is often used to analyse the convergence properties of EAs [3,2,16,20]. A most popular Markov chain analysis of genetic algorithms is made by Rudolph [19].…”

“…Therefore, for the population Markov chain { X(t)} ∞ t=0 generated by LDSE, the optimal state set * is a closed set. (2) Note that all of the reproduction operators in basic LDSE, including reflection, contraction and last struggle are linear. Once all individuals in the population are degenerated to a subspace of R n , then no offspring individual can get out of this subspace, even though the evolutionary process continues infinitely.…”

“…To reduce its required population size, we try to (1) use nonlinear operators to replace the linear ones, (2) implement simplex operations in low-dimensional spaces, and (3) enhance LDSE's local search ability. Specifically, the three techniques are called low-dimensional reproduction (LDR), normal struggle (NS) and variable dimension (VD).…”