Chapter 4: Theoretical Foundations of GAs

Section 4.3: Exact Mathematical Models of Simple GAs

Vose and Liepins, 1991

Algorithm

1. Calculate f(x)
2. Choose with replacement 2 parents using fitness based selection
3. Perform single point crossover with p_c, select one offspring randomly to add to the next generation
4. Mutate each bit with p_m
5. Until the next generation is complete, go to step 2
6. Go to step 1

Definitions

• Use a bit representation of length L for each chromosome
• Encode integers from 0 through 2^L - 1
• Let p_i(t) be the proportion of string i in generation t
• Let p(t) be the vector of all p_i(t)
• Let f_i(t) be the fitness of string i in generation t
• Let F(t) be the fitness matrix. F_i,i(t) = f_i(t) and F_i,j(t) = 0 for i ≠ j
• Let s_i(t) be the probability that string i is selected in generation t
• Let s(t) be the vector of all s_i(t), s(t) = F(t) p(t) / Σ F_j,j(t) p_j(t)
• Let G be an operator such that s(t+1) = G(s(t))
• Vose analyzes G in the context of an infinite population with a simplified GA (e.g. mutation only, selection only, etc.)
• Vose is interested in finding the fixed points, e.g. G(s(t)) yields s(t)

Nix and Vose, 1991

Overview

• The goal is to model a GA with a finite population
• Use Markov Chains where the probability of the state at time t is solely a function of the state at time t-1
• Let a state be a particular finite population
• Let Z be a matrix where each column is a population vector
• Let Q be a matrix that shows the probability of going from any one state to any other state

Example

• Let the length of a chromosome, L = 2
• Let the population size, n = 2
• P₀ = {00, 00}
• P₁ = {00, 01}
• P₂ = {00, 10}
• P₃ = {00, 11}
• P₄ = {01, 01}
• P₅ = {01, 10}
• P₆ = {01, 11}
• P₇ = {10, 10}
• P₈ = {10, 11}
• P₉ = {11, 11}
• Z is a 4 by 10 matrix
• Q is a 10 by 10 matrix

Results

• For large n, the infinite population model closely mimics the finite population model
• If the GA has one or more fixed points, the GA asymptotically spends all of its time at these fixed points

Section 4.4: Statistical Mechanical Approaches

• The idea is to use macro statistics such as the mean fitness value to understand the system