On the Convergence of Biogeography-Based Optimization for Binary Problems

Haiping Ma,Minrui Fei,Dan Simon

doi:10.1155/2014/147457

Abstract

Biogeography-based optimization (BBO) is an evolutionary algorithm inspired by biogeography, which is the study of the migration of species between habitats. A finite Markov chain model of BBO for binary problems was derived in earlier work, and some significant theoretical results were obtained. This paper analyzes the convergence properties of BBO on binary problems based on the previously derived BBO Markov chain model. Analysis reveals that BBO with only migration and mutation never converges to the global optimum. However, BBO with elitism, which maintains the best candidate in the population from one generation to the next, converges to the global optimum. In spite of previously published differences between genetic algorithms (GAs) and BBO, this paper shows that the convergence properties of BBO are similar to those of the canonical GA. In addition, the convergence rate estimate of BBO with elitism is obtained in this paper and is confirmed by simulations for some simple representative problems.

Highlights

Mathematical models of biogeography describe the immigration and emigration of species between habitats
The convergence properties and convergence rates derived here are not surprising in view of previous evolutionary algorithms (EAs) convergence results, but this paper represents the first time that such results have been formalized for Biogeography-based optimization (BBO)
In this paper we modeled BBO as a homogeneous finite Markov chain to study convergence, and we obtained new theoretical results for BBO

Summary

Introduction

Mathematical models of biogeography describe the immigration and emigration of species between habitats. We present a review of the biogeography-based optimization (BBO) algorithm with migration and mutation (Section 2.1) and provides a review of the Markov transition probability of BBO populations (Section 2.2). Based on the previously derived Markov chain model for BBO [11], the probability that migration results in the kth candidate solution yk are equal to xi at generation t + 1 is given by. Where 10 is the indicator function on the set 0 (i.e., 10(a) = 1 if a = 0, and 10(a) = 0 if a ≠ 0), r denotes the index of the candidate solution feature (i.e., the bit number), λk denotes the immigration rate of candidate solution yk, μj denotes the emigration rate of candidate solution xj, and Vj denotes the number of xj individuals in the population. Where q is the number of bits in each candidate solution and Hij represents the Hamming distance between bit strings xi and xj

Convergence of Biogeography-Based Optimization

Simulation Results

Conclusion