# The Global Convergence Analysis of the Bat Algorithm Using a Markovian Framework and Dynamical System Theory

## TL;DR

This paper provides a theoretical analysis of the convergence and stability of the bat algorithm using Markov chain and dynamical system theories, demonstrating its ability to achieve global optimality on benchmark functions.

## Contribution

It introduces a Markov model for the bat algorithm and proves its global convergence, also designing an updated model to enhance convergence performance.

## Key findings

- BA can achieve global convergence on benchmark functions
- The stability parameter ranges for BA are identified
- BA demonstrates efficient global optimality on tested functions

## Abstract

The bat algorithm (BA) has been shown to be effective to solve a wider range of optimization problems. However, there is not much theoretical analysis concerning its convergence and stability. In order to prove the convergence of the bat algorithm, we have built a Markov model for the algorithm and proved that the state sequence of the bat population forms a finite homogeneous Markov chain, satisfying the global convergence criteria. Then, we prove that the bat algorithm can have global convergence. In addition, in order to enhance the convergence performance of the algorithm, we have designed an updated model using the dynamical system theory in terms of a dynamic matrix, and the parameter ranges for the algorithm stability are then obtained. We then use some benchmark functions to demonstrate that BA can indeed achieve global optimality efficiently for these functions.