Global convergence analysis of the bat algorithm using a markovian framework and dynamical system theory

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 and to identify the possible effect of parameter settings on convergence, we have designed an updated model in terms of a dynamic matrix. Subsequently, we have used the stability theory of discrete-time dynamical systems to obtain the stable parameter ranges for the algorithm. Furthermore, we use some benchmark functions to demonstrate that BA can indeed achieve global optimality efficiently for these functions.