  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.
