Stability analysis of chemotaxis dynamics in bacterial foraging optimization over multi-dimensional objective functions

Abstract

Bacterial foraging optimization (BFO) has been proved to be an efficient optimization method and successfully applied to a variety of fields in the real world. In BFO, the chemotaxis process is a complex and close combination of swimming and tumbling and plays a crucial role in searching better solutions. A previous study has modeled the dynamics of the chemotaxis mechanism mathematically and investigated the stability and convergence behavior of the chemotaxis dynamics over the one-dimensional objective function by Lyapunov stability theorem. However, this study appears to be very limited from a practical point of view, and how to extend their study to the multi-dimensional objective function is a challenge. To solve it, we present a stability analysis of chemotaxis dynamics in BFO over the multi-dimensional objective function in this paper. First, the general mathematical model of the chemotaxis mechanism over the multi-dimensional objective function is created. Secondly, this paper uses the general descent search to analyze the general mathematical model and points out two necessary conditions for avoiding the bacterium to trap into a non-optimal solution. And then, the stability and convergence of the chemotaxis dynamics, represented by the general mathematical model, are proved by using Lyapunov stability theorem. Finally, empirical research is conducted to validate the above theoretical analysis.