Convergence And Stability Analysis Of Artificial Bee Colony Algorithm

Artificial bee colony algorithm(ABC)is a swarm intelligence optimization algorithm simulating bee colony cooperative foraging behavior proposed by Karaboga,a scholar in Turkey in 2005.In recent years,with the continuous research of experts and scholars,ABC algorithm is becoming more and more important in the field of intelligent algorithms because of its simple calculation and operation,less control parameters and easy implementation.At the same time,the application of the algorithm is becoming more and more widely in the field of function and engineering.With the further development of intelligent computing,people's requirements for algorithm become higher.Compared with other intelligent algorithms like ant colony algorithm and particle swarm optimization algorithm,the basic research of the mathematical theory of ABC algorithm is relatively low.the research of convergence is not thorough.the stability research is still blank.So it is urgent to study the mathematical theory foundation of ABC algorithm more deeply.In view of the convergence and stability of algorithm theory,this paper makes a deep research and proof for ABC algorithm.First,the convergence of ABC algorithm is proved by the combination of number and shape.Then this method is applied to an improved ABC algorithm—— GABC algorithm,which proves that this method is widely effective.Finally,the stability of ABC algorithm is proved by Lyapunov stability theory.The existing methods of convergence analysis for artificial bee colony algorithm(ABC)are based on the analysis of global convergence.But these convergence analysis methods can't show the convergence change in the convergence process of ABC.This paper adopts the method of combination of number and shape,and combines the objective function diagram to divide the convergence process of ABC into the global search stage and the optimal region search stage.Then the convergence process and changes of each stage are analyzed one by one based on the transferring character that the artificial bees follow a certain degree of average distribution.Finally,the convergence results and change of ABC are obtained.This method can clearly show the convergence advantages and defects of the ABC algorithm,and reveal the changing process of the convergence probability of the algorithm.The position updating formula of the GABC algorithm is similar to that of the ABC algorithm.Therefore,the convergence of the GABC algorithm is proved by the same method,and the wide applicability of this method is also shown.At present,the stability of intelligent algorithms is mostly prove by writing the characteristic equation of the algorithm position updating formula,calculating the solution of the characteristic equation,then using the sufficient and necessary conditions of the discrete system stability or the Routh criterion.But the control parameters of ABC algorithm are so few that it is difficult to write the characteristic equation of location update formula,so this method cannot be used.Using the basic definition of Lyapunov stability theory,this paper proves the stability of the ABC algorithm by proving the existence of the equilibrium state of the ABC algorithm,and in accordance with the requirements of the stability of Lyapunov stability and the uniform asymptotic stability.