A SUFFICIENT CONDITION FOR CHAOS IN THE GRADIENT MODEL WITH PERTURBATION METHOD FOR GLOBAL OPTIMIZATION

Abstract

The chaotic system has been exploited in metaheuristic methods of solving global optimization problems having a large number of local minima. In those methods, the selection of chaotic system is significantly important to search for solutions extensively. Recently, a novel chaotic system, the gradient model with perturbation methods (GP), was proposed, which can be regarded as the steepest descent method for minimizing an objective function with additional perturbation terms, and it is reported that chaotic metaheuristic method with the GP model has a good performance of solving some benchmark problems through numerical experiments. Moreover, a sufficient condition of parameter was theoretically shown for chaoticity in a simplified GP model where the descent term for the objective function is removed from the original model. However, the shown conditions does not provide enough information to select parameter values in the GP model for metaheuristic methods. Therefore, in this paper, we theoretically derive a sufficient condition under which the original GP model is chaotic, which can be usefully exploited for an appropriate selection of parameter values. In addition, we examine the derived sufficient condition by calculating the Lyapunov exponents of the GP model, and analyze its bifurcation structure through numerical experiments.