A Perturbation Based Chaotic System Exploiting the Quasi-Newton Method for Global Optimization

Abstract

The chaotic system has been exploited in metaheuristic methods of solving continuous global optimization problems. Recently, the gradient method with perturbation (GP) was proposed, which was derived from the steepest descent method for the problem with additional perturbation terms, and it was reported that chaotic metaheuristics with the GP have good performances of solving some benchmark problems. Moreover, the sufficient condition of its parameter values was theoretically shown under which its updating system is chaotic. However, the sufficient condition of its chaoticity and the width of strange attractor around each local minimum, which are important properties for exploiting the chaotic system in optimization, deeply depend on the eigenvalues of the Hessian matrix of the objective function at the local minimum. Thus, if the eigenvalues of different local minima are widely different from each other, or if it is different in different problems, such properties can cause the difficulty of selecting appropriate parameter values for an effective search. Therefore, in this paper, we propose modified GPs based on the quasi-Newton method instead of the steepest descent method, where their chaoticities and the width of strange attractor do not depend on the eigenvalue of the Hessian matrix at any local minimum due to the scale invariant of the quasi-Newton method. In addition, we empirically demonstrate that the parameter selection of the proposed methods is easier than the original GP, especially with respect to the step-size, and the chaotic metaheuristics with the proposed methods can find better solutions for some multimodal functions.