Modified Perturbation-Based Chaotic System Using the Quasi-Newton Method with the Symmetric Rank-One Formula for Global Optimization

Recently, the chaotic systems have been exploited in metaheuristic methods of solving continuous global optimization problems. We focus on the perturbation-based chaotic system using the quasi-Newton method (PQN) based on the damped BFGS formula. The PQN has useful chaotic properties for solving the problem such that its chaoticity and the sizes of its strange attractors are independent of the objective function due to the scale invariance of the quasi-Newton method, which means that the balance between intensification and diversification of the search can be basically adjusted only by the parameter selection. In this paper, from a practical viewpoint, we introduce the PQN based on the SSR1 formula, which requires less computational time than the BFGS one, and we propose some techniques which make it easier to select parameter values of the PQN, an adaptive selection of its step-size and a scale transformation of the decision variables. In addition, by using the techniques, we derive another chaotic metaheuristic method of keeping the chaoticity of the PQN and deceasing gradually the sizes of strange attractors, which also make it easier to select the parameter selection of the metaheuristic method. Through numerical experiments, we compare the existing and proposed metaheuristic methods with modified PQNs by applying them to some multimodal functions.