Particle Swarm Optimization (PSO) is one of the most effective optimization methods for the black-box optimization problem. PSO involves a large number of particles sharing information with each other to search for the optimal solution. The method in which a large number of search individuals cooperate to search for the optimal solution is called swarm intelligence optimization. In the group intelligence optimization, the balance between exploration and exploitation is important. However, in PSO, it is unclear to what extent each parameter affects exploration and exploitation. Therefore, we proposed a deterministic PSO without probabilistic elements and analyzed the dynamics of PSO using dynamical systems theory. Each particle in deterministic PSO has its motion determined by its eigenvalue. In order to make this motion clearer, a canonical deterministic PSO on a regularized phase space was proposed. The results of these analyses clarified what is attributed to the parameters for exploration and exploitation, i.e., global and local search capabilities. Based on this fact, we proposed a nonlinear map optimization (NMO) with improved local search capability. In this paper, we present the background of our proposal and consider the solution-search capability of nonlinear map optimization.
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