In our previous study, a piecewise-linear particle swarm optimizer (PPSO) has been proposed. The dynamics of each particle in PPSO can be controlled by two thresholds for convergence and divergence modes. Each threshold can be determined by the particle's trajectory and the system parameters. However, the conditions of these thresholds for finding desired solutions have not been clarified. In this paper, these conditions are investigated by the theoretical analysis and numerical experiments. Furthermore, the solving performances for threshold patterns are compared in the numerical experiments.