Particle swarm optimization (PSO) is a swarm intelligence algorithm and has good search performance and simplicity in implementation. Because of its properties, PSO has been applied to various optimization problems. However, the search performance of the classical PSO (CPSO) depends on reference frame of solution spaces for each objective function. CPSO is an invariant algorithm through translation and scale changes to reference frame of solution spaces but is a rotationally variant algorithm. As such, the search performance of CPSO is worse in solving rotated problems than in solving non-rotated problems. In the reference frame invariance, the search performance of an optimization algorithm is independent on rotation, translation, or scale changes to reference frame of solution spaces, which is a property of preferred optimization algorithms. In our previous study, piecewise-linear particle swarm optimizer (PPSO) has been proposed, which is effective in solving rotated problems. Because PPSO particles can move in solution spaces freely without depending on the coordinate systems, PPSO algorithm may have rotational invariance. However, theoretical analysis of reference frame invariance of PPSO has not been done. In addition, although behavior of each particle depends on PPSO parameters, good parameter conditions in solving various optimization problems have not been sufficiently clarified. In this paper, we analyze the reference frame invariance of PPSO theoretically, and investigated whether or not PPSO is invariant under reference frame alteration. We clarify that control parameters of PPSO which affect movement of each particle and performance of PPSO through numerical simulations.

In this paper, we propose a new paradigm of deterministic PSO, named piecewise-linear particle swarm optimizer (PPSO). In PPSO, each particle has two search dynamics, a convergence mode and a divergence mode. The trajectory of each particle is switched between the two dynamics and is controlled by parameters. We analyze convergence condition of each particle and investigate parameter conditions to allow particles to converge to an equilibrium point through numerical experiments. We further compare solving performances of PPSO. As a result, we report here that the solving performances of PPSO are substantially the same as or superior to those of PSO.

In this article, we consider a hysteresis oscillator which includes periodic thresholds. This oscillator relates to a model of human's sleep-wake cycles. Deriving a one dimensional return map rigorously, we can clarify existence regions of various periodic attractors in some parameter subspace. Also, we clarify co-existence regions of periodic attractors and existence regions of quasi-periodic attractors. Some of theoretical results are confirmed by laboratory measurements.