Particle swarm optimizer (PSO) is a stochastic global optimization technique based on a social interaction metaphor. Because of the complexity, dynamics and randomness involved in PSO, it is hard to theoretically analyze the mechanism on which PSO depends. Statistical results have shown that the probability distribution of PSO is a truncated triangle, with uniform probability across the middle that decreases on the sides. The "truncated triangle" is also called the "Maya pyramid" by Kennedy. However, very little is known regarding the sampling distribution of PSO in itself. In this paper, we theoretically analyze the "Maya pyramid" without any assumption and derive its computational formula, which is actually a hybrid uniform distribution that looks like a trapezoid and conforms with the statistical results. Based on the derived density function of the hybrid uniform distribution, the search strategy of PSO is defined and quantified to characterize the mechanism of the search strategy in PSO. In order to show the significance of these definitions based on the derived hybrid uniform distribution, the comparison between the defined search strategies of the classical linear decreasing weight based PSO and the canonical constricted PSO suggested by Clerc is illustrated and elaborated.

Kennedy has proposed the bare bones particle swarm (BBPS) by the elimination of the velocity formula and its replacement by the Gaussian sampling strategy without parameter tuning. However, a delicate balance between exploitation and exploration is the key to the success of an optimizer. This paper firstly analyzes the sampling distribution in BBPS, based on which we propose an adaptive BBPS inspired by the cloud model (ACM-BBPS). The cloud model adaptively produces a different standard deviation of the Gaussian sampling for each particle according to the evolutionary state in the swarm, which provides an adaptive balance between exploitation and exploration on different objective functions. Meanwhile, the diversity of the swarms is further enhanced by the randomness of the cloud model itself. Experimental results show that the proposed ACM-BBPS achieves faster convergence speed and more accurate solutions than five other contenders on twenty-five unimodal, basic multimodal, extended multimodal and hybrid composition benchmark functions. The diversity enhancement by the randomness in the cloud model itself is also illustrated.

This paper presents an inertial estimator learning automata scheme by which both the short-term and long-term perspectives of the environment can be incorporated in the stochastic estimator – the long term information crystallized in terms of the running reward-probability estimates, and the short term information used by considering whether the most recent response was a reward or a penalty. Thus, when the short-term perspective is considered, the stochastic estimator becomes pertinent in the context of the estimator algorithms. The proposed automata employ an inertial weight estimator as the short-term perspective to achieve a rapid and accurate convergence when operating in stationary random environments. According to the proposed inertial estimator scheme, the estimates of the reward probabilities of actions are affected by the last response from environment. In this way, actions that have gotten the positive response from environment in the short time, have the opportunity to be estimated as “optimal”, to increase their choice probability and consequently, to be selected. The estimates become more reliable and consequently, the automaton rapidly and accurately converges to the optimal action. The asymptotic behavior of the proposed scheme is analyzed and it is proved to be ε-optimal in every stationary random environment. Extensive simulation results indicate that the proposed algorithm converges faster than the traditional stochastic-estimator-based S ERI scheme, and the deterministic-estimator-based DGPA and DPRI schemes when operating in stationary random environments.

This paper presents an adaptive magnification transformation based particle swarm optimizer (AMT-PSO) that provides an adaptive search strategy for each particle along the search process. Magnification transformation is a simple but very powerful mechanism, which is inspired by using a convex lens to see things much clearer. The essence of this transformation is to set a magnifier around an area we are interested in, so that we could inspect the area of interest more carefully and precisely. An evolutionary factor, which utilizes the information of population distribution in particle swarm, is used as an index to adaptively tune the magnification scale factor for each particle in each dimension. Furthermore, a perturbation-based elitist learning strategy is utilized to help the swarm's best particle to escape the local optimum and explore the potential better space. The AMT-PSO is evaluated on 15 unimodal and multimodal benchmark functions. The effects of the adaptive magnification transformation mechanism and the elitist learning strategy in AMT-PSO are studied. Results show that the adaptive magnification transformation mechanism provides the main contribution to the proposed AMT-PSO in terms of convergence speed and solution accuracy on four categories of benchmark test functions.