This paper presents an analysis of the relationship of particle velocity and convergence of the particle swarm optimization. Its premature convergence is due to the decrease of particle velocity in search space that leads to a total implosion and ultimately fitness stagnation of the swarm. An improved algorithm which introduces a velocity differential evolution (DE) strategy for the hierarchical particle swarm optimization (H-PSO) is proposed to improve its performance. The DE is employed to regulate the particle velocity rather than the traditional particle position in case that the optimal result has not improved after several iterations. The benchmark functions will be illustrated to demonstrate the effectiveness of the proposed method. 1. Introduction Algorithms to tackle optimization problems include not only classical techniques such as dynamic programming, branch-and-bound, and gradient-based methods, but also more recent techniques such as metaheuristics . Among the existing metaheuristic algorithms, the particle swarm optimization (PSO) algorithm is a population-based optimization technique developed by Kennedy and Eberhart in 1995 . The PSO has resulted in a large number of variants of the standard PSO. Some variants are designed to deal with specific applications [3–6], and others are generalized for numerical optimization [7–10]. A hierarchical version of PSO (H-PSO) has been proposed by Janson and Middendorf . In H-PSO, all particles are arranged in a tree that forms the hierarchy. A particle is influenced by its own best position and the best position particle in its neighborhood. It was shown that H-PSO performed very well compared to the standard PSO on unimodal and multimodal test functions [10, 11]. H-PSO presents the advantage of being conceptually very simple and requiring low computation time. However, the main disadvantage of H-PSO is the risk of a premature search convergence, especially in complex multiple peak search problems. A number of algorithms combined various algorithmic components, often originating from algorithms of other research areas on optimization. These approaches are commonly referred to as hybrid meta-heuristics . The surveys on hybrid algorithms that combine the PSO and differential evolution (DE)  were presented recently [14, 15]. These PSO-DE hybrids usually employ DE to adjust the particle position. But the convergence performance is dependent on the particle velocity. Limiting the velocity can help the particle to get out of local optima traps [16, 17]. In this paper, we will combine
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