Quantum-behaved particle swarm optimization (QPSO), motivated by concepts from quantum mechanics and particle swarm optimization (PSO), is a probabilistic optimization algorithm belonging to the bare-bones PSO family. Although it has been shown to perform well in finding the optimal solutions for many optimization problems, there has so far been little analysis on how it works in detail. This paper presents a comprehensive analysis of the QPSO algorithm. In the theoretical analysis, we analyze the behavior of a single particle in QPSO in terms of probability measure. Since the particle's behavior is influenced by the contraction-expansion (CE) coefficient, which is the most important parameter of the algorithm, the goal of the theoretical analysis is to find out the upper bound of the CE coefficient, within which the value of the CE coefficient selected can guarantee the convergence or boundedness of the particle's position. In the experimental analysis, the theoretical results are first validated by stochastic simulations for the particle's behavior. Then, based on the derived upper bound of the CE coefficient, we perform empirical studies on a suite of well-known benchmark functions to show how to control and select the value of the CE coefficient, in order to obtain generally good algorithmic performance in real world applications. Finally, a further performance comparison between QPSO and other variants of PSO on the benchmarks is made to show the efficiency of the QPSO algorithm with the proposed parameter control and selection methods.

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