Allocation of local and global search capabilities of particle in canonical PSO

This paper analyzes theoretically the exact sampling distribution of the particle swarm optimization (PSO) without any assumption imposed by all current analyses. The distribution of particles in the PSO in one-step transition is analyzed in details. Especially, local and global search capabilities of particles in the PSO are defined implicitly, and are allocated adaptively to show how the PSO works by several experiments. In essence, the PSO works by just allocating each particle in the swarm to finish two jobs in probability, locally searching the range around the current best positions and globally searching whole solution space. According to our definitions and analyses, the exact probabilities of the two jobs can be measured by theoretic derivations and experiments whilst how the PSO allocate the particles' search capabilities can be recognized clearly. So we can look into the PSO in depth.