Social Best (i.e. Global Best or Neighborhood Best)
Each particle in the swarm also accelerates iteratively (i.e. with each time step) toward a socially determined best. In Gbest PSO, the global best (i.e. the best position encountered yet by the swarm) is iteratively communicated to all particles for their consideration. In Lbest PSO, each particle iteratively communicates with the other particles in its own topological neighborhood to determine its local best rather than having knowledge of the global best. The term social best is here used to encompass both variations of the algorithm and affects particles’ velocities via the third term of the velocity update equation presented below.
Acceleration
Particles are physical conceptualizations that accelerate according to cognitive and social influences. While maintaining inertia, each particle accelerates (i) cognitively toward its own personal best, as per the second term of the velocity update equation below, and (ii) socially toward the social best, as per the third term below. Each term is a vector pointing in a different direction, and the vector sum determines a particle’s overall velocity at each iteration. The cognitive and social acceleration coefficients (c1 and c2 respectively) determine the importance each particle assigns to its experiential and socially derived knowledge. These must be balanced like ingredients of a recipe to produce a quality optimizer. As in life, there is seeming randomness at each step.
Velocity Update Equation
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is the n-dimensional position vector of particle i at iteration k,
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is the n-dimensional velocity vector of particle i at iteration k,
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is the n-dimensional personal best of particle i found through iteration k,
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is the n-dimensional social best of particle i found through iteration k: for Gbest PSO, the social best is the same for all particles so that subscript i can be dropped.
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is the inertia weight between 0 and 1, which simulates friction
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c1
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is the cognitive acceleration coefficient,
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c2
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is the social acceleration coefficient,
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are pseudo-random vectors with each dimension sampled from U(0, 1)
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denotes element -wise multiplication between vectors.
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