Inertia Weight
Shi and Eberhart found that introducing the inertia weight to the first term of the velocity update equation improved performance over the previous model, which carried over 100% of the previous iteration’s velocity [3]. Since only a percentage of the velocity is retained from the previous iteration, the inertia weight may be thought of as a simplified form of friction, which reduces velocities over time – thus helping particles settle onto a good solution without excessive wandering. Perhaps more analogously to the social and cognitive aspects of the algorithm, one may think of the inertia weight as controlling the rate at which particles forget previous information in order to focus primarily on the presumably higher quality bests of later iterations. The latter perspective was utilized in thesis [4].
Velocity Clamping
Velocities are iteratively clamped to lie within [-vmax, vmax] on each dimension – the same range used for initialization. The velocity clamping value, vmax, is a reasonable percentage of the range of the search space per dimension. This helps particles comb the search space in a fashion analogous to logical reasoning rather than potentially taking iterative leaps that would increase the probability of important information going unconsidered.
Position Updates
Once each particle's velocity vector has been updated and its components clamped, the resulting velocity vector is added to the previous iteration’s position vector to determine its new position according to the position update equation below. Velocity is considered to be displacement per iteration rather than displacement per real-time unit, and the implicit iterative multiplication of the velocity by t = 1 iteration drops from the equation - just as multiplication by one iteration was implicit on the previous page when converting from acceleration to velocity.
Premature Convergence
When the global best does not improve for a considerable number of iterations, repeated accelerations toward it cause the swarm to collapses around it. When the global best converged to does not approximate a global minimizer, premature convergence is said to have occurred (i.e. often stagnation in PSO literature). Fig’s 7 & 1 of the HESTEC poster illustrate this condition, which is the primary difficulty of many population-based search heuristics and the problem addressed in thesis [4, 5].
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