As can be seen from Table V of [1], Type 1" outperformed Type 1 on Ackley, Rastrigin, and Rosenbrock, while Type 1 outperformed Type 1” on Schaffer’s f6 and Griewangk.  Though it had been hoped that the constriction coefficient would eliminate the need for velocity clamping, standard PSO with velocity clamping (i.e. “E&S”) performed more consistently than both Type 1 and Type 1” without velocity clamping.  Approaches other than Type 1", Type 1, and E&S were highly inconsistent and therefore are not worth mentioning.  Standard PSO with velocity clamping provided more consistent performance than the constriction models.

I think the interesting question Clerc raised was whether or not well-balanced parameters could alleviate the tendency of particles to take unnecessarily large steps.  If huge steps could be prevented via proper parameter selection, velocity clamping might become unnecessary. So from an inside-looking-out perspective, attempting to eliminate the parameter made sense. But from an outside-looking-in perspective, velocity clamping merely ensures that step sizes are never unreasonably large, so it might as well be kept until robust parameters are discovered—especially in light of the results of Table V [1].

Empirically searching for well-performing parameters might eventually tell whether velocity clamping becomes unnecessary once the "drunkard's walk" is eliminated or is a critical part of the algorithm analogous to limiting in magnitude each individual's emotive responses. Table III-3 of thesis recommends parameters that appear to work well in general despite the fact that parameter selection is always somewhat problem-dependent [2]. Pederson independently searched for quality parameters during his thesis studies as well [3].

[1] M. Clerc and J. Kennedy, "The particle swarm - explosion, stability, and convergence in multidimensional complex space", IEEE Transactions on Evolutionary Computation, vol. 6, pp. 58-73, Feb. 2002. (see especially part 2)
- Ctrl + F, "5.3" to find Equation 5.3
- Ctrl + F, "empirical results" to find Table V

[2] G. Evers, “An Automatic Regrouping Mechanism to Deal with Stagnation in Particle Swarm Optimization,” M.S. thesis, The University of Texas – Pan American, Edinburg, TX, 2009
- Ctrl + F, "Type 1" to find the discussion of the Clerc’s equivalents
- Ctrl + F, “Table III-3” to find the results of an empirical search for quality parameters
- Ctrl + F, “Table V-2” to find the same alongside RegPSO results

[3] Pedersen, M.E.H. (2010). Tuning & Simplifying Heuristical Optimization (PhD thesis). University of Southampton, School of Engineering Sciences, Computational Engineering and Design Group.
- Ctrl + F, “Table 5.10” to find the results of an empirical search for quality parameters

To contact me:
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Concerning the Constriction Models

George Evers, MSE

Electrical Engineer