Basic PSO Parameters in .NET framework

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164 Basic PSO Parameters
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con dence a particle has in itself, while c2 expresses how much con dence a particle has in its neighbors With c1 = c2 = 0, particles keep ying at their current speed until they hit a boundary of the search space (assuming no inertia) If c1 > 0 and c2 = 0, all particles are independent hill-climbers Each particle nds the best position in its neighborhood by replacing the current best position if the new position is better Particles perform a local search On the other hand, if c2 > 0 and c1 = 0, the entire swarm is attracted to a single point, y The swarm turns into one stochastic hill-climber Particles draw their strength from their cooperative nature, and are most e ective when nostalgia (c1 ) and envy (c2 ) coexist in a good balance, ie c1 c2 If c1 = c2 , particles are attracted towards the average of yi and y [863, 870] While most applications use c1 = c2 , the ratio between these constants is problemdependent If c1 >> c2 , each particle is much more attracted to its own personal best position, resulting in excessive wandering On the other hand, if c2 >> c1 , particles are more strongly attracted to the global best position, causing particles to rush prematurely towards optima For unimodal problems with a smooth search space, a larger social component will be e cient, while rough multi-modal search spaces may nd a larger cognitive component more advantageous Low values for c1 and c2 result in smooth particle trajectories, allowing particles to roam far from good regions to explore before being pulled back towards good regions High values cause more acceleration, with abrupt movement towards or past good regions Usually, c1 and c2 are static, with their optimized values being found empirically Wrong initialization of c1 and c2 may result in divergent or cyclic behavior [863, 870] Clerc [134] proposed a scheme for adaptive acceleration coe cients, assuming the social velocity model (refer to Section 1635): c2 (t) = c2,max c2,min e mi (t) 1 c2,min + c2,max + + m (t) 2 2 e i +1 (1635)
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where mi is as de ned in equation (1630) The formulation of equation (1630) implies that each particle has its own adaptive acceleration as a function of the slope of the search space at the current position of the particle Ratnaweera et al [706] builds further on a suggestion by Suganthan [820] to linearly adapt the values of c1 and c2 Suganthan suggested that both acceleration coe cients be linearly decreased, but reported no improvement in performance using this scheme [820] Ratnaweera et al proposed that c1 decreases linearly over time, while c2 increases linearly [706] This strategy focuses on exploration in the early stages of optimization, while encouraging convergence to a good optimum near the end of the optimization process by attracting particles more towards the neighborhood best (or global best) positions The values of c1 (t) and c2 (t) at time step t is calculated as c1 (t) = (c1,min c1,max ) t + c1,max nt t c2 (t) = (c2,max c2,min ) + c2,min nt (1636) (1637)
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16 Particle Swarm Optimization where c1,max = c2,max = 25 and c1,min = c2,min = 05
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A number of theoretical studies have shown that the convergence behavior of PSO is sensitive to the values of the inertia weight and the acceleration coe cients [136, 851, 863, 870] These studies also provide guidelines to choose values for PSO parameters that will ensure convergence to an equilibrium point The rst set of guidelines are obtained from the di erent constriction models suggested by Clerc and Kennedy [136] For a speci c constriction model and selected value, the value of the constriction coe cient is calculated to ensure convergence For an unconstricted simpli ed PSO system that includes inertia, the trajectory of a particle converges if the following conditions hold [851, 863, 870, 937]: 1>w> 1 ( 1 + 2 ) 1 0 2 (1638)
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and 0 w < 1 Since 1 = c1 U (0, 1) and 2 = c2 U (0, 1), the acceleration coe cients, c1 and c2 serve as upper bounds of 1 and 2 Equation (1638) can then be rewritten as 1 (1639) 1 > w > (c1 + c2 ) 1 0 2 Therefore, if w, c1 and c2 are selected such that the condition in equation (1639) holds, the system has guaranteed convergence to an equilibrium state The heuristics above have been derived for the simpli ed PSO system with no stochastic component It can happen that, for stochastic 1 and 2 and a w that violates the condition stated in equation (1638), the swarm may still converge The stochastic trajectory illustrated in Figure 166 is an example of such behavior The particle follows a convergent trajectory for most of the time steps, with an occasional divergent step Van den Bergh and Engelbrecht show in [863, 870] that convergent behavior will be observed under stochastic 1 and 2 if the ratio, ratio = is close to 10, where crit = sup | 05 1 < w, (0, c1 + c2 ] (1641) crit c1 + c2 (1640)
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It is even possible that parameter choices for which ratio = 05, may lead to convergent behavior, since particles spend 50% of their time taking a step along a convergent trajectory
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