Velocity Models in .NET

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Velocity Models
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Kennedy [446] investigated a number of variations to the full PSO models presented in Sections 1611 and 1612 These models di er in the components included in the velocity equation, and how best positions are determined This section summarizes these models
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Cognition-Only Model
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The cognition-only model excludes the social component from the original velocity equation as given in equation (162) For the cognition-only model, the velocity update
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163 Basic Variations changes to vij (t + 1) = vij (t) + c1 r1j (t)(yij (t) xij (t))
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(1633)
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The above formulation excludes the inertia weight, mainly because the velocity models in this section were investigated before the introduction of the inertia weight However, nothing prevents the inclusion of w in equation (1633) and the velocity equations that follow in this section The behavior of particles within the cognition-only model can be likened to nostalgia, and illustrates a stochastic tendency for particles to return toward their previous best position From empirical work, Kennedy reported that the cognition-only model is slightly more vulnerable to failure than the full model [446] It tends to locally search in areas where particles are initialized The cognition-only model is slower in the number of iterations it requires to reach a good solution, and fails when velocity clamping and the acceleration coe cient are small The poor performance of the cognitive model is con rmed by Carlisle and Dozier [107], but with respect to dynamic changing environments (refer to Section 1663) The cognition-only model was, however, successfully used within niching algorithms [89] (also refer to Section 1664)
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Social-Only Model
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The social-only model excludes the cognitive component from the velocity equation: vij (t + 1) = vij (t) + c2 r2j (t)( j (t) xij (t)) y for the gbest PSO For the lbest PSO, yj is simply replaced with yij For the social-only model, particles have no tendency to return to previous best positions All particles are attracted towards the best position of their neighborhood Kennedy empirically illustrated that the social-only model is faster and more e cient than the full and cognitive models [446], which is also con rmed by the results from Carlisle and Dozier [107] for dynamic environments (1634)
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Sel ess Model
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The sel ess model is basically the social model, but with the neighborhood best solution only chosen from a particle s neighbors In other words, the particle itself is not allowed to become the neighborhood best Kennedy showed the sel ess model to be faster than the social-only model for a few problems [446] Carlisle and Dozier s results show that the sel ess model performs poorly for dynamically changing environments [107]
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16 Particle Swarm Optimization
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Basic PSO Parameters
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The basic PSO is in uenced by a number of control parameters, namely the dimension of the problem, number of particles, acceleration coe cients, inertia weight, neighborhood size, number of iterations, and the random values that scale the contribution of the cognitive and social components Additionally, if velocity clamping or constriction is used, the maximum velocity and constriction coe cient also in uence the performance of the PSO This section discusses these parameters The in uence of the inertia weight, velocity clamping threshold and constriction coe cient has been discussed in Section 163 The rest of the parameters are discussed below: Swarm size, ns , ie the number of particles in the swarm: the more particles in the swarm, the larger the initial diversity of the swarm provided that a good uniform initialization scheme is used to initialize the particles A large swarm allows larger parts of the search space to be covered per iteration However, more particles increase the per iteration computational complexity, and the search degrades to a parallel random search It is also the case that more particles may lead to fewer iterations to reach a good solution, compared to smaller swarms It has been shown in a number of empirical studies that the PSO has the ability to nd optimal solutions with small swarm sizes of 10 to 30 particles [89, 865] Success has even been obtained for fewer than 10 particles [863] While empirical studies give a general heuristic of ns [10, 30], the optimal swarm size is problem-dependent A smooth search space will need fewer particles than a rough surface to locate optimal solutions Rather than using the heuristics found in publications, it is best that the value of ns be optimized for each problem using cross-validation methods Neighborhood size: The neighborhood size de nes the extent of social interaction within the swarm The smaller the neighborhoods, the less interaction occurs While smaller neighborhoods are slower in convergence, they have more reliable convergence to optimal solutions Smaller neighborhood sizes are less susceptible to local minima To capitalize on the advantages of small and large neighborhood sizes, start the search with small neighborhoods and increase the neighborhood size proportionally to the increase in number of iterations [820] This approach ensures an initial high diversity with faster convergence as the particles move towards a promising search area Number of iterations: The number of iterations to reach a good solution is also problem-dependent Too few iterations may terminate the search prematurely A too large number of iterations has the consequence of unnecessary added computational complexity (provided that the number of iterations is the only stopping condition) Acceleration coefficients: The acceleration coe cients, c1 and c2 , together with the random vectors r1 and r2 , control the stochastic in uence of the cognitive and social components on the overall velocity of a particle The constants c1 and c2 are also referred to as trust parameters, where c1 expresses how much
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