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Scaling factor: The scaling factor, (0, ), controls the ampli cation of the di erential variations, (xi2 xi3 ) The smaller the value of , the smaller the mutation step sizes, and the longer it will be for the algorithm to converge Larger values for facilitate exploration, but may cause the algorithm to overshoot good optima The value of should be small enough to allow di erentials to explore tight valleys, and large enough to maintain diversity As the population size increases, the scaling factor should decrease As explained in Section 1311, the more individuals in the population, the smaller the magnitude of the difference vectors, and the closer individuals will be to one another Therefore, smaller step sizes can be used to explore local areas More individuals reduce the need for large mutation step sizes Empirical results suggest that large values for both ns and often result in premature convergence [429, 124], and that = 05 generally provides good performance [813, 164, 19] Recombination probability: The probability of recombination, pr , has a direct in uence on the diversity of DE This parameter controls the number of elements of the parent, xi (t), that will change The higher the probability of recombination, the more variation is introduced in the new population, thereby increasing diversity and increasing exploration Increasing pr often results in faster convergence, while decreasing pr increases search robustness [429, 164] Most implementations of DE strategies keep the control parameters constant Although empirical results have shown that DE convergence is relatively insensitive to di erent values of these parameters, performance (in terms of accuracy, robustnes, and speed) can be improved by nding the best values for control parameters for each new problem Finding optimal parameter values can be a time consuming exercise, and for this reason, self-adaptive DE strategies have been developed These methods are discussed in Section 1333.

5/14/2001

2 x1

Figure 163 Illustration of lbest PSO f move towards e as illustrated in Figure 163(b) (only part of the solution space is illustrated) The blocks represent the previous positions Note that e remains the best solution for neighborhood 2 Also note the general movement towards the minimum More in-depth analyses of particle trajectories can be found in [136, 851, 863, 870]

10/27/1997

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A few aspects of Algorithms 161 and 162 still need to be discussed These aspects include particle initialization, stopping conditions and de ning the terms iteration and function evaluation With reference to Algorithms 161 and 162, the optimization process is iterative Repeated iterations of the algorithms are executed until a stopping condition is satis ed One such iteration consists of application of all the steps within the repeatuntil loop, ie determining the personal best positions and the global best position, and adjusting the velocity of each particle Within each iteration, a number of function evaluations (FEs) are performed A function evaluation (FE) refers to one calculation of the tness function, which characterizes the optimization problem For the basic PSO, a total of ns function evaluations are performed per iteration, where ns is the total number of particles in the swarm The rst step of the PSO algorithm is to initialize the swarm and control parameters In the context of the basic PSO, the acceleration constants, c1 and c2 , the initial velocities, particle positions and personal best positions need to be speci ed In addition, the lbest PSO requires the size of neighborhoods to be speci ed The importance of optimal values for the acceleration constants are discussed in Section 164

2/14/1994

Usually, the positions of particles are initialized to uniformly cover the search space It is important to note that the e ciency of the PSO is in uenced by the initial diversity of the swarm, ie how much of the search space is covered, and how well particles are distributed over the search space If regions of the search space are not covered by the initial swarm, the PSO will have di culty in nding the optimum if it is located within an uncovered region The PSO will discover such an optimum only if the momentum of a particle carries the particle into the uncovered area, provided that the particle ends up on either a new personal best for itself, or a position that becomes the new global best Assume that an optimum needs to be located within the domain de ned by the two vectors, xmin and xmax , which respectively represents the minimum and maximum ranges in each dimension Then, an e cient initialization method for the particle positions is: x(0) = xmin,j + rj (xmax,j xmin,j ), where rj U (0, 1) The initial velocities can be initialized to zero, ie vi (0) = 0 (1610) j = 1, , nx , i = 1, , ns (169)

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