According to these principles, each particle investigates the search spaceanalyzing its own flying experience and that of the other members of the swarm.The PSO algorithm is composed of the following four steps: Step 0. (Initialization) Distribute a set of particles xi0 inside the design space withrandom distribution and random initial velocities. Set k = 1.Step 1. (Compute velocity) Calculate a velocity vector vifor each particle, using theparticle’s memory and the knowledge gained by the swarm according tovik = χ[wkvik−1 +c1r1(pi−xik−1)+c2r2(pbk−1 − xik−1)], (14)where χ is a constriction factor, w is called inertia weight, c1 and c2 arepositive constants, r1 and r2 are random numbers equally distributed between0 and 1, piis the best position found by the particle i and pbk−1 is the bestposition found by the swarm up to iteration k − 1.Step 2. (Update position) Update the position of each particle, xi, using the velocityvector and previous positionxik = xik−1 + vik. (15)Step 3. (Check convergence) Set k = k + 1. Go to Step 1 and repeat until conver-gence.Literature reports that fine tuning of the parameters in (14) is crucial for the opti-mization process, and that the final solution and the calculation time are strictly linkedto the parameters setting. The inertia weight w regulates the trade-off between theglobal (wide-ranging) and local (nearby) exploration abilities of the swarm. A largeinertia weight facilitates global exploration (searching new areas), while a small onefacilitates local exploration. A suitable value for w usually provides balance betweenglobal and local exploration abilities and consequently results in a reduction of thenumber of iterations required to locate the optimal solution. Experimental results in-dicate that it is better to initially set the inertia to a large value, in order to promoteglobal exploration of the design variables space and gradually decrease it to get amore refined solution (Venter and Sobieszczanski-Sobieski 2003). For these reasons,an initial value for w is set and the decrease rate is calculated bywk = wk−1gw, (16)where wk is the new value for the inertia weight, wk−1 is the previous one and gw isa constant chosen between 0 and 1. In Venter and Sobieszczanski-Sobieski (2003)itis suggested to use 0.35 <w< 1.4, and gw = 0.975.The constriction factor χ is used alternatively to w to limit the maximum velocity.The major difference between the two is that while the inertia w is employed tocontrol the impact of the previous history of velocities on the current one, χ offers tothe user the chance to select the search resolution. Quantitatively, if box constraintsare given, χ takes a value equal to a fraction of the characteristic dimension of thebox.The constants c1 and c2 are called cognitive and social parameter, respectively.The cognitive parameter indicates how much confidence the particle has in itself,while the social parameter indicates how much confidence it has in the swarm. Properfine-tuning of these coefficients may result in faster convergence and alleviation of local minima. In basic PSO algorithm (Kennedy and Eberhart 1995) the authors pro-pose c1 = c2 = 2, so that the mean of stochastic multipliers of (14) is 1. In Venter andSobieszczanski-Sobieski (2003) different values for the two coefficients are used. Inparticular, c1 = 1.5 and c2 = 2.5 work well in their examples.In Campana et al. (2006a, 2006b) a more rigorous analysis is carried out.
In thesepapers a generalized PSO iteration is described by means of a dynamic linear system,whose properties are analyzed. The influence of the particles’ starting points and theuse of deterministic or stochastic parameters are investigated and some partial con-vergence results are given. In particular, the PSO parameters are selected by imposingthat the particles trajectories are confined in a suitable compact set.3.2 A new PSO algorithm: DDFPSOThis section focuses on the development of an enhanced PSO version, named Deter-ministic Derivative-Free Particle Swarm Optimization (DDFPSO). In the followingall the modifications are described.(i) Initialization of the swarm: The use of GO algorithms based on expensive analy-sis tools imposes a substantial reduction of the particles number. Instead of usinga random distribution, a deterministic one is proposed. In particular, at Step 0the initial swarm is built with one particle at the center of each face of the n-dimensional hyper-cube which represents the design space. As a consequence,the total number of particles is 2n. Moreover, the initial velocity vector, definedin (14), is set equal to 0.(ii) Boundary search phase: According to (14) and (15) each particle is attractedby the others. As a consequence, they cannot escape from the dashed regionindicated in Fig.1 unless the inertia term is sufficiently strong, i.e., the cornersof the design space cannot be reached by any particle of the swarm. 源]自{751^`论\文}网·www.751com.cn/ 船舶设计问题上的新全局优化英文文献和中文翻译(4):http://www.751com.cn/fanyi/lunwen_64778.html