16 Particle Swarm Optimization

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where Pc (0, 1) is the probability of crossover, and > 0 is a scaling factor The position of the particle is only replaced if the o spring is better That is, xi (t+1) = xi (t + 1) only if f (xi (t + 1)) < f (xi (t)), otherwise xi (t + 1) = xi (t) (assuming a minimization problem) Hendtlass applied the above DE process to the PSO by executing the DE crossover operator from time to time [360] That is, at speci ed intervals, the swarm serves as population for the DE algorithm, and the DE algorithm is executed for a number of iterations Hendtlass reported that this hybrid produces better results than the basic PSO Kannan et al [437] applies DE to each particle for a number of iterations, and replaces the particle with the best individual obtained from the DE process Zhang and Xie [954] followed a somewhat di erent approach where only the personal best positions are changed using the following operator: yij (t + 1) = yij (t) + j yij (t) if U (0, 1) < Pc and j = U (1, nx ) otherwise (1674)

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where is the general di erence vector de ned as, j = y1j (t) y2j (t) 2 (1675)

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with y1j (t) and y2j (t) randomly selected personal best positions Then, yij (t + 1) is set to yij (t + 1) only if the new personal best has a better tness evaluation

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Sub-Swarm Based PSO

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A number of cooperative and competitive PSO implementations that make use of multiple swarms have been developed Some of these are described below Multi-phase PSO approaches divide the main swarm of particles into subgroups, where each subgroup performs a di erent task, or exhibits a di erent behavior The behavior of a group or task performed by a group usually changes over time in response to the group s interaction with the environment It can also happen that individuals may migrate between groups The breeding between sub-swarms developed by L vberg et al [534, 536] (refer to Section 1653) is one form of cooperative PSO, where cooperation is implicit in the exchange of genetic material between parents of di erent sub-swarms Al-Kazemi and Mohan [16, 17, 18] explicitly divide the main swarm into two subswarms, of equal size Particles are randomly assigned to one of the sub-swarms Each sub-swarm can be in one of two phases: Attraction phase, where the particles of the corresponding sub-swarm are allowed to move towards the global best position Repulsion phase, where the particles of the corresponding sub-swarm move away from the global best position

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165 Single-Solution Particle Swarm Optimization For their multi-phase PSO (MPPSO), the velocity update is de ned as [16, 17]: vij (t + 1) = wvij (t) + c1 xij (t) + c2 yj (t)

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(1676)

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The personal best position is excluded from the velocity equation, since a hill-climbing procedure is followed where a particle s position is only updated if the new position results in improved performance Let the tuple (w, c1 , c2 ) represent the values of the inertia weight, w, and acceleration coe cients c1 and c2 Particles that nd themselves in phase 1 exhibit an attraction towards the global best position, which is achieved by setting (w, c1 , c2 ) = (1, 1, 1) Particles in phase 2 have (w, c1 , c2 ) = (1, 1, 1), forcing them to move away from the global best position Sub-swarms switch phases either when the number of iterations in the current phase exceeds a user speci ed threshold, or when particles in any phase show no improvement in tness during a userspeci ed number of consecutive iterations In addition to the velocity update as given in equation (1676), velocity vectors are periodically initialized to new random vectors Care should be taken with this process, not to reinitialize velocities when a good solution is found, since it may pull particles out of this good optimum To make sure that this does not happen, particle velocities can be reinitialized based on a reinitialization probability This probability starts with a large value that decreases over time This approach ensures large diversity in the initial steps of the algorithm, emphasizing exploration, while exploitation is favored in the later steps of the search Cooperation between the subgroups is achieved through the selection of the global best particle, which is the best position found by all the particles in both sub-swarms Particle positions are not updated using the standard position update equation Instead, a hill-climbing process is followed to ensure that the tness of a particle is monotonically decreasing (increasing) in the case of a minimization (maximization) problem The position vector is updated by randomly selecting consecutive components from the velocity vector and adding these velocity components to the corresponding position components If no improvement is obtained for any subset of consecutive components, the position vector does not change If an improvement is obtained, the corresponding position vector is accepted The value of changes for each particle, since it is randomly selected, with U (1, max ), with max initially small, increasing to a maximum of nx (the dimension of particles) The attractive and repulsive PSO (ARPSO) developed by Riget and Vesterstr m [729, 730, 877] follows a similar process where the entire swarm alternates between an attraction and repulsion phase The di erence between the MPPSO and ARPSO lies in the velocity equation, in that there are no explicit sub-swarms in ARPSO, and ARPSO uses information from the environment to switch between phases While

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