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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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with r3j (t) U (0, 1) The velocity, vij (t), is now a probability for xij (t) to be 0 or 1 For example, if vij (t) = 0, then prob(xij (t + 1) = 1) = 05 (or 50%) If vij (t) < 0, then prob(xij (t + 1) = 1) < 05, and if vij (t) > 0, then prob(xij (t + 1) = 1) > 05 Also note that prob(xij (t) = 0) = 1 prob(xij (t) = 1) Note that xij can change even if the value of vij does not change, due to the random number r3j in the equation above It is only the calculation of position vectors that changes from the real-valued PSO The velocity vectors are still real-valued, with the same velocity calculation as given in equation (162), but including the inertia weight That is, xi , yi , y Bnx while vi Rnx The binary PSO is summarized in Algorithm 1613 Algorithm 1613 binary SO Create and initialize an nx -dimensional swarm; repeat for each particle i = 1, , ns do if f (xi ) < f (yi ) then yi = xi ; end y if f (yi ) < f ( ) then y = yi ; end end for each particle i = 1, , ns do update the velocity using equation (162); update the position using equation (1690); end until stopping condition is true;.

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A very simple approach to cope with constraints is to reject infeasible particles, as follows: Do not allow infeasible particles to be selected as personal best or neighborhood lobal best solutions In doing so, infeasible particles will never in uence other particles in the swarm Infeasible particles are, however, pulled back to feasible space due to the fact that personal best and neighborhood best positions are in feasible space This approach can only be e cient if the ratio of number of infeasible particles to feasible particles is small If this ratio is too large, the swarm may not have enough diversity to e ectively cover the (feasible) space Reject infeasible particles by replacing them with new randomly generated positions from the feasible space Reinitialization within feasible space gives these particles an opportunity to become best solutions However, it is also possible that these particles (or any other particle for that matter), may roam outside the boundaries of feasible space This approach may be bene cial in cases where the feasible space is made up of a number of disjointed regions The strategy will allow particles to cross boundaries of one feasible region to explore another feasible region Most applications of PSO to constrained problems make use of penalty methods to penalize those particles that violate constraints [663, 838, 951] Similar to the approach discussed in Section 1261, Shi and Krohling [784] and Laskari et al [504] converted the constrained problem to an unconstrained Lagrangian Repair methods, which allow particles to roam into infeasible space, have also been developed These are simple methods that apply repairing operators to change infeasible particles to represent feasible solutions Hu and Eberhart et al [388] developed an approach where particles are not allowed to be attracted by infeasible particles The personal best of a particle changes only if the tness of the current position is better and if the current position violates no constraints This ensures that the personal best positions are always feasible Assuming that neighborhood best positions are selected from personal best positions, it is also guaranteed that neighborhood best positions are feasible In the case where the neighborhood best positions are selected from the current particle positions, neighborhood best positions should be updated only if no constraint is violated Starting with a feasible set of particles, if a particle moves out of feasible space, that particle will be pulled back towards feasible space Allowing particles to roam into infeasible space, and repairing them over time by.

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