Single-Solution Particle Swarm Optimization in .NET framework

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165 Single-Solution Particle Swarm Optimization
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At xed intervals, as is done in the mass extinction PSO developed by Xie et al [923, 924] As discussed above, xed intervals may prematurely reinitialize a particle Probabilistic approaches, where the decision to reinitialize is based on a probability In the dissipative PSO, Xie et al [922] reinitialize velocities and positions based on chaos factors that serve as probabilities of introducing chaos in the system Let cv and cl , with cv , cl [0, 1], be respectively the chaos factors for velocity and location Then, for each particle, i, and each dimension, j, if rij U (0, 1) < cv , then the velocity component is reinitialized to vij (t + 1) = U (0, 1)Vmax,j Also, if rij U (0, 1) < cl , then the position component is initialized to xij (t + 1) U (xmin,j , xmax,j ) A problem with this approach is that it will keep the swarm from reaching an equilibrium state To ensure that an equilibrium can be reached, while still taking advantage of chaos injection, start with large chaos factors that reduce over time The initial large chaos factors increase diversity in the rst phases of the search, allowing particles to converge in the nal stages A similar probabilistic approach to reinitializing velocities is followed in [765, 766] Approaches based on some convergence condition, where certain events trigger reinitialization Using convergence criteria, particles are allowed to rst exploit their local regions before being reinitialized Venter and Sobieszczanski-Sobieski [874, 875] and Xie et al [923] initiate reinitialization when particles do not improve over time Venter and SobieszczanskiSobieski evaluate the variation in particle tness of the current swarm If the variation is small, then particles are centered in close proximity to the global best position Particles that are two standard deviations away from the swarm center are reinitialized Xie et al count for each xi = y the number of times that f (xi ) f ( ) < When this count exceeds a given threshold, the corresponding y particle is reinitialized Care should be taken in setting values for and the count threshold If is too large, particles will be reinitialized before having any chance of exploiting their current regions Clerc [133] de nes a hope and re-hope criterion If there is still hope that the objective can be reached, particles are allowed to continue in their current search directions If not, particles are reinitialized around the global best position, taking into consideration the local shape of the objective function Van den Bergh [863] de ned a number of convergence tests for a multi-start PSO, namely the normalized swarm radius condition, the particle cluster condition and the objective function slope condition (also refer to Section 1616 for a discussion on these criteria) L vberg and Krink [534, 535] use self-organized criticality (SOC) to determine when to reinitialize particles Each particle maintains an additional variable, Ci , referred to as the criticality of the particle If two particles are closer than a threshold distance, , from one another, then both have their criticality increased by one The larger the criticality of all particles, the more uniform the swarm becomes To prevent criticality from building up, each Ci is reduced by a fraction in each iteration As soon as Ci > C, where C is the global criticality limit, particle i is reinitialized Its criticality is distributed to its immediate neighbors
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16 Particle Swarm Optimization and Ci = 0 L vberg and Krink also set the inertia weight value of each particle to wi = 02 + 01Ci This forces the particle to explore more when it is too similar to other particles
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The next issue to consider is which particles to reinitialize Obviously, it will not be a good idea to reinitialize the global best particle! From the discussions above, a number of selection methods have already been identi ed Probabilistic methods decide which particles to reinitialize based on a user-de ned probability The convergence methods use speci c convergence criteria to identify particles for reinitialization For example, SOC PSO uses criticality measures (refer to Algorithm 1611), while others keep track of the improvement in particle tness Van den Bergh [863] proposed a random selection scheme, where a particle is reinitialized at each tr iteration (also refer to Algorithm 1610) This approach allows each particle to explore its current region before being reinitialized To ensure that the swarm will reach an equilibrium state, start with a large tr < ns , which decreases over time Algorithm 1610 Selection of Particles to Reinitialize; indicates the index of the global best particle Create and initialize an nx -dimensional PSO: S; sidx = 0; repeat if sidx = then Sxidx U (xmin , xmax ); end sidx = (sidx + 1) mod tr ; for each particle i = 1, , ns do if f (Sxi ) < f (Syi ) then Syi = Sx; end y if f (Syi ) < f (S ) then S = Syi ; y end end Update velocities; Update position; until stopping condition is true; Finally, how are particles reinitialized As mentioned earlier, velocities and/or positions can be reinitialized Most approaches that reinitialize velocities set each velocity component to a random value constrained by the maximum allowed velocity Venter and Sobieszczanski-Sobieski [874, 875] set the velocity vector to the cognitive component after reinitialization of position vectors Position vectors are usually initialized to a new position subject to boundary constraints; that is, xij (t + 1) U (xmin,j , xmax,j ) Clerc [133] reinitializes particles on the basis of estimates of the local shape of the objective function Clerc [135] also proposes alternatives, where a particle returns to its previous best position, and from
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165 Single-Solution Particle Swarm Optimization Algorithm 1611 Self-Organized Criticality PSO Create and initialize an nx -dimensional PSO: S; Set Ci = 0, i = 1, , ns ; repeat Evaluate tness of all particles; Update velocities; Update positions; Calculate criticality for all particles; Reduce criticality for each particle; while i = 1, , ns such that Ci > C do Disperse criticality of particle i; Reinitialize xi ; end until stopping condition is true;
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there moves randomly for a xed number of iterations A di erent approach to multi-start PSOs is followed in [863], as summarized in Algorithm 1612 Particles are randomly initialized, and a PSO algorithm is executed until the swarm converges When convergence is detected, the best position is recorded and all particles randomly initialized The process is repeated until a stopping condition is satis ed, at which point the best recorded solution is returned The best recorded solution can be re ned using local search before returning the solution The convergence criteria listed in Section 1616 are used to detect convergence of the swarm
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