One of the dynamic nonlinear inertia weights is represented by (18) and selleck Volasertib (19), while the other is represented by (20) and (21). They are meant to achieve tradeoff between exploration and exploitation:dn=dnmin?+(dnmax??dnmin?)(iteritermax?),(18)��=��min?+(��max??��min?)(iteritermax?)dn,(19)dn=dnmax??(dnmax??dnmin?)(iteritermax?),(20)��=��max??(��max??��min?)(iteritermax?)dn,(21)where dn is the dynamic nonlinear factor, �� is the inertia weight, ��max and ��min are the maximum and minimum values of ��, respectively, dnmax and dnmin are the maximum and minimum values of dn, respectively, and iter and itermax are the current iteration numbers and the maximum iteration number, respectively.

A dynamic logistic chaotic map in (4) was incorporated into the PSO variant inertia weight as shown in (23) to enrich searching behaviors and avoid being trapped into local optima:��=��max??(��max??��min?)(iteritermax?),(22)��=��+(1?��)Lmap,(23)where �� is the dynamic chaotic inertia weight adjustment factor, ��max and ��min are the maximum and minimum values of ��, respectively, and Lmap is the result of logistic chaotic map. In this variant, using (19) and (23) was labeled DLPSO1, while using (21) and (23) was captioned DLPSO2.For the purpose of achieving a balance between global exploration and local exploitation and also avoiding premature convergence, (19), (21), and (23) were mixed together to dynamically adjust the inertia weight of the variant as shown in Algorithm 4, where fi is the fitness value of particle i and favg is the average fitness value of the swarm.

Experiments and comparisons showed that the DLPSO2 outperformed several other well-known improved particle swarm optimization algorithms on many famous benchmark problems in all cases.Algorithm 43.7. DiscussionsLDIW-PSO is relatively simple to implement and fast in convergence. When [4] experimentally ascertained that LDIW-PSO is prone to premature convergence, especially when solving complex multimodal optimization problems, a new area of research was opened up for improvements on inertia weight strategies in PSO, and LDIW-PSO became a popular yard stick for many other variants.From the variants described previously, there are ample expectations that they should outperform LDIW-PSO judging by the various additional strategies introduced into the inertia weight strategies used by them.

For example, CDIW-PSO introduced chaotic optimization characteristic, REPSO introduced a combined effort of simulated annealing idea and fitness variance of particles, DAPSO introduced Anacetrapib a dynamic adaptive strategy based on swarm diversity function, APSO introduced an adaptive mutation to the particle positions and made the inertia weight dynamic based on the best global fitness, while DLPSO2 used different formulas coupled with chaotic mapping.