Multimodal function optimizations with multiple maximums and multiple minimums using an improved PSO algorithm - Chang, Wei-Der (Applied soft computing, 2017)
[Display omitted] •This paper utilizes an improved PSO algorithm to solve the function optimization with multiple maximums and minimums.•The original population needs to be divided into two main groups.•One group is to tackle the maximum optimization and the other focuses on the function minimum optimization.•Each main group is further split up into a certain number of subgroups.•Every subgroup can individually search for one function optimum. In this paper, a multimodal function optimization problem consisting of multiple maximums and multiple minimums is solved using an improved particle swarm optimization (PSO) algorithm. In the proposed scheme, the original population needs to be randomly divided into two main groups in the first stage. One group is to tackle the maximum optimization of the multimodal function and the other then focuses on the function minimum optimization. In the second stage, each group is split up into several subgroups in order to seek for function optimums simultaneously. There is no relation among subgroups and each subgroup can individually seek for one of function optimums. To achieve that, it is necessary to enroll the best particle information of each subgroup. It means that the proposed structure contains a number of best particles, not a single global best particle. The third stage is to modify the velocity updating formula of the algorithm where the global best particle is simply replaced by the best particle of each subgroup. Under the proposed scheme, multiple maxima and minima of the multimodal function can probably be solved separately and synchronously. Finally, many different kinds of multimodal function problems are illustrated to certify the applicability of the presented method, including one maximum and one minimum, two maximums and two minimums, multiple maximums and multiple minimums, and a complex engineering optimization problem with inequality conditions.