Analytics on Fireworks Algorithm Solving Problems with Shifts in the Decision Space and Objective Space

Fireworks algorithms for solving problems with the optima shift in decision space and/or objective space are analyzed in this paper. The standard benchmark problems have several weaknesses in the research of swarm intelligence algorithms for solving single objective problems. The optimum is in the center of search range, and is the same at each dimension of the search space. The optimum shift in decision space and/or objective space could increase the difficulty of problem solving. A mapping strategy, modular arithmetic mapping, is utilized in the original fireworks algorithm to handle solutions out of search range. The solutions are implicitly guided to the center of search range for problems with symmetrical search range via this strategy. The optimization performance of fireworks algorithm on shift functions may be affected by this strategy. Four kinds of mapping strategies, which include mapping by modular arithmetic, mapping to the boundary, mapping to stochastic region, and mapping to limited stochastic region, are compared on problems with different dimensions and different optimum shift range. From experimental results, the fireworks algorithms with mapping to the boundary, or mapping to limited stochastic region obtain good performance on problems with the optimum shift. This is probably because the search tendency is kept in these two strategies. The definition of population diversity measurement is also proposed in this paper, from observation on population diversity changes, the useful information of fireworks algorithm solving different kinds of problems could be obtained.