Abstract
Using a geometric framework for the interpretation of crossover of recent introduction, we show an intimate connection between particle swarm optimisation (PSO) and evolutionary algorithms. This connection enables us to generalise PSO to virtually any solution representation in a natural and straightforward way. The new Geometric PSO (GPSO) applies naturally to both continuous and combinatorial spaces. We demonstrate this for the cases of Euclidean, Manhattan and Hamming spaces and report extensive experimental results. We also demonstrate the applicability of GPSO to more challenging combinatorial spaces. The Sudoku puzzle is a perfect candidate to test new algorithmic ideas because it is entertaining and instructive as well as being a nontrivial constrained combinatorial problem. We apply GPSO to solve the Sudoku puzzle.
Highlights
Particle swarm optimization (PSO) is a relatively recently devised population-based stochastic global optimization algorithm [1]
There is a number of extensions of PSO to combinatorial spaces with various degrees of success [2, 3]. (Notice that applications of traditional PSO to combinatorial optimization problems cast as continuous optimization problems are not extensions of the PSO algorithm.) every time a new solution representation is considered, the PSO algorithm needs to be rethought and adapted to the new representation
In order to test the efficacy of the geometric PSO (GPSO) algorithm on the Sudoku problem, we ran several experiments in order to thoroughly explore the parameter space and variations of the algorithm
Summary
Particle swarm optimization (PSO) is a relatively recently devised population-based stochastic global optimization algorithm [1]. We show formally how a general form of PSO (without the inertia term) can be obtained by using theoretical tools developed for evolutionary algorithms with geometric crossover and geometric mutation. These are representation-independent operators that generalize many pre-existing search operators for the major representations, such as binary strings [4], real vectors [4], permutations [5], syntactic trees [6], and sequences [7].
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