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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.

Geometric crossover is a representation-independent generalization of the traditional crossover defined using the distance of the solution space. By choosing a distance firmly rooted in the syntax of the solution representation as a basis for geometric crossover, one can design new crossovers for any representation. Using a distance tailored to the problem at hand, the formal definition of geometric crossover allows us to design new problem-specific crossovers that embed problem-knowledge in the search. The standard encoding for multiway graph partitioning is highly redundant: each solution has a number of representations, one for each way of labeling the represented partition. Traditional crossover does not perform well on redundant encodings. We propose a new geometric crossover for graph partitioning based on a labeling-independent distance that filters out the redundancy of the encoding. A correlation analysis of the fitness landscape based on this distance shows that it is well suited to graph partitioning. A second difficulty with designing a crossover for multiway graph partitioning is that of feasibility: in general recombining feasible partitions does not lead to feasible offspring partitions. We design a new geometric crossover for permutations with repetitions that naturally suits partition problems and we test it on the graph partitioning problem. We then combine it with the labeling-independent crossover and obtain a much superior geometric crossover inheriting both advantages.