Geometric particle swarm optimization

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.

[1]  Maurice Clerc,et al.  Discrete Particle Swarm Optimization, illustrated by the Traveling Salesman Problem , 2004 .

[2]  Riccardo Poli,et al.  Geometric landscape of homologous crossover for syntactic trees , 2005, 2005 IEEE Congress on Evolutionary Computation.

[3]  M. Clerc.  When Nearer is Better , 2007 .

[4]  Heinz Mühlenbein,et al.  The parallel genetic algorithm as function optimizer , 1991, Parallel Comput..

[5]  Riccardo Poli,et al.  Topological crossover for the permutation representation , 2005, GECCO '05.

[6]  T. Yato,et al.  Complexity and Completeness of Finding Another Solution and Its Application to Puzzles , 2003, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[7]  Donald E. Knuth,et al.  Dancing links , 2000, cs/0011047.

[8]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

[9]  Yue Shi,et al.  A modified particle swarm optimizer , 1998, 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360).

[10]  Kenneth Alan De Jong,et al.  An analysis of the behavior of a class of genetic adaptive systems. , 1975 .

[11]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[12]  Riccardo Poli,et al.  Product Geometric Crossover , 2006, PPSN.

[13]  Russell C. Eberhart,et al.  A discrete binary version of the particle swarm algorithm , 1997, 1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation.

[14]  Julian Togelius,et al.  Product Geometric Crossover for the Sudoku Puzzle , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[15]  Panos M. Pardalos,et al.  Handbook of applied optimization , 2002 .

[16]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[17]  Riccardo Poli,et al.  Geometric Crossover for Sets, Multisets and Partitions , 2006, PPSN.

[18]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[19]  Riccardo Poli,et al.  Geometric Crossover for Biological Sequences , 2006, EuroGP.

[20]  Tim Jones.  Evolutionary Algorithms, Fitness Landscapes and Search , 1995 .

[21]  Riccardo Poli,et al.  Topological Interpretation of Crossover , 2004, GECCO.

[22]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..