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Searching the Solution Space in Constructive Geometric Constraint Solving with Genetic Algorithms

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Abstract

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one instance such that besides fulfilling the geometric constraints, exhibits some additional properties. Selecting a solution instance amounts to selecting a given root every time the geometric constraint solver needs to compute the zeros of a multi valuated function. The problem of selecting a given root is known as the Root Identification Problem.

In this paper we present a new technique to solve the root identification problem. The technique is based on an automatic search in the space of solutions performed by a genetic algorithm. The user specifies the solution of interest by defining a set of additional constraints on the geometric elements which drive the search of the genetic algorithm. The method is extended with a sequential niche technique to compute multiple solutions. A number of case studies illustrate the performance of the method.

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References

  1. W. Bouma, I. Fudos, C. Hoffmann, J. Cai, and R. Paige, “Geometric constraint solver,” Computer Aided Design, vol. 27, no. 6, pp. 487–501, 1995.

    Google Scholar 

  2. L. Brisoux-Devendeville, C. Essert-Villard, and P. Schreck, “Exploration of a solution space structured by finite constraints,” in ECAI 14th European Conference on Artificial Intelligence. Workshop on Modelling and Solving Problems with Constraints, Berlin, 2000, pp. F:1–18.

  3. C. Essert-Villard, P. Schreck, and J.-F. Dufourd, “Skecth-based pruning of a solution space within a formal geometric constraint solver,” Artificial Intelligence, vol. 124, pp. 139–159, 2000.

    Google Scholar 

  4. M. Luz#x000F3;n, “Resoluci#x000F3;n de restricciones geométricas. selecci#x000F3;n de la soluci#x000F3;n deseada,” Ph.D. thesis, Dept. Informática, Universidad de Vigo. Written in Spanish, 2001.

  5. D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, 1989.

  6. Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, 1996.

  7. N. Mata, “Solving incidence and tangency constraints in 2D,” Technical Report LSI-97-3R, Department LiSI, Universitat Politècnica de Catalunya, 1997.

  8. G. Laman, “On graphs and rigidity of plane skeletal structures,” Journal of Engineering Mathematics, vol. 4, no. 4, pp. 331–340, 1970.

    Google Scholar 

  9. L. Lovász and Y. Yemini, “On generic rigidity in the plane,” SIAM Journal on Algebraic and Discrete Methods, vol. 3, no. 1, pp. 91–98, 1982.

    Google Scholar 

  10. B. Brderlin, “Rule-based geometric modelling,” Ph.D. thesis, Institut für Informatik der ETH Zürich, 1988.

  11. I. Fudos and C. Hoffmann, “Correctness proof of a geometric constraint solver,” International Journal of Computational Geometry and Applications, vol. 6, no. 4, pp. 405–420, 1996.

    Google Scholar 

  12. R. Joan-Arinyo and A. Soto, “A correct rule-based geometric constraint solver,” Computer & Graphics, vol. 21, no. 5, pp. 599–609, 1997.

    Google Scholar 

  13. Kleene, S., Mathematical Logic, New York: John Wiley and Sons, 1967.

    Google Scholar 

  14. C. Durand, “Symbolic and numerical techniques for constraint solving,” Ph.D. thesis, Computer Science, Purdue University, 1998.

  15. R. Joan-Arinyo, A. Soto-Riera, S. Vila-Marta, and J. Vilaplana, “Declarative characterization of a general architecture for constructive geometric constraint solvers,” in The Fifth International Conference on Computer Graphics and Artificial Intelligence, edited by D. Plemenos, Limoges, France, Université de Limoges, 2002, pp. 63–76.

    Google Scholar 

  16. R. Joan-Arinyo and A. Soto-Riera, “Combining constructive and equational geometric constraint solving techniques,” ACM Transactions on Graphics, vol. 18, no. 1, pp. 35–55, 1999.

    Google Scholar 

  17. I. Fudos and C. Hoffmann, “A graph-constructive approach to solving systems of geometric constraints,” ACM Transactions on Graphics, vol. 16, no. 2, pp. 179–216, 1997.

    Google Scholar 

  18. N. Mata, “Constructible geometric problems with interval parameters,” Ph.D. thesis, Dept. LiSI, Universitat Politécnica de Catalunya, 2000.

    Google Scholar 

  19. R. Joan-Arinyo and N. Mata, “A data structure for solving geomegtric cosntraint problems with interval parameters,” Technical Report LSI-00-24-R, Department LiSI, Universitat Politécnica de Catalunya, 2000.

    Google Scholar 

  20. A. Eiben and Z. Ruttkay, “Constraint-satisfaction problems,” in Handbook of Evolutionary Computation, edited by T. Bäck, D. Fogel, and Z. Michalewicz, Institute of Physics Publishing Ltd and Oxford University Press, Chapt. C5.7, 1997, pp. C5.7:1–C5.7:5.

  21. J.H. Holland, Adaptation in Natural and Artificial Systems, Ann Arbor: The University of Michigan Press, 1975.

  22. T. Bäck and H. Schwefel, “Evolution strategies I: Variants and their computational implementation,”Genetic Algorithms in Engineering and Computer Science, 1995, pp. 111–126.

  23. H. Schwefel, Evolution and Optimum Seeking. Sixth-Generation Computer Technology Series, John Wiley and Sons, 1995.

  24. D. Fogel, System Identification Trough Simulated Evolution. A Machine Learning approach, Ginn Press, 1991.

  25. T. Bäck, D. Fogel, and Z. Michalewicz (eds.), Handbook of Evolutionary Computation, Institute of Physics Publishing Ltd and Oxford University Press, 1997.

  26. Koza, J., Genetic Programming: On the Programming of Computer by Means of Natural Selection, MIT Press: Cambridge, MA, 1992.

    Google Scholar 

  27. J. Koza, Genetic Programming II: Automatic Discovery of Reusable Programs, MIT Press: Cambridge, MA, 1994.

    Google Scholar 

  28. H. Bremermann, J. Roghson, and S. Salaff, “Global properties of evolution processes,” in Natural Automata and Useful Simulations, edited by H. Pattee, E. Edelsack, L. Fein, and A. Callahan, Macmillan, 1966, pp. 3–42.

  29. J. Richardson, M. Palmer, G. Liepins, and M. Hilliard, “Some guidelines for genetic algorithms with penalty functions,” in Third IEEE International Conference on Genetic Algorithms, edited by J. Schaffer, San Mateo, CA, Morgan Kauffmann, 1989, pp. 191–197.

  30. A. Eiben and Z. Ruttkay, “Self-adaptivity for constraint satisfaction: Learning penalty functions,”in Third IEEE World Conference on Evolutionary Computation, Nagoya, Japan, 1996, pp. 258–261.

    Google Scholar 

  31. V. Petridis, S. Kazarlis, and A. Bakirtzis, “Varying quality functions in genetic algorithm constrained optimization: The cutting stock and unit commitment problems,” IEEE Transactions on Systems, Man and Cybernetics, vol. 28, Part B, no. 5, pp. 1998.

  32. H. Mühlenbein and M.G.-S. nad O. Krämer, “Evolution algorithm in combinatorial optimization,” Parallel Computing, vol. 7, pp. 65–85, 1988.

    Google Scholar 

  33. J. Grefenstette, “Proportional selection and samplig algorithms,” in Handbook of Evolutionary Computation, edited by T. Bäck, D. Fogel, and Z. Michalewicz, Institute of Physics Publishing Ltd and Oxford University Press, Chapt. C2.2, 1997a, pp. C2.2:1–C2.2:7.

  34. J. Grefenstette, “Rank-based selection,” in Handbook of Evolutionary Computation, edited by T. Bäck, D. Fogel, and Z. Michalewicz, Institute of Physics Publishing Ltd and Oxford University Press, Chapt. C2.4, 1997b, pp. C2.4:1–C2.4:6.

  35. J.E. Baker, “Reducing bias and inefficiency in the selection algorithm,” in Proc. Second International Conference on Genetic Algorithms (ICGA’87), 1987, pp. 14–21.

  36. L. Booker, D. Fogel, D. Whitley, and P. Angeline, “Recombination,” in Handbook of Evolutionary Computation, edited by T. Bäck, D. Fogel, and Z. Michalewicz, Institute of Physics Publishing Ltd and Oxford University Press, Chapt. C3.3, 1997, pp. C3.3:1–C3.3:10.

  37. K. Deb, D.G., “An investigation of niche and species formation in genetic function optimization,” in Proc. of the Second International Conference on Genetic Algorithms, Hillsdale, NJ, Lawrence Erlbaum, 1989, pp. 42–50.

  38. D. Beasly, D. Bull, and R. Martin, “A sequential niche technique for multimodal function optimization,” Evolutionary Computation, vol. 1, no. 2, pp. 101–125, 1993.

    Google Scholar 

  39. R. Hamming, “Error detecting and error correcting codes,” Bell Systems Technical Journal, vol. 29, pp. 147–160, 1950.

    Google Scholar 

  40. A. Pétrowski, “A cleaning procedure as a niching method for generic algorithms,” in First IEEE International Conference on Evolutionary Computaion, 1996, pp. 798–803.

  41. J. Grefenstette, “Optimization of control parameters for genetic algorithms,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-16, no. 1, pp. 122–128, 1986.

    Google Scholar 

  42. K. Ogata, State Space Analysis of Control Systems, Prentice Hall: N.J. Englewood Cliffs, 1967.

    Google Scholar 

  43. A. Eiben, P.-E. Raué, and Z. Ruttkay, “GA-easy and GA-hard constraint satisfaction problems,” in Constraint Processing, edited by M. Meyer, LNCS Series 923. Heidelberg: Springer-Verlag, 1995, pp. 267–284.

    Google Scholar 

  44. L.J. Fogel, A.J. Owens, M.W., Artificial Intelligence Through simulated evolution, John Wiley and Sons, 1966.

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Authors and Affiliations

  1. Escuela Superior de Ingeniera Informática, Universidade de Vigo, Av. As Lagoas s/n, E-32004, Ourense

    M. V. Luzón & J. F. Gálvez

  2. Escola Técnica Superior d’Enginyeria Industrial de Barcelona, Universitat Politécnica de Catalunya, Av. Diagonal 647, 8a, E-08028, Barcelona

    A. Soto & R. Joan-Arinyo

Authors
  1. M. V. Luzón
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  2. A. Soto
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  3. J. F. Gálvez
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  4. R. Joan-Arinyo
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Correspondence to M. V. Luzón.

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Luzón, M.V., Soto, A., Gálvez, J.F. et al. Searching the Solution Space in Constructive Geometric Constraint Solving with Genetic Algorithms. Appl Intell 22, 109–124 (2005). https://doi.org/10.1007/s10489-005-5600-1

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