Computational Optimization and Applications

, Volume 45, Issue 3, pp 521–541 | Cite as

A penalized nonparametric method for nonlinear constrained optimization based on noisy data

  • Ronaldo Dias
  • Nancy L. Garcia
  • Adriano Z. Zambom


The objective of this study is to find a smooth function joining two points A and B with minimum length constrained to avoid fixed subsets. A penalized nonparametric method of finding the best path is proposed. The method is generalized to the situation where stochastic measurement errors are present. In this case, the proposed estimator is consistent, in the sense that as the number of observations increases the stochastic trajectory converges to the deterministic one. Two applications are immediate, searching the optimal path for an autonomous vehicle while avoiding all fixed obstacles between two points and flight planning to avoid threat or turbulence zones.


Autonomous vehicle B-splines Consistent estimator Confidence ellipses Constrained optimization Nonparametric method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antoniadis, A.: Wavelet methods for smoothing noisy data. In: Wavelets, Images, and Surface Fitting, Chamonix-Mont-Blanc, 1993, pp. 21–28. A.K. Peters, Wellesley (1994) Google Scholar
  2. 2.
    Asseo, S.J.: In-flight replanning of penetration routes to avoid threat zones of circular shapes. In: Proceedings of the IEEE 1998 National Aerospace and Electronics Conference (NAECON 1998), pp. 383–391 (1998) Google Scholar
  3. 3.
    Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman Advanced Publishing Program. Boston, Pitman (1984) MATHGoogle Scholar
  4. 4.
    Barraquand, J., Latombe, J.-C.: Nonholonomic multibody mobile robots: controllability and motion planning in the presence of obstacles. Algorithmica 10(2–4), 121–155 (1993). (Computational robotics: the geometric theory of manipulation, planning, and control) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodin, P., Villemoes, L.F., Wahlberg, B.: Selection of best orthonormal rational basis. SIAM J. Control Optim. 38(4), 995–1032 (2000) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Burgardand, W., Kavraki, L., Thrun, S.: Principles of Robot Motion: Theory, Algorithms and Implementations. MIT Press, Cambridge (2005) MATHGoogle Scholar
  7. 7.
    Cremean, L.B., Foote, T.B., Gillula, J.H., Hines, G.H., Kogan, D., Kriechbaum, K.L., Lamb, J.C., Leibs, J., Lindzey, L., Rasmussen, C.E., Stewart, A.D., Burdick, J.W., Murray, R.M.: ALICE: An information-rich autonomous vehicle for high-speed desert navigation. J. Field Robotics 23(9), 777–810 (2006) CrossRefGoogle Scholar
  8. 8.
    de Boor, C.: A Practical Guide to Splines. Springer, New York (1978) MATHGoogle Scholar
  9. 9.
    De Vore, R., Petrova, G., Temlyakov, V.: Best basis selection for approximation in L p. Found. Comput. Math. 3(2), 161–185 (2003) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dias, R.: Density estimation via hybrid splines. J. Stat. Comput. Simul. 60, 277–294 (1998) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dias, R.: Sequential adaptive non parametric regression via H-splines. Commun. Stat. Comput. Simul. 28, 501–515 (1999) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dias, R.: A note on density estimation using a proxy of the Kullback–Leibler distance. Brazilian J. Probab. Stat. 13(2), 181–192 (2000) MathSciNetGoogle Scholar
  13. 13.
    Laumond, J.-P. (ed.): Robot Motion Planning and Control. Lecture Notes in Control and Information Science, vol. 229. Springer, New York (1998). Available online: Google Scholar
  14. 14.
    Grundel, D., Murphey, R., Pardalos, P., Prokopyev, O. (eds.): Cooperative Systems, Control and Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 588. Springer, New York (2007) MATHGoogle Scholar
  15. 15.
    Hirsch, M.J., Pardalos, P., Murphey, R., Grundel, D. (eds.): Advances in Cooperative Control and Optimization. Lecture Notes in Control and Information Sciences, vol. 369. Springer, New York (2008). Papers from a meeting held in Gainesville, FL, 31 January–2 February 2007 Google Scholar
  16. 16.
    Jabri, Y.: The Mountain Pass Theorem: Variants, Generalizations and Some Applications. Cambridge University Press, Cambridge (2003) MATHCrossRefGoogle Scholar
  17. 17.
    Kohn, R., Marron, J.S., Yau, P.: Wavelet estimation using Bayesian basis selection and basis averaging. Stat. Sinica 10(1), 109–128 (2000) MATHMathSciNetGoogle Scholar
  18. 18.
    Kooperberg, C., Stone, C.J.: A study of logspline density estimation. Comput. Stat. Data Anal. 12, 327–347 (1991) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lavalle, S.: Planning Algorithms. Cambridge University Press, Cambridge (2006) MATHCrossRefGoogle Scholar
  20. 20.
    Luo, Z., Wahba, G.: Hybrid adaptive splines. J. Am. Stat. Assoc. 92, 107–116 (1997) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Reif, U.: Orthogonality of cardinal B-splines in weighted Sobolev spaces. SIAM J. Math. Anal. 28(5), 1258–1263 (1997) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1986) MATHGoogle Scholar
  23. 23.
    Tiwari, A., Chandra, H., Yadegar, J., Wang, J.: Constructing optimal cyclic tours for planar exploration and obstacle avoidance: A graph theory approach. In: Advances in Variable Structure and Sliding Mode Control. Springer, Berlin (2007) Google Scholar
  24. 24.
    Vidakovic, B.: Statistical Modeling by Wavelets. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York (1999) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ronaldo Dias
    • 1
  • Nancy L. Garcia
    • 1
  • Adriano Z. Zambom
    • 1
  1. 1.Departamento de EstatísticaUniversidade Estadual de Campinas (UNICAMP)CampinasBrazil

Personalised recommendations