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


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.

This is a preview of subscription content, access via your institution.


  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)

  3. 3.

    Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman Advanced Publishing Program. Boston, Pitman (1984)

    MATH  Google 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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Google 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)

    Article  Google Scholar 

  8. 8.

    de Boor, C.: A Practical Guide to Splines. Springer, New York (1978)

    MATH  Google 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)

    Article  MathSciNet  Google Scholar 

  10. 10.

    Dias, R.: Density estimation via hybrid splines. J. Stat. Comput. Simul. 60, 277–294 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Dias, R.: Sequential adaptive non parametric regression via H-splines. Commun. Stat. Comput. Simul. 28, 501–515 (1999)

    MATH  Article  MathSciNet  Google 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)

    MathSciNet  Google 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: http://www.laas.fr/~jpl/book.html

    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)

    MATH  Google 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)

    MATH  Book  Google 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)

    MATH  MathSciNet  Google Scholar 

  18. 18.

    Kooperberg, C., Stone, C.J.: A study of logspline density estimation. Comput. Stat. Data Anal. 12, 327–347 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  19. 19.

    Lavalle, S.: Planning Algorithms. Cambridge University Press, Cambridge (2006)

    MATH  Book  Google Scholar 

  20. 20.

    Luo, Z., Wahba, G.: Hybrid adaptive splines. J. Am. Stat. Assoc. 92, 107–116 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    Reif, U.: Orthogonality of cardinal B-splines in weighted Sobolev spaces. SIAM J. Math. Anal. 28(5), 1258–1263 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  22. 22.

    Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1986)

    MATH  Google 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)

    MATH  Book  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Ronaldo Dias.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dias, R., Garcia, N.L. & Zambom, A.Z. A penalized nonparametric method for nonlinear constrained optimization based on noisy data. Comput Optim Appl 45, 521–541 (2010). https://doi.org/10.1007/s10589-008-9185-6

Download citation


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