Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

An efficient algorithm for range computation of polynomials using the Bernstein form

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We present a novel optimization algorithm for computing the ranges of multivariate polynomials using the Bernstein polynomial approach. The proposed algorithm incorporates four accelerating devices, namely the cut-off test, the simplified vertex test, the monotonicity test, and the concavity test, and also possess many new features, such as, the generalized matrix method for Bernstein coefficient computation, a new subdivision direction selection rule and a new subdivision point selection rule. The features and capabilities of the proposed algorithm are compared with those of other optimization techniques: interval global optimization, the filled function method, a global optimization method for imprecise problems, and a hybrid approach combining simulated annealing, tabu search and a descent method. The superiority of the proposed method over the latter methods is illustrated by numerical experiments and qualitative comparisons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berchtold J., Bowyer A.: Robust arithmetic for multivariateBernstein-form polynomials. Comput. Aided Geom. Des. 32, 681–689 (2000)

    Google Scholar 

  2. Berchtold, J., Voiculescu, I., Bowyer, A.: Multivariate Bernstein form polynomials. Technical Report 31/98, School of Mechanical Engineering (1998)

  3. Cox, D., Little, J., O’Shea, D.: Using algebraic geometry. In: Graduate Texts in Mathematics, vol. 185. Springer-Verlag, New York (1998)

  4. Farouki R.T., Rajan V.T.: On the numerical condition of polynomials in Bernstein form. Comput. Aided Geom. Des. 4, 191–216 (1987)

    Article  Google Scholar 

  5. Garloff J.: The Bernstein algorithm. Interval Comput. 2, 154–168 (1993)

    Google Scholar 

  6. Garloff J.: The Bernstein expansion and its applications. J. Am. Romanian Acad. 25–27, 80–85 (2003)

    Google Scholar 

  7. Garloff J., Smith A.P.: Solution of systems of polynomial equations by using Bernstein expansion. In: Alefeld, G., Rohn, J., Rump, S., Yamamoto, T. (eds) Symbolic Algebraic Methods and Verification Methods, pp. 87–97. Springer, New York (2001)

    Google Scholar 

  8. Garloff J., Smith A.P.: A comparison of methods for the computation of affine lower bound functions for polynomials. In: Jermann, C., Neumaier, A., Sam, D. (eds) Global Optimization and Constraint Satisfaction: 2nd International Workshop, COCOS 2003, Lecture Notes in Computer Science, pp. 71–85. Springer, Berlin (2005)

    Google Scholar 

  9. Garloff J., Jansson C., Smith A.P.: Lower bound functions for polynomials. J. Comput. Appl. Math. 157(1), 207–225 (2003)

    Article  Google Scholar 

  10. Ge R.P., Qin Y.F.: A class of filled functions for finding global minimizers of a function of several variables. J. Optim. Theory Appl. 54(2), 241–252 (1987)

    Article  Google Scholar 

  11. Ge R., Qin Y.: The globally convexized filled functions for global optimization. Appl. Math. Comput. 35, 131–158 (1990)

    Article  Google Scholar 

  12. Hammer R., Hocks M., Kulisch U., Ratz D.: Numerical Toolbox for Verified Computing I. Springer Verlag, Heidelberg, New York (1993)

    Google Scholar 

  13. Hansen E.R.: Nonlinear equations and optimization. Comput. Math. Appl. 25(10/11), 125–145 (1993)

    Article  Google Scholar 

  14. Hansen E.R., Walster G.W.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (2004)

    Google Scholar 

  15. Henrion D., Lasserre J.B.: Gloptipoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Soft. 29, 165–194 (2003)

    Article  Google Scholar 

  16. Horst R., Pardalos P.M.: Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  17. Jibetean D., Laurent M.: Semidefinite approximations for global unconstrained polynomial optimization. SIAM J. Optim. 16(2), 490–514 (2005)

    Article  Google Scholar 

  18. Kearfott R.B.: Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  19. Li H.L., Chang C.T.: An approximate approach of global optimization for polynomial programming problems. Eur. J. Oper. Res. 107, 625–632 (1998)

    Article  Google Scholar 

  20. Lorenz C.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953)

    Google Scholar 

  21. Malan, S., Milanese, M., Taragna, M., Garloff, J.: B3 algorithm for robust performance analysis in presence of mixed parametric and dynamic perturbations. In: Proceedings of the 31st Conference on Decision and Control, pp. 128–133. Tucson, Arizona (1992)

  22. Moore R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Google Scholar 

  23. Nataraj P.S.V., Kotecha K.: An improved interval global optimization algorithm using higher order inclusion function forms. J. Global Optim. 32(1), 35–63 (2005)

    Article  Google Scholar 

  24. Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. In: Basu, S., Gonzales-Vega, L. (eds.) Algorithmic and Quantitative Real Algebraic Geometry, vol. 60 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science (2003)

  25. Pinter J.D .: Global optimization: software, test problems, and applications. In: Pardalos, P.M., Romeijn, H.E. (eds) Handbook of Global Optimization, vol. 2, pp. 515–569. Kluwer Academic Publishers, London (2002)

    Google Scholar 

  26. Ratschek H., Rokne J.: Computer Methods for the Range of Functions. Ellis Horwood, New York (Chichester) (1984)

    Google Scholar 

  27. Ray, S.: A new appraoch to range computation of polynomials using the Bernstein form. PhD Thesis, System and Control Engineering, Indian Institute of Technology, Bombay, India (2007)

  28. Salhi S., Queen N.M.: A hybrid algorithm for detecting global and local minima when optimizing functions with many minima. Eur. J. Oper. Res. 155, 51–67 (2004)

    Article  Google Scholar 

  29. Smith, A.P.: Fast construction of constant bound functions for sparse polynomials. J. Global Optim. July (2007, published online)

  30. Sun Microsystems, Palo Alto, CA, USA. Forte FORTRAN 95 User Manual (2001)

  31. Verschelde, J.: The PHC pack, the database of polynomial systems. Technical Report, University of Illinois, Mathematics Department, Chicago, USA (2001)

  32. Vrahatis M.N.: A generalized bisection method for large and imprecise problems. In: Alefeld, G., Frommer, A., Lang, B. (eds) Scientifc Computing and Validated Numerics, pp. 186–192. Akademie Verlag, Berlin (1996)

    Google Scholar 

  33. Vrahatis M.N., Sotiropoulos D.G., Triantafyllou E.C.: Global optimization for imprecise problems. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds) Developments in Global Optimization, pp. 37–54. Kluwer, The Netherlands (1997)

    Google Scholar 

  34. Wolfe M.A.: Interval methods for global optimization. Appl. Math. Comput. 75(2–3), 179–206 (1996)

    Google Scholar 

  35. Zettler M., Garloff J.: Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Trans. Automatic Control 43(3), 425–431 (1998)

    Article  Google Scholar 

  36. Zhang L.S., Ng C.K., Li D., Tian W.W.: A new filled function method for global optimization. J. Global Optim. 28, 17–43 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Systems and Control Engineering Group, ACRE Building, Indian Institute of Technology, Bombay, 400 076, India

    Shashwati Ray & P. S. V. Nataraj

Authors
  1. Shashwati Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. S. V. Nataraj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. S. V. Nataraj.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ray, S., Nataraj, P.S.V. An efficient algorithm for range computation of polynomials using the Bernstein form. J Glob Optim 45, 403–426 (2009). https://doi.org/10.1007/s10898-008-9382-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-008-9382-y

Keywords

Search

Navigation