Structural and Multidisciplinary Optimization

, Volume 35, Issue 4, pp 327–339 | Cite as

Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions

Open Access
Research Paper


Polynomials, trigonometric polynomials, and rational functions are widely used for the discrete approximation of functions or simulation models. Often, it is known beforehand that the underlying unknown function has certain properties, e.g., nonnegative or increasing on a certain region. However, the approximation may not inherit these properties automatically. We present some methodology (using semidefinite programming and results from real algebraic geometry) for least-norm approximation by polynomials, trigonometric polynomials, and rational functions that preserve nonnegativity.


(Trigonometric) polynomials Rational functions Semidefinite programming Regression (Chebyshev) approximation 


  1. Barlow R, Bartholomew R, Bremner J, Brunk H (1972) Statistical inference under order restrictions. Wiley, ChichesterMATHGoogle Scholar
  2. Bates D, Watts D (1988) Nonlinear regression analysis and its applications. Wiley, New YorkMATHGoogle Scholar
  3. Benson S, Ye Y, Zhang X (2000) Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J Optim 10:443–461MATHCrossRefMathSciNetGoogle Scholar
  4. Croarkin C, Tobias P (eds) (2005) NIST (National Institute of Standards and Technology)/SEMATECH e-handbook of statistical methods.
  5. Cuyt A, Lenin R (2002) Computing packet loss probabilities in multiplexer models using adaptive rational interpolation with optimal pole placement. Technical report, University of AntwerpGoogle Scholar
  6. Cuyt A, Lenin R, Becuwe S, Verdonk B (2006) Adaptive multivariate rational data fitting with applications in electromagnetics. IEEE Trans Microwave Theor Tech 54:2265–2274CrossRefGoogle Scholar
  7. Fassbender H (1997) On numerical methods for discrete least-squares approximation by trigonometric polynomials. Math Comput 66(218):719–741MATHCrossRefMathSciNetGoogle Scholar
  8. Floater MS (2005) On the convergence of derivatives of bernstein approximation. J Approx Theory 134:130–135MATHCrossRefMathSciNetGoogle Scholar
  9. Forsberg J, Nilsson L (2005) On polynomial response surfaces and Kriging for use in structural optimization of crashworthiness. Struct Multidiscipl Optim 29:232–243CrossRefGoogle Scholar
  10. den Hertog D, de Klerk E, Roos K (2002) On convex quadratic approximation. Stat Neerl 563:376–385CrossRefGoogle Scholar
  11. Hilbert D (1888) Über die Darstellung definiter Formen als Summe von Formenquadraten. Math Ann 32:342–350CrossRefMathSciNetGoogle Scholar
  12. Hubbard A, Robinson W (1950) A thermal decomposition study of Colorado oil shale. Rept. Invest. 4744, US Bureau of MinesGoogle Scholar
  13. Jansson T, Nilsson L, Redhe M (2003) Using surrogate models and response surfaces in structural optimization – with application to crashworthiness design and sheet metal forming. Struct Multidiscipl Optim 25:129–140CrossRefGoogle Scholar
  14. Jibetean D, de Klerk E (2006) Global optimization of rational functions: a semidefinite programming approach. Math Program A 106(1):93–109MATHCrossRefGoogle Scholar
  15. de Klerk E, Elabwabi GE, den Hertog D (2006) Optimization of univariate functions on bounded intervals by interpolation and semidefinite programming. CentER discussion paper 2006-26. Tilburg University, TilburgGoogle Scholar
  16. Kuijt F (1998) Convexity preserving interpolation – stationary nonlinear subdivision and splines. Dissertation, University of Twente, Enschede, The NetherlandsGoogle Scholar
  17. Kuijt F, van Damme R (2001) A linear approach to shape preserving approximation. Adv Comput Math 14:25–48MATHCrossRefMathSciNetGoogle Scholar
  18. Lofberg J, Parrilo P (2004) From coefficients to samples: a new approach to SoS optimization. Working paperGoogle Scholar
  19. Montgomery D, Peck E (1992) Introduction to linear regression analysis. Wiley, New YorkMATHGoogle Scholar
  20. Nesterov Y (2000) Squared functional systems and optimization problems. In: Frenk H, Roos K, Terlaky T (eds) High performance optimization, chap. 17, vol. 13. Kluwer, The Netherlands, pp 405–440Google Scholar
  21. Powell M (1981) Approximation theory and methods. Cambridge University Press, CambridgeMATHGoogle Scholar
  22. Powers V, Reznick B (2000) Polynomials that are positive on an interval. Trans Am Math Soc 352(10):4677–4692MATHCrossRefMathSciNetGoogle Scholar
  23. Putinar M (1993) Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42:969–984MATHCrossRefMathSciNetGoogle Scholar
  24. Reznick B (2000) Some concrete aspects of Hilbert’s 17th problem. Contemp Math 253:251–272MathSciNetGoogle Scholar
  25. Robertson T, Wright F, Dykstra R (1988) Order restricted statistical inference. Wiley, ChichesterMATHGoogle Scholar
  26. Schweighofer M (2005) Optimization of polynomials on compact semialgebraic sets. SIAM J Optim 5(3):805–825CrossRefMathSciNetGoogle Scholar
  27. Sturm J (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11–12:625–653CrossRefMathSciNetGoogle Scholar
  28. Watson G (1980) Approximation theory and numerical methods. Wiley, ChichesterMATHGoogle Scholar
  29. Yeun Y, Yang Y, Ruy W, Kim B (2005) Polynomial genetic programming for response surface modeling. Part 1: a methodology. Struct Multidisc Optim 29:19–34CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Department of Econometrics and Operations Research/Center for Economic Research (CentER)Tilburg UniversityTilburgThe Netherlands

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