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Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

  • Alex Y. D. Siem
  • Dick den Hertog
  • Aswin L. Hoffmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3982)

Abstract

The main contents of this paper is two-fold. First, we present a method to approximate multivariate convex functions by piecewise linear upper and lower bounds. We consider a method that is based on function evaluations only. However, to use this method, the data have to be convex. Unfortunately, even if the underlying function is convex, this is not always the case due to (numerical) errors. Therefore, secondly, we present a multivariate data-smoothing method that smooths nonconvex data. We consider both the case that we have only function evaluations and the case that we also have derivative information. Furthermore, we show that our methods are polynomial time methods. We illustrate this methodology by applying it to some examples.

Keywords

Multiobjective Optimization Derivative Information Naval Research Logistics Isotonic Regression Pareto Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Kuijt, F.: Convexity preserving interpolation – Stationary nonlinear subdivision and splines. PhD thesis, University of Twente, Enschede, The Netherlands (1998)Google Scholar
  2. 2.
    Siem, A.Y.D., de Klerk, E., den Hertog, D.: Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions. CentER Discussion Paper 2005-73, Tilburg University, Tilburg (2005)Google Scholar
  3. 3.
    den Hertog, D., de Klerk, E., Roos, K.: On convex quadratic approximation. Statistica Neerlandica 563, 376–385 (2002)CrossRefGoogle Scholar
  4. 4.
    Burkard, R.E., Hamacher, H.W., Rote, G.: Sandwich approximation of univariate convex functions with an application to separable convex programming. Naval Research Logistics 38, 911–924 (1991)MATHMathSciNetGoogle Scholar
  5. 5.
    Fruhwirth, B., Burkard, R.E., Rote, G.: Approximation of convex curves with application to the bi-criteria minimum cost flow problem. European Journal of Operational Research 42, 326–338 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Rote, G.: The convergence rate of the Sandwich algorithm for approximating convex functions. Computing 48, 337–361 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Siem, A.Y.D., den Hertog, D., Hoffmann, A.L.: A method for approximating univariate convex functions using only function evaluations. Working paper, Tilburg University, Tilburg (2005)Google Scholar
  8. 8.
    Yang, X.Q., Goh, C.J.: A method for convex curve approximation. European Journal of Operational Research 97, 205–212 (1997)MATHCrossRefGoogle Scholar
  9. 9.
    Cullinan, M.P.: Data smoothing using non-negative divided differences and ℓ2 approximation. IMA Journal of Numerical Analysis 10, 583–608 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Demetriou, I.C., Powell, M.J.D.: Least squares smoothing of univariate data to achieve piecewise monotonicity. IMA Journal of Numerical Analysis 11, 411–432 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Demetriou, I.C., Powell, M.J.D.: The minimum sum of squares change to univariate data that gives convexity. IMA Journal of Numerical Analysis 11, 433–448 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Barlow, R.E., Bartholomew, R.J., Bremner, J.M., Brunk, H.D.: Statistical inference under order restrictions. Wiley, Chichester (1972)MATHGoogle Scholar
  13. 13.
    Ben-Tal, A., Nemirovski, A.: Robust optimization - methodology and applications. Mathematical Programming, Series B 92, 453–480 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Charnes, A., Cooper, W.W.: Programming with linear fractional functional. Naval Research Logistics Quarterly 9, 181–186 (1962)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  16. 16.
    Hoffmann, A.L., Siem, A.Y.D., den Hertog, D., Kaanders, J.H.A.M., Huizenga, H.: Dynamic generation and interpolation of pareto optimal IMRT treatment plans for convex objective functions. Working paper, Radboud University Nijmegen Medical Centre, Nijmegen (2005)Google Scholar
  17. 17.
    den Boef, E., den Hertog, D.: Efficient line searching for convex functions. CentER Discussion Paper 2004-52. Tilburg University, Tilburg (2004)Google Scholar
  18. 18.
    Chinneck, J.W.: Discovering the characteristics of mathematical programs via sampling. Optimization Methods and Software 17(2), 319–352 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alex Y. D. Siem
    • 1
  • Dick den Hertog
    • 1
  • Aswin L. Hoffmann
    • 2
  1. 1.Department of Econometrics and Operations Research/Center for Economic Research (CentER)Tilburg UniversityTilburgThe Netherlands
  2. 2.Department of Radiation OncologyRadboud University Nijmegen Medical CentreNijmegenThe Netherlands

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