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On convex quadratic approximation

Author

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  • Dick Den Hertog
  • Etienne De Klerk
  • Kees Roos

Abstract

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques.
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Suggested Citation

  • Dick Den Hertog & Etienne De Klerk & Kees Roos, 2002. "On convex quadratic approximation," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(3), pages 376-385, August.
    • Handle: RePEc:bla:stanee:v:56:y:2002:i:3:p:376-385
    DOI: 10.1111/1467-9574.t01-1-00075
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    Cited by:

    1. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2005. "Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing," Other publications TiSEM ad31ef2c-fc29-46c1-9b8f-6, Tilburg University, School of Economics and Management.
    2. Zhichao Zheng & Karthik Natarajan & Chung-Piaw Teo, 2016. "Least Squares Approximation to the Distribution of Project Completion Times with Gaussian Uncertainty," Operations Research, INFORMS, vol. 64(6), pages 1406-1421, December.
    3. Siem, A.Y.D. & den Hertog, D. & Hoffmann, A.L., 2005. "Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing," Discussion Paper 2005-132, Tilburg University, Center for Economic Research.
    4. Stinstra, Erwin & den Hertog, Dick, 2008. "Robust optimization using computer experiments," European Journal of Operational Research, Elsevier, vol. 191(3), pages 816-837, December.
    5. Siem, A.Y.D. & de Klerk, E. & den Hertog, D., 2005. "Discrete Least-norm Approximation by Nonnegative (Trigonomtric) Polynomials and Rational Functions," Discussion Paper 2005-73, Tilburg University, Center for Economic Research.
    6. Siem, A.Y.D. & de Klerk, E. & den Hertog, D., 2005. "Discrete Least-norm Approximation by Nonnegative (Trigonomtric) Polynomials and Rational Functions," Other publications TiSEM 43a1152a-8130-4e42-851b-e, Tilburg University, School of Economics and Management.

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