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Loss functions for finite sets

Author

Listed:
  • Jiawang Nie

    (University of California San Diego)

  • Suhan Zhong

    (Texas A &M University)

Abstract

This paper studies loss functions for finite sets. For a given finite set S, we give sum-of-square type loss functions of minimum degree. When S is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When S is approximately given by a sample set T, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.

Suggested Citation

  • Jiawang Nie & Suhan Zhong, 2023. "Loss functions for finite sets," Computational Optimization and Applications, Springer, vol. 84(2), pages 421-447, March.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:2:d:10.1007_s10589-022-00420-9
    DOI: 10.1007/s10589-022-00420-9
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    References listed on IDEAS

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    1. Pooriya Beyhaghi & Ryan Alimo & Thomas Bewley, 2020. "A derivative-free optimization algorithm for the efficient minimization of functions obtained via statistical averaging," Computational Optimization and Applications, Springer, vol. 76(1), pages 1-31, May.
    2. Frank Schorfheide, 2000. "Loss function-based evaluation of DSGE models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 15(6), pages 645-670.
    3. Christoffersen, Peter & Jacobs, Kris, 2004. "The importance of the loss function in option valuation," Journal of Financial Economics, Elsevier, vol. 72(2), pages 291-318, May.
    4. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
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