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A Geometric Programming Framework for Univariate Cubic L 1 Smoothing Splines

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  • Hao Cheng
  • Shu-Cherng Fang
  • John Lavery

Abstract

Univariate cubic L 1 smoothing splines are capable of providing shape-preserving C 1 -smooth approximation of multi-scale data. The minimization principle for univariate cubic L 1 smoothing splines results in a nondifferentiable convex optimization problem that, for theoretical treatment and algorithm design, can be formulated as a generalized geometric program. In this framework, a geometric dual with a linear objective function over a convex feasible domain is derived, and a linear system for dual to primal conversion is established. Numerical examples are given to illustrate this approach. Sensitivity analysis for data with uncertainty is presented. Copyright Springer Science + Business Media, Inc. 2005

Suggested Citation

  • Hao Cheng & Shu-Cherng Fang & John Lavery, 2005. "A Geometric Programming Framework for Univariate Cubic L 1 Smoothing Splines," Annals of Operations Research, Springer, vol. 133(1), pages 229-248, January.
  • Handle: RePEc:spr:annopr:v:133:y:2005:i:1:p:229-248:10.1007/s10479-004-5035-9
    DOI: 10.1007/s10479-004-5035-9
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    References listed on IDEAS

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    1. Elmor L. Peterson, 1977. "The Duality between Suboptimization and Parameter Deletion," Mathematics of Operations Research, INFORMS, vol. 2(4), pages 311-319, November.
    2. Elmor L. Peterson, 1977. "The Duality Between Suboptimization and Parameter Deletion," Discussion Papers 273, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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    Cited by:

    1. Chiu, Nan-Chieh & Fang, Shu-Cherng & Lavery, John E. & Lin, Jen-Yen & Wang, Yong, 2008. "Approximating term structure of interest rates using cubic L1 splines," European Journal of Operational Research, Elsevier, vol. 184(3), pages 990-1004, February.
    2. Lu, Hao-Chun, 2020. "Indicator of power convex and exponential transformations for solving nonlinear problems containing posynomial terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    3. Qingwei Jin & Lu Yu & John Lavery & Shu-Cherng Fang, 2012. "Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties," Computational Optimization and Applications, Springer, vol. 51(2), pages 575-600, March.
    4. Han-Lin Li & Hao-Chun Lu, 2009. "Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables," Operations Research, INFORMS, vol. 57(3), pages 701-713, June.

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