Geometric dual formulation for first-derivative-based univariate cubic L1 splines

Authors:
Y. B. Zhao;S. -C. Fang;J. E. Lavery
Affiliations:
Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China 100080;Industrial Engineering and Operations Research, North Carolina State University, Raleigh, USA 27695-7906 and Departments of Mathematical Sciences and Industrial Engineering, Tsinghua University, B ...;Mathematics Division, Army Research Office, Army Research Laboratory, Research Triangle Park, USA 27709-2211
Venue:
Journal of Global Optimization
Year:
2008

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Abstract

With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based $$\mathcal{C}^1$$ -smooth univariate cubic L 1 splines. An L 1 spline minimizes the L 1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L 1 spline is a nonsmooth non-linear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.