, Volume 37, Issue 3, pp 409-425
Date: 17 Mar 2007

Regularity and well-posedness of a dual program for convex best C 1-spline interpolation

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Abstract

An efficient approach to computing the convex best C 1-spline interpolant to a given set of data is to solve an associated dual program by standard numerical methods (e.g., Newton’s method). We study regularity and well-posedness of the dual program: two important issues that have been not yet well-addressed in the literature. Our regularity results characterize the case when the generalized Hessian of the objective function is positive definite. We also give sufficient conditions for the coerciveness of the objective function. These results together specify conditions when the dual program is well-posed and hence justify why Newton’s method is likely to be successful in practice. Examples are given to illustrate the obtained results.

The work was supported by EPSRC grant EP/D502535/1 for the first author and by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5141/01E) for the second author.