Regularity and well-posedness of a dual program for convex best C1-spline interpolation

Authors:
Houduo Qi;Xiaoqi Yang
Affiliations:
School of Mathematics, University of Southampton, Southampton, UK SO17 1BJ;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
Venue:
Computational Optimization and Applications
Year:
2007

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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.