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

It has been known that an efficient approach to the determination of the convex best C-spline interpolant to a set of given data is to solve its unconstrained dual program by standard numerical methods (e.g., Newton’s method.) Regularity and well-posedness of the dual program, which are two important issues but not well-addressed in the literature, are focus of the paper. Our well-posedness result characterizes when the objective function is coercive; and our regularity results characterize when the generalized Hessian of the objective function is positive definite. These results justify why Newton’s method is likely to be successful in practice. Examples are given to illustrate the obtained results.