Monotone Data Smoothing by Quadratic Splines via Dualization

Abstract

For monotone smoothing of univariate data sets an objective function is used which represents a trade‐off between curvature minimization and interpolation. The monotonicity condition is taken into the constraints. Using quadratic splines this smoothing problem is discretized by a quadratic program, and this in turn can be dualized in such a way that an unconstrained program results. In addition, a return‐formula holds true by means of which the optimal spline is directly computed from a solution of the dual problem. Thus, from a numerical point of view this approach is attractive. Here the dualization is performed via Fenchel's theory. For computing the needed Fenchel conjugates it is convenient to fix one of the variables.