Constrained Smoothing Splines by Optimal Control

In this letter, we consider the problem of constructing constrained smoothing splines, which are of great importance in data science. The novelty of this work is to formulate the problem as an optimal control problem, and we mathematically analyze the optimal smoothing spline with intermediate constraints using first-order optimality condition from a nonstandard version of the Pontryagin maximum principle. We also propose a novel algorithm to compute the optimal constrained splines based on Newton-Raphson iterative scheme combined with stochastic approximation. A numerical example is shown to illustrate the advantages of the proposed method.