Constrained data smoothing via optimal control

Summary: In this letter, we address the problem of constructing constrained smoothing splines in data science. The problem is formulated as an optimal control problem, and the optimal smoothing splines with intermediate constraints are analyzed using first-order optimality condition from a nonstandard version of the Pontryagin maximum principle. We propose a novel algorithm to compute the optimal constrained splines based on Newton-Raphson iterative scheme combined with stochastic approximation. A numerical example is provided to illustrate the advantages of the proposed method.