Shape restricted smoothing splines via constrained optimal control and nonsmooth Newton's methods

Shape restricted smoothing splines receive growing attention, motivated by a wide range of important applications in science and engineering. In this paper, we consider smoothing splines subject to general linear dynamics and control constraints, and formulate them as finite-horizon constrained linear optimal control problems with unknown initial state and control. By exploring techniques from functional and variational analysis, optimality conditions are developed in terms of variational inequalities. Due to the control constraints, the optimality conditions give rise to a nonsmooth B-differentiable equation of an optimal initial condition, whose unique solution completely determines the shape restricted smoothing spline. A modified nonsmooth Newton’s algorithm with line search is exploited to solve this equation; detailed convergence analysis of the proposed algorithm is presented. In particular, using techniques from nonsmooth analysis, polyhedral theory, and switching systems, we show the global convergence of the algorithm when a shape restricted smoothing spline is subject to a general polyhedral control constraint.