Quadratic Programming for Convex Control Theoretic Smoothing Splines

Abstract

Over the recent years constrained smoothing splines
have drawn much attention and significant amount of research is
being done in this area. In this paper producing control theoretic
smoothing splines with constraints on the derivative is discussed.
Producing monotone control theoretic splines for general linear
systems have been solved by formulating the problem as a
semidefinite quadratic programming problem. The main focus of
this paper will be on producing convex control theoretic splines.
The linear system considered is two dimensional and therefore
would give rise to a convex cubic control theoretic spline. A new
basis functions are defined and a discretization technique based
on fast discretization was adopted. Then the problem is reduced
to a finite dimensional quadratic programming and can be solved
easily by numerical computation. The construction of the optimal
smooth convex curve is based on minimization of a quadratic cost
functional using sequential quadratic programming.