Monotone Smoothing Splines Using General Linear Systems

TL;DR

This paper introduces a novel approach for monotone smoothing splines using general linear systems, extending previous methods beyond second-order integrators by discretizing a semi-infinite quadratic programming problem.

Contribution

It formulates the monotone spline problem as a semi-infinite quadratic program and proves convergence of the discretized solution to the true solution, broadening applicability.

Findings

The discretized quadratic programming solution satisfies the infinite-dimensional monotonicity constraint.

The method converges as the discretization grid size approaches zero.

An example demonstrates the effectiveness of the proposed approach.

Abstract

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second-order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval which has the cardinality of the continuum. To solve this problem, we first formulate the problem as a semi-infinite quadratic programming, and then we adopt a discretization technique to obtain a finite-dimensional quadratic programming problem. It is shown that the solution of the finite-dimensional problem always satisfies the infinite-dimensional monotonicity constraint. It is also proved that the approximated solution converges to the exact solution as the discretization grid-size tends to zero. An example is presented to show the effectiveness of the proposed method.