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Characterization theorem for best polynomial spline approximation with free knots

Nadezda Sukhorukova, Julien Ugon

TL;DR

A characterization theorem for the polynomial splines with fixed tails is obtained, that is the value of the spline is fixed in one or more knots (external or internal).

Abstract

In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.

Characterization theorem for best polynomial spline approximation with free knots

TL;DR

A characterization theorem for the polynomial splines with fixed tails is obtained, that is the value of the spline is fixed in one or more knots (external or internal).

Abstract

In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.

Paper Structure

This paper contains 25 sections, 9 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and formulations
  4. Existing work and motivation
  5. Characterization theorems and necessary optimality conditions
  6. Algorithms
  7. Quasidifferentiable functions
  8. Characterization through quasidifferentiability
  9. Quasidifferential of the objective function
  10. Quasidifferential of the spline functions
  11. Case when the point $t$ is not a knot
  12. Case when point $t$ is a knot
  13. Quasidifferential of function $\gamma_l^t$:
  14. Quasidifferential of function $\zeta_l^t$:
  15. Combine $\gamma_l^t$ and $\zeta_l^t$:
  16. ...and 10 more sections

Key Result

Proposition 2.1

If the function $f$ is continuous, problem eq:optimisation_problem admits a solution

Figures (1)

  • Figure 1: An example where the algorithm fails to reach a locally optimal solution

Theorems & Definitions (31)

  • Definition 2.1: Polynomial Spline
  • Definition 2.2: multiplicity
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Example 2.1
  • Example 2.2
  • ...and 21 more