Knot Locating in Piecewise Linear Approximation

TL;DR

This paper addresses the problem of optimally placing knots in piecewise linear approximations of nonlinear functions to improve approximation quality, using advanced optimization methods and numerical experiments.

Contribution

It introduces formulations for optimal knot placement and applies sequential quadratic programming and spectral projected gradient methods to solve them.

Findings

The proposed methods effectively optimize knot locations.

Numerical experiments demonstrate improved approximation accuracy.

The approaches outperform existing heuristics in test cases.

Abstract

Many separable nonlinear optimization problems can be approximated by their nonlinear objective functions with piecewise linear functions. A natural question arising from applying this approach is how to break the interval of interest into subintervals (pieces) to achieve a good approximation. We present formulations to optimize the location of the knots. We apply a sequential quadratic programming method and a spectral projected gradient method to solve the problem. We report numerical experiments to show the effectiveness of the proposed approaches.