Structure-preserving function approximation via convex optimization

TL;DR

This paper introduces a convex optimization framework for approximating functions while preserving structural properties like positivity, monotonicity, and convexity, through flexible algorithms tested on univariate functions.

Contribution

It develops a formalism and algorithms for structure-preserving function approximation using convex feasibility, handling complex constraints efficiently.

Findings

Algorithms successfully preserve structural properties in function approximation.

Flexible approach handles multiple constraints simultaneously.

Demonstrated effectiveness on univariate function approximation problems.

Abstract

Approximations of functions with finite data often do not respect certain "structural" properties of the functions. For example, if a given function is non-negative, a polynomial approximation of the function is not necessarily also non-negative. We propose a formalism and algorithms for preserving certain types of such structure in function approximation. In particular, we consider structure corresponding to a convex constraint on the approximant (for which positivity is one example). The approximation problem then converts into a convex feasibility problem, but the feasible set is relatively complicated so that standard convex feasibility algorithms cannot be directly applied. We propose and discuss different algorithms for solving this problem. One of the features of our machinery is flexibility: relatively complicated constraints, such as simultaneously enforcing positivity, monotonicity, and convexity, are fairly straightforward to implement. We demonstrate the success of our algorithm on several problems in univariate function approximation.