Steklov Regularization and Trajectory Methods for Univariate Global Optimization

TL;DR

This paper presents a novel Steklov regularization technique that convexifies univariate functions and constructs a trajectory-based algorithm to find global minimizers, demonstrated on polynomials and non-polynomial functions.

Contribution

Introduction of the Steklov regularization method for univariate global optimization and proof of its effectiveness on quartic polynomials.

Findings

Steklov regularization convexifies coercive functions

Trajectory method finds global minimizers for quartic polynomials

Numerical experiments validate the approach on various functions

Abstract

We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.