Steklov Convexification and a Trajectory Method for Global Optimization of Multivariate Quartic Polynomials

Abstract

The Steklov function ${\mu }_f(\cdot ,t)$ is defined to average a continuous function $f$ at each point of its domain by using a window of size given by $t>0$. It has traditionally been used to approximate $f$ smoothly with small values of $t$. In this paper, we first find a concise and useful expression for ${\mu }_f$ for the case when $f$ is a multivariate quartic polynomial. Then we show that, for large enough $t$, ${\mu }_f(\cdot ,t)$ is convex; in other words, ${\mu }_f(\cdot ,t)$ convexifies $f$. We provide an easy-to-compute formula for $t$ with which ${\mu }_f$ convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving ${\mu }_f$, a trajectory $x(t)$ emanating from the minimizer of the convexified $f$ and ending at $x(0)$, an estimate of the global minimizer of $f$. For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.