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.