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

The Steklov function μf(·,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 μf for the case when f is a multivariate quartic polynomial. Then we show that, for large enough t, μf(·,t) is convex; in other words, μf(·,t) convexifies f. We provide an easy-to-compute formula for t with which μf convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving μf, 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.